NSW 11 Standard 2026- Complete Edition by Mathspace

Contents 1

Formula and equations

1.01

Simplify algebraic expressions

1.02

Substitution

1.03

Evaluate the subject of a formula

1.04

Speed, distance and time

1.05

BAC (Blood Alcohol Content)

1.06

Medication doses

1.07

Other formulas

1.08

Change the subject of a formula

1.09

Write and solve equations

Investigation: Spreadsheets and formulas

Chapter 1 review

Population and sample

Investigation: Statistical investigation process 2.01

Census or survey

2.02

Samples and populations

2.03

Sampling techniques

2.04

Survey design

Investigation: Conduct a survey 2.05

Privacy, bias and ethics

103

Investigation: Misrepresentation of results Chapter 2 review

112

Contents mathspace.co

iii

Data classification and display

116

3.01

Classify data

118

3.02

Tables and bar charts

126

3.03

Sector and line graphs

145

3.04

Dot plots and stem-and-leaf plots

163

3.05

Histograms and grouped frequency tables

177

3.06

Cumulative frequency tables and graphs

195

3.07

Shape of distribution

207

3.08

Misleading graphs and appropriate displays

216

Investigation: Appropriate choice of graph for data Investigation: Spreadsheets to tabulate and graph data Investigation: Infographics Chapter 3 review

235

Practicalities of measurement

242

4.01

Multiplication and division by powers of 10

244

4.02

Units of length and area

251

4.03

Units of volume and capacity

258

4.04

Units of mass

266

4.05

Scientific notation and significant figures

271

Chapter 4 review

280

Perimeter, area and volume

284

5.01

Pythagoras’ theorem

286

5.02

Perimeter

297

5.03

Perimeter of composite shapes

307

5.04

Area

320

5.05

Area of composite shapes

336

5.06

Surface area

341

5.07

Surface area of composite solids

355

5.08

Volume and capacity

362

5.09

Volume of composite solids

378

5.10

Trapezoidal rule

388

Chapter 5 review

399

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Earning money

408

6.01

Salaries and wages

410

6.02

Overtime

419

6.03

Commission, piecework and royalties

428

6.04

Government allowances

437

6.05

Annual leave loading

458

Investigation: Spreadsheets and income Chapter 6 review

464

Taxation 470 7.01

Allowable deductions

472

7.02

Tax tables

478

7.03

PAYG tax

484

7.04

Medicare levy

499

7.05

Net earnings

504

7.06

Yearly tax liability

511

Investigation: Spreadsheets and taxation Chapter 7 review

519

Networks, paths and trees

526

8.01

Introduction to networks

528

8.02

Network representations

544

8.03

Weighted graphs

558

8.04

Spanning trees

571

8.05

Prim’s algorithm

583

8.06

Shortest path

598

Chapter 8 review

609

Contents mathspace.co

Time and location

618

9.01

Units of time

620

9.02

Time intervals

627

9.03

Elapsed time applications

632

9.04

Latitude and longitude

645

9.05

International time zones and time differences

656

9.06

Australian time zones and daylight savings

667

Investigation: Geocaching Chapter 9 review

Measures of centre and spread

675

680

10.01

Measures of centre

682

10.02

Measures of spread

694

10.03

Compare datasets

Investigation: Statistical reports in the media 706

Investigation: Spreadsheets and centre and spread 10.04

Quartiles and interquartile range

722

Chapter 10 review

739

Box plots, clusters and outliers

744

11.01

Five-number summaries and box plots

746

11.02

Parallel box plots

762

11.03

Histograms, dot plots and box plots

777

11.04

Outliers

792

11.05

Identify clusters and gaps

804

Chapter 11 review

823

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Linear relationships

828

12.01

Straight line graphs

830

12.02

Gradient and intercept

841

12.03

Gradient-intercept form

853

12.04

Modelling linear relationships

864

12.05

Direct variation

887

Chapter 12 review

896

Purchases and budgets

902

Investigation: Spreadsheets and linear relationships

13.01

Percentage increase and decrease

904

13.02

Profit and loss

914

13.03

Purchase options

919

13.04

Purchase a car

930

13.05

On-road and running costs of a car

941

Investigation: Getting on the road Investigation: Choose the right car Investigation: Car costs with spreadsheets 13.06

Household bills

948

13.07

Prepare a personal budget

959

Investigation: Create a budget Chapter 13 review

966

Answers

972

Contents mathspace.co

vii

Big ideas Algebraic manipulation and formula application enable accurate problem-solving across diverse contexts, using substitution, rearrangement, and equations to model real-world scenarios.

1 Formula and equations Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09

Simplify algebraic expressions Substitution Evaluate the subject of a formula Speed, distance and time BAC (Blood Alcohol Content) Medication doses Other formulas Change the subject of a formula Write and solve equations Investigation: Spreadsheets and formulas Chapter 1 review

4 13 20 25 32 39 45 51 56 61

1.01 Simplify algebraic expressions After this lesson, you will be able to… • identify components of an algebraic expression, including term, pronumeral, coefficient, and constant. • recognise like terms as those having identical pronumeral parts, including their respective powers. • simplify algebraic expressions by combining like terms through the addition and subtraction of coefficients. • simplify algebraic expressions by multiplying or dividing algebraic terms, applying index laws to pronumerals where appropriate.

Simplify algebraic expressions Algebraic expression A mathematical statement formed by combining numbers and algebraic symbols using arithmetic operations. Example: a2 + 3ab − 2b2 Pronumeral A letter or symbol that is used to represent a value in a problem that can vary or change. Also known as a variable. Example: In 5a + b, a and b are pronumerals. Like terms Terms that contain the exact same pronumerals, including the same powers, regardless of their coeﬃcients.

Index (power)

3 x2 Coefficient

Constant

Pronumeral

Addition and subtraction of algebraic terms involve combining like terms by adding or subtracting their coeﬃcients: 2a + 3a = 5a.

Mathspace New South Wales – Year 11 Standard mathspace.co

Multiplication and division of algebraic terms do not require like terms. Coeﬃcients are multiplied or divided, and index laws may apply: 8a4 ÷ 2a = 4a3. Simplifying algebraic expressions follows the order of operations, which can include non-linear terms with higher powers or multiple variables, as well as decimals and fractions.

Interactive exploration Discover this concept in action online

mathspace.co

Example 1 Simplify: a 9a + 3b + 2a − 4b

Create a strategy Group the like terms, then combine.

Apply the idea 9a + 3b + 2a − 4b = 9a + 2a + 3b − 4b

Group like terms

= 11a − b

Combine like terms

Reﬂect and check The expression contains terms in a and b. The simpliﬁed result has one term in a and one in b.

b 5wy2 − 7w2 y + wy − 2w2 y + wy2

Create a strategy Group the like terms, then combine, leaving unlike terms unchanged.

Apply the idea 5wy2 − 7w2 y + wy − 2w2 y + wy2 = 5wy2 + wy2 − 7w2 y − 2w2 y + wy 2

= 6wy − 9w y + wy

Group like terms Combine like terms

Reﬂect and check The original expression has three term types: wy2, w2 y, and wy. The simpliﬁed result retains these three term types.

1.01 Simplify algebraic expressions mathspace.co

Example 2 Simplify: a 5r × (−7)

Create a strategy Separate the coefficients and pronumerals, then multiply.

Apply the idea 5r × (−7) = 5 × (−7) × r = −35r

Separate coeﬃcients and pronumerals Multiply coeﬃcients and simplify

b 3c2 × 5d3

Create a strategy Separate the coefficients and pronumerals, then multiply.

Apply the idea 3c2 × 5d3 = 3 × 5 × c2 × d3 2 3

= 15c d

Separate coeﬃcients and pronumerals Multiply coeﬃcients and simplify

Reﬂect and check Since c2 and d3 have different bases, index laws do not apply.

c 8ab2 × 3abc

Create a strategy Separate the coefficients and pronumerals, multiply, and apply index laws.

Apply the idea 8ab2 × 3abc = 8 × 3 × a × a × b2 × b × c 2 3

= 24a b c

Separate coeﬃcients and pronumerals Multiply coeﬃcients and apply index laws

Reﬂect and check • Evaluating the coeﬃcients: 8 × 3 = 24 • Simplifying the pronumerals: a × a = a2, b2 × b = b3, and one c

Mathspace New South Wales – Year 11 Standard mathspace.co

Create a strategy Multiply the fractions and pronumerals separately, then apply index laws.

Apply the idea Separate coeﬃcients and pronumerals

Multiply coeﬃcients and apply index laws

Reﬂect and check The fractional coeﬃcients simplify directly, and index laws combine like bases, resulting in a single term with higher powers.

Example 3 Simplify: a 30m ÷ 5m

Create a strategy Write as a fraction, then remove the common factors from the coefficients and pronumerals.

Apply the idea Write as a fraction

Simplify and remove common factors

Create a strategy Factorise the coefficients and pronumerals, then remove the common factors.

Apply the idea Factorise coeﬃcients and pronumerals

Simplify and remove common factors

1.01 Simplify algebraic expressions mathspace.co

Example 4 Simplify: a 3a2 + 5a × 7a

Create a strategy Multiply ﬁrst, then combine the like terms.

Apply the idea 3a2 + 5a × 7a = 3a2 + 35a2 2

= 38a

Multiply 5a × 7a Combine like terms

Reﬂect and check Multiplication creates like terms, enabling further simpliﬁcation.

b (7a − 2a) × 3ab

Create a strategy Combine the like terms in brackets, then multiply.

Apply the idea (7a − 2a) × 3ab = 5a × 3ab 2

= 15a b

Combine like terms in brackets Multiply

Create a strategy Combine the like terms in the numerator, multiply the denominator, then simplify.

Apply the idea Combine like terms and evaluate the multiplication

Simplify

Create a strategy Multiply the terms in the second expression, then check for like terms to combine.

Mathspace New South Wales – Year 11 Standard mathspace.co

Practice Ex 1

Simplify: a

5x + 8x

7c + 3c − c

6c + 3d + 3c − d

3x − 4x − x

7x + 4y − 2x + y

2x + 7y + 3x + z

3x + 7 − 2y + x

5x − (−3x) + 7 + (−2)

3ab + 9cd − (−2ab) − (−cd)

x2 + 6x + 5x + 10

5ab + 2bc − 2ab + 3cb

5a2 b + 7ab2 + 2a2 b − 3ab2

x − (−3x) − 8x + x2

7r × 4

o 4

xy + 2x y − 7 − 2xy + 3xy

Simplify: a

9 × 5u

2 × x × y × 3

4r × 6s

(−3c) × (−5d)

5e × (−5f )

3u3 × 5v5

12p × 5q2

(−3x) × (−10y)

6x2 × 2y2 × 3z2

9a × 3a × (−2a)

5a × 2b × a × 2b

3w3 x2 y × (−3)wx2 y3 × 2xy

−3e7 f 2 × 5e6 f 3 × e × f 2

3 2

4 3

5 3

m 5a b × 4a b o Ex 3

7c d × 3c × 2d

Simplify: a

6m ÷ 36

10ab ÷ 50ab

15x6 ÷ 5x6

(−32p7) ÷ 8p7

(9cd) ÷ 12de

(−12fg) ÷ (−3g2)

Simplify: a c e g i k

10y − 7y

m x − 4x + 4x − 16

Ex 2

5u × 8v ÷ 4w 25g ÷ 5 × 7f (8y ÷ 4x) × (12y ÷ 3x)

Mathspace New South Wales – Year 11 Standard mathspace.co

d f

(−3a) × (−5b) ÷ 10a2

h j

11xy × 5z ÷ x2 y2

(9a × 4b) ÷ (5 × 2c)

Ex 4

Simplify: a

(7c − 3c) × 2c

(19t + 5t) ÷ 4u

5f − 3 × 4f + 7f

(12a − 4a) ÷ (8 × 4a2)

2(4x2 + x × 2x) ÷ 16x

5(3a × 2b) + 4a × (−3b)

o 8

3x5 × 7y4 − 2x3 y2 × 3x2 y2

Simplify: 0.5a2 × 4ab

3.2x3 y − 1.8x3 y + 2.5x2 y2 − 0.7x2 y2

a 9

Simplify: a

2.7a4 b − 1.5a4 b +

a3 b2 × 2a

Simplify: a

2.75x4 y2 − 1.25x4 y2 + 0.9x3 y3 + 0.6x3 y3

3.1a3 b2 × (−2.4a2 b)

1.6w3 z4 − 0.9w3 z4 +

w2 z × 3wz2

Simplify: a

1.01 Simplify algebraic expressions mathspace.co

Extend your thinking 12

An isosceles triangle has two sides of length 3b cm and a third side of 2c cm. Write a simpliﬁed expression for the perimeter.

A rectangle has length 3x + y cm and width 2x − 2y cm. Find the perimeter in terms of x and y.

Are there values of a for which 3a × 2a = 5a2 holds? Provide examples or explain why not.

Simplify 7a + 5a × 2 in two ways: • Multiply first, then simplify. • Add first, then simplify. Explain the necessity of order of operations in algebraic simplification.

Addition pyramids assign the sum of two lower boxes to the box above. The example illustrates this with numbers: 13 9 4

4 −1

Complete the addition pyramids: a

4a + b a + 2b

3a − b

5x − 2y 3x + 2y

4a + 3b

3x − y

A rectangle has a length four times its width, with width 3a cm. Find an expression for the area.

Emma bakes x batches of chocolate chip cookies, each with y cookies. a

Determine the total number of chocolate chip cookies baked.

If each cookie has an average of y chocolate chips and she bakes the same number of cookies on 3 consecutive days, how many chocolate chips are used?

A piece of paper, with dimensions 8.5 cm by 11 cm, has a square of side x cut from each corner. Write a simpliﬁed expression for the perimeter of the resulting shape.

Mathspace New South Wales – Year 11 Standard mathspace.co

1.02 Substitution After this lesson, you will be able to… • substitute numerical values into algebraic expressions and equations to evaluate expressions and verify solutions. • substitute given values into formulas to determine an unknown quantity, including those with various number types (integers, decimals, fractions). • apply the order of operations correctly when performing substitutions. • recognise the importance of using correct units when substituting into formulas and stating results.

Substitution into expressions and equations Substitution is the process of replacing variables or pronumerals in algebraic expressions or equations with given numerical values. Substitution into expressions or equations can be used to calculate the value of them or to check if a number is a correct solution for an equation. Equation An expression showing the equality of 2 quantities, using the = sign between them. A mathematical formula asking for a solution so that the 2 expressions in that variable are equal. Example: x2 − 1 = x Expressions A mathematical statement formed by combining numbers and algebraic symbols using arithmetic operations. Example: a2 + 3ab − 2b2

Example 1 Substitute and evaluate: a r2, where r = −2

Create a strategy Substitute r = −2 then calculate the power.

Apply the idea r2 = (−2)2 =4

Substitute r = −2 Evaluate

1.02 Substitution mathspace.co

Reﬂect and check Squared values are positive, so 4 is correct. When applying powers to negative values, use brackets to ensure the power applies to the entire term, including the negative sign. For instance, (−2)2 = 4, whereas omitting brackets yields −22 = −4 by order of operations.

b 14.5 − 4x, where x = 4.2

Create a strategy Substitute x = 4.2 then multiply and subtract.

Apply the idea 14.5 − 4x = 14.5 − 4 × 4.2

Substitute x = 4.2

= 14.5 − 16.8

Evaluate 4 × 4.2

= −2.3

Evaluate

Reﬂect and check Since 16.8 > 14.5, a negative result is expected.

c 3x + 5, where x =

Create a strategy Substitute x =

then multiply and add.

Apply the idea Substitute x =

Evaluate 3 ×

Rewrite 5 as

Evaluate

Reﬂect and check The result

= 6.5 is positive, as expected since both terms are positive.

Mathspace New South Wales – Year 11 Standard mathspace.co

1.02 Practice questions What do you remember? 1

Explain why the order of operations matters when substituting into 3x + 4y2.

What is the difference between substituting into an expression like 2x + 3 and an equation like 2x + 3 = 7?

Why must units be considered when substituting into formulas like A = l × w?

Explain why the order of operations is important when evaluating an expression like 2 + 3 × 42. Show the correct evaluation and what would happen if the expression were evaluated left to right without following the order of operations.

Evaluate the expressions by applying the order of operations: a

(5 + 3) × 22

10 ÷ 2 + 3 × 4

2 × (3 + 5 ÷ 5)2

Practice Ex 1

Ex 2

Substitute and evaluate: a

a + b + c, where a = 10, b = 15 and c = 11

kr, where k = 5 and r = 6

y2, where y = 4

r2, where r = −4

4r × 3s + 38, r = −8 and s = 5

6r × 4s, where r =

8p2 + 5q3, where p = 4 and q = −1

5y − z2 + 9, where y = 7 and z = 4

The formula for the area of a triangle is A =

and s = 7

bh. Determine the area for:

Base = 7 cm and height = 12 cm

Base = 7 cm and height = 5 cm

Base = 25 cm and height = 16 cm

Base = 10 cm and height = 9 cm

Substitute the given values into geometric formulas to determine the perimeter or area: a

The formula for the perimeter of a square with side length a is P = 4a. Determine P if each side is 9 cm.

The formula for the area of a square with side length s is A = s2. Determine A if each side is 6 cm.

The formula for the area of a rectangle is A = l × w. Determine the area if length is 2 cm and width is 3 cm.

The formula for the perimeter of a triangle with sides x, y, z is P = x + y + z. Determine P if x = 6 cm, y = 3 cm, z = 7 cm.

The formula for the perimeter of a rectangle is P = 2(l + w). Determine the perimeter if width is 10 cm and length is 5 cm.

1.02 Substitution mathspace.co

The formula for the area of a rhombus is A = Determine the area for the diagonal lengths:

xy, where x, y are diagonal lengths.

Diagonals 6 cm and 8 cm

Diagonals 3.5 cm and 7.2 cm

Diagonals

Diagonals 4 mm and 10 mm

cm and

The formula for the volume of a rectangular prism is V = l × w × h. Determine the volume for the dimensions: a

Length 5 cm, width 3 cm and height 7 cm

Length 2.4 cm, width 6.5 cm and height 4.1 cm

Length

Length 10 mm, width 15 mm and height 20 mm

cm, width

cm and height

The equation of a line is y = mx + c. Determine y if m = 6, x = −4 and c = 9.

Complete the table of values for the formula: a

q = 2p − 3

q = −2p − 3

p p 0 1 2 3 4 q 13

Verify the solution to the equations by substitution: a

2x + 5 = 11, where x = 3

x2 − 4 = 5, where x = 3

The Celsius to Fahrenheit conversion formula is F =

The density formula is D =

The arithmetic sequence term formula is T = a + (n − 1)d. Find T if a = 6, n = 5 and d = 8.

Evaluate x2 − 9x + 18 for: a

x=2

+ 32. Determine F if C = 15.

. Determine the density if mass is 6.16 g and volume is 5.6 cm3.

x=4

x = −1

6m2 + 4n3, where m = 2 and n = −3

x=0

Substitute and evaluate the expressions: a

5x × 2y + 15, where x = −3 and y = 4

Verify if x = −2 is a solution to 5x + 7 = −3 by substitution.

The simple interest formula is I =

The formula for the surface area of a rectangular prism is S = 2(lw + wh + lh). Determine the surface area if length is 8 cm, width is 7 cm, and height is 9 cm.

The formula for the sum of n terms in an arithmetic sequence is S = Determine S if n = 10, a = 3 and d = 9.

Mathspace New South Wales – Year 11 Standard mathspace.co

. Determine the interest if P = $1000, R = 6 and T = 7.

(2a + (n − 1) d).

The formula to determine the number of edges of a 3D shape is E = V + F − 2, where V is vertices and F is faces. Determine the edges for: a

7 vertices and 7 faces

8 vertices and 6 faces (F − 32). Convert 86°F to Celsius.

The Fahrenheit to Celsius conversion formula is C =

Newton’s second law is F = ma, where F is force (Newtons), m is mass (kg), and a is acceleration (m/s2). Calculate the force for: a

Mass 0.83 kg and acceleration 9 m/s2

Mass 0.66 kg and acceleration 13 m/s2

Extend your thinking 26

Solve the equation 3x − 5 = 10 for x, then verify the solution by substitution.

The vertical position of a ball launched from a cliff is y = 14.7t − in seconds. a

Determine the position after: i

t2 metres, where t is time

2 seconds

3 seconds

iii

8 seconds

Which time has the ball above the launch point?

The range of a projectile launched at an angle is given by R =

, where R is the

range (metres), v is the initial velocity (m/s), θ is the launch angle, and g = 9.8 m/s2 is the acceleration due to gravity. Determine the range for each scenario rounded to two decimal places: a

v = 20 m/s and θ = 30°

v = 15.5 m/s and θ = 45°

v = 25 m/s and θ = 30°

v = 30 km/h and θ = 60°

1.02 Substitution mathspace.co

Solve for a pronumeral with substitution and algebra To ﬁnd the pronumeral (letter) that is not on the left, substitute the known numbers and use the inverse operations to solve. For formulas with fractions, multiply or divide to clear the fraction before solving. Rearrange step-by-step to get the letter alone, keeping units consistent.

Example 2 The formula to determine the perimeter of a rectangle is P = 2(a + b). Find b when P = 22 m and a = 2 m.

Create a strategy Substitute the numbers, and solve for b using inverse operations.

Apply the idea P = 2(a + b)

Write the formula

22 = 2(2 + b)

Substitute P = 22 and a = 2

22 = 4 + 2b

Expand the brackets

18 = 2b

Subtract 4 from both sides

9=b

Divide both sides by 2

b=9m

Make b the subject

Reﬂect and check To verify the answer, substitute back a = 2 and b = 9 to check if the result is still P = 22. P = 2(a + b)

Write the formula

= 2(2 + 9)

Substitute a = 2 and b = 9

= 2 × 11

Evaluate the addition

= 22

Evaluate

This conﬁrms b = 9 is correct.

Example 3 The formula to determine the area of a triangle is A = b = 5 mm.

. Find h when A = 30 mm2 and

Create a strategy Substitute the numbers, multiply both sides by 2, then divide by b.

1.03 Evaluate the subject of a formula mathspace.co

Practice Ex 1

Use the given formulas to calculate the requested values: a

A for A = l × w, where l = 7 cm and w = 4 cm

V for V = IR, where I = 2 A and R = 12 Ω

A for A =

s for s =

E for E =

S for S = 2(ab + bh + ah), where a = 3 cm, b = 4 cm and h = 5 cm

, where b = 10 mm and h = 6 mm , where d = 120 km and t = 2 h mv2, where m = 8 kg and v = 5 m/s

Ex 2

Calculate b for P = 2(a + b), where P = 30 m and a = 5 m.

Ex 3

Calculate h for A =

Use the given formulas to calculate the requested values:

u for v = u + at, where v = 45, a = 3 and t = 4

m for B =

r for C = 2π r, where C = 31.4 cm, using π = 3.14

d for s =

he formula for the circumference of a circle is C = 2π r. Calculate r given that T C = 47.1 cm and using π = 3.14.

Using the radius from part (a), calculate the area using the formula, A = π r2. Round to one decimal place.

The formula for velocity is v = u + at. Calculate a given that v = 4.8, u = 5 and t = 10.

Using the acceleration from part (a), calculate the distance using the formula, s = ut +

, where b = 8 mm and A = 48 mm2.

, where B = 21.5 and h = 1.6 m, rounded to two decimal places

, where s = 50 km/h and t = 3 h

at2, where u = 5 and t = 10. mv2. Calculate v given that E = 400 J, m = 10 kg,

he formula for kinetic energy is E = T rounded to two decimal places.

If the mass doubles to m = 20 kg, calculate the new velocity for the same energy E = 400 J, rounded to two decimal places.

The formula for depreciation is S = V (1 − r)n. Calculate V, given that S = 8000, r = 0.05 and n = 4. Round your answer to the nearest whole number.

The formula for surface area of a prism is S = 2(ab + bh + ah). Calculate h given that a = 4 mm, b = 8 mm and S = 304 mm2.

The formula for arithmetic series sum is S = d = 4 and S = 840.

(2a + (n − 1) d). Calculate a given that n = 20,

1.03 Evaluate the subject of a formula mathspace.co

The formula for Newton’s second equation is s = ut + u = 300 and t = 20.

at2. Calculate a given that s = 4000,

Extend your thinking 14

+ . Calculate b given that a = 12 and c = 27,

he formula for parallel resistance is T rounded to two decimal places.

If a third resistor d = 18 is added in parallel, calculate the new equivalent resistance anew, rounded to two decimal places.

The thin lens equation relates the focal length ( f ) of a lens to the object distance (do) and the image distance (di) by the formula:

If a lens has a focal length f = 10 cm and an object is placed at a distance do = 30 cm from the lens, calculate the image distance di.

Rearrange the thin lens equation to make the focal length f the subject. Then, use your rearranged formula to calculate f if an object is placed at do = 20 cm and its image is formed at di = 20 cm.

The period (T ) of a simple pendulum is given by the formula T = 2π length of the pendulum and g is the acceleration due to gravity.

, where L is the

Calculate the length L (in metres) of a pendulum that has a period T = 2.0 s on Earth, where g ≈ 9.8 m/s2. Use the approximation π ≈ 3.14. Round your answer for L to two decimal places.

The acceleration due to gravity on Mars is gMars ≈ 3.71 m/s2. What would be the period of the pendulum from part (a) if it were on Mars? Use the value of L that you calculated and rounded in part (a), and continue to use π ≈ 3.14. Round your ﬁnal answer for the period on Mars to two decimal places.

Mathspace New South Wales – Year 11 Standard mathspace.co

1.04 Speed, distance and time After this lesson, you will be able to… • calculate speed, distance, or time given the other two quantities using the formula s =

and its variations.

• convert units of speed (e.g., km/h to m/s) as required for calculations. • calculate reaction distance, braking distance, and total stopping distance for a vehicle. • understand the concept of relative speed for objects moving towards each other. • apply these formulas to solve practical problems related to motion and vehicle safety.

Speed, distance and time Speed The absolute value of an object’s velocity. It represents how fast the object is moving. Distance The length between two points. Distance is a positive scalar quantity. Time The duration of motion, measured in seconds, minutes, or hours.

The core relationship between speed (s), distance (d), and time (t) is given by the formula, s = This can be rearranged to ﬁnd distance as d = s × t or time as t =

To remember these variations, use these formula triangles: D S

D T

÷ T

÷ S

Speed

D T

Time

S × T Distance

Ensure that units are consistent, for example, km/h with kilometre and hours, or m/s with metres and seconds. For relative speed, if two vehicles travel toward each other, their relative speed is the sum of their speeds. Relative distance or time can be calculated using this relative speed. 1.04 Speed, distance and time mathspace.co

Stopping distances Stopping distances of vehicles are calculated using the formula: Stopping distance = Reaction distance + Braking distance The reaction distance is determined by Reaction distance = s × t, where s is the speed (in m/s) and t is the reaction time (in seconds). The braking distance is typically given by ks2 (in metres), where k is a braking coeﬃcient. To ensure accuracy, speeds often need to be converted to m/s using the relationship s =

, such as converting from km/h to m/s.

In scenarios involving multiple vehicles, comparing their stopping distances helps understand how speed and reaction time affect safety outcomes.

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Example 3 A car travels at 60 km/h with a reaction time of 1.5 s and braking distance coeﬃcient k = 0.05. Calculate the stopping distance (in metres) rounded to two decimal places.

Create a strategy Convert speed to m/s and calculate reaction and braking distances to use the stopping distance formula.

Apply the idea Convert the speed from km/h to m/s rounded to two decimal places: Write the initial speed Convert km/h to m/s: 1 km = 1000 m and 1 h = 3600 s

Simplify Evaluate Calculate the reaction distance: Reaction distance = s × t

Write the formula

= 16.67 × 1.5

Substitute s = 16.67 and t = 1.5

= 25.005 m

Evaluate

Calculate the braking distance rounded to two decimal places: Braking distance = ks2

Write the formula 2

= 0.05 × 16.67

Substitute k = 0.05 and s = 16.67

= 13.89 m

Evaluate and round

1.04 Speed, distance and time mathspace.co

Calculate the stopping distance: Stopping distance = Reaction distance + Braking distance

Write the formula

= 25.005 + 13.89

Substitute the values

= 38.895 m

Evaluate

Example 4 Two cars travel at 50 km/h and 70 km/h with reaction times of 1 s and 1.2 s, respectively, and k = 0.04. Compare their stopping distances.

Create a strategy For each car, convert speed to m/s and calculate the reaction and braking distances to use the stopping distance formula.

Apply the idea Convert the speed of the ﬁrst car from km/h to m/s rounded to two decimal places: Write the initial speed

Convert km/h to m/s: 1 km = 1000 m and 1 h = 3600 s

Simplify

Evaluate

Calculate the first car’s reaction distance: Reaction distance1 = s × t

Write the formula

= 13.89 × 1

Substitute s = 13.89 and t = 1

= 13.89 m

Evaluate

Calculate the first car’s reaction distance rounded to two decimal places: Braking distance1 = ks2

Write the formula 2

= 0.04 × 13.89

Substitute k = 0.04 and s = 13.89

= 7.72 m

Evaluate and round

Calculate the first car’s stopping distance: Stopping distance1 = Reaction distance + Braking distance

Write the formula

= 13.89 + 7.72

dd the reaction and A braking distances

= 21.61 m

Evaluate

Mathspace New South Wales – Year 11 Standard mathspace.co

1.04 Practice questions What do you remember? 1

Match each term to its description: i

Speed

Distance

How far something travels

How far something travels in a given time

Calculate each speed in metres per second, given in kilometres per hour: a

120 km/h

60 km/h

90 km/h

30 km/h

Describe how the relative speed of two vehicles approaching each other is calculated, and why it is used in distance problems.

Explain two factors that increase a vehicle’s stopping distance and how each affects the reaction or braking distance.

Practice Ex 1

Ex 2

Calculate the speed: a

A cyclist travels 135.5 km in 2.5 hours

A car travels 240 km in 4 hours

A bus covers 180 km in 3 hours

A train travels 315 km in 3.5 hours

A runner covers 12 km in 0.8 hours

A motorbike travels 208 km in 2.6 hours

Calculate the distance: a

A runner moves at 10.5 m/s for 25.5 seconds

A car travels at 72.5 km/h for 4.8 hours

A cyclist rides at 15 km/h for 3 hours

A swimmer moves at 2 m/s for 120 seconds

A bus travels at 55 km/h for 2.4 hours

A motorbike rides at 90 km/h for 1.5 hours

Calculate the time: a

A bus travels 187.5 km at 62.5 km/h

A car covers 360 km at 90 km/h

A cyclist travels 60 km at 20 km/h

A train covers 240 km at 80 km/h

A runner travels 15 km at 12 km/h

A motorbike covers 180 km at 72 km/h

For these vehicles approaching each other: a

Two cars approach each other at 60 km/h and 80 km/h. Calculate their relative speed.

Two trains, 500 km apart, approach each other at 120 km/h and 80 km/h. Calculate the time until they meet.

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 3

Calculate the stopping distance: a

A car travels at 70 km/h with a reaction time of 1.2 s and braking coeﬃcient k = 0.04.

A truck travels at 20 m/s with a reaction time of 1.5 s and a braking distance calculated using

Ex 4

, where s is speed in m/s.

Two motorcycles travel at 45 km/h and 80 km/h with reaction times of 0.9 s and 1.3 s, respectively, and k = 0.035. Compare their stopping distances.

James and Valentina travel in opposite directions from an airport at 48 km/h and 39 km/h. Calculate their distance apart after 1.8 hours.

Carl and Amelia travel in the same direction. Carl travels at 18 km/h for 180 km. Amelia travels at 54 km/h. Calculate the time for Amelia to catch up with Carl.

Extend your thinking 13

A round trip averages 75 km/h over 2.4 hours there and 2.8 hours back. Calculate the total distance.

Katrina and Ned travel toward a store. Ned leaves 2.5 hours after Katrina at 80 km/h. They arrive after 3.5 hours of Ned’s travel. Calculate the distance Ned travels and Katrina’s average speed, rounded to one decimal place.

Professor Lopez is observing two robotic rovers, TerraTrek and NovaGlide, on a simulated Martian landscape. They both start from the same base station and travel along the same linear path. TerraTrek, the slower rover, begins its exploration ﬁrst. It travels at a constant speed of 5 m/min and has already covered a distance of 150 m from the base. At this moment, the faster rover, NovaGlide, is dispatched from the base station and travels at a speed of 20 m/min to catch up with TerraTrek. Calculate the total distance TerraTrek will have travelled from the base station by the time NovaGlide catches up to it.

A round trip averages 100 km/h for 1.2 hours on a motorway, 40 km/h for 1.8 hours on suburban streets, and 25 km/h for 1.2 hours in school zones. Calculate the total distance.

A truck travels at 20 m/s with a reaction time of 1.5 s and a braking distance of 20 m. It needs to stop within 45 m to avoid a collision. Will the truck stop in time, and if not, by how much will it miss the target distance?

1.04 Speed, distance and time mathspace.co

1.05 BAC (Blood Alcohol Content) After this lesson, you will be able to… • calculate the number of standard drinks in an alcoholic beverage. • apply the appropriate formula to estimate Blood Alcohol Content (BAC) for males and females. • calculate the estimated time required for BAC to reach zero. • interpret BAC values in the context of legal driving limits in NSW. • identify and explain limitations of the methods used to estimate BAC.

Standard drinks A standard drink in Australia contains 10 g of pure alcohol. Calculate the number of standard drinks (N ) using:

V A

is the volume in millilitres is the alcohol percentage (alc/vol) as a decimal

The constant 0.789 is the density of alcohol in grams per millilitre. Division by 10 accounts for 10 grams per standard drink. Drink volumes are typically given in millilitres, for example, 375 mL for a can of beer, so no unit conversion is needed. An alternative formula, N = litres to millilitres. Since

, uses volume in litres (VL), where 1000 converts = 100, this simpliﬁes to 0.789 × VL × A × 100, but the millilitre-based

formula is simpler as drink labels use millilitres.

Example 1 Calculate the number of standard drinks in a 375 mL can of beer with 3.5% alc/vol, rounded to one decimal place.

Create a strategy Use N =

, with volume in millilitres and percentage as a decimal, and calculate.

Mathspace New South Wales – Year 11 Standard mathspace.co

The term 10N converts drinks to grams, 7.5H accounts for metabolism (0.015 g /100 mL per hour), and denominators reﬂect body composition. Negative BAC is set to zero. Time to zero BAC is:

is the hours to reach zero BAC

NSW legal BAC limits are 0.00 for learner/provisional drivers and 0.05 for full licence holders. BAC above 0.08 significantly impairs driving.

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Example 2 A female weighing 60 kg consumes 5 standard drinks over 3 hours. a Calculate her BAC, rounded to three decimal places.

Create a strategy Use BACfemale =

, substitute values, and then calculate.

Apply the idea Write the formula

Substitute N = 5, H = 3 and M = 60

Simplify numerator and denominator

Evaluate and round

Her BAC is approximately 0.083.

b Calculate the time to reach zero BAC, rounded to two decimal places.

Create a strategy Use t =

, substitute BAC, and calculate.

Mathspace New South Wales – Year 11 Standard mathspace.co

Apply the idea Write the formula

Substitute BAC = 0.083

Evaluate and round

She must wait approximately 5.53 hours.

Reﬂect and check Verify that after 5.53 hours, BAC reaches zero. BAC reduction = t × 0.015

Write the formula

= 5.53 × 0.015

Substitute t = 5.53

= 0.08295

Evaluate

The reduction of 0.08295 matches the initial BAC of 0.083, conﬁrming the time to zero.

Example 3 A male weighing 80 kg consumes 4 standard drinks over 2 hours. Calculate his BAC, rounded to three decimal places.

Create a strategy Use BACmale =

, substitute values, and then calculate.

Apply the idea Write the formula

Substitute N = 4, H = 2 and M = 80

Simplify numerator and denominator

Evaluate and round

His BAC is approximately 0.046.

Reﬂect and check Verify that 0.046 is below the NSW full licence limit of 0.05. BAC = 0.046

State calculated BAC

< 0.05

Compare to legal limit

The BAC is below 0.05, indicating legal driving ability but potential mild impairment.

1.05 BAC (Blood Alcohol Content) mathspace.co

The BAC formulas are:

Why are there different formulas for males and females? 4

In NSW, what is the legal BAC limit for learner or provisional drivers?

Name one factor that the BAC formulas do not account for that can affect actual blood alcohol content.

Practice Ex 1

Ex 2

Ex 3

Calculate the number of standard drinks for each scenario, rounded to one decimal place: a

A 330 mL can of beer with 4.0% alc/vol

A 750 mL bottle of wine with 12.5% alc/vol

A 30 mL shot of vodka with 40% alc/vol

A female weighing 65 kg consumes 4 standard drinks over 2 hours. Calculate: a

Her BAC, rounded to three decimal places

The time to reach zero BAC, rounded to two decimal places

A male weighing 75 kg consumes 3 standard drinks over 1 hour. Calculate his BAC, rounded to three decimal places.

Calculate the time to reach zero BAC, rounded to two decimal places, for: a

A person with BAC of 0.03

A person with BAC of 0.10

Calculate the number of standard drinks, rounded to one decimal place, for: a

A 150 mL glass of champagne with 12% alc/vol

A 700 mL bottle of wine with 13.5% alc/vol

Calculate BAC, rounded to three decimal places, for: a

A male, 87 kg, consumes 4 standard drinks in 4 hours

A male, 70 kg, consumes 6 standard drinks in 2 hours

Lachlan (male, 81 kg) drinks two 375 mL bottles of light beer (2.7% alc/vol) in the ﬁrst hour and one every hour for the next three hours. Calculate: a

The total number of standard drinks to one decimal place

His BAC to three decimal places

A male (70 kg) consumes a 750 mL bottle of spirits (40% alc/vol) over 5 hours. Calculate: a

The number of standard drinks, rounded to two decimal places

His BAC, rounded to two decimal places

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 4

A female weighing 55 kg consumes 2 standard drinks over 1 hour after a high-fat meal. Calculate her BAC using the standard formula, then estimate the actual BAC if food reduces absorption by 20%. Round your answer to three decimal places.

Extend your thinking 15

Explain why the BAC formula might underestimate the actual blood alcohol content for some individuals.

A male (85 kg) and a female (65 kg) each consume 5 standard drinks over 2 hours. a

Calculate their BAC values, rounded to three decimal places.

Explain the difference in their BAC values.

In NSW, full licence holders have a BAC limit of 0.05. Calculate the maximum number of standard drinks a female (60 kg) can consume in 2 hours without exceeding this limit.

A provisional driver (BAC limit 0.00) has a BAC of 0.08 at 10:00 p.m. Calculate the earliest time they can legally drive.

1.06 Medication doses After this lesson, you will be able to… • identify the appropriate formula (Fried’s, Young’s, or Clark’s) for calculating a child’s medication dosage based on age or weight. • substitute values correctly into Fried’s, Young’s, and Clark’s formulas to calculate dosages for infants and children based on age and weight. • round calculated dosages to a practical level of precision (e.g., nearest milligram).

Fried’s formula for infants Fried’s formula calculates medication dosages for infants aged 1 to 2 years based on age in months:

is the dosage for infants aged 1 to 2 years

Limitations include its reliance on age alone, which may not account for weight variations in infants, potentially leading to under- or overdosing in atypical cases.

1.06 Medication doses mathspace.co

Ex 3

Using Clark’s formula:

Calculate the dosage for a 5-year-old weighing 35 kg with an adult dosage of 600 mg, rounding to the nearest milligram. 7

Using Fried’s formula:

Calculate the dosage for a 13-month-old child needing paracetamol with an adult dose of 400 mg, rounding to the nearest milligram. 8

An 8-year-old child weighing 25 kg needs ibuprofen with an adult dosage of 500 mg. Calculate: a

The dosage using Young’s formula, rounded to the nearest milligram

The dosage using Clark’s formula, rounded to the nearest milligram

Using the table of ideal masses for children: Age

Mass of females (kg)

17.9

Using Clark’s formula:

Calculate the dosage for a 5-year-old girl with an adult dosage of 600 mg, rounding to the nearest milligram. 10

Using Fried’s formula:

Calculate the following dosages, rounding to the nearest milligram:

Find the dosage for a 12-month-old child needing prednisolone with an adult dose of 300 mg.

Find the dosage for a 20-month-old child needing paracetamol with an adult dose of 500 mg.

Using Clark’s formula:

Calculate the following dosages, rounding to the nearest milligram: a

Find the dosage for a 7-year-old with a mass of 20 kg and an adult dosage of 400 mg.

Find the dosage for a 10-year-old with a mass of 30 kg and an adult dosage of 600 mg.

1.06 Medication doses mathspace.co

Extend your thinking 12

An 16-month-old child needs prednisolone with an adult dose of 180 mg. a

Calculate the dosage using Fried’s formula, rounding to the nearest milligram.

Calculate the dosage using Young’s formula (assuming 0.92 years), rounding to the nearest milligram.

Explain why the results differ between the two formulas.

Using the table of ideal masses for children: Age

Mass of males (kg)

Mass of females (kg)

12.5

12.0

14.0

14.2

16.3

15.4

18.4

17.9

20.6

19.9

22.9

22.4

25.6

25.8

28.6

28.1

32.0

31.9

35.6

36.9

39.9

41.5

Calculate the Clark’s formula dosage for a 4-year-old girl with an adult dosage of 500 mg, rounding to the nearest milligram.

Determine the adult dosage for a 3-year-old boy with a dosage of 220 mg using Clark’s formula.

A child’s mass is 16 kg at age 5, and they need ibuprofen with an adult dose of 200 mg. a

Calculate dosages using Clark’s and Young’s formulas, rounding to the nearest milligram.

Discuss why a doctor might prefer one formula over the other in this case.

A medication’s adult dosage is 1.2 g daily, given in tablets of 300 mg. A 10-year-old child (mass 32 kg) needs a dosage based on Clark’s formula. a

Calculate the child’s dosage in milligrams, rounding to the nearest milligram.

Calculate the number of tablets the child should take daily.

Explain the practical adjustment from the calculated dosage to the tablet amount.

Mathspace New South Wales – Year 11 Standard mathspace.co

1.07 Other formulas After this lesson, you will be able to… • substitute given values into various formulas from different contexts (e.g., finance, health, science). • evaluate the subject of a given formula after substitution. • solve for a specified pronumeral in a variety of formulas after substituting known values, using inverse operations. • manage calculations involving decimals, fractions, squares, square roots, and exponential terms (like ex) within formulas. • ensure consistency of units and apply appropriate rounding in the context of the problem.

Other formulas Formulas are used in diverse ﬁelds to solve practical problems. Examples include calculating interest in ﬁnance, body mass index in health, volumes in geometry, and population growth in biology. Accurate substitution ensures reliable results, while complex formulas, such as those with exponents, model dynamic systems like population or investment growth.

Example 1 The simple interest formula is I = PRT , where I is interest ($), P is principal ($), R is annual rate (decimal), and T is time (years). a Calculate I for P = 1000, R = 0.05, and T = 2 years.

Create a strategy Substitute P = 1000, R = 0.05, and T = 2 into I = PRT and calculate.

Apply the idea I = PRT

Write the formula

= 1000 × 0.05 × 2

Substitute P = 1000, R = 0.05 and T = 2

= $100

Evaluate

1.07 Other formulas mathspace.co

b Calculate T, rounded to the nearest year for I = 350, P = 4000, and R = 0.03.

Create a strategy Substitute I = 350, P = 4000, R = 0.03, and into I = PRT then use the inverse operation.

Apply the idea

Write the formula Substitute I = 350, P = 4000 and R = 0.03

Evaluate the multiplication

Divide both sides by 120

Evaluate and round

Make T the subject

Example 2 The body mass index formula is BMI = and h is height (m).

, where BMI is body mass index (kg/m2), m is mass (kg),

a Calculate BMI for a person with m = 70 kg and h = 1.75 m, rounded to three decimal places.

Create a strategy Substitute m = 70 and h = 1.75 into BMI =

Apply the idea Write the formula

Substitute m = 70 and h = 1.75

Evaluate the index

Evaluate and round

b Calculate h for BMI = 21 kg/m2 and m = 47 kg, rounded to two decimal places.

Create a strategy Substitute BMI = 21 and m = 47 into BMI =

, then use the inverse operations.

Mathspace New South Wales – Year 11 Standard mathspace.co

Apply the idea Write the formula

Substitute BMI = 21 and m = 47

Multiply both sides by h2

Divide both sides by 21

Take the square root of both sides

Evaluate and round

c Calculate m for BMI = 19 kg/m2 and h = 1.45 m, rounded to two decimal places.

Create a strategy Substitute BMI = 19 and h = 1.45 into BMI =

, then use the inverse operations.

Apply the idea Write the formula

Substitute BMI = 19 and h = 1.45

Evaluate the power

Evaluate and round

Make m the subject

Example 3 The population growth formula is P = P0 ert , where P is the ﬁnal population, P0 is the initial population, r is the growth rate (per year), t is time (years), and e ≈ 2.718. a Calculate P for P0 = 1000, r = 0.03 and t = 5 years, rounded to the nearest person.

Create a strategy Substitute P0 = 1000, r = 0.03 and t = 5, into P = P0 ert.

1.07 Other formulas mathspace.co

1.07 Practice questions What do you remember? 1

Identify the number of variables in each formula: a

I = PRT

BMI =

P = P0 ert

Identify the key variables in these formulas, specifying their units of measurement: a

Simple interest

Population growth

Body mass index

Practice Ex 1

Ex 2

The simple interest formula is I = PRT, where I is interest ($), P is principal ($), R is annual rate (decimal), and T is time (years). a

Calculate I for P = 2500, R = 0.04, and T = 3 years.

Calculate T for I = 400, P = 5000, and R = 0.04.

Calculate P for I = 600, R = 0.05, and T = 2 years.

Calculate R for I = 750, P = 10 000, and T = 3 years.

Calculate I for P = 3000, R = 0.06, and T = 18 months.

Calculate T in years for I = 200, P = 4000, and R = 0.05.

Calculate P for I = 360, R = 0.03, and T = 24 months.

Calculate R for I = 480, P = 8000, and T = 36 months.

The body mass index formula is BMI =

, where BMI is body mass index (kg/m2), m is mass

(kg), and h is height (m), rounded to two decimal places.

Ex 3

Calculate BMI for a person with m = 85 kg and h = 1.8 m.

Calculate h for BMI = 24 kg/m2 and m = 70 kg.

Calculate m for BMI = 20 kg/m2 and h = 1.65 m.

Calculate BMI for a person with m = 75 kg and h = 1.6 m.

Calculate h for BMI = 22 kg/m2 and m = 60 kg.

Calculate m for BMI = 18 kg/m2 and h = 1.5 m.

Calculate BMI for a person with m = 72 000 g and h = 1.7 m.

Calculate h for BMI = 23 kg/m2 and m = 65 000 g.

Calculate m in kg for BMI = 21 kg/m2 and h = 170 cm.

The population growth formula is P = P0 ert , where P is the ﬁnal population, P0 is the initial population, r is the growth rate (per year), t is time (years), and e = 2.718. a

Calculate P for P0 = 800, r = 0.03 and t = 5 years, rounded to one decimal place.

Calculate P0 for P = 1200, r = 0.03 and t = 3 years, rounded to the nearest integer.

1.07 Other formulas mathspace.co

The volume of a sphere formula is V = Use π ≈ 3.1416.

π r3, where V is volume (m3) and r is radius (m).

Calculate V for a sphere with r = 0.5 m, rounded to three decimal places.

Calculate r for a sphere with V = 33.510 m3, rounded to two decimal places.

Newton’s second law is F = ma, where F is force (N), m is mass (kg), and a is acceleration (m/s2). a

Calculate F for m = 10 kg and a = 2 m/s2.

Calculate m for F = 50 N and a = 5 m/s2.

Calculate a for F = 100 N and m = 20 kg.

Calculate F for m = 5000 g and a = 3 m/s2.

Calculate m in kg for F = 80 N and a = 400 cm/s2.

Calculate a in

for F = 60 N and m = 15 000 g.

Extend your thinking 8

The simple interest formula is I = PRT , where I is interest ($), P is principal ($), R is annual rate (decimal), and T is time (years). a

Calculate I for P = 5000, R = 0.04, and T = 3 years.

Compare the result with P = 4000, R = 0.06, T = 2 years, and explain which variable (P, R, or T ) has the greatest impact on the interest.

The volume of a sphere formula is V = Use π ≈ 3.1416.

π r3, where V is volume (m3) and r is radius (m).

A spherical water tank has a radius of 1.2 m. Calculate the volume, rounded to two decimal places.

If the tank is ﬁlled with water, and 1 cubic metre of water weighs 1000 kg, calculate the total weight of the water in the tank, rounded to the nearest kilogram.

Mathspace New South Wales – Year 11 Standard mathspace.co

1.08 Change the subject of a formula After this lesson, you will be able to… • identify the current subject of a given formula. • rearrange a linear formula to make a specified variable the subject, using inverse operations. • rearrange a non-linear formula of the form y = ax2 + c to make x the subject. • apply inverse operations correctly and in the appropriate order to isolate a variable. • substitute values into a rearranged formula to solve for the new subject.

Change the subject of a formula Changing the subject of a formula involves rearranging it to isolate a speciﬁc variable using inverse operations applied in reverse order. This skill is essential for solving problems in contexts like physics, economics, and engineering, where formulas model real-world relationships. Equation An expression showing the equality of two quantities, using the = sign. A mathematical formula asking for a solution so that the two expressions in that variable are equal, for example x2 − 1 = x. Linear Something that can be represented or modelled by a straight line. Linear equation An equation involving linear expressions. The general form of a linear equation in one variable is ax + b = c where a, b and c are constants. Non-linear Functions or graphs that cannot be represented by a straight line or a linear function.

A subject is a pronumeral isolated on one side of a formula, expressed in terms of other variables. For example, in y = mx + c, y is the subject. To change the subject, apply inverse operations, as you would when solving a linear equation. • Addition and subtraction are inverse operations • Multiplication and division are inverse operations • Power of 2, and square root are inverse operations

1.08 Change the subject of a formula mathspace.co

Example 1 In a production model, the total cost formula C = k + nx relates cost C ($) to ﬁxed cost k ($), number of units x (units), and cost per unit n($/unit). Make x, the number of units produced, the subject of the formula, then determine its value given C = $500, k = $200, and n = $10/unit.

Create a strategy Change the subject to x by subtracting k and dividing by n.

Apply the idea Write the formula

Subtract k from both sides

Divide both sides by n

Make x the subject

Substitute C = 500, k = 200 and n = 10

Evaluate

Reﬂect and check Verify by substituting x = 30: C = k + nx

Write the formula

= 200 + 10 × 30

Substitute k = 200, n = 10 and x = 30

= 200 + 300

Evaluate the multiplication

= $500

Evaluate

The result matches C = $500, conﬁrming accuracy. Producing 30 units is feasible for small-scale operations.

Mathspace New South Wales – Year 11 Standard mathspace.co

1.08 Practice questions What do you remember? 1

Identify the pronumeral that is the subject of each formula: a

y = 3x + 5

z = 4w − 7

k = m2 + n

Identify the ﬁrst inverse operation to change the subject to x in each formula: a

a=x+b

a = bx

a = x2 + c

Practice 3

Change the subject to x: a

y = 2x + c

m = h − 3x

a = 5x + b

p = 4x − q

k=x+n

t = 7x

w = 2x − m

z = 6x + 1

Change the subject to the speciﬁed pronumeral in each formula: a

A = P (1 + r)t, change to P

y = 3x2 + 4, change to x

y = −4x2 + 10, change to x

y = 6mx − 9, change to m

, change to x

− 4, change to k

Change the subject to x in the following formulas, justifying each step with properties of equality: a

8y = x + 2z

=h−k

The perimeter formula for a rectangle is P = 2l + 2w, where P is the perimeter (cm), l is the length (cm), and w is the width (cm). Make w the subject of the formula, then calculate w for P = 20 cm and l = 7 cm.

Ex 1

In a manufacturing model, the total cost formula C = f + cx relates cost C ($) to ﬁxed cost f ($), number of items x (items), and cost per item c ($/item). Make x the subject then determine the number of items produced given C = $800, f = $300, and c = $5/item.

Ex 2

The height of a bridge arch is modelled by y = −3x2 + 12, where y is height (in m) and x is horizontal distance from the centre (in m). Make x the subject then determine the distance from the centre when the height is y = 9 m.

In a retail business, the profit formula P = sx − c relates profit P ($) to selling price per item s ($/item), number of items sold x (items), and fixed costs c ($). Make x the subject then determine the number of items sold given P = $1200, s = $30/item, and c = $600.

Mathspace New South Wales – Year 11 Standard mathspace.co

The height of a parabolic fountain stream is modelled by y = −4x2 + 16, where y is the height (m) and x is the horizontal distance from the centre (m). Make x the subject then determine the distance from the centre when the height is y = 12 m.

Extend your thinking 11

A trapezium-shaped table surface has an area of 50 cm2, a height of 5 cm, and an upper base, a = 8 cm. Using the trapezium area formula A= (a + b)h, make b the subject then determine the length of the lower base. Explain why the solution is suitable for the table’s dimensions.

A gas storage tank has a pressure of P = 200 kPa, volume of V = 0.5 m3, and contains n = 2 mol of gas, with gas constant R = 8.31 J/(mol × K). Using the ideal gas law PV = nRT , make T the subject then determine the temperature inside the tank rounded to two decimal places. Explain a scenario where this calculation is critical.

A production process follows the rate model y =

, where y is the production rate (items

per hour) and x is time (hours). Make x the subject then determine the time required to achieve a rate of y = 2 items per hour and verify the solution. Discuss how this time impacts production planning. 14

Ruﬁno has submitted the following work to write the equation for the mass of an object, m, given the kinetic energy, KE, and velocity, v:

Determine whether Ruﬁno is correct by explaining each step of his working if the step is correct, otherwise ﬁx his error.

1.08 Change the subject of a formula mathspace.co

1.09 Write and solve equations After this lesson, you will be able to… • identify unknown quantities, constants, and coefficients in a word problem. • translate verbal statements into linear algebraic equations. • solve linear equations using inverse operations, including those with variables on both sides. • interpret the solution of an equation in the context of the original word problem. • verify the solution by substituting it back into the formulated equation or checking against the problem’s conditions.

Write and solve equations To write a linear equation from a word problem, identify the pronumeral (unknown quantity), constants (fixed numbers), and coefficients (multipliers). Translate words into mathematical operations using key terms: “and” or “more than” (+), “less than” (−), “times” or “per” (×), “divided by” (÷), and “is” (=). Linear equations typically take the form ax + b = c or ax + b = cx + d, where x is the pronumeral, and a, b, c, d are constants. Solve the equation using inverse operations. For example, addition/subtraction, multiplication/ division, to isolate the pronumeral. Ensure each step maintains equality by applying operations to both sides. Interpret the solution’s practical significance in the problem’s context, such as budgeting, inventory management, or geometry, considering whether the solution is realistic, for example, positive integers for quantities. For budgeting scenarios, equations often follow the form C = f + cn, where C is total cost, f is fixed cost, c is cost per item, and n is the number of items. Solving for n helps determine affordable quantities within budget constraints.

Example 1 A small business budgets $150 for supplies, including a ﬁxed delivery fee of $30 and $5 per item. How many items, n, can the business purchase within the budget?

Create a strategy Identify the pronumeral, constants and operations. Write the equation, then use inverse operations to solve for n.

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Apply the idea The pronumeral is the number of items, n. Constants are 30 (delivery fee), 5 (cost per item), and 150 (budget). “Including” and “and” suggest addition, “per item” indicates multiplication, and the budget implies equality. So, the equation is: 30 + 5n = 150 Delivery fee plus cost per item times n equals budget 30 + 5n = 150

Write the equation

5n = 120

Subtract 30 from both sides

n = 24 items

Divide both sides by 5

Reﬂect and check Verify by substituting n = 24: LHS = 30 + 5n

Write the left-hand side

= 30 + 5 × 24

Substitute n = 24

= 30 + 120

Evaluate the multiplication

= 150

Evaluate

= RHS The solution n = 24 is correct.

Example 2 An inventory count ﬁnds that the number of boxes in a warehouse, when increased by 8, equals twice the number of boxes minus 4. Find the number of boxes, n.

Create a strategy Identify the pronumeral, constants and operations. Write the equation, then use inverse operations to solve for n.

Apply the idea The pronumeral is the number of boxes, n. Constants are 8 (added amount) and 4 (subtracted amount). “Increased by” indicates addition, “twice” means multiplication by 2, “minus” suggests subtraction, and “equals” implies equality. So, the equation is: n + 8 = 2n − 4 Number of boxes plus 8 equals twice the number minus 4 n + 8 = 2n − 4

Write the equation

8=n−4

Subtract n from both sides

12 = n

Add 4 to both sides

n = 12 boxes

Make n the subject

1.09 Write and solve equations mathspace.co

For each keyword, write the correct mathematical operation for writing equations from word problems. a

“More than”

“Per”

“Decreased by”

“Is”

Match the phrase to the correct operation. a

“Twice as many”

“Split equally”

×2

“Reduced to one-third”

iii

“Three more than”

÷3

−3

Practice 5

Write an equation for each situation and solve for the pronumeral. a

A number increased by 7 is equal to 12. Let the number be n.

A subscription costs $10 per month plus a $25 signup fee. If the total cost is $55, how many months, m, were subscribed?

An art supply store charges $2 per paint tube plus a $15 delivery fee. If the total cost is $27, how many tubes, t, were ordered?

A rectangle’s length is 5 cm more than its width. If the perimeter is 26 cm and the width is w, ﬁnd w.

Ex 1

A catering service budgets $180 for food, including a $40 preparation fee and $7 per meal. How many meals, m, can be prepared?

Ex 2

A stock check shows that the number of crates, when increased by 11, equals three times the number minus 5. Find the number of crates, c.

A coffee shop sells cups at $4 each, with a $10 daily overhead, earning $50 in a day. How many cups, c, were sold?

A number tripled then decreased by 4 equals 17. Let the number be n. Find n.

A phone plan charges $0.5 per minute plus a $20 monthly fee. If the bill is $35, how many minutes, m, were used?

A number, when doubled and then increased by , equals 10.5. Let the number be x. Find x.

A bookshop sells books at $15 each, with a $30 shipping fee. If the total cost is $90, how many books, b, were ordered?

A car’s speed, increased by 15 km/h, is three times its original speed minus 10 km/h. Find the original speed, s.

1.09 Write and solve equations mathspace.co

Extend your thinking 14

A company spends $15 000 on equipment and $200 per unit produced. Each unit sells for $300. How many units, u, must be sold to break even?

Determine if the student’s equation for “A delivery service charges a $20 base fee plus $3 per package, totaling $32” as 20 + 3(p + 1) = 32 is correct.

A gym charges a $100 membership fee plus $30 per class with a

discount on the class fee.

If the total cost is $340, how many classes, c, were attended? 17

A bakery’s daily cost is $200 plus $5 per cake, and each cake sells for $15. Find the number of cakes, k, needed for a proﬁt of $400.

A number, when multiplied by

and then increased by 5.5, equals 10. Let the number be n.

Find n.

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Did you know?

When you shop online, you’re doing maths without even realising it! Every time you see a “20% off” sale, add shipping fees, or calculate taxes, you’re solving equations in the background. Understanding how these parts fit together helps you compare deals, avoid overspending, and make smarter choices before you hit “checkout.”

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1 Chapter review 1

A train travels 315 km in 3.5 hours. What is its average speed? A

80 km/h

90 km/h

95 km/h

A female weighing 70 kg consumes 5 standard drinks over 3 hours. Using the formula BACfemale = A

85 km/h

, her BAC is approximately:

0.065

0.071

0.078

0.083

Using Young’s formula:

Calculate the dosage for a 6-year-old child needing an antibiotic with an adult dose of 450 mg, rounding to the nearest milligram. A 4

120 mg

135 mg

150 mg

165 mg

Simplify: a

9a + 4b + 2a − b

y2 + 5y + 4y + 20

6m × 5n

4x2 y3 × 3xy4

(15pq) ÷ 20qr

5(3p × 2q) + 4p × (−5q)

0.6x3 × 5xy2

An equilateral triangle has a side length of 4x cm. Write a simpliﬁed expression for its perimeter.

A rectangular field has a length that is five times its width. If the width is 2w metres, find a simplified expression for the area of the field.

Substitute and evaluate: a

mn, where m = 7 and n = 4

p , where p = 5

q2, where q = −3

3a × 2b + 25, where a = −4 and b = 3

5x × 3y, where x =

g 8

x + y + z, where x = 12, y = 18 and z = 5

4k + 2j , where k = 3 and j = −2

and y = 9

6c − d + 10, where c = 5 and d = 3

The formula for the area of a rectangle is A = l × w. Determine the area for: a

Length = 8 cm and width = 5 cm

Length = 12 m and width = 6 m

Length = 4.5 cm and width = 2 cm

Length = 10 km and width = 3.5 km Chapter 1 review mathspace.co

Complete the table of values for the formula: a

b = 3a + 2

y = −4x + 1

a 0 1 2 3 4 b 10

⬚

The Celsius to Fahrenheit conversion formula is F =

+ 32:

Convert 20°C to Fahrenheit.

Convert 35°C to Fahrenheit.

Convert 23°F to Celsius.

Convert 212°F to Celsius.

The range of a projectile launched at an angle is given by R =

, where R is the

range (metres), v is the initial velocity (m/s), θ is the launch angle, and g = 9.8 m/s2 is the acceleration due to gravity. Determine the range for each scenario, rounded to two decimal places: a

v = 25 m/s and θ = 30°

v = 18 m/s and θ = 45°

v = 30 m/s and θ = 60°

v = 40 km/h and θ = 50°

Calculate m for F = ma with F = 75 N, a = 5 m/s2.

Calculate r for the area of a circle formula A = π r2 with A = 78.5 cm2, using π = 3.14. (Assume r > 0).

The formula for the kinetic energy is E =

mv2:

Calculate m (mass in kg) if E = 600 J and v = 10 m/s, rounded to one decimal place.

Calculate v (velocity in m/s, assume v > 0) if E = 450 J and m = 4 kg, rounded to one decimal place.

The simple interest formula is I = P × r × t, where I is interest, P is principal, r is the annual interest rate (as a decimal), and t is time in years. a

Calculate the principal P if the interest earned is $360, the rate is 0.04, and the time is 3 years.

Calculate the time t (in years) if the interest earned is $500, the principal amount is $2500, and the rate is 0.05.

Calculate the stopping distance for a vehicle under the following conditions: a

A car travels at 60 km/h with a driver reaction time of 1.5 s. The braking distance is given by 0.007 × (speed in km/h)2 metres. Calculate the total stopping distance in metres, rounded to one decimal place.

A truck travels at 25 m/s with a driver reaction time of 1.8 s. The braking distance is calculated using the formula

, where s is speed in m/s. Calculate the total stopping

distance in metres, rounded to one decimal place.

Mathspace New South Wales – Year 11 Standard mathspace.co

Two cars are 450 km apart and start driving towards each other at the same time. Car A travels at an average speed of 70 km/h and Car B travels at an average speed of 80 km/h: a

Calculate their relative speed.

Calculate how long it will take for them to meet, in hours.

How far will Car A have travelled when they meet?

Calculate the BAC, rounded to three decimal places, for: a

A male, 90 kg, consumes 5 standard drinks in 3 hours.

A female, 58 kg, consumes 3 standard drinks in 1.5 hours.

Sarah (female, 62 kg) drinks three 150 mL glasses of wine (13% alc/vol) over 4 hours. a

Calculate the total number of standard drinks Sarah consumed, rounded to one decimal place.

Calculate her BAC at the end of the 4 hours, rounded to three decimal places.

A provisional driver (BAC limit 0.00) has a BAC of 0.06 at 11:30 p.m. Calculate the earliest time they can legally drive.

A 6-year-old child weighing 20 kg needs an antihistamine. The adult dosage is 250 mg. Calculate the child’s dosage, rounding to the nearest milligram, using: a

Clark’s formula

Using the table of ideal masses for children: Age (years)

Mass of males (kg)

Mass of females (kg)

22.9

22.4

28.6

28.1

Calculate the Clark’s formula dosage for a 7-year old male if the adult dosage of a cough syrup is 350 mg, rounded to the nearest milligram.

Determine the adult dosage of an antibiotic if a 9-year-old female receives a Clark’s formula dosage of 150 mg. Round the adult dosage to the nearest milligram.

The population growth formula is P = P0 ert , where P is the ﬁnal population, P0 is the initial population, r is the growth rate (per year, as a decimal), t is time (years), and e ≈ 2.718 is Euler’s number. a

Calculate the ﬁnal population P if the initial population P0 = 1200, the growth rate r = 0.025, and the time t = 10 years. Round to the nearest whole number.

Calculate the initial population P0 if the ﬁnal population P = 2500, the growth rate r = 0.03, and the time t = 8 years. Round to the nearest whole number.

π r3, where V is volume and r is radius. Use π ≈ 3.1416: A spherical balloon has a radius of 0.8 m. Calculate its volume, rounded to two decimal places. If the balloon is ﬁlled with helium, and 1 cubic metre of helium has a mass of 0.1785 kg, calculate the total mass of the helium in the balloon, rounded to two decimal places.

The volume of a sphere formula is V = a b

Young’s formula

Change the subject to x: a

a = 3x − d

n = k + 4x

b = 6x − c

q = 5x + r Chapter 1 review mathspace.co

Change the subject to the speciﬁed pronumeral in each formula: a c

FV = PV (1 + i)n, change to PV 2

h = −5t + c, change to t

A = π r2 + π rs, change to s

E = mc2 + k, change to m

In a physics experiment, the displacement s of an object is given by s = ut +

at2, where u

is initial velocity, t is time, and a is acceleration. If u = 0, the formula simpliﬁes to s =

at2:

Rearrange the simpliﬁed formula to make acceleration a the subject.

An object starting from rest (u = 0) travels 100 m in 5 s. Calculate its acceleration a using the rearranged formula.

Ruﬁno is trying to make v the subject of the formula for kinetic energy, KE = His working is shown:

mv2.

Identify the step where Ruﬁno made an error, explain the error, and provide the correct step and ﬁnal correct formula for v (assuming v > 0). 29

Write an equation for each situation and solve for the pronumeral: a

A number decreased by 9 is equal to 15. Let the number be x.

A gym membership costs $50 per month plus a $75 joining fee. If the total cost for a period was $325, for how many months, m, was the membership held?

A ﬂorist charges $8 per rose plus a $12 delivery fee. If a bouquet costs $60, how many roses, r, were in the bouquet?

A rectangle’s length is 7 m longer than its width. If the perimeter is 50 m and the width is w, ﬁnd w.

A certain number of apples in a basket, when doubled, is 8 more than the original number of apples. Find the original number of apples, a.

A local concert venue has ﬁxed costs of $12 000 for an event. Each ticket sold generates $75 in revenue, but there’s a $15 cost per attendee (for services, security, etc.).

If x tickets are sold, write an expression for the total revenue.

Write an expression for the total variable costs associated with x attendees.

Write an expression for the total costs.

Write an equation to determine the number of tickets x that must be sold for the venue to break even.

Solve the equation to ﬁnd the break-even number of tickets.

If the venue wants to make a proﬁt of $3000, how many tickets must be sold?

Mathspace New South Wales – Year 11 Standard mathspace.co

Big ideas • Sampling concepts and methods enable accurate representation of populations through careful selection of representative samples to minimise bias. • Survey design facilitates effective data collection by crafting clear, unambiguous, and ethical questions to address statistical inquiries. • Survey reliability and ethical considerations ensure accurate data interpretation by minimising errors and addressing potential misrepresentations.

2 Population and sample Chapter outline 2.01 2.02 2.03 2.04 2.05

Investigation: Statistical investigation process Census or survey Samples and populations Sampling techniques Survey design Investigation: Conduct a survey Privacy, bias and ethics Investigation: Misrepresentation of results Chapter 2 review

68 73 81 94 103 112

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2.01 Census or survey After this lesson, you will be able to… • distinguish between a population and a sample in a given context. • define census and survey, and identify their key characteristics. • explain the practical advantages and disadvantages of using a census versus a survey. • determine whether a census or a survey is more appropriate for a given scenario. • recognise that surveys aim to estimate characteristics of a population.

Census or survey A population includes all members of a group of interest, such as every student in a school or all cars in a city. A census collects data from the entire population. In contrast, a sample includes only a subset of the population. A survey collects data from a sample, usually to estimate characteristics of the whole population.

Exploration

Sample 1

Sample 4

Sample 5

Sample 2 Central container Sample 3

Sample 6

Imagine the central container represents a population and each sample container is a survey. Consider these questions: 1. How do the samples differ from the population and from each other? 2. Why might a sample’s proportions differ from those of the population?

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Example 1 Determine whether each scenario uses a census or a survey: a A teacher records the attendance of all 30 students in her class.

Create a strategy

Apply the idea

Identify the population and check if all members are included.

The population is the 30 students in the class. Since all are included, this is a census.

b A researcher asks 50 randomly selected residents in a city of 10 000 about park usage.

Create a strategy

Apply the idea

Identify the population and check if all members are included.

The population is all 10 000 residents. Only 50 are asked, so this is a survey.

Example 2 A town of 5000 people wants to decide on a new library location. Explain why a survey of 200 residents might be more suitable than a census.

Create a strategy

Apply the idea

Compare the practicality and accuracy of a census vs. a survey for this population size.

A census would involve all 5000 residents, ensuring high accuracy but requiring signiﬁcant time and cost. A survey of 200 is faster and cheaper, and can still provide a reliable estimate for a town this size. Thus, a survey is more suitable here.

Example 3 A tech startup with 25 employees wants to assess staff satisfaction with their new coffee machine. Explain why a census might be more suitable than a survey.

Create a strategy

Apply the idea

Assess the population size and the need for precision in this context.

The population is only 25 employees, a small group. A census ensures every opinion is captured accurately, avoiding sampling error, and is manageable in time and cost. A survey might miss key perspectives in such a small group, making a census more suitable.

Mathspace New South Wales – Year 11 Standard mathspace.co

Practice Ex 1

Identify whether each scenario describes a census or a survey: a

A librarian counts all 40 books on a shelf.

A town polls 150 out of 20 000 residents about recycling habits.

A chef tests every dish prepared during a shift.

A researcher interviews 5% of a city’s pet owners about pet care.

Liam checks the quality of all 50 chairs in a store.

Soﬁa surveys 10% of a town’s 8000 residents about bus services.

A manager reviews all 12 employee timecards.

A lab tests 20 out of 500 batteries produced.

Counting all buses in a small town’s ﬂeet.

Estimating coffee consumption in a city of 100 000.

Recording grades of 20 students in a class.

Estimating the number of cyclists in a state.

m Measuring satisfaction of 15 staff in a small oﬃce. n Ex 2

Estimating average commute distance in a large city.

A regional supermarket chain with 6000 loyalty card members wants to gauge interest in a new organic produce section. They plan to email a survey to 300 randomly selected members. Explain why surveying 300 members might be more suitable than conducting a census of all 6000 loyalty card members in this situation.

Ex 3

A small startup company with 18 employees is considering a signiﬁcant change to their health insurance plan. Explain why conducting a census (getting feedback from all 18 employees) might be more suitable than surveying a sample of the employees in this situation.

A village of 3000 residents plans a new playground: a

Is a survey of 300 residents or a census more suitable? Explain.

If the village had only 60 residents, would your answer change? Why?

A school randomly surveys 80 of its 960 students about lunch options: a

Is this a census or survey?

If 70% of the sample prefers pizza, estimate how many students school-wide agree.

A factory tests all 200 widgets produced in a shift: a

Is this a census or survey?

If only 25 widgets were tested, would it be a census or survey? Why?

Mathspace New South Wales – Year 11 Standard mathspace.co

Extend your thinking 12

A nurse surveys 60 students at a cafeteria, ﬁnding 75% drink soda daily. Can this result be generalised to all students at the school? Explain why or why not.

Compare a census and survey for a town of 500 000 residents studying bike lane usage:

List one advantage and one disadvantage of each method.

Which method is more suitable? Justify your choice.

A census shows 32% of a region uses bicycles weekly: a

A survey of 4000 ﬁnds 1260 cyclists. Calculate the survey proportion and round your answer to three decimal places.

Compare this to the census proportion. Is the survey reliable? Why?

List three reasons why a survey may sometimes provide more useful results than a census.

A survey of 200 people ﬁnds 12 are over 60 years old, while another survey of 3000 people ﬁnds 570. Which is a better estimate of the population proportion, and why?

2.02 Samples and populations After this lesson, you will be able to… • define a representative sample and explain its importance. • identify key factors that affect the representativeness of a sample. • recognise potential sources of bias in sampling.

Samples and populations Population The complete set of individuals, objects, places, etc., that we want information about. In statistics it is the entire dataset from which a statistical sample may be drawn. Sample A subset of a population used to estimate characteristics of the population. For example, a randomly selected group of 8-year-olds (sample) selected to estimate the height of 8-year-olds in Australia (population).

2.02 Samples and populations mathspace.co

Population

Sample

A census provides the most accurate data but is often costly or impractical. Sampling is faster and less expensive, but the sample must be representative to draw valid conclusions about the population. A sample is representative if its demographic diversity (e.g., age, gender) and relevant traits (e.g., study habits for a study on learning) match the population. The sampling method determines representativeness. Key factors affecting representativeness include: • Demographic diversity: Reflect the population’s age, gender, or ethnicity. • Relevant characteristics: Include a range of traits pertinent to the study. • Random selection: Use methods like random sampling or stratified sampling to minimise bias. • Sample size: Balance representativeness with efficiency; too small risks missing diversity, too large is inefficient. Common sampling methods include: • Random sampling: Equal chance of selection for all members. • Stratified sampling: Proportional sampling from population subgroups. • Convenience sampling: Selecting accessible individuals, often biased.

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Example 1 A researcher surveys study habits by interviewing 50 students in the school library during lunch. The school has 800 students, 60% in Years 7 to 9 and 40% in Years 10 to 12. Justify whether this sample is representative of the school population.

Create a strategy Evaluate the sampling method and its impact on demographic representation.

Apply the idea The sample uses convenience sampling, selecting students in the library during lunch. This method may over-represent students who use the library, such as those in higher years who study during lunch, and under-represent others, like younger students who may be outside. The sample is unlikely to be representative, as it does not ensure proportional representation of year groups (60% Years 7-9, 40% Years 10-12) and introduces bias based on location and time.

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Reﬂect and check A random or stratiﬁed sample, selecting students proportionally from each year group, would better ensure representativeness.

Example 2 A town with 5000 residents (50% under 30 years, 50% over 30) is surveyed about park usage. Method A randomly selects 100 residents. Method B selects 100 residents from one neighborhood. Justify which method produces a more representative sample.

Create a strategy Compare the sampling methods based on their ability to reﬂect the population’s age distribution.

Apply the idea Method A uses random sampling, giving each resident an equal chance of selection. This method is likely to produce a sample with approximately 50% under 30 and 50% over 30, reﬂecting the population’s age distribution. Method B uses convenience sampling, selecting residents from one neighbourhood. Neighbourhoods often have skewed demographics (e.g., mostly younger or older residents), so the sample may not reﬂect the 50% − 50% age split. Method A is more likely to produce a representative sample, as it minimises bias and better captures the population’s age diversity.

Reﬂect and check Method B’s sample could be skewed if the neighborhood is not demographically diverse, highlighting the limitations of non-random sampling.

Example 3 A company with 300 employees (200 full-time, 100 part-time) surveys job satisfaction by randomly selecting 30 full-time and 15 part-time employees. Justify whether this sample is representative of the company’s employees.

Create a strategy Check if the sample proportions match the population’s employment type distribution.

2.02 Samples and populations mathspace.co

Apply the idea The sample uses stratiﬁed sampling, dividing employees into full-time and part-time strata.

Write the proportion

Substitute values

Simplify

Convert to percentage and round Write the sample proportion

Substitute values

Evaluate

Simplify

Convert to percentage and round

The sample has 66.67% full-time employees, matching the population’s 66.67%. Random selection within strata reduces bias. The sample is representative, as it proportionally reflects the employment type distribution.

Reflect and check Stratified sampling ensures key demographics are represented, unlike convenience sampling, which might over-represent one group.

Example 4 A gym with 2000 members (1200 males, 800 females, aged 16 to 70 years) surveys exercise habits using two methods: • Method A: Select the first 100 members entering on Monday morning. • Method B: Randomly select 60 males and 40 females. Which method better ensures a representative sample? Justify.

Create a strategy Compare methods based on demographic alignment and bias.

Apply the idea Method B uses stratiﬁed sampling, selecting 60 males and 40 females, matching the 60% male and 40% female proportions. Random selection reduces bias. Method A, a convenience sample, may over-represent Monday morning visitors, skewing demographics.

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Ex 2

Ex 3

A study examines dietary habits in a city. Which characteristics are important for a representative sample? a

Favourite book genre

Preferred music

Age group

The population is all employees in a company. Are these subsets samples of that population? a

All employees in the marketing department

The ﬁrst 15 employees arriving at work

Contractors working temporarily

A sample of 40 people is chosen from individuals aged 16 to 70. Could this sample reasonably have come from each of the following populations? a

Shoppers at a mall

Attendees at a music festival

Residents of a nursing home

For each population, suggest a survey question and a sample that would provide biased results: a

All customers at a coffee shop

All pet owners in a town

All voters in a city

A city planner surveys residents about public transport use. Is each sample biased or suﬃcient for representativeness? a

All commuters at a bus station

10 residents in one neighbourhood

150 randomly selected residents

A sample of 2500 residents from a population of 25 000

A factory produces 40% of its products with recyclable packaging. A sample of 300 products is tested, and 132 have recyclable packaging: a

Deﬁne the population in this context.

What proportion of the sample has recyclable packaging?

Is the sample representative?

A vineyard ﬁnds 70% of its grapes are Cabernet. A sample of 400 grapes is tested, and 288 are Cabernet: a

Deﬁne the population in this context.

What proportion of the sample is Cabernet?

Is the sample representative?

Are these survey questions biased or fair? a

Do you want soup for lunch or a sandwich?

What do you do on a Sunday morning?

Do you think the government should be allowed to cut down some of the oldest trees in the area to construct a metro railway line in the city?

Do you eat at least the recommended number of servings of fruits and vegetables to ensure a healthy and long life?

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Identify the type of bias (leading or loaded) that could be present in these biased survey questions: a

Do you prefer the natural beauty of hardwood ﬂoors in your home?

Why has TV news become so out of touch with the problems of ordinary people?

Don’t you think this newspaper is biased?

Do you prefer the look and feel of thick lush carpet in your living room?

After the government decided to increase the minimum retirement age, a survey is to select a group of people to ask their opinions on the changes. Should each of these groups be included in the survey?

Politicians and policy makers

Members in the community that are directly impacted by the changes

Children under the age of 10

A question is inappropriate for a survey given it fulﬁlls any of the following criteria: • Question is unclear • Emotional language • Question is too personal Using the criteria list, determine why each of the questions is inappropriate:

Ex 4

Many people have worked incredibly hard and even died making this bridge. Do you like the bridge?

How many large electronic devices are in your home?

How many people have died in your family?

A researcher surveys every person at a pet store to study pet owners in a city. Discuss the advantages and disadvantages of this sampling method.

Extend your thinking 18

A population has 25 000 members. Is a sample of 50 suﬃcient to be representative? Explain why.

A pollster calls 800 households from 3:00 p.m. to 6:00 p.m., getting responses from 320 people:

How could random sampling be used to select households?

Why might the results be inaccurate?

A health company trials a supplement with 800 participants in Brisbane, ﬁnding 15% experienced mild side effects. They plan to distribute it to Brisbane’s 2 500 000 residents: a

Estimate how many residents might experience side effects.

Is the sample representative of Brisbane’s population?

Why is household size relevant or irrelevant for the sample’s representativeness?

2.02 Samples and populations mathspace.co

A conservationist tags 300 fish in a lake. In a follow-up, they catch 500 fish, 45 tagged. In another round, they catch 500 fish, 42 tagged. This capture-recapture method estimates the population by assuming the proportion of tagged fish in a sample equals the proportion of tagged fish in the population: a

Estimate the ﬁsh population using both samples.

Why are the estimates different?

Determine whether each question is likely to give biased results or not. If you think results will be biased, state why. a

Damon asked four of his friends, “What is your favourite subject at school?”

Lulu is interviewing people outside of a train station. She asks, “Should the government spend more money on public transportation?”

Lys is interviewing people about how shark nets are affecting shark attack numbers. He says, “Shark nets cost $1 000 000 per year and barely help. Do you support shark nets?”

Explain why each survey question would give unreliable results. Rewrite each question to provide more reliable results: a

The Prime Minister believes that taxes are too high, do you think taxes are too high?

Do you have a Snapchat or Twitter account?

Do your siblings ever drive you to school?

Do you often waste time on social media?

What brand of shoe polish do you use every day?

Patricia surveys her class about their favourite music. Her results are shown in the table: Genre

Country

Pop

No. students

According to the survey, which genre is most popular among her class?

This was her survey question:

“The coolest kids like country music, and nobody likes pop. Do you like country music or pop music?” Do you trust the results of Patricia’s survey? Explain your answer.

Mathspace New South Wales – Year 11 Standard mathspace.co

2.03 Sampling techniques After this lesson, you will be able to… • describe the stages of the sampling process: target population, sampling frame, sample, and participant group. • explain common non-random sampling methods (volunteer, convenience, quota) and their limitations. • explain common random sampling methods (simple random, systematic, stratified) and their advantages. • compare and contrast different sampling techniques in terms of potential bias and practicality. • calculate necessary figures for systematic and stratified sampling (e.g., interval, number per stratum).

Sampling process Exploration A local council mayor needs to decide which intersections in the city require stop signs, traﬃc lights, or conversion to roundabouts. To inform these decisions, the mayor decides to conduct a survey to determine public sentiment regarding current road conditions. Intersections causing the most frustration for drivers will be prioritised for improvements. Survey scenarios: • The mayor asks the principal of the local high school to distribute the survey to 100 randomly selected students who drive themselves to school. • The mayor gives the survey to everyone in her neighbourhood. • The mayor sets up a booth at a local footy game and asks anyone who drives and is interested to ﬁll out a survey. • The mayor randomly selects 100 members from the electoral roll and conducts phone surveys. • The mayor asks staff members at the local Roads and Maritime Services (RMS) centre to ask everyone who comes in that day if they’d be willing to participate in the survey. 1. Decide whether the results from the following surveys accurately represent the population. 2. If they do not accurately represent the population, explain why.

2.03 Sampling techniques mathspace.co

Target population All the members of the group relevant to the study Sampling frame Possible members of the target population Sample Members selected to participate Participant group Members who actually respond

The target population is the particular group that a researcher aims to study. Identifying the target population ensures the study is relevant and accurate. A sampling frame is a list of all elements (individuals, households, organisations, etc.) in the target population from which the sample is drawn. It can contain individual members or groups, depending on the sampling method used. Ideally, the sampling frame would cover the entire target population; however, it is not always possible to identify every individual. The sample is then selected from the sampling frame. However, not all individuals chosen for the sample will agree to participate, a phenomenon known as non-response. Non-response can introduce bias into the study, as the individuals who do not respond might differ in significant ways from those who do. The individuals who do respond form the participant group.

Example 1 Mathspace High conducted a study to understand the technology use habits of its students. They used their enrolment database to randomly select 100 students to complete a survey, of which 85 students responded: a Identify the target population and the sampling frame.

Create a strategy

Apply the idea

The target population is the broad group of interest, whilst the sampling frame is a detailed list of individuals from that group.

The target population is all students at Mathspace High. The sampling frame is all students listed on Mathspace High’s enrolment database at the time the sample was taken.

b Suggest a reason why the sampling frame may not be identical to the target population.

Create a strategy

Apply the idea

Examine the sampling frame for potential gaps, such as missing or outdated entries, that could exclude some population members.

New students may not have been added to the database and departed students may not have been removed.

Mathspace New South Wales – Year 11 Standard mathspace.co

c Identify the size of the sample.

Create a strategy Locate the number of individuals chosen from the sampling frame to participate in the survey.

Apply the idea The sample size was 100 students.

d How many students did not respond?

Create a strategy Subtract the number of respondents from the total sample size to ﬁnd the number of nonrespondents.

Apply the idea Non-response = 100 − 85 = 15 students

Write the equation Evaluate

Example 2 For a statistical survey the target population is deemed to be all people in a city who play in any organised sporting competition. Identify whether these are samples of the target population. Justify your response: a 500 spectators chosen from a weekend sports match

Create a strategy Decide if the group consists entirely of members who play in an organised sporting competition.

Apply the idea The members of this group are spectators, not players. Therefore, this is not a sample of the target population.

b The members of 3 teams chosen from the local hockey tournament

Create a strategy Decide if the group consists entirely of members who play in an organised sporting competition.

Apply the idea Every member of this group plays in their local hockey tournament, which is an organised sporting competition. Therefore, this is a sample of the target population.

2.03 Sampling techniques mathspace.co

Disadvantages: • The sampling frame requires a complete list of every individual in the population with their own unique identifier and detailed demographic information. • More complex, time-consuming and expensive. • Overlapping or poorly defined strata can lead to errors, such as double-counting individuals or excluding them entirely. • Another sampling method is required to randomly select individuals within each stratum. Difficult if the population is highly diverse or lacks sufficient information

Example 4 Identify the type sampling method used: a In a manufacturing plant, quality control involves selecting every 50th product off the assembly line for inspection.

Create a strategy Consider how the participants are selected.

Apply the idea There is a process, every 50th product is inspected. This is systematic sampling.

b A university is conducting a student satisfaction survey. They randomly select students using computer-generated numbers from a complete list of all enrolled students, identiﬁed by their unique student numbers.

Apply the idea Participants are selected using a random number generator. This is simple random sampling.

c A political party wants to understand voting intentions across different age groups. They divide the population into age brackets and then randomly select individuals from each age group. The proportion of each age group in the sample is the same as in the population.

Apply the idea The population is divided into groups based on their age. Individuals are then selected proportionally from each group. This is stratiﬁed sampling.

2.03 Sampling techniques mathspace.co

Example 5 A machine produces 400 items a day. At what interval should an item be selected in order to obtain a systematic sample of 25 items?

Create a strategy Divide the population size by the sample size.

Apply the idea Write the equation Evaluate Every 16th item should be selected.

Example 6 In a group of 360 students, 90 are primary students and 270 are secondary students. A stratiﬁed sample of 120 is to be selected from the group based on year level.

Create a strategy Determine the proportion of primary students that is equal to the propotion of total students selected.

Apply the idea First, identify the proportion of total students selected. Express the 120 as a fraction of 360 So

Simplify

of the primary students should be selected. Write the equation

Evaluate

Reﬂect and check Alternatively, use: Write the equation

Substitute the values

Evaluate

Mathspace New South Wales – Year 11 Standard mathspace.co

Practice Ex 1

A school librarian surveys 150 randomly selected students from Bluewater High School’s enrolment database about reading preferences, with 120 responding. Answer the questions about the survey:

Ex 2

Identify the target population and sampling frame.

Identify the sample size.

Deﬁne non-response in this context.

Identify the sampling method used in each scenario: a

A researcher surveys the ﬁrst 60 commuters at a train station about travel habits.

A student posts a survey link on social media about canteen food, allowing open responses.

A factory tests every 20th phone off the production line.

A researcher surveys the ﬁrst 40 males and the ﬁrst 40 females at a community centre about park facilities.

Identify the target population for each scenario: a

A council assesses the need for public transport improvements.

A teacher evaluates a new maths app’s effectiveness.

A sports club gauges interest in a new ﬁtness program.

A hospital surveys patient satisfaction with emergency services.

A school surveys 150 randomly selected students, with 120 responding. Calculate for the survey: a

The number of non-respondents.

The response rate as a percentage.

Calculate the interval for systematic sampling in each scenario: a

A factory produces 2400 tablets daily and needs a sample of 80 tablets.

A mayor surveys 5000 drivers, selecting a sample of 100 drivers.

Identify the students selected using simple random sampling: a

A teacher selects students from a list of 30 students, numbered 1 to 30, using random numbers: 5, 12, 19, 27.

A school selects students from a list of 50 students, numbered 1 to 50, using random numbers: 3, 15, 28, 42.

Calculate the number of students to sample from each stratum using stratiﬁed sampling: a

A school with 1200 students (50% junior years, Years 7–9, and 50% senior years, Years 10–12) selects 240 students.

A council surveys 8000 residents (40% aged 18–35, 30% aged 36–50, 30% aged 51+), selecting 400 residents.

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 3

Analyse the sampling method and response for each survey: a

Ex 4

Ex 6

Identify the sampling method.

Calculate the response rate.

A researcher stands outside a shopping centre and surveys the ﬁrst 100 passers-by, with 85 responding. i

Identify the sampling method.

Calculate the response rate.

Analyse the sampling method and response for each survey: a

Ex 5

A researcher posts an online questionnaire about transport usage, targeting 500 people, and receives 200 responses.

A teacher uses a computer to randomly select 90 students from a course of 300 to complete a feedback form, and all selected students respond. i

Identify the sampling method.

Calculate the response rate.

A company selects every 5th customer from a list of 500, and receives 470 responses. i

Identify the sampling method.

Calculate the response rate.

For a survey targeting all employees in a company who work remotely, identify whether these are samples of the target population. Justify your response: a

200 employees attending an in-person training session.

50 employees randomly selected from the company’s remote worker database.

All employees participating in a company-wide online meeting.

100 employees randomly selected from the company’s full employee list.

A library surveys 200 randomly selected patrons about digital resource usage, with 160 responding. Calculate for the survey: a

The number of non-respondents.

The response rate as a percentage.

Extend your thinking 16

A council plans a survey on recycling habits for 300 residents from 6000. Describe how to implement stratiﬁed and systematic sampling, including calculations where needed. Discuss one advantage and one disadvantage of each method.

Compare and contrast systematic, self-selected, random, and stratiﬁed sampling in terms of bias, ease of implementation, and population representation.

2.03 Sampling techniques mathspace.co

A school surveys study habits using stratiﬁed sampling across year levels (25% Year 7, 25% Year 8, 20% Year 9, 30% Years 10–12). The total population is 1000 students, and a sample of 200 is needed.

Answer the questions about the survey: a

Calculate the number of students to sample from each stratum.

Explain why stratiﬁed sampling is suitable here compared to self-selected sampling.

A hospital surveys patient satisfaction with 400 patients from 10 000 (40% surgical, 30% medical, 30% outpatient). A stratiﬁed sample of 200 is needed.

Answer the questions about the survey:

Calculate the number of patients to sample from each stratum.

If only 160 patients respond, calculate the response rate and discuss how non-response might introduce bias.

A city council wants to assess public opinion on a new park development, targeting all adult residents. Design a sampling strategy using one random and one non-random method. Justify your choices and discuss potential biases for each.

2.04 Survey design After this lesson, you will be able to… • differentiate between open, closed, and partially closed survey questions. • identify principles of effective questionnaire design, including clarity, conciseness, and avoiding bias. • recognise and improve poorly worded or structured survey questions. • understand the importance of layout, question sequencing, and ethical considerations in survey design. • describe potential faults in survey design that can lead to unreliable data.

Survey design Questionnaires are commonly used to collect data through printed forms, online platforms, or interviews. There are three main types of questions: • Closed questions offer fixed response options, making them easy to answer and analyse but requiring careful design to cover all possibilities. • Open questions allow detailed responses, providing rich insights but taking more time to complete and interpret. • Partially closed questions combine set options with an “Other” field, balancing structure with flexibility.

Mathspace New South Wales – Year 11 Standard mathspace.co

Effective questionnaires are clear, concise, and engaging: • Length: Keep surveys short and focused to avoid fatigue. • Clarity: Use plain language and avoid jargon or emotive terms. • Precision: Ask one clear question at a time to prevent confusion. • Bias: Avoid leading questions that could influence responses. Design matters in both physical and digital formats: • Use legible fonts, clean layouts, and high contrast. • Align response boxes clearly, with enough space for answers. Structure also influences quality: • Start with simple questions to build comfort. • Reserve sensitive items for later in the survey. • Use filter questions to guide respondents efficiently. • Sequence questions to avoid influencing responses. Ethical considerations underpin trust: • Transparency: Explain the survey’s purpose and process clearly. • Comfort: Allow respondents to skip questions or withdraw freely. • Confidentiality: Store data securely and anonymise where possible.

Interactive exploration Discover this concept in action online

mathspace.co

Example 1 Identify issues in the following questions and suggest improved versions: a Do you frequently visit your doctor?

Create a strategy Evaluate the question for ambiguity and speciﬁcity. Rephrase to ensure clarity and measurable responses.

Apply the idea The term ‘frequently’ is subjective, leading to inconsistent responses. Improved question: How many times have you visited a doctor in the past 12 months?

Reflect and check The improved question uses a specific time frame (12 months) and quantifiable response, reducing ambiguity.

2.04 Survey design mathspace.co

b Which doctor is the most experienced and kind?

Create a strategy Identify combined attributes in the question. Separate into distinct questions for clarity.

Apply the idea The question combines ‘experienced’ and ‘kind,’ which may not align, causing confusion. Improved questions: Which doctor do you consider the most experienced? Which doctor do you consider the kindest?

Reﬂect and check Separating the qualities ensures respondents can evaluate each attribute independently, improving response accuracy.

c How satisﬁed are you with our exceptional healthcare services?

Create a strategy Check for leading language. Rephrase to eliminate bias.

Apply the idea The word ‘exceptional’ implies high quality, introducing bias. Improved question: How satisﬁed are you with our healthcare services?

Reﬂect and check Removing ‘exceptional’ ensures neutrality, allowing unbiased responses.

d What is your predominant method for alleviating stress?

Create a strategy Assess the complexity of language. Simplify for accessibility.

Apply the idea ‘Predominant’ and ‘alleviating’ are complex terms that may confuse respondents. Improved question: What is your main method for reducing stress?

Reﬂect and check Simpliﬁed language enhances accessibility for a broader audience.

Mathspace New South Wales – Year 11 Standard mathspace.co

e For identiﬁcation purposes, what is your mobile number and home address?

Create a strategy Evaluate privacy concerns. Modify to request minimal personal information.

Apply the idea Requesting full mobile numbers and home addresses compromises privacy and may deter participation. Improved question: For identiﬁcation purposes, please provide your postcode and the last 3 digits of your mobile number.

Reﬂect and check Limiting personal information respects privacy while maintaining identiﬁcation functionality.

Where do you work?

Create a strategy Address ambiguity in the question. Clarify the intended information.

Apply the idea The question is unclear, as ‘where’ could refer to location or employer. Improved questions (depending on intent): What is the name of your employer? OR: In which suburb is your workplace located?

Reﬂect and check Specifying the intent reduces confusion and ensures relevant responses.

g Don’t you oppose not allowing employees to work from home?

Create a strategy Identify confusing language and bias. Simplify and neutralise the question.

Apply the idea The double negative (‘don’t’ and ‘not’) is confusing, and the phrasing may lead respondents. Improved question: Do you support or oppose allowing employees to work from home?

Reﬂect and check The revised question is clear, neutral, and offers balanced response options.

2.04 Survey design mathspace.co

Ex 2

Identify whether the survey questions are biased or fair. For biased questions, provide a fair rewrite: a

Are you satisﬁed with our exceptional hospital services?

How often do you visit the local community centre?

Why is our school’s canteen the best in NSW?

Explain why the survey questions are ambiguous and suggest an improved version: a

How often do you see a doctor?

Is the program good?

Do you travel often?

Which of the following response scales would be appropriate for measuring satisfaction with internet speed in a user experience survey? A

☐

Unacceprtable

☐

Below average

☐

Average

☐

Above Average

☐

Excellent

☐

Never

☐

Rarely

☐

Sometimes

☐

Often

☐

Always

☐

Very dissatisfied

☐

Dissatisfied

☐

Neutral

☐

Satisfied

☐

Very satisfied

Rewrite these survey questions as two separate questions: a

Is the school library spacious and well-resourced?

Are the park facilities clean and safe?

Rewrite the survey questions to remove double negatives: a

Don’t you think it’s not necessary to upgrade the school facilities?

Do you never not attend community events?

Isn’t it true that you don’t oppose the new policy?

Rewrite the survey questions to make them more respectful and appropriate for sensitive topics: a

Are you too lazy to exercise regularly?

What is your exact monthly income?

Do you agree that it’s selﬁsh not to volunteer in your community?

A school survey begins with: “Do you believe too much screen time harms learning?”, then later asks: “How many hours do you spend on screens each day?”. Explain what could be improved in the structure of this survey.

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Extend your thinking 12

Explain the importance of pretesting a survey on recycling habits for NSW residents. Outline two steps involved and one potential issue that pretesting could identify.

A survey on NSW public transport usage lists activities in this order: train travel, bus travel, cycling, walking. Results show train and bus travel as most popular. Identify a potential fault in the survey design and suggest how to address it.

A survey claims 75% of NSW students support a new school policy based on 60 responses from one school. Explain why this claim might be misleading and suggest an improvement to the survey design.

In a digital survey, one response option is displayed in a large, colourful button, while the others appear in small, plain text. Explain what type of bias this might introduce and suggest how the survey design could be improved to enhance fairness and reliability.

A researcher is surveying students about their experiences with bullying. To boost participation, they offer entry into a prize draw for completing all questions, including optional ones about emotional impact. Explain one ethical concern raised by this strategy and suggest a way to improve the ethical integrity of the survey while still encouraging participation.

Investigation: Conduct a survey Investigate online

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2.05 Privacy, bias and ethics After this lesson, you will be able to… • distinguish between a fault in a survey procedure and an error in survey results. • identify and describe common types of errors, including measurement error and sampling error. • recognise common methods used to misrepresent statistical results. • explain how issues like bias, privacy, and ethics can affect survey integrity. • critically evaluate claims made based on survey data, considering potential faults and misrepresentations.

Faults and errors Exploration

Collect da ta

e d ata

ign es

ly s

Census/Survey Procedure

Research a nd d

The procedure for conducting a census or survey is shown in this diagram.

lis

h re

s u lt s

1. Why is it important to follow a systematic procedure in a census or survey to minimise faults and errors? 2. How might faults in the research and design stage lead to ethical issues, such as breaching privacy? 3. Which stage might introduce bias, and how could this affect the accuracy of the survey results? 4. Describe how faults in data collection could disrespect diverse groups or cultures.

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A fault is a potential defect in the procedure that could lead to inaccurate results. Faults affecting data collection include leading or biased questions, non-responses, inadequate sample sizes, or poor sampling planning. For example, conducting a survey about work-life balance right after a major deadline may bias responses due to heightened stress. Faults can compromise privacy if sensitive personal data is collected without consent or stored insecurely. Ethics are violated when respondents are misled about the survey’s purpose or when data is used in ways that harm participants. An error occurs when a fault impacts the results, leading to unreliable data. Examples of errors include: • Measurement error: Unintentional bias introduced by the device or method, such as a poorly calibrated tool skewing results. • Sampling error: Variation between the true population value and the sample estimate, often due to non-representative sampling. Bias in surveys can arise from faults like leading questions or sampling only accessible groups, skewing results and undermining fairness.

Example 1 For each scenario, identify the type of error: a A TV station wants to know the most popular type of music, so they ask listeners to contact them and vote for their favourite type of music.

Create a strategy

Apply the idea

The sample is from self-selecting participants, meaning only those who make an effort to respond will be included, which may not be representative of the entire population.

This is a sampling error.

b Adults attending a local cinema were asked this question: “How many times did you see a movie at this cinema last year?”

Create a strategy

Apply the idea

This question relies on respondents’ memory over a long period, which could lead to inaccurate recording of information.

This is a measurement error.

c A researcher is measuring blood pressure in a study and uses a faulty blood pressure cuff that consistently overestimates the readings by 10 mmHg.

Create a strategy

Apply the idea

The faulty equipment leads to consistently inaccurate measurements.

This is a measurement error.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Example 2 Identify the cause of misrepresentation in each of these scenarios: a A political poll reports that 60% of people support a new policy. However, the poll only sampled people from a particular region known to favour the policy.

Create a strategy Evaluate whether the sample represents the entire population and if the results are generalised inappropriately.

Apply the idea Overgeneralisation and selection bias. The sample only includes people from a region known to favour the policy, making it unrepresentative of the general population. By reporting that 60% of people support the policy based on this biased sample, the poll misleads readers about the level of nationwide support.

b The daily sales of a small bookstore increased from 2 books to 5 books. The marketing company responsible for the advertising campaign claims that their latest promotion increased sales by 150%.

Create a strategy Assess whether the reported percentage increase accurately reﬂects the impact of the promotion or misleads due to the scale of change.

Apply the idea Exaggeration. The 150% increase is mathematically correct but misleading due to the small base value, exaggerating the promotion’s impact. The actual change is from 2 to 5 books, a small numerical increase of 3 books. The marketing company’s claim of a 150% sales boost makes the promotion seem more effective than it is.

c A ﬁtness company studied their new workout program, where one participant lost 30 kg. The mean weight loss was 9 kg and the median was 4 kg. The company claims the program leads to an average weight loss of 9 kg in one month.

Create a strategy Compare the mean and median to determine if the chosen measure of average misrepresents the typical outcome.

Apply the idea Strategic choice of measures. The company uses the mean, heavily inﬂuenced by an outlier, to overstate the effectiveness of the program. One participant lost 30 kg (outlier), which skews the mean weight loss to 9 kg, while the median of 4 kg better represents the typical experience. The company’s claim of an average 9 kg weight loss is misleading.

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d A university surveyed student satisfaction in three different departments. The results showed that Department A had 85% satisfaction, Department B had 88% satisfaction, and Department C had 55% satisfaction. The university claims, “Our students have over 85% satisfaction.”

Create a strategy Check if the reported statistic selectively uses data to present a more favourable outcome.

Apply the idea Cherry-picking data. The university selectively highlights the higher satisfaction rates while omitting the unfavourable data from one department. By highlighting the satisfaction rates in Departments A and B, while excluding Department C, with only 55% satisfaction, the university creates a misleadingly positive impression of overall satisfaction.

e An electronics company compares the battery life of their new smartphone to that of a competitor’s older model. The results show that the new smartphone has a battery life of 12 hours while the competitor’s older model has a battery life of 8 hours. The company advertises, “Our smartphone has 50% longer battery life than the competitor’s.”

Create a strategy Determine if the comparison uses similar or dissimilar datasets to evaluate fairness.

Apply the idea Comparing dissimilar datasets. The company compares a new smartphone to an older model, which is an unfair comparison. Comparing a new smartphone with advanced technology to an outdated model creates a biased comparison that misleads customers about the product’s superiority.

Example 3 Lachlan asked 120 Year 12 students at his school how much time they typically spend on homework per night. He found that 78 of the students said they do more than 3 hours. At a meeting of the student council, Lachlan reports, “65% of students at this school do too much homework.” Identify three issues with this survey that explain why this interpretation is misleading.

Create a strategy Evaluate whether the sample represents the population, the survey question is unbiased, and the conclusion aligns with the data.

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107

Identify the ‘misrepresentation strategy’ described by each deﬁnition: • Procedural issues • Cherry-picking data • Strategic choice of measures of centre and spread • Over-generalisation and selection bias • Exaggeration • Confusing correlation and causation • Comparing dissimilar datasets • Misleading graphs • Lack of context

Selective use of favourable results while omitting unfavourable data to present a misleading conclusion.

Choosing a statistical measure, like mean over median, to give a desired impression, often inﬂuenced by outliers.

Making ﬁndings seem more signiﬁcant than they are, often by emphasising percentages with small base values.

Assuming a relationship between two variables implies one causes the other, ignoring other factors or coincidence.

Comparing non-comparable data, like new and outdated products, to create a biased impression.

Using distorted scales, omitted data, or inappropriate graph types to mislead viewers.

Omitting background information on sample selection or methods to obscure the true reliability of results.

Match the scenario with the correct type of measurement error: a

A scale records weights of 3.45 kg, 3.42 kg, and 3.48 kg for the same item.

Faulty instrument calibration.

A sensor consistently reports temperatures 1.5° C below the actual value.

Inaccurate reading by the observer.

A stopwatch only measures to the nearest 0.1 s.

iii

Limited device precision.

A researcher records 72 cm instead of 27 cm due to a reading mistake.

Random variations in measurements.

Practice Ex 1

For each scenario, identify the type of error: a

A radio station asks listeners to call in and vote for their favourite TV show.

A survey asks cinema-goers, “How many movies did you watch here last year?”

A study on heart rate uses a monitor that overestimates by 5 bpm.

Sarah collects coffee prices from three cafes and one supermarket.

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Ex 2

Identify the cause of misrepresentation in each scenario: a

A survey reports 70% support for a new law, but only sampled urban areas favouring it.

A cafe’s sales rise from 10 to 25 pastries, claimed as a 150% increase.

A diet program reports an average 10 kg weight loss, but the median is 3 kg due to one 50 kg loss.

A college claims 90% student satisfaction, ignoring a department with 50% satisfaction.

A new laptop’s speed is compared to a 5-year-old model.

A researcher surveys 80 shoppers at a mall to determine the average weekly grocery spending in a city. Determine whether these are reasons why her sampling method might give biased results:

Ex 3

The sample does not include people from rural areas.

The mall attracts higher-income shoppers.

The sample size is too small for a large city.

Random sampling always ensures unbiased results.

Sarah surveyed 150 Year 10 students at her school about their daily screen time. She found that 90 students reported spending more than 2 hours on screens. At a parent meeting, Sarah claims, “60% of students at this school have excessive screen time.” Identify three issues with this survey that explain why this interpretation is misleading.

Identify the type of sampling error for each scenario from these options: • The sample is too small. • The sample is not random. • The sample does not represent the population. • The sample is from self-selecting participants.

A library surveys 25 members of its book club to determine reading habits of all patrons.

A website invites users to submit feedback on a new feature via an online form.

A study on public transport use interviews 15 people at one bus stop.

A researcher interviews the ﬁrst 60 people entering a park to study outdoor activity preferences.

Identify the privacy or ethical issue in each scenario: • Collecting sensitive data without consent. • Using leading or biased questions. • Disregarding cultural diversity. • Publishing identiﬁable participant data.

110

A survey asks, “Do you agree our fantastic new app improves your life?” with options “Agree” or “Strongly Agree.”

A study collects participants’ phone numbers and incomes without explaining data usage.

A dietary survey assumes all respondents eat meat, ignoring vegetarian or cultural diets.

A researcher shares a dataset with participants’ names and survey responses online.

Mathspace New South Wales – Year 11 Standard mathspace.co

Explain why these survey questions may lead to response bias: a

Do you dislike using public transport?

How much alcohol do you consume weekly?

Would you choose this eco-friendly product over cheap, polluting alternatives?

A company surveys 250 customers about product satisfaction immediately after a promotional event offering free gifts: a

Identify one fault in the survey design.

Explain how this fault could lead to an error.

A study claims 65% of residents support a new recycling program based on a survey of 100 people at an environmental fair: a

Identify the sampling error.

Explain how this affects the study’s validity.

A survey on community health asks, “Do you follow a healthy lifestyle to live longer?” Identify the fault and suggest an improvement to ensure cultural responsiveness.

Extend your thinking 15

A store reports that its new discount campaign increased sales by 300% based on an increase from 20 to 80 daily transactions: a

Is the percentage increase accurate?

Identify the misrepresentation strategy.

A ﬁtness app claims, “Users burn an average of 500 kcal per session.” The mean is 500 kcal, but the median is 300 kcal due to a few users burning 1000 kcal: a

Identify the misrepresentation strategy.

Why is this misleading?

A survey on internet usage collects data from 400 users, but only 55% respond. The company reports, “100% of users love our platform.” Discuss the ethical and bias issues in this claim, and suggest one way to improve responsiveness to diverse groups.

Investigation: Misrepresentation of results Investigate online

mathspace.co

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111

2 Chapter review 1

Which response scale is most appropriate for measuring user agreement with a statement in a customer feedback survey for a new software feature? A

Very slow, Slow, Average Speed, Fast, Very fast

Daily, Weekly, Monthly, Yearly, Never

Strongly disagree, Disagree, Neutral, Agree, Strongly agree

Yes, No

A prize wheel with 4 equal sections (Red, Blue, Green, Yellow) is spun 150 times, landing on Blue 45 times. What is the most effective action to better estimate the true population proportion for landing on Blue? A

Spin the wheel 10 more times.

Decrease the number of sections on the wheel.

Increase the sample size signiﬁcantly (e.g., spin 500 more times).

Ensure each spin is done by a different person.

A national census shows 25% of households own a pet. A survey of 1000 households ﬁnds 240 own a pet. Which statement is true about the sample and population proportions? A

The sample proportion (0.24) equals the population proportion (0.25), so the sample is perfectly representative.

The sample proportion (0.24) differs from the population proportion (0.25), indicating deﬁnite bias.

The sample proportion (0.24) estimates the population proportion (0.25), with some expected sampling variation.

A census is always less accurate than a large sample survey.

Identify whether each scenario describes a census or a survey: a

A quality inspector checks every 50th smartphone from a production line of 2000 units.

A school principal reviews the academic records of all 85 Year 12 students.

A research company calls 500 randomly selected adults in a city of 500 000 about voting intentions.

A café owner asks all 15 staff members for feedback on a new work roster.

A local council with 25 000 registered voters plans to survey 500 randomly selected voters about a proposed sports complex. Why might this be more suitable than a census of all 25 000 voters?

A national park census shows 45% native forest coverage.

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An aerial survey of 5000 hectares ﬁnds 2150 hectares of native forest. Calculate the survey proportion, rounded to three decimal places.

Compare the survey proportion to the census proportion. Is the survey a reliable estimate? Explain.

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A city of 800 000 plans to upgrade public transport and needs commuter preferences. a

List one advantage and one disadvantage of a census, and one of a survey.

Which method is more suitable? Justify.

Identify the population and sample in each scenario: a

A university food services manager questions 150 students at the main cafeteria about menu choices.

A car manufacturer tests battery life of 75 electric cars from a batch of 1500.

A journalist interviews 40 ﬁlm festival attendees about their favourite movie.

A clothing factory aims for 95% defect-free t-shirts. A sample of 250 t-shirts has 230 defectfree. a

Deﬁne the population.

What is the defect-free sample proportion as a percentage?

Is the sample representative of the factory’s aim? Explain.

An online news platform polls 5000 readers on a government policy via a yes/no click poll. a

Could random sampling be used for a more representative sample of citizens? How?

Why might the click poll be inaccurate or biased?

A tech company tests software with 600 Sydney beta testers; 18% report usability issues. They plan to launch to 3 000 000 Australian users. a

Estimate how many Australian customers might face usability issues.

Is the sample representative of the Australian customer base? Explain.

Is the testers’ primary operating system relevant for representativeness?

Identify the sampling method in each scenario: a

A market researcher interviews shoppers at a supermarket every Monday morning about cereal packaging.

A university emails a survey to all students about online learning platforms.

A health inspector selects every 15th patient record for review.

A government agency surveys proportional random samples from age groups about retirement planning.

Calculate the sampling interval for systematic sampling: a

A streaming service surveys 250 of 12 500 new subscribers.

A bakery tests 30 of 900 daily loaves.

A council seeks feedback on a community garden from 5000 residents using two strategies: • Strategy 1: Randomly select 200 residents from the electoral roll for a mailed survey. • Strategy 2: Collect 200 responses at a weekend market stall. For each, describe the sampling method, justify the label, and discuss one potential bias.

A university with 20 000 students (Arts: 30%, Science: 25%, Engineering: 20%, Business: 25%) uses stratiﬁed sampling for a 400-student survey on career aspirations. a

Calculate the number of students sampled from each faculty.

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Classify each survey question as open, closed, or partially closed: a

Primary mode of transport? Train, Bus, Car, Bicycle, Walk

Suggestions for improving customer service?

Streaming services subscribed to? Netﬂix, Stan, Disney+, Prime Video, Other (specify)

Explain why each survey question is ambiguous and suggest a clearer version: a

Do you exercise regularly?

How much time do you spend on your computer?

Is the local park satisfactory?

A council plans a survey on library services satisfaction. Explain the importance of pretesting, outline two pretesting steps, and identify one wording issue pretesting could correct.

A survey on green waste bin use lists options: (A) Weekly, (B) Fortnightly, (C) Monthly, (D) Rarely, (E) Never. Weekly and Fortnightly are most selected. Identify a design fault and suggest a ﬁx.

A university surveys students on teaching quality: • Course A (120 surveyed, 48 Excellent rating) • Course B (80 surveyed, 36 Excellent rating) • Course C (50 surveyed, 20 Excellent rating)

Calculate the overall proportion rating teaching Excellent.

Estimate Excellent ratings for 600 enrolled students.

Excluding biased Course B data, recalculate the proportion for Courses A and C.

A research institute surveys 40 employees per company on work-life balance satisfaction. Sample proportions are shown in the graph:

Frequency

10 9 8 7 6 5 4 3 2 1 0

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

Satisfactory rate

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How many samples were taken?

Estimate the population proportion, rounded to three decimal places.

Mathspace New South Wales – Year 11 Standard mathspace.co

A survey of 200 smartphone users ﬁnds 120 prefer Brand X. A simulation assumes a 55% population proportion with 500 samples of size 200. a

What is the approximate range of sample proportions expected?

How would increasing sample size to 1000 affect the spread?

What is the approximate mean of the sample proportions?

Is the survey’s 60% plausible given 55%? Explain.

Identify whether each scenario involves sampling or measurement error: a

A survey on exercise habits only includes gym members.

Respondents underreport income due to discomfort.

A thermometer reads 0.5° C too high.

A teacher measures only front-row students’ heights.

Identify the misrepresentation or poor practice in each scenario: a

A company reports a 200% traﬃc increase from 50 to 150 daily visitors.

A poll claims national representation but only surveys the capital city.

A drug company publishes only positive trial results.

Ice cream sales correlate with crime, implying causation.

Identify the primary privacy/ethical issue from: Lack of informed consent, Leading questions, Failure to ensure anonymity, Cultural insensitivity. a

A health survey sells medical data to insurers without informing participants.

A survey asks, “Only irresponsible citizens oppose this project. Do you support it?”

A company links employee satisfaction quotes to names without permission.

A family traditions survey is translated into only one minority language.

A streaming service survey with 30% response rate claims “95% of users highly satisﬁed” based on respondents. Discuss ethical/bias issues and suggest an improvement for diverse groups.

A streaming service claims users watch 20 hours weekly (mean), but the median is 8 due to heavy users. a

Identify the misrepresentation strategy.

Why is this claim misleading?

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Big ideas Data classification and visualisation enable effective representation and analysis of datasets through appropriate graph types and design techniques, highlighting trends, comparisons, and distributions.

3 Data classification and display Chapter outline 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08

Classify data Tables and bar charts Sector and line graphs Dot plots and stem-and-leaf plots Histograms and grouped frequency tables Cumulative frequency tables and graphs Shape of distribution Misleading graphs and appropriate displays Investigation: Appropriate choice of graph for data Investigation: Spreadsheets to tabulate and graph data Investigation: Infographics Chapter 3 review

118 126 145 163 177 195 207 216

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3.01 Classify data After this lesson, you will be able to… • classify variables as categorical or numerical. • distinguish between nominal and ordinal categorical variables. • distinguish between discrete and continuous numerical variables. • explain how the phrasing of a question can determine the type of data collected from a variable. • identify the type of data (and its sub-types) produced by a given variable or context.

Types of data In statistics, data refers to a collection of observations or measurements gathered from a study or survey. Data Facts or units of information collected together. Categorical data Data associated with a categorical variable. Also known as qualitative data. Numerical data Data associated with a numerical variable. Also known as quantitative data.

A variable is a characteristic or property that can take different values across observations. For example, in a survey, the “height of a student” is a variable because it varies from student to student. Variable Something measurable or observable that is expected to change either over time or between individual observations. For example, the age of students, their hair colour or a playing ﬁeld’s length or its shape. Data can be classiﬁed into two main types based on the nature of the variable: categorical data and numerical data. A categorical variable produces categorical data, which is non-numerical and describes qualities or characteristics. For example, the variable “favourite colour” produces categorical data like “blue” or “red.” In contrast, categorical data is the actual set of responses collected, while the categorical variable is the characteristic being measured.

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Categorical variable A variable whose values belong to exactly one of a number of categories. A categorical variable describes a quality or characteristic of something. Sometimes called a discrete variable. There are 2 types of categorical variables: nominal and ordinal. For example, your home state or blood type are categorical variables. A numerical variable produces numerical data, which can be counted, ordered, or measured. Numerical variable Variables whose values are numbers, and for which arithmetic processes such as adding and subtracting, or calculating an average, make sense. A discrete numerical variable is a numerical variable, each of whose possible values is separated from the next by a deﬁnite ‘gap’. The most common numerical variables have the counting numbers 0, 1, 2, 3, … as possible values. Others are prices, measured in dollars and cents. For example, the number of children in a family or the number of days in a month. Numerical data can be divided into two subcategories based on the type of numerical variable: • A discrete variable produces discrete data, involving only whole numbers starting from 0 (e.g., 0, 1, 2, …). For example, the variable “number of students in a class” might yield data like 20 students, as there cannot be 20.5 students. • A continuous variable produces continuous data, which can take any value, including decimals or negative numbers. For example, the variable “foot length” might yield data like 25.3 cm, “temperature” might yield −4 or 25.5° C. Discrete variable Individual and countable items that can be listed. Also known as a discrete random variable. Continuous variable A numerical variable that can take any value that lies within an interval. The values taken are subject to the accuracy of the measuring instrument used to obtain these values. For example, height, reaction time to a stimulus, and systolic blood pressure.

Categorical data can be divided into two subcategories based on the type of categorical variable: • A nominal variable produces nominal data, where categories have no natural order. For example, the variable “favourite subject” might yield data like “Maths” or “English.” • An ordinal variable produces ordinal data, where categories have a natural order. For example, the variable “race position” yields data like “first,” “second,” etc.

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Context dependent data classification The same variable can produce numerical data or categorical data depending on how survey questions are phrased.

Exploration Work with a partner to design survey questions about exercise habits: 1. Write a survey question for a numerical variable that yields numerical data (e.g., a measurable quantity like time or distance). 2. Rewrite the question for a categorical variable to yield categorical data (e.g., categories like yes/no or types of exercise). 3. Discuss how the variable type affects the data collected and its analysis.

Example 3 A school with 500 students conducts a survey using random sampling. Classify the variable as producing numerical data or categorical data for each question. a How many minutes does it take you to travel to school?

Create a strategy Determine if the variable produces a measurable quantity or a category.

Apply the idea

Reﬂect and check

The variable “travel time” produces a time (e.g., 15.5 minutes), a measurable quantity. It is a numerical variable producing numerical data, speciﬁcally a continuous variable yielding continuous data.

A numerical variable like travel time allows calculations, such as the average travel time for the sample.

b What is your primary mode of transport to school (walk, car, bus)?

Create a strategy Check if the variable produces categories or a measurable value.

Apply the idea

Reﬂect and check

The variable “mode of transport” produces categories (e.g., walk, car, bus), with no natural order. It is a categorical variable producing categorical data, speciﬁcally a nominal variable yielding nominal data.

A categorical variable allows us to count frequencies, e.g., the number of students using each transport mode.

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3.01 Practice questions What do you remember? 1

True or false: a

A nominal variable produces numerical data.

A numerical variable can be discrete or continuous.

A categorical data is also known as qualitative data.

A numerical variable produces categorical data.

Fill in the blank with the correct term (numerical, categorical, discrete, continuous, nominal, ordinal, context-dependent): a b c d

A variable that represents measurable quantities, like height, is a ⬚ variable.

A categorical variable with categories that have no order, like favourite colour, is a ⬚ variable. A numerical variable that can take any value in a range, like time, is a ⬚ variable.

A variable that can produce numerical or categorical data based on how the question is asked is associated with ⬚ classiﬁcation.

Write a statement that best describes each concept: a

Categorical variable

Numerical variable (discrete)

Categorical variable (ordinal)

Numerical variable (continuous)

Context-dependent variable

Practice 4

Classify each variable as numerical or categorical: a

Amount owing on layby

Types of vegetables

Brands of tablets

Colours of folders

Brands of phones

Maximum snowfall

Daily UV index

Types of dogs

Favourite colours

Cost of a computer game

Types of fruits

For each categorical variable, identify whether it is nominal or ordinal: a

Ex 1

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Ice cream ﬂavour

Smartphone application

Clothing size

Blood type

Customer satisfaction rating

Pain level

Membership rank

Cuisine preference

For each numerical variable, identify whether it is discrete or continuous: a

Number of classrooms at a school

Daily humidity

The time taken to run 375 metres

The number of people taller than you

The number of coins in a coin collection

The amount of snowfall in a city

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 2

Ex 3

Ex 4

For each survey question, classify the variable as numerical (discrete or continuous) or categorical (nominal or ordinal): a

How far do you travel to school each day?

How many siblings do you have?

What modes of transport do you use to get to school?

What kinds of pets do you own?

How many pets do you own?

How tall are you?

What level of education have you completed?

A community centre with 1000 members conducts a survey using random sampling. Classify the variable as numerical (discrete or continuous) or categorical (nominal or ordinal) for each question: a

How many hours do you spend at the community centre each week?

What is your preferred activity at the centre (e.g., swimming, yoga, gym)?

How satisfied are you with the centre’s facilities (very satisfied, satisfied, unsatisfied)?

A company with 200 employees uses stratiﬁed sampling to survey 50 employees, proportional to team sizes. For each question, classify the variable as numerical (discrete or continuous) or categorical (nominal or ordinal) and suggest an alternative question to produce the opposite data type: a

How many minutes is your daily commute to work?

What is your work stress level (low, medium, high)?

Extend your thinking 10

Design two survey questions about dietary habits: one that yields a categorical variable and one that yields a numerical variable. Classify the variable type (including subtypes: nominal, ordinal, discrete, continuous) for each.

Explain why the variable “age” can be classiﬁed as either a discrete or continuous numerical variable, and provide two survey questions—one yielding a discrete numerical variable and one yielding a continuous numerical variable.

A city council wants to understand the commuting habits of its residents. Suggest two variables to collect, classify each as numerical (discrete or continuous) or categorical (nominal or ordinal), and explain why each is useful.

You are part of a school council tasked with ﬁnding out the most popular after-school activities among students: a

Describe how and where you would carry out this survey, including which groups of people you would focus on.

What type of variable would you be collecting? Include an additional question about time spent on activities and classify its variable type.

How would you ensure that your data collection methods are unbiased and accurate?

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You are a researcher studying the relationship between hours of sleep and academic performance among high school students: a

What type of variable would you need to collect to investigate this? Suggest an alternative question about sleep that yields a categorical variable.

How would you go about collecting this data?

3.02 Tables and bar charts After this lesson, you will be able to… • construct and interpret frequency tables for categorical data, including the use of tallies. • interpret information presented in horizontal bar graphs and vertical column graphs. • construct column graphs from given categorical data or frequency tables. • interpret two-way frequency tables. • interpret side-by-side bar charts and understand their relationship to two-way tables.

Bar graphs Bar graph is a generic name for any graph that displays information using rectangular or cylindrical bars. The terms bar graph and bar chart are used interchangeably. The height or length of the bars is proportional to the values they represent. Bar charts are a popular choice because they are easy to create and interpret.

Example 1 The sales (in thousands) of different products are shown in this horizontal bar graph. Product A Product B Product C Product D Product E Product F 0

Sales (thousands of units) 126

Mathspace New South Wales – Year 11 Standard mathspace.co

Column graphs A column graph is the name for a speciﬁc type of bar graph that uses vertical bars, so that they appear like columns.

Example 2 A survey of the preferred sport was done for a group of boys and the results are shown in this bar graph:

Number of boys

Preferred sport 10 9 8 7 6 5 4 3 2 1 0

Football

Tennis

Rugby Sport

Basketball

Hockey

a How many boys prefer football to other sports?

Create a strategy Look across to the number that matches the height of the bar for football.

Apply the idea The height of the bar for football goes up to the line next to the number 6 on the left. 6 boys prefer football to other sports.

b Which type of sport is the most popular?

Create a strategy Choose the sport that has the tallest column in the graph.

Apply the idea The most popular type of sport means the most boys preferred it. Hockey has the tallest column in the graph, indicating it is the most preferred sport. The most popular type of sport is hockey.

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Side-by-side bar charts Also known as clustered bar charts or grouped bar charts, these are useful for displaying information about different subgroups within the main categories. Each subgroup is coloured or shaded differently to distinguish between them, and a legend is used to indicate the subgroup that each colour represents. The data for a side-by-side bar chart may come from a two-way table. The table shows the amount (in kilotonnes) of various categories of waste in Australia that were either recycled or dumped into landﬁll. Recycled

Landﬁll

Totals

Paper and cardboard

3361

2230

5591

Plastics

334

2182

2516

Glass

6612

4467

11 079

Organics

7461

6710

14 171

Totals

11 768

11 589

23 357

Notice that the ﬁnal row and column of the two-way table contain the totals for each row and column. The table was used to create this side-by-side bar chart:

Percentage of waste within each category

100.0

Waste management in Australia 2016/17 financial year

75.0

Recycled Landfill

50.0

25.0 0

Paper and cardboard

Plastics

Glass

Organics

Notice that the vertical axis represents the percentage amount allocated to either recycling or landﬁll, rather than the actual amounts in kilotonnes to make the total waste of the categories comparable. For example, the percentage of paper and cardboard that was recycled was calculated as follows: Set up the calculation Evaluate and round to one decimal place

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3.02 Practice questions What do you remember? 1

True or false: a

A bar graph uses bars to represent numerical data only.

Column graphs are a type of bar graph with vertical bars.

Frequency tables show how often each category occurs.

Side-by-side bar charts cannot display data from two-way tables.

Fill in the blank with the correct term (bar graph, column graph, frequency table, side-by-side bar chart): a b c

A ⬚ uses vertical bars to display categorical data.

A ⬚ organises categorical data with tallies and counts.

A ⬚ shows subgroups within categories using different colours.

Match each term to its key feature: a

Bar graph

Lists categories and their occurrence counts

Compares subgroups within categories

iii

Bars proportional to category values and with gaps

Frequency table

Side-by-side bar chart

Practice Ex 1

The ticket sales for various amusement park rides are shown in this horizontal bar graph: Roller Coaster Ferris wheel Log ride Drop tower Bumper cars Carousel 0

9 10 11 12 13 14 15 16 17 18

Tickets sales (in thousands)

134

Which is the lowest-selling ride?

How many tickets for all rides were sold in total?

If the best-selling ride’s tickets were sold for $10 each, calculate the revenue generated by the best-selling ride alone.

Mathspace New South Wales – Year 11 Standard mathspace.co

Construct a two-way table for each set these data: a

Miss Merryweather recorded the hair type and colour of her students in this graph: 10 9

Number of students

8 7 6

Straight Curly

5 4 3 2 1 0

Red

Brown

Blonde

Black

Rhett surveyed a group of people about their preferred shopping methods for clothing purchases. The results are shown in this graph:

Number of people

32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

Men Women

Local shops

Online marketplaces

Malls

Second-hand markets

Shopping methods c

A survey was conducted about people’s favourite movie genre. The results were plotted in this graph: 50 45

Number of people

Ex 4

40 35 30

Women Men

25 20 15 10 5 0

Comedy

Action

Drama

Horror

Movie genre

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Members of a gym were surveyed about what kind of training they do. Each responder only did one kind of training and the results are given in the table: Cardio

Weight

Male

Female

Determine the total number of gym members surveyed.

Determine the number of members who undertake weight training.

Calculate the percentage of members who undertake weight training.

Mr. Tobit asked the students in his class to pick their favourite subject. He displayed the results in the given two-way table: Maths

Music

Science

English

Boys

Girls

How many girls did not pick Maths as their favourite subject?

How many students picked Music?

Did boys prefer Music or Science?

The table shows the house points earned by four different houses at their swimming carnival:

House

Points

Construct a column graph for the data.

Yellow

Which house earned the most points?

Blue

Which house earned as many points as the Yellow house and Blue house combined?

Red

Green

The column graph illustrates the genres of movies on Alex’s favourite streaming site:

Number

Movies

138

500 450 400 350 300 250 200 150 100 50 0

Fiction

Sci-Fi

Comedy

Horror Genre

How many movies in total are there on the site?

What percentage of the movies are comedy?

Mathspace New South Wales – Year 11 Standard mathspace.co

Thriller

Action

The horizontal bar chart shows the cost of 1 GB of mobile data in selected countries: a

Which country has the cheapest mobile internet data?

How much is the price of 1 GB of mobile internet data in Australia?

How much would it cost for 17 GB of data in the United Kingdom?

How many gigabytes of data could you get in India for the same price as 1 GB of data in Australia?

International mobile data prices India $0.26 $ Italy $1.73 Nigeria $2.22 Australia $2.47 France $2.99 $3.50 Brazil $6.66 UK China USA

$9.89 $11.71 0 2 4 6 8 10 12

Cost of 1 GB of data ($/GB) The table and bar chart shows the travel method used by Sydney residents according to their age: Transport

Number of trips (thousands)

Vehicle driver

8310

Vehicle passenger

3830

Train

945

Bus

1058

Walk only

3076

Other

385

10 000

Thousands of trips

Number of trips (in thousands) by mode of transport Sydney 2012/13

7500 5000 2500 0

Driver Passenger Train

Bus

Walk

Other

Mode of transport a

Which mode of public transport is the most popular?

How many trips were covered in total by the survey?

What percentage of the total number of trips were made by walking? Give your answer as a percentage, rounded to one decimal place.

What percentage of the total number of trips were a driver or passenger? Give your answer as a percentage, rounded to one decimal place. 3.02 Tables and bar charts mathspace.co

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This stacked bar chart shows different types of internet traﬃc on a website over a three-month period:

Website traffic

Determine the total number of visits to the website in November. Round your answer to the nearest 100.

During October, how many visitors to the website were new visitors?

What proportion of traﬃc during October was returning visitors?

Approximately how many new visitors visited the website during November? Round your answer to the nearest 100.

Number of visitors

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

Oct

New visitors

Nov

Returning visitors

The table and stacked bar chart shows the travel method used by Sydney residents according to their age: Age

Car (driver)

Car (passenger)

Train

Bus

Walk

Other

Total

0 − 10

76.6

0.7

17.3

1.4

100

11 − 20

12.8

42.7

9.1

17.8

14.9

2.7

100

21 − 30

47.3

12.6

10.4

7.4

19.7

2.6

100

31 − 40

59.6

7.8

6.3

4.2

19.9

2.2

100

41 − 50

69.6

7.3

2.6

14.3

2.2

100

51 − 60

65.7

9.4

4.2

3.2

15.8

1.7

100

61 − 70

58.9

13.5

3.4

3.9

18.7

1.6

100

Choice of travel method by age Proportion of population (%)

100 90 80

Car (driver)

Car (passenger)

Train

50 40

Bus

Walking

Other

10 0

0-10 11-20 21-30 31-40 41-50 51-60 61-70

Age

140

Dec

Which age group has the highest proportion of car travel as a passenger?

Which age group uses public transport (bus or train) the most?

What percentage of people aged 11 − 20 travel by public transport?

Which age group does the least amount of walking?

Mathspace New South Wales – Year 11 Standard mathspace.co

Extend your thinking

What is the most expensive vehicle type to run?

How much would it cost to travel 100 km in a small hatch?

How much would it cost to travel 100 km in a large sedan?

How much cheaper is it to travel 100 km in a 2WD ute compared to a 4WD ute?

Vehicle type

Cost (cents/km)

Small hatch

55.4

Medium sedan

70.6

Large sedan

86.5

People mover

83.8

SUV medium

71.5

SUV large

80.7

2WD ute

75.4

4WD ute

84.8

This bar graph shows the changes in tourism rates in different cities during 2011 and 2012: 100 90 80 70 60 50 40 30 20 10

e Ro m

ai gh

n an

Lo n

ko k

yo Ba

2011

To k

Pa r

Ist

i ba Du

Yo r

Consider the running cost, in cents per kilometre, of different vehicle types outlined in the table provided:

Percentages of tourism (%)

2012

What is the difference between Shanghai’s tourism percentages for 2011 and 2012?

Which cities had the greatest difference in percentages of tourism over 2011 and 2012?

Which city had the smallest difference in minimum and maximum percentages of tourism over 2011 and 2012?

What is the difference between the minimum and maximum percentages of tourism of all the cities in the graph over the two years?

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This table shows the number of people in NSW, aged 15 years and older, who attended selected cultural events during the 2013/14 ﬁnancial year: Event type

Population

Art galleries

1540

Museums

1559

Zoos and aquariums

1909

Botanic gardens

2079

Libraries

1883

Classical music concerts

548

Popular music concerts

1951

Theatre performances

890

Dance performances

601

Musicals and operas

862

Cinemas

3821

Total

17 643

The table was then used to create a horizontal bar chart where each bar represents the percentage of the population that attended a particular event: Attendance of cultural events NSW 2013 Art galleries Museums Zoos and aquariums Botanic gardens Libraries Classical music concerts Popular music concerts Theatre performances Dance performances Musicals and operas Cinemas 0

Percentage of population attendance (%) a

Which was the most popular cultural event?

How many cultural events were attended by less than 20% of the population?

A NSW town funds these six events: • The town library • The wildlife zoo • The royal botanical garden

• The local art gallery • The annual theatre production • The science museum

A local mayor has decided to reduce funding to the two least popular events in the town. Which two events should they cut?

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The horizontal bar graph shows the number of crashes by type for NSW during 2017: Crashes in NSW 2017 Rear-end Off-path (on straight) Off-path (on curve) Intersection Side swipe Opposite direction Pedestrian Manoeuvring Head-on On-path Overtaking Other 0

Metropolitan Country

500 1000 1500 2000 2500 3000

Number of crashes a

In which region did most accidents occur?

Which was the most common accident type in country areas?

Which accident types had over double the incidents on metropolitan roads compared to country roads?

Identify whether each statement is true or false based on the bar graph provided: i

There were more off-path crashes (both on straight and curved metropolitan roads) than rear-end crashes.

There were about the same number of metropolitan pedestrian crashes as head-on and on-path crashes across all roads.

iii

There were more rear-end crashes on country roads than manoeuvring crashes.

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The table and graph show the average annual cost of comprehensive car insurance for NSW drivers: Proﬁle

Cost

Under 25 years old, Male

$2601

Under 25 years old, Female

$2315

25 − 29 years old, Male

$1202

25 − 29 years old, Female

$1298

30 − 44 years old

$1006

45 + years old

$868

Average annual comprehensive car insurance premium Annual insurance premium ($)

3000 2500 2000 1500 1000 500 0

< 25 M

< 25 F

25-29 F

25-29 M 30-44 M/F 45+ M/F

Age profile

144

Which age group pays the most for comprehensive car insurance?

How much more does a male under 25 pay than a female under 25?

How much would a 27 year old female expect to pay for comprehensive car insurance?

If insurance companies offer lower premiums to safer drivers, identify whether these statements are supported by the data given: i

Female drivers in the age range 25 − 29 are statistically safer drivers than men in this age range.

An under 25 female driver is a statistically safer driver than a 25 − 29 male driver.

iii

Drivers are statistically safer the older they get.

At any age, female drivers are statistically safer drivers than men.

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3.03 Sector and line graphs After this lesson, you will be able to… • interpret information presented in pie graphs (sector graphs), including percentages and fractions. • calculate sector angles and construct simple pie graphs from given data. • interpret information presented in line graphs, identifying trends, peaks, and troughs. • construct line graphs from data presented in tables. • interpret and understand the purpose of divided bar graphs in representing proportions.

Pie graphs A pie graph divides a circle into sectors to represent the different parts that make up a whole. Each sector, or slice, represents a speciﬁc percentage or proportion of the total. The larger the sector, the greater the percentage of data it represents. The total percentages in a pie graph sum to 100%, and the central angles of the sectors total to 360°. The information in the table shown can be presented as a pie graph: Sport

Frequency

Football

Fraction

Percentage frequency 35%

10% 10%

Basketball

25%

Netball

20%

Tennis

10%

Swimming

10%

Total

100%

25%

20% 35%

Basketball

Football

Netball

Tennis

Swimming

To draw a pie graph: 1. Use a compass to draw a circle and mark the centre. From the circumference of the circle, draw a straight line to the centre to establish a starting point for measuring angles. 2. Measure the angle of the ﬁrst sector using a protractor. Mark the point and draw a line from the centre to the circumference towards this point. 3. Repeat this process for all sectors until the pie graph is complete. 4. Label each sector with its category name or use a legend.

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To calculate the angle size ( θ ) for each sector, express the frequency of each category as a fraction of the total frequency and multiply by 360°.

θ = Fraction × 360°

Example 1 A florist counted the roses in the store in each colour and displayed the results in a pie graph:

10% 15% 50% 25%

Red

Orange

White

Pink

a What is the most common rose colour?

Create a strategy

Apply the idea

Identify the colour with the largest sector.

The most common colour is red.

b What percentage of roses are pink?

Create a strategy

Apply the idea

Examine the pink sector of the graph.

10% of the roses are pink.

c What fraction of the roses are orange?

Create a strategy Convert the percentage of orange roses to a fraction.

Apply the idea

Reﬂect and check

The orange sector represents 25% of the

If the percentage was not given, it is clear that

roses. 25% is equivalent to .

the orange sector takes up

of the circle.

d Determine the number of white roses given that the total number of roses is 200.

Create a strategy Identify the percentage of roses that are white and determine this percentage of 200 roses.

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Apply the idea Number of white roses = 15% of 200

Write the equation

= 0.15 × 200

Express percentage to a decimal

= 30

Evaluate

Example 2 Emma and Lucas surveyed car colours at a busy intersection for one hour, counting a total of 500 cars. Their results are shown in this table: Colour

Frequency

Blue

Red

White

150

Black

125

Other

100

Total

500

Fraction

Angle size

a Complete the table by providing the entries for the fraction and angle size columns.

Create a strategy Express the frequency of each colour car as a fraction of the total number of cars and simplify. To determine the angle size, multiply the fraction by 360°.

Apply the idea Colour

Frequency

Blue

Red

White

150

Black

125

Other

100

Total

500

Fraction

Angle size

360°

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b Construct a pie graph to represent the results.

Create a strategy Draw a circle and use a protractor to measure and mark the angles for each sector as calculated in part (a).

Apply the idea

Use a compass to draw a circle and mark the centre. From the circumference of the circle, draw a straight line to the centre.

Measure the angle of the ﬁrst sector using a protractor.

Car colours

Blue

Red

White

Black

Other Mark the point and draw a line from the centre to the circumference towards this point.

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Repeat the process until all sectors are complete. Label each sector with its category name and percentage.

b What was the highest number of ice creams sold at any time?

Create a strategy

Apply the idea

Determine the number of ice creams sold when the line graph reaches its highest point.

The highest number of ice creams sold at any time was 24.

c There are two peak hours for ice cream sales: at 1:00 p.m. (lunchtime) and at 6:00 p.m. (evening) What was the difference in sales between these two peak times?

Create a strategy Calculate the difference between the number of ice creams sold at 1:00 p.m. and the number of ice creams sold at 6:00 p.m.

Apply the idea The number of ice creams sold was 24 at 1:00 p.m. and 23 at 6:00 p.m. Difference = 24 − 23 = 1 ice cream

Write the equation Evaluate

Example 4 The monthly visitor numbers to a popular Australian beach resort over the course of a year is shown in the table: a Construct a line graph to represent the data. Month

Visitors

Month

Visitors

January

1500

July

500

February

1200

August

550

March

1300

September

700

April

1100

October

900

May

800

November

1300

June

600

December

1800

Create a strategy Time is generally displayed on the horizontal axis. Plot the points from the table and join them with straight lines.

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Divided bar graphs A divided bar graph is a graph in which the bar represents the whole dataset and the bar is divided into several segments to represent the proportional size of each category. A bar graph can be any length, but it can be helpful to think about what length could make the data easier to divide up - multiples of 5 or 10 are often good. Maintaining an appropriate length is essential. Consider a collection of 100 jellybeans sorted by colour: 30 are green, 28 are pink, 28 are orange, and 14 are white. With a total of 100 jellybeans, the divided bar graph can be made 10 cm long. This choice creates a convenient ratio where 1 cm represents 10 jellybeans. To determine each colour’s segment length on the bar graph. Each colour should be written as a fraction of the whole, then evaluate this fraction of the line. For example, jellybeans are green and

of the

× 10 = 3. This means that 3 cm of the 10 cm bar graph should be

given to the green jellybeans. Similarly

× 10 = 2.8, so 2.8 cm should be given to both pink and

orange and 1.4 cm should be given to white. The calculations can be checked by adding up the length values:

30 jellybeans

28 jellybeans

14 jellybeans

3 + 2.8 + 2.8 + 1.4 = 10

Exploration Consider this table showing the distribution of 100 jellybeans by colour, and its representation in a sector graph, divided bar graph, and line graph. 1. What’s similar or different about how they show the data?

Colour

Frequency

2. Are any of the graphs more suited or less suited for this scenario?

Green

Pink

Orange

White

Total

100

3. Which graph is most effective for quickly comparing the exact frequencies of each colour, and why?

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This pie graph shows the results of a class survey where students were asked to nominate their favourite food: a

Which food was the most popular?

Which two foods were equally popular?

Favourite food Nuggets Pizza

Noodles Burger

Which brand is the most popular?

Which brand is the least popular?

Which brand is approximately preferred by one-quarter of the students?

Noodles

Nuggets

14% 18% 45% 25%

If 56 students selected Hot Cheezos, how many students were surveyed?

Tortillos

Hot cheezos

Snakis

Wrinkles

Roald has a weekly income of $1200. This pie graph shows a break-down of his spending: a

Burger

This pie graph represents the popularity of each chip brand amongst Grade 12 students:

Pizza

Determine how much money Roald spends weekly on each of these: i

Leisure

Clothes

iii

Food

Housing

30%

If he sticks to his savings plan, how much will he have saved after 5 weeks?

Food Savings Housing

25% 5% 10%

Leisure Clothes

The pie graph represents the market share of four brands: a

Given that Brand C is twice as popular as Brand A, determine the angle at the centre for Brand C.

What should be the total angle for all sectors at the centre of the circle?

Determine the angle at the centre for Brand D.

72°

90°

Brand A

Brand B

Brand C

Brand D

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This pie graph represents government spending: a

What percentage is spent on Education?

What percentage is spent on Healthcare?

What percentage is spent on Defence?

90° 147.6°

54°

Ex 2

The number of songs on Maria’s iPod, sorted by genre, is displayed in this table:

b Ex 3

Healthcare

Education

Defence

Roads and transport

Genre

Number of songs

Rock

Pop

Soundtracks

RnB

Complete the table by ﬁlling in the entries for the fraction and angle. Genre

Frequency

Rock

Pop

Soundtracks

RnB

Total

180

Fraction

Angle

Construct a pie graph for this data.

This line graph shows the seal population at Rocky Point over ﬁve years: 55 Number of seals 50 45 40 35 30 25 20

156

Years 2013

2014

2015

2016

2017

Which year had the lowest seal population?

What was the highest seal population over the ﬁve years?

By how much did the seal population increase over the ﬁve years?

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The highest daily temperature in Sommersville over a week is shown in this line graph: Daily temperature

Temperature (°C)

26 24 22 20 18 16 14

Monday Tuesday Wednesday Thursday Friday Saturday Sunday Day

Which day had the warmest temperature?

Which two days had the same temperature?

How much warmer was Friday than Monday?

Manuela is an international student who is feeling homesick. She formulates the question: “How long are international students usually homesick for?” She surveys everyone currently on exchange through her program. She organises the data in this table: Time

Number of students

Fraction

Angle

Less than a week

220

90°

1–2 weeks

264

108°

3–4 weeks

132

54°

1–2 months

176

72°

3–4 months

36°

Construct a pie graph to represent this data.

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This line graph shows the height of a tree over four years: 11 10 9 8 7 6 5 4 3 2 1

Height (m)

Years 0.5

1.5

2.5

Determine the initial height of the tree?

Determine the height of the tree after 2.5 years?

How much did the tree grow during the ﬁrst year?

How much did the tree grow during the second year?

Did the tree grow more in the ﬁrst or the second year?

Determine the total growth of the tree over the 4 years.

3.5

The price of cans of soft drink over a year is shown on this line graph: $3.00

Soft drink prices

$2.75

Price

$2.50 $2.25 $2.00 $1.75 $1.50 $1.25 $1.00

158

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Month

Determine the lowest price of the cans of soft drink.

Identify the months in which the price was the lowest.

Calculate the total value of the sales during the months identiﬁed in part (b) if 250 cans of soft drink are sold each month.

Determine the highest price of the cans of soft drink.

Identify the months in which the price was the highest.

Calculate the total value of the sales during the months identiﬁed in part (e) if 250 cans of soft drink are sold each month.

Calculate the difference in the value of sales between the months with the highest price and the months with the lowest price.

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Customers at Pareto’s burritos can select the hotness of their burritos on the Pareto hotness scale from

to . The hotness that customers chose over a day is shown on the line graph: 18 16 14 12 10 8 6 4 2

Number of customers

Pareto hotness scale 0 5

1 5

2 5

3 5

4 5

5 5

Complete this table: Pareto hotness scale Number of customers

Express the scores from the Pareto hotness scale in order from most popular (highest number of customers) to least popular (lowest number of customers).

How many more customers chose the most popular score than the second most popular score?

How many more customers chose the second least popular score than the least popular score?

How many more customers chose the most popular score than the least popular score?

How many customers participated in the survey?

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This graph shows monthly rainfall data for Berlin and London: Monthly rainfall (mm) in major cities

70 65 60 55 50 45 40 35 30

Ex 4

Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Berlin London

Determine the highest monthly rainfall recorded in Berlin over the year.

Determine the highest monthly rainfall recorded in London over the year.

Identify which city records the highest monthly rainfall over the year.

Calculate the range in monthly rainfall for Berlin between June and October.

In the months between March and July (inclusive), determine the highest monthly rainfall for London.

This table shows the Australian mean monthly rainfall for 2018: a

Construct a line graph to represent this data.

Describe the trend of the rainfall over the year.

Rainfall (mm)

Jan

105

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

The table shows the number of different vegetables Susana ate on each day of the week: Day

Sun

Mon

Tue

Wed

Thu

Fri

Sat

Number of vegetables

Construct a line graph for this information.

160

Month

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A divided bar graph shows the percentage of total books in a library by category. Fiction books number 36 000 and represent 20% of the total books: a

What is 1% of the total books?

Find the total number of books in the library.

This graph represents the number of iTunes sales (in millions) and iPod sales (in hundreds of thousands) every 3 months between 2003 and 2007: 4000 3500 3000 2500

Sales

Ex 5

2000 1500 1000 500 0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 2007 2006 2003 2004 2005 Quarter iPod

iTunes

How many more iTunes sales were there than iPod sales in the ﬁrst quarter of 2006? 23

This table shows the UV rating for a summer’s day in Sydney: Time (in 24 hours)

0600

0800

1000

1200

1400

1600

1800

UV index

0.9

6.3

10.5

11.7

7.2

2.4

Plot a line graph for the data.

Determine at approximately what time was the UV index highest.

A very high rating of UV is considered to be 7.2 or above. Identify the approximately range of hours should a person stay inside to avoid being in the sun above this UV index.

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Extend your thinking 24

This question is being explored through the data cycle: “What activities do students do after school? Does it vary by school level?” Random samples of students were taken from several elementary schools, middle schools, and high schools. Students were asked about the ﬁrst activity they do after school. These pie graphs summarise the data: Elementary after school activities

Middle after school activities

High after school activities

Play outside

Video games

Homework

Sport team

Artistic activity

Work for family business

Others

Perform a detailed analysis of how after school activities differ amongst elementary, middle, and high school students. Provide insights into how this data can be used to make informed decisions about after school programs. 25

Lucille formulated the question “What dairy products are the most popular amongst Americans?”. She collected data from her classmates and presented it in pie graph A. Her friend DeShaun used data from a national survey of 10 000 people and summarised the results in pie graph B. Pie graph A

Pie graph B

Milk

Yogurt

Milk

Yogurt

Cheese

Cottage cheese

Cheese

Cottage cheese

DeShaun believes that his graph is more representative of the American population. Evaluate the reasonableness of this belief. Justify your decision using mathematical reasoning. 26

A kitchen sink starts with 20 L of water in it. It empties at a rate of 4 L per minute and then after 4 minutes the drain gets blocked. No water empties for 2 more minutes while it gets unblocked, and then the remaining water is drained out in 1 minute. Construct a line graph to represent this scenario.

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Describe the change in temperature of the cup of tea over the 15-minute period as shown in this graph, including an analysis of the rate of cooling for each segment.

100 °C 90 80 70 60 50 40 30 0

Time (mins) 5

3.04 Dot plots and stem-and-leaf plots After this lesson, you will be able to… • construct and interpret dot plots for discrete numerical data. • construct and interpret stem-and-leaf plots, including the use of a key. • understand and construct split stem-and-leaf plots to better display data distribution. • construct and interpret back-to-back stem-and-leaf plots to compare two datasets. • identify key features such as spread and clustering from these plots.

Dot plots A dot plot displays the frequency of discrete numerical data using dots, with each possible outcome shown along a horizontal axis. Each individual data value is represented by a single dot stacked vertically above the axis, making it ideal for small to medium-sized datasets. The height of each stack corresponds to the frequency of that particular outcome. Features to note: • There is no vertical axis, only a horizontal axis. • Outcomes are evenly spaced along the horizontal axis. • Outcomes are in consecutive, ascending order including all values between the minimum and maximum. • Dots are even spaced in columns above the line. • The axis is labelled appropriately. 0

Days per week spent exercising

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Example 1 This dot plot displays the number of goals scored in each of Bob’s soccer games.

Number of goals a Determine the number of games in which zero goals were scored.

Create a strategy

Apply the idea

Count the dots above the number 0.

There are 9 dots above the number 0, indicating there were 9 games where zero goals were scored.

Reﬂect and check It is important to understand what each component of the plot represents. A common misconception is focusing on the column with no dots. Here, the column with no dots is two, indicating Bob had no games where two goals were scored. It is also important to note that outcome 2 is included in the dot plot even if there are no dots above it, this is because numerical data must be arranged in consecutive, ascending order including all values between the minimum and maximum. b Identify the most common number of goals scored per game.

Create a strategy

Apply the idea

Identify the tallest stack of dots.

The tallest stack of dots is above the number 3, indicating the most common number of goals scored per game is 3.

c Determine the total number of games played.

Create a strategy Count the total number of dots in the graph.

Apply the idea Add the number of dots in each column. Number of games = 9 + 7 + 10 + 6 + 3 = 35

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Write the equation Evaluate

3.05 Histograms and grouped frequency tables After this lesson, you will be able to… • explain why some datasets need to be grouped into class intervals. • construct and interpret grouped frequency tables for numerical data. • calculate class centres (midpoints) for grouped data. • construct and interpret histograms for ungrouped and grouped numerical data. • identify the modal class and estimate the mean and median from grouped data.

Histograms Histograms are used to display the frequency of both discrete (grouped or ungrouped) and continuous (grouped) numerical data. Unlike column charts, histograms have no gaps between the columns. A class interval can have a frequency of zero, but this differs from having gaps between the bars; it simply indicates that no data falls within that speciﬁc interval. In a histogram, the position of the values on the horizontal axis depends on whether the data is grouped or ungrouped. • Ungrouped data: The values are placed at the centre of each column. • Grouped data: For grouped data, the lower boundary of each class interval aligns with the left edge of the column. This is covered at the end of this section. Distribution of running time for a 10 km race 72 runners

20 18 16 14 12 10 8 6 4 2 0

Frequency

Number of pets owned by the students of Mr. Rodriguez’s class

Number of pets Ungrouped data

35 30 25 20 15 10 5 0

45 50 55 60 65 70

Running time (minutes) Grouped data

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Exploration Histograms and column graphs look very similar. The ﬁrst graph is a histogram, the second is a column graph. Distribution of vehicle speeds across highway traffic

14 Frequency

12 10 8 6 4 2 0

100

110

120

130 140 Speed (km/h)

150

160

170

Waste generated in Australia by industry 2016/17 financial year

Amount of waste (millions of tonnes)

25 20 15 10 5 0

Households Construction Commercial Coal powered electricity and and and local generation industrial demolition councils

1. Compare the data type used for each graph. 2. What can be observed about the axes for the histogram and column graph? 3. Is it possible to change the order of the columns in either of the graphs?

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The key features of a histogram are: • The horizontal axis is a numerical scale where class intervals are adjacent, with the boundaries of each interval marking the edges of the columns. • The vertical axis is the frequency of each data value or group of values. • There are no gaps between the columns because the horizontal axis is a continuous scale. It is possible for a class interval to have a frequency of zero, but this is not the same as having gaps between each column. • When creating a histogram, leave a half-column-width gap between the vertical axis and the ﬁrst column.

Example 1 A government agency records how long people wait on hold to speak to their representatives. The results are displayed in this histogram:

Time on hold 12

Frequency

10 8 6 4 2 0

Length of hold (in minutes) a Complete the corresponding frequency table: Length of hold (minutes)

Frequency

1 2 3 4 5

Create a strategy List the corresponding frequency of each length of hold (minutes).

Apply the idea Length of hold (minutes)

Frequency

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b How many phone calls were made?

Create a strategy Add all the frequencies from part (a).

Apply the idea Number of calls = 11 + 12 + 11 + 2 + 4 = 40

Write the equation Evaluate

c How long in total did these people wait on the hold?

Create a strategy Multiply each hold time by its frequency, and add the results together.

Apply the idea Total time = (1 × 11) + (2 × 12) + (3 × 11) + (4 × 2) + (5 × 4)

Multiply each time by the frequency

= 11 + 24 + 33 + 8 + 20

Evaluate the multiplication

= 96 minutes

Evaluate the addition

d What was the mean wait time? Give your answer as a decimal.

Create a strategy To determine the mean wait time, divide the total hold time by the total calls.

Apply the idea Write the equation Evaluate

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Grouped frequency tables Grouped frequency tables can be helpful when collecting continuous numerical data or when the data is more spread out. It shows the number of values within a class interval or group. The frequency of a group is found by adding the frequencies of all results contained within it. It is also required that each class interval be the same size. This grouped frequency table shows the time (minutes) taken, x, for 126 people to complete a 5 kilometre walking track: Class interval

Frequency

45 ≤ x < 50

50 ≤ x < 55

55 ≤ x < 60

60 ≤ x < 65

65 ≤ x < 70

For continuous data, a class interval is used to capture the full range of possible scores.

The ﬁrst class interval includes the walking times for 9 different people. Each of their times fall within 45 to 50 minutes, including 45 but not including 50. The class centre is the midpoint or mean of each class interval. The class centre for the ﬁrst class interval would be: Class interval

Class centre

Frequency

45 ≤ x < 50

47.5

50 ≤ x < 55

52.5

55 ≤ x < 60

57.5

60 ≤ x < 65

62.5

65 ≤ x < 70

67.5

To ﬁnd the measures of centre and spread in the grouped frequency table: • The modal class in a grouped frequency table is the group that has the greatest frequency. If there are multiple groups that share the greatest frequency, then there will be more than one modal class. The modal class is 65 to 70. • The range, in grouped frequency tables, can be found by calculating the difference between the highest and lowest possible values. The range is 70 − 45 = 25. • The median is the middle value of ordered data. For a sample of size n = 126, the position of the median is calculated using the formula

= 63.5. Thus, the median lies between

the 63rd and 64th positions when the data are arranged in ascending order. The median is found in the 60–65 class interval, and its value is approximated by the class centre of this interval, which is 62.5. • Class centres are also used to calculate the mean. The mean is found by ﬁrst multiplying the class centre of each group of scores by its corresponding frequency, then adding our results, and dividing it by the total frequency. The mean is approximately 62.46, rounded to two decimal places.

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The disadvantage of a grouped frequency table is that the data becomes less precise, since multiple data points are observed in groups rather than as individual points.

Example 2 Consider the following table: Score

Frequency

1–4

5–8

9–12

13–16

17–20

Note: This is discrete data so we display the intervals as discrete values, 1–4, then 5–8. It implies that there are no scores possible between 4 and 5.

a Use the class centre (midpoint) of each class interval to determine an estimate for the mean of the sample distribution. Round your answer to one decimal place.

Create a strategy To ﬁnd the midpoint of each class, ﬁnd the average of the two end points.

Apply the idea For the ﬁrst class, get the midpoint (or class centre) to be each class, get the following midpoints:

= 2.5. Using the same process for

Score

Class centre

Frequency

Class centre × Frequency

1–4

2.5

5–8

6.5

45.5

9 – 12

10.5

157.5

13 – 16

14.5

72.5

17 – 20

18.5

299

Total

To ﬁnd the mean, multiply each class centre by the frequency, add them together, and divide by the number of scores. The number of scores can be found by adding the frequencies. Write the formula

Substitute the values

Evaluate and round

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b Find the modal class of the data.

Create a strategy Choose the class that has the highest frequency.

Apply the idea The table shows that the modal class is the group 9 – 12, since it has the highest frequency.

Example 3 The scores of a basketball team are collected over a season: 65, 41, 53, 76, 37, 48, 68, 34, 59, 72, 45, 50, 32, 78, 62, 69, 56, 76, 75, 47 a Determine a set of ﬁve class intervals that could be used to separate this data.

Create a strategy Identify the maximum and minimum values in the dataset and ﬁnd appropriate class intervals. Each class interval must be the same size.

Apply the idea Divide the range by the desired number of classes:

To ensure that all data values are included and easy to work with, round this up to 10. Therefore, a class interval of 10 is appropriate. Five class intervals could be: 30–39, 40–49, 50–59, 60–69, 70–79.

b Construct a grouped frequency table for the data.

Create a strategy Sort the data, and determine which class intervals they belong to. Then, count the number of values in each class interval. Note: This is discrete data, it is not possible to achieve a score of 39.5, so the classes are set as 30–39, 40–49, etc.

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A histogram is a visual representation of a frequency table. It displays the frequency distribution of continuous data. In a histogram, the class frequency is the height of the rectangle. This frequency distribution table and histogram shows the time taken, for 73 runners, to complete a 10 kilometre race: Class interval

Class centre

Frequency

45 ≤ x < 50

47.5

50 ≤ x < 55

52.5

55 ≤ x < 60

57.5

60 ≤ x < 65

62.5

65 ≤ x < 70

67.5

Note: For continuous data, inequality notation is used to capture the full range of scores. Distribution of running times for a 10 km race 73 runners

35 30

Frequency

The median, which is the middle score, is the 37th in the list of 73 runners. Adding the frequencies for the first three intervals (9 + 7 + 20 = 36) shows that the 37th score falls within the 60-65 minute interval. Since the data is grouped into intervals, the exact median value cannot be determined directly, this is called the median class. The mode is the most common class interval (tallest column). The modal class is 60−65.

20 15 10 5 0 45

Running time (minutes) 70 − 45 = 25 Range

The mean can be calculated in a way similar to the method for a frequency table. First multiply the class centre, of each group of scores, by its corresponding frequency. Then sum those up and divide by the total frequency. The mean is approximately 58.80. The range is the difference between the highest and lowest score.

The class centre (midpoint) is not typically used in the histogram itself but can be useful for statistical calculations, like estimating means.

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Example 4 Some people were asked how many of their high school friends they remained in contact with after high school. The results were grouped into categories, presented in the frequency distribution table: Score

Frequency

0–9

10–19

20–29

30–39

40–49

Note: The score is a count of people, which is discrete. It is not possible to have a score of 9.5 people. a Construct a histogram for the data.

Create a strategy Examine and use the frequency table to determine how many times each range of scores occurred.

Apply the idea

There are no gaps between the columns.

Frequency

The vertical axis as the frequency of each data value or group of values.

10 9 8 7 6 5 4 3 2 1 0

Half-column-width gap between the vertical axis and the first column.

10 20 30 40 50

No. of friends

The horizontal axis is a continuous numerical scale, where intervals are placed on either side of the columns.

b Find the modal class of the data.

Create a strategy Examine the graph or use the frequency table to assess which range of scores occurs the most (has the highest frequency).

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Apply the idea Based on the given frequency table and the graph from part (a), the modal class is 10–19 as this is the range of scores that occurs the most.

Example 5 The heights of students in a class were recorded. The frequency table shows the ﬁndings. Height (in cm)

Frequency

140.9 < h ≤ 150.9

150.9 < h ≤ 160.9

160.9 < h ≤ 170.9

170.9 < h ≤ 180.9

180.9 < h ≤ 190.9

Note: Heights are continuous data, so we write the classes using inequalities to capture the full range of scores.

a What could be the height of the tallest student in the class?

Create a strategy Identify the highest class interval value in the last row.

Apply the idea From the last row, 180.9 < h ≤ 190.9, the highest class interval value is 190.9

b How many students in the class are taller than 160.9 cm?

Create a strategy Identify the total frequency in the corresponding class intervals.

Apply the idea The frequencies from the following class intervals must be added: 160.9 < h ≤ 170.9, 170.9 < h ≤ 180.9, 180.9 < h ≤ 190.9 10 + 2 + 3 = 15 So, there are 15 students who are taller than 160.9 cm.

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3.05 Practice questions What do you remember? 1

For each survey, state whether the data found should be grouped or ungrouped when constructing a frequency table: a

A survey conducted of 1000 people, asking them how many languages they speak.

A survey conducted of 1000 people, asking them how many different countries they know the names of.

A survey conducted of a class of 30 students, asking them how many pets they have.

A survey conducted of a class of 30 students, asking them how many minutes they spend on their phone each week.

Identify if the statements are true or false: a

The modal class is the one with the least frequency.

The class centre is found by calculating the range of the class interval values.

The median in a grouped frequency table is found by identifying the middle position and ﬁnding the corresponding class centre.

The mean in a grouped frequency table is found by ﬁrst multiplying the class centre of each group of scores by its corresponding frequency, then adding these results, and dividing it by the total frequency.

Deﬁne the class centre in a grouped frequency table, and provide the formula to calculate it for a class interval.

Practice 4

The following histogram shows the scores out of 30 in a maths exam for a cohort of Year 11 students:

Frequency

Ex 1

22 20 18 16 14 12 10 8 6 4 2

Score

Frequency

23 24 25 26 27 28

Score

190

Complete the following frequency table:

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How many students are there in the cohort?

Complete the frequency table based on the dataset: 36, 50, 66, 46, 78, 63, 58, 39, 84, 40, 81, 45, 86, 51, 68, 43, 64, 67, 37, 68, 48, 65, 60, 82, 75

Class

Frequency

30–39 40–49 50–59 60–69 70–79 80–89

The masses (in grams) of 30 laboratory rats are shown: 401, 402, 402, 403, 403, 403, 404, 404, 404, 405, 405, 405, 405, 406, 406, 406, 406, 407, 407, 407, 407, 408, 408, 408, 408, 409, 409, 409, 410, 410.

Ex 2

Ex 3

Create a frequency table for this data, showing the count of each unique value.

Construct a histogram to represent the frequency of each unique value.

Based on the frequency table of scores: a

Estimate the mean using class centres, rounded to one decimal place.

Determine the modal class of the data.

Score

Frequency

1–9

10 – 18

19 – 27

28 – 36

37 – 45

Score

Frequency

1–5

6–10

11–15

16–20

21–25

26–30

For the frequency table: a

Estimate the mean by calculating the class centre for each row. Round your answer to one decimal place.

State the modal group of scores.

Construct a grouped frequency table for the data. a

46, 54, 35, 23, 24, 28, 26, 11, 19, 17, 32

83, 68, 39, 42, 86, 66, 64, 76, 63, 43, 65, 83, 63, 67, 49, 51, 32, 55, 38, 65, 41, 73, 35, 36, 74

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Ex 5

A survey was conducted on the number of daily coffee purchases.

Number of coffee purchases

Frequency

The results were grouped into categories, presented in the frequency distribution table:

0–1

Note: The number of coffees is a count, which is discrete. It is not possible to purchase 1.5 coffees.

2–3

4–5

Create a histogram to represent the data.

6–7

Identify the modal class of the data.

As part of a fuel watch initiative, the price of petrol, p, at a service station was recorded each day for 21 days. The frequency table shows the ﬁndings: a

Determine the highest price that could have been recorded.

How many days was the price above 130.9 cents?

Create a histogram for the petrol prices over the 21 days.

If an additional column is added to the right side of the histogram, what would the new class interval be?

Price (in cents per litre)

Frequency

120.9 < p ≤ 125.9

125.9 < p ≤ 130.9

130.9 < p ≤ 135.9

135.9 < p ≤ 140.9

The histogram shows the number of hours that students in a particular class had slept for the night before: 12 11 10 9

Frequency

Ex 4

8 7 6 5 4 3 2 1 0

Time (hours)

192

How many students are in the class?

How many students had at least 8 hours of sleep that night?

What percentage of students had less than 6 hours of sleep?

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Extend your thinking

How many leaves less than 60 mm were collected?

What was the most common interval of leaf length collected?

Are leaves more likely to be at least 60 mm in length?

Can we conclude from the table that there were no leaves collected with a length less than 5 mm? Explain your answer.

Leaf length

Frequency

0 ≤ x < 20

20 ≤ x < 40

40 ≤ x < 60

60 ≤ x < 80

80 ≤ x < 100

Sylvia and Lee create two different histograms for the same dataset. The data collected shows the years of life expectancy after patients were diagnosed with lung cancer: Lee’s histogram

Sylvia’s histogram

60 50

Frequency

The frequency table shows the data distribution for the length of leaves (in millimetres) collected from a species of tree in the botanical gardens:

Frequency

15 10 5 0

40 30 20 10

5 10 15 20 25 30

Years

Explain which histogram is more useful to analyse the dataset.

Lee claims that their histogram shows that 51 patients lived 30 years after being diagnosed with lung cancer. Is this correct? Explain your answer.

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The frequency table shows the average time spent travelling to work for 50 people: Commute time (minutes)

Frequency

0 ≤ Time < 20

20 ≤ Time < 40

40 ≤ Time < 60

60 ≤ Time < 80

80 ≤ Time < 100

Total

Construct a histogram to display the data shown in the frequency table.

State whether the statements about the data are accurate. i

The data shows that most people travel to work by car or by walking, since most travel times are fairly short, and only a few people travel by bus or train.

The data suggests that people prefer a shorter commute to work. A majority live within 40 minutes travel, and in general the longer the commute the less people there are in that category.

iii

The data suggests that people don’t care too much about how far away from work they live. Roughly equal portions of people live less than 40 minutes away and more than 40 minutes away.

The data shows that everyone lives within an hour travel from their work, with the peak amount of people living between 20 and 40 minutes away.

A battery manufacturer tests their batteries in a portable music player, to ﬁnd out how long the batteries last under constant use. The histogram shows the data from their tests: 35 30

Frequency

25 20 15 10 5 0

400

500

600

700

800

900

1000 1100 1200 1300

Battery life (mins)

194

Create a frequency table for this set of data.

What percentage of batteries last longer than 10 hours? Round your answer to one decimal place.

If the average length of a song is 4 minutes, what percentage of batteries will last for an average of 250 songs or more? Round your answer to one decimal place.

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3.06 Cumulative frequency tables and graphs After this lesson, you will be able to… • define cumulative frequency and understand its calculation as a running total. • construct a cumulative frequency column for a given frequency distribution table. • construct and interpret cumulative frequency histograms and polygons (ogives). • use cumulative frequency graphs to estimate values such as the median.

Cumulative frequency tables Cumulative frequency The accumulating total of frequencies within an ordered dataset. To calculate cumulative frequency, an additional column is added to the frequency table. The frequencies of data values can then be used to create a histogram. The cumulative frequencies can also be plotted to create another type of chart, called a cumulative frequency graph. This graph can be used for ﬁnding values such as the median and interquartile range from a set of grouped data more easily. Cumulative frequency is a “running total” of the frequencies. To calculate it, we add an additional column to the frequency distribution table: Class interval

Frequency

Cumulative frequency

50 ≤ t < 55

55 ≤ t < 60

5 + 10 = 15

60 ≤ t < 65

15 + 25 = 40

65 ≤ t < 70

40 + 26 = 66

70 ≤ t < 75

66 + 40 = 106

75 ≤ t < 80

106 + 49 = 155

80 ≤ t < 85

155 + 28 = 183

Total

183

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• The first value in the cumulative frequency column will always be the same as the first value in the frequency column. • To get the second cumulative frequency value we add the second frequency to the first cumulative frequency value, 5 + 10 = 15. • The second cumulative frequency value tells us there are 15 values in the interval 50 ≤ t < 60. • The third cumulative frequency value tells us there are 40 values in the interval 50 ≤ t < 65, and so on. • The final cumulative frequency value is always equal to the sum of the frequencies. In this case, there are 183 values in the entire dataset, represented by 50 ≤ t < 85.

Example 1 A principal wants to investigate the performance of students at his school in Performing Arts. To do this, he has the marks of each student studying Performing Arts collected into groups and put into a frequency table. Each group of marks is assigned a grade: Grade

Score (x)

Frequency ( f )

0 ≤ x < 20

20 ≤ x < 40

40 ≤ x < 60

60 ≤ x < 80

80 ≤ x < 100

Cumulative frequency (cf )

a Complete the cumulative frequency column.

Create a strategy On any row, the cumulative frequency is the total sum of the frequency on that row and all frequencies in the previous rows. The cumulative frequency in the ﬁrst row is the frequency of the ﬁrst group of data. For the following rows, add the frequency for that row to the cumulative frequency of the previous row.

Apply the idea

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Grade

Score (x)

Frequency ( f )

Cumulative frequency (cf )

0 ≤ x < 20

20 ≤ x < 40

40 ≤ x < 60

60 ≤ x < 80

150

80 ≤ x < 100

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Cumulative frequency graphs Cumulative frequency histogram A visual representation of data using bars to represent the class boundaries and the cumulative frequencies. The area of each bar is proportional to the cumulative frequency of the observations up to the end of that class. Cumulative frequency polygon A series of straight lines representing the cumulative frequency for a given dataset. Sometimes called the ‘ogive’.

Using the values in the cumulative frequency column, we can create a cumulative frequency histogram. Class interval

Frequency

Cumulative frequency

50 ≤ t < 55

55 ≤ t < 60

60 ≤ t < 65

65 ≤ t < 70

70 ≤ t < 75

106

75 ≤ t < 80

155

80 ≤ t < 85

183

Total

183

Cumulative frequency

Notice that the columns in a cumulative frequency histogram will always increase in size from left to right. The frequency represented by any particular column will be equal to the difference in height between that column and the one before it.

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200 180 160 140 120 100 80 60 40 20 0

Cumulative frequency distribution Global life expectancy, 2016

Life expectancy (years)

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Cumulative frequency

In this histogram, colours show the “stacking” effect, where each column’s height sums all previous frequencies plus its own. The height difference between consecutive columns represents that range’s frequency.

200 180 160 140 120 100 80 60 40 20 0

Cumulative frequency distribution Global life expectancy, 2016

Class interval (t)

Cumulative frequency

A cumulative frequency polygon, also known as an ogive, is a line graph connecting cumulative frequencies at the upper endpoint of each class interval. Sometimes the cumulative frequency histogram and polygon are displayed together:

200 180 160 140 120 100 80 60 40 20 0

Cumulative frequency distribution Global life expectancy, 2016

Life expectancy (years)

Cumulative frequency

The cumulative frequency polygon can also be displayed on its own.

200 180 160 140 120 100 80 60 40 20 0

Cumulative frequency distribution Global life expectancy, 2016

Life expectancy (years)

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Example 2 For the given frequency table: Score (x)

Frequency ( f )

1−5

6 − 10

11 − 15

16 − 20

21 − 25

26 − 30

Cumulative frequency (cf )

a Calculate the cumulative frequency to complete the table.

Create a strategy Add the frequency of the current interval to the cumulative frequency of the previous interval, starting with the ﬁrst frequency.

Apply the idea Score (x)

Frequency ( f )

Cumulative frequency (cf )

1−5

6 − 10

15 + 26 = 41

11 − 15

41 + 18 = 59

16 − 20

59 + 14 = 73

21 − 25

73 + 7 = 80

26 − 30

80 + 2 = 82

b Draw the cumulative frequency histogram, including the polygon.

Create a strategy Plot the histogram with the x-axis showing the class intervals (1 to 30) and the y-axis showing the cumulative frequencies, ensuring bars are adjacent with heights matching the cumulative frequencies. Draw the polygon by plotting points at the upper endpoint of each interval (e.g., 5, 10, … , 30) with their corresponding cumulative frequencies, starting from (1, 0), and connect them with straight lines.

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Apply the idea 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

5 10 15 20 25 30

c Calculate the total frequency.

Create a strategy Identify the last value in the cumulative frequency column, as it represents the sum of all frequencies in the table.

Apply the idea From the last value in the cumulative frequency column, the total frequency is 82.

d Approximately half of the scores recorded are greater than what score? Score (x)

Frequency ( f )

Cumulative frequency (cf )

1−5

6 − 10

11 − 15

16 − 20

21 − 25

26 − 30

Create a strategy To find the median score, first calculate 50% of the total frequency to determine the position of the median. Then, identify the class interval where the cumulative frequency first exceeds this value. The upper endpoint of that interval is the median score.

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Practice Ex 1

Ex 2

A survey collected data on the number of weekly public transport trips. The results were grouped into categories with assigned usage levels, as shown in this frequency table: Usage Level

Trips (x)

Frequency ( f )

Minimal

0≤x<5

Low

5 ≤ x < 10

Moderate

10 ≤ x < 15

Frequent

15 ≤ x < 20

Heavy

20 ≤ x < 25

Complete the cumulative frequency column.

Calculate the total frequency.

Identify the class size.

Complete the sentence:

Cumulative Frequency (cf )

Approximately three-quarters of the adults made more than ⬚ trips.

For the given frequency table: a

Calculate the cumulative frequency to complete the table.

Score (x)

Frequency (f )

Draw the cumulative frequency histogram, including the polygon.

20 − 24

Calculate the total frequency.

25 − 29

Approximately one third of the scores recorded are greater than what score?

30 − 34

35 − 39

40 − 44

45 − 49

50 − 54

Consider this table: a

Calculate the total number of scores recorded.

Determine the number of times a score of 14 occurred.

Determine the number of times a score less than 13 occurred.

Cumulative frequency (cf )

Score (x)

Cumulative frequency (cf )

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204

Score (x)

Frequency (f )

Cumulative frequency (cf )

⬚

Construct the cumulative frequency table for the data.

Draw the cumulative frequency histogram, including the polygon.

Calculate the total frequency. Determine the score above which approximately half of the scores lie.

⬚

Score (x)

Frequency (f )

Cumulative frequency (cf )

1−4

5−8

9 − 12

13 − 16

17 − 20

For the given frequency table: a

Complete the cumulative frequency column.

Calculate the total frequency.

Calculate the class size.

Construct a frequency table for the data represented in the given cumulative frequency histogram:

Cumulative frequency

A teacher records student scores on a quiz, grouped into this frequency table:

Consider this cumulative frequency histogram: a

Calculate the total number of scores recorded.

Determine the number of times a score of 46 occurred.

Determine the number of times a score of 45 occurred.

Determine the percentage of scores that were 43 or less. Round your answer to one decimal place.

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Cumulative frequency

20 18 16 14 12 10 8 6 4 2 0

⬚ ⬚ ⬚

Score

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Score

This graph shows the ogive of the masses of ﬁsh caught in a ﬁshing competition:

Cumulative frequency (375, 169)

170 160

(375, 169)

(275, 158)

Construct a frequency histogram to depict the 150 distribution of the masses of the ﬁsh.

(225, 146)

140 130

(175, 122)

120 110 100

(125, 89)

90 80 70 60

(75, 52)

50 40 30 20 10

Mass (g) 0

50 100 150 200 250 300 350 400 450

Extend your thinking 13

A pair of dice are rolled 50 times and the numbers appearing on the uppermost face are added to give a score. The results are recorded in the given table:

Score (x)

Frequency (f )

Identify the lowest possible score when a single pair of dice are rolled.

Identify the highest possible score when a single pair of dice are rolled.

Complete the table by ﬁnding the cumulative frequency values.

Determine the number of times a score of 8 occurred.

Determine the number of times a score more than 9 occurred.

Determine the number of times a score of at most 6 occurred.

Cumulative frequency (cf )

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The number of sightings of the Northern Lights were recorded across various Canadian locations over a period of 1 month. This list represents the number of sightings at each location:

Number of sightings

11, 10, 10, 9, 7, 8, 8, 12, 12, 12, 12, 12, 12, 9, 9, 12, 9, 9, 8, 8 a

Complete the table.

In how many locations were there at least 8 sightings?

8 9 10 11 12

In how many locations were there less than 11 sightings?

A 1500 m swimmer records her time over several training sessions. Her times are recorded in this histogram: a

Construct a cumulative frequency table for the data using the given intervals.

Calculate the total number of training sessions she completed.

Determine the number of times she recorded a swim time faster than 16 : 40.

Calculate the percentage of swims that were less than 16 : 30.

10 9 8 7 6 5 4 3 2 1 0 :0 0 16 :10 16 :2 0 16 :3 0 16 :4 0 16 :5 0 17 :0 0

Cumulative frequency (cf )

Number of locations (f )

Frequency

Time recorded 16

The heights of 22 boys in a class are listed: 164, 167, 158, 159, 150, 166, 150, 146, 149, 161, 164, 163, 152, 161, 157, 157, 153, 157, 156, 165, 162, 161

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Construct a cumulative frequency histogram for the data. Use the discrete intervals of 146 − 150, 151 − 155, etc.

How many students are taller than 155 cm?

How many students are at most 155 cm tall?

How many students are taller than 150 cm but shorter than 156 cm?

Calculate the centre of the class 151 − 155.

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3.07 Shape of distribution After this lesson, you will be able to… • describe a dataset’s distribution as symmetrical, positively skewed, or negatively skewed. • identify the key features of symmetrical distributions, including the approximate equality of mean, median, and mode. • identify the key features of positively and negatively skewed distributions, including the direction of the tail. • understand the typical relationship between mean, median, and mode in skewed distributions. • explain why the median is often a better measure of central tendency for skewed data.

Shape of distribution Symmetrical distribution When the 2 sides of the distribution are a mirror image of each other. A normal distribution is a true symmetric distribution of observed values. Skewed A distribution is negatively skewed when its tail is on the left side. A distribution is positively skewed when its tail is on the right side.

A dataset’s distribution can be described as symmetrical, positively skewed, or negatively skewed, based on the shape of its frequency distribution. A bell-shaped curve, often called a normal distribution, represents a symmetrical distribution where data clusters around a central value.

Key features of a symmetrical distribution include: • Data clusters around a central value. • Approximately 50% of data lies above and below the mean. • The mean, median, and mode are approximately equal in a symmetrical distribution. A positively skewed distribution has a longer tail on the right (higher values). In a positively skewed distribution, the mean is typically greater than the median, which is greater than the mode, due to the inﬂuence of the right tail.

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A negatively skewed distribution has a longer tail on the left (lower values). In a negatively skewed distribution, the mean is typically less than the median, which is less than the mode, due to the inﬂuence of the left tail.

Exploration Consider the time taken to complete a task. Discuss with a partner: Would you expect the distribution of completion times to be symmetrical, positively skewed, or negatively skewed? Why? For example, think about tasks like running a race versus solving a complex puzzle.

A frequency histogram with an approximately symmetrical shape can have a bell-shaped curve overlaid to represent the data’s general pattern.

Percentage

30 25 20 15 10 5 75 80 85 90 95 100 105 110

To describe a distribution’s shape, the data need not be perfectly symmetrical; the focus is on the pattern it most closely resembles: symmetrical, positively skewed, negatively skewed, or, in some cases, no clear pattern is present.

Weight

Example 1 Determine whether the scores in each data display are best described as positively skewed, negatively skewed, or symmetrical.

Frequency

10 9 8 7 6 5 4 3 2 1 0

Score

Create a strategy

Apply the idea

Examine the shape of the distribution, focusing on the position of the bulk of the data and the length of the tails.

The high-frequency columns are centred, with similar frequencies on both sides, indicating a symmetrical distribution.

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Practice 5

Identify if the graph is positively skewed, negatively skewed, or symmetric.

Frequency

10 9 8 7 6 5 4 3 2 1 0

Class

Frequency

10 9 8 7 6 5 4 3 2 1 0

Class c

Frequency

Ex 1

10 9 8 7 6 5 4 3 2 1 0

Class

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Frequency

Score

14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Frequency

10 9 8 7 6 5 4 3 2 1 0

15 10 5 0

Score

Frequency

25 20 15 10 5 0

Score

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Extend your thinking 12

A histogram shows the time (in minutes) taken by 50 students to complete a quiz. Most students ﬁnished between 10–15 minutes, with fewer taking up to 25 minutes. Describe the shape of the distribution.

A company records the number of sales made by its salespeople over a month. The data is heavily skewed to the right. Describe what can be inferred about the sales performance of the team based on the heavily right-skewed data.

A teacher wants to compare the test scores of two different classes. One class has a symmetric distribution, while the other has a positively skewed distribution. Describe how the teacher might interpret and compare the performance of the two classes based on their distribution shapes.

A dataset records the wait times (in minutes) for 30 customers at a service desk: 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 15, 18, 20, 25, 30, 35, 40, 45, 50, 60, 75 Determine the shape of the distribution, recommend whether the mean or median should be used to report the typical wait time, and explain what the distribution suggests about the service process.

Did you know?

The size of apples in a harvest often forms a symmetrical distribution. Most apples are mediumsized, with only a few very small or very large ones. This predictable pattern is useful for farmers and retailers because it helps them sort apples efficiently into size categories for packaging and pricing. Supermarkets can also use this information to manage stock and meet customer demand for “standard” sizes. Understanding this distribution is a real-life application of statistics — by knowing the mean, median, and mode of apple sizes, farmers can make practical decisions, such as estimating yields or planning how many boxes they’ll need for shipping. 3.07 Shape of distribution mathspace.co

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3.08 Misleading graphs and appropriate displays After this lesson, you will be able to… • identify common features that make graphs misleading (e.g., manipulated scales, omitted data). • explain how misleading graphs can lead to incorrect interpretations of data. • select the most appropriate type of graph (e.g., pie chart, bar/column graph, dot plot, histogram, line graph) to represent a given dataset. • justify the choice of a particular graphical display based on the data type and the information to be conveyed. • critically evaluate graphical representations of data encountered in various contexts.

Misleading graphs Misleading graphs can distort data intentionally or due to poor design, leading to incorrect conclusions. Common misleading features include: • Manipulating the scale by not starting at zero, using a non-uniform scale or omitting the scale entirely. • Manipulating intervals that could exaggerate the distance between data points, making variations seem less or more important than they really are. • Omitting certain data points, such as outliers or values that do not align with the desired conclusion. • Omitting important information in titles and labels. • Using pictures or three-dimensional graphics that distort results. • Choosing a graphical display that does not best represent the data. Always analyse graphs in context to avoid misinterpretation. Graphs in media often compare data or show trends but can mislead if poorly designed or intentionally biased.

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Exploration

78 77 76 75 74 73 72 71 70 69 68

In favour of bill X Percent in favour

Percent in favour

Graph 1 and Graph 2 show support for a bill among political parties.

Liberal Labor Other Political party Graph 1

80 70 60 50 40 30 20 10 0

In favour of bill X

Liberal Labor Other Political party Graph 2

1. Do both graphs display the same data? Do they convey the same message? Explain. 2. Identify differences between the graphs and explain their signiﬁcance. 3. How might each graph inﬂuence media viewers’ perceptions?

Example 1 Shawnte played 17 soccer games this season. The dot plot shows goals scored per game:

Shawnte’s soccer season

Goals scored a Identify the misleading feature in this dot plot.

Create a strategy Check for common misleading features, such as omitted data or manipulated scales.

Apply the idea The dot plot shows 14 dots but the season had 17 games, indicating omitted data. The scale, intervals, and labels are appropriate, and the dot plot suits the data type. Thus, the misleading feature is omitted data.

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b Explain why this is misleading.

Create a strategy Consider the impact of omitted data on interpretation.

Apply the idea Missing 3 games’ data (possibly zero goals) creates an incomplete view of Shawnte’s performance, potentially overestimating their scoring consistency.

Example 2 Consider this dataset of monthly sales (in thousands): January (50), February (52), March (55), April (60). a Create a misleading column graph.

Create a strategy Use a common misleading feature, such as a non-zero y-axis scale.

Apply the idea

Monthly sales (in thousands)

A column graph with a y-axis starting at 40 instead of 0, exaggerates differences between months, making sales growth appear more dramatic.

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60 58 56 54 52 50 48 46 44 42 40

Jan

Feb

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Mar

Apr

Appropriate graph displays Exploration This table shows the number of student birthdays by season: Seasons

Number of students

Winter

Spring

Summer

Fall

Two displays represent this data: Number of students Number of students

Number of students vs. Season

2 8

Winter

Spring

Summer

Fall

8 6 4 2 0

Winter

Spring Summer Season

Fall

Which season had the most birthdays? Which display did you use?

Which season had the fewest birthdays? Which display did you use?

How do Winter and Fall compare? Which display did you use?

Choose a graph type based on the data and the feature you want to emphasise. Below are common graph types, their suitable data, and pros/cons: Pie Graph: Shows proportions of a whole using circle sectors. Data type

10.8% 24.3%

13.5%

Advantages

10.8% 40.5%

Disadvantages

Categorical or numerical data  Visually compares proportions  Shows contribution to the whole  Hard to compare without labelled percentages  Cluttered with many categories

Cheese

Pepperoni

Vegetarian

Shawarma

Hawaiian

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Column graph: Uses vertical bars to show frequencies. 15

Cheese

Pepperoni Vegetarian Shawarma

Hawaiian

Data type

Categorical or discrete numerical data

Advantages

Identifies extremes and handles large datasets

Disadvantages

 Cluttered with many categories  Misleading if y-axis does not start at 0

Histogram: Uses adjacent bars to show frequency distributions. Age of first car purchase 6

Frequency

5 4 3 2 1 0

10-14

15-19

20-24 25-29 30-35 40-44

Age Data type

Numerical data

Advantages

Shows distribution and handles large datasets

Disadvantages

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 Misleading if scale is manipulated  Loses raw data in intervals

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b Suggest a better graph type.

Create a strategy Choose a graph that handles multiple selections and percentages.

Apply the idea A column graph shows each topping’s percentage without implying a single whole, suitable for multiple-choice data. Students’ preferred pizza topping 75% 70% 65% 60% 55% 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0%

i s s s n h n e m le rs ron Olive nion ppe usag Ha app room inac Baco icke h p e O Pe Sa h S C Pin Mus

e pp

Example 5 Choose a graph to display this dataset of weekly study hours: 5, 7, 10, 12, 15, 20. Justify your choice. a Select and justify a graph type.

Create a strategy

Apply the idea

Match the data type and purpose to a graph’s strengths.

A histogram is ideal for numerical data to show frequency distribution across intervals, revealing patterns like clusters or gaps.

b Compare the histogram to a dot plot for this data. Why is the histogram more suitable?

Create a strategy

Apply the idea

Evaluate the advantages and disadvantages of both graph types for numerical data.

A dot plot shows individual data points but is less effective for visualising distribution in larger datasets. The histogram groups data into intervals, making it easier to identify frequency patterns and trends, which suits the study hours’ range and purpose.

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Practice Ex 1

Analyse this line graph:

Sales

Identify one feature that makes this graph misleading.

Explain how this feature leads to misinterpretation of the data.

Month 0

Select the most appropriate graph for each scenario and justify your choice: • Dot plot • Pie graph • Line graph • Histogram

A survey of 100 people to ﬁnd the most common eye colour.

An athlete tracking heartbeat patterns during a workout.

A teacher surveying 25 students on their favourite subject.

A survey of 10 000 people on their preferred soap brand.

An 8th grader surveying 30 classmates on favourite juice.

A study measuring foot sizes of 200 people.

A trainer analysing daily step counts of clients over weeks.

A business owner surveys 180 employees to choose a retreat city from ﬁve options. Are these displays appropriate for the data? Justify your answers: a

Ex 2

Dot plot

Bar graph

Picture graph

Pie graph

Consider this dataset of the number of days with freezing temperatures: 2017 (45) and 2018 (48). a

Create a misleading column graph.

Suggest a correction and create the correct graph.

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Kenau’s histogram shows daily step counts over a month: 2000, 4000, 8000, 8000, 8000, 8500, 8500, 9000, 9500, 9500, 10 000, 10 500, 11 000, 11 500, 11 500, 12 000, 12 000, 12 500, 13 000, 13 500, 14 000, 14 000, 14 000, 15 000, 15 000, 16 000, 17 000, 17 000, 18 000, 19 000

Frequency

Steps in a month 10 9 8 7 6 5 4 3 2 1 0

8000

10000 12000 14000 16000 18000 20000

Number of steps Explain why this histogram may lead to misinterpretation of the data. 11

This pie graph shows viewership for sports watched on television. Most popular sports to watch on TV

180

120

Football

Basketball 35

Boxing 15

Hockey 30

Racing 15

Tennis 20

Golf 10

Volleyball 20

Cricket 5

Basketball

Soccer Explain why this pie graph might lead to misinterpretation based on the viewer counts. 12

This histogram shows occupancy rates (%) for 28 hotel customers last week.

Frequency

Hotel occupancy 12 10 8 6 4 2 0

100

Occupancy rate(%) Identify errors in the horizontal scale that could lead to misinterpretation. 228

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Participants recalled 30 names after one minute. The results are shown in this dot plot:

Names remembered Identify errors in the horizontal scale that could lead to misinterpretation. 14

Two line graphs show the same data but with different horizontal axis spacing. Discuss how this affects data interpretation. Line graph 1: 9000 8000 7000 6000 5000 4000 3000 2000 1000

Sales ($)

Month

Line graph 2: 9000

Sales ($)

8000 7000 6000 5000 4000 3000 2000 1000 0

Month 1

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Two line graphs show the same data but with different vertical axis scales. Discuss how this affects interpretation.

50 45 40 35 30 25 20 15 10 5 0

Global average temperature

1880 1884 1888 1892 1896 1900 1904 1908 1912 1916 1920 1924 1928 1932 1936 1940 1944 1948 1952 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012 2016

Temperature (°C)

Line graph 1:

Year Line graph 2: Global average temperature

14.5 13 13.5 13 1880 1885 1890 1895 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

Temperature (°C)

Year

A survey recorded the number of students selecting their favourite extracurricular activities. The data is shown in a histogram and a line graph: Favourite extracurricular activities 50

Number of students

Ex 3

40 30 20 10 0

Debate

Sports

Music Activities

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Art

Coding

Favourite extracurricular activities Number of students

50 40 30 20 10 0

Debate

Sports

Music

Art

Coding

Activities

Ex 4

Which graph better shows activities with around 20 students?

Which graph better shows the most popular activity?

Analyse this pie graph of election candidate support:

Presidential run 2012

Why is this pie graph inappropriate?

Create a bar graph for the data and explain why it might be preferred for comparing candidate support.

70% 63% 60%

Supports Palin Supports Huckabee Supports Romney 18

A website claims, “Customers are happiest when using our product daily,” using this graph. Explain why this display may lead to misinterpretation.

Customer usage and satisfaction Satisfaction rating 5 4 3 2 1

Number of time used per week 1

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Ex 5

The frequency distribution shows Deana’s monthly book reading data: a

Month

Number of books

Choose and create the best display for this data and justify your choice, considering how it supports trend analysis.

January

February

Explain why your chosen display is the most suitable.

March

April

May

June

July

August

September

October

November

December

Goals scored by Team 1 and Team 2 in a football tournament: Create the best display to compare this data and justify your choice.

Match

Team 1

Team 2

Extend your thinking 21

An Argentinian news service claimed, “Argentina’s Covid-19 testing rate is almost as high as the United States,” using this graph: Testeos por million de habitantes

7.000 258

14.100

15.700

22.300

330

Brasil Argentina USA Italia Alemania Noruega Ante la presencia de sintomas Ilmar al 107 San Luis Assess the claim’s accuracy, explaining how the graph’s features lead to misinterpretation.

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An affordable housing initiative requires 10% of apartments to cost $250 000 or less. Apartment prices are: $250 000, $250 000, $250 000, $600 000, $605 000, $620 000, $630 000, $640 500, $645 000, $650 000, $660 000, $662 500, $668 000, $675 000, $680 000, $695 000, $700 000, $710 000, $715 000, $725 000, $740 000, $744 000, $750 000, $755 000, $759 000, $760 500, $765 000, $770 000, $772 000, $775 000 The marketer claims, “Most apartments are under $630 000,” using this box plot: Apartment prices (in thousands)

250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775

Assess the claim’s accuracy, considering the box plot’s limitations. Felicia polled 6th graders on their class captain vote. Results are shown in two displays: Class representative poll

Class representative poll

15 10

Votes

5 0

l a a n tin nie him an so All Bron stan Da Ebra n Co Candidate

Alanna

Bronson

Constantina

Daniel

Ebrahim a

If you were Constantina, which display would you use? Explain, considering interpretation limitations.

If you were Alanna, which display would you use? Explain, considering interpretation limitations.

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Using the provided monthly sales data for a small bakery, create three distinct data displays. Each graph should be a different type and should highlight unique aspects of the sales data. Describe the speciﬁc insight each visualisation provides about the bakery’s sales trends. Monthly sales data for a small bakery (in units sold) Month

Cakes

Bread

Pastries

January

130

150

110

February

120

140

105

March

140

160

115

April

125

100

155

120

May

150

110

170

130

June

135

120

165

140

July

145

130

175

135

August

160

140

180

150

September

155

135

190

145

October

170

150

195

155

November

165

145

185

160

December

175

155

200

170

Investigation: Appropriate choice of graph for data Investigate online

Investigation: Spreadsheets to tabulate and graph data Investigate online

mathspace.co

Investigation: Infographics Investigate online

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3 Chapter review 1

Categorical (Nominal)

Categorical (Ordinal)

Numerical (Discrete)

Numerical (Continuous)

The histogram shows the distribution of hours spent by students on a school project. What is the shape of this distribution?

Frequency

A survey asks participants: “How many cups of coffee do you typically drink per day?” What type of variable is being measured?

10 9 8 7 6 5 4 3 2 1 0

Class

Positively skewed

Negatively skewed

Symmetric

Uniform

The horizontal bar graph shows the number of books borrowed from a library in a week. Which category had the most books borrowed? Fiction Mystery Sci-fi Non-fiction 0

Books borrowed from a library in a week A

Fiction

Mystery

Sci-Fi

Non-Fiction

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Classify each variable as numerical or categorical: a

The distance from home to the nearest supermarket.

The brand of smartphone a person owns.

The number of songs on a playlist.

A person’s preferred genre of music.

The time taken to complete a jigsaw puzzle.

Jason is designing a survey about students’ exercise habits. a

How would he design a survey question yielding a categorical (ordinal) variable for exercise frequency?

How would he design a survey question yielding a numerical (continuous) variable for exercise duration?

A group of Year 11 students were asked about their preferred exam revision method. The results are shown in the two-way table:

Female

Practice Papers

Summarising Notes

How many female students preferred summarising notes?

How many male students preferred practice papers?

Did male students prefer practice papers or summarising notes more?

The table shows the number of visitors (in thousands) to a national park during four seasons: a

Construct a column graph to represent this data.

Which season had the most visitors?

If the park aims to increase Winter visitors by 20% next year, how many Winter visitors are they aiming for?

Season

Visitors (thousands)

Spring

Summer

Autumn

Winter

A family with a monthly income of $4500 has spending shown in the pie chart: a

Determine the monthly spending on each category: i

236

Male

Housing iv

ii Food Utilities

iii

Transport

If they reduce Other spending by $50 and add it to Savings, what is the new monthly savings?

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8% 12%

30%

10% 15%

25%

Housing

Food

Transport

Utilities

Savings

Other

The histogram shows the ages of players in a chess club:

Frequency

20 18 16 14 12 10 8 6 4 2 0

10-19

20-29

30-39

40-49

50-59

60-69

Age (years) a

Complete the frequency table based on the histogram: Age Group

Frequency

10 − 19

⬚

20 − 29

⬚

30 − 39

⬚

40 − 49

⬚

50 − 59

⬚

60 − 69 b 13

238

⬚

How many members are in the chess club?

The table shows the time (in minutes) taken by athletes to complete a 5 km run: Time (minutes)

Frequency

15 − 19

20 − 24

25 − 29

30 − 34

35 − 39

Estimate the mean time to complete the run, using class centres, rounded to one decimal place.

Determine the modal class for the running times.

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The frequency table shows hours students spent watching TV in a week: Hours (x)

Frequency ( f )

Cumulative Frequency (cf )

0−4

5−9

⬚

10 − 14

15 − 19

20 − 24

⬚

⬚ ⬚ ⬚

Complete the cumulative frequency column.

How many students were surveyed?

What is the class size for the Hours intervals?

What percentage of students watched TV for 9 hours or less per week?

The frequency table shows customer call durations (in minutes) at a call centre: Call Duration (mins)

Frequency ( f )

0–<5

5 – < 10

10 – < 15

15 – < 20

20 – < 25

Construct the cumulative frequency table.

Draw the cumulative frequency polygon (ogive), plotting points at the upper end of each class interval.

How many calls were recorded?

Estimate how many calls lasted less than 15 minutes.

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Big ideas • The metric system leverages base-10 relationships to enable accurate measurement and conversion of length, area, volume, capacity, and mass in real-world contexts. • Scientific notation and significant figures facilitate precise representation and communication of numerical data in scientific contexts.

4 Practicalities of measurement Chapter outline 4.01 4.02 4.03 4.04 4.05

Multiplication and division by powers of 10 Units of length and area Units of volume and capacity Units of mass Scientific notation and significant figures Chapter 4 review

244 251 258 266 271 280

4.01 Multiplication and division by powers of 10 After this lesson, you will be able to… • multiply or divide whole numbers by powers of 10. • multiply or divide decimals by powers of 10. • relate multiplication and division by powers of 10 to shifts in place value or movement of the decimal point.

Multiply and divide by powers of 10 When a number is multiplied or divided by a power of 10 (such as 10, 100, or 1000), the decimal place is shifted a number of times corresponding to that power of 10. Multiplying by a power of 10n would move the decimal place n places to the right, adding zeros as placeholders. Dividing by a power of 10n would move the decimal place n places to the left.

Multiply by 10n : move decimal point right by n places n

the power of 10

Divide by 10n : move decimal point left by n places n

the power of 10

327.0 0. ÷

Interactive exploration Discover this concept in action online

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Example 1 Fill in the boxes with the missing numbers. 8 × ⬚ = 80

8 × ⬚ = 800

8 × ⬚ = 8000

8 × ⬚ = 80 000

Apply the idea

• To get 80 from 8, move the decimal point, one position right so ⬚ = 10. 8 × ⬚ = 80

Write the equation

= 8 × 10

Write 80 as 8 × 10

Therefore, the missing number is 10. • To get 800 from 8, move the decimal point, two positions right so ⬚ = 100. 8 × ⬚ = 800

Write the equation

= 8 × 100

Write 800 as 8 × 100

Therefore, the missing number is 100. • To get 8000 from 8, move the decimal point, three positions right so ⬚ = 1000. 8 × ⬚ = 8000

Write the equation

= 8 × 1000

Write 8000 as 8 × 1000

Therefore, the missing number is 1000. • To get 80 000 from 8, move the decimal point, four positions right so ⬚ = 10 000. 8 × ⬚ = 80 000

Write the equation

= 8 × 10 000

Write 80 000 as 8 × 10 000

Therefore, the missing number is 10 000.

Reﬂect and check Looking at the same but now with regards to powers of 10, we will have: 8 × 10

80 = 8 × 101

8 × 100

800 = 8 × 102

8 × 1000

8000 = 8 × 103

8 × 10 000

80 000 = 8 × 104

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Example 2 Evaluate: a 5 × 10

Apply the idea Shift the decimal point, one position right, as multiplying by 10 corresponds to 101. 5 × 10 = 50

Evaluate

b 20 × 100

Apply the idea Shift the decimal point, two positions right, as multiplying by 100 corresponds to 102. 20 × 100 = 2000

Evaluate

c 36 × 1000

Apply the idea Shift the decimal point, three positions right, as multiplying by 1000 corresponds to 103. 36 × 1000 = 36 000

Evaluate

Example 3 Evaluate: a 180 ÷ 10

Apply the idea Shift the decimal point, one position left, as dividing by 10 corresponds to 101. 180 ÷ 10 = 18

Evaluate

b 250 ÷ 100

Apply the idea Shift the decimal point, two positions left, as dividing by 100 corresponds to 102. 250 ÷ 100 = 2.5

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Evaluate

4.01 Practice questions What do you remember? 1

Determine whether these statements are true or false: a

Multiplying by 10n shifts digits to the left by n places.

Dividing by 10n shifts digits to the right by n places.

State the step to convert each number as shown: a

720 = 7.2

3.45 = 3450

5800 = 58

0.62 = 62

For each number, ﬁnd the exponent n such that the number equals a value times 10n: a

600

8000

0.07

0.0009

Practice Ex 1

Determine the missing numbers in the equations, expressing your answers as powers of 10: a c e g

Ex 2

Ex 3

Ex 5

15 × ⬚ = 150

3 × ⬚ = 300

b d f h

Evaluate:

9 × ⬚ = 9000

4 × ⬚ = 40 000

15 × ⬚ = 15 000

6 × ⬚ = 60 000

6 × 10

8 × 100

12 × 1000

51 × 10 000

43 × 10

45 × 1000

10 × 100

95 × 10 000

400 ÷ 100

Evaluate: a

Ex 4

7 × ⬚ = 70

4 × ⬚ = 400

80 ÷ 10

6400 ÷ 1000

78 000 ÷ 10 000

270 ÷ 10

9200 ÷ 100

7000 ÷ 1000

50 000 ÷ 10 000

Evaluate: a

0.82 × 10

3.7 × 100

0.048 × 1000

1.25 × 10 000

0.006 × 100

9.4 × 1000

0.013 × 10 000

6.8 × 100

Evaluate: a

5.28 ÷ 10

17.94 ÷ 100

0.0037 ÷ 1000

0.96 ÷ 10

0.619 ÷ 10

2.405 ÷ 100

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g 9

c e

0.6 × ⬚ = 600

0.9 × ⬚ = 9000

1.6 × ⬚ = 160

0.045 × ⬚ = 450

28 × ⬚ = 280 000

0.73 × ⬚ = 730

90 ÷ ⬚ = 9

1200 ÷ ⬚ = 12

15 000 ÷ ⬚ = 15

81 ÷ ⬚ = 0.81

d f

360 000 ÷ ⬚ = 36

540 ÷ ⬚ = 0.054

2700 ÷ ⬚ = 2.7

63 000 ÷ ⬚ = 6.3

72 × 102

0.31 ÷ 101

0.007 × 103

980 ÷ 102

3400 ÷ 103

0.45 × 100 000

7200 ÷ 1 000 000

0.734 ÷ 1000

Fill in the blanks to make the equations true: a

0.0081 ÷ 1000

Evaluate:

0.056 × 10

Evaluate: a c

8.1 × 10 000 000

0.0093 × 100 000

15 000 ÷ 10 000 000

0.007 × 1 000 000

620 000 ÷ 100 000

0.12 × 10 000 000

Extend your thinking 13

A bacterium has a mass of 0.00000015 grams. Given there are 103 mg in 1 g, convert the mass of the bacterium to mg.

A model scales a cell’s diameter from 25 micrometres to 2500 micrometres. Determine the power of 10 by which the diameter has been scaled.

A number multiplied by 103 gives 4500. Find the number and then calculate the quotient after dividing it by 102.

A distance is measured as 0.00042 kilometres. Convert this to millimetres (1 km = 1 000 000 mm).

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4.02 Units of length and area After this lesson, you will be able to… • identify and convert between the metric units of length: millimetres (mm), centimetres (cm), metres (m) and kilometres (km) • identify and convert between metric units of area: square millimetres (mm2), square centimetres (cm2), square metres (m2), square kilometres (km2), and hectares (ha) • understand that area conversion factors are the square of the corresponding length conversion factors • apply conversion skills to solve practical problems involving length and area

Conversion of length units Metric units are used to measure length. Common length units are: • Millimetres (mm): e.g., a grain of sand is 1 mm. • Centimetres (cm): e.g., an ant is 1 cm. • Metres (m): e.g., a guitar is 1 m. • Kilometres (km): e.g., Sydney Harbour Bridge is 1 km. The conversion factors are summarised in this conversion chart: ÷101

÷102

×101

÷103

×102

×103

Example 1 Convert 6.5 cm to mm.

Create a strategy Use 1 cm = 10 mm and multiply.

Apply the idea 6.5 cm = 6.5 × 10 = 65 mm

Multiply by 10 Evaluate

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Conversion of area units Metric units are used to measure area. Common area units are: • Square millimetre (mm2): A square millimetre is about the size of the head of a small pin. • Square centimetre (cm2): A square centimetre is roughly the size of a fingernail on a pinky finger. • Square metres (m2): A square metre is roughly the floor space occupied by an adult standing with arms spread wide (approximately 1 metre by 1 metre). • Square kilometres (km2): A square kilometre is a massive area, about the size of 100 football fields. Given units of area are in two dimensions, apply the conversion factors in two dimensions also. This can be done by multiplying or dividing by the conversion factor for length twice, or multiply or divide by the conversion factor squared. For example, converting m to cm requires a conversion factor of ×100. Converting m2 to cm2 requires a conversion factor of ×1002. ÷102

mm2

÷1002

cm2

×102

÷10002

×1002

km2

×10002

Hectare A common unit of land measure in the metric system equal to 10 000 square metres (approximately 2.47 acres).

÷10 000

×10 000

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4.02 Practice questions What do you remember? 1

Identify the appropriate unit of measurement for each: a c

The thickness of a book = 2.5 ⬚

b d

Fill in the blanks for length and area conversions: a b

The length of a bus = 12 ⬚

The distance between two cities = 13 ⬚

The length of an ant = 3.7 ⬚

There are ⬚ mm in 1 cm, ⬚ cm in 1 m, and ⬚ m in 1 km.

Order these units of area from smallest to largest: km2, cm2, mm2, m2.

Fill in the boxes to show relationships between these units: a c e

⬚ m = ⬚ km

⬚ cm = ⬚ mm

⬚ mm2 = ⬚ km2

⬚ cm = ⬚ m

⬚ mm = ⬚ m

⬚ km2 = ⬚ cm2

Practice Ex 1

Ex 2

Convert these lengths: a

3 cm to millimetres

60 mm to centimetres

6 m to centimetres

800 cm to metres

6 km to metres

9000 m to kilometres

45 mm to centimetres

7.2 cm to millimetres

12.5 m to centimetres

350 cm to metres

2.8 km to metres

15 000 m to kilometres

3484 mm to metres

Convert these lengths: a

24 m to millimetres

6 km to centimetres

866 800 cm to kilometres

15 km to millimetres

7500 mm to metres

3.2 m to millimetres

12 500 cm to kilometres

Calculate differences in length: a

A desk is 1.39 m long. A rug is 120 cm long. Calculate the difference in centimetres.

A car is 4 m long. A truck is 934 cm long. How many metres longer is the truck?

A surfboard is 2.1 m long. A paddleboard is 320 cm long. How much longer is the paddleboard in metres?

A bridge is 1.5 km long. A tunnel is 1200 m long. Calculate the difference in metres.

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Ex 3

Convert these areas: a

3 m2 to square centimetres

0.56 km2 to square metres

6 ha to square metres

20 000 m2 to hectares

450 cm2 to square metres

1.2 m2 to square centimetres

15 000 m2 to square kilometres

0.08 km to hectares

Calculate the area of a square with sides 290 cm in square metres.

A park has an area of 19 acres. Given 1 acre is approximately 4000 m2, calculate the area in square metres.

Calculate the area of a rectangle with length 5.5 m and width 300 cm in square metres.

A paddock is 2.5 ha. Convert this to square kilometres.

Extend your thinking 12

The table shows the heights of some of the tallest buildings from around the world: Name

Height (metres)

One World Trade Center

541.3

Shanghai Tower Petronas Twin Towers

0.632 451.9

Zifeng Tower Burj Khalifa

Height (kilometres)

0.45 828

Complete the conversions table

Order the buildings in increasing height order

There are 100 cm in 1 m, but the conversion from square metres to square centimetres requires multiplying by 10 000. Explain why.

A bridge’s support is 100 m above the riverbed. Erosion lowers the riverbed by 150 mm per year. How many years until the riverbed lowers by 100 m?

A young tree in a national park is 2 m tall and grows at 500 mm per year. A trail is 10 m away. How many years until the tree’s height reaches 10 m, posing a falling risk to the trail?

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A sample of rectangular farms around Australia were measured and their dimensions, in metres, are recorded in the table: Farm

Length, m

Width, m

200

100

350

100

400

100

Determine the area of each farm in square metres.

Which farms have an area of more than 1 ha?

Which farm has an area of exactly 1 ha?

Area, m2

A property is 7.22 ha. Given 1 ha = 10 000 m2 and 1 acre is approximately 4000 m2, calculate: a

The area in square metres

The approximate area in acres

A rectangular pool is 10 m by 6 m with a circular ﬂoat of diameter 200 cm centred in it. Calculate the distance from the ﬂoat to the pool edges.

A coral reef is 40 m long, forming a square. If the reef grows at 50 cm per side annually, how many years until its area exceeds 2 ha?

A path 1.5 m wide surrounds a rectangular garden of area 120 m2 and length 12 m. Calculate the total area including the path in square metres.

Bob owns a paddock with an area of 290 acres. He wants to advertise his eggs as free range, with no more than 400 chickens per hectare. Given that 1 acre is approximately 0.4 ha, determine the maximum number of chickens that Bob can keep in his paddock.

A farm covers an area measuring 340 acres. Given that 1 acre is approximately 0.4 ha, and 1 ha is 10 000 m2, calculate: a

The area of the farm in hectares

The area of the farm in square metres

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Convert units of volume The following diagram shows the conversion factor between different units of volume: ÷103

mm3

÷106

cm3

×103

÷109

×106

km3

×109

The conversion factors for volume involve very large numbers, so index notation is used in this conversion chart. 103 = 1000, which can also be described as “1 followed by 3 zeros”. However, rather than remembering these factors it is useful to understand that converting units of volume involves converting the three length dimensions of a cube unit. This can be done by multiplying or dividing by the conversion factor for lengths three times - that is multiply or divide by the conversion factor cubed. This means we only have to recall the conversion factor for lengths. When converting between units of volume: • Multiply if converting to a smaller unit - more smaller cubes will be needed to ﬁll the same space. • Divide if converting to a larger unit - fewer larger cubes will be needed to ﬁll the same space. • Multiply or divide by the conversion factor for lengths three times (or cubed). ÷10

÷100

×10

÷1000

×100

×1000

Example 1 Convert 8.12 m3 into cubic centimetres.

Create a strategy Use the fact 1 m3 = 1 000 000 cm3.

Apply the idea m3 × 1 000 000 = 8.12 × 1 000 000 3

= 8 120 000 cm

Multiply 8.12 by 1 000 000 Evaluate

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Convert: a

68 000 mL to L

52 L to mL

88 000 L to kL

63 kL to L

1 cm3 to mL

1 mL to cm3

36 cm3 to mL

720 mL to cm3

1 m3 to kL

1 kL to m3

83 m3 to kL

270 kL to m3

m 0.4 L to mL

3.8 mL to L

1.67 kL to L

84.6 L to kL

A container has a volume of 1 cubic metre. Determine its capacity in: a

Ex 4

Litres

Kilolitres

Millilitres

A rectangular prism has dimensions 3 m, 8 m and 6 m. a

Calculate the volume of the prism in cubic metres.

Calculate the capacity of the prism in kilolitres.

A small pond contains 4300 L of water. What is the volume of the pond in m3?

A baby bottle holds 100 mL of milk.

Determine its volume in cm3.

Show that the bottle has an equivalent volume to a rectangular prism with height of 10 cm, width of 2 cm and length of 5 cm.

Determine the dimensions of another rectangular prism with the same volume.

Consider a cube with side length a m. a

Explain a process for converting a volume in m3 into a capacity in L.

Determine an expression for the volume of the cube in L.

Determine an expression for the volume of the cube in kL.

What does this indicate about the conversion between m3 and kL?

Calculate the volume of this rectangular prism in mm3 in two different ways. Method 1: First convert each side into mm and then calculate the volume. Method 2: First calculate the volume in cm3 and then use a conversion factor.

15 cm 4 cm 6 cm

A rectangular tank has a volume of 120 000 cm3. Explain why its capacity in litres is 120 L.

Quiana bought 12 bottles of soft drink, each containing 400 mL. How many litres of soft drink did she buy in total?

This juice box contains 1600 mL of juice:

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Determine its volume in cm3.

Calculate the length of the juice box if it is 8 cm tall and 5 cm wide.

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Extend your thinking 21

A box has a capacity of 30 litres. If the box is ﬁlled with a liquid, how many 200 mL cups can be ﬁlled from the box?

Scott has a tank that can hold 1 kL of water which is currently empty. He has a bucket with capacity 2000 mL and his sister has a jug with volume 500 cm3, which they use to fill the tank. They fill their containers and empty them into the tank the same number of times. How many times do they need to fill and empty their containers?

This swimming pool is composed of a trapezoidal prism joined to a half cylinder: a

Calculate the volume of the pool in cubic metres. Round your answer to three decimal places.

How many litres of water can ﬁt in the pool? Round your answer to the nearest litre.

3m If the pool is ﬁlled to a height 10 cm below the top, 0.8 m how many litres of water are in the pool? Round your answer to the nearest litre.

After construction works at a neighbouring property, a crack opens in the bottom of the pool and water begins to leak from the pool. It is observed that the height of the surface of the water in the pool is decreasing by 7 cm each week. Determine the amount of water that is leaking out each week, rounded to the nearest litre.

Assuming that water continues to leak at this rate, determine how many whole weeks it will take to empty the pool.

Xavier has been hired by a live performance group that are famous for their strange and demanding requests. He must transport 228 323 L of edible slime from a production facility and deliver it to a nearby concert hall. If even a single litre is missing, he will not be paid. Xavier has a truck with a large, adjustable tank. The base of the tank has an area of 9.5 m2 , and to avoid collisions on his route the top of the tank must be no more than 3.94 m from the base. How many trips must Xavier make from the slime production facility to the concert hall?

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Example 1 Convert 0.493 tonnes to kg.

Create a strategy Use the conversion factor 1 tonne = 1000 kg.

Apply the idea To move from larger mass units to smaller mass units, multiply by the conversion factor. 1 tonne = 1000 kg

Write the conversion factor

0.493 tonnes = 1000 kg × 0.493

Multiply both sides by 0.493

= 493 kg

Evaluate

Example 2 Calculate the total mass of 12 cans of soup, in kg, if each has a mass of 350 grams.

Create a strategy Determine total mass and change the units to kg.

Apply the idea Multiply the single mass by the number of cans to get the total mass. Total mass = 12 × 350 g = 4200g

Multiply number of cans by the mass of each can Evaluate

Since the conversion is from a smaller mass unit (g) to a larger mass unit (kg), division by the conversion factor is required. Conversion factor: 1 kg = 1000 g Divide both sides by 1000

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Evaluate

Ex 2

Valerie has fruit everyday for dinner. Today she consumed 45 cherries, which weighed a total of 180 g. How many milligrams did the average cherry weigh?

Rewrite 2468 g in terms of the number of whole kilograms (kg) and grams (g) left over.

Buzz’s bag weighs

Calculate the total mass of 11 bottles of water, in kilograms, if each has a mass of 500 grams.

The cost of mining gold for a mining company is $840 per gram. How much will it cost the company to mine 0.2 kg of gold?

If 11 apples have a total mass of 3.19 kg, calculate the average mass of those apples in grams.

A truck is transporting 8 horses. Each horse in the truck weighs 800 kg.

of a kilogram. How many grams does the bag weigh?

How much do all the horses in the truck weigh altogether?

If the truck weighed 4000 kg before the horses were loaded into it, how many kilograms does the loaded truck weigh now?

James buys a 4000 g box of washing powder. If he uses 2.3 kg of the washing powder, how many grams of powder is left in the box?

Christa buys a 8 kg bag of rice. If she uses 2800 g, how many kilograms of rice is left?

Passengers have a limit of 20 kg of luggage on a plane. If James’s bag was over the weight limit by 1800 g, how much does his bag weigh in kilograms?

If each carrot has a mass of 380 g and each lemon has a mass of 350 g, how much would 6 carrots and 9 lemons weigh altogether? Give your answer in kilograms.

If 25 bags of ﬂour have a mass of 2 kg, calculate the mass of 5 bags, in grams.

Extend your thinking 18

If a truck can carry 720 kg of soil in one load, how many loads does it need to take in order to empty a ﬁeld of 23.04 tonnes of soil?

The nutritional information on a brand of corn chips states that there is 6 mg of sodium in every 100 g of corn chips. How many grams of sodium would there be in 10 packets of corn chips if each packet weighs 230 g?

An empty box has a mass of 750 g. It is packed with 8 bottles of drink, each of mass 1.25 kg. Calculate the total mass in grams of the packed box.

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A logistic company transports rectangular prism shaped steel beams with dimensions of 3.2 m, 0.25 m and 0.15 m. The density of the steel used is 7850 kg/m3. The truck carrying these beams has a maximum load capacity of 9 tonnes. a

Calculate the volume of a single steel beam in m3.

Determine the mass of a single steel beam in kg.

Calculate the maximum number of beams that can be safely loaded onto the truck without exceeding its load capacity.

If each beam costs $4500 and the truck can make only one trip, calculate the total cost of all the beams that can be transported in one trip.

4.05 Scientific notation and significant figures After this lesson, you will be able to… • express numbers in scientific notation (a × 10n, where 1 ≤ a < 10). • convert numbers from scientific notation to standard decimal form. • identify the number of significant figures in a given measurement. • round numbers to a specified number of significant figures. • apply scientific notation to represent numbers involving standard prefixes.

Scientific notation and significant figures Scientiﬁc notation A way of writing numbers that are too big or too small to be written in an accessible way.

In scientiﬁc notation, numbers are written in the form a × 10n, where a is any number between 1 and 10, and n is either a positive or negative integer. • A positive index indicates how many places the decimal point should be moved to the right. • A negative index indicates how many places the decimal point should be moved to the left. For example: • The sun has a mass of approximately 1.98 × 1030 kg which is much easier to write than 1 980 000 000 000 000 000 000 000 000 000 kg. • The mass of an atom of Uranium (one of the heaviest elements) is only approximately 3.95 × 10−22 g. That is 0.000 000 000 000 000 000 000 395 g.

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Significant figures Each of the digits of a number that are used to express it to the required degree of accuracy, starting from the first non-zero digit.

Significant figures are the digits in a number that convey meaningful information about its precision. Questions often specify the expected number of significant figures in the answer. In science, it is common practice to give answers to the same number of significant figures as the measurement that has the fewest significant figures. Digits that are significant are: • All non-zero digits • Zeros appearing between two non-zero digits • Trailing zeros in a number containing a decimal point

Exploration Explore this illustration of digits that are signiﬁcant: Zero between non-zeros are significant

7004.040200 All non-zero digits are significant

Trailing zeros to the right of the decimal point are significant

1. In the number 7004.040 200, how many significant figures are there? 2. In the number 7004 how many significant figures are there? Explain why each digit is considered significant or not. 3. Compare the significance of trailing zeros in the numbers 1500 and 1500 (with a decimal point). How does the presence or absence of the decimal point affect the number of significant figures in each case?

Consider an example where a concert reported having 60 000 attendees, when in fact it had 59 759, it is impossible to tell if the report has been given to one or two significant figures, since both would round to 60 000. Often the number of significant figures are indicated after an answer to be clear. For example, 60 000 (2 significant figures) or 60 000 (2 s.f.) for short.

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Example 1 Express each number rounded to three signiﬁcant ﬁgures: a 10 432

Create a strategy Count the number of significant figures from the first non-zero digit. If the next digit is greater or equal to 5 round up, otherwise round down.

Apply the idea Starting from 1 count the first 3 significant figures as 104. The zero between the 1 and the 4 is counted as significant. The next digit is 3, then round down and replace the remaining digits by zeros: 10 432 ≈ 10 400

b 2.4983

Create a strategy Count the number of significant figures from the first non-zero digit. If the next digit is greater or equal to 5 round up, otherwise round down.

Apply the idea Starting from 2 count the first 3 significant figures as 2.49. The next digit is 8, so round up and remove the remaining digits: 2.4983 ≈ 2.50 Remember to state the trailing zero if necessary.

c 6.53126 × 107

Create a strategy Count the number of significant figures from the first non-zero digit. If the next digit is greater or equal to 5 round up, otherwise round down.

Apply the idea Starting from 6 count the first 3 significant figures as 6.53. The next digit is 1, so round down: 6.53126 × 107 ≈ 6.53 × 107

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Example 2 Express 0.0003 in scientiﬁc notation, where the difference between two sprinters is measured to be 0.0003 seconds.

Create a strategy Use the scientiﬁc notation form a × 10n, where n is a negative integer.

Apply the idea To determine the first part of scientific notation, place the decimal point after the first non-zero number, so a = 3. The original number 0.0003 requires the decimal to move 4 places to the right to reach 3, indicating a negative exponent. 0.0003 = 3 × 10−4 s

Example 3 If rounded to one significant figure, sound travels at a speed of approximately 0.3 kilometres per second, while light travels at a speed of approximately 300 000 kilometres per second: a Express the speed of sound in kilometres per second in scientific notation.

Create a strategy Use the scientiﬁc notation form a × 10n , where n is a positive integer.

Apply the idea 3 needs to be multiplied by 10−1 to get 0.3: 0.3 = 3 × 10−1 km/s

b Express the speed of light in kilometres per second in scientiﬁc notation.

Create a strategy Use the scientiﬁc notation form a × 10n , where 1 ≤ a < 10 and n is a positive integer.

Apply the idea 3 needs to be multiplied by 100 000 or 105 to get 300 000, so: 300 000 = 3 × 105 km/s

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4.05 Practice questions What do you remember? 1

Complete this table by expressing each measurement in scientific notation using the specified prefix: Measurement

0.0000000072 m

nano-

4500 g

kilo-

0.065 s

milli-

12000000 bytes

mega-

0.00000045 m

micro-

9800000000 bytes

giga-

15000000000000 bytes

tera-

Scientiﬁc Notation

Determine whether each statement about signiﬁcant ﬁgures is true or false: a

In the number 0.00 780, there are three signiﬁcant ﬁgures.

Zeros between non-zero digits are not signiﬁcant.

The number 5000 has four signiﬁcant ﬁgures when written without a decimal point.

Identify the number of signiﬁcant ﬁgures in each measurement: a

Preﬁx

0.00450 cm

7200 kg

6.0700 × 104 m

800.0 g

Express each measurement in scientific notation, rounded to the specified number of significant figures: a

0.0000000567 s (2 signiﬁcant ﬁgures)

3 456 000 W (3 signiﬁcant ﬁgures)

0.008234 m (2 signiﬁcant ﬁgures)

987 654 321 bytes (4 signiﬁcant ﬁgures)

4.13 × 104

Practice 5

Evaluate: a

2.49 × 100

8.97 × 10

1.3008 × 107

2 × 107 4

5.03 × 10

3.014 × 10

8.36 × 10−1

7.91 × 106

0.005 = ⬚ × 10−3

9 × 10−3

8 × 106

7.14 × 10−3

Complete these statements: a c

276

300 = ⬚ × 102

= 4 × 10⬚

Mathspace New South Wales – Year 11 Standard mathspace.co

1 × 10−2 × 2 × 10−5 = 2 × ⬚

Determine whether each of these numbers is equivalent to 0.00045: i

b Ex 1

45 × 10−4

iii

4.5 × 10−6

0.45 × 10−3

Of the expressions selected in part (a), which one is in scientiﬁc notation?

340 211

2.00434

12 008

0.000456056

7 ÷ 100

6.23 ÷ 103

884 000

Express in scientiﬁc notation: a

3 × 100

4.5 × 10 000

2000

0.002

84 626 000

6.14

0.000347

0.764

2 000 000

Identify the smallest number.

m 0.07 10

Express each number rounded to four signiﬁcant ﬁgures: a

Ex 2

4.5 × 10−4

For each set of numbers: i

Identify the largest number.

• 7.27 × 10−7 • 2.22 × 10−7 • 4.76 × 10−7

b • 1.25 × 10−2 • 1.25 × 10−6 • 1.25 × 102

• 9.37 × 10−4 • 6 × 10−4 • 6 × 10−9

d • 7.31 × 10−9 • 5.13 × 10−9 • 8.31 × 10−9

Use a calculator to evaluate each expression and express the answer in scientific notation, rounded to four significant figures: a

(6.20 × 10−2)2

0.000512 × 0.00814

For each expression: i

Identify the number of signiﬁcant ﬁgures given in the least precise measurement.

Calculate the result, rounded to the least number of signiﬁcant ﬁgures.

9.17 m × 9.790 m × 4.70 m

418 mL × 5.7 mL ÷ 934.5 mL

Consider the equation 9n = 11: a

Determine the closest whole number to n.

The value of n is given by n =

. Using the log button on the calculator, determine

the value of n. Round the answer to three signiﬁcant ﬁgures. c

Using the same process in part (b), solve 8n = 13. Round the answer to two signiﬁcant ﬁgures.

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The world’s oceans hold approximately 1 332 000 000 cubic kilometres of water: a

Express this volume of water in scientiﬁc notation.

Given 1 km3 = 1 000 000 000 000 L, how many litres of water do the world’s oceans hold? Express the answer in scientiﬁc notation.

Convert the volume of water to cubic terametres (Tm3) and express in scientiﬁc notation.

Convert the volume of water to gigalitres (GL) and express in scientiﬁc notation.

Bone marrow in a person’s body produces 2.3 × 106 red blood cells each second: a

b Ex 3

Which value would be the closest rounded value for this number? A

20 million each second

2 million each second

2 hundred thousand each second

20 thousand each second

Estimate the number of red blood cells produced each minute.

Rounded to one signiﬁcant ﬁgure, the distance from Earth to the Moon is approximately 400 000 kilometres, and the distance from Earth to the Sun is approximately 0.15 billion kilometres: a

Express the distance to the Moon in kilometres in scientiﬁc notation.

Express the distance to the Sun in kilometres in scientiﬁc notation.

How many times farther is the Sun from Earth than the Moon?

The mass of the largest mammal on Earth is approximately 1.5 × 103 times greater than the mass of an average adult human who weighs 90 kg. According to this information, determine the approximate mass of the largest mammal on Earth. Express your answer as a basic numeral.

Evaluate each expression, giving your answers in scientiﬁc notation: a c e

2 × 106 × 3 × 105 (9 × 109) × (5 × 10−6)

4 × 10−3 × 4 × 10−5

d f

(7 × 105) × (3 × 104)

11 000 000 × 0.004 × 0.0005 × 20 000

Extend your thinking 19

278

One mole (mol) of an element consists of approximately 6.02 × 1023 atoms. a

Determine the number of atoms in 3.5 mol of element X. Express your answer in scientiﬁc notation:

If 3.5 mol of element X weighs 31.5 g, determine the mass of 1 mol of element X.

Determine the approximate mass of one atom of element X. Express your answer in scientific notation, rounded to four significant figures.

Mathspace New South Wales – Year 11 Standard mathspace.co

The Earth orbits the Sun at an approximate speed of 8333 metres per second. Determine how far it travels in 23 hours. Express your answer in scientific notation, rounded to one significant figure.

A calculator displays 3.35E5 as the correct ﬁnal answer when solving the given problem. Determine the missing values in the problem to produce the given answer. “At any particular time, approximately 2.32 × 10⬚ aircraft are registered as being in ﬂight over ⬚ × 1010 square metres of air space. On average, how much air space does each aircraft have?”

A number, when rounded to one signiﬁcant ﬁgure, is 3000. Determine the difference between the largest possible value and smallest possible value of the original number.

Asteroids A and B are currently 4 × 1014 km apart. If they continue in a straight line towards one another, travelling at 3857 km/h and 7938 km/h respectively, how many days will it take for them to collide? Express your answer in scientific notation, rounded to five significant figures.

Planet A has a radius of 8.2 × 1018 km, while planet B has a radius of 4.1 × 1023 km. How many times greater than the volume of planet A is the volume of planet B? Express your answer in scientiﬁc notation. The volume of a sphere is given by V =

π r3.

Calculate the volume of a cylindrical wire in an electric circuit with a radius of 2.57 × 10−4 m and a length of 9.13 × 10−2 m. Express your answer in scientific notation rounded to three significant figures.

An ant has a body length of 25 mm and can move at a speed of 0.08 m/s. The Earth has a circumference of 40 075 km: a

Calculate the number of body lengths the ant travels per second. Express your answer in scientiﬁc notation.

Calculate the number of days it would take for an ant to walk non-stop around the Earth’s equator, rounded to two decimal places.

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4 Chapter review 1

A scientific instrument magnifies an object’s length from 0.042 mm to 420 mm. By what power of 10 has the length been magnified? A

104

105

0.40 cm

4.0 cm

40 cm

400 cm

5.7 × 105

5.7 × 106

5.8 × 106

57 × 105

6.17 ÷ 10

23.85 ÷ 100

0.0042 ÷ 1000

0.81 ÷ 10

45.6 ÷ 100

0.073 ÷ 10

128.4 ÷ 1000

9.05 ÷ 100

0.7 × ⬚ = 7000

0.035 × ⬚ = 350

Fill in the blanks to make the equations true: e

9 × ⬚ = 90

4 × ⬚ = 400

b f

Convert: a

Evaluate:

103

The estimated population of a city is 5 720 000 people. How is this number expressed in scientific notation, rounded to two significant figures? A

A wooden plank is 2.35 m long. A metal bar is 195 cm long. What is the difference in their lengths, in centimetres? A

102

16 × ⬚ = 1600

0.12 × ⬚ = 1200

18 m to millimetres

25 × ⬚ = 250

4250 mm to metres

0.008 × ⬚ = 80

4 km to centimetres

75 000 cm to kilometres

9.5 m to centimetres

1200 mm to metres

Convert: a

5 m2 to square centimetres

0.75 km2 to square metres

8 ha to square metres

35 000 m2 to hectares

2.5 m2 to square centimetres

12 000 m2 to hectares

Calculate the area of a rectangular paddock with length 6.2 m and width 250 cm. Give your answer in square metres.

Convert to cubic millimetres:

4.2 cm3

2.5 m3

1.8 cm3

0.75 m3

0.5 m3

48 kL to litres

Convert: a

280

75 000 mL to litres 3

52 cm to millilitres

65 m3 to kilolitres

12 500 mL to litres

33 cm3 to millilitres

Mathspace New South Wales – Year 11 Standard mathspace.co

23 cm3

Convert: a

7.152 kg to grams

820 g to kilograms

15 t to kilograms

7200 kg to tonnes

950 g to kilograms

3.8 t to kilograms

Calculate the total mass of 15 cans of soup, in kilograms, if each has a mass of 400 g.

Evaluate: a

5.09 × 104

7 × 10−3

2.76 × 105

0.00032178

Express each number rounded to three signiﬁcant ﬁgures: a

3.18 × 100

450 812

3.00725

15 006

The value of X ÷ 102

Given X × 104 = 58 000, determine: a

The value of X

An elevator has a maximum load capacity of 1.2 tonnes. If the average mass of a person using the elevator is 75 kg, what is the maximum number of people that can safely use the elevator at one time?

Light travels at approximately 3.00 × 108 m/s. The distance from the Sun to Mars is approximately 2.28 × 1011 m. How long does it take for light from the Sun to reach Mars? Express your answer in seconds, in scientific notation, rounded to two significant figures.

A single grain of sand has a mass of approximately 5.0 × 10−5 g. A small bag contains 2.5 × 104 grains of sand. What is the total mass of the sand in the bag, in grams?

A sample of rectangular garden plots were measured, and their dimensions, in metres, are recorded in the table: Plot

Length (m)

Width (m)

150

250

Area (m2)

Determine the area of each plot in square metres.

Which plots have an area greater than 0.8 ha? (1 ha = 10 000 m2)

A ﬁeld has an area of 150 acres. If fruit trees are planted with a density of no more than 250 trees/ha, and 1 acre ≈ 0.405 ha, determine the maximum number of trees that can be planted.

A container has a capacity of 45 L. How many 250 mL glasses can be ﬁlled from it?

A cereal contains 8 mg of iron per 50 g. How many grams of iron are in 5 boxes, each weighing 375 g?

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Micro-organism X has a mass of 3.1 × 10−12 kg and micro-organism Y has a mass of 7.75 × 10−9 kg. How many times more massive is Y than X?

One mole of a substance contains approximately 6.022 × 1023 particles. For 2.5 mol of water: a

Determine the number of molecules, in scientific notation, rounded to three significant figures.

If 2.5 mol weighs 45.0 g, determine the mass of 1 mol.

Determine the approximate mass of one water molecule, in scientific notation, rounded to three significant figures.

Did you know?

Did you know a single drop from a medicine dropper is about 0.05 mL? That means there are roughly 20 drops in just 1 mL of liquid. Precision in measuring these tiny volumes is extremely important. For example, many medications — such as liquid antibiotics, pain relievers, or paediatric medicines — are prescribed in very small doses. Even a slight error could mean the difference between an effective dose and an unsafe one. Pharmacists also rely on exact measurements when preparing custom medications, ensuring the right balance of ingredients in every mixture. This attention to precision isn’t limited to pharmacies — scientists in laboratories also use droplet measurements for experiments, where accuracy is crucial to producing reliable results.

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Big ideas • Measurement in two dimensions involves quantifying length (perimeter) and surface (area). Applying specific formulas is essential for standard shapes, while problem-solving strategies, such as decomposition for composite figures or approximation using the trapezoidal rule for irregular figures, are required for more complex scenarios. Pythagoras’ theorem serves as a fundamental tool for determining unknown lengths, a critical step in many geometric calculations. • Extending measurement principles into three dimensions involves calculating surface area (the total external area) and volume (the space occupied). The analysis of composite solids requires a distinct approach for each measure: volume is found by simple addition or subtraction of component parts, whereas calculating surface area demands the careful identification and exclusion of any internal surfaces. The concept of volume is directly linked to capacity, enabling practical applications.

5 Perimeter, area and volume Chapter outline 5.01 5.02 5.03 5.04 5.05 5.06 5.07 5.08 5.09 5.10

Pythagoras’ theorem Perimeter Perimeter of composite shapes Area Area of composite shapes Surface area Surface area of composite solids Volume and capacity Volume of composite solids Trapezoidal rule Chapter 5 review

286 297 307 320 336 341 355 362 378 388 399

5.01 Pythagoras’ theorem After this lesson, you will be able to… • identify the hypotenuse of a right-angled triangle. • apply Pythagoras’ theorem to find the length of the hypotenuse. • apply Pythagoras’ theorem to find the length of a shorter side. • use Pythagoras’ theorem to test if a triangle is right-angled. • solve practical problems involving right-angled triangles.

Pythagoras’ theorem Hypotenuse In a right-angled triangle, the largest angle is 90°. The side opposite the right angle is the longest side, called the hypotenuse. Right angle

All three sides of a right-angled triangle are related by the equation shown.

a2 + b2 = c2

The two shorter sides are labelled a and b, and the hypotenuse (the longest side) is labelled c.

a2 + b2 = c2 a, b, c

are the sides of a right-angled triangle

a, b

are the lengths of the shorter sides

is the length of the hypotenuse

Pythagoras’ theorem The square of the length of the hypotenuse, c, of a right-angled triangle equals the sum of the squares of the lengths of the other two sides, a and b, such that c2 = a2 + b2.

Pythagoras’ theorem can be used to: • Check whether a triangle is a right-angled triangle • Find the length of the hypotenuse • Find the length of one of the shorter sides in a right-angled triangle 286

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Interactive exploration Discover this concept in action online

mathspace.co

Example 1 Use Pythagoras’ theorem to determine whether this is a right-angled triangle.

16 9

a Let a and b represent the two shorter side lengths. Find the value of a2 + b2.

Create a strategy Substitute the lengths of the two shorter sides and add them.

Apply the idea a2 + b2 = 92 + 162

Substitute the values of a and b

= 81 + 256

Evaluate the squares

= 337

Evaluate the sum

b Let c represent the length of the longest side. Find the value of c2.

Create a strategy Substitute the length of the longest side.

Apply the idea c2 = 182 = 324

Substitute the value of c Evaluate

c Is the triangle a right-angled triangle?

Create a strategy Compare the results from parts (a) and (b) to determine if a2 + b2 = c2.

Apply the idea 337 ≠ 324 The triangle is not a right-angled triangle because a2 + b2 ≠ c2.

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Example 2 Find the length of the unknown side as an exact value: a

Create a strategy Use Pythagoras’ theorem: c2 = a2 + b2

Apply the idea c2 = a2 + b2 2

Write the formula 2

c = 12 + 16

Substitute a and b

c = 144 + 256

Evaluate the squares

c2 = 400

Evaluate the sum

Take the square root of both sides

= 20

Evaluate the square root

Reﬂect and check The hypotenuse, c, should always be the longest side in a right-angled triangle. Make sure that 20 is larger than the smaller sides. If it is not, you may have done a mistake in your working out.

Create a strategy Use Pythagoras’ theorem: a2 + b2 = c2

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5.01 Practice questions What do you remember? 1

Write Pythagoras’ theorem in a form you could use to ﬁnd: a

The length of r

The length of p r

p 2

Calculate the value of a, rounded to two decimal places when necessary. All values of a are positive: a

a2 = 72 + 242

132 = 52 + a2

11 =

Can a triangle with sides of length 9 cm, 12 cm, and 15 cm be a right-angled triangle? Use Pythagoras’ theorem to justify your answer.

Practice Ex 1

Use Pythagoras’ theorem to determine whether this is a right-angled triangle: a

Let a and b represent the two shorter side lengths. Find the value of a2 + b2.

Let c represent the length of the longest side. Find the value of c2.

Is the triangle a right-angled triangle?

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Ex 2

Find the length of the unknown side as an exact value: a

6 c

d s

12 e

f u

20 24

A right-angled triangle has one side of length 14 and another side of length 22. Find the length of the hypotenuse.

A right-angled triangle has a hypotenuse of length 30 and another side of length 7. Find the length of the third side.

Two poles, one of length 6 m and the other of length 10 m, stand 12 m apart on level ground. Find the length of wire required to join the tops of the poles, rounded to two decimal places.

10 m 6m 12 m

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293

Find the length of a diagonal of each shape, rounded to one decimal place where appropriate: a

12 cm

12 mm 6 mm

12 cm

28 m

12 cm 5 cm

21 m 18 cm

Find the distance between given points, rounded to one decimal place: a

Points A and B

Points P and Q

6 cm P

16 mm

6 cm 8 cm

B 8 mm 6 cm

6 mm

Ex 3

16 mm

4 mm

Two buildings, one of height 400 m and the other of height 550 m, stand a distance apart on level ground. If a cable joining the tops of the buildings is 250 m long, ﬁnd, to the nearest metre, the distance between the buildings.

250 m 550 m

400 m ?

Ex 4

294

Determine whether each set of values are Pythagorean triads: a

(6, 8, 10)

(6, 9, 11)

(8, 16, 20)

(9, 12, 15)

(12, 16, 20)

(10, 11, 14)

(20, 21, 29)

(5, 12, 14)

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Find the length of the diameter CD of the circle.

Find the exact length y. 15 cm O

A house has outer dimensions as shown in the diagram:

D 11 cm

8 cm

A drainpipe runs from the vertex of the roof, point B, vertically downwards until it is 75 cm above the ground. Find the length of pipe needed in metres, rounded to two decimal places.

5.8 m

3.5 m

10 m 13

Find the value of y, rounded to one decimal place. a

8m y

6 cm

Extend your thinking 14

A triangle with sides 5, 12, and 13 units is a right-angled triangle. If the triangle is enlarged by a factor of 4, does Pythagoras’ theorem still hold? Justify your answer.

Is a triangle with these sides a right-angled triangle? a

20, 48, 52

25, 60, 70

5k, 12k, 13k

15a, 36a, 46a

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If XB = 12 cm, XY = 15 cm and ∠XBY is a right angle, show that B is the midpoint of Y Z.

Y 17

A factory produces thin plastic right-angled triangles. a

If the triangles need edging material costing 5 cents/ cm, how much, in dollars, will it cost to edge 200 triangles?

If the triangles need a coating costing 4 cents/cm2, how much, in dollars, will it cost to coat 200 triangles?

28 cm

53 cm

The diagram shows four squares. The outer square has side lengths of 20 cm. The next square is formed by joining the midpoints of the outer square, the third by joining the midpoints of the second, and the fourth by joining the midpoints of the third. Calculate, to the nearest centimetre, the total length of wire needed to outline all four squares.

Two hikers start from the same point. Hiker A walks 900 m north and then 500 m west. Hiker B walks 1200 m north and then 300 m west. If they return to the starting point in a straight line, who walks further, and by how much, to the nearest metre?

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5.02 Perimeter After this lesson, you will be able to… • calculate the perimeter of polygons by summing side lengths. • calculate the circumference of a circle given its radius or diameter. • calculate the arc length of a sector. • calculate the perimeter of sectors and semicircles. • solve practical problems involving perimeter and circumference.

Perimeter of polygons The perimeter of a polygon is the total length of all its sides. To calculate the perimeter, add the lengths of all sides. For regular polygons (where all sides are equal), the perimeter can be found using the formula:

P=n×s P is the perimeter n is the number of sides s

is the length of each side

For irregular polygons, sum the lengths of all sides, which may require interpreting diagrams. Length

Width

For a rectangle with length l and width w, the perimeter is given by the formula:

P = 2 × (l + w) P

is the perimeter

is the length

is the width

To calculate the perimeter accurately, all side lengths must be expressed in the same unit.

Interactive exploration Discover this concept in action online

mathspace.co

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Circumference The circumference of a circle is the distance around its boundary. The ratio of a circle’s circumference to its diameter is the constant π, approximately 3.141592 …

C = πd C

is the circumference

is the diameter

C = 2π r C

is the circumference

is the radius

For a fraction of a circle (e.g., a semicircle), multiply the circumference by the fraction and add any straight edges for the perimeter.

Example 3 A circle has a radius of 22 cm. Find its exact circumference.

Create a strategy Use the formula C = 2π r.

Apply the idea C = 2π r

Write the formula

= 2π × 22

Substitute r = 22

= 44π cm

Simplify

The exact circumference is 44π cm.

Example 4 Find the perimeter of a semicircle with a diameter of 10 m, rounded to two decimal places.

Create a strategy Calculate the arc length of the semicircle (half the circumference) and add the diameter.

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5.02 Practice questions What do you remember? 1

Complete the unit conversions: a c

4 metres = ⬚ centimetres

6 kilometres = ⬚ metres

b d

12 centimetres = ⬚ millimetres

3.2 metres = ⬚ millimetres

A regular pentagon has five equal sides. One way to calculate the perimeter is to add the five side lengths. What is another way to find the perimeter of this shape? a

What is the diameter of this circle in terms of y?

Write the circumference of the circle in terms of π and y. y

Are these statements true about π ? a b c d

π is equal to 3.141. π is equal to 3.14 when rounded to two decimal places. π can be expressed as a fraction using whole numbers. The formula for the circumference of a circle with diameter d is C = π d.

Practice 5

302

Identify the shape given the information: a

Perimeter of 24 cm and sides of length 6 cm each

Perimeter of 80 m and sides of length 35 m, 22 m, and 23 m

Perimeter of 28 mm and sides of length 7 mm, 7 mm, 7 mm, and 7 mm

Perimeter of 150 cm and sides of length 15 cm each

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Find the perimeter of each shape with the given dimensions: a

10 cm

6 cm

8 cm

15 cm

5 cm

7 cm

12 cm e

10 cm 6 cm

10 cm 20 cm 12 cm Ex 1

Find the perimeter of a rectangle with the given dimensions: a

Length: 15 m, Width: 8 m

Length: 12.7 m, Width: 9.3 m

Find the perimeter, given that each square is 1 unit in length: a

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Ex 2

Find the perimeter of the irregular hexagon shown. 9 cm 6 cm

10 cm

11 cm

8 cm 13 cm

Ex 3

Find the perimeter of a regular hexagon with each side length of 14 cm.

Write the diameter of the semicircle in terms of x.

What is the length of the semicircular arc in terms of π and x?

Write the perimeter of the semicircle in terms of π and x.

Write an algebraic expression for the perimeter of a parallelogram with sides 2x + 5 and x + 3.

Find the circumference of a circle with: a

Radius: 8 cm (exact answer)

Diameter: 15.6 m (rounded to two decimal places)

A circle has a circumference of 40 cm. Calculate the radius, rounded to one decimal place.

Find the perimeter of: a

An equilateral triangle with a side length of 12 cm

A scalene triangle with side lengths 14 mm, 27 mm and 19 mm

An isosceles triangle where the 2 equal side lengths are 30 mm and the third side measures 4 cm

A rectangle has a perimeter of 110 cm. If it has a length of 26 cm, ﬁnd its width.

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Find the side length of a square with perimeter of 52 m:

P = 52 m

Ex 4

For each value of r: i

Find the length of the quarter circle arc.

Find the perimeter of the quarter circle.

Round all answers to two decimal places. a

r = 16 cm

r = 8.2 m

r = 9.5 cm

r = 12 km

Ex 5

The angle ∠Y XZ is equal to 80°: a

Find the fraction of the whole circle that lies on the arc Y Z

Find the exact length of the arc Y Z

80° 9 cm

A sector-shaped pizza slice has a crust (arc) length of 4π cm and a radius of 8 cm. Find the angle of the sector and the perimeter of the slice, rounded to two decimal places.

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Find the perimeter of each sector. Round your answer to two decimal places: a

9 cm 4.6 cm

240°

280°

d 4.8 m

30 mm

250°

A sector with a radius of 10 cm and angle of 80°.

A sector with a radius of 5 m and angle of 120°.

285°

Extend your thinking 22

A rectangle has a perimeter of 48 cm. Find all possible whole number dimensions of the rectangle. Explain your method.

A square and a triangle have the same perimeter. If the triangle is equilateral with side length 12 m, ﬁnd the side length of the square and write an expression for their perimeter relationship.

Determine if each statement is true or false. Justify your answer: a

The perimeter of a square with side length 8 cm equals the perimeter of a rectangle with length 10 cm and width 6 cm.

A regular octagon with a perimeter of 56 cm has each side length of 7 cm.

The circumference of a circle is always greater than its radius.

A rectangular garden is 25 m long and 10 m wide. A fence is built around it, with posts every 5 m. How many posts are needed?

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5.03 Perimeter of composite shapes After this lesson, you will be able to… • identify the external boundary of a composite shape. • calculate unknown straight side lengths using addition and subtraction. • apply Pythagoras’ theorem to find unknown side lengths within composite shapes. • calculate the perimeter of composite shapes that include circular or sector components. • solve practical problems involving the dissection of irregular shapes to find perimeter.

Perimeter of composite shapes Composite shape A shape that is formed by combining other plane shapes. Composite shapes are often described as ‘complex’ when they are made up of many and different shapes. The perimeter of a composite shape is found by adding the lengths of the sides around the boundary of the shape. Sometimes a length may need to be calculated using addition or subtraction, Pythagoras’ theorem or the arc length formula.

Example 1 Find the perimeter of the shape, rounding your answer to two decimal places.

17 cm 6 cm

Create a strategy The perimeter of a ﬁgure is the sum of the sections making up its boundary.

17 cm 3 cm

6 cm

This shape has two straight sections, a semicircle of radius 3 cm, and a quarter circle of radius 6 cm. The circumference of a circle is given by C = 2π r.

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Apply the idea From the figure, the top straight edge is 17 cm, the bottom straight edge is 17 + 6 cm the semicircle and the quarter circle is

. Write the perimeter

Evaluate and round

Example 2 An outline of a block of land is shown:

17 m xm

8m 32 m

a Find the length of the side labelled x m.

Create a strategy Visualise the block of land as a combination of triangle and rectangle: 17 m xm

8m 32 m

Use Pythagoras’ theorem to ﬁnd the length x.

Apply the idea The two sides of the triangular part of the land would be 8 m and 32 − 17 = 15 m. Write the formula of Pythagoras’ theorem

Substitute x, 8 and 15 into the formula

Take the square root of both sides

Evaluate the squares

Evaluate the addition

Evaluate the square root

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b Find the perimeter of the block of land in metres.

Create a strategy Add up the lengths of all sides.

Apply the idea Perimeter = 17 + 17 + 8 + 32

Add the side lengths

= 74 m

Evaluate

Example 3 Find the perimeter of the shape, rounding your answer to two decimal places.

12 cm 35°

Create a strategy The perimeter is equal to the sum of three arcs and six straight edges. Since the three sectors are identical, focus on ﬁnding the perimeter of one sector then multiply by 3. This consists of an arc and two radii. The arc length of a sector is given by

× 2π r.

Apply the idea rite the formula to find the perimeter of one W sector and multiply it by 3

Evaluate and round

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Practice Ex 1

For each composite shape: i

What are the basic shapes that make up the composite shape?

Find the perimeter, rounded to two decimal places, where appropriate.

26 cm

13 cm

18 cm 12 cm 18 cm c

18 cm

18 cm 5

Calculate the perimeter: 3 mm y 7 mm 16 mm

Calculate the perimeter: a

24 m 6m

13 m

18 m

x 4m y

3m 18 m

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Calculate the perimeter: 18 m 3m 14 m 13 m

Find the perimeter: a

20 cm

5 mm

10 cm

7 mm

12 mm

5 cm

5 mm

13 cm

2 mm 13 mm

10 cm

12 cm

9 cm

10 cm

11 cm

8 cm

14 cm

9 cm e

9 mm

14 m

19 m

5 mm

11 m 20 mm

23 mm

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30 m

Find the perimeter of the arc, rounded to two decimal places:

16 10

A rectangle has two semicircles with a radius of 4 m cut out at each end. Find the perimeter of the shape, rounded to two decimal places.

Ex 2

32 14 m

An outline of a block of land is pictured: a

Find the value of y.

Find the perimeter of the block of land.

15 m ym

5m 27 m

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Ex 3

Find the perimeter, rounded to one decimal place: a

9 cm

8 cm

6 cm e

76.4 4m g

38 cm

50 cm 28 cm

22 cm 19 cm

48°

14 cm 50°

30°

30° 28 cm

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Ivan is calculating the perimeter of the ﬁgure. He writes down this calculation: P = 9 + 20 + 9 + 20 + π × 20 Ivan has made two mistakes. What are they?

9 cm 20 cm 14

Determine if each statement is true or false. Justify your answer: a

The perimeter of a composite shape is always equal to the sum of the perimeters of its simple shapes.

If two composite shapes have the same perimeter, they must have the same number of component shapes.

The perimeter of a composite shape with a hole in the middle is always greater than the perimeter of a solid shape with the same outer dimensions.

A rectangular block of land is 14 m long and 7 m wide. A fence is to be constructed around the perimeter. What is the length of the fence?

Find the perimeter of an outline of a block of land:

28 m 28 m

23 m 44 m 17

Use two different methods to calculate the perimeter of the shape, rounded to the nearest centimetre.

10 cm

20 cm

14 cm 8 cm

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Alice noticed that these three sectors make up a semicircle. She thinks that the ﬁgure will have the same perimeter as a semicircle with a radius of 8 cm: 60°

60°

Is Alice correct? Explain your answer.

Find the exact perimeter of the shape.

8 cm 60°

The diagram shows a plot of land taken by a drone. All measures are in millimetres. Calculate the perimeter of the plot of land, rounded to the nearest metre, if the scale is 3 : 5000. treet Cedar S 138

Line of wall

Main Street

10.5 41

Building

19 Property Line

16 142

Ivy is fencing a paddock. The length across the paddock is 100 m: a

Find the value of y, rounded to two decimal places.

What length of fencing does Ivy need in total?

If the cost of fencing is $21.04 per metre, how much will the fencing cost?

100

120

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A farmer wants to build a fence around the entire perimeter of his land. The fencing costs $36 per metre: a

Find the value of x, rounded to two decimal places.

Find the value of y, rounded to two decimal places.

How many metres of fencing does the farmer require?

How much will it cost to build the fence?

A 2m F x

6m 10 m

y C 5m 22

The diagram shows the outer walls of a house that is to be treated for insects. Insecticide needs to be sprayed along the bottom of the walls, where 1 L of insecticide covers 18 m of wall: a

Find the value of x. Round your answer to one decimal place.

Find the amount of insecticide needed to treat the whole perimeter of the house in litres, rounded to one decimal place.

The insecticide costs $19 per litre, the labour cost for the job is $72, and there is a call-out fee of $40. Find the total cost of the treatment.

x 7.5 m 1.7 m 6m

3.8 m

8m 5m

6m 10 m

A landscape architect designs a pond for his backyard. The pond is surrounded by a brick wall made up of a straight section and a circular arc with a radius of 6 m. The wall will cost $216 per metre to build.

Calculate the total cost of the wall. Round your answer to two decimal places.

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A standard 300 m running track consists of a straight section with a length of 63.29 m and two semicircular sections. The radius of the inner edge of Lane 1 is 27.60 m and the radius of the inner edge of Lane 2 is 28.52 m:

28.52 m 27.60 m

Lane 1 Lane 2 Start

63.29 m

Calculate the inside perimeter of Lane 1 in metres. Round your answer to two decimal places.

Calculate the inside perimeter of Lane 2 in metres. Round your answer to two decimal places.

If athletes must stay in their lanes, how much further would an athlete in Lane 2 run in a 5 km race? Round your answer to two decimal places.

Use your answer in part (c) to explain why Lane 2 has a further starting point than Lane 1.

Consider the following section of a street map: Addition

89-95

Ave

85-87

81-83

2-8 e

26 24

162

23 160

d bra R

Alge

17a 17b 17c 19a 19b 19c t etry S

1 Geom

Trigonometry St

14-18

d io R Rat

Calculus Av

156

158

d Rhombus R

154

150-152 7m

Use the scale to estimate the perimeter of the shaded block of land in metres. 318

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The diagram shows an aerial view of a swimming pool. The pool is surrounded by rectangular pavers.

Each paver is 120 cm long and 60 cm wide: a

What is the perimeter of the pool? Use the outside edge and round your answer to two decimal places.

As part of his daily exercise routine, Lachlan does 500 m of walking.

How many laps around the pool will Lachlan have to walk to reach his daily exercise goal?

5.04 Area After this lesson, you will be able to… • recall and apply area formulas for common polygons like triangles and quadrilaterals. • calculate the area of a circle given its radius or diameter. • calculate the area of a sector of a circle. • distinguish between exact and approximate answers for area. • solve practical problems involving area and unit conversions, including hectares.

Area of polygons Area is the amount of space enclosed inside a two-dimensional shape. From smallest to largest, the common units used for measuring area are: • square millimetres (mm2) • square centimetres (cm2)

• square metres (m2) • square kilometres (km2)

A hectare is a special unit of area often used to measure the area of a piece of land. 1 ha = 10 000 m2. A summary of the area formulas for common polygons: Rectangle

Square

Parallelogram

A = lw

A = s2

A = bh

Rhombus

Kite

Triangle

Trapezium

a w l

h b

x y

To ensure accurate area calculations, convert all measurements to the same unit. 320

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h b

Area of circles The area of a circle can be calculated using the formula:

A = π r2 A

is the area

is the radius of the circle

When required to give the exact value of the area of a circle, leave the answer in terms of π. To find the area of a semicircle (half a circle), find the total area, then find a half of that by dividing by 2. To find the area of a quarter of a circle, find the total area, then find a quarter of that by dividing by 4.

Example 2 Find the area of the circle as an exact value.

3 cm

Create a strategy Use the formula for the area of a circle: Area = π r2

Apply the idea Area = π r2

Write the formula 2

=π×3

Substitute r = 3 2

= 9π cm

Evaluate, leaving your answer as an exact value

Reﬂect and check It can be helpful when using exact values to ﬁnd an approximate answer. A rough estimate for π is 3, so a rough estimate for this area would be 9 × 3 = 27 cm2.

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Area of a sector Imagine you have a round pizza in front of you. When you take a slice, you are actually taking a ‘sector’ from the whole pizza.

A sector is the region inside a circle between two radii.

Similar to how the arc length is simply a fraction of the circumference, the area of a sector is simply a fraction of the circle’s area. The area of a sector can be calculated by ﬁnding the area of the circle they are a part of and then taking the appropriate fraction. For example, the area of a semicircle is half the area of the full circle. Area of semicircle =

× π r2

Alternatively, the area of a semicircle can be expressed as r

Area of semicircle =

× π r2

since a semicircle corresponds to a 180° angle out of the full 360°.

A formula for the area of any sector can be made depending on the contained angle θ that subtends the arc at the centre.

θ r

A sector with contained angle θ corresponds to a fraction

of a full circle and so its area is given by: A=

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× π r2

Example 4 The sector in the diagram has an angle of 30° and a radius of 6 cm. Q

30° 6 cm

a What fraction of the circle’s area is covered by this sector?

Create a strategy To ﬁnd the fraction, make the angle of the sector the numerator and 360 the denominator.

Apply the idea Divide the angle of arc by 360

Simplify the fraction

b Find the exact area of the sector.

Create a strategy Use the formula A = π r2 and multiply the area by

Apply the idea Write the formula

Multiply the area by

Substitute the radius

Evaluate the product

Simplify

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A circle has a diameter of 8 m: i

Is each area value correct or incorrect?

Is each area value exact or approximate?

8π m2

50.3 m2

100.5 m2

4 mm to centimetres

40 000 m2 to hectares

16π m2

Convert to the unit indicated: a c

5 m to centimetres 2

0.12 km to square metres

Find the fraction of the circle’s area covered by each sector: a

155°

84°

295°

143°

f 90°

72°

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Find the exact area of each sector: a

8 mm

10 cm

14 mm

d 10 cm

Practice Ex 1

Find the area of these shapes: a

25 mm

6 cm

11 cm c

18 mm

d 7m 15 cm 11 m

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42 cm

14 cm 33 cm g

h 18 cm

7m 24 cm 9m i

j 20 cm

11 cm

20 m 12 m

l 5 cm

26 mm

22 cm

14 mm

m 30 mm

n 7 mm

5 mm 18 mm

11 mm

18 mm

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Complete the table for bases, heights, and areas of parallelograms.

Base

Height

7 mm

4 mm 96 cm2

12 cm 6 cm

60 cm2 36 m2

9m 8

Area

A harbour has a pier in the shape of a trapezium with a perpendicular height of 6 m. One base of the pier has a length of 8 m and the other has a length of 4 m. Using a labelled diagram, ﬁnd the area of the pier.

Buzz used some scrap paper to make a birthday card in the shape of a parallelogram. The base of the card is 12 cm long, and the perpendicular height is 10 cm. Using a labelled diagram, ﬁnd the area of the card.

Ex 2

Find the area of these triangles with its given side lengths, rounded to one decimal place: a

Base length of 6 cm and a perpendicular height of 7 cm

Base length of 4.2 cm and a perpendicular height of 2 cm

Base length of 8 cm and a perpendicular height of 5 cm

Base length of 15.6 cm and a perpendicular height of 8.4 cm

The faces on a 4-sided die are all triangular. Each face has a base length of 10 mm and a perpendicular height of 16 mm. Find the area of one face of the die.

Find the exact area of these circles: a

b 30 mm

15 cm

8.5 m 11 cm

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Ex 3

Find the area of these circles, rounded to one decimal place: a

22 cm

28 mm

26 cm 18 m

The radius of a circular baking tray is 12 cm. Find its area, rounded to two decimal places.

Find the radius of a circle that has an area of: a

Ex 4

25π cm2

144π mm2

625π cm2

36π m2

For each circle: i

Find the fraction of the circle’s area covered by the sector.

Find the exact area of the sector.

90° X 18 cm

b Q

45° 20 cm

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Ex 5

Calculate the area of each sector, rounding your answers to one decimal place: a

108°

11 mm

150°

18.4 m

7.2 cm

240° 6 cm 35°

The area of the circle is 30 cm2. Find the area of the shaded sector. Round your answer to two decimal places.

75°

Find the exact area of a semicircle with a radius of 10 cm.

Find the area of the sector of a circle of radius 14 cm if the sector subtends an angle of 60° at the centre. Round your answer to two decimal places.

The diagram shows an arc JK of a circle, with centre O. The radius of the circle measures 12 cm and the arc measures 10 cm in length: a

Calculate ∠JOK, rounded to the nearest degree.

Calculate the exact area of the sector.

K 12 cm O

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In the diagram, O is the centre of a circle with radius 10 cm. Arc JK has a length measuring 5π cm.

Determine the exact area of sector OJK. K 10 cm O 23

The given circle, centred at O, has radius 10 cm and arc AB has length 8 cm: Find the area of the sector OAB. O 10 cm A

8 cm

The area of the sector shown is 2000 m2: a

Find the length of the radius, rounded to one decimal place.

Find the perimeter of the sector, rounded to one decimal place.

60°

Consider a rectangular park that is 420 m long and 280 m wide: a

Calculate the area of the park in square metres.

Calculate the area of the park in hectares. (Note: 10 000 m2 = 1 ha)

280 m

420 m

Extend your thinking 26

A circle has a circumference of 20π. Explain how to ﬁnd the area of the circle.

A formula to ﬁnd the perimeter of a sector when the angle θ is given in degrees is: P = 2r + a

πr

If the perimeter of the following sector is 26.28 m, ﬁnd the size of the angle θ, to the nearest degree.

10 m

Find the area of the sector. Round your answer to the nearest integer.

If we did not have the information from part (a), could we still solve part (b)? Justify your answer. 5.04 Area mathspace.co

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A large 20 m long sprinkler is placed in a crop ﬁeld, with one end ﬁxed and the other end free to move. As it rotates, it waters everything underneath it: a

If the sprinkler has rotated 120° since the farmer left, ﬁnd the area of the crop ﬁeld it has watered. Round your answer to one decimal place.

Explain how the rotation angle of the sprinkler affects the area of the crop ﬁeld it waters.

20 m

120°

A security light shines on a fence at a distance of 16 metres opposite the light bulb: a

Find the area on which this security light shines. Round your answer to one decimal place.

The security light is replaced with a different model that has a beam angle twice as wide. Explain how this change affects the illuminated area.

Five farms around Australia were measured and their dimensions, in metres, were recorded in the table:

100°

Farm

Length (m)

Width (m)

250

Find the area of each farm in square metres.

Which farms have an area of more than 1 ha?

Which farms have an area of less than 1 ha?

300

Which farm has an area of exactly 1 ha?

120

What is the average area of the farms in hectares?

350

150

Katrina is digging a rectangular garden bed to plant some new hedging: a

If the garden bed is 0.6 m wide and 10 m long, calculate the total area of the garden bed.

Each hedge plant ﬁlls an area that is the equivalent of 1.2 square metres. How many hedge plants are needed to ﬁll Katrina’s garden bed?

Sharon has purchased a rectangular piece of fabric that is 10 m long and 6 m wide. Find the area of the largest triangular piece she can cut out from it.

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Area (m2)

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An area measuring 1800 cm2 is to be paved with identical tiles in the shape of parallelograms. Each tile measures 15 cm along the base, and has a perpendicular height of 8 cm: a

Determine the area that each tile covers.

Calculate the number of tiles needed to cover the whole area.

8 cm

15 cm

Some car parks require the cars to park at an angle as shown. The dimensions of the car park are as given, where each individual parking space has a perpendicular length of 5.5 m and a width of 4.2 m.

4.2 m 5.5 m

Determine the area needed to create an angled carpark suitable for 6 cars. 35

A circular lake has a diameter of 40 metres. There’s a small circular island in the lake with a diameter of 8 metres. Calculate the area of the lake water, not including the island.

The design shown is made using a large circle and two smaller circles of diameter 2.5 cm. Find the area of the shaded region, rounded to one decimal place.

2.5 cm

The diagram shows a piece of jewellery made out of gold: a

Find the area of the piece. Round your answer to the nearest whole number.

If the gold costs $6 per square millimetre, ﬁnd the cost of the piece of jewellery.

The diagram shows the sectors of two concentric circles with common centre O, where ∠O = 60°, OR = 6 cm, and OQ = 10 cm:

Calculate the area of the shaded region between the two circular arcs, rounded to one decimal place. O

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5.05 Area of composite shapes After this lesson, you will be able to… • identify the simple shapes that form a composite shape. • calculate the area of a composite shape by adding the areas of its component shapes or subtracting the area of a smaller shape from a larger one. • determine unknown dimensions within a composite shape. • solve practical problems involving the area of composite shapes.

Area of composite shapes A composite shape is one made from multiple smaller shapes. This shape comprises a rectangle, square, triangle, and parallelogram.

This shape combines a square with side length 6 cm and two semicircles, equivalent to one circle with diameter 6 cm. Area = Area of square + Area of circle 6 cm To calculate the area of a composite shape, use one of two methods: • Addition method — Divide the composite shape into basic shapes, calculate each shape’s area, then sum them. • Subtraction method — Calculate the area of a larger shape enclosing the composite shape, then subtract the areas of smaller shapes as needed.

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5.05 Practice questions What do you remember? 1

State the two methods used to calculate the area of a composite shape.

A composite shape is formed by a rectangle and a semicircle. Complete the statement to describe how to ﬁnd the area: Total area = Area of ⬚ + Area of ⬚

Which basic shapes can be combined to form a composite shape? List at least four examples.

True or False: The area of a composite shape is always the sum of the areas of its component shapes. Justify your answer.

Practice Ex 1

For each of these composite shapes: i

What basic shapes make up the composite shape?

Find the area, rounded to two decimal places.

6 cm

8 cm

6 cm 7 cm 6 cm

14 cm

16 cm

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Ex 2

Find the shaded area in the diagrams. Round your answer to one decimal place. a

10 cm 13 cm

15 cm c

d 6 cm

12 cm

30 cm

24 cm 7

Find the shaded area, rounded to two decimal places: a

12 cm

7 cm 16 cm c

18 cm

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12 cm

A window is designed with a rectangular base 120 cm by 80 cm and a semicircular top with diameter 120 cm. Calculate the total area of the window, rounded to two decimal places.

Extend your thinking 9

An annulus is formed by two concentric circles with radii 10 cm and 6 cm. a

Calculate the area of the annulus, rounded to one decimal place.

Derive a general formula for the area of an annulus with outer radius R and inner radius r.

A garden bed is designed as a square with side length 4 m and a circular pond with radius 1 m inscribed in it. Calculate the area of the garden bed excluding the pond, rounded to two decimal places.

A composite shape is formed by a square with side length

cm and four equilateral

triangles, one on each side, forming the net of a square-based pyramid. The area of an equilateral triangle is A =

× side2. Derive an expression for the total area of the net.

5.06 Surface area After this lesson, you will be able to… • calculate the surface area of prisms by summing the area of each face. • apply formulas to find the surface area of rectangular prisms and cubes. • calculate the surface area of a cylinder and a sphere using their respective formulas. • solve practical problems involving surface area of 3D objects.

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Surface area The measure of the total area of the surface(s) of a 3-dimensional shape or object. For example, the surface area of a cube with side length 5 units is 150 square units.

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The surface area of a prism is the sum of the areas of all the faces.

To ﬁnd the surface area of a prism, we can determine the basic shapes we need to add together. The net of a prism will show all of the faces as 2D shapes.

Another way to calculate the surface area for rectangular prisms and cubes is to use the formula. For any faces that are the same, we can ﬁnd the area of one face and multiply it by how many times that face appears. A rectangular prism has surface area: SA = 2 × top + 2 × front + 2 × side = 2lw + 2wh + 2lh = 2 (lw + lh + wh) For length, l, height, h, and width, w. A cube has surface area: SA = 6 × Area of front = 6s2 where s is the side length of the cube. The surface area of a triangular prism can be found by ﬁnding the sum of the areas of the two triangular faces and the three rectangular faces. The units for surface area will be in square units like mm2, cm2, and m2.

Example 1 Find the surface area of the triangular prism shown.

4 cm 3 cm

5 cm

19 cm

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Cylinder An object that has parallel circular discs of equal radius at the ends that are joined by a curved surface. A cylinder has three faces: two identical circular bases and a curved surface that joins the two bases together. The surface area of a cylinder is the sum of the areas of these three faces.

The area of a circle is A = π r2. Now we need to ﬁnd the area of the curved face that connects the two circles. When unrolled, the curved surface is a rectangle. One side length is equal to the height of the cylinder and the other side length is the circumference of the base circle. The area of the curved part of a cylinder is 2π rh, where r is the radius and h is the height.

2π r

To ﬁnd the surface area of the whole cylinder, we need to add the area of the top and bottom circles to the area of the curved part. Surface area of a cylinder = 2 × (Area of circular base) + Area of curved face = 2 × ( π × r2) + 2π r × h

SAcylinder = 2π r2 + 2π rh

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is the radius of the cylinder

is the height of the cylinder

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Example 2 Consider the following cylinder: 7m 4m 4m 7m

Find the area of the curved part of the cylinder, rounded to two decimal places.

Create a strategy Use the curved part area formula of the cylinder: Curved part area = 2π rh

Apply the idea Curved part area = 2π rh

Write the formula

= 2π × 4 × 7 ≈ 175.93 m

Substitute the values of r and h Evaluate and round

Example 3 Consider the cylinder shown in the diagram.

5 cm

6 cm a Find the surface area of the cylinder in square centimetres. Round your answer to one decimal place.

Create a strategy Find the radius then use the formula for the surface area of a cylinder.

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Surface area of a sphere A sphere is a solid consisting of all points at a ﬁxed distance from a central point. The central point of a sphere is often just called its centre. The distance of each point on the sphere from the centre is called the radius, just like for a circle. Unlike the solids we have seen so far, we cannot unwrap a sphere to get a 2D net to calculate its area, so the surface area of a sphere is approached in a different way. Archimedes discovered that the area of the curved face of the cylinder is equal to the surface area of the sphere.

We saw that the curved surface of a cylinder ﬂattens out into a rectangle. Since the height of this cylinder is twice the radius of the inscribed sphere, which is also the radius of the circular ends, the area of the resulting rectangle is 2π r × 2r = 4π r2. We therefore have our formula for the surface area of a sphere.

h = 2r

SA = 4π r2

Example 4 Find the surface area of this sphere. Round your answer to two decimal places.

9 cm

Create a strategy Use the formula for the surface area of a sphere SA = 4π r2.

Apply the idea The radius of the sphere is r = 9 cm. SA = 4π r2

Write the formula 2

= 4π (9)

= 1017.88 cm

Substitute the value of the radius 2

Evaluate

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The surface area of a sphere is directly proportional to the square of its radius.

Practice 6

The cube has a side length of 8 cm: a

Draw the net.

Find the surface area of the cube.

8 cm 8 cm 7

Find the surface area of the following: a

4 cm c

14 cm

10 cm 14 m

4m 6m

5 cm 8 cm

f 11 mm

20 m

12 mm

10 mm

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Ex 1

Find the surface area of the following: a

4 cm

17 cm

3 cm 5 cm

20 cm

23 cm 8 cm

13 cm

5 cm

10 cm

6 cm

28 cm

21 cm 8 cm

24 cm 9

15 cm

A birthday gift is placed inside the box shown: Find the minimum amount of wrapping paper needed to wrap this gift.

8 cm 11 cm

24 cm 10

Sally is building a storage chest in the shape of a rectangular prism. The chest will be 104 cm long, 90 cm deep, and 20 cm high. Find the surface area of the chest.

Find the surface area rounded to two decimal places when necessary. a

11 cm 7 cm 6 cm

12 m

14 cm

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Ex 2

For each pair of cylinders and their nets, ﬁnd the curved surface area of the cylinder. Round your answers to two decimal places. a

8m 4m 6m

10 m

6m 8m

10 m 13

For each cylinder: i

Find the curved surface area of the cylinder, rounded to two decimal places.

Find the total surface area of the cylinder, rounded to two decimal places.

2 cm

3m 6 cm

4m 8m 9m

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Ex 3

Find the surface area of each cylinder, rounded to two decimal places. a

5 cm

13 cm

3m c

23 cm

5 cm 87 cm 10 cm

f 200 mm

55 cm

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800 mm 53 cm

Ex 4

Find the surface area for each sphere, rounded to two decimal places: a

4 cm

35.06 mm

11.5 cm

Calculate the exact surface areas of these spheres with: a

Radius = 3 cm

Radius = 6 cm

Radius =

Radius = 16.7 cm

Diameter = 12 cm

Diameter = 26 mm

For each hemisphere, ﬁnd the total surface area rounded to three decimal places. a

80.12 m

30.08 cm

A cube has surface area of 1350 cm2. a

Find the area of one of the square faces.

Find the length of one side of the cube, rounded to two decimal places.

A cylindrical can of radius 8 cm and height 12 cm is open at one end. Find the external surface area of the can. Round your answer to two decimal places.

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5.07 Surface area of composite solids

After this lesson, you will be able to… • identify the exposed faces of a composite solid. • calculate the surface area of composite solids formed from prisms, cylinders, and hemispheres. • identify and exclude shared surfaces that are not part of the total surface area. • solve multi-step practical problems involving the surface area of composite solids.

Surface area of composite solids A composite solid is a combination of multiple solids. To ﬁnd the surface area, add the areas of all exposed faces, excluding any shared surfaces where solids join to avoid double-counting. For composite solids including cylinders or spheres, use the relevant parts of their surface area formulas, ensuring the correct faces are included. The surface area of a sphere with radius r is: SA = 4π r2 The surface area of a cylinder with radius r and height h is: SA = Area of two circular ends + Area of curved surface = 2π r2 + 2π rh

2π r

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Example 1 Calculate the surface area of the solid shown, rounding your answer to one decimal place.

55 cm

34 cm

Create a strategy Calculate the area of each exposed face, then add these.

Apply the idea The surface area of a closed cylinder is SA = 2π r2 + 2π rh. The hemisphere covers one circular base of the cylinder, so include only one circular base and the curved surface. SA (cylindrical part) = π r2 + 2π rh

rite the modiﬁed formula for the exposed W parts of the cylinder

= π (34)2 + 2π (34)(55)

Substitute r = 34 and h = 55

= 1156π + 3740π

Simplify the terms

= 4896π cm

Combine the terms

The other surface is the curved part of a hemisphere (half a sphere). Write the formula for the hemisphere’s curved surface

Substitute r = 34 and simplify

Evaluate

The total surface area is the sum of the two parts. SATotal = 4896π + 2312π = 7208π ≈ 22 644.6 cm

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Add the surface areas Sum the terms

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Evaluate and round to one decimal place

Example 2 Determine the surface area of this house rounded to two decimal places:

6m 9m 5m

Create a strategy Add all the faces together. The diagram shows which faces are equal. Use Pythagoras’ theorem, c2 = a2 + b2, to determine the width of the roof panels.

Apply the idea Add the areas and multiply by 2

Substitute the values of the dimensions

Evaluate

Side walls = 9 × 6 × 2

Multiply the length and width and multiply by 2

= 108 m

Base = 5 × 9

Evaluate Multiply the length and width

= 45 m

Evaluate

Using Pythagoras’ theorem, we can determine the width of the roof panels. On the diagram, we can see two right-angled triangles where the shorter sides are a = 3 and b =

= 2.5:

Write the Pythagoras’ theorem

Substitute the values

Evaluate

Take the square root of both sides Determine the area of the 2 roof sides

Evaluate Add all the areas

Evaluate and round 2

The surface area of the house is 298.29 m .

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Practice 4

The solid shown is a hemisphere on a cylinder: a

Ex 1

Find the exact surface area of each of these faces: i

The circular base.

The curved part of the cylinder.

iii

The hemisphere.

11 mm 28 mm

Find the total surface area of the solid, rounded to two decimal places.

Calculate the surface area of the solid shown, rounding your answer to two decimal places.

6 cm

14 cm

Ex 2

Determine the surface area of this solid:

5 cm

7 cm

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Calculate the surface area of these composite solids, rounded to two decimal places: a

12 cm 1 cm

1 cm

1 cm 7 cm 4 cm

13 cm

1 cm

4 cm 10 cm c

9 mm

2 cm

16 mm 3 cm

10 mm

3 cm 36 mm

53 mm 10 cm

The solid was constructed by removing a hemisphere from a cylinder: a

Determine the curved surface area of the hemisphere, rounded to two decimal places.

Determine the surface area of the open cylinder including the base. Round your answer to two decimal places.

Determine the total surface area of the shape, rounded to two decimal places.

The solid shown is constructed by cutting out one-eighth of a sphere from a cube. The side length of the cube is 12 cm and the radius of the sphere is 6 cm: a

List all of the faces of the solid with its dimensions.

Find the surface area of the solid.

22 cm

15 cm

6 cm

12 cm

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Consider the hollow pipe shown: a

Determine the external surface area of the curved surface, rounded to two decimal places.

Determine the total surface area of the front and back, rounded to two decimal places.

Determine the internal surface area, rounded to two decimal places.

Determine the total surface area of the solid, rounded to two decimal places.

24 cm

11 cm 10 cm

Extend your thinking 11

A triangular tunnel is made through a rectangular prism: Explain the process of determining the surface area of the solid formed, including the inside of the tunnel.

3 cm

6 cm 8 cm

3 cm 7 cm 12

The base of a water feature is made of a rectangular prism of stone from which a hemisphere has been removed. The entire surface of the solid, except the underside which will sit on the ground, must be polished: a

Find the surface area of the solid, excluding the underside. Round your answer to two decimal places.

The cost of polishing the stone is $214/ m2. Determine the total cost of polishing the water feature base.

3m 4m

A company manufactures nuts shaped like regular hexagonal prisms, with cylindrical boltholes cut out of the centre: a

The total surface area of a nut before the bolthole is drilled is 14.7 cm2. Determine the surface area of a single nut after the bolthole is drilled out, including the inside area of the hole. Round your answer to one decimal place. Each nut that is manufactured requires a zinc coating to prevent corrosion. If 1 kg of zinc is enough to coat a surface area of 1 m2, how many complete nuts can be coated with 1 kg of zinc?

0.5 cm 1 cm 1.2 cm

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Volume and capacity of prisms and cylinders Volume The amount of space occupied by an object. Volume is the amount of space a three-dimensional object occupies. A prism is a three-dimensional object with two identical ends joined by rectangular faces. For all prisms, the volume is calculated as:

V = Abase × h Abase

is the area of the base

is the height of the prism

Triangular prism

Rectangular prism

Height

Cylinder

Height

Base

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A cylinder is very similar to a prism with a circular base. The volume of a cylinder can be found in a similar way: Volume = Area of base × Height Height

Remember the area of a circle is equal to π r2 . So, the volume of the cylinder can be written as: Volume = Area of base × Height of prism = π r2 × h

Base

= π r2 h

Radius

Volume = π r2 h r

is the radius

is the height of the cylinder

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Example 2 Find the volume of the following prism:

A = 35 cm2

8 cm

Create a strategy Apply the volume formula for a prism.

Apply the idea V = Abase × h = 35 × 8 = 280 cm

Write the formula Substitute the base area and height

Evaluate

Example 3 Calculate the volume of a cylinder, rounded to one decimal place, with a radius of 5 cm and a height of 13 cm.

Create a strategy Apply the formula for the volume of a cylinder.

Apply the idea Given a radius of 5 cm and a height of 13 cm, substitute into the formula. V = π r2 h

Write the formula 2

= π × (5) × 13

Substitute the values

= π × 25 × 13

Evaluate the square

= 325π

Calculate the exact value

= 1021.0 cm3

Evaluate and round

Reﬂect and check The exact value is 325π cubic centimetres, which approximates to 1021.0 cm3 when rounded.

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Example 4 Calculate the volume of a solid that is a quarter of a cylinder, rounded to one decimal place, with a radius of 5 cm and a height of 14 cm.

14 cm

5 cm

Create a strategy Multiply the volume of a cylinder by one-quarter.

Apply the idea Given a radius of 5 cm and a height of 14 cm. Multiply the formula by one-quarter

Substitute the values

Evaluate and round

Example 5 A rectangular prism-shaped water tank has a base area of 24 m2 and a height of 3 m. Calculate its capacity in litres, rounded to the nearest litre.

Create a strategy Use the volume formula for a prism and convert cubic metres to litres.

Apply the idea For the volume: V = Abase × h

Write the volume formula for a prism

= 24 × 3

Substitute the base area and height

= 72 m

Evaluate

For the capacity: Capacity = 72 × 1000 = 72 000 L

Convert cubic metres to litres (1 m3 = 1000 L) Evaluate

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Volume and capacity of pyramids, cones, and spheres A pyramid is a shape with a polygon base and triangular faces that meet at a point (apex). Its volume is one-third that of a prism with the same base and height, as the cross-sectional area decreases linearly from base to apex. The volume of a pyramid is:

is the volume of the pyramid

is the area of the pyramid’s base

is the perpendicular height of the pyramid

A cone is similar to a cylinder with a circular base that tapers to a point (apex). Its volume is one-third that of a cylinder with the same base and height, as the cross-sectional area decreases linearly from base to apex. The volume of a cone is:

is the volume of the cone

is the radius of the cone’s base

is the perpendicular height of the cone

The volume of a sphere is:

is the radius of the sphere

Radius

A hemisphere is the word we use to describe half a sphere, so to ﬁnd the volume of a hemisphere, ﬁnd the volume of the sphere and halve it.

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Example 7 Calculate the volume of the square-based pyramid shown. 3 cm 9 cm

Create a strategy

Apply the idea

Use the formula for the volume of a pyramid, where the area of a square is the side length squared.

Given a base side length of 9 cm and a height of 3 cm. Write the formula

Substitute the values

Evaluate

Example 8 Consider the cone:

34 cm

32 cm a Find the perpendicular height of the cone.

Create a strategy

Apply the idea

Apply Pythagoras’ theorem.

Given the slant height of 34 cm and a diameter of 32 cm, the radius is 16 cm. Write Pythagoras’ theorem

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Substitute the values

Evaluate the squares

Subtract 256 from both sides

Take the square root

Evaluate

b Calculate the volume of the cone, rounded to the nearest cubic cm.

Create a strategy

Apply the idea

Use the formula for the volume of a cone.

Given a radius of 16 cm and a height of 30 cm. Write the formula

Substitute the values

Evaluate the square

Evaluate and round

c If the cone is used as a container, calculate its capacity in millilitres.

Create a strategy Use the volume formula for a cone and convert cubic cm to mL.

Apply the idea Capacity = 8042 × 1 = 8042 mL

Convert cubic cm to mL (1 cm3 = 1 mL) Evaluate

Example 9 Calculate the volume of a sphere with a radius of 4 cm, rounded to two decimal places. 4 cm

Create a strategy

Apply the idea

Use the formula for the volume of a sphere.

Write the formula

Substitute the radius

Evaluate the cube

Evaluate and round

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Example 10 A hemispherical bowl has a radius of 10 cm. Calculate its capacity in millilitres, rounded to the nearest millilitre.

Create a strategy Use the volume formula for a sphere, halve it for the hemisphere, and convert to millilitres.

Apply the idea For the volume of the sphere: Write the volume formula for a sphere

Substitute the radius

Evaluate the cube

Evaluate in exact value

For the volume of the hemisphere: Halve the sphere volume for the hemisphere

Simplify

Evaluate and round

For the capacity: Capacity = 2094 × 1 = 2094 mL

370

Convert cubic cm to mL (1 cm3 = 1 mL) Evaluate

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5.08 Practice questions What do you remember? 1

Identify the base shape and the height for these prisms: a

10 cm 12 cm 5 cm 15 cm c

d 11 cm 4 cm

5 cm

3 cm 6 cm 2

4 cm

13 cm

Complete the table for volume and capacity conversions: Volume 3

1 cm

75 cm3 ⬚ cm 1m 3

372

Are these statements true or false?

Capacity ⬚ mL ⬚ mL

500 mL ⬚L

The volume of a prism is calculated by multiplying the area of the base by the height of the prism.

A rectangular prism with the same height and base area as a triangular prism will have the same volume.

The volume of a cone is one-third the volume of a cylinder with the same base and height.

The volume of a sphere is calculated using the formula V = πr3.

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What is the shape of the base of the solid?

7 cm

12 cm

Practice Ex 1

A cylindrical water tank has a radius of 3 m and a height of 6 m. Calculate its capacity in litres, rounded to the nearest litre. 3m

Ex 2

Find the volume of each prism: a

Area = 25 cm2

9 cm

10 m

Area = 30 m2

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Ex 3

Find the volume of each cylinder, rounded to one decimal place: a

6 cm

12 cm

16 cm

Ex 4

40 cm

Find the volume of each partial cylinder, rounded to one decimal place: a

15 cm

12 cm

5 cm

6 cm Ex 5

Find the capacity of each rectangular prism in millilitres: a

7 cm 10 cm

Area of base = 56 cm2

Ex 6

Area of base = 72 cm2

A cylindrical storage silo has a radius of 4 m and a height of 12 m. Calculate its capacity in litres, rounded to the nearest litre, if it is ﬁlled to 80% capacity.

A box is 1.2 m long, 35 cm high, and 50 cm wide. What is the volume of the box in cubic centimetres?

35 cm 50 cm

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1.2 m

Ex 7

Calculate the volume of a cylinder with a radius of 8 cm and a height of 10 cm, rounded to one decimal place.

Find the volume of each pyramid, rounded to two decimal places: a

7 cm 5 cm 8 cm

12 cm

10 cm 14

Find the volume of each cylinder as an exact value: a

b 15 cm

10 mm

30 mm

25 cm

Find the volume of each cone, rounded to one decimal place: a

b 10 cm

4 cm 8 cm

3 cm

Find the exact volume of each cone: a

Radius 5 cm, height 9 cm

Radius 6 cm, height 7 cm

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Ex 8

Consider the cone shown:

26 cm

24 cm

Ex 9

Find the perpendicular height of the cone, rounded to two decimal places.

Calculate the volume of the cone, rounded to the nearest cubic cm.

If the cone is used as a container, calculate its capacity in litres, rounded to the nearest litre.

Find the volume of the sphere, rounded to two decimal places:

10 cm

Ex 10

A cone with a radius of 7 cm and a height of 9 cm is used as a container. Calculate its capacity in millilitres, rounded to the nearest millilitre.

Find the exact volume of each sphere: a

r = 10 m

d = 30 mm

Find the volume of each hemisphere, rounded to three decimal places: a

8 units

b 100 cm

A garden bed is 12 m in length, 5 m in width, and 50 cm in height. Find the volume of soil in m3 needed to ﬁll the garden bed.

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A bottle of orange juice has a volume of 600 cm3. What is its capacity in litres?

A prism has a volume of 480 cm3 and a base area of 80 cm2. Find the height of the prism.

Extend your thinking 25

If a cube and a rectangular prism have the same volume, will they also have the same surface area? Justify your answer with an example.

A storage box needs a volume of 192 780 cm3, a height of 65 cm and a width of 90 cm. Find the length of the box to the nearest cm.

The volume of the triangular prism is 180 m3. Find the value of y.

11 m

7.5 m

ym 8m

A cylindrical lip gloss container has a volume of 50 cm3 and a radius of 2.5 cm. Find the height, rounded to one decimal place.

A cylindrical spa has a radius of 2 m and water depth of 180 cm. Find the volume of water in litres, rounded to two decimal places.

A paperweight is a square-based pyramid and is made of glass. Find the volume of glass needed for 3000 paperweights.

3 cm

5 cm 31

A cylindrical candle has a radius of 50 mm and a height of 0.15 m. Find the exact volume in cubic centimetres.

An aquarium is a rectangular prism with dimensions 130 cm by 70 cm by 80 cm. Find the water needed to ﬁll it to 95% capacity, in litres.

A labourer is building a concrete pathway made of 15 identical rectangular sections. Each section is 2 m long, 1 m wide, and 10 cm deep. Concrete costs $250 per cubic metre, and the labourer charges a ﬂat fee of $50 to lay each section. Calculate the total cost to build the entire pathway. 5.08 Volume and capacity mathspace.co

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5.09 Volume of composite solids After this lesson, you will be able to… • identify the individual solids that form a composite solid. • calculate the volume of a composite solid by adding the volumes of its component solids or subtracting the volume of a removed part. • determine unknown dimensions within composite solids to facilitate volume calculations. • solve practical problems related to the volume and capacity of composite solids.

Volume of composite solids Composite solids are three-dimensional objects formed by combining two or more geometric solids, such as prisms, pyramids, cylinders, cones, spheres, or hemispheres. The volume is calculated by adding or subtracting the volumes of the component solids, depending on whether they are joined or removed. To calculate the volume of a composite solid, use one of two methods: • Addition method - Divide the composite solid into basic solids, calculate each solid’s volume, then sum them. • Subtraction method - Calculate the volume of a larger solid enclosing the composite solid, then subtract the volumes of smaller solids as needed.

This solid is a composite object that combines three cylinders. A smaller cylinder connects two other cylinders, with one attached to each of its ends, creating a single, uniﬁed form. The total volume of the solid is calculated by summing the individual volumes of each of the three cylinders.

This solid is a rectangular prism with a half-cylinder removed. The volume is the prism’s volume minus the half-cylinder’s volume.

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To calculate the volume of composite solids, follow these steps: • Identify the component solids. • Determine the dimensions of each component. • Add or subtract the volumes as required.

Exploration Consider a composite solid formed by attaching a square pyramid to a cube. 1. What challenges might arise if the pyramid’s base is smaller than the cube’s face? 2. Suggest another composite solid combining a prism and a curved solid, and describe how to ﬁnd its volume.

Example 1 A storage unit consists of a square prism with a half-cylindrical roof. Find the volume, rounded to 2 decimal places.

10 m 4m

Create a strategy Add the volume of the square prism to the volume of the half-cylinder.

Apply the idea The half-cylinder has a diameter of 4 m, so the radius is 2 m . Write the formula for a half-cylinder

Substitute r = 2, h = 10

Evaluate and round to 2 decimal places

Square prism volume = s2 h

Write the formula for a square prism

= 4 × 10

Substitute s = 4, h = 10

= 160 m

Evaluate

Total volume = 62.83 + 160 3

= 222.83 m

Add the volumes Evaluate

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Reﬂect and check Alternatively, calculate the base area (square plus semicircle) and multiply by the height. Write the formula

Substitute r = 2

Evaluate

Base area = s2 + 2π

Add square and semicircle areas

= 4 + 2π

Substitute s = 4

= 16 + 2π

Evaluate

Volume = (16 + 2π ) × 10

Multiply by height h = 10

≈ 222.83 m

Evaluate and round

Example 2 A solid consists of a large rectangular prism with a smaller rectangular prism removed. Find the volume.

13 cm 2 cm

6 cm

11 cm

15 cm

Create a strategy Subtract the volume of the smaller rectangular prism from the volume of the larger rectangular prism.

Apply the idea Large prism volume = l × w × h = 13 × 15 × 6 3

= 1170 cm

Small prism volume = l × w × h = 11 × 15 × 2 = 330 cm

Total volume = 1170 − 330 = 840 cm

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Write the formula Substitute l = 13, w = 15, h = 6 Evaluate Write the formula Substitute l = 11, w = 15, h = 2 Evaluate Subtract the volumes Evaluate

Example 3 A composite solid consists of a cone and a hemisphere joined at their bases. Find the volume, rounded to 2 decimal places.

10 cm 4 cm

Create a strategy Add the volumes of the cone and the hemisphere.

Apply the idea Write the formula

Substitute r = 4, h = 10

Evaluate Write the formula

Substitute r = 4

Evaluate Add the volumes

Simplify

Evaluate and round

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Example 4 A truncated cone is formed by removing a smaller cone from the top of a larger cone: 6

2 3 3 a Find the volume of the removed cone as an exact value.

Create a strategy Apply the cone volume formula V =

π r2 h.

Apply the idea The removed cone has radius r = 2 cm and height h = 6 cm . Write the formula

Substitute r = 2, h = 6

Evaluate

b Find the volume of the original cone as an exact value.

Create a strategy Add the heights of the two sections to ﬁnd the total height of the original cone, then apply the volume formula.

Apply the idea The original cone has a radius of r = 3 cm. The total height is the sum of the height of the removed cone (6 cm) and the height of the truncated base (3 cm), which is 9 cm. Write the formula for the volume of a cone

Substitute r = 3 and h = 9

Evaluate

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c Find the volume of the truncated cone, rounded to 2 decimal places.

Create a strategy Subtract the volume of the removed cone from the original cone’s volume.

Apply the idea Volume = 27π − 8π

Subtract the volumes

= 19π cm

= 59.69 cm

Simplify 3

Evaluate and round

Example 5 A composite solid consists of a cylindrical base with a radius of 4 m and height of 6 m, topped with a conical roof with the same base radius and a height of 3 m. Calculate the capacity of the solid if it is ﬁlled to 90% capacity, in litres, rounded to the nearest litre.

Create a strategy Calculate the volume of the cylinder and cone, sum them, adjust for 90% capacity, and convert to litres.

Apply the idea For the volume of the cylinder: Vcylinder = π r2 h

Write the volume formula for a cylinder 2

= π × (4) × 6

Substitute the radius and height

= π × 16 × 6

Evaluate the square

= 96π m3

Calculate the exact volume

For the volume of the cone: Write the volume formula for a cone

Substitute the radius and height

Evaluate the square

Simplify

For the total volume: Vtotal = 96π + 16π 3

= 112π m

= 351.86 m

Sum the volumes Combine like terms

Approximate using π ≈ 3.1416

For the capacity: Capacity = 0.9 × 351.86 × 1000 Adjust for 90% capacity and convert to litres (1 m3 = 1000 L) = 316 674 L

Evaluate and round to the nearest litre

5.09 Volume of composite solids mathspace.co

383

Practice Ex 1

Find the volume of these solids. Round your answers to one decimal place when necessary: a

4 cm

6 cm

10 cm 16 cm

4 cm

6 cm

9 cm 10 cm c 7 cm

10 cm Ex 2

A composite solid consists of a large rectangular prism (16 cm × 14 cm × 8 cm) with a smaller rectangular prism (12 cm × 14 cm × 3 cm) removed. Find the volume.

16 cm 8 cm

3 cm

12 cm Ex 3

14 cm

A composite solid consists of a cone (radius 5 cm, height 12 cm) and a hemisphere (radius 5 cm) joined at their bases. Find the volume, rounded to one decimal place.

12 cm 5 cm

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Ex 4

Ex 5

A truncated cone is formed by removing a smaller cone from the top of a larger cone. The large cone has a radius of 6 cm and a height of 18 cm. The removed cone has a radius of 4 cm and a height of 12 cm: a

Find the volume of the removed cone as an exact value.

Find the volume of the original cone as an exact value.

Find the volume of the truncated cone, rounded to one decimal place.

12 cm 18 cm 4 cm

6 cm

A composite solid consists of a cylindrical base with a radius of 5 m and height of 8 m, topped with a conical roof with the same base radius and a height of 4 m. Calculate the capacity of the solid if it is ﬁlled to 90% capacity, in litres, rounded to the nearest litre.

Two identical rectangular prisms, each with dimensions 10 cm × 8 cm × 6 cm, are joined with an overlap of 4 cm × 8 cm × 6 cm. Find the volume of the composite solid.

8 cm 10 cm 4 cm

6 cm

8 cm 10

Two identical cylinders, each with radius 7 cm and height 10 cm, are joined at their bases. A cylindrical hole with diameter 2 cm is cut through the centre along the entire length: a

Find the volume of the composite solid before the hole is cut, in exact form.

Find the volume of the solid after the hole is cut, rounded to one decimal place.

7 cm

2 cm

10 cm 20 cm

A composite solid consists of a cube with side length 8 cm and a hemisphere with radius 4 cm attached to one face. Calculate the capacity of the solid if it is ﬁlled to 100% capacity, in millilitres, rounded to the nearest millilitre.

A composite solid consists of a rectangular prism with dimensions 12 m × 10 m × 6 m and a cylindrical hole with radius 2 m and height 6 m removed through its length. Calculate the capacity of the solid if it is ﬁlled to 80% capacity, in litres, rounded to the nearest litre.

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A water storage tank consists of a cylindrical base with radius 4 m and height 5 m, topped with a conical section of radius 4 m and height 2 m. Calculate the capacity of the tank if it is ﬁlled to a depth of 5 m, in litres, rounded to the nearest litre.

A chemical reactor consists of a large cylindrical vessel with radius 3 m and height 6 m, containing a smaller concentric cylindrical core with radius 1 m and height 6 m:

Find the volume of the outer cylinder as an exact value.

Find the volume of the inner cylindrical core as an exact value.

Calculate the capacity of the reactor if it is ﬁlled to 90% capacity, in litres, rounded to the nearest litre.

A decorative fountain consists of a square prism with base 6 m × 6 m and height 4 m, a cylindrical column with radius 1 m and height 4 m, and a hemispherical bowl with radius 1 m. Water ﬁlls the bowl and the column to a height of 2 m. Calculate the capacity of the ﬁlled portion, in litres, rounded to the nearest litre.

Extend your thinking 16

A storage shed is in the shape of a rectangular prism (12 m × 10 m × 8 m), and the greenhouse is in the form of a triangular prism (base 10 m × 6 m, height 12 m). Find the volume of the entire building.

12 m

10 m 8m 12 m 10 m 17

A cylindrical tank with diameter 4 m is placed in a 2.5 m deep circular hole so that there is a gap of 50 cm between the side of the tank and the hole. The top of the tank is level with the ground: a

What volume of dirt was removed to make the hole? Give your answer to the nearest cubic metre.

Find the capacity of the tank to the nearest litre.

2.5 m

50 cm

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Alice and Bob share a half-cylinder-shaped ﬂower bed (radius 2 m, length 10 m) and divide it into two parts with a 1 : 3 ratio, where Alice gets the smaller part and Bob gets the larger part:

a b

Calculate the volume of Alice’s and Bob’s portions, rounded to two decimal places.

10 m

2m Compost costs $12.50 for a 103 litre bag. How much does Alice and Bob each need to pay if they split the cost in the ratio of ownership? Round your answer to the nearest cent.

5.10 Trapezoidal rule After this lesson, you will be able to… • apply the trapezoidal rule to approximate the area under a curve or of irregular shapes using single or multiple applications. • identify the correct values for variables in the trapezoidal rule formula from a diagram or description. • distinguish between the first, last, and intermediate measurements in the multiple application formula. • solve practical problems involving area, volume, and capacity using the trapezoidal rule.

Trapezoidal rule for a single trapezium Trapezoidal rule A method of approximating the area of an irregular shape, or a region bounded by a curve and an axis, by slicing the area up into trapeziums (trapezia) of equal height. Parallel Two distinct lines, rays or line segments in the same plane that have no points of intersection and so necessarily have the same gradient (slope). Perpendicular Two lines, rays, line segments, vectors, planes or other objects that intersect at a 90° angle (a right angle).

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Trapezoidal rule for multiple trapeziums To approximate the area of an irregular shape, divide it into multiple trapeziums. For n trapeziums, each of width h, the trapezoidal rule is:

is the approximate total area

is the width of each trapezium

is the height of the shape at the nth yn vertical line This formula can be understood by applying the trapezoidal rule for a single trapezium multiple times. Consider an irregular shape divided into n trapeziums, each with width h. The total area is the sum of the areas of each trapezium. For the ﬁrst trapezium (between y1 and y2), the area is ( y1 + y2). For the second (between y2 and y3), it is (between yn and yn + 1), which is

( y2 + y3), and so on, up to the last trapezium

( yn + yn + 1). Summing these areas: Sum the areas of each trapezium

Factor out

Combine like terms, noting y2 to yn appear twice This derivation shows that the formula accounts for each intermediate height being used twice, except for the first and last heights.

Interactive exploration Discover this concept in action online

mathspace.co

Example 2 A garden is 49 m long. At 7 m intervals, the width of the garden is given by these measurements in metres: 0, 2.9, 5.2, 6.6, 5.6, 4.3, 3, 2.5 Using the trapezoidal rule, approximate the area of the garden rounded to two decimal places.

Create a strategy Apply the trapezoidal rule for 7 trapeziums with width h = 7 m, using the formula A≈

( y1 + 2y2 + 2y3 + ⋯ + 2yn + yn + 1).

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Explain how the trapezoidal rule approximates the area of a trapezium.

Match the terms with their deﬁnitions in the context of the trapezoidal rule: a

The length of the ﬁrst parallel side

The perpendicular distance between parallel sides

iii

The length of the last parallel side

Practice Ex 1

Use one application of the trapezoidal rule to approximate these areas. Give your answer in square centimetres: a

b 16 cm

22 cm

28 cm

12 cm

19 cm

17 cm

15 cm Ex 2

Use the trapezoidal rule to approximate the area of the garden in square metres.

10 m

8m 4m 7

A river has its depths marked out at equal intervals of 9 m. The depths are given by these measurements in metres: 0, 12, 14, 17, 5, 0 Find the approximate area of the cross-section of the river.

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These pieces of land have straight boundaries on the east, west and south borders and follows a creek at the north. For each of these pieces of land: i

Approximate Area 1 using one application of the trapezoidal rule.

Approximate Area 2 using one application of the trapezoidal rule.

iii

Hence, ﬁnd the approximate area of the piece of land.

Is the actual area of the land greater than or less than this calculated area? Explain your answer

206 m

596 m

252 m Area 1

322 m

Area 1

Area 2

668 m

452 m 226 m

576 m

530 m

334 m

226 m

334 m

The diagram shows the cross-section of a river at the exact point where it feeds into a dam. The depths of the river are marked at 2 m intervals. 2m

0.5 m

0.5 m 5.6 m

7.3 m

Using three applications of the trapezoidal rule, approximate the area of the cross-section of the river rounded to one decimal place. 10

Use four applications of the trapezoidal rule to approximate the area of the cross-section of this river:

6m 7m

11 m 13 m

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393

This shape has measurements given in metres: Use the trapezoidal rule to ﬁnd the area in hectares (1 ha = 10 000 m2). Round your answer to two decimal places. 120 140

40 12

170 160 150 100

The elevation values of a mountain are recorded at equal intervals of 250 m. The heights are shown in the diagram:

3548 m

1160 m

1558 m 510 m 55 m

55 m 250 m Find the approximate area of the cross-section of the mountain. 13

This piece of land has straight boundaries on the east, west and south borders and follows a creek at the north. The land has been divided into two sections so use the trapezoidal rule to approximate the area: a

Find the approximate area of the piece of land by using two applications of the trapezoidal rule. Give your answer in square metres.

During a heavy storm, 35.2 mm of rain fell. Find the volume of water that lands on this property in cubic metres. Give your answer rounded to the nearest cubic metre.

318 m

364 m

322 m

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398 m

322 m

Find the approximate area of the piece of land by using two applications of the trapezoidal rule. Give your answer in square metres.

During a heavy storm, 15.5 mm of rain fell. Find the volume of water that lands on this property. Give your answer rounded to the nearest cubic metre.

676 m

442 m

396 m

Area 1

Area 2

724 m 362 m 15

A surveyor provided this diagram with measurements for a property she was mapping out: a

Find the approximate total area of the property by using three applications of the trapezoidal rule. Give your answer in square metres.

82 m 87 m

The average weekly rainfall is 34 mm. Find the total volume of water that falls on the land in cubic metres. Give your answer in cubic metres, rounded to two decimal places.

89 m

43 m 43 m 43 m

41 m 16

362 m

99 m

A surveyor provided this diagram with measurements for a property she was mapping out: a

Find the approximate total area of the property by using three applications of the trapezoidal rule. Give your answer in square metres.

82 m

The surveyor is reading a meteorological report that lists the average monthly rainfall in the region. According to the report, August sees 13.8 cm of rainfall on average.

Find the volume of rainfall that the surveyor can expect to fall over the property next August. Give your answer to the nearest cubic metre. c

81 m

83 m

46 m 46 m 46 m

44 m

88 m

Convert the volume of rainfall from part (b) into litres, given that 1 m3 = 1000 L. Round your answer to the nearest litre.

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395

This diagram shows the measurements of a plot of land: 10.3 m

10.3 m

21.8 m

21.9 m

21.1 m

21.2 m

22.7 m

Find the approximate total area of the block of land by using four applications of the trapezoidal rule. Give your answer in square metres, rounded to one decimal place.

The average weekly rainfall is 42.2 mm. Find the total volume of water that falls on the land in cubic metres. Give your answer rounded to one decimal place.

Convert the total volume of water from part (b) into litres, given that 1 m3 = 1000 L. Round your answer to one decimal place.

A large company recently purchased a block of land to build a warehouse on. This is a surveyor’s diagram of the block: a

Find the approximate total area of the block of land by using ﬁve applications of the trapezoidal rule. Give your answer in square metres, rounded to one decimal place.

72.1 m

The average annual rainfall is 900 mm. Find the total volume of water that falls on the land in cubic metres. Give your answer rounded to two decimal places. Convert the total volume of water from part (b) into litres, given that 1 m3 = 1000 L. Round your answer to the nearest litre.

72.4 m

19 m

66.4 m

19 m

70.1 m

19 m

68.3 m

19 m

65.3 m

19 m 50 m

Consider this tent: a

Use the trapezoidal rule to ﬁnd the area of the front of the tent. Give your answer in square metres.

If the tent is 8 m deep, approximate the volume of the tent. Give your answer in cubic metres. 3.2 m 1.8 m 2.1 m

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2.1 m

Five measurements across a pool were taken at 3 m intervals: a

Using four applications of the trapezoidal rule, ﬁnd the area of surface of the pool. Give your answer in square metres, rounded to one decimal place.

3m 2m

2.5 m

1.2 m

The pool is a constant 2 m deep. Find the capacity of the pool. Round your answer to the nearest kilolitre.

Consider the driveway with these measurements: a

Use two applications of the trapezoidal rule to ﬁnd the area of the driveway. Round your answer to the nearest square metre.

Find the approximate cost of sealing the driveway if sealer costs $45 per litre and 1 L covers 11 m2.

3.5 m

A section of the ground is removed for a road to be built. At every 7 m, the area of the cross-sections are measured in square metres, as follows: 0, 2.8, 3.3, 4.4, 4.1, 2.3, 0 Using the trapezoidal rule, approximate the volume of the ground removed. Give your answer in cubic metres, rounded to one decimal place.

A boat sits partly in the water. At every 2 m, the area of the cross-section of the boat submerged underwater is measured in square metres, as follows: 1.94, 2.98, 3.3, 4.13, 2.63, 1.14, 0.86 Using the trapezoidal rule, approximate the volume of the submerged part of the boat. Give your answer in cubic metres, rounded to two decimal places.

Extend your thinking 24

Consider this cross-section of a river: a

Use the trapezoidal rule to approximate the area of the cross-section. Give your answer in square metres, rounded to one decimal place. The river ﬂows at 0.6 m/s. Approximate the volume of water that passes through the cross-section after 9 seconds. Give your answer in cubic metres, rounded to two decimal places.

1.5 m 2.2 m

2.4 m

2.5 m 2.7 m

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397

Consider this shape: a

c 26

Find the area of the shape using one application of the trapezoidal rule. Give your answer in square metres.

14 m

Find the area of the shape using two applications of the trapezoidal rule. Give your answer in square metres.

18 m

Which approximation is more accurate? Justify your answer.

The diagram shows the cross-section of a river at the exact point where it feeds into a dam. The depths of the river are marked at 2 m intervals: 2m

2m 0.5 m

0.5 m

3.9 m

11 m

5.5 m

Using three applications of the trapezoidal rule, approximate the area of the crosssection of the river to one decimal place.

Water ﬂows down the river at a rate of 3.4 m/s. Find the volume of water that ﬂows into the dam each day. Round your answer to the nearest cubic metre.

A children’s toy company is creating couches that are ﬁlled with foam in the shape shown:

36 cm

27 cm

22 cm 22 cm 23 cm 71 cm

30 cm 39 cm

43 cm 76 cm 79 cm

69 cm

23 cm

398

Use the trapezoidal rule to approximate the area of the cross-section of the couch.

To minimise the cost of production, the company calculates that each couch should use 1 553 510 cm3 of foam to ﬁll the couch. Find the length of each couch to the nearest centimetre.

Mathspace New South Wales – Year 11 Standard mathspace.co

5 Chapter review 1

A ﬂagpole is 12 m tall. A support wire is attached from the top of the ﬂagpole to a point on the ground 5 m from the base of the pole. What is the length of the support wire? A

10 m

12 m

16 m

24 m

P = 48 m

23 cm2

46 cm2

60 cm2

120 cm2

14 m2

20 m2

40 m2

50 m2

14 cm

21 cm

28 cm

49 cm

5.0 cm

7.0 cm

7.1 cm

10.0 cm

What is the area of a circle with a radius of 3 cm, using π ≈ 3.14 and rounded to one decimal place? 18.8 cm2

28.3 cm2

37.7 cm2

56.5 cm2

What is the perimeter of a sector with a radius of 10 cm and an angle of 60°, rounded to one decimal place? 20.0 cm

26.3 cm

30.5 cm

36.7 cm

What is the volume of a rectangular prism with length 5 m, width 3 m, and height 2 m? A

169 m

What is the length of the diagonal of a square with a side length of 5 cm, rounded to one decimal place?

A 9

What is the perimeter of a square with a side length of 7 cm?

A 8

17 m

What is the area of a rectangle with a length of 10 m and a width of 4 m?

A 7

A triangular section of a quilt has a base length of 15 cm and a perpendicular height of 8 cm. What is its area?

A 6

13 m

A 5

A square garden bed has a perimeter of 48 m, as shown in the diagram where P represents the perimeter. What is the length of one side of the garden bed?

A 4

11 m

10 m3

15 m3

20 m3

30 m3

Using the trapezoidal rule with two applications, approximate the area of a shape with heights 0 m, 4 m, 6 m, and 0 m at intervals of 3 m. Round to one decimal place. A

12.0 m2

18.0 m2

24.0 m2

36.0 m2

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399

What is the area of a triangle with a base of 6 cm and a height of 4 cm? A

18 cm2

24 cm2

13 m

26 m

40 m

64 m

96 cm2

What is the surface area of a cube with a side length of 4 cm? A

12 cm2

What is the perimeter of a rectangle with length 8 m and width 5 m? A

10 cm2

24 cm2

48 cm2

64 cm2

Find the length of the diagonal for each shape, rounded to one decimal place where appropriate: a

10 cm

15 mm

7 mm 10 cm

20 m

10 cm 4 cm

15 m 15 cm

400

A factory produces thin plastic right-angled triangle components: a

If the components need a decorative edging strip costing 6 cents per cm, how much, in dollars, will it cost to edge 250 components?

If the components need a special coating costing 3 cents per cm2, how much, in dollars, will it cost to coat 250 components?

Mathspace New South Wales – Year 11 Standard mathspace.co

25 cm

60 cm

Find the perimeter of each sector. Round your answer to two decimal places: a

A sector with a radius of 15 cm and angle of 95°.

A sector with a radius of 5.5 cm and angle of 70°.

12 cm

270°

4.6 cm

290°

A rectangular sports ﬁeld is 100 m long and 60 m wide. A running track is built around it, with posts every 4 m. How many posts are needed?

A rectangular garden has a perimeter of 56 cm. Find all possible whole number dimensions (length and width) of the rectangle. Explain your method.

An outline of a park area is shown: a

Find the value of y (the length of the slanted side).

Find the perimeter of the park area.

30 m ym

23 m

39 m 20

A farmer wants to build a fence around the entire perimeter of his land, shaped as shown. The fencing costs $40 per metre:

Calculate the value of x, rounded to two decimal places.

Calculate the value of y, rounded to two decimal places.

How many metres of fencing does the farmer require?

How much will it cost to build the fence?

x F 2m A

G 7m

13 m 21

Calculate the perimeter of the composite shape shown:

12 mm 7 mm

5 mm

Chapter 5 review mathspace.co

401

In the diagram, O is the centre of a circle with radius 15 cm. Arc MN has a length measuring 6π cm: a

Determine the angle ∠MON subtended by the arc MN at the centre O, in degrees.

Determine the exact area of sector OMN.

6π cm

15 cm O 23

A large 30 m long automated irrigation sprinkler arm is ﬁxed at one end and rotates to water a circular area of crops: a

If the sprinkler rotates through an angle of 135°, calculate the area of the crop ﬁeld it has watered. Round your answer to one decimal place.

The farmer adjusts the sprinkler to cover a smaller angle but wants to ensure the same total area is watered by increasing the length of the sprinkler arm.

30 m

135°

If the new angle is 90°, what would the new length of the sprinkler arm need to be, to water the same area as calculated in part (a)? Round your answer to one decimal place. 24

For the composite shape shown, made of a central rectangle and two semicircles on its shorter sides. The rectangle has a length of 14 cm. The diameter of each semicircle (and thus the width of the rectangle) is 6 cm: a

What are the basic shapes that make up the composite shape?

Find the area rounded to two decimal places.

14 cm 6 cm

An annulus is formed by two concentric circles with radii 15 cm and 9 cm: a

Calculate the area of the annulus, rounded to one decimal place.

If the cost of gold leaf to cover the annulus is $0.75 per square centimetre, what is the total cost to cover the annulus?

A composite shape is formed by a square with side length

units and four equilateral

triangles, one on each side, forming a star. The area of an equilateral triangle is given by the formula A =

× (side length)2. Derive an expression for the total area of this star shape in

terms of x and y.

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Determine the surface area of the prisms shown: a

19 m

5m 7m 11 cm c

8 cm 6 cm 10 cm

9 cm

15 cm 25 cm 12 cm

21 cm

16 cm

7 cm 71 cm 12 cm

6.8 m

9.7 cm

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403

For the cylinder shown:

5 cm

Find the curved surface area of the cylinder, rounded to two decimal places.

Find the total surface area of the cylinder, rounded to two decimal places. 10 cm

Write an expression for the surface area of this pentagonal prism. Justify your formula.

L Ab s

404

The solid shown is made up of a cone and a sphere, joined at their circular bases. The cone has a radius of 4 cm and a height of 10 cm. The sphere also has a radius of 4 cm: a

Calculate the slant height of the cone, rounded to two decimal places.

Find the surface area of the exposed part of the cone, rounded to two decimal places.

Find the surface area of the exposed part of the sphere, rounded to two decimal places.

Find the total surface area of the solid, rounded to the nearest square centimetre.

A sculptor creates an artwork by taking a cube with side length 15 cm and removing a square-based pyramid from its interior. The pyramid’s base is one face of the cube, and its apex is at the centre of the opposite face: a

Find the slant height of the triangular faces of the internal pyramid, rounded to two decimal places.

Find the surface area of the composite solid. Round your answer to one decimal place.

Mathspace New South Wales – Year 11 Standard mathspace.co

10 cm

4 cm

15 cm

A small square-based pyramid was removed from the top of a larger square-based pyramid to form a frustum: a

Find the length of the slant height of the trapezoidal sides of the frustum formed. Round your answer to two decimal places.

8 cm 6 cm

Find the total surface area of the frustum, rounded to one decimal place.

16 cm

18 cm 33

Find the volume of each prism shown: a

b 7.5 cm

7 cm

9 cm c

19 cm

4 cm

6 cm

15 cm

16 cm

20 cm 24 cm 6.5 cm 34

Consider the cone shown, which has a diameter of 20 m and a slant height of 29 m:

29 cm

10 cm

Find the perpendicular height of the cone.

Calculate the volume of the cone, rounded to the nearest cubic metre.

Chapter 5 review mathspace.co

405

A glass company manufactures decorative square-based pyramid paperweights. The base is 6 cm by 6 cm, and the height of the pyramid is 4 cm. Find the volume of glass needed to make 2000 such paperweights.

4 cm 6 cm 6 cm

Find the volume of the composite solid shown, which is a cylinder (radius 7 cm, height 12 cm) with a hemisphere (radius 7 cm) on top. Round your answer to one decimal place.

12 m 7m

The diagram shows the cross-section of a canal at the exact point where it meets a reservoir. The depths of the canal are marked at 2.5 m intervals. The depths (from left to right) are 0 m, 1.5 m, 2.8 m, 3.5 m, 2.2 m, 0 m. 2.5 m

2.5 m

0m 1.5 m

2.8 m

3.5 m

2.2 m

Using ﬁve applications of the trapezoidal rule, approximate the area of the cross-section of the canal rounded to one decimal place.

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Mathspace New South Wales – Year 11 Standard mathspace.co

The following piece of land has straight boundaries on three sides and follows a winding river on the north side:

250 m

389 m

310 m

190 m

200 m

Approximate the area of Section 1 using one application of the trapezoidal rule.

Approximate the area of Section 2 using one application of the trapezoidal rule.

Hence, ﬁnd the approximate total area of the piece of land.

Is the actual area of the land likely greater than or less than this calculated area, based on the curvature shown in the diagram?

Did you know?

Perimeter, area, and volume are essential for sustainable building design! Engineers calculate area to install solar panels efficiently and volume to design proper ventilation systems. By mastering these measurements, they can create eco-friendly buildings that save energy and resources.

Chapter 5 review mathspace.co

407

Big ideas Income calculation and modelling enable accurate financial management by integrating diverse earnings components and spreadsheet tools to analyse and plan personal and business finances.

6 Earning money Chapter outline 6.01 6.02 6.03 6.04 6.05

Salaries and wages Overtime Commission, piecework and royalties Government allowances Annual leave loading Investigation: Spreadsheets and income Chapter 6 review

410 419 428 437 458 464

6.01 Salaries and wages After this lesson, you will be able to… • define and differentiate between a salary and a wage. • calculate earnings based on an hourly wage rate, including consideration of hours worked and unpaid breaks. • convert an annual salary into weekly, fortnightly, or monthly earnings. • convert weekly, fortnightly, or monthly earnings into an equivalent annual salary. • solve problems involving salaries and wages in practical contexts.

Salaries and wages Wage An amount of money that is paid for work or services, based on the time spent or the work done. It is usually calculated by the hour, day or week, and paid at regular intervals, such as daily, weekly or fortnightly.

In Australia, a normal full-time working week is 38 hours. Often a higher rate of pay, known as overtime, is paid for work completed above the standard 38 hours. Examples of workers who are paid wages include retail and hospitality workers, clerical workers, hairdressers, mechanics and workers in the manufacturing and construction industries. Most countries in the world specify a minimum wage for workers to ensure that everyone is paid appropriately for the work they do. In Australia, as of 1 July 2024, the current national minimum wage is $24.103 per hour, or $915.90 for a 38-hour week.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Exploration Time conversion summary: = 4 quarters (of 3 months each) = 12 months = 26 fortnights

1 year

= 52 weeks = 365 days 1 fortnight

= 2 weeks

1 month

≠ 4 weeks

1. Are there exactly 52 weeks in a year? Multiply 7 days by 52 weeks and compare the result to the number of days in a standard year (365 days). What do you notice about the difference? 2. Why is 52 weeks used as a standard measure for a year rather than the precise number you calculated in the previous question? 3. Explain why one month is not exactly equal to four weeks.

Salary A ﬁxed regular payment made by an employer to an employee for their work or services. The amount of salary paid is usually expressed as an annual sum, which is then divided into equal payments over the course of the year such as weekly, fortnightly or monthly.

Examples of jobs that earn a salary include teachers, architects, engineers and managers.

Example 1 Calculate the annual salary of a worker whose weekly pay is $307. Use the approximation of 52 weeks in a year.

Create a strategy Multiply the worker’s weekly wage by the number of weeks there are in a year.

Apply the idea Salary = 307 × 52 = $15 964

Write the equation Evaluate

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Example 2 Jordan works from 9:00 a.m. to 5:00 p.m., Monday to Friday. He has an unpaid break of 45 minutes. If his hourly rate is $24.60: a Determine how many hours Jordan gets paid for each day.

Create a strategy Determine the number of hours between 9:00 a.m. to 5:00 p.m., then subtract 45 minutes.

Apply the idea There are 8 hours between 9:00 a.m. and 5:00 p.m. To subtract the lunch break, recall that 45 minutes = 0.75 hours Number of paid hours = 8 − 0.75 = 7.25 hours

Write the equation Evaluate

Reﬂect and check Alternatively, consider subtracting the lunch break off the ﬁnish time. Jordan gets paid from 9:00 a.m. to 4:15 p.m. which is 7 hours and 15 minutes or 7.25 hours.

b Calculate his daily wage.

Create a strategy Multiply the number of hours worked by the rate of pay.

Apply the idea Daily wage = 7.25 × 24.60 = $178.35

Write the equation Evaluate

c Calculate his weekly wage.

Create a strategy Jordan works Monday to Friday, so 5 days a week. Multiply his daily wage found in part (b) by 5.

Apply the idea Weekly wage = 178.35 × 5 = $891.75

Write the equation Evaluate

Reﬂect and check Alternatively, calculate the number of hours he works each week, 7.25 × 5 = 36.25 hours. Then multiply by his hourly rate, 36.25 × 24.60 = $891.75

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Example 3 Bill earns a salary of $41 000 per year: a What is Bill’s weekly income rounded to the nearest cent?

Create a strategy Divide the annual salary by the number of weeks in a year.

Apply the idea There are 52 weeks in a year. Write the equation

Evaluate and round

b What is his fortnightly income rounded to the nearest cent?

Create a strategy Divide the annual salary by the number of fortnights in a year.

Apply the idea There are 26 fortnights in a year. Write the equation

Evaluate and round

Reﬂect and check Alternatively, there are 2 weeks in a fortnight. Multiply his weekly wage by 2. 788.46 × 2 = $1576.92 c What is his monthly income rounded to the nearest cent?

Create a strategy Divide the annual salary by the number of months in a year.

Apply the idea There are 12 months in a year. Write the equation

Evaluate and round

Reﬂect and check It might be tempting to multiply Bill’s weekly wage by 4 to get his monthly income, but this method is inaccurate because not every month consists of exactly four weeks.

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d Assuming a working week of 38 hours, what is Bill’s hourly income rounded to the nearest cent?

Create a strategy Divide the annual salary by the number of hours worked in a year.

Apply the idea There are 52 weeks in a year, each with 38 working hours. Therefore, there is a total of 52 × 38 = 1976 working hours in a year. Write the equation

Evaluate and round

Reﬂect and check Alternatively, divide the weekly wage found in part (a) by 38. Write the equation

Evaluate

Example 4 Joe works in a cafe. He is paid a wage of $25.10 per hour: a How much would Joe earn for working a 35-hour week?

Create a strategy Multiply the number of hours worked by the rate of pay.

Apply the idea Weekly earnings = 35 × 25.10 = $878.50

Write the equation Evaluate

b What would he earn over a full year?

Create a strategy Multiply Joe’s weekly earnings by 52, the number of weeks in a year.

Apply the idea Annual income = $878.50 × 52 = $45 682

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Write the equation Evaluate

6.01 Practice questions What do you remember? 1

Identify how many weeks are there in a year.

How many pay periods are there in a year if an employee is paid: a

Weekly?

Fortnightly?

Monthly?

What is the formula to calculate pay per period from an annual salary?

Are these statements true or false? a

An annual salary of $52 000 paid weekly is $1000 per week.

A fortnightly pay of $2000 results in an annual salary of $48 000.

A monthly pay of $4000 equals an annual salary of $48 000.

Practice

Ex 1

Noah’s annual salary is $90 282. If Noah gets paid monthly, calculate his monthly income.

Calculate the annual income in the following scenarios. Use the approximation of 52 weeks in a year:

Ex 2

Ex 3

A worker whose weekly pay is $320.

$20 per hour, working 35 hours per week.

Find the weekly wage for the following scenarios. Use the approximation of 52 weeks in a year: a

Christa earns $21 per hour working as a receptionist. She works 19 hours per week.

Michael’s annual salary is $73 398.

Sarah’s monthly income is $1586.

Elena works from 8:00 a.m. to 4:00 p.m., Monday to Friday. She has an unpaid break of 30 minutes. If her hourly rate is $26.80: a

Determine how many hours Elena gets paid for each day.

Calculate her daily wage.

Calculate her weekly wage.

Lucy is an ambulance oﬃcer. If she earns $979.60 per week, calculate her income for 15 weeks.

Roxanne’s monthly salary is $8142.50. If Roxanne gets paid fortnightly, calculate her fortnightly pay rate. Use the approximation of 52 weeks in a year.

Sarah earns a salary of $37 000 per year, and works 38 hours per week. Calculate:

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Her weekly income

Her fortnightly income

Her monthly income

Her hourly income

Mathspace New South Wales – Year 11 Standard mathspace.co

Han earns $18.00 per hour from 9:00 a.m. to 5:00 p.m. (business hours), and $27.00 per hour outside business hours. Calculate Han’s income for each shift: 9:00 a.m. to 5:00 p.m.

11:00 a.m. to 4:00 p.m.

6:00 a.m. to 4:00 p.m.

10:00 a.m. to 11:00 p.m.

5:00 a.m. to 10:00 p.m.

Ex 4

Neil works from 10:00 a.m. to 3:00 p.m., Monday to Friday, and from 8:00 a.m. to 4:00 p.m., on weekends. His weekly income is $1927: a

How many hours does Neil work per week?

Calculate his hourly rate.

Buzz has a job that pays him $27 per hour. He has a working week of 38 hours and works 52 weeks in a year. Calculate: a

His weekly income

His fortnightly income

His annual salary

His monthly income

Dave’s contract is based on a 23-hour working week and an hourly rate of $18. Valentina’s contract is based on an annual salary of $28 543. Use the approximation of 52 weeks in a year: a

What is Dave’s weekly income?

What is Valentina’s weekly income?

Who has the higher weekly income?

Find the difference in their weekly incomes.

Jack works as a receptionist at a medical centre and earns $22.04 per hour. His weekly timesheet is displayed in the given table. He takes a 30-minute unpaid lunch break each day. Calculate Jack’s weekly wage.

Start time

Finish time

Monday

9:00 a.m.

5:00 p.m.

Tuesday

8:30 a.m.

2:30 p.m.

Wednesday

8:30 a.m.

3:00 p.m.

Thursday

10:00 a.m.

6:00 p.m.

Friday

9:00 a.m.

4:30 p.m.

Aoife is paid according to a ﬂat daily wage. If she earned $2910 in September when she worked for 20 days, calculate how much would she earn in October if she works for 22 days.

Extend your thinking 18

Robert is an architect who works, on average, 43 hours a week. He earns a salary of $62 070. Calculate his equivalent hourly rate.

Hermione wants to calculate what her new weekly income would be if her annual salary of $63 000 was increased by 4.3%: a

Calculate the new annual salary.

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Jimmy is considering three different job offers, and wants to choose the one that will pay him the most, but they are all given at different rates: • Offer 1: $4142.50 per month • Offer 2: $621.12 per week • Offer 3: $48 000.00 per year Which job offer should Jimmy accept? Explain your answer.

Maria is considering three different job offers, and wants to choose the one that will pay her the most, but they are all given at different rates: • Offer 1: $3965.00 per month • Offer 2: $1023.00 per week • Offer 3: $47 008.00 per year Which job offer should Maria accept? Explain your answer.

For each scenario: i

Find the hourly wage.

Calculate the increase in the annual salary if the equivalent hourly wage was increased by $4. Use the approximation of 52 weeks in a year.

Han earned $53 030 in one year, and worked an average of 25 hours per week.

Dimitri earned $53 625 in one year, and worked an average of 25 hours per week.

An employee earns $4200 monthly and receives a bonus of $3000 at year-end. If they switch to weekly payments with the bonus evenly distributed, calculate their weekly earnings. Compare this to their original monthly pay without the bonus.

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6.02 Overtime After this lesson, you will be able to… • define overtime and penalty rates. • calculate time and a half and double time rates from a normal hourly rate. • calculate total earnings for a work period that includes normal hours and hours at various penalty rates. • determine an equivalent hourly rate from an annual salary to calculate overtime payments. • calculate weekly, fortnightly, monthly, and yearly earnings including regular overtime.

Overtime In Australia, a normal full-time working week is 38 hours. Often a higher rate of pay, known as overtime, is paid for work completed above the standard 38 hours. Overtime Time worked before or after regular scheduled working hours. A higher rate of pay is often earnt during this time, known as penalty rates. Penalty rate A rate of pay determined by an award, higher than the usual rate, in compensation for working outside normal hours, such as time-and-a-half, and double time.

The two most common overtime or penalty rates are time and a half rate and double time rate. Time and a half rate = Normal rate × 1.5 Double time rate = Normal rate × 2

Example 1 Clint is a web designer who runs his own small business and charges an hourly rate of $145. For services provided between 5:00 p.m. and 11:00 p.m. on weekdays, he applies a time and a half rate. On weekends and public holidays, his rate increases to double time. a What is the time and a half rate?

Create a strategy Time and a half means one and a half times the normal hourly rate.

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Apply the idea Time and a half rate = Normal rate × 1.5

Write the equation

= 145 × 1.5

Substitute Normal rate = 145

= $217.50 per hour

Evaluate

Reﬂect and check Alternatively, half of $145 is $72.50. So time and a half is 145 + 72.50 = $217.50 per hour.

b What is the double time rate?

Create a strategy Double time means twice the normal hourly rate.

Apply the idea Double time rate = Normal rate × 2

Write the equation

= 145 × 2

Substitute Normal rate = 145

= $290 per hour

Evaluate

c How much would Clint charge for a call-out between 10:00 a.m. and 5:00 p.m. on a Sunday?

Create a strategy Clint charges double time on a Sunday. Multiply the number of hours in the call-out by the double time rate.

Apply the idea The number of hours in the call-out from 10:00 a.m. to 5:00 p.m. is 7 hours. Amount charged = 7 hours × Normal rate × 2 Write the equation = 7 × 145 × 2

Substitute the values

= $2030

Evaluate

Reﬂect and check Alternatively, use the double time rate found in part (b): Amount charged = 7 hours × Double time rate

420

Write the equation

= 7 × 290

Substitute Double time rate = 290

= $2030

Evaluate

Mathspace New South Wales – Year 11 Standard mathspace.co

Example 2 For an hourly rate of $30, calculate the income earned for working these hours: a 35 hours at normal rates and 6 hours at time and a half

Create a strategy Multiply the number of hours worked at normal rates by the hourly rate, multiply the number of hours worked at time and a half by 1.5 times the hourly rate, and add the two amounts together.

Apply the idea Income = (35 × 30) + (6 × 30 × 1.5)

Write the equation

= 1050 + 270

Evaluate the multiplication

= $1320

Evaluate

b 28 hours at normal rates and 5 hours at double time

Create a strategy Multiply normal hours by the hourly rate, double time hours by twice the hourly rate, and add the results.

Apply the idea Income = (28 × 30) + (5 × 30 × 2)

Write the equation

= 840 + 300

Evaluate the multiplication

= $1140

Evaluate

c 29 hours at normal rates, 9 hours at time and a half and 7 hours at double time

Create a strategy Multiply normal hours by the hourly rate, time and a half hours by 1.5 times the rate, and double time hours by 2 times the hourly rate, then add all three amounts.

Apply the idea Income = (29 × 30) + (9 × 30 × 1.5) + (7 × 30 × 2)

Write the equation

= 870 + 405 + 420

Evaluate the multiplication

= $1695

Evaluate

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Example 3 Emma is a retail worker with an annual salary of $62 400. She works 38 normal hours per week, with overtime paid at time and a half and public holiday shifts paid at double time. Calculate using the approximation of 52 weeks in a year: a Equivalent hourly rate for normal hours, assuming she works 38 hours per week. Round your answer to the nearest cent.

Create a strategy Divide the annual salary by the total number of normal hours worked in a year to ﬁnd the hourly rate.

Apply the idea For total annual hours: Write the equation

Evaluate

For hourly rate: Write the equation

Substitute the values

Evaluate and round

b Time and a half rate for overtime hours.

Create a strategy Multiply the normal hourly rate by 1.5 to ﬁnd the time and a half rate.

Apply the idea Time and a half rate = Normal rate × 1.5

Write the equation

= 31.58 × 1.5

Substitute Normal rate = 31.58

≈ $47.37 per hour

Evaluate

c Double time rate for public holiday shifts.

Create a strategy Multiply the normal hourly rate by 2 to ﬁnd the double time rate.

Apply the idea Double time rate = Normal rate × 2

422

Write the equation

= 31.58 × 2

Substitute Normal rate = 31.58

≈ $63.16 per hour

Evaluate

Mathspace New South Wales – Year 11 Standard mathspace.co

6.02 Practice questions What do you remember? 1

What are the standard full-time working hours per week in Australia?

What does “time and a half” mean in terms of an hourly rate?

What does “double time” mean in terms of an hourly rate?

Are these statements true or false? a

A time and a half rate is always higher than a double time rate.

If the normal rate is $20 per hour, the time and a half rate is $30 per hour.

Double time means the worker earns twice as many hours as they work.

Penalty rates only apply to work done on weekends.

Practice Ex 1

424

Sarah is a plumber who charges an hourly rate of $120 for services provided during regular hours. For services between 4:00 p.m. and 10:00 p.m. on weekdays, she applies a time and a half rate. On weekends, her rate increases to double time. a

What is Sarah’s time and a half rate?

What is Sarah’s double time rate?

How much would Sarah charge for a job from 9:00 a.m. to 3:00 p.m. on a Saturday?

For each of these normal pay rates, calculate the: i

Time and a half rate

Double time rate

$26 per hour

$23.60 per hour

$27.20 per hour

Calculate the earnings for each shift, if they were paid at: i

Time and a half rate

4 hours at a base rate of $26.50 per hour

3 hours at a base rate of $28.00 per hour

6 hours at a base rate of $23.75 per hour

8 hours at a base rate of $24.50 per hour

Double time rate

Calculate the equivalent number of hours worked at normal pay: a

8 hours at time and a half

3.5 hours at double time

Mathspace New South Wales – Year 11 Standard mathspace.co

4 hours at double time

Ex 2

Ex 3

Calculate the equivalent number of hours worked at normal pay for working: a

33 hours at normal rates and 10 hours at time and a half

37 hours at normal rates and 5 hours at double time

21 hours at normal rates, 10 hours at time and a half and 8 hours at double time

7 hours at time and a half and 10 hours at double time

For an hourly rate of $28, calculate the income earned for working: a

32 hours at normal rates and 4 hours at time and a half

27 hours at normal rates and 6 hours at double time

20 hours at normal rates, 5 hours at time and a half and 5 hours at double time

For each week, Christa earns a wage of $24.50 per hour for the ﬁrst 23 hours, time and a half for the next 9 hours, and double time after that. If Christa works 37 hours in one week: a

How many hours did she work at her normal wage?

How many hours did she work at time and a half?

How many hours did she work at double time?

Calculate her earnings for that week.

Liam is a hospitality worker with an annual salary of $56 160. He works 38 normal hours per week, with overtime paid at time and a half and public holiday shifts paid at double time. Calculate using the approximation of 52 weeks in a year: a

Equivalent hourly rate for normal hours, assuming he works 38 hours per week, rounded to the nearest cent

Time and a half rate for overtime hours

Double time rate for public holiday shifts

Weekly earnings if he works 38 normal hours, 4 overtime hours, and 3 hours on a public holiday in a week

Annual earnings if he works 38 normal hours, 4 overtime hours, and 3 hours on a public holiday each week

Avril works part-time as a bar attendant and is paid $23.89 per hour. She is rostered to work 6 hours each day from Tuesday to Friday. If Avril works over her rostered hours from Monday to Friday, she is paid time and a half for the ﬁrst 2 hours of overtime and double time for any further overtime. Any work she does on the weekend is paid at double time. Complete the missing entries in the timesheet: TIMESHEET

Name: Avril Smith

Day

Start

Finish

Total hours

Normal hours

Time and a half hours

Double time hours

Tue

11:00 a.m.

5:00 p.m.

−

Wed

11:00 a.m.

8:00 p.m.

⬚

Thu

11:00 a.m.

6:00 p.m.

Fri

11:00 a.m.

8:00 p.m.

⬚

Sat

1:00 p.m.

5:00 p.m.

⬚

Total

Earnings

⬚ ⬚ ⬚

⬚

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Maria is a customer service representative and is paid $22.50 per hour. She typically works 7 hours each day from Monday to Thursday. For any work done beyond her scheduled hours from Monday to Thursday, Maria is paid time and a half for the ﬁrst 1.5 hours of overtime and double time for any additional hours. Work done on Sunday is paid at double time. Complete the missing entries in Maria’s timesheet: TIMESHEET

Name: Maria

Day

Start

Finish

Mon

8:30 a.m.

3:30 p.m.

Tue

9:00 a.m.

5:00 p.m.

Wed

8:00 a.m.

4:00 p.m.

Thu

9:00 a.m.

7:00 p.m.

Sun

10:00 a.m.

2:00 p.m.

Normal hours

Time and a half hours

Double time hours

Earnings

⬚

Total hours

⬚

Total

⬚ ⬚

David is a maintenance worker who is paid $19.80 per hour. He is assigned to work 8 hours each day from Monday to Friday. Overtime from Monday to Friday is paid at time and a half for the ﬁrst 4 hours and double time thereafter. Any work done on Saturday is paid at double time. Complete the missing entries in David’s timesheet: TIMESHEET

Name: David Total hours

Normal hours

Time and a half hours

Double time hours

Earnings

6:00 p.m.

⬚

⬚ 8

⬚

⬚ −

⬚

7:00 a.m.

5:00 p.m.

−

Thu

8:00 a.m.

4:00 p.m.

⬚

9:00 a.m.

1:00 p.m.

⬚

Sat

⬚

Day

Start

Finish

Mon

7:00 a.m.

4:00 p.m.

Tue

8:00 a.m.

Wed

⬚ ⬚

⬚

Total

⬚

Sandra is paid a normal rate of $42 per hour. How many hours would she have to work to earn $798 at a: a

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⬚

Normal rate

Time and a half rate

Mathspace New South Wales – Year 11 Standard mathspace.co

Double rate

Asmaa earns $24 per hour as a sales assistant. If she works on a Sunday, she is paid time and a half. For Asmaa to earn $540 per week: a

How many hours would she need to work at her normal rate?

How many hours would she need to work on a Sunday only?

How many hours would she need to work on a Sunday, if she also works for 12 hours at her normal rate?

Extend your thinking 18

Skye’s monthly salary is $5748.50. If Skye works for 37 hours every week, with 3 hours being paid at double rate for 44 weeks per year, calculate her hourly pay rate.

Stavros earns $910 per week. Calculate his normal hourly rate of pay if he works:

34 hours at normal rates and 10 hours at time and a half

35 hours at normal rates and 10 hours at double time

34 hours at normal rates, 7 hours at time and a half and 6 hours at double time

Matt needs to earn $739 this week to cover his expenses. His normal hourly rate is $23.90 and he is rostered to work 24 normal hours from Monday to Friday. Calculate how many hours Matt must work on Saturday, when he will earn time and a half, in order to cover his expenses.

An employee’s normal pay rate is $h per hour. They work for a hours at their normal hourly rate of pay and b hours at double time. Assuming they work the same hours every week for 50 weeks of the year, write an expression to predict their annual earnings.

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6.03 Commission, piecework and royalties After this lesson, you will be able to… • define and differentiate between piecework, commission, and royalties. • calculate earnings from piecework given a rate per unit and number of units. • calculate earnings from commission, including scenarios with a retainer and varying sales values. • calculate earnings from royalties based on a percentage of revenue and sales volume. • solve problems involving these earning methods in practical contexts, including calculating weekly and annual income.

Piecework Piecework Employment where a worker is paid a ﬁxed rate for each item produced or action performed regardless of the time taken.

However, piecework does not typically include beneﬁts such as sick leave, holiday leave or superannuation, and earnings stop when work stops. Earnings from piecework are calculated as: Earnings = Number of units × Rate per unit

Example 1 Max is a dog groomer and charges a set amount of $55 per dog wash: a How much does Max earn for washing 20 dogs?

Create a strategy Multiply the number of dogs washed by piecework rate.

Apply the idea Income = 20 × 55 = $1100

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Write the equation Evaluate

b How many dogs does Max need to wash to earn $2000?

Create a strategy To determine the number of dogs, ﬁrst divide the target income by the piecework rate. If the division results in a decimal, round up to the next whole number of dogs as Max cannot wash a fraction of a dog.

Apply the idea Write the equation

Evaluate

Since Max cannot wash partial dogs and needs to earn at least $2000, the calculated value must be rounded up to the next whole number. Washing 36 dogs would result in an income of 36 × $55 = $1980, which is less than $2000. Therefore, Max must wash 37 dogs, which will earn 37 × $55 = $2035, meeting the requirement. c In one week, Max offered a discounted rate as part of an advertising campaign. He washed 28 dogs and earned $1400. Determine the piecework rate per dog during this week.

Create a strategy Divide the income by the number of dogs washed.

Apply the idea Write the equation

Evaluate

d Calculate Max’s earnings over a fortnight if he washes 20 dogs in the ﬁrst week and 25 dogs in the second week at the standard rate of $55 per dog.

Create a strategy Calculate the earnings for each week and sum them to ﬁnd the fortnightly earnings.

Apply the idea For the ﬁrst week: First week earnings = 20 × 55 = $1100

Write the equation Evaluate

For the second week: Second week earnings = 25 × 55 = $1375

Write the equation Evaluate

For the total fortnightly earnings: Fortnightly earnings = 1100 + 1375 = $2475

Add the weekly earnings Evaluate

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6.03 Practice questions What do you remember? 1

What is the difference between commission, piecework, and royalties?

Are these statements about piecework and commission true or false?

Piecework earnings depend on the number of hours worked.

A commission can include a retainer for ﬁnancial stability.

Piecework typically includes beneﬁts like sick leave.

Commission is always a ﬁxed percentage of sales.

Determine whether these payment scenarios represent commission, piecework, or royalties: a

A salesman earns 5% of the value of each car he sells.

A factory worker is paid $2 for each toy assembled.

An author receives 8% of the revenue from book sales.

A real estate agent earns $500 plus 3% of each house sale.

A carpenter is paid $150 per custom chair built.

A musician earns $0.10 every time their song is streamed.

A furniture store employee earns 2% of total weekly sales.

A seamstress receives $6 for each dress sewn.

Calculate the income for these piecework tasks: a

10 items at $5 each

25 units at $3 each

8 jobs at $12 each

15 pieces at $4 each

A musician earns a royalty of 8% on album sales, where each album is priced at $15. Calculate the yearly royalty earnings if 1500 albums are sold in a year.

Practice 6

Calculate the income earned for these amounts of piecework: a

Washing 24 cars at $10 per car

Baking 26 croissants at $7 per croissant

Sewing 84 shirts at $8 per shirt

Painting 15 rooms at $50 per room

A craftsperson receives $3 for each handmade item produced. Determine how many items a craftsperson must create to earn: a

$87

$120

$250

$500

A freelance graphic designer charged a client $700 for editing 500 images: a

Determine the piecework rate per image.

How much would the designer earn if they edited 3750 images for a client?

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Ex 1

Sophie is a baker and charges $6 per bread loaf: a

How much does Sophie earn for baking 20 loaves?

How many loaves must Sophie bake to earn $1800?

In one week, Sophie earns $1000 by baking 25 loaves. Determine the rate per loaf.

Calculate Sophie’s earnings over a fortnight if she bakes 20 loaves in the ﬁrst week and 30 loaves in the second week at $6 per loaf.

Sam has a holiday job picking oranges. He is paid $32 per bin and each bin holds 400 kg of oranges. He can pick at a rate of 0.8 bins per hour: a

Calculate how much Sam receives for one hour’s work.

If Sam works 10 hours per day for 6 days a week, calculate his gross weekly income.

Calculate how many kilograms of oranges Sam picks in a week.

During the month of August, Amelia has a job pruning grape vines. She prunes at a rate of 47 vines per hour. She worked 191 hours over the entire month and earned a total of $5565.74. Calculate Amelia’s piecework rate per vine.

Ex 2

Oliver is paid a piecework rate of $5.20 per kilogram of strawberries he picks. He currently picks at an average rate of 3.70 kg of strawberries per hour. Lucy also picks strawberries but is paid an hourly rate of $26.30 rather than a piecework rate: a

Calculate how much Oliver is paid per hour.

How many kilograms of strawberries would Oliver need to pick each hour, so that his hourly pay was the same as Lucy’s? Round your answer to two decimal places.

Sally is a salesperson who receives a weekly retainer of $400 and a commission of 5% on the value of goods she sells. In one week, she sells goods worth $15 000: a

Calculate the commission earned.

Determine her total income for the week.

If a commission of 20% is paid on all sales, determine the total value of sales if the gross income is $238.

Calculate the income:

James is paid a monthly retainer of $600 and a commission of 2% of the value of the products he sells. His sales for the month are $261 000.

Allan is paid a base monthly salary of $800 and a commission of 4% of the value of the products he sells. His sales for the month are $299 000.

Solve: a

Han is paid a commission of 3% of the value of trucks sold in excess of $2100. Calculate his income if he sells trucks to the value of $6600.

Valentina is paid a retainer of $200 per week plus 5% commission on sales in excess of $2100. In one week, she sells goods to the value of $5800. Calculate her income for that week.

A musician earns a royalty of 8% on album sales, where each album is priced at $15. Calculate her yearly royalty earnings if 1500 albums are sold in a year.

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Ex 3

A writer earns a royalty on e-book sales, with each e-book priced at $10. If 2500 e-books are sold annually to earn $3000 in royalties, what is the royalty percentage?

A photographer earns a royalty of 12% on prints sold at $20 each:

Calculate the yearly royalty earnings if 2500 prints are sold in a year.

How many prints must be sold to earn $7200 in royalties for the year?

An auctioneer of artwork is paid 2% of the value of a sale up to $500 000, and 5% of any amount by which the sale value exceeds $500 000: a

Calculate the commission she would earn on a sale of $500 000.

The auctioneer sells an artwork for more than $500 000. If she receives commission of $13 050, determine how much the artwork was sold for.

For each scenario, calculate: i

The commission earned

The commission rate as a percentage of the sales

Lyra is paid a retainer of $210 per week plus a commission based on her weekly sales. In one week, she sells $9800 worth of lighting. Her total pay for the week was $504.

Ben is paid a retainer of $280 per week plus a commission based on his weekly sales. In one week, he sells $7800 worth of lighting. His total pay for the week was $670.

Dave sold motorbikes to the value of $66 000 in one week. His pay for the week was $2434, which comprised a retainer plus a commission of 2.9% on his sales: a

Calculate the amount of commission Dave was paid.

Calculate the amount of retainer Dave was paid.

Extend your thinking 23

Mia is a real estate agent who earns a sliding scale commission: 3% on sales up to $50 000 and 5% on sales above $50 000, plus a weekly retainer of $250. In one week, she generates $60 000 in sales: a

Calculate Mia’s commission for the week.

Calculate Mia’s total income for the week, including her retainer.

A painter earns a royalty of 6% on sales of her artwork, each sold for $50: a

Calculate her royalty earnings if 400 artworks are sold in a year.

Determine her total earnings if she sells 1200 artworks in a year, including a ﬂat annual fee of $500 from her gallery.

If she earned $3600 in royalties for the year, how many artworks were sold?

Neil is a sole trader running a photography business, earning $2000 per week from client projects but incurring weekly expenses of $300 for equipment and studio costs: a

Calculate Neil’s net income for the week.

Calculate Neil’s net income over a fortnight.

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Amarah earns a weekly retainer of $300 plus a commission of 4.6% based on sales of airline tickets. If she is aiming to earn $1183.20 each week, determine the value of ticket sales she must achieve.

Arona earns a commission of 3.5% of the total value of her sales, plus a weekly retainer. In a week where she sold items with a total value of $25 200, her weekly income was $1122. Determine the value of Arona’s weekly retainer.

A professional window cleaner is paid a set amount of $80 per house cleaned. What is the range of annual income expected for the window cleaner if they can expect to clean between 25 and 40 houses per month?

Elna is considering two job offers: • Offer A pays a weekly retainer of $500 and a commission of 10% on weekly sales. • Offer B pays a weekly retainer of $1000 and a commission of 5% on weekly sales. Elna expects she can make about $16 000 monthly sales. Calculate her yearly earnings for each offer and justify which one she should choose.

Cythia works as a financial adviser and earns a commission of 0.5% of the value of her clients’ profits. Her retainer is $800 per week. What would her clients’ total weekly profits need to be for Cythia to earn an annual income of $70 200?

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6.04 Government allowances After this lesson, you will be able to… • identify common types of government income support payments and their general purpose. • interpret tables to determine eligibility and maximum fortnightly payments for allowances like Youth Allowance and Austudy based on given circumstances. • calculate reductions in government allowances based on an individual’s earned income using specified income test rules. • understand the principles of the income test and assets test for the Age Pension and determine the final pension amount. • calculate annual government payment amounts from fortnightly figures.

Income support Government payments Payments made by the government to individuals to provide ﬁnancial support.

Government income support aids those unable to work due to sickness, age, or disability, and some full-time students in approved courses or Australian Apprenticeships. An income support payment from the government is called an allowance or pension. It is provided by Services Australia through the Centrelink program. Eligibility for income support depends on the circumstances of the individual. Students in full-time study or apprenticeships receive these payments for daily expenses: Government allowance

Description

Youth Allowance

Full-time students, under 25 years of age, studying an approved course or doing an Australian Apprenticeship

Austudy

Full-time students, aged 25 years or older, studying an approved course or doing an Australian Apprenticeship

ABSTUDY

Aboriginal or Torres Strait Islander students, studying an approved course or doing an Australian Apprenticeship

There are additional allowances students can also claim to help cover speciﬁc costs like travel fares and rent assistance. Visit Service Australia’s website for details.

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This table shows some of the other types of income support provided by the government: Government allowance or pension

Description

Age Pension

People who have reached the pension age

Disability Support Pension

People with a permanent condition that prevents them from working

Family Tax Benefit

Assists families with the cost of raising children

JobSeeker Payment

Unemployed people aged 22 or over who are looking for work, or temporarily unable to work or study due to illness, injury, or disability

Government allowances and pensions come with a set of eligibility criteria that an applicant must meet before they can receive any payments. These could include being a certain age, earning less than a certain amount, or being enrolled in an approved course. Tables, like those in the worked examples, are used to determine how much government income support a person is entitled to receive.

Exploration How might the amount of a government allowance change if a student’s part-time income increases? Discuss how this could affect their total income and study commitments. 1. How might a student balance the need to increase part-time work hours to cover living expenses with the potential reduction in their government allowance? Consider the impact on their ﬁnancial planning. 2. What strategies could a student employ to optimise their total income (part-time earnings plus government allowance) while maintaining eligibility for support? Discuss the role of income thresholds in these decisions. 3. How might changes in government allowance rates or eligibility criteria affect a student’s decision to pu rsue full-time study versus part-time study with increased work hours? Explore the trade-offs involved.

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Example 1 This table is used to determine the maximum amount of Youth Allowance available each fortnight for students and Australian Apprentices (as of January 1, 2025): Maximum fortnightly payment

Circumstances Single, no children, younger than 18 years old and live at their parents’ home

$410.30

Single, no children, younger than 18 years old and need to live away from their parents’ home to study, train or look for work

$663.30

Single, no children, 18 years or older and live at their parents’ home

$472.50

Single, no children, 18 years or older and required to live away from their parents’ home

$663.30

Single with children

$836.60

Member of a couple with no children

$663.30

Member of a couple with children

$718.10

Consider each case assuming that the students or their guardians are not earning above the thresholds that will impact these payments. a Olivia is 17 years old, single with no children, and lives at home with her parents. She is entitled to Youth Allowance because she is studying full-time in an approved course. Determine the maximum amount she will receive from her Youth Allowance each fortnight.

Create a strategy Match her circumstances with the appropriate row in the table.

Apply the idea The ﬁrst row in the table matches Olivia’s description of being single, with no children, 17 years old, and living at home with parents, therefore Olivia is entitled to $410.30.

b Emily is 18 years old and needs to live away from her parents’ home in order to study an approved course. She is single and has no children. What is the maximum fortnightly amount of Youth Allowance she can receive?

Apply the idea The fourth row matches Emily’s description of being single, with no children, 18 years old, and living away from parents’ home, therefore Emily is entitled to $663.30.

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Example 2 Marge is 20 years old, single, no children, lives with parents, works part-time, earns $218 fortnightly, and seeks full-time work. Eligibility basics: • Aged 16 to 21 and looking for full-time work • Aged 18 to 24 and studying full-time • Aged 16 or 17 and studying full-time and have completed Year 12 or equivalent • Aged 16 or 17, studying full-time, and either independent or need to live away from home to study • Aged 16 to 24 and undertaking a full-time Australian Apprenticeship Circumstances

Maximum fortnightly payment

Single, no children, younger than 18 years old and live at their parents’ home

$410.30

Single, no children, younger than 18 years old and need to live away from their parents’ home to study, train or look for work

$663.30

Single, no children, 18 years or older and live at their parents’ home

$472.50

Single, no children, 18 years or older and required to live away from their parents’ home

$663.30

Single with children

$836.60

Member of a couple with no children

$663.30

Member of a couple with children

$718.10

For a student on Youth Allowance Earnings

Fortnightly payment reduction

Between $195 − $323

50 cents for each dollar earned over $195

More than $323

$64 plus 60 cents for each dollar earned over $323 For a job-seeker on Youth Allowance

Earnings

440

Fortnightly payment reduction

Between $154 – $262

50 cents for each dollar earned over $154

More than $262

$54 plus 60 cents for each dollar earned over $262

Mathspace New South Wales – Year 11 Standard mathspace.co

a Calculate her fortnightly Youth Allowance rounded to two decimal places.

Create a strategy Determine the maximum payment using the circumstances table, then apply the income reduction for a job-seeker based on her earnings.

Apply the idea Marge is 20 years old, single, with no children, and lives at her parents’ home, matching the third row of the circumstances table. The maximum payment she can receive is $472.50 per fortnight. Since Marge earns $218 per fortnight, which is between $154 and $262, her Youth Allowance is reduced by 50 cents for each dollar over $154. Fortnightly Youth Allowance = 472.50 − 0.50 × (218 − 154)

Write the equation

= 472.50 − 0.50 × 64

Evaluate inside the brackets

= 472.50 − 32

Evaluate the multiplication

= $440.50

Evaluate

Reﬂect and check Notice that $440.50 is the income support payment from Centrelink. Her total earnings per fortnight will be 218 + 440.50 = $658.50.

b Calculate the annual Youth Allowance for Marge using the approximation of 52 weeks per year rounded to two decimal places.

Create a strategy Convert the fortnightly payment to an annual amount by multiplying by the number of fortnights in a year, where 52 weeks corresponds to 26 fortnights.

Apply the idea Marge’s fortnightly Youth Allowance is $440.50. To calculate the annual amount, multiply by 26 fortnights. Annual Youth Allowance = 440.50 × 26 = $11 453.00

Write the equation Evaluate

Reﬂect and check To verify, note that 52 weeks divided by 2 weeks per fortnight gives 26 fortnights. The calculation 440.50 × 26 = $11 453.00 conﬁrms the annual amount.

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Example 3 This table is used to determine the maximum amount of Austudy available each fortnight for students, assuming that a student is not earning above additional thresholds that may further impact these payments: Circumstances

Maximum fortnightly payment

Single, no children

$663.30

Single with children

$836.60

Member of a couple with no children

$663.30

Member of a couple with children

$718.10

Sarah is 25 years old, single, with no children, and needs to live away from her parents’ home to study an approved course. She works part-time and earns $400 per fortnight: For a student on Austudy Earnings

Fortnightly payment reduction

Between $492 − $583

50 cents for each dollar earned over $492

More than $583

$45.50 plus 60 cents for each dollar earned over $583

a Calculate her fortnightly Austudy payment rounded to two decimal places.

Create a strategy Determine the maximum payment using the circumstances table, then apply the income reduction based on her earnings.

Apply the idea Sarah is 25 years old, single, with no children, and lives away from her parents’ home, matching the ﬁrst row of the circumstances table. The maximum payment she can receive is $663.30 per fortnight. Since Sarah earns $400 per fortnight, which is below $492, her Austudy payment is not reduced. Fortnightly Austudy payment = 663.30

No reduction applies

Reﬂect and check To verify, Sarah’s total fortnightly income is 400 + 663.30 = $1063.30. Since her earnings are below the threshold, she receives the full payment.

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Age pension An Age Pension is given to people who have reached retirement age. For a person to be eligible for an Age Pension in Australia, they must ﬁrst meet the residency and age requirements. A person who is 67 years old or older, has Australian citizenship or is a permanent resident, and has lived in Australia for at least ten years may be eligible to receive the Age Pension. When calculating the amount of Age Pension an individual or a couple will receive, their age, income, and assets are taken into consideration. Services Australia will conduct an assets test and an income test that may reduce the maximum pension payable. The outcomes of both tests are compared, and the lesser payment is given. Table 1 shows the maximum Age Pension for various situations. It also includes other supplement payments that pensioners may be eligible for: Table 1: Maximum pension rates (2025) Single

Couple each

Couple combined

Couple apart due to ill health

Maximum basic rate

$1051.30

$792.50

$1585.00

$1051.30

Maximum pension supplement

$83.60

$63.00

$126.00

$83.60

Energy supplement

$14.10

$10.60

$21.20

$14.10

Total

$1149.00

$866.10

$1732.20

$1149.00

Per fortnight

The income test assesses all sources of income such as dividend payments, interest earnings from savings, superannuation, rental income, or any employment. If earning above a speciﬁed amount, known as the free area, the maximum pension amount is reduced by a speciﬁc rate. Currently in Australia, these rates are 50 cents for every $1 over the free area for singles, and 25 cents per $1 for each member of a couple. However, it is always important to check the information as it may vary. These tables show the current income rules for pensioners: Table 1: Single person Income per fortnight

Amount of pension reduction

Up to $210 (free area)

Over $210

50 cents for each dollar over $210

Table 2: Couple living together or apart due to ill health Combined income per fortnight

Amount each member of the couple’s pension reduction

Up to $367 (free area)

Over $367

25 cents for each dollar over $367

The assets test assesses all sources of any assets owned, excluding their own home. If the value of assets exceeds a speciﬁc limit, the maximum pension is reduced by a speciﬁc rate. Currently in Australia, these rates are $3 for every $1000 over the limit. However, this information may vary and should be verified.

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Table 3 shows the current asset limits for pensioners: Table 3: Asset limits for pensioners (2025) Situation

Homeowner

Non-homeowner

Single

$314 000

$566 000

A couple, combined

$470 000

$722 000

A couple, separated due to illness, combined

$470 000

$722 000

A couple, one partner eligible, combined

$470 000

$722 000

Example 4 Carol, aged 68, and Tom, aged 70, are a couple residing in their own home, and are eligible for the pension based on age and citizenship. The couple earn $450 per fortnight from various sources and own $515 000 worth of assets excluding their own home. a Calculate the Age Pension payment under the income test.

Create a strategy Use the tables to determine the maximum possible pension, then calculate the reductions based on the income rules.

Apply the idea Carol and Tom are a couple eligible for the basic pension rate. Using Table 1, the maximum pension they can receive is $1732.20 combined. The couple earns $450 per fortnight in income, which exceeds the free area outlined in Table 2. A reduction of 25 cents per $1 over the free area needs to be calculated for each member. To calculate the reduced pension under the income test: Reduction per person = 0.25 × (450 − 367)

Write the equation

= 0.25 × 83

Evaluate inside the brackets

= 20.75

Evaluate the multiplication

Total reduction = 20.75 × 2 = 41.50 Reduced pension = 1732.20 − 41.50 = $1690.70

Multiply by two for the couple Evaluate Subtract from maximum pension Evaluate

The maximum pension under the income test is $1690.70 (combined) per fortnight.

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6.04 Practice questions What do you remember? 1

What is the purpose of these government allowances? a

Austudy

Youth Allowance

Age Pension

JobSeeker Payment

What is the minimum age requirement for eligibility for these allowances? a

Austudy

Youth Allowance (for students)

Age Pension

JobSeeker Payment

Determine if these individuals meet the minimum age requirement for the speciﬁed allowance: a

Sarah, aged 23, applying for Austudy

Tom, aged 19, applying for Youth Allowance as a student

Lisa, aged 66, applying for Age Pension

Mark, aged 30, applying for JobSeeker payment

Determine if each individual meets the eligibility requirements for the speciﬁed government allowance: a

Amy, aged 20, single, studying full-time at university, applying for Youth Allowance

Ethan, aged 24, studying part-time at TAFE, applying for Austudy

Noah, aged 18, Aboriginal student, full-time apprentice mechanic, applying for ABSTUDY

Liz, aged 66, single, retired with no income, applying for Age Pension

Alan, aged 35, with a permanent vision impairment preventing work, applying for Disability Support Pension

Meg, aged 30, single, no children, unemployed, applying for Family Tax Beneﬁt

Laurie, aged 28, unemployed, seeking full-time work, applying for JobSeeker Payment

Ava, aged 17, single, studying full-time at home, applying for Youth Allowance

Using this simpliﬁed table of fortnightly allowance payments, calculate the maximum payment for each case: Circumstances

Payment per fortnight

Single, under 18, at home (Youth Allowance)

$410.30

Single, 25 or older, studying (Austudy)

$663.30

Single, with children (JobSeeker Payment)

$756.90

Single, over 67 (Age Pension)

$1149.00

A 17-year-old job-seeker living with parents

A 26-year-old full-time student

A 35-year-old single parent seeking work

A 70-year-old retiree

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Practice Ex 1

To be eligible for Youth Allowance, individuals must satisfy one of these conditions: • Be 16 to 21 years of age and looking for full-time work • Be 16 to 24 years of age and studying full-time • Be 16 or 17 years of age, studying full-time, and have completed Year 12 or equivalent • Be 16 or 17 years of age, studying full-time, and either independent or need to live away from home to study • Be 16 to 24 years of age and undertaking a full-time Australian Apprenticeship This table shows the fortnightly payments that eligible people could receive on Youth Allowance: Maximum fortnightly payment

Circumstances Single, no children, younger than 18 years old and live at their parents’ home

$410.30

Single, no children, younger than 18 years old and need to live away from their parents’ home to study, train or look for work

$663.30

Single, no children, 18 years or older and live at their parents’ home

$472.50

Single, no children, 18 years or older and required to live away from their parents’ home

$663.30

Single with children

$836.60

Member of a couple with no children

$663.30

Member of a couple with children

$718.10

These tables show how much Youth Allowance a person can receive when they already have an income: For a student on Youth Allowance Earnings

Fortnightly payment reduction

Between $195 – $323

50 cents for each dollar earned over $195

More than $323

$64 plus 60 cents for each dollar earned over $323 For a job-seeker on Youth Allowance

Earnings

Fortnightly payment reduction

Between $154 – $262

50 cents for each dollar earned over $154

More than $262

$54 plus 60 cents for each dollar earned over $262

Consider each case and determine the fortnightly income they earn given their Youth Allowance payment: a

448

Amanda, aged 22, is single, with no children. She lives away from her parents’ home, is seeking full-time work, and receives a Youth Allowance payment of $629.80.

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Ex 2

Karen is a member of a couple with a child, studying full-time. She receives a Youth Allowance payment of $598.90.

Joe, aged 21, is single with one child. He is currently seeking full-time employment and receives a Youth Allowance payment of $705.10.

Circumstances Single, no children, younger than 18 years old and live at their parents’ home

$410.30

Single, no children, younger than 18 years old and need to live away from their parents’ home to study, train or look for work

$663.30

Single, no children, 18 years or older and live at their parents’ home

$472.50

Single, no children, 18 years or older and required to live away from their parents’ home

$663.30

Single with children

$836.60

Member of a couple with no children

$663.30

Member of a couple with children

$718.10

These tables show how much Youth Allowance a person can receive when they already have an income: For a student on Youth Allowance Earnings

Fortnightly payment reduction

Between $195 − $323

50 cents for each dollar earned over $195

More than $323

$64 plus 60 cents for each dollar earned over $323 For a job-seeker on Youth Allowance

Earnings

Fortnightly payment reduction

Between $154 − $262

50 cents for each dollar earned over $154

More than $262

$54 plus 60 cents for each dollar earned over $262

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Consider each case and calculate:

Ex 3

The fortnightly Youth Allowance rounded to two decimal places

The annual Youth Allowance using the approximation of 52 weeks per year rounded to two decimal places

Anne, aged 19, is living with her parents, working part-time and seeking full-time work. She earns $207 a fortnight.

Bill is required to live independently, aged 21, working part-time but looking for full-time work. He earns $569 per fortnight.

Charlene, aged 17, is required to live independently and is studying full-time. She earns $235 per week.

To be eligible for Austudy you must: • Be 25 years old or older, and • Be studying full-time in an approved educational institution, or • Be undertaking a full-time Australian Apprenticeship This table shows the fortnightly payments that eligible people could receive on Austudy: Circumstances

Payment per fortnight

Single, no children

$663.30

Single, with children

$836.60

In a couple, no children

$663.30

In a couple, with children

$718.10

Consider each case and calculate:

The fortnightly Austudy payment rounded to two decimal places

The annual Austudy payment using the approximation of 52 weeks per year rounded to two decimal places

Cassie, aged 28, a single parent and is undertaking a full-time apprenticeship as a hairdresser.

Jenny, aged 32, married without children and is studying full-time at TAFE.

Christa, aged 40, married with 2 children and is studying full-time.

These tables provide a summary of the maximum fortnightly payments through ABSTUDY for those who are dependent and those living away from home: Dependent living at home Age

Payment per fortnight

Younger than 16 in tertiary education

$41.90

Aged 16 to 17

$410.30

Aged 18 to 21

$472.50

Aged 22 or older

$663.30

Living away from home Age

Payment per fortnight

Younger than 16

$434.20

Aged 16 to 21

$663.30

Aged 22 or older

$663.30

Based on the information provided, what would be the maximum fortnightly payment for a 20-year-old student who is living away from home? 450

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To be eligible for Age Pension you must: • Be at least 67 years old • Meet an income test and an assets test This table shows the fortnightly payments that eligible people could receive on Age Pension: The person is single

The person is in a couple

Maximum basic rate

$1051.30

$792.50

Maximum pension supplement

$83.60

$63.00

Energy supplement

$14.10

$10.60

Total

$1149.00

$866.10

Use the information to determine the Age Pension they are entitled to each fortnight:

Helen is 62 and has no income. She qualiﬁes for all the supplements.

Rosa is 82, single and does not qualify for any supplements.

Justin is 81, single and qualiﬁes for a maximum pension supplement.

Derek and Tara are a couple who live together. Derek, aged 77, and Tara, aged 79, qualify for one energy supplement and currently have no income.

Consider this information that outlines the Family Tax Beneﬁt (FTB) payable to families with one child, based on their level of income: Combined annual income

Family tax beneﬁt for the year

Up to $55 900

$9286.64

$55 900 to $101 524

$9286.64 minus 20 cents for each dollar over $55 900

Over $101 524

Nil

• For 1 child aged 0-12: Maximum benefit is $178.64 per fortnight • For 1 child aged 13-15: Maximum benefit is $213.24 per fortnight Determine the annual Family Tax Benefit paid to a family with one child aged 9, meeting the FTB requirements and with a combined annual income of $50 000. Assume there are 26 fortnights in a year. 12

Consider this information that outlines the Family Tax Beneﬁt (FTB) payable to families with one child, based on their level of income: Combined annual income

Family tax beneﬁt for the year

Up to $55 900

$9284.00

$55 900 to $101 524

$9284.00 minus 20 cents for each dollar over $55 900

Over $101 524

• For 1 child aged 0-12: Maximum benefit is $187.60 per fortnight • For 1 child aged 13-15: Maximum benefit is $244.06 per fortnight Determine the annual Family Tax Benefit paid to a family with one child aged 14, meeting the FTB requirements, with a combined annual income of $85 208. Assume there are 26 fortnights in a year. 6.04 Government allowances mathspace.co

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Consider this table regarding JobSeeker Payment: Fortnightly payment

Payment reaches Nil once income reaches this per fortnight

Single, no children

$781.10

$1438.50

Single, principal carer, with a dependent child

$836.50

$1541.83

Single, 55 or older, after 9 continuous months on payment

$836.50

$1541.83

Single, partial capacity to work <15 hours/ week

$836.50

$1541.83

$715.10 each

$1304.50

Applicant details

Partnered Your income per fortnight

452

Reduces your fortnightly payment by

Up to $154

$154 to $262

50 cents for each dollar over $154

Over $262

$54 plus 60 cents for each dollar over $262

Partners, Katrina and Adam, are trying to calculate how much JobSeeker Payment Katrina is entitled to. Katrina earns $200 per fortnight. Calculate her fortnightly JobSeeker Payment in dollars.

Uther is trying to calculate how much JobSeeker Payment he is entitled to. Uther is single with no children and earns $800 per fortnight. Calculate his fortnightly JobSeeker Payment in dollars.

Jeremy is single and is the principal carer for his daughter. He earns $500 per week. Calculate his fortnightly JobSeeker Payment in dollars.

Mathspace New South Wales – Year 11 Standard mathspace.co

The person is in a couple

Maximum basic rate

$1051.30

$792.50

Maximum pension supplement

$83.60

$63.00

Energy supplement

$14.10

$10.60

Total

$1149.00

$866.10

If a person already has an income, the Age Pension may be reduced according to the table: Marital status of pensioner

Income per fortnight

Age Pension payment reduction

Single person

Up to $210

Single person

Over $210

50 cents for each dollar over $210

A couple

Up to $367

A couple

Over $367

25 cents for each dollar over $367

Use the tables to answer these questions: a

Kevin and Bea are a couple who live together. They are both aged over 67 and qualify for an energy supplement. The couple sells vegetables from their garden at a Thursday market, where they make $125 a week. How much Age Pension do they receive, as a couple, each fortnight?

Ray and Eileen are a couple who live together. They are both aged over 67 and neither qualiﬁes for a supplement. Eileen works part-time and earns $191 a week. How much Age Pension do they receive, as a couple, each fortnight?

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Ex 4

The fortnightly maximum Age Pension available for different circumstances: Single

Couple each

Couple combined

Couple apart due to ill health

Maximum basic rate

$1051.30

$792.50

$1585.00

$1051.30

Maximum pension supplement

$83.60

$63.00

$126.00

$83.60

Energy supplement

$14.10

$10.60

$21.20

$14.10

Total

$1149.00

$866.10

$1732.20

$1149.00

Per fortnight

The income rules for pensioners are outlined in these tables: Single person

Couple living together or apart due to ill health Amount each member of the couple’s pension reduction

Income per fortnight

Amount of pension reduction

Up to $210 (free area)

Up to $367 (free area)

Over $210

50 cents for each dollar over $210

Over $367

25 cents for each dollar over $367

Combined income per fortnight

The maximum Age Pension is reduced by $3 for every $1000 over the asset limits outlined in the table: Situation

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Homeowner

Non-homeowner

Single

$314 000

$566 000

A couple, combined

$470 000

$722 000

A couple, separated due to illness, combined

$470 000

$722 000

A couple, one partner eligible, combined

$470 000

$722 000

Calculate the fortnightly Age Pension under the income test.

Calculate the fortnightly Age Pension under the assets test.

iii

By comparing the outcomes of both tests, determine the Age Pension will be received.

David is a single pensioner eligible for the Age Pension and all supplements. He earns $620 per fortnight and owns his own home plus $390 000 in assets.

Jane and Paul are a couple eligible for the Age Pension and the energy supplement. They have a combined income of $2100 per fortnight and own their own home plus $550 000 in assets.

Alice and Bob are a couple, separated due to illness, eligible for the Age Pension and all supplements. They have a combined income of $330 per fortnight and combined assets of $170 000 and are non-homeowners.

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Job-seekers living at home and under the age of 18 qualify for a government allowance of $410.30 per fortnight, so long as they do not earn more than $154 in any given fortnight. Your income per fortnight

Reduces your fortnightly payment by

Up to $154

$154 to $262

50 cents for each dollar over $154

Over $262

$54 plus 60 cents for each dollar over $262

Determine the allowance paid to each of these young job-seekers:

Anna, who earns $90 per week

Belinda, who has a part-time job paying $8.50 per hour for 16 hours a fortnight

Deanne, who is paid $21.10 per hour for working two hours a day for six days a week

Use the tables regarding JobSeeker Payment to answer these questions: Fortnightly payment

Payment reaches Nil once income reaches this per fortnight

Single, no children

$781.10

$1438.50

Single, principal carer, with a dependent child

$836.50

$1541.83

Single, 55 or older, after 9 continuous months on payment

$836.50

$1541.83

Single, partial capacity to work <15 hours/week

$836.50

$1541.83

$715.10 each

$1304.50

Applicant details

Partnered Your income per fortnight

Reduces your fortnightly payment by

Up to $154

$154 to $262

50 cents for each dollar over $154

Over $262

$54 plus 60 cents for each dollar over $262

Jonathan is a single, principal parent of 2 children. He receives a fortnightly JobSeeker Payment of $711.60. What is his weekly income from his job?

Tracey is single with no children. She receives a fortnightly JobSeeker Payment of $539.30. What is her annual income from her job?

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ABSTUDY provides ﬁnancial assistance to eligible students. The maximum fortnightly payments are shown: Per fortnight

Single

Couple each

Couple combined

Couple apart due to ill health

Maximum basic rate

$1051.30

$792.50

$1585.00

$1051.30

Maximum pension supplement

$83.60

$63.00

$126.00

$83.60

Energy supplement

$14.10

$10.60

$21.20

$14.10

Total

$1149.00

$866.10

$1732.20

$1149.00

The income rules for students are outlined in these tables: Single person

Couple living together or apart due to illness Amount each member of the couple’s pension reduction

Income per fortnight

Amount payment reduction

Up to $210 (free area)

Up to $367 (free area)

Over $210

50 cents for each dollar over $210

Over $367

25 cents for each dollar over $367

Combined income per fortnight

Consider each case and calculate: i

The maximum fortnightly ABSTUDY payment including all supplements, rounded to two decimal places.

The payment reduction due to fortnightly income, rounded to two decimal places.

iii

The actual fortnightly ABSTUDY payment received, rounded to two decimal places.

Chen is a 19-year-old single student, earns $350 per fortnight from a part-time job.

Ella is a 20-year-old student, in a couple, lives together, with combined income of $400 per fortnight.

Liam is a 22-year-old single student with one child, earns $250 per fortnight from a parttime job.

Extend your thinking 19

A young person qualiﬁes for Youth Allowance of $410.30 per fortnight, so long as they do not earn more than $154 before tax in that time. In any fortnight that they do earn more than $154, their allowance will be reduced by 50 cents in the dollar for earnings over $154 and up to $262 and reduced by 60 cents in the dollar for earnings over $262. The young person has a part-time job that pays $16 per hour. Determine, to the nearest hour, how many hours the young person can work in a fortnight before they are no longer paid any Youth Allowance.

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Ben earns $23 per hour and works 17 hours a week. He receives a Youth Allowance of $78 per fortnight. The reductions on his Youth Allowance are outlined in this table: Income

Fortnightly Youth Allowance reduction

Up to $154

$154 to $262

50 cents for each dollar over $154

Over $262

$54 plus 60 cents for each dollar over $262

What is the maximum fortnightly allowance he is entitled to? 21

Ella is a university student who works part-time for a company based in the United States. She receives her income in US dollars (USD). Ella works 12 hours per week and earns 21 USD per hour. To apply for Youth Allowance in Australia, Ella must convert her income into Australian dollars (AUD). The current exchange rate is 1 USD = 1.55 AUD. Ella’s maximum Youth Allowance payment is 472.50 AUD per fortnight. However, her payment is reduced by 60 cents for every dollar she earns above 262 AUD per fortnight:

Calculate her fortnightly Youth Allowance payment.

What does the exchange rate need to be for Ella to receive no Youth Allowance payment at all?

A student named Maya is 25 years old and enrolled full-time in university. She is eligible for Austudy, which provides a maximum payment of $663.30 per fortnight. She also earns money from a part-time tutoring job, and is planning her budget for the semester. a

Calculate Maya’s total earnings per fortnight from her tutoring job, given she earns $20 per hour for 12 hours per week.

Determine how much her Austudy payment is reduced by, given reductions are 50 cents per dollar between $492 and $583, and $45.50 plus 60 cents per dollar over $583, and what she actually receives from Austudy.

What is her total fortnightly income, combining work and Austudy?

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6.05 Annual leave loading After this lesson, you will be able to… • define annual leave and annual leave loading. • calculate an employee’s normal pay for a specified period of annual leave. • calculate the annual leave loading amount (17.5%) based on the normal pay for the leave period. • determine the total holiday pay by summing the normal pay and the leave loading. • solve problems involving annual leave loading in various contexts, such as when leave is split.

Annual leave loading In Australia, employees, excluding casual workers, are entitled to 4 weeks (20 days) of paid annual leave each year. This is based on normal working hours and rates of pay. Some employees also receive an additional payment known as annual leave loading, calculated as a percentage of their normal pay. Annual leave loading is an extra payment applied to regular annual leave, typically at a rate of 17.5% of the normal or base pay over the leave period. Eligibility depends on the employment contract, award, or agreement. This payment, sometimes called holiday loading, compensates employees during their leave. Employees may split their 4 weeks of annual leave into multiple periods, such as two separate fortnights. In such cases, the annual leave loading is calculated and distributed proportionally across each period.

Example 1 An employee earns an annual salary of $85 200 and is entitled to 4 weeks of annual leave with a loading of 17.5%. Assume 52 weeks in a year: a Calculate the normal weekly pay rounded to two decimal places.

Create a strategy Divide the annual salary by the approximated number of weeks per year, 52.

Apply the idea Divide annual salary by 52

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Evaluate and round

b Calculate the normal pay over 4 weeks.

Create a strategy Multiply the weekly pay by 4.

Apply the idea 4-week pay = 1638.46 × 4 = $6553.84

Multiply the weekly pay by 4 Evaluate

c Calculate the annual leave loading for 4 weeks rounded to two decimal places.

Create a strategy Multiply the 4-week normal pay by 17.5%.

Apply the idea Annual leave loading = 6553.84 × 17.5%

Multiply the normal pay by 17.5%

= 6553.84 × 0.175

Convert percentage to decimal

= $1146.92

Evaluate and round

d Calculate the total holiday pay for 4 weeks.

Create a strategy Add the normal pay for 4 weeks to the annual leave loading.

Apply the idea Holiday pay = 6553.84 + 1146.92 = $7700.76

Add the normal pay to annual leave loading Evaluate

Example 2 Brolly earns $720 per week and takes 2 weeks of annual leave with a loading of 17.5%. Calculate his total holiday pay for this period.

Create a strategy Determine the normal pay for 2 weeks by multiplying the weekly pay by 2, calculate the leave loading as 17.5% of that amount, and add the two together to ﬁnd the total holiday pay.

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Practice 5

An employee is entitled to 4 weeks annual leave, with a leave loading percentage of 17.5%. Calculate the leave loading amount on the base fortnightly salary of: a

Ex 1

Ex 2

$1900

$2500

$3200

Sophia earns an annual salary of $72 800 and is entitled to 4 weeks of annual leave with a loading of 17.5%. Using the approximation of 52 weeks in a year: a

Calculate her normal weekly pay.

Calculate her normal pay over 4 weeks.

Calculate her annual leave loading for 4 weeks.

Calculate her total holiday pay for 4 weeks.

Xavier earns $22.80 per hour in a 38-hour week. Calculate his holiday loading if it is 17.5% of 4 weeks of normal pay rounded to two decimal places.

Patricia earns $55 900 per year. Calculate her leave loading if it is 17.5% of 4 weeks of normal pay rounded to two decimal places. Use the approximation of 52 weeks in a year.

Michael normally receives $1100 per fortnight as pay. He is taking holidays for 4 weeks and is to be paid for the 4 weeks as well as receiving a leave loading of 17.5%. Calculate his gross pay for the 4-week period rounded to two decimal places.

Tracy is paid $1096 per week. She decides to use some of her annual leave and take a 3-week holiday. If Tracy is entitled to 17.5% holiday loading, calculate her total pay for the 3 weeks rounded to two decimal places.

Jade earns $950 per week and takes 2 weeks of annual leave with a loading of 17.5%. Calculate her total holiday pay for this period rounded to two decimal places.

Roxanne is a qualiﬁed ﬁtness instructor. She earns $898.20 per week. The Fitness Industry Award entitles her to 4 weeks annual leave and an annual leave loading of 17.5%. Roxanne decides to split her annual leave. She takes 1 week holiday in April and 3 weeks holiday in June. Calculate rounding to two decimal places: a

Her holiday pay for April

Her holiday pay for June

Andrei is an hourly-paid employee earning $25 per hour, working 25 hours per week, taking 1 week of leave with 17.5% loading. Calculate his total holiday pay.

Fred’s annual salary is $50 000. At Christmas time, Fred takes half of his annual leave and is paid 2 weeks normal pay plus a holiday leave loading of 17.5% on this amount. Use the approximation of 52 weeks in a year. Calculate rounding to two decimal places: a

His holiday leave loading

His total holiday pay

Danielle earns $4810 per month. Calculate her holiday loading if it is 17.5% of 4 weeks of normal pay. Use the approximation of 52 weeks in a year.

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Kate’s gross fortnightly salary is $2515. At Christmas, she receives 4 weeks pay plus a holiday leave loading of 17.5%. Calculate Kate’s annual salary including the annual leave loading rounded to two decimal places. Assume that 1 year comprises 26 fortnights.

Over the 4-week Christmas period, casual staff are given a bonus of $566 and permanent staff are given 17.5% holiday loading.

Calculate the holiday loading for permanent staff rounded to two decimal places given that the normal weekly wage is $908.

Calculate the difference between the holiday loading given to permanent employees and the holiday bonus given to casuals.

Roxanne receives a 4-week annual leave loading of $923 given at a rate of 17.5%. Calculate her fortnightly salary rounded to two decimal places.

Extend your thinking 19

For the ﬁrst 6 months of the year, Quiana’s gross monthly salary is $5990, during which time she takes 2 weeks annual leave with a leave loading of 17.5%. Use the approximation of 52 weeks in a year. a

Calculate the leave loading received for the 2 weeks holiday in the ﬁrst half of the year rounded to two decimal places.

After 6 months, Quiana receives a pay rise and her new gross monthly salary is $7290. Calculate the leave loading received for the 2 weeks holiday she takes in the second half of the year rounded to two decimal places.

Calculate her total annual salary including the annual leave loading.

Consider the two different salary options. Use the approximation of 52 weeks in a year. • Salary A: $64 240 per year (no annual leave loading). • Salary B: $63 300 per year and leave loading of 17.5% on 4 weeks of normal pay.

For Salary B, calculate the annual leave loading amount rounded to two decimal places.

For Salary B, calculate the total annual salary amount rounded to two decimal places.

Which is the higher gross salary?

Amelia earns a salary of $68 698. She is currently entitled to 4 weeks annual pay and 17.5% leave loading. Next year, Amelia’s employer is removing annual leave loading as an entitlement and replacing it with a 1.8% increase in annual salary. Determine whether Amelia is better off with the annual leave loading or the salary increase.

William is an airline pilot. He is paid $6706.43 for 4 weeks holiday pay, which includes an annual leave loading of 17.5%. If his holiday pay is based on a 38-hour week, calculate his hourly rate of pay rounded to two decimal places.

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6 Chapter review 1

Olivia’s annual salary is $75 660. If Olivia gets paid monthly, her monthly income is: A

$6305

$3152.50

$7566

A gardener’s normal hourly rate is $28. What is their time-and-a-half rate? A

$12 610

$35

$42

$56

$28.50

A worker is paid to assemble 30 ﬂat-pack chairs at $12 per chair. What is their total earnings for this task? A

$120

$300

$360

$420

Gerald earns $24.50 per hour working as a barista. He works 22 hours per week. Calculate his weekly wage.

Sam works from 9:00 a.m. to 5:30 p.m., Monday to Friday. He has an unpaid break of 45 minutes each day. His hourly rate is $23.50:

Determine how many hours Sam gets paid for each day.

Calculate his daily wage.

Calculate his weekly wage.

Leo’s annual salary of $58 500 was increased by 3.5%. Calculate, rounded to the nearest cent (using 52 weeks in a year): a

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His new annual salary.

His new weekly salary.

Layla earned $61 200 in one year, and worked an average of 30 hours per week. Use the approximation of 52 weeks in a year: i

Find her equivalent hourly wage, rounded to the nearest cent.

Calculate the increase in her annual salary if her equivalent hourly wage was increased by $3.50.

A factory worker earns an hourly rate of $25. Calculate their income for working: a

30 hours at normal rates and 5 hours at time and a half.

35 hours at normal rates and 3 hours at time and a half.

33 hours at normal rates and 6 hours at double time.

30 hours at normal rates, 3 hours at time and a half, and 4 hours at double time.

Peter has an annual salary of $59 280. He works 40 normal hours per week. Overtime is paid at time and a half, and public holiday shifts are paid at double time. Use the approximation of 52 weeks in a year: a

Calculate his weekly earnings if he works 40 normal hours, 6 overtime hours, and 5 hours on a public holiday in one week.

What would his total annual earnings be?

Mathspace New South Wales – Year 11 Standard mathspace.co

Carlos makes custom keychains and charges $4.50 per keychain: a

How much does Carlos earn for making 30 keychains?

How many keychains must Carlos make to earn $1350?

In one week, Carlos earns $800 by making 100 keychains. Determine the rate per keychain.

Calculate Carlos’s earnings over a fortnight if he makes 40 keychains in the ﬁrst week and 55 keychains in the second week at $4.50 per keychain.

Raj is a salesperson who receives a weekly retainer of $350 and a commission of 4% on the value of goods he sells. In one week, he sells goods worth $12 500: a

Calculate the commission earned.

Determine his total income for the week.

Liam sells electronics and earns a sliding scale commission: 2.5% on sales up to $30 000 and 4.5% on sales above $30 000. He also receives a weekly retainer of $300. In one week, he generates $45 000 in sales: a

Calculate Liam’s commission for the week.

Calculate Liam’s total income for the week, including his retainer.

Chloe earns a weekly retainer of $280 plus a commission of 3.8% on software sales. If she aims to earn $1350 each week, determine the value of software sales she must achieve.

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Ben (19, single, lives with his parents) is a job-seeker eligible for Youth Allowance. Use the tables shown to determine his Youth Allowance payment if he earns $200 per fortnight from a part-time job. To be eligible for Youth Allowance, individuals must satisfy one of these conditions: • Be 16 to 21 years of age and looking for full-time work • Be 16 to 24 years of age and studying full-time • Be 16 or 17 years of age, studying full-time, and have completed Year 12 or equivalent • Be 16 or 17 years of age, studying full-time, and either independent or need to live away from home to study • Be 16 to 24 years of age and undertaking a full-time Australian Apprenticeship This table shows the fortnightly payments that eligible people could receive on Youth Allowance: Circumstances

Maximum fortnightly payment

Single, no children, younger than 18 years old and live at their parents’ home

$410.30

Single, no children, younger than 18 years old and need to live away from their parents’ home to study, train or look for work

$663.30

Single, no children, 18 years or older and live at their parents’ home

$472.50

Single, no children, 18 years or older and required to live away from their parents’ home

$663.30

Single with children

$836.60

Member of a couple with no children

$663.30

Member of a couple with children

$718.10

These tables show how much Youth Allowance a person can receive when they already have an income: For a student on Youth Allowance Earnings

Fortnightly payment reduction

Between $195-$323

50 cents for each dollar earned over $195

More than $323

$64 plus 60 cents for each dollar earned over $323 For a job-seeker on Youth Allowance

Earnings

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Fortnightly payment reduction

Between $154-$262

50 cents for each dollar earned over $154

More than $262

$54 plus 60 cents for each dollar earned over $262

Mathspace New South Wales – Year 11 Standard mathspace.co

David, aged 30, is single with one child and is studying full-time. He is eligible for Austudy. Use the payment table shown to calculate his annual Austudy payment, assuming 52 weeks in a year. This table shows the fortnightly payments that eligible people could receive on Austudy: Circumstances

Payment per fortnight

Single, no children

$663.30

Single, with children

$836.60

In a couple, no children

$663.30

In a couple, with children

$718.10

Maya is single with no children and qualiﬁes for JobSeeker Payment. Her maximum fortnightly payment is $781.10. Use the income test rules to determine her JobSeeker Payment if she earns $250 per fortnight from her casual job. Relevant JobSeeker Payment information: Fortnightly payment

Payment reaches Nil once income reaches this per fortnight

Single, no children

$781.10

$1438.50

Single, principal carer, with a dependent child

$836.50

$1541.83

Single, 55 or older, after 9 continuous months on payment

$836.50

$1541.83

Single, partial capacity to work <15 hours/week

$836.50

$1541.83

$715.10 each

$1304.50

Applicant details

Partnered Your income per fortnight

Reduces your fortnightly payment by

Up to $154

$154 to $262

50 cents for each dollar over $154

Over $262

$54 plus 60 cents for each dollar over $262

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Henry is single and qualiﬁes for the Age Pension. His maximum fortnightly Age Pension (including all supplements based on his circumstances) is $1149.00. Henry earns $500 per fortnight and owns his home plus $350 000 in other assets. The income rules for pensioners are outlined in these tables: Single person

Couple living together or apart due to ill health Amount each member of the couple’s pension reduction

Income per fortnight

Amount of pension reduction

Up to $210 (free area)

Up to $367 (free area)

Over $210

50 cents for each dollar over $210

Over $367

25 cents for each dollar over $367

Combined income per fortnight

The maximum Age Pension is reduced by $3 for every $1000 over the asset limits outlined in the table: Situation

Homeowner

Non-homeowner

Single

$314 000

$566 000

A couple, combined

$470 000

$722 000

A couple, separated due to illness, combined

$470 000

$722 000

A couple, one partner eligible, combined

$470 000

$722 000

Calculate Henry’s Age Pension based on the income test.

Calculate Henry’s Age Pension based on the asset test.

What is the actual Age Pension Henry will receive?

A student qualiﬁes for an allowance of $380 per fortnight. Their job pays $19 per hour. The allowance is reduced by 60 cents for every dollar earned over $180 per fortnight. If income is $180 or less, there is no reduction. Determine how many hours the student can work in a fortnight before their allowance reduces to zero, rounded to the nearest hour.

Liam earns an annual salary of $68 640 and is entitled to 4 weeks of annual leave with a loading of 17.5%. Use the approximation of 52 weeks in a year: a

Calculate his normal weekly pay.

Calculate his normal pay over 4 weeks.

Calculate his annual leave loading for 4 weeks.

Calculate his total holiday pay for 4 weeks.

Aisha earns $26.50 per hour in a 35-hour week. Calculate her holiday loading if it is 17.5% of 4 weeks of normal pay, rounded to two decimal places.

Josh is paid $1250 per week. He takes a 2-week holiday. If Josh is entitled to 17.5% holiday loading, calculate his total pay for the 2 weeks, rounded to two decimal places.

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For the ﬁrst 6 months of the year, David’s gross monthly salary is $6200. He takes 2 weeks annual leave with 17.5% loading during this time. For the next 6 months, his gross monthly salary increases to $6800, and he takes another 2 weeks leave with loading. Use the approximation of 52 weeks in a year for weekly conversion, and round ﬁnal answers to two decimal places: a

Calculate the leave loading received for the 2 weeks holiday in the ﬁrst half of the year.

Calculate the leave loading received for the 2 weeks holiday in the second half of the year.

Calculate his total annual income including all leave loading.

Sarah received $3100 for 3 weeks holiday pay, which included an annual leave loading of 17.5%. If her holiday pay is based on a 36-hour week, calculate her hourly rate of pay, rounded to two decimal places.

Did you know?

Technology has opened up new ways to earn money, like freelancing online, creating digital products, or even becoming a content creator. The internet has made global opportunities accessible to anyone with a computer and the right skills! These digital opportunities also allow individuals to turn their passions into income streams, whether through art, gaming, teaching, or other talents. By embracing technology, Australians can create flexible and innovative ways to earn a living in today’s digital economy.

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Big ideas Taxation calculations and management enable precise financial planning by determining taxable income, applying progressive tax rates, and modelling scenarios to ensure compliance and optimise net earnings.

7 Taxation Chapter outline 7.01 7.02 7.03 7.04 7.05 7.06

Allowable deductions Tax tables PAYG tax Medicare levy Net earnings Yearly tax liability Investigation: Spreadsheets and taxation Chapter 7 review

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7.01 Allowable deductions After this lesson, you will be able to… • define gross income, allowable deductions, and taxable income. • identify common types of allowable work-related deductions. • calculate total allowable deductions from a list of expenses. • calculate taxable income using the formula: Taxable income = Gross income − Allowable deductions. • calculate gross income given taxable income and total allowable deductions.

Allowable deductions Deduction An expense incurred by a worker in relation to their job or profession and which can be taken away from annual earnings to obtain the taxable income. Deductions form part of an individual’s or a company’s tax return. Wage An amount of money that is paid for work or services, based on the time spent or the work done. It is usually calculated by the hour, day or week, and paid at regular intervals, such as daily, weekly or fortnightly. Salary A ﬁxed regular payment made by an employer to an employee for their work or services. The amount of salary paid is usually expressed as an annual sum, which is then divided into equal payments over the course of the year such as weekly, fortnightly or monthly. Tax A compulsory monetary contribution required by a government for its support, levied on incomes, property value, goods purchased. Taxable income The amount of yearly income that is used to calculate an individual’s or company’s payable income tax. Overtime Time worked before or after regular scheduled working hours. A higher rate of pay is often earnt during this time, known as penalty rates.

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To understand allowable deductions, it is first important to understand the concept of gross income. Gross income refers to the total income earned by an individual over a financial year before any deductions or taxes are applied. This can include wages, salaries, bonuses, or income from investments. Allowable deductions are specific work-related expenses that can be subtracted from gross income to reduce the amount of income subject to tax. These deductions are recognised by the Australian Taxation Office (ATO) and must meet certain criteria to be claimed. Common allowable deductions include work-related expenses such as: • Travel costs for work-related purposes (e.g. attending a conference, excluding daily commuting) • Home-office expenses (e.g. electricity for work-from-home setups) • Work-specific clothing and laundry (e.g. distinctive compulsory uniforms, registered non-compulsory uniforms, or protective clothing) • Self-education costs related to current employment • Tools and equipment required for work Other allowable deductions may include union fees, charitable donations, and costs associated with tax preparation, such as accountant fees. Taxable income refers to the portion of an individual’s gross income that remains after allowable deductions are subtracted. It represents the amount of income on which tax is calculated and may include earnings from employment, investments, or other sources. Taxable income = Gross income − Allowable deductions

Exploration Consider a list of expenses an employee might incur over a year. Which of these could be claimed as allowable deductions? Discuss why some expenses might not qualify. 1. $1200 spent on a new laptop for work-from-home tasks 2. $50 weekly fuel costs for commuting to the workplace 3. $200 donated to a registered charity

Example 1 Liam earns $80 000 annually as a graphic designer and receives an additional $5600 from freelance projects. During the ﬁnancial year, he claims the following allowable deductions: • $350 on design software subscriptions • $500 donated to a registered charity • $180 on travel to a design workshop Calculate Liam’s taxable income for the ﬁnancial year.

Create a strategy Subtract his total allowable deductions from his gross income.

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Apply the idea Calculate the total gross income: Gross income = 80 000 + 5600

Add salary and freelance income

= $85 600

Evaluate

Calculate the total allowable deductions: Total deductions = 350 + 500 + 180

Add all allowable deductions

= $1030

Evaluate

Calculate the taxable income: Taxable income = Gross income − Total deductions

Write the formula

= 85 600 − 1030

Substitute the values

= $84 570

Evaluate

Liam’s taxable income for the ﬁnancial year is $84 570.

Reflect and check Confirm that the charitable donation qualifies as an allowable deduction, as it must be to a registered organisation per ATO rules. The travel expense is valid only if the workshop is directly related to Liam’s employment or freelance work.

Example 2 Priya is a nurse who incurs the following allowable work-related deductions during the financial year: • $600 on uniforms • $200 on union fees • $150 on work-related travel (not commuting) Her taxable income for the financial year is $69 050. Calculate Priya’s gross income for the financial year.

Create a strategy Sum the allowable deductions then add this total to the taxable income to calculate the gross income.

Apply the idea Calculate the total allowable deductions: Total deductions = 600 + 200 + 150 = $950

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um the deductions for uniforms, union fees, S and travel Evaluate

Practice 5

Emily earns an annual salary of $58 900 as a nurse. During the ﬁnancial year, she incurs the following expenses: • $320 on medical supplies • $150 on union fees • $90 on tax agent fees Calculate Emily’s taxable income for the ﬁnancial year.

Ex 1

Dave earned a gross income of $60 500 last ﬁnancial year. He gave $160 to various charities, spent $137 on union fees, and spent $119 on uniform cleaning expenses. Calculate to the nearest dollar: a

Dave’s total allowable deductions for the last ﬁnancial year

Dave’s taxable income for the last ﬁnancial year

Aisha earns $45 200 annually as a teacher and receives an additional $3800 from tutoring. She claims the following allowable deductions: • $280 on teaching resources • $450 donated to a charity • $120 on travel to a teaching conference Calculate Aisha’s taxable income for the ﬁnancial year rounded to the nearest dollar.

Ex 2

Jordan incurs the following allowable work-related deductions during the financial year: • $450 on protective work clothing • $180 on union fees • $120 on work-related travel (not commuting) His taxable income for the financial year is $64 270. Calculate Jordan’s gross income for the financial year.

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Maria claims $2 per week in laundry expenses and $21.60 per fortnight in dry cleaning expenses for her uniform. Her taxable income for the year is $42 000. a

Calculate her total annual deductions for laundry and dry cleaning. Assume there are 52 weeks in a year.

If she has no other deductions, calculate Maria’s gross income for the year.

Dylan has a gross income of $92 749 and claims $877 in allowable deductions. He wants to reduce his taxable income to $90 000 by making voluntary superannuation contributions. a

Calculate Dylan’s taxable income before extra contributions.

Calculate the amount he needs to contribute to his superannuation.

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Extend your thinking 11

Yuri’s gross annual income is $51 120. He wants to reduce his taxable income to $50 820 by making superannuation contributions (assuming no other deductions). a

Calculate the amount he needs to contribute to his superannuation.

Calculate the monthly superannuation contribution needed.

Noah has prepared a list of expenses to review with his accountant. For each, justify whether it can be claimed as an allowable deduction: a

The cost of catching public transport to work

The cost of travelling to a work meeting away from his oﬃce

The cost of purchasing work-related books

Kate earns a gross annual salary of $127 857. She donates $22 per month to charity, attends 3 seminars at $600 each, and spends $168 on clinic stationery. Next year, she plans to attend 1 more seminar (same cost) and double her monthly donations. What will her new taxable income be?

Priya earns $68 000 annually. She claims $1200 in work-related deductions. If her taxable income must be below $66 000 to qualify for a tax beneﬁt, how much more must she donate to charity?

Raj earns $85 000 annually and claims $2500 in allowable deductions. He wants to reduce his taxable income below $80 000 to qualify for a grant, using only charitable donations and self-education expenses. The ATO limits self-education claims to $1000. a

Calculate Raj’s taxable income before any additional deductions.

How much must his taxable income decrease to be below $80 000?

If he claims the maximum $1000 for self-education, how much must he donate to charity to achieve this?

What are his total allowable deductions after adding these amounts?

Lena has a gross income of $72 300 and claims $1800 in deductions for union fees and travel. She plans to attend a course costing $2500, but the ATO allows only 60% of self-education expenses over $1000 in the ﬁrst year. She also wants to donate to charity to make her taxable income exactly $67 000. a

Calculate the allowable deduction for the course in the ﬁrst year.

Calculate her taxable income after adding the course deduction to her existing deductions.

How much must she donate to charity to reach a taxable income of $67 000?

What are her total allowable deductions for the year?

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7.02 Tax tables After this lesson, you will be able to… • identify the tax-free threshold and its significance. • interpret information presented in a standard income tax table. • identify the correct tax bracket for a given taxable income. • calculate income tax payable using the rules specified in a tax table. • calculate taxable income given the income tax payable (working backwards).

Tax tables The tax-free threshold is the amount of gross income an individual can earn before being required to pay income tax. For Australian residents in the 2025-2026 ﬁnancial year, this threshold is $18 200, meaning no tax is payable on income up to this amount. The income tax rates for Australian residents, applied to taxable income above the tax-free threshold, are outlined in this table: Resident tax rates (2025-2026) Taxable income

Tax on this income

0 –$18 200

Nil

$18 201 – $45 000

16c for each $1 over $18 200

$45 001 – $135 000

$4288 plus 30c for each $1 over $45 000

$135 001 – $190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

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To calculate the tax payable, identify the income bracket where the taxable income lies, then apply the rule: Income tax payable = Constant amount of tax + Amount over threshold × Variable tax rate For a taxable income of $100 000, the bracket is $45 000 – $135 000. The “Constant amount of tax” is $4288. The “Amount over threshold” is found by evaluating $100 000 – $45 000. The “Variable tax rate” is 30c for each $1, or

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= 30%.

Tax return A tax return is an annual statement of all income, allowable deductions, PAYG tax paid and other personal financial information so as to allow the Australian Taxation Office to calculate the amount of income tax an individual should pay for the financial year.

At the end of the financial year, individuals must file a tax return to determine whether they have paid the correct amount of tax over the year through the PAYG system, which withholds an amount of tax each payslip. Normally, the amount withheld will be approximately the right amount to cover your tax liability at the end of the financial year. If too much has been withheld, you will receive a refund when you lodge your tax return. If not enough has been withheld, you will have a tax liability.

Example 1 Nathan’s annual taxable income for the 2025-2026 ﬁnancial year is $72 500. He claims the tax-free threshold. Use the following resident tax table to calculate his income tax payable: Resident tax rates (2025-2026) Taxable income

Tax on this income

0 – $18 200

Nil

$18 201 – $45 000

16c for each $1 over $18 200

$45 001 – $135 000

$4288 plus 30c for each $1 over $45 000

$135 001 – $190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Create a strategy Locate the income bracket where Nathan’s taxable income lies, and apply the rule: Income tax payable = Constant amount of tax + Amount over threshold × Variable tax rate

Apply the idea The taxable income of $72 500 falls within the bracket $45 001 – $135 000, where the tax is $4288 plus 30 cents for each $1 over $45 000. Income tax payable = 4288 + (72 500 − 45 000) × 0.30

Substitute the values

= 4288 + 27 500 × 0.30

Calculate amount over threshold

= 4288 + 8250

Multiply to ﬁnd additional tax

= $12 538

Evaluate

Nathan’s income tax payable for the ﬁnancial year is $12 538.

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Reﬂect and check This could also have been done one step at a time: 1. Determine the tax bracket: $72 500 falls in the $45 001 – $135 000 bracket. 2. Determine the amount above the threshold. The $4288 covers tax for the ﬁrst $45 000. Now, identify the amount above the threshold: 72 500 − 45 000 = $27 500 3. Determine the variable amount of income tax. The tax paid on this $27 500 is 30c in each $1, which is calculated as: 0.30 × 27 500 = $8250 4. Calculate the total amount of income tax. Therefore, the total income tax paid on $72 500 is: Total income tax = 4288 + 8250 Total income tax = $12 538

Example 2 Sanjay wants his income tax payable to be exactly $15 000 in 2025-2026. He does not claim the tax-free threshold. Use the 2025-2026 resident tax table to determine his taxable income: Resident tax rates (2025-2026) Taxable income

Tax on this income

0 − $18 200

Nil

$18 201 − $45 000

16c for each $1 over $18 200

$45 001 − $135 000

$4288 plus 30c for each $1 over $45 000

$135 001 − $190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Create a strategy Since Sanjay does not claim the tax-free threshold, tax is payable on all income. Test tax brackets to ﬁnd the taxable income where the tax payable equals $15 000 using the formula: Income tax payable = Constant amount of tax + Amount over threshold × Variable tax rate

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Practice Use this 2025-2026 resident tax table to answer questions 5-16. Resident tax rates (2025-2026) Taxable income

Tax on this income

0 – $18 200

Nil

$18 201 – $45 000

16c for each $1 over $18 200

$45 001 – $135 000

$4288 plus 30c for each $1 over $45 000

$135 001 – $190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Ex 1

Sophie’s annual taxable income for 2025-2026 is $68 300. She claims the tax-free threshold. Use the resident tax table to calculate her income tax payable income.

Ex 2

Priya wants her income tax payable to be exactly $10 000 in 2025-2026. She does not claim the tax-free threshold. Use the 2025-2026 resident tax table to determine her taxable income.

Carl works 4 days a week for 40 weeks, earning $349 per day. Use the 2025-2026 resident tax table to determine: a

Carl’s annual gross income.

His income tax payable assuming Carl has no allowable deductions.

Hermione’s taxable income last year was $203 400. Use the 2025-2026 resident tax table to calculate her income tax payable.

Emma’s taxable income is $97 589. Use the 2025-2026 resident tax table to calculate her income tax payable.

Mia’s annual taxable income is $145 670. Use the 2025-2026 resident tax table to calculate her income tax payable.

Carmen’s taxable income last year was $197 955. Use the 2025-2026 resident tax table to calculate her income tax payable.

Jake earns taxable income from his job and a bonus. The tax on his $40 000 salary is $3488.64. His bonus increases his total tax by an additional $900. Using the 2025-2026 resident tax table, calculate Jake’s total taxable income rounded to two decimal places.

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Extend your thinking 13

Lena earns taxable income from two jobs. Her primary job has a tax payable of $5000 with the tax-free threshold claimed. Her secondary job increases her total tax payable to $10 000. Use the 2025-2026 resident tax table to calculate Lena’s total taxable income from her two jobs.

Stewie, a resident taxpayer, calculated his 2025-2026 income tax using the taxable income ﬁgure of $65 000: Tax = $4288 + 0.30 × ($65 000 − $45 000) = $4288 + 0.30 × $20 000 = $4288 + $7000 = $11 288

Identify and explain Stewie’s error.

Calculate the correct tax payable using the 2025-2026 resident tax table.

A proposal increases the tax-free threshold to $20 000, with 20c per $1 over $20 000 up to $45 000, keeping tax for incomes over $45 001 unchanged. Test this with a taxable income of $50 000: a

Calculate the original tax payable.

Calculate the proposed tax payable.

Who does the proposal beneﬁt?

A tax reform suggests a ﬂat rate of 25c per $1 over $18 200 up to $135 000. Alex has a taxable income of $80 000. Determine Alex’s: a

Original tax payable

Reformed tax payable

Explain which system beneﬁts Alex more.

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Income tax is a tax levied on an individual’s taxable income, collected by the Australian Taxation Office (ATO) to fund public services such as education and healthcare. Pay-As-You-Go (PAYG) tax A system for making regular tax instalments which are removed from gross pay towards the expected income tax liability for that financial year. The amount of tax deducted depends on the worker’s taxable income for the pay period, which is the gross pay minus allowable deductions. The ATO provides PAYG tax tables to determine the exact amount to withhold, available in weekly, fortnightly, and monthly formats. PAYG tax tables account for the tax-free threshold, which exempts the first $18 200 of annual income from tax if claimed. Workers typically claim this threshold for their primary job, reducing the tax withheld per pay period. For secondary jobs, it is not claimed, resulting in tax being withheld from all earnings in that job. To calculate the tax to be deducted, the taxable income for the pay period is matched to the appropriate amount in the PAYG tax table, based on whether the tax-free threshold is claimed. These tables list whole-dollar earnings and corresponding tax amounts. Note: When using PAYG withholding tables, always round the income down to the nearest whole dollar before using the table.

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Example 3 The PAYG tax amount, y, that an employer must withhold each week, in January 2025, was given by the formula: y = ax − b where x is the number of whole dollars of weekly income, plus 99 cents, and a and b are values obtained from this table: Weekly earnings (x) less than

$355

$422

0.1493

53.1493

$528

0.1887

69.8113

$711

0.2186

85.6380

$1282

0.2872

134.4827

$1730

0.3192

175.5196

$3400

0.39

357.3861

$3652 and over

0.47

648.9053

Note: This table is based on tax withheld with the tax-free threshold. Jeremy’s gross weekly earnings are $959.50. Calculate the amount of PAYG tax his employer will need to withhold from his wage each week.

Create a strategy To calculate the amount of PAYG tax, use the table and substitute the appropriate values of a and b into the formula.

Apply the idea Notice that the number of whole dollars in Jeremy’s weekly income is 959. Adding 99 cents gives x = $959.99. This amount is greater than $711, but less than $1282, so from the ﬁfth row of the table, use the values a = 0.2872 and b = 134.4827. y = ax − b

Write the formula

= 0.2872 × 959.99 − 134.4827

Substitute a, x, b

= 141.4306

Evaluate

= $141

Round to the nearest dollar

Tax withholding amounts are always rounded to the nearest dollar, so Jeremy’s employer would deduct $141 in income tax from his gross weekly pay.

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Ex 3

Ex 4

Alex earns $800 weekly with a PAYG tax withheld using the formula y = ax − b, where x is whole dollars plus 99 cents. Use the table to calculate the amount of PAYG tax his employer will need to withhold from his wage each week, rounded to the nearest dollar. Weekly earnings (x) less than

$355

$422

0.1493

53.1493

$528

0.1887

69.8113

$711

0.2186

85.6380

$1282

0.2872

134.4827

$1730

0.3192

175.5196

$3400

0.39

357.3861

$3652 and over

0.47

648.9053

Lena earns $1200 weekly. Her employer withholds $307 in PAYG tax using the formula y = ax − b, where x is whole dollars plus 99 cents. If the tax-free threshold is not claimed, b is adjusted by subtracting 54.86. Use the table to determine: Weekly earnings (x) less than

$355

$422

0.19

67.4635

$528

0.29

109.7327

$711

0.21

135.3985

$1282

0.3477

165.4423

$1730

0.345

161.9808

$3461

0.39

239.8654

$3461 and over

0.47

516.7885

PAYG tax if threshold is claimed.

PAYG tax if threshold is not claimed.

Does she claim the tax-free threshold?

Maria earns a weekly taxable income of $2737 from her second job and does not claim the tax-free threshold. Using the ATO PAYG weekly tax table, calculate the amount of tax that should be withheld from her weekly pay.

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Sophie’s weekly taxable income increases from $2730 to $3080. She claims the tax-free threshold. Use the tax table to calculate the change in her PAYG tax withheld: a

Original PAYG tax withheld

New PAYG tax withheld

Change in PAYG tax withheld

Carl’s fortnightly PAYG tax withheld is $1684 with the tax-free threshold. Use the weekly tax table to determine: a

Weekly PAYG tax withheld

Weekly taxable income range

Fortnightly taxable income range

Jeremy earns $5827 weekly across two jobs: 53% from his primary job with the tax-free threshold, and 47% from his secondary job without it. Use the weekly tax table to find his total fortnightly PAYG tax withheld: a

Weekly tax from primary job

Weekly tax from secondary job

Total fortnightly PAYG tax withheld

Did you know?

Parking metres are small but important tools in local taxation! Fees collected from parking help councils fund road maintenance, public spaces, and community services, keeping cities functional and accessible for everyone. They also encourage turnover in busy areas, making it easier for more people to find parking, while promoting the use of public transport and reducing traffic congestion. Properly managed parking systems can even improve local business by increasing customer access and convenience.

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7.04 Medicare levy After this lesson, you will be able to… • explain the purpose of the Medicare levy. • identify the standard Medicare levy rate (e.g., 2%). • calculate the Medicare levy given a taxable income and the standard rate. • verify the correctness of a withheld Medicare levy amount. • understand that the Medicare levy is proportional to taxable income at the standard rate.

Medicare levy Medicare levy A tax that ﬁnances the national health insurance program, Medicare, for all Australians, calculated as a percentage of taxable income. The standard Medicare levy for the 2025-2026 ﬁnancial year is 2%.

For most individuals, it is calculated as a ﬁxed percentage, 2% of their taxable income, provided they meet certain conditions. This rate applies unless speciﬁc exemptions or reductions are applicable, such as for low-income earners or those with certain family circumstances, which are not covered in this lesson.

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Example 1 Calculate the Medicare levy for a taxable income of $48 500 in the 2025-2026 ﬁnancial year, assuming the standard rate applies.

Create a strategy Multiply the taxable income by 2% to ﬁnd the Medicare levy.

Apply the idea Medicare levy = 48 500 × 0.02 = 970

Multiply taxable income by 2% Evaluate

The Medicare levy is $970.

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7.04 Practice questions What do you remember? 1

What is the purpose of the Medicare levy in Australia?

What is the standard Medicare levy rate for the 2025-2026 ﬁnancial year?

How is the Medicare levy calculated?

Deﬁne taxable income in the context of the Medicare levy.

Practice Ex 1

Ex 2

Calculate the Medicare levy for a taxable income of $36 700 in the 2025-2026 ﬁnancial year, assuming the standard rate applies.

Sophie has a taxable income of $55 000. Calculate her Medicare levy for the 2025-2026 ﬁnancial year.

Liam earns a taxable income of $28 400:

What is his Medicare levy?

If Liam’s taxable income increases by $5000, what is his new Medicare levy?

Amelia has a taxable income of $61 200 last 2025-2026 ﬁnancial year. Her employer withholds $1224 for the Medicare levy. The resident tax rates for 2025-2026 are shown: Resident tax rates (2025-2026) Taxable income

Tax on this income

0 – $18 200

Nil

$18 201 – $45 000

16c for each $1 over $18 200

$45 001 – $135 000

$4288 plus 30c for each $1 over $45 000

$135 001 – $190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Verify if the withheld amount is correct, assuming the standard Medicare levy rate applies. 9

Ethan’s taxable income is $42 500. His employer withholds $850 for the Medicare levy. Is the withheld amount correct? Explain your answer.

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Olivia earns a taxable income of $95 000 in the 2025-2026 ﬁnancial year and receives a taxable bonus of $12 000: a

Calculate her total Medicare levy, including the bonus.

Another employee, Sam, has a taxable income of $107 000 and no bonus. How much more is Olivia’s Medicare levy compared to Sam’s?

A part-time worker has a taxable income of $35 465. Calculate the Medicare levy then express it as a percentage of their taxable income.

Emma has a taxable income of $90 000 in the 2025-2026 ﬁnancial year. Her employer withholds $1810 for the Medicare levy. Verify if the withheld amount is correct, assuming the standard Medicare levy rate applies.

Extend your thinking 13

Zara’s employer withholds $2100 for her Medicare levy, but the correct amount should be $2000. What is Zara’s correct taxable income, and by how much was the withholding incorrect? Use this 2025-2026 resident tax table to answer questions 14 and 15. Resident tax rates (2025-2026) Taxable income

Tax on this income

0 – $18 200

Nil

$18 201 – $45 000

16c for each $1 over $18 200

$45 001 – $135 000

$4288 plus 30c for each $1 over $45 000

$135 001 – $190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Noah has a taxable income of $68 400 in the 2025-2026 ﬁnancial year. He receives a promotion that changes his taxable income in the next year. The resident tax rates for 2025-2026 are shown: a

Calculate Noah’s Medicare levy for his current taxable income.

After his promotion, Noah’s taxable income increases by 15%. Calculate his new Medicare levy.

By what percentage does Noah’s total tax payable increase due to the promotion?

A small business owner has a taxable income in the 2025-2026 financial year. Their total tax payable, Medicare levy included is $37 288. Calculate their taxable income.

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Two colleagues, Ava and Ben, have taxable incomes of $50 000 and $75 000, respectively. Show that the ratio of their Medicare levies is equal to the ratio of their taxable incomes.

Consider Charlotte’s taxable income of $80 000: a

She claims her Medicare levy was calculated incorrectly as $1800. What is the correct levy?

Explain the likely error in Charlotte’s calculation.

Did you know?

The Medicare levy ensures Australians have access to free or affordable hospital care! By contributing just a small percentage of income, taxpayers help maintain a healthcare system that supports millions of people every year.

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7.05 Net earnings After this lesson, you will be able to… • define net earnings and its components. • recall and apply the formula for taxable income. • calculate income tax payable based on taxable income and tax tables. • calculate the Medicare levy based on taxable income. • calculate net earnings by subtracting all relevant amounts from gross income.

Net earnings Net earnings The amount of gross pay remaining after tax and other deductions have been made.

Gross income is the total income earned before deductions or taxes, including wages and additional earnings. Allowable deductions are speciﬁc expenses (like work-related costs, union fees) subtracted from gross income to determine taxable income. Income tax is calculated on taxable income using resident tax rates and withheld progressively via the PAYG system. The Medicare levy, typically 2% of taxable income, is also deducted from Gross income. The formula for net earnings is: Net earnings = Gross income − Allowable deductions − Income tax − Medicare levy The resident tax rates for 2025-2026, effective from 1 July 2025, are shown: Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

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Example 1 Emma earns an annual gross income of $35 000 as a part-time retail worker in the 2025-2026 ﬁnancial year. She has no allowable deductions. Using the resident tax rates provided and a Medicare levy of 2%, calculate her net earnings.

Create a strategy Since there are no deductions, taxable income equals gross income. Determine income tax using the tax table, compute the Medicare levy as 2% of taxable income, and subtract income tax and the levy from gross income to ﬁnd net earnings.

Apply the idea Taxable income is $35 000, which falls in the $18 201–$45 000 bracket. Income tax is 16c per $1 over $18 200. Amount over threshold = 35 000 − 18 200 = 16 800 Income tax = 16 800 × 0.16 = 2688

Calculate excess over $18 200 Evaluate Multiply by 16c per $1 Evaluate

Income tax is $2688. Calculate the Medicare levy: Medicare levy = 35 000 × 0.02 = 700

Multiply taxable income by 2% Evaluate

Medicare levy is $700. Calculate net earnings: Net earnings = 35 000 − 2688 – 700 Subtract income tax and Medicare levy from gross income = 31 612

Evaluate

Emma’s net earnings are $31 612.

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Example 2 Sarah earns a gross income of $100 000 annually as a project manager in the 2025-2026 ﬁnancial year. She claims allowable deductions totaling $2000 for work-related expenses. Using the resident tax rates provided and a Medicare levy of 2%, calculate her net earnings: Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Create a strategy Subtract allowable deductions from gross income to ﬁnd taxable income. Use the tax table to calculate income tax, compute the Medicare levy as 2% of taxable income, and subtract deductions, income tax, and the levy from gross income to ﬁnd net earnings.

Apply the idea Calculate taxable income: Taxable income = 100 000 – 2000 = 98 000

Subtract deductions from gross income Evaluate

Taxable income is $98 000, which falls in the $45 001–$135 000 bracket. Income tax is $4288 plus 30c per $1 over $45 000. Amount over threshold = 98 000 − 45 000 = 53 000 Variable tax = 53 000 × 0.30 = 15 900 Income tax = 4288 + 15 900 = 20 188

Calculate excess over $45 000 Evaluate Multiply by 30c per $1 Evaluate Add ﬁxed tax and variable tax Evaluate

Income tax is $20 188. Calculate the Medicare levy: Medicare levy = 98 000 × 0.02 = 1960

Multiply taxable income by 2% Evaluate

Medicare levy is $1960. Calculate net earnings: Net earnings = 100 000 − 2000 − 20 188 – 1960 Subtract deductions, income tax, and Medicare levy from gross income = 76 252 Sarah’s net earnings are $76 252.

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Evaluate

Explain why allowable deductions can signiﬁcantly affect net earnings, even if the gross income remains the same.

Liam earns a gross income of $80 000 in the 2025-2026 ﬁnancial year and adjusts his allowable deductions over time. Using the resident tax rates and a Medicare levy of 2%:

Calculate his net earnings with $5000 in deductions.

If Liam’s deductions increase to $10 000 with the same gross income, calculate his new net earnings.

By what percentage does Liam’s net earnings increase due to the additional deductions?

Explain why the net earnings increase is not proportional to the deduction increase.

Emma and Lucas have gross incomes of $60 000 and $65 000, respectively, with no deductions in the 2025-2026 ﬁnancial year. Using the resident tax rates and a Medicare levy of 2%: a

Calculate the difference in their net earnings.

If Emma claims $5000 in deductions, how does this affect the difference in their net earnings?

Did you know?

Taxes on imported goods help protect local industries and jobs! They make domestic products more competitive and support the national economy. These taxes, called tariffs, are especially important for industries like agriculture and manufacturing. By reducing reliance on foreign products, they help maintain Australia’s economic stability. This also encourages local businesses to grow and innovate, creating more opportunities for workers.

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7.06 Yearly tax liability After this lesson, you will be able to… • define yearly tax liability. • calculate total tax liability by summing income tax and Medicare levy. • explain the purpose of a tax return in reconciling tax paid with tax owed. • compare total tax liability with PAYG withholdings to determine if a refund is due or additional tax is payable. • calculate the amount of a tax refund or additional tax payable.

Yearly tax liability Yearly tax liability represents the total amount of income tax and Medicare levy an individual must pay based on their taxable income for the ﬁnancial year. Comparing this to tax withheld through the PAYG system determines whether a refund is owed or additional tax is payable upon ﬁling a tax return. Total tax liability is the sum of income tax and the Medicare levy: Total tax liability = Income tax + Medicare levy A tax return reconciles PAYG withholdings with the total tax liability. If withholdings exceed the liability, a refund is owed; if less, additional tax is payable. The resident tax rates for 2025-2026, effective from 1 July 2025, are shown: Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Interactive exploration Discover this concept in action online

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Example 1 Kenneth earns an annual gross income of $68 500 as an IT technician in the 2025-2026 ﬁnancial year. He claims allowable deductions of $600 for professional subscriptions and $400 for work-related training. His employer withheld $10 500 in PAYG tax. Using the resident tax rates table and a Medicare levy of 2%: a Calculate Kenneth’s taxable income, income tax, and Medicare levy.

Create a strategy Subtract total allowable deductions from gross income to ﬁnd taxable income. Use the tax table to calculate income tax, then compute the Medicare levy as 2% of taxable income.

Apply the idea Calculate total allowable deductions: Total deductions = 600 + 400 = 1000

Add all allowable deductions Evaluate

Calculate taxable income: Taxable income = 68 500 – 1000 = 67 500

Subtract deductions from gross income Evaluate

Taxable income is $67 500, which falls in the $45 001–$135 000 bracket. Income tax is $4288 plus 30c per $1 over $45 000. Amount over threshold = 67 500 − 45 000 = 22 500 Variable tax = 22 500 × 0.30 = 6750 Income tax = 4288 + 6750 = 11 038

Calculate excess over $45 000 Evaluate Multiply by 30c per $1 Evaluate Add ﬁxed tax and variable tax Evaluate

Income tax is $11 038. Calculate the Medicare levy: Medicare levy = 67 500 × 0.02 = 1350 Medicare levy is $1350.

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Multiply taxable income by 2% Evaluate

b Determine if Kenneth owes additional tax or is entitled to a refund.

Create a strategy Calculate total tax liability by adding income tax and Medicare levy. Compare with PAYG withholdings to determine a refund or liability.

Apply the idea Calculate total tax liability: Total tax liability = 11 038 + 1350 = 12 388

Add income tax and Medicare levy Evaluate

Total tax liability is $12 388. Compare with PAYG withholdings: Refund or liability = 10 500 − 12 388 Subtract total tax liability from PAYG withholdings = −1888

Evaluate

Since PAYG withholdings ($10 500) are less than the total tax liability ($12 388), Kenneth owes additional tax of $1888.

Reﬂect and check The tax liability arises because PAYG withholdings underestimated Kenneth’s total tax, possibly due to higher income not fully accounted for in withholdings. Verify deduction eligibility: subscriptions and training must be directly work-related per ATO rules.

Example 2 Sarah has an annual gross income of $42 000 as a part-time receptionist in the 2025-2026 ﬁnancial year, with no allowable deductions. Her employer withheld $4800 in PAYG tax. Using the resident tax rates table and a Medicare levy of 2%, determine whether Sarah owes additional tax or is entitled to a refund.

Create a strategy Calculate taxable income, then determine income tax using the tax table and the Medicare levy as 2% of taxable income. Sum these to ﬁnd total tax liability. Compare with PAYG withholdings to determine a refund or liability.

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7.06 Practice questions What do you remember? 1

What is meant by yearly tax liability in Australia’s taxation system?

Write the formula for calculating total tax liability.

How does a tax return determine whether a refund or additional tax is owed?

If you have a PAYG tax withheld of $5000, determine how much you would owe or receive as a refund for each total tax liability: a

$4500

$6700

$3800

$5500

Practice Use this 2025-2026 resident tax table to answer questions 5-7. Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Emma earns a gross income of $38 000 as a retail assistant, with no deductions. Her employer withheld $4200 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%, determine whether Emma owes additional tax or is entitled to a refund.

Oliver earns a gross income of $50 000 with no deductions. His employer withheld $6800 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%, determine if he owes additional tax or is entitled to a refund.

Lily has a taxable income of $45 000 and a reported total tax liability of $5188. Using the resident tax rates and a Medicare levy of 2%, verify the components of her total tax liability: a

Conﬁrm her income tax amount is correct.

Conﬁrm her Medicare levy and total tax liability are correct.

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Use this 2025-2026 resident tax table to answer questions 8-15. Resident tax rates (2025-2026) Taxable income

Ex 1

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Ava’s gross income is $72 000 with $1500 in deductions for union fees. Her employer withheld $12 000 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%, determine if she owes additional tax or is entitled to a refund.

Zara’s taxable income is $80 000. She paid $14 000 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%, determine if she owes additional tax or is entitled to a refund.

Lucas earns a gross income of $90 300 as a mechanic. He claims deductions of $500 for tools and $300 for work-related courses. His employer withheld $9200 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%:

Calculate Lucas’s taxable income, income tax, and Medicare levy.

Determine if Lucas owes additional tax or is entitled to a refund.

Amelia and Ben have taxable incomes of $50 000 and $60 000, respectively, with no deductions. Using the resident tax rates and a Medicare levy of 2%: a

Calculate the difference in their total tax liabilities.

If both had PAYG withholdings of $8000, who owes more additional tax?

Ethan and Kim have taxable incomes of $60 000 and $65 000, respectively. Their PAYG withholdings are $9000 for Ethan and $10 000 for Olivia. Using the resident tax rates and a Medicare levy of 2%: a

Calculate the difference in their total tax liabilities.

Who receives a larger refund or owes less tax?

Sophie’s taxable income is $44 000. She wants to know how much additional income she can earn before entering the next tax bracket. Using the resident tax rates: a

Determine the additional income she can earn without entering the next tax bracket.

If she earns an additional $2000, calculate the increase in her income tax compared to her current income tax.

Grace earns a gross income of $58 000 as a librarian in the 2025-2026 ﬁnancial year, with $1500 in deductions for professional development. Her employer withheld $8700 in PAYG tax, based on an estimated withholding rate of 15% of her gross income. Using the resident tax rates and a Medicare levy of 2%. Verify if her PAYG withholdings match the estimated 15% of her gross income.

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Ex 2

Jessie earns a gross income of $42 000 as a receptionist in the 2025-2026 ﬁnancial year, with no allowable deductions. Her employer withheld $4800 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%, verify if she is entitled to a refund of $152.

Extend your thinking Use this 2025-2026 resident tax table to answer questions 16-18. Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Jack’s gross income is $75 000 with PAYG withholdings of $12 500. He wants to claim enough deductions to break even (no refund or additional tax owed). Using the resident tax rates and a Medicare levy of 2%: a

Calculate his total tax liability with no deductions.

Determine the deductions needed to make his total tax liability equal his PAYG withholdings.

Hannah earns a gross income of $60 000 in the 2025-2026 ﬁnancial year as a graphic designer. Her employer withheld $10 000 in PAYG tax. She can claim deductions for work-related expenses up to $4000. Using the resident tax rates and a Medicare levy of 2%: a

Determine whether she has refund or additional tax owed with no deductions.

Determine how much her total tax liability decreases for each $1000 in deductions claimed, assuming taxable income remains in the $45 001–$135 000 bracket.

Calculate the minimum deduction amount needed to achieve a refund of at least $500, or state if it’s not possible within the deduction limit.

Shirley earns a gross income of $85 000 with varying deductions this ﬁnancial year. Her employer withheld $14 000 in PAYG tax. Using the resident tax rates and a Medicare levy of 2%: a

Calculate her total tax liability with $2500 in deductions.

Does she owe additional tax or get a refund with $2500 in deductions?

If her deductions increase to $5000 without changing gross income, recalculate her total tax liability.

With the new deductions and same PAYG withholdings, does she now owe tax or get a refund?

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Explain why PAYG withholdings may not match an individual’s total tax liability, even if their income and deductions are accurately reported.

Liam earns a gross income of $100 000 with $3000 in deductions this year. His PAYG withholdings were incorrectly calculated as $20 000. Using the resident tax rates and a Medicare levy of 2%: Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Calculate his actual total tax liability.

Determine if he owes additional tax or is entitled to a refund.

Explain the likely error in the PAYG withholdings calculation.

Investigation: Spreadsheets and taxation Investigate online

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7 Chapter review 1

Sarah earns an annual salary of $62 500 as a graphic designer. During the ﬁnancial year, she incurs the following allowable work-related expenses: • $250 on new design software subscription • $120 on professional association fees • $80 on fees paid to her tax agent What is her taxable income for the ﬁnancial year? A

$62 950

$62 250

$62 050

$62 130

Tim’s taxable income for the 2025-2026 ﬁnancial year is $48 000. He claims the tax-free threshold. Using the resident tax table provided, which of the following is his income tax payable? Resident tax rates (2025-2026) Taxable income

A 3

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

$4288

$5000

$5188

If Chen’s taxable income for the 2025-2026 ﬁnancial year is $75 000, what is his Medicare levy, assuming the standard rate of 2% applies? A

$900

Tax on this income

$750

$1500

$15 000

$3000

Michael earned a gross income of $70 200 last ﬁnancial year as a consultant. He made $200 in donations to registered charities, spent $150 on professional association fees, and paid $100 for work-related journal subscriptions. Calculate: a

Michael’s total allowable deductions for the last ﬁnancial year.

Michael’s taxable income for the last ﬁnancial year.

Aisha earns an annual salary of $72 000. She currently claims $1500 in general work-related deductions. To qualify for a speciﬁc government rebate, her taxable income must be below $69 000. How much more must she contribute to her superannuation fund (assume this is an allowable deduction in this scenario) to meet this requirement?

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Use this 2025-2026 resident tax table to answer questions 6 and 7. Resident tax rates (2025-2026) Taxable income

520

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Ben works as a contractor 5 days a week for 25 weeks of the year, earning a daily rate of $380. Use the 2025-2026 resident tax table provided to determine: a

Ben’s annual gross income.

His income tax payable for the year, assuming Ben has no additional allowable deductions.

A government proposal aims to change the tax rate for the $18 201 − $45 000 income bracket to 18c for each $1 over $18 200. Other brackets from the 2025-2026 resident tax table remain unchanged. For an individual with a taxable income of $40 000: a

Calculate the income tax payable under the current 2025-2026 rates.

Calculate the income tax payable under the proposed new rate for that bracket.

Would this individual pay more or less tax under the proposal, and by how much would their tax change?

Mathspace New South Wales – Year 11 Standard mathspace.co

James earns a weekly taxable income of $945 from his second job and does not claim the tax-free threshold. Use the provided excerpt from the weekly tax table to determine the tax withheld from his weekly pay: Amount to be withheld Weekly earnings ($)

With tax-free threshold ($)

No tax-free threshold ($)

944.00

50.00

210.00

946.00

50.00

212.00

948.00

52.00

212.00

950.00

52.00

212.00

952.00

52.00

212.00

954.00

52.00

214.00

956.00

54.00

214.00

958.00

54.00

214.00

960.00

54.00

214.00

962.00

56.00

214.00

964.00

56.00

216.00

966.00

56.00

216.00

968.00

56.00

216.00

970.00

58.00

216.00

972.00

58.00

218.00

974.00

58.00

218.00

976.00

60.00

218.00

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Use this table to answer questions 9 and 10.

522

Amount to be withheld

Weekly With tax-free No tax-free earnings ($) threshold ($) threshold ($)

2726.00

777.00

914.00

3068.00

910.00

1,047.00

2727.00

777.00

914.00

3069.00

911.00

1,047.00

2728.00

778.00

914.00

3070.00

911.00

1,048.00

2729.00

778.00

915.00

3071.00

911.00

1,048.00

2730.00

778.00

915.00

3072.00

912.00

1,048.00

2731.00

779.00

915.00

3073.00

912.00

1,049.00

2732.00

779.00

916.00

3074.00

913.00

1,049.00

2733.00

780.00

916.00

3075.00

913.00

1,050.00

2734.00

780.00

917.00

3076.00

913.00

1,050.00

2735.00

780.00

917.00

3077.00

914.00

1,050.00

2736.00

781.00

917.00

3078.00

914.00

1,051.00

2737.00

781.00

918.00

3079.00

915.00

1,051.00

2738.00

782.00

918.00

3080.00

915.00

1,052.00

2739.00

782.00

919.00

3081.00

915.00

1,052.00

2740.00

782.00

919.00

3082.00

916.00

1,052.00

2741.00

783.00

919.00

3083.00

916.00

1,053.00

2742.00

783.00

920.00

3084.00

917.00

1,053.00

2743.00

784.00

920.00

3085.00

917.00

1,054.00

2744.00

784.00

921.00

3086.00

917.00

1,054.00

2745.00

784.00

921.00

3087.00

918.00

1,054.00

2746.00

785.00

921.00

3088.00

918.00

1,055.00

3066.00

910.00

1,046.00

3089.00

918.00

1,055.00

3067.00

910.00

1,047.00

3090.00

919.00

1,055.00

Kevin earns a gross fortnightly income of $6300.80 and claims the tax-free threshold. His allowable fortnightly deductions are $150. Use the provided weekly tax tables to calculate: a

Kevin’s weekly taxable income.

His total fortnightly PAYG tax withheld.

Fatima earns a total of $5830 weekly from two jobs. She claims the tax-free threshold on her primary job, which accounts for 53% of her total weekly income. She does not claim the taxfree threshold on her secondary job. Use the weekly tax tables provided to ﬁnd: a

The PAYG tax withheld from her primary job.

The PAYG tax withheld from her secondary job.

Her total weekly PAYG tax withheld.

Mathspace New South Wales – Year 11 Standard mathspace.co

David earns a taxable income of $32 500 for the 2025-2026 ﬁnancial year. The Medicare levy is 2%: a

What is his Medicare levy?

If David’s taxable income increases by $7000 in the following year, what will his new Medicare levy be?

Marcus has a taxable income of $72 500 in the 2025-2026 ﬁnancial year. The standard Medicare levy is 2%. Refer to this table for resident tax rates (2025-2026) for any relevant questions: Resident tax rates (2025-2026) Taxable income

Tax on this income

0–$18 200

Nil

$18 201–$45 000

16c for each $1 over $18 200

$45 001–$135 000

$4288 plus 30c for each $1 over $45 000

$135 001–$190 000

$31 288 plus 37c for each $1 over $135 000

$190 001 and over

$51 638 plus 45c for each $1 over $190 000

Calculate Marcus’s Medicare levy for his current taxable income.

After a pay rise, Marcus’s taxable income increases by 12%. Calculate his new Medicare levy.

By what percentage does Marcus’s total tax payable increase due to the pay rise? (Show your calculations for old and new total tax).

Chloe earns a gross annual income of $58 900 and has no allowable deductions. Calculate her net earnings for the 2025-2026 ﬁnancial year using the resident tax rates and a Medicare levy of 2%.

Daniel earns a gross income of $55 000 as a graphic designer and an additional $5200 from freelance projects in the 2025-2026 ﬁnancial year. He claims allowable deductions of $750 for software subscriptions and $300 for professional membership fees. Use the resident tax rates and a Medicare levy of 2% to calculate Daniel’s net earnings.

Priya’s taxable income is $70 000. Her allowable deductions were $2500. For the 2025-2026 ﬁnancial year using the resident tax rates and a Medicare levy of 2%, calculate: a

Her gross income.

Her net earnings.

Brian earns a gross annual income of $41 500 as a shop assistant and has no allowable deductions. His employer withheld $4706 in PAYG tax during the 2025-2026 ﬁnancial year. Using the resident tax rates and a Medicare levy of 2%, determine whether Brian owes additional tax or is entitled to a refund, and state the amount.

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Laura earns a gross annual income of $95 800 as an engineer. She claims allowable deductions of $600 for safety equipment and $450 for a professional training course. Her employer withheld $22 216 in PAYG tax for the 2025-2026 ﬁnancial year. Using the resident tax rates and a Medicare levy of 2%: a

Calculate Laura’s taxable income, income tax payable, and Medicare levy.

Determine if Laura owes additional tax or is entitled to a refund, and state the amount.

Samuel’s gross annual income is $82 000. His employer withheld $15 000 in PAYG tax for the 2025-2026 ﬁnancial year. He wants to claim enough allowable deductions to receive a tax refund of exactly $500. Using the resident tax rates and a Medicare levy of 2%: a

What would Samuel’s total tax liability need to be for him to receive a $500 refund?

Determine the taxable income that would result in this total tax liability.

Calculate the total amount of allowable deductions Samuel would need to claim.

Liam has a gross income of $75 500 and currently claims $2000 in deductions for union fees and professional development. He plans to undertake a self-education course costing $3000. The Australian Taxation Office (ATO) allows 50% of self-education expenses exceeding $800 to be claimed as a deduction in the first year. Liam also wants to make a charitable donation so that his final taxable income is exactly $70 000: a

Calculate the allowable deduction for the self-education course in the ﬁrst year.

Calculate his taxable income after adding the allowable course deduction to his existing deductions (before considering the charitable donation).

How much must Liam donate to charity (which is an allowable deduction) to reduce his taxable income to exactly $70 000?

What are his total allowable deductions for the year?

Ken earns a gross annual income of $85 000 in the 2025-2026 ﬁnancial year. He is exploring how allowable deductions impact his net earnings. Use the resident tax rates and a Medicare levy of 2%: a

Calculate Ken’s net earnings if he claims $4000 in allowable deductions.

If Ken increases his allowable deductions to $9000, calculate his new net earnings.

By what percentage did Ken’s net earnings increase due to the additional $5000 in deductions?

Explain brieﬂy why the percentage increase in net earnings is less than the percentage increase of the deductions relative to his initial net earnings or taxable income.

Mathspace New South Wales – Year 11 Standard mathspace.co

Big ideas • Networks provide a powerful visual and mathematical language for modelling complex systems of connections. By representing components as vertices and their relationships as edges, which can have properties like weight and direction, intricate real-world scenarios can be translated into structured diagrams and tables, allowing for systematic analysis. • Once a problem is modelled as a network, systematic methods and algorithms can be applied to find optimal solutions to practical questions. Key applications include determining the most efficient way to connect all points in a system (the minimum spanning tree) and identifying the shortest or lowest-cost path between two points, providing powerful tools for decision-making and optimisation.

8 Networks, paths and trees Chapter outline 8.01 8.02 8.03 8.04 8.05 8.06

Introduction to networks Network representations Weighted graphs Spanning trees Prim’s algorithm Shortest path Chapter 8 review

528 544 558 571 583 598 609

8.01 Introduction to networks After this lesson, you will be able to… • describe a network as a collection of vertices and edges. • define and use network terminology: vertex, edge, and degree. • identify features of a network including directed edges, weighted edges, and loops. • determine the degree of a vertex in a network.

Graphs and networks What do roads, human ancestry, currency exchange, planning a wedding, and the internet have in common? They can all be represented with a network diagram also called a graph. Network A set of points (vertices or nodes) some of which are joined by lines or curves (edges) that sometimes enclose regions (faces). For example, road networks, a family tree or the edges lining a tennis court.

Pink

Belconnen

y Po

4 Arda

Cambray

Fernam

Mordan Range 3

528

l La

l She

Green

Eastfarthing 16

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Dolby Tiller’s

Gilgandra

The roads and towns in this map form a graph/network. Daniela Bruno

Pierre

Lina

Michelle

Jason This social network illustrates the relationships between individuals. Network diagrams are graphs that represent systems in the real-world. The terms graph and network are interchangeable. Vertex (networks) A vertex is a point in a network diagram at which lines of pathways (called edges) intersect or branch. Also called a node. A vertex (or node) is the fundamental building block of a graph and represents a single object or idea. It is drawn as a dot or circle. The plural of vertex is vertices. Edge (networks) A line that joins vertices to each other in a network diagram.

An edge is a line segment that begins and ends at a vertex. An edge represents a relationship between the objects or ideas that it links together, and the kind of relationship depends on the context. Edge Vertex

Vertex

On the left is a single vertex. On the right are two vertices with one edge between them. A graph is a collection of vertices with edges drawn between them. Edges do not have to be straight. The following graphs are a collection of vertices and edges. 7

8 9 4

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Let’s take a moment and notice a few things about the graphs: • Some edges have direction (they look like arrows). • Sometimes, a vertex is connected to itself by an edge. • Sometimes, vertices are connected to another vertex by more than one edge. • The edges can have a numerical value. The following networks have labelled vertices. The labels can be something speciﬁc such as the name of a person or city or the labels can be more abstract like capital letters. B

Chronos Rhea

Athens

Coeus Phoebe

A C

Florence Zeus Amsterdam

Berlin

Leto

Paris D

Artemis Palermo

Oxford

An edge that starts and ends at the same vertex is called a loop. When there are two or more edges that start at the same vertex and ﬁnish at the same vertex, we refer to these as multiple edges. C

D This graph has a loop at vertex A and multiple edges between vertices D and F.

A F B A valid network has:

• At least one vertex • All edges must start at one vertex and end at one vertex Valid

Invalid B

D B

C This is an invalid network because the edges here do not end at another vertex.

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A connected network is one where all the vertices are connected (either directly or indirectly). Connected

Not connected A

B A

E C E

C D The vertex which is not connected is an “isolated vertex”.

Example 1 For this graph: A

E a Which vertex is isolated?

Apply the idea Choose the vertex that is not connected to the other vertices. Vertex B is isolated, as it is not connected to any other vertices by any edges.

b How many edges are there in the network?

Apply the idea There are 7 edges.

c How many vertices are there?

Apply the idea There are 5 vertices.

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If there are directed edges in the graph, it is called a directed graph. If there are no directed edges, we call it an undirected graph. A graph that is shown with edges that are a mix of arrows and lines is still a directed graph, where the lines represent a two-way connection. “Mixed” graphs like this are not common. It makes things easier if either all of the edges are lines, or all of the edges are arrows. We can redraw such a graph by replacing two-way edges with a pair of directed edges. Weighted edges A weighted edge is an edge of a network diagram that has a number assigned to it which implies some numerical value such as cost, distance or time.

1.10 1.34

1.10 1.10

1.23 1.43 0.96

A graph with numerical values on its edges, called weights, is referred to as a weighted graph. The weight of the graph is the sum of all the values. The values written on the edges here, could represent distances or costs.

Some graphs can be both weighted and directed.

Example 2 Consider this weighted and directed network:

P 13 14

S T

a What is the weight of the edge from S to Q?

Apply the idea The weight is 7.

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Example 3 Consider this network:

A B

D C a What is the degree of vertex C?

Apply the idea Vertex C is only connected by one edge. The degree of vertex C is 1.

b What is the degree of vertex D?

Apply the idea Vertex D is connected to three edges. The degree of vertex D is 3.

c What is the degree of vertex B?

Apply the idea Vertex B is connected to three edges in which one edge is a loop that is counted twice. The degree of vertex B is 4.

d Which vertices are odd?

Apply the idea Vertex A is connected to two edges, so it has a degree of 2, which is not an odd number. Based on our answers from parts (a) to (c), vertices C and D have a degree of an odd number.

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Are these networks valid or non-valid? a

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538

How many edges and vertices does each network have? a

Match each term to its correct description in the context of network diagrams. a

Edge

Vertex

Weight

Network

Mathspace New South Wales – Year 11 Standard mathspace.co

dot used to represent a location or A object.

line connecting two vertices, A representing a path or connection.

iii

number on an edge representing a A value like distance, cost, or time.

collection of vertices and edges A representing a system of connections.

Practice Ex 1

How many vertices and edges does the network have?

Is the network directed or undirected?

How many vertices are there?

How many edges are there?

Which vertex has a loop?

Between which two vertices are there multiple edges?

Is the network directed or undirected?

What is the degree of vertex S?

S P U Q

How many vertices and edges does the network have?

How many edges need to be added for the isolated vertex to be connected?

Do these networks have multiple edges? a

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Are these networks directed or undirected? a

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Ex 2

i ii

ind the weight of the edge connecting F Q and S.

Calculate the weight of the entire network. 7 Q 2 9 4 S

5 T

Find the weight of the edge from S to Q.

Calculate the weight of the entire network.

P Q

11 16

4 7

S T

How many loops does each network contain? a

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Ex 3

For each network: i

Identify the degree for each vertex.

State if the network is connected. A B

A C

D C

E c

A C

Extend your thinking 14

Jonah says the network has 6 vertices and 8 edges because there are 6 circles and 8 line segments. What mistake has he made and what is the correct answer?

Is it possible to draw one edge so that every vertex is even in this network? If so, what edge would need to be A added? If not, explain why not.

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Would a directed or undirected network be appropriate for these situations? a

Countries that border each other

The results of an elimination-style sports tournament

The animal food chain

Your parents and their ancestors

Ways to get from one classroom to another at school

How parts of the body are connected

How many edges begin at vertex Z?

What is the weight of edge from B to C?

What is the difference in the total weighting if you travel from A to B to C to A in a clockwise direction compared to anticlockwise?

11 39

41 6

53 18

How many edges end at vertex T ?

Calculate the total weight of all edges that end at vertex V.

6 S

Is it possible to travel along the edges from vertex W to P ?

Show that the sum of the degrees of the vertices for the network is equal to double the number of edges. Why does this rule always apply for undirected networks?

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8.02 Network representations After this lesson, you will be able to… • identify and describe simple and complete graphs. • recognise that a network can be represented by different but equivalent diagrams. • interpret information from a network diagram in a practical context.

Simple and complete graphs A simple graph is a network where: • the edges have no direction • there are no multiple edges • there are no loops a

d This is a simple graph.

This is not a simple graph.

The second image is not a simple graph because it has multiple edges between vertices b and d, and a loop at vertex b. A complete graph is a simple graph where each pair of vertices, in the network, are connected by an edge. These graphs show complete graphs with n nodes for Kn.

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Read network diagrams There are many contexts where network graphs can be used such as social networks, traﬃc ﬂow networks, supply chain networks and communication infrastructure. Explore some situations that can be represented by networks.

Example 3 A beach volleyball competition team can have 3 players on the court at any time. The team has 5 players at the tournament. The given network diagram shows the combination of players that have already played together in their ﬁrst three games:

Lucy

Ryan Beth Jimmy Ellie

a What do the vertices represent in this network diagram?

Apply the idea The vertices represent the ﬁve different players on the team.

b What does an edge between two vertices represent?

Apply the idea An edge between two players represents that the players in those vertices have played together in one of the ﬁrst three games.

c Is this a simple graph?

Apply the idea A simple graph has no direction on the edges, no multiple edges, and no loops. Yes, this is a simple graph.

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Example 4 The given network diagram has 3 vertices that represent three rides at an amusement park: Carousel (C), Pirate ship (P) and Haunted house (H). The weighted edges represent the amount of time, in minutes, it takes to travel from one ride to the next. The loops represent the time it takes to queue up for each ride.

3 C

a How long does it take to travel from the Carousel to the Pirate ship?

H 11

Create a strategy

Apply the idea

Carousel is represented by vertex C and Pirate ship is represented by vertex P. We are looking for the weight of the edge between them.

The weight of the edge between C and P is 5, so it takes 5 minutes to travel from the Carousel to the Pirate ship.

b How long does it take to queue up at the Pirate ship?

Create a strategy

Apply the idea

The Pirate ship is represented by vertex P. We are looking for the weight of the loop at this vertex.

The weight of the loop at P is 13, so it takes 13 minutes to queue up at the Pirate ship.

c Sam got in line for the Haunted house at 10 a.m. It takes 7 minutes to complete the tour of the Haunted house. He had his turn in the Haunted house, then travelled to the Pirate ship. What time was it when he was able to board the Pirate ship?

Create a strategy We can add the weights of the edges for each step of his journey.

Apply the idea We can list out each step: • Waiting in line at Haunted house: 11 minutes • Toured the Haunted house: 7 minutes • Travel from Haunted house to Pirate ship: 6 minutes • Waiting in line at the Pirate ship: 13 minutes Adding up all the step: 11 + 7 + 6 + 13 = 37 minutes It was 10:37 a.m. when he boarded the Pirate ship.

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Select the network graph which represents the same information as the given network.

Practice Ex 1

Describe these networks as simple or not simple: a

B C D Ex 2

Considering the map, draw a graph that represents the major roads connecting towns.

Young

Temora

A41

Boorowa

B94 Harden Cootamundra

B94

B85 A41

B81

M31

Yass

Junee A41

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A25 Lake Burrinjuck

Ex 3

A group of 5 students is playing 3-on-3 basketball against another group. After one team gets to 21 points or 10 minutes have passed, they change who is playing, and start a new game. This network graph shows who has played together after three games: a

Have James and Jenny played together? Explain why or why not?

They are going to play one more game before going home. Which players should play so that every team member has played with every other team member at least once?

Lisa

James Belinda Jenny Hannah

A group of journalists have been asked to work in pairs to collaborate on an article. This network graph represents the journalists who have worked together: a

Have Xanthe and Tina worked together? Explain how you know.

Name the journalists who still need to collaborate with someone else so that each journalist will have collaborated with every other journalist at least once.

Derek Xante Neil Tina Xavier

Over the course of a year, students work in pairs to complete poster projects. A group of 4 friends have already completed four projects together as shown with the network graph: a

Who has Aaron worked with?

Name the pairs of students that should work together on the next poster project so that all 4 friends have worked with each other.

Aaron

Rochelle

Mario

Bianca

The buses that run between these four towns are represented with this network graph: a

Which town can you not get to by bus?

Which town can you not leave by bus?

Describe a directed edge that could be added so you could get from any town to any other town by bus.

Teribithia

Nod

Narnia

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When converting between two currencies a bank charges a fee as a percentage. Let (A) represent Australian Dollars, (E) represent Euros, (N) represent New Zealand Dollars, (U) represent US Dollars, and (Y) represent Yen. The value on an edge represents the percentage Y fee that is charged to convert between those currencies: a

Which conversion has the highest fee percentage?

Is this graph a complete graph?

Why are there no loops?

U 4

2 4 4

E Ex 4

The given network diagram has three vertices that represent three rides at an amusement park: Roller coaster (R), Bumper cars (BC) and Big dipper (D). The weighted edges represent the amount of time, in minutes, it takes to travel from one ride to the next. The loops represent the time it takes to queue up for each ride: a

How long does it take to travel from the Roller coaster to the Big dipper?

How long is the wait for the Bumper cars?

Each ride is two minutes long. If you started at the Roller coaster, then did the Bumper cars, then the Big dipper, how long would it take from the time you got in line for the Roller coaster to the time you got off the Big dipper?

8 BC

3 R

7 4

D 9

The given graph represents the number of likes that four friends gave to each other’s social media posts over a week: a

How many likes did Fan receive in total?

How many likes did Zoya give in total?

Describe the type of network graph in two ways.

11 39

32 53

F 9

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554

12 Z

For each pair of networks, could one network be redrawn as the other with different labels for the vertices? a

E c

D A B

C A

X Y

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Extend your thinking 14

If a network is not simple, can it be redrawn in a different way so that it is simple? Explain.

Can you draw a simple graph that has one vertex and one edge?

How many simple graphs can you draw with three vertices?

The given network represent social media connections on a particular app: Brad Sharon Adam

Joanne

Edward

Is the app one where two people are friends or is it one where people follow and get followed? Explain.

Would it be possible for this graph to become a non-simple graph as more people connect on this app?

Would it be possible for this graph to become a non-simple graph using a different app? Explain.

The two given network graphs represent the same information. If vertex A maps to vertex 1, write out a mapping for the other five vertices. 1

A B

F 5 2

3 E C

4 6

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Consider the image of a train network showing the line colours and stations:

Willamette River

Expo Centre

Airport

Rose Quarter Pioneer Square

Gateway

Hillsboro Gresham Beaverton

PSU Clackmas Town Centre

Draw a graph that represents the train network.

List the stations that would be passed if travelling from the airport to PSU.

Draw a directed graph to represent each of these food chain descriptions. Draw the arrows pointing from the organisms that consumes to the organisms that is consumed: a

Within a water system: • The kingfisher, a skilful fishing bird, survives on eating small fish and frogs. • Small fish eat the tadpoles, whilst frogs survive on water beetles and snails. • Tadpoles, water beetles and snails all survive by eating algae.

Rabbits and squirrels both eat plants. Foxes and hawks eat both rabbits and squirrels.

The killer whale depends on tuna as a primary food source. The tuna feeds on ﬁsh called mackerel. For mackerel to survive, they depend on microscopic organisms collectively known as zooplankton.

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8.03 Weighted graphs After this lesson, you will be able to… • identify weighted edges and their meaning in various contexts. • calculate the total weight of a network, a subnetwork, or a path. • construct a weighted network diagram from a table or map. • solve practical problems by interpreting weighted networks.

Weighted graphs from tables Information about connections between locations or objects is often presented in a table, such as distances between towns or travel times. This can be represented as a weighted network diagram. A weighted graph is a network where each edge is assigned a numerical value, or weight. This allows us to model real-world scenarios where connections have properties like distance, time, or cost. For example, the network shown models an underground system connecting plant roots. A 14 B

10 D

These underground connections are created and maintained by the fungus to connect and nurture the roots of plants (represented here by the vertices) under the ground. The weights represent distance in metres.

H 21

15 C

26 8

F 10

7 G

14 E

In these networks: • Vertices (dots) represent locations or objects. • Edges (lines) represent connections. • Weights (numbers on edges) represent values like distance or time. A table representing a network is called a distance matrix. Rows and columns are labelled with vertices, and cell values give the weight of the edge between vertices.

558

Brisbane

Sydney

Canberra

Melbourne

Brisbane

−

915

1200

1650

Sydney

915

−

286

878

Canberra

1200

286

−

663

Melbourne

1650

878

663

−

Mathspace New South Wales – Year 11 Standard mathspace.co

A dash (−) or 0 indicates no direct connection. Symmetrical tables (e.g., Sydney to Brisbane same as Brisbane to Sydney) represent undirected graphs.

Exploration Consider this table of travel times (minutes) between theme park locations: Dino-Land, Pirate-Cove, Space-Zone, Aqua-World. Dino-Land

Pirate-Cove

Space-Zone

Aqua-World

Dino-Land

−

Pirate-Cove

−

Space-Zone

−

Aqua-World

−

Discuss: 1. Number of vertices and their labels? 2. Furthest direct travel time between two locations? 3. Location with no direct path to Space-Zone? 4. How to start drawing the network diagram?

Example 1 Table of travel times (minutes) for roads connecting ﬁve towns: Avon, Boro, Ceto, Dale, Eton. Avon

Boro

Ceto

Dale

Eton

Avon

−

Boro

−

Ceto

−

Dale

−

Eton

−

Construct a weighted network diagram and answer the following: a Identify the vertices and draw the weighted network diagram.

Create a strategy Identify towns as vertices, draw them, read table for connections and weights, draw labelled edges.

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Network diagram problems Weighted network diagrams are used to model real-world scenarios by assigning a numerical value, or weight, to each edge. These weights can represent quantities such as distance, time, or cost, allowing us to solve practical optimisation problems like determining the shortest path, the most cost-effective route, or the maximum possible flow through a system. Examples of weighted networks include: • Approximate travel times, in hours, between European capitals. Moscow

Copenhagen 9

Amsterdam

14 Berlin

Paris 6

Budapest

Bern

Kiev

12 15

10 10

Warsaw

Bucharest

Rome Madrid

Ankara Athens

• Average bond lengths in acetic acid (CH3 COOH), measured in Angstrom: 1.10 1.10

1.23

1.34

1.43

1.10

0.96

• A circuit diagram with resistances in Ohms: Switch 0

440

Diode

Battery

LED

220

560 220

900 0

Sensor 0

LED

120

Junction

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The total weight of a network or subnetwork is the sum of its edge weights. A subnetwork is a subset of vertices and edges. A walk’s weight (its length) is the sum of the weights of its edges. 4 7

11 3

6 5

4 5 Start

11 3

End

• Network total weight: 4 + 5 + 7 + 3 + 6 + 6 + 11 + 5 + 16 + 21 = 84. • Orange subnetwork weight: 4 + 5 + 7 + 3 + 6 = 25. • Red walk weight: 3 + 5 + 16 + 11 = 35.

Example 2 The network diagram shows five towns, P, Q, R, S, and T. The weights on the edges represent the travel time in minutes between the towns.

7 Q 2

9 4 S

5 T

a A tourist wants to travel from town P to town T. Identify all possible paths they could take without visiting the same town twice.

Create a strategy Trace all unique routes from the starting vertex P to the destination vertex T, ensuring no vertex is repeated in any single path.

Apply the idea By tracing the connections from P to T, the possible paths are: • P→R→T • P→Q→T • P→Q→S→T

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8.03 Practice questions What do you remember? 1

The following table shows the direct travel times in minutes between four locations: A, B, C, and D: a

What is the weight of the edge connecting B and D?

Which two locations are not directly connected?

How many edges would the corresponding network diagram have? Which connection has the lowest weight?

−

Complete the sentences about converting a table of connections into a network diagram: a b c d

The labels for the rows and columns in the table become the ⬚ of the network. The numerical values in the cells of the table become the ⬚ on the edges.

A symmetrical table, where the value in row A, column B is the same as row B, column A, represents a(n) ⬚ network. A dash (–) in a cell typically means there is no direct ⬚ between the two vertices.

Identify each of the following statements about weighted networks as true or false: a

The weight of an edge must always represent a physical distance.

In a network represented by a symmetrical table, the connection from vertex A to vertex B has the same weight as the connection from B to A.

The weight of a path is determined by the number of vertices it passes through.

A dash (–) in a distance matrix means the weight of the edge is zero.

Practice Ex 1

The following network diagram shows a series of hiking trails between landmarks in a national park. The weights on the edges represent the hiking time in hours: W

A 2

1 B

3 3 D

4 5

564

4 Z

What is the hiking time for the trail connecting landmark C and landmark X ?

What is the total time required to hike every trail in the park?

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Ex 2

The table shows the travel times in minutes between ﬁve cities: F, G, H, I, J. Construct a weighted network diagram and answer the following: F

−

Identify the vertices and draw the weighted network diagram.

Which two cities have the longest direct travel time?

Which city has no direct path to F ?

Calculate the weight of these paths in this network: A

8 2

2 7

The path Y, W, X

The path Z, Y, D, C

The path A, B, E, Y, W

Does travelling to more vertices always mean you will end up with a larger weight? Justify.

The table shows the costs in dollars to lay cables between ﬁve suburbs: P, Q, R, S, and T. Construct a weighted network diagram to represent this information. P

−

100

150

−

200

100

−

120

−

150

120

−

110

200

−

110

−

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For each table, construct a network representing the given information: a

The following table lists the amount of time it takes to queue up for three rides at an amusement park, and the amount of time it takes to travel between them: Carousel

Pirate Ship

Haunted House

Carousel

Pirate Ship

Haunted House

The following table lists the number of termites observed to be moving within and between three sites in 1 hour: Nest

Plains

Forest

Nest

3570

325

1240

Plains

475

965

Forest

410

535

230

The following table lists the number of likes that four friends give to each other’s social media posts over a week: Rick Rick

Zoya

Aarav

Fan

Zoya

Aarav

23 2

The following table lists the fees (as a percentage) charged by a bank to convert between ﬁve types of currency: USD USD

566

Fan

AUS

NZD

EUR

YEN

AUS

NZD

EUR

YEN

Mathspace New South Wales – Year 11 Standard mathspace.co

3 3

The map shows several campsites and the lengths of the hiking trails (in km) connecting them: a b

Lookout

Create a distance table to represent the information from the map.

Waterfall

Draw the corresponding weighted network diagram.

2 4

Ridge

Cave 7

Base camp

For each real-world network diagram provided, construct a table to represent the weights of the connections. Use a dash (−) for connections that do not exist: a

Flight costs (in hundreds of dollars)

Signal delay (in ms) from a central

hub Q to remote sites.

between four cities. A B 5

6 C

S c

Friendship ratings (out of 10) in a

Cycling times (in minutes) between

social group.

park features.

U 9

5 2

M 6 3 5 P

1 N X

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The table shows the schedule for a local basketball tournament, where a number indicates that two teams played each other. Draw a network diagram to show which teams have played each other. (The weight of each edge can be considered 1). Eagles

Hawks

Lions

Tigers

Eagles

−

Hawks

−

Lions

−

Tigers

−

Consider this network of campus locations connected by electrical cables: A $400

$800 $300

B $600

$300

$250 C

$600

$750

A: Admin building B: Library C: Science lab D: Cafeteria E: Gymnasium F: Auditorium

$850 F

What is the smallest total cost to connect all the buildings with electrical cables?

If the library and all its connecting cables are removed from the network, determine the new total cost to connect the remaining buildings.

The network represents an irrigation system, with weights showing water flow rates in litres per minute: a

Calculate the total flow rate of the subnetwork connecting only the junctions P, Q, R and S.

Calculate the total flow rate of the subnetwork formed by removing junctions P, Q, R and their connecting pipes.

12 S 7

8 T

14 U

Three friends, Sam, Tia, and Leo, live near each other. The cycling time between Sam and Tia is 8 minutes. The cycling time between Tia and Leo is 6 minutes. The cycling time directly between Sam and Leo is 12 minutes. Draw a weighted graph to represent this information.

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A courier business has to make multiple pick-ups from several stores. This map displays the routes between the stores: a

Calculate the weight of the path A, E, F, C.

Calculate the weight of the path A, E, C, D.

Calculate the weight of the path A, B, D, C.

Determine the shortest path from store A to store C.

F 4

11 8 5

9 7

Extend your thinking 16

An incomplete table and an incomplete network diagram are shown. They are supposed to represent the same information. Complete both the table and the diagram by finding the values of x and y, when the total weight is 43. A

−

A 7 B

x 10 C

9 12 D

A delivery truck must travel between warehouses A, B, C, D, and E. The table shows the maximum weight limit in tonnes for the roads connecting them: A

−

Draw a weighted network diagram for this information.

Can a truck weighing 10 tonnes travel directly from A to E? Explain why or why not.

Identify a valid path from warehouse A to warehouse E for the 10-tonne truck.

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Two tables show the cost ($) and time (hours) for travel between four holiday islands: Cost ($)

−

120

−

120

−

Draw a single network diagram that shows both the cost and time for each connection. Label each edge with a pair of numbers representing (cost, time).

Using your diagram or the tables, identify the cheapest path and the fastest path from island P to island S. Are these paths the same?

This network represents a delivery drone’s routes between depots. Each edge is weighted with two values: (time in minutes, battery usage in %):

Determine the path from A to E that is the fastest (minimum time). What is its total time and battery usage?

b c

D (8, 10)

(20, 25)

(5, 8)

(10, 15)

Determine the path from A to E that is the most energy-eﬃcient (minimum battery usage). What is its total battery usage and time?

(18, 20)

(15, 12)

(12, 10)

A The drone starts with 100% battery. A return trip must be completed without recharging. If the drone must travel from A to D and back to A, which route would you recommend and why?

The network shows the travel time in minutes between suburbs during off-peak hours. During peak hours (7:00 a.m. to 9:00 a.m.), the travel time for the edges A–B and C–D doubles due to traﬃc: a

570

Time (hours)

What is the weight of the same path, A–B–C–D, during peak hours? A commuter wants to travel from A to D during peak hours. Is the path via E (path A–E–D) faster than the path via B and C (path A–B–C–D)? Justify your answer with calculations.

Mathspace New South Wales – Year 11 Standard mathspace.co

What is the weight of the path A–B–C–D during offpeak hours?

12 B E

8 D

A courier has a van with a limited fuel range. The van can travel a total of 50 km before needing to refuel. The network shows the distances in km between warehouses: a

The courier needs to travel from Start to W3, then to W5. Can this trip be completed on a single tank of fuel using the path Start→ W2 → W3 → W5?

Can the same trip (Start to W3 to W5) be completed using the path Start → W1 → W3 → W5?

The courier must make a round trip starting at ‘Start’, visiting W4, then W3, and then returning to ‘Start’. Can the path Start → W2 → W4 → W3 → W1 → Start be completed on a single tank of fuel? Justify your answer.

Start 10 W1

W2 8

14 9

W3 18

The courier must travel from ‘Start’ to W5, stopping at W4 on the way. Find a path that allows this trip to be completed within the 50 km range and state its total distance.

8.04 Spanning trees After this lesson, you will be able to… • define and identify a tree, a spanning tree and a minimum spanning tree. • explain the properties of a spanning tree, such as having n − 1 edges for n vertices. • determine the minimum spanning tree of a weighted network by inspection. • use minimum spanning trees to solve minimal connector problems.

Spanning trees Tree (networks) A tree is an undirected network in which any two vertices are connected by exactly one path.

These networks have a distinctly tree-like appearance:

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Once a vertex is left on any of these networks, it cannot be returned to without repeating an edge. Any connected network can have edges removed, taking care not to disconnect the network, until the result is a subnetwork that is a tree - this is then called a spanning tree. Spanning tree A spanning tree of an undirected network diagram is a tree which includes all the vertices of the original network connected together, but not necessarily all the edges of the original network diagram. A network can have many different spanning trees.

Here’s an example:

The network in the centre has its eight different spanning trees arranged around it. It is still possible to get from one vertex to any other, but a return path cannot be made without repeating an edge. Therefore, a spanning tree has n − 1 edges, where n is the number of vertices in the network. As shown in the image, each network graph is connected and cycle-free, meaning once a vertex is left, it cannot be returned to without repeating an edge. This tree-like structure is achieved by removing certain edges to avoid any loops, while still connecting all vertices. A connected network that isn’t already a tree can have multiple spanning trees. This network has 11 possible spanning trees: This network has weight 4 + 6 + 5 + 2 + 3 + 10 = 30.

6 4 10

5 2 3

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Here are all the spanning trees, with their weights. 6

3 Weight 24 4

4 6

5 10

3 Weight 18 6 4

Weight 25 6 4 2 3 Weight 15

3 Weight 22 6

3 Weight 23 6 4

2 3 Weight 21

Weight 22

5 2

Weight 21

5 2 3 Weight 14

2 10 3 Weight 20

The last spanning tree is special, since it has the lowest weight out of all of them. This is called the minimum spanning tree. Sometimes there can be two or more spanning trees with equal lowest weight, in which case they are all minimum spanning trees. Minimum spanning tree A tree of minimum weight in a connected, undirected network. It connects all the vertices together with the minimum total weighting for the edges. Here’s a network with two minimum spanning trees: 3

6 5

Finding both can be quite hard and may take a long time.

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Example 1 Determine whether each graph is a tree. Justify the answer. a

Create a strategy Recall that a tree must be a connected graph and contain no cycles. Examine the graph for these properties.

Apply the idea A graph is a tree if it is connected and contains no cycles. This graph is not a tree for two reasons: 1. It is disconnected: There are pairs of vertices (for example, vertex A and vertex D) that are not connected by any path. 2. It contains a cycle: The component involving vertices D, E, and F forms a cycle. Therefore, this graph is not a tree.

Apply the idea A graph is a tree if it is connected and contains no cycles. This graph is connected, as every vertex can be reached from every other vertex. However, it contains a cycle (for example, the central triangular cycle). Therefore, this graph is not a tree because it has a cycle.

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Apply the idea This graph contains no cycles; there is no path that starts and ends at the same vertex without retracing edges. Therefore, this graph is a tree.

Example 2 Determine the minimum spanning tree and its total weight for the network shown. E

3 4

C 2

Create a strategy To find the minimum spanning tree, we must connect all the vertices together with the lowest possible total weight. A reliable method is to select the edges with the smallest weights in ascending order, ensuring that we do not create a cycle at any stage.

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Apply the idea First, list all the edges in the network from lowest weight to highest: D − E(1), B − C(2), C − A(3), A − B(4), B − D(5), C − E(6), D − F(7), C − D(8), E − F(9) Now, add edges to the tree one by one, starting with the cheapest, and skip any edge that would form a cycle. • Add edge D − E, weight 1. • Add edge B − C, weight 2. • Add edge C − A, weight 3. • Next is A − B (weight 4). Adding this edge would create a cycle A − B − C − A. Reject this edge. • Add edge B − D, weight 5. This connects the two separate parts of our growing tree. • Next is C − E (weight 6). Adding this would create a cycle C − B − D − E − C. Reject this edge. • Add edge D − F, weight 7. All vertices are now connected. We stop here. The minimum spanning tree consists of the edges D − E, B − C, C − A, B − D, and D − F. The total weight is the sum of the weights of these edges: 1 + 2 + 3 + 5 + 7 = 18. E

A 6

3 4

C 2

1 D

Reflect and check The final tree has 6 vertices and 5 edges, which satisfies the n − 1 rule for trees. No cycles were formed. By systematically choosing the cheapest available edges and rejecting any that would form a loop, we ensure the final spanning tree has the minimum possible weight.

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Example 3 An energy company needs to connect a main power station (P ) to 5 substations (A, B, C, D, E). The network diagram shows the possible connection paths, with the weights representing the cost in thousands of dollars to lay the cable.

P 10

12 5

B 6

8 C

15 D

4 9

7 E Determine the minimum cost required to connect all substations to the power station.

Create a strategy This is a minimal connector problem. The goal is to find the cheapest way to connect all vertices in the network, which means finding the minimum spanning tree. We can build this minimum-cost network by inspecting the graph and selecting the cheapest connections (edges) one by one, making sure not to form a loop, until all locations are connected.

Apply the idea We will add the cheapest edges to our network, ensuring no cycles are created. • The cheapest connection is C − D with a cost of 4. Add this edge. • The next cheapest is A − B with a cost of 5. Add this edge. • The next cheapest is B − C with a cost of 6. This connects our two separate parts. Add this edge. • The next cheapest is C − E with a cost of 7. Add this edge. • The next cheapest is A − C (cost 8), but adding it would create a loop (A − B − C − A), so we ignore it. • The next cheapest is D − E (cost 9), but adding it would create a loop (D − C − E − D), so we ignore it. • The next cheapest is P − A with a cost of 10. Add this edge. Now all substations and the power station are connected in a single network with no cycles. The edges included are C − D, A − B, B − C, C − E, and P − A. The minimum cost is the sum of the weights of these edges: Minimum Cost = 4 + 5 + 6 + 7 + 10 Minimum Cost = 32 Since the costs are in thousands of dollars, the minimum cost to connect all substations is 32 × $1000 = $32 000.

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Practice Ex 1

Determine whether each graph is a tree. If it is not a tree, explain why: a

A C

F B

C A

E A

B B e

A B

D 6

Sarah has three children: Kate, Matt and Tom. Kate has four children: Lisa, Sandy, Amy and Robert. Matt has two sons: Joseph and Chris. Both Tom and Lisa have one child each: Barbara and Alex, respectively. If the starting vertex represents Sarah, create a tree to represent the parent-child relationships in this family.

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Mitchell is tracing his family lineage back three generations: • His mother’s name is Rebecca, and his father’s name is Connor. • Rebecca’s parents are Mabel and Patrick, and Connor’s parents are Juliet and Mark. • Mitchell does not know Mabel’s parents’ names, but he knows that Patrick’s parents are April and Todd. • On the other side, Mitchell knows that Juliet’s parents are Daisy and Leonard, and Mark’s dad is called Richard. • Mitchell does not know the name of Mark’s mother, however. If the initial vertex represents Mitchell, create a family tree of the last three generations from the information above. Make sure to include any ancestors that he doesn’t know the name of using the word “Unknown”.

Ex 2

Find the minimum spanning tree for the following network. State the total weight of the tree.

A 3

5 B

8 6 2

C 9

For the network shown, determine the minimum spanning tree and its weight.

5 11

3 T

7 14

S 10

Find the minimum length of edges required to connect all vertices in the network below.

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Ex 3

An office building needs to connect several servers to a main router. The costs (in hundreds of dollars) of laying ethernet cables between locations are shown in the network. What is the minimum cost to connect all the servers to the network?

Router 7

4 2

3 E

A company is planning to build bridges to connect a group of 5 islands. The costs, in millions of dollars, for building a bridge between pairs of islands are shown. Determine the minimum cost to connect all the islands.

H 15

12 8

10 9

S 13

An irrigation system is being installed in a community garden to connect a water pump to 5 garden beds. The diagram shows the cost of laying pipes between points. Find the minimum cost to ensure all garden Roses beds are connected to the water pump.

Pump 20

35 10

Veggies

15 Herbs

Berries

Fruit

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Extend your thinking 14

Consider this weighted network representing the cost, in thousands of dollars, to build paths between new community facilities:

B 10 A

Draw two different spanning trees for this network.

Calculate the total cost (weight) for each of your spanning trees.

By inspecting the graph, can you ﬁnd a spanning tree with a lower total cost than the ones you drew? If so, determine and calculate its cost.

12 14

A municipal council plans to connect ﬁve towns with a new water pipe system. The network shows the cost in millions of dollars for laying pipes along possible routes. To be cost-effective, they need to connect all towns using a spanning tree: a b

Explain why connecting the towns using all the routes shown is not an eﬃcient use of resources. By inspection, ﬁnd the most cost-effective spanning tree (the one with the minimum total weight) and calculate its total cost.

E 7

6 8

5 C

A connected network has 8 vertices and 12 edges: a

How many edges must be in any spanning tree of this network?

How many edges must be removed from the original network to form any spanning tree?

Find the minimum spanning tree for the network below and state its total weight.

B 3

C 6

5 D 18

A resort developer is connecting several bungalows with walkways. Some walkways (shown as dashed lines) have already been built. The costs for building new walkways are shown in thousands of dollars. What is the minimum additional cost to ensure all bungalows are connected?

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582

8.05 Prim’s algorithm After this lesson, you will be able to… • describe the steps of Prim’s algorithm for finding a minimum spanning tree. • apply Prim’s algorithm to a weighted network to determine the minimum spanning tree and its total weight. • describe Kruskal’s algorithm as an alternative method for finding a minimum spanning tree. • apply Kruskal’s algorithm to determine the minimum spanning tree. • explain why a systematic algorithm is necessary for solving problems with complex networks.

Prim’s algorithm Sometimes, it is going to be very impractical to ﬁnd a minimal spanning tree by eye. This network represents a mycorrhizal nutrient transfer system: A 14 26

15 C

12 10

12 D

H 21

F 10

7 G

A severe drought is affecting the area, so only the most essential connections can be sustained by the fungus. What is the shortest total distance that the mycorrhizal network needs to spread itself across, while still making sure that every plant is connected to every other one? The minimal spanning tree needs to be found. This particular network has 416 spanning trees. Determining which one is minimal by finding all of them and adding up all their weights would take a very long time. The Greedy algorithm is a problem-solving approach that makes the locally optimal choice at each step with the hope of finding a global optimum. At each stage, the algorithm selects the best option available at that moment, without considering the overall future consequences beyond the immediate step. The Prim’s algorithm is a greedy algorithm used to find a minimal spanning tree for a weighted, connected graph. It builds the tree one vertex at a time, by adding the cheapest possible connection from the existing tree to a vertex not yet in the tree.

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Prim’s algorithm is an example of a greedy algorithm. While this approach does not guarantee the best solution for all problems, it works effectively for finding minimal spanning trees. Another greedy algorithm for finding minimal spanning trees is Kruskal’s algorithm. Prim’s algorithm was first created by Vojtech Jarník, a Czech mathematician, in 1930. Both Robert Prim (in 1957) and Edsger Dijkstra (in 1959) independently rediscovered it, though Prim’s work popularised its use. Prim’s algorithm (from a network): 1. Pick any vertex to start the spanning tree. 2. Locate the lowest weight edge that connects a vertex currently in the spanning tree to a vertex not yet in the spanning tree. 3. Reject any edge that connects two vertices already in the spanning tree. 4. If there is more than one edge with the lowest weight, pick any of them. 5. Add this edge and the vertex at the other end to the spanning tree. If all vertices are part of the spanning tree, stop. Otherwise, repeat from step 2. A 14 26

15 C

12 10

12 D

H 21

F 10

Before starting, note that since this network has 8 vertices, the minimal spanning tree will have 7 vertices. Begin by picking a vertex; any vertex can be chosen. For instance, if F. is picked, the lowest weight edge coming out of F. is then highlighted.

7 G

This edge, and the vertex on its other side, are then added to the spanning tree.

A 14 26

15 C

12 10

12 D

H 21

F 10

7 G

The same process is repeated. Highlight the edge of lowest weight coming out of the current spanning tree. The vertex F was just a starting point; the lowest weight edge connected to the spanning tree now comes out of E. That edge, and its vertex, are then added to the spanning tree.

A 14 26

15 C

12 10

12 D

584

H 21

F 10

7 G

Once again, the lowest weight edges coming out of the spanning tree are highlighted. This time, there are two candidates. Both DC and DF have weight 12.

Mathspace New South Wales – Year 11 Standard mathspace.co

Kruskal’s algorithm An alternative method for ﬁnding the minimum spanning tree of a weighted, connected graph is Kruskal’s algorithm. Unlike Prim’s algorithm, which starts from a single vertex and grows outwards, Kruskal’s algorithm focuses on adding the edges with the lowest weight ﬁrst, regardless of where they are in the network. The core principle is to add edges in increasing order of weight, as long as they do not form a cycle (a closed loop). An edge is rejected if its two vertices are already connected to each other through a path made of previously selected edges. The steps for applying Kruskal’s algorithm are as follows: 1. List all the edges in the network in ascending order of their weights. 2. Select the edge with the smallest weight and add it to the spanning tree. This edge will form the initial part of the tree. 3. Continue selecting the next-lightest edge. Before adding it, check if its two endpoints are already connected by a path within the tree being built. If they are, adding the edge would create a cycle, so it must be discarded. If they are not yet connected, add the edge. 4. Repeat step 3 until there are n − 1 edges in the spanning tree, where n is the number of vertices. The resulting connected graph is the minimum spanning tree.

Example 2 Use Kruskal’s algorithm to ﬁnd the minimum spanning tree for the network shown. Determine the total weight of the minimum spanning tree.

4 C

3 E

5 2 F

Create a strategy To apply Kruskal’s algorithm, ﬁrst list all edges from the lightest to the heaviest. Then, add each edge to the spanning tree in this order, making sure to skip any edge that would form a cycle. The process stops when 5 edges have been added (since there are 6 vertices, and a spanning tree must have n − 1 edges).

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Apply the idea Step 1: List all edges in ascending order of weight. Edge

Weight

Step 2: Add edges to the tree, checking for cycles. A spanning tree for a graph with 6 vertices must have 6 − 1 = 5 edges. Go through the sorted list and add each edge if it does not form a cycle with the edges already selected. • Edge DE (Weight 2): Add to the tree. Vertices D and E are now connected. • Edge EF (Weight 2): This connects F to the existing tree {D, E}. Add to the tree. • Edge BC (Weight 3): This connects B and C, which are currently separate from the other part of the tree. Add to the tree. • Edge CE (Weight 3): This connects the {B, C} part of the tree to the {D, E, F} part. Add to the tree. All vertices except A are now connected. • Edge CD (Weight 4): Check if C and D are already connected. Yes, a path exists: C-E-D. Adding this edge would form a cycle. Reject. • Edge DF (Weight 5): Check if D and F are already connected. Yes, a path exists: D-E-F. Adding this edge would form a cycle. Reject. • Edge BD (Weight 6): Check if B and D are already connected. Yes, a path exists: B-C-E-D. Adding this edge would form a cycle. Reject. • Edge AB (Weight 7): This connects the isolated vertex A to the rest of the tree via B. Add to the tree. The tree is now complete as it has 5 edges, which is the required number for 6 vertices. Any further edges, such as AC, will necessarily form a cycle and must be rejected. A

7 3

D C

3 E

2 F

Step 3: Calculate the total weight. Total Weight = 2 + 2 + 3 + 3 + 7 = 17

Sum the weights of the edges Evaluate

The minimum spanning tree consists of edges AB, BC, CE, DE and EF, with a total minimum weight of 17.

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Which of the following statements are true or false? a

Prim’s algorithm can be started from any vertex in the network.

Kruskal’s algorithm always adds the edge connected to the current tree with the lowest weight.

If a graph’s edge weights are all unique, it can only have one unique minimum spanning tree.

A minimum spanning tree for a given graph will always have the same total weight, regardless of the algorithm used (Prim’s or Kruskal’s).

Practice Ex 1

A group of environmental scientists are setting up monitoring stations at 6 locations across a conservation zone. The potential paths between these locations, along with the number of hours needed to prepare each path, are shown in the given graph:

Q 5 4

To eﬃciently connect all monitoring stations, they need to construct a minimal spanning tree of the graph.

Ex 2

Construct/highlight the minimal spanning tree.

Find the minimum total time required to establish the network connecting all locations.

Use Kruskal’s algorithm to determine the set of roads that should be built to form a minimum spanning tree.

What is the minimum total length of road required to connect all the towns?

How many edges will be a part of a spanning tree for this graph?

A B

8 7 C

5 F

500 C

720

740 A

If the cost of building a road is $2000 per kilometre, what is the minimum cost to connect all the towns?

The direct ﬂight routes between 5 major cities of the same country are shown in the graph. Each edge is marked with the distance (in km) travelled on each route:

592

To optimise the investment placed into these routes, the operating airline wants to determine the minimum spanning tree for the graph.

A network of towns needs to be connected by a new road system. The distances (in kilometres) for possible road connections are shown in the network below. Use Kruskal’s algorithm to ﬁnd the most cost-effective way to connect all the towns:

3 7

520

640

720 E

560

Construct/highlight the minimum spanning tree.

If the cost of maintaining each ﬂight route is $5500 per kilometre per year, what is the minimum annual cost to connect all cities?

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A university is connecting its campus buildings with ﬁbre optic cables. To minimise cost, they need to ﬁnd the network with the minimum total cable length. The weights are in metres: a

Construct the network that represents the minimum cable length.

What is the minimum length of cable required?

If the ﬁbre optic cable costs $85 per metre, what is the minimum cost for the project?

G 12 F

4 5

3 10

2 H

K 4

Water pipes need to be laid down in a new area that is undergoing construction. This pipe map was suggested:

D E

G F

To minimise the cost, the council wants to connect every house to water, minimising the total length of water pipes used. Construct a graph that would achieve this.

C 14

B 24

A D

The bus company decided to redesign its travel routes after some new roads were built. The map shows the routes connecting the towns and the respective times (in minutes) taken to cover the route:

The bus company wants to connect all towns with the most time eﬃcient route.

Construct the graph that will achieve this.

The council wants to lay down new electric cables between the houses in a particular area. The graph displays the map of the area: To minimise the cost of cables, the council wants to connect the houses in such a way that they minimise the cable length. Construct a graph that would achieve this.

10 10

45 31

A 11

B 10

3 A

6 D 2

4 G

5 J

C F 9

4 I

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The following weighted network shows possible connections (and associated costs, in millions of dollars) for connecting rural towns to a common cabled internet service.

A 12

To minimise the cost of connecting each town to the network, we want to ﬁnd the minimal spanning tree:

How many edges will be required for the minimal spanning tree?

Construct the minimum spanning tree.

Find the minimum cost of connecting all of the towns to the service.

2 7

E 2

6 F

J F

Since the project has a limited budget, the engineers have to design a connection layout that uses the minimum cable length.

In a certain area, new underground cables need to be laid to connect electric substations. The following graph displays the current connection layout:

Construct the graph that represents the connection layout with the minimum amount of cable.

17 16

G 10

14 11

A 14

An aeroplane company wants to redesign their flight routes to make it cost-efficient. The following map displays the current flight routes:

G 111

In order for the ﬂight routes to be eﬃcient, each city needs to be connected and the sum of the route distances needs to be minimised.

126

Construct the graph that will achieve this.

137

148

160

151 175 A

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100

117

134

594

153

A large amusement park is implementing a new security system connecting all security posts with electric cabling. The map shows possible connections and D their lengths in metres. To minimise the budget, engineers must ﬁnd the minimum length of cabling required: 29 a Construct the graph that represents the

Each cable connection requires two junction boxes costing $75 each. If the cable itself costs $40 per metre, what is the total minimum cost of the project?

The water authority needs to lay pipes to connect ﬁre hydrants at every node on the map. The numbers represent the pipe length in metres. To minimise operational costs, the total length of the pipes must be minimised: a

What is the minimum length of cable required to connect all security posts?

What is the minimum total length of pipe needed?

A construction team can lay 5 metres of pipe per hour at a labour cost of $350 per hour. Calculate the total minimum labour cost for the project.

Construct the graph that represents the most distance-eﬃcient network.

What is the minimum total length of road required?

Road construction costs are tiered: $3000 per metre for the ﬁrst 100 metres of the project, and $3500 per metre for any additional length. Calculate the total minimum construction cost.

B E

27 33

31 H

D 15

E 21 H

A 22

11 G

Urban planners are designing a road network to connect houses in a new area. The map shows possible routes and their distances in metres. They want to ﬁnd the most distance-eﬃcient network that connects every house: a

17 18

Construct the graph that shows the minimum length of pipe required.

connections using the least amount of cable. b

A 20

B C 19

11 E G

13 17

26 J

D 26 F

17 27

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Extend your thinking 18

Two park rangers, Alex and Ben, are planning a network of walking trails to connect 6 key landmarks (A, B, C, D, E, F ) in a national park. They want to ﬁnd a Minimum Spanning Tree (MST) to ensure all landmarks are connected with the minimum total length of cleared trails. The available potential trails and their lengths (in km) are given in the table: Trail

Length (km)

A–B

A–C

A–F

B–C

B–D

C–D

C–E

C–F

D–E

E–F

Both rangers decide to use Prim’s algorithm as described in their training materials.

596

Alex starts Prim’s algorithm from landmark A. List the trails (edges) Alex selects in the order he adds them to form his MST. What is the total length of Alex’s MST?

Ben starts Prim’s algorithm from landmark F. List the trails (edges) Ben selects in the order he adds them to form his MST. What is the total length of Ben’s MST?

Compare the ﬁnal set of trails selected by Alex and Ben. Are their MSTs identical in terms of the trails included? Explain why this might or might not always be the case when Prim’s algorithm is applied from different starting vertices on the same graph.

After their initial planning, a new survey reveals an additional potential trail between landmarks D and F, with a length of 6 km. If Alex were to re-run Prim’s algorithm (still starting from landmark A) including this new D − F trail option, would the D − F trail be part of his new MST? Justify your answer by describing the state of Alex’s MST construction at the point the D − F trail would be considered.

Considering the original graph (without the D − F trail from part d), suppose the cost to clear trail D − F (originally 9 km) is uncertain. What is the maximum possible length for trail C − F such that it could be included in an MST constructed by Prim’s algorithm ( for example, starting from A)? Explain your reasoning, detailing how Prim’s algorithm would make its selection.

Mathspace New South Wales – Year 11 Standard mathspace.co

Urban planners are designing a new subdivision with 5 key locations (A, B, C, D, E). The possible direct road connections and their lengths (in metres) are: • A − B : 120 m • A − C : 150 m • A − D : 200 m • B − C : 90 m • B − E : 180 m • C − D : 100 m • C − E : 220 m • D − E : 130 m The urban planners wish to create roads that are most distance eﬃcient, and allow them to connect each location. Represent this information as a weighted graph. Then, construct the minimum spanning tree that is the most distance eﬃcient and state its total length.

A telecommunications company is planning to lay new ﬁbre optic cables to connect 5 remote data centres (P, Q, R, S, T ). The costs (in thousands of units) for laying cable directly between certain pairs of centres are as follows: Connection

Cost (thousands)

P−Q

P−R

P−T

120

Q−R

Q−S

R−S

R−T

S−T

Note: Some pairs like P − S and Q − T have no direct link listed, meaning the cost is prohibitive or not feasible for a direct connection. To make a network of cables in the most cost-effective way, the company wants to ﬁnd a minimal spanning tree. a

Represent this information as a weighted graph. Then, construct the minimal spanning tree to connect all data centres.

Calculate the minimum cost to connect all data centres.

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8.06 Shortest path After this lesson, you will be able to… • define a path and the shortest path in a weighted network. • find the shortest path between two vertices by systematically inspecting all possible paths. • explain the difference between the shortest path and the best path in a practical context. • describe a circumstance in which a shortest path is not contained in a minimum spanning tree.

Shortest path A common problem in networks is to find the path of lowest total weight between two specified vertices. This can involve finding the fastest, shortest, or cheapest route between two locations. Path (networks) In a network diagram, it is a walk in which all of the edges and all the vertices are different. A path that starts and finishes at different vertices is said to be open, while a path that starts and finishes at the same vertex is said to be closed. There may be multiple paths between the same two vertices. Shortest path A shortest path in a network diagram is a path between two vertices in a network where the sum of the weights of its edges are minimised.

An example of a shortest path is if path A, X, B has a total weight of 10, and path A, Y, B has a total weight of 8, and path A, Z, B has a total weight of 12, then A, Y, B is the shortest path from A to B. To find the shortest path by inspection in a small network: 1. Identify the starting and ending vertices. 2. Systematically list all distinct simple paths (paths that do not revisit any vertex) between these two vertices. 3. For each path, sum the weights of all edges along that path to find its total weight (or length). 4. Compare the total weights of all the paths found. The path with the smallest total weight is the shortest path. It is important to be systematic in listing paths to ensure no possibilities are missed. This method differs from simply picking the edge with the smallest weight at each step (a greedy approach), which does not guarantee finding the true shortest path between two specific points.

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Consider this network of towns: A 4

2 C

The weights represent travel time in minutes. The objective is to travel from A to B via the optimal (fastest) path. Three possible paths are:

9 4

3 E

B A 4

• The red path A, B takes 9 minutes. • The blue path A, C, E, B takes 2 + 4 + 4 = 10 minutes. • The orange path A, D, B takes 4 + 4 = 8 minutes.

2 C

9 4

This last path is close, but town D can be reached from A even faster by taking a path through C.

B A 4

2 C

This path, A, C, D, B, takes 2 + 1 + 4 = 7 minutes, and is the quickest way to get from A to B.

4 E

The same systematic inspection strategy can be applied to directed networks. A 6 B

11 6 5

Here is a directed graph. In this case, the connections are all one-way.

E 5

C F

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D Consider a few paths from A to G:

11 B

6 5

E 5

6 5

Further paths can be checked by considering variations of known paths. Starting from the path A, D, G, (the best found so far at 16 minutes), the trip from D to G can be improved by going through E.

11 B

E 5

C F

• The blue path A, F, G takes 10 + 11 = 21 minutes. • The orange path A, D, G takes 4 + 12 = 16 minutes. • The red path A, B, C, E, G takes 6 + 1 + 5 + 5 = 17 minutes.

Going along the path A, D, E, G takes 4 + 6 + 5 = 15 minutes, and by systematic inspection of all other simple paths, this is found to be the fastest route.

Using trial and error by listing paths is an appropriate method for solving shortest path problems by inspection. A systematic approach is crucial to ensure all relevant paths are considered and the true shortest path is found.

Example 1 A rock band is planning a tour across several cities. The following table represents the routes and distance between the cities at which they will be performing.

600

Starting City

Ending City

Distance

Mathspace New South Wales – Year 11 Standard mathspace.co

Best path While the shortest path provides the route with the minimum total weight, it is not always the best path in a real-world context. The ‘best’ path depends on specific criteria beyond the numerical weight. Consider these scenarios: • GPS Navigation: The shortest route in distance might pass through a school zone with a low speed limit, heavy traffic, or numerous traffic lights. A GPS might determine the ‘best’ path as a longer route that is faster in time. Similarly, the ‘best’ path could be one that avoids toll roads, even if it is longer or slower. • Tourism: A tourist might prefer a scenic route that visits certain landmarks or lookouts. This ‘best’ path would be deliberately longer than the most direct route. • Accessibility: A person with a pram or wheelchair needs a path that avoids stairs or rough terrain. The ‘best’ path for them is the shortest one that meets these accessibility requirements, which may be significantly longer than the absolute shortest path available. It is also important to understand that the shortest path between two vertices is not necessarily contained within a minimum spanning tree (MST). An MST connects all vertices in a network with the minimum possible total edge weight, optimising for the entire network’s connectivity. A shortest path problem only optimises the route between two specific vertices.

Example 2 The diagram shows a map of walking tracks in a national park. The weights on the edges represent the length of the track in hundreds of metres. Some tracks involve steep stairs, as indicated by the S.

Picnic area

a Find the shortest path and its length from the Carpark to the Waterfall.

Car park

4 S

3 S Waterfall Lookout

Create a strategy List all possible paths from the Carpark (C) to the Waterfall (W) and calculate their total lengths. The shortest path is the one with the minimum length.

Apply the idea The possible paths from C to W are: • Path C-L-W: Length = 5 + 3 = 8 • Path C-P-W: Length = 4 + 7 = 11 • Path C-L-P-W: Length = 5 + 6 + 7 = 18 • Path C-P-L-W: Length = 4 + 6 + 3 = 13 Comparing the lengths, the minimum length is 8. The shortest path is C-L-W with a length of 800 m.

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Are these statements true or false? a

The shortest path is always the path with the fewest edges.

The ‘best’ path might be longer than the shortest path.

The shortest path between two vertices must be part of the minimum spanning tree.

To ﬁnd the shortest path by inspection, you must check all possible paths between the two vertices.

A network represents roads between towns, with weights as distances in kilometres. Describe one reason why the shortest path might not be the best path for a driver.

Practice 5

A few colleagues decided to carpool on the way to work. The following map displays the routes between their homes:

Starting from house A, determine the shortest path that visits every house exactly once.

17 12

14 B

Ex 1

A logistics company needs to optimise its shipping routes between its major distribution hubs. The following table represents the available direct routes and the average shipping time in hours: a

Construct a weighted graph that represents the table.

Determine the shortest route and its total time (in hours) from hub S to hub A by inspection.

Start hub

End hub

Time (hours)

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The following map shows all possible routes in Mohamad’s town. Mohamad travels from his home to his friend’s house on a daily basis.

If Mohamad’s house is vertex C, and his friend’s house is vertex I, identify the shortest path between the two places and its weight.

16 B

23 16 15

The grocery store has to make a delivery across town. The store is known for its fast delivery. This graph represents a map of the town: If the store is at vertex A, and the delivery has to be made to vertex H, identify the shortest path between both vertices and its weight.

F B 23

21 21

I 7

22 26

13 25

19 H

F 9

A group is organising a tour in the park that ends with a picnic. The graph displays the map of the park: Since the tour is for the elderly, the group wants the distance from the start of the park (at vertex A), to the picnic area (at vertex J ) to be as short as possible. Identify the shortest route between these two vertices and its weight.

I 6

C The police department receives a call about a robbery in progress at a grocery store. This graph shows the map of the town:

If the grocery store is at vertex I, and the nearest police car is located at vertex D, identify the shortest route for the police to follow and its weight.

C 5

Mathspace New South Wales – Year 11 Standard mathspace.co

25 10

29 11 A

606

12 H

23 10

F 12 24

13 7

10 J

Ex 2

The table shows travel times in minutes for a train network. Some routes are express (‘E’) and require a more expensive ticket: Start

End

Time (mins)

Central

Market

Central

Uptown

Market

Uptown

Market

Stadium

Uptown

Museum

Stadium

Museum

Notes

Express (E)

Find the shortest path and its time from Central to the Museum.

A passenger wants the cheapest trip, avoiding all express routes. Find the best path for them and its total time.

A school bus has to pick up students from a certain area. This map shows the routes connecting the students’ houses: The school bus driver wants to save as much time as possible on his route. If he starts his route at vertex D, identify the shortest path for him to take if passing by each house only once.

D 3

5 9 B

A canned goods company has to make drops at every store in an area. This map displays the routes between the stores:

A 13

E 17

Starting from store B, identify the shortest path and its weight to take if passing by every store only once.

5 C 13

12 9

8.06 Shortest path mathspace.co

607

The delivery man has to drop a newspaper at every house in a certain area. This graph shows the map of that area:

The delivery man wants to be time eﬃcient. If he enters the area at house B and leaves at house G, identify the shortest path and its weight for him to travel if he passes by each house only once.

3 15

9 12 A

17 10

17 B

Extend your thinking 15

A local delivery service needs to plan routes between several towns. The following table represents the routes and distances (in km) between the towns.

Town 1

Town 2

Distance (km)

Construct a weighted graph to represent this information, and then determine the shortest route for a van to take if it starts at town S and must visit every other town exactly once.

A technician needs to visit several client sites. The table shows the estimated travel times (in minutes) between sites. Construct a weighted graph representing this information. Then, determine the quickest route for the technician if they start at site D, must make their ﬁrst visit to site E, and then visit every other site exactly once.

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Starting site

Ending site

Time

8 Chapter review 1

Which of the following statements is true for a tree with n vertices? A

It has n edges.

It has n − 1 edges.

It has n + 1 edges.

It must contain at least one cycle.

Consider the network shown. What is the degree of vertex C? B A C

E A 3

For the weighted network shown, what is the weight of the path P –R–T ? P

6 7

R Q

9 4 S

A 4

T C

How many edges and vertices does each network have? a

Chapter 8 review mathspace.co

609

For each network, identify the degree of each vertex and state if the network is connected: a

2 4

4 3

5 7

E 6

For the directed network showing one-way ﬂight paths: a

How many ﬂights depart from city Z?

What is the distance of the ﬂight from B to C?

11 39

Z 2

41 6

This map represents the travel distance for an ambulance to get between the seven townships it services. Draw a simple network graph that represents the map shown. Start at Arda and ﬁnish at Gilgandra:

Belconnen

Pinky Pond

Cambray

Arda

Fernam

Mordan Range 3

Dolby

Shell Lake

Tiller’s Green

610

Eastfarthing 16

Mathspace New South Wales – Year 11 Standard mathspace.co

Gilgandra

A group of 5 colleagues collaborates on various projects. This network graph shows who has worked together: a

Have David and Fiona worked on a project together? Explain why or why not.

Which colleagues should be paired up for the next project so that every member has worked with every other member at least once?

Chloe

Fiona

Ben

David 9

Grace

This network represents one-way streets in a town centre: a

Which location can you not drive to?

Which location can you not drive away from?

C B

D 10

The table shows the travel times, in hours, for a regional train network. Construct a weighted network diagram to represent this information. City A

City B

City C

City D

City A

−

City B

−

City C

−

City D

−

For the network diagram provided, construct a table to represent the weights of the connections. A

7 13 D

9 C

Chapter 8 review mathspace.co

611

Calculate the weight of the following paths in this network: W

A 2 C

X 3

3 3

Path A, C, X

Path W, E, D, C, X, Z

4 Z

Path B, A, C, D

The network shows cycling times in minutes. Find the shortest path from house A to house C by inspection. E

19 13

15 17 12 A

612

17 10

Determine whether each graph is a tree. If it is not, explain why: a

Mathspace New South Wales – Year 11 Standard mathspace.co

A connected network has 10 vertices and 15 edges: a

How many edges must be in any spanning tree of this network?

How many edges must be removed from the original network to form a spanning tree?

A new railway system is being built. The map shows possible routes and their construction costs in millions of dollars. Use an appropriate algorithm to ﬁnd the minimum cost to connect all stations. A 20 22 D 31 B C 24

11 E

27 10

H 21

26 J

The map shows travel times between checkpoints in an adventure race. C

A 16

19 B

24 20

16 15 D

21 I

If a racer starts at checkpoint A and must ﬁnish at checkpoint I, identify the shortest path and its total time. 18

The table shows travel times for a bus network. Some routes are scenic (‘S’) but have a higher ticket price: Start

End

Time (mins)

Notes

Scenic (S)

Find the shortest path and its time from A to E.

A tourist wants the most scenic trip, taking at least one scenic route. Find the best path for them from A to E and its total time. Chapter 8 review mathspace.co

613

A salesperson must visit every client in a city. The map shows the routes and travel times in minutes. E 17

5 C 13

12 9

10 D

Starting from client A, identify the shortest path that visits every other client exactly once, and state its total time. 20

Would a directed or undirected network be more appropriate for modelling these situations? a

Friendships on a social media platform.

The ﬂow of water through a city’s pipe system.

The hierarchy of management in a company.

International ﬂight paths between airports.

A student says the network shown has 5 vertices and 6 edges. What mistake has the student made, and what are the correct numbers?

Draw a simple graph with four vertices that is connected but not complete.

The network represents a one-way system of hiking trails, with weights representing the diﬃculty rating of each trail section. Draw the network, using arrows to show the direction.

614

Start

Lookout

Waterfall

Summit

Start

−

Lookout

−

Waterfall

−

Summit

−

Mathspace New South Wales – Year 11 Standard mathspace.co

A delivery van has a maximum load capacity. The network shows available routes, with weights representing the height limit (in metres) of bridges on each road. D 4.5

4.0 3.8

4.0

4.2 4.5

Depot

If the van is 4.1 metres high, what is the best path it can take from the Depot to location D? Explain your reasoning. 25

A company is setting up a computer network between its 6 oﬃces. The costs, in hundreds of dollars, are shown: Q 5

8 P 6 U

2 S 5

Find the minimum cost to connect all the oﬃces by ﬁnding the minimum spanning tree.

If the link between oﬃces Q and T is unavailable, what is the new minimum cost?

A city council is connecting tourist sites with a free shuttle bus. The table lists the travel times in minutes between sites. Start

End

Time (mins)

Museum

Gallery

Museum

Park

Gallery

Park

Gallery

Wharf

Park

Wharf

Park

Tower

Wharf

Tower

Construct a weighted graph and use an algorithm to ﬁnd the minimum spanning tree for the bus route. What is the minimum total time to form a connected network?

Chapter 8 review mathspace.co

615

The network represents paths in a garden, with weights as distances in metres: G E

7 18

26 D

F 12

10 J

Find the shortest path from the Entrance (A) to the Fountain (J ).

A visitor is at the Bridge (B) and wants to go to the Rose Garden (H). Find the shortest path and its distance.

This network shows ﬂight costs between cities. The edge A − D is a premium-class-only ﬂight: A 200

150

500

180 250

100

What is the shortest path (minimum cost) from city A to city E?

An economy traveller wants the best path from A to E, avoiding the premium ﬂight. What is their best path and its cost?

Two algorithms, Prim’s and Kruskal’s, are used to ﬁnd a Minimum Spanning Tree (MST). If a network has several edges with the same lowest weight, could the two algorithms potentially produce different MSTs? Explain why or why not.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Big ideas • Time measurement and calculations enable effective scheduling and planning in everyday and global contexts. • Latitude and longitude provide a global coordinate system for precise location identification and time difference calculations.

9 Time and location Chapter outline 9.01 9.02 9.03 9.04 9.05 9.06

Units of time Time intervals Elapsed time applications Latitude and longitude International time zones and time differences Australian time zones and daylight savings Investigation: Geocaching Chapter 9 review

620 627 632 645 656 667 675

9.01

Units of time After this lesson, you will be able to… • convert between seconds, minutes, and hours. • represent time in 12-hour format (using a.m. and p.m.) and 24-hour format. • convert between 12-hour and 24-hour time formats. • calculate elapsed time between two time points. • solve a variety of practical problems by calculating elapsed time.

Time unit conversions Time is measured in units such as seconds, minutes, and hours, with fixed relationships: 1 minute equals 60 seconds, and 1 hour equals 60 minutes.

1 minute = 60 seconds minute

is a unit of time, where 1 minute is 60 seconds

seconds

is the smallest unit of time, where 60 seconds make 1 minute

1 hour = 60 minutes hour

is a unit of time, where 1 hour is 60 minutes

minutes

is a unit of time, where 60 minutes make 1 hour

Conversions between these units are performed using multiplication or division. To convert minutes to seconds, multiply by 60. To convert hours to minutes, multiply by 60. To convert seconds to minutes or minutes to hours, divide by 60.

Seconds

Minutes

x60

620

÷24

÷60

Hours

x60

Mathspace New South Wales – Year 11 Standard mathspace.co

Days

x24

12-hour and 24-hour time Interactive exploration Discover this concept in action online

mathspace.co

Time can be expressed in 12-hour or 24-hour formats. The 12-hour time divides a day into two 12-hour blocks, using a.m. (before midday) and p.m. (after midday). The 24-hour time uses a single 24-hour block, without using a.m. or p.m., and is written as HH:MM (for example, 14:30). To convert from 12-hour to 24-hour time: • For a.m. times, keep the hour or add a leading zero (for example, 9:00 a.m. becomes 09:00). • For p.m. times, add 12 to the hour (for example, 3:00 p.m. becomes 15:00). • Exceptions: 12:00 a.m. is 00:00 and 12:00 p.m. is 12:00. To convert from 24-hour to 12-hour time: • For hours 00:00 to 11:59, append a.m. (for example, 09:00 is 9:00 a.m.). • For hours 12:00 to 23:59, subtract 12 and append p.m. (for example, 15:00 is 3:00 p.m.). 24-hour time 00:00

04:00

08:00

12:00

16:00

20:00

00:00

12:00 a.m.

4:00 a.m.

8:00 a.m.

12:00 p.m.

4:00 p.m.

8:00 p.m.

12:00 a.m.

Midnight

Midday 12-hour time

Example 3 Convert 3:41 p.m. to 24-hour time.

Create a strategy Since the time is p.m., add 12 to the hour, keeping the minutes unchanged.

Apply the idea 3:41 p.m. = 3:41 + 12:00 = 15:41 Therefore, 3:41 p.m. is 15:41 in 24-hour time.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Add 12 hours Evaluate

Write the time shown on each clock in 24-hour time: a

Ex 4

Ex 5

Convert these 24-hour times to 12-hour time: a

07:45

15:30

00:00

12:00

09:15

23:50

A train departs Sydney at 09:15 and arrives in the Blue Mountains at 11:45. Calculate the elapsed time.

A ferry leaves Circular Quay at 10:30 and arrives at Manly at 11:05. Calculate the elapsed time.

A movie starts at 6:45 p.m. and ends at 9:15 p.m.. Calculate the elapsed time.

Add: 2 hours 40 minutes + 1 hour 25 minutes

Subtract: 5 hours 10 minutes − 2 hours 35 minutes

Extend your thinking 17

A Vivid Sydney light show starts at 18:00 and ends at 22:30. If there is a 15-minute break halfway through, calculate the total elapsed time excluding the break.

A cyclist completes 3 laps of a track in 15 minutes 45 seconds. Calculate the duration of one lap in seconds.

A student’s study session spans from 11:50 a.m. to 2:20 p.m. Calculate the elapsed time, explaining why using 24-hour time simplifies the calculation.

A relay race involves four runners, each running for: 1 minute 52 seconds, 1 minute 48 seconds, 2 minutes 5 seconds, and 1 minute 55 seconds. Calculate the total time in seconds.

A traveller flies from Sydney to Perth with a stopover in Melbourne. The flight departs Sydney at 08:30 and arrives in Melbourne at 10:05. After a stopover, the next flight departs Melbourne at 12:30 and arrives in Perth at 15:35. Calculate:

626

The flight time from Sydney to Melbourne

The stopover duration in Melbourne

The flight time from Melbourne to Perth

The total travel time, including the stopover, in hours and minutes Practice

Mathspace New South Wales – Year 11 Standard mathspace.co

9.02

Time intervals

After this lesson, you will be able to… • calculate a future or past time by adding or subtracting a given time interval. • determine the duration (elapsed time) between two specified time points. • use timelines as a strategy to accurately calculate time intervals, managing transitions across midday and midnight.

Time intervals Interactive exploration Discover this concept in action online

mathspace.co

Calculating time intervals involves finding the duration between two times or determining a new time after adding or subtracting a duration. A timeline is an effective tool to break intervals into manageable segments, such as minutes to the next hour, full hours, and remaining minutes. • For addition, sum the hours and minutes, converting excess minutes (over 60) to hours. • For subtraction, adjust by borrowing 1 hour (60 minutes) if needed. • Handle transitions across midday (e.g., from a.m. to p.m.) or midnight carefully, ensuring accurate hour and minute calculations. 10:50 a.m.

11:00 a.m.

+10 minutes

1:00 p.m.

+2 hours

1:45 p.m.

+45 minutes

Example 1 Calculate the time difference between 13:32 and 20:14.

Create a strategy Use a timeline to break the interval into segments: minutes to the next hour, full hours, and remaining minutes. Sum the segments to find the total duration.

9.02 Time intervals mathspace.co

627

Apply the idea 13:32

14:00

+28 minutes

20:00

+6 hours

20:14

+14 minutes

From 13:32 to 14:00 is 60 − 32 = 28 minutes. From 14:00 to 20:00 is 20 − 14 = 6 hours. From 20:00 to 20:14 is 14 minutes. Total = 28 minutes + 6 hours + 14 minutes

Sum the segments

= 6 hours + (28 + 14) minutes

Combine minutes

= 6 hours + 42 minutes

Evaluate

The time difference is 6 hours and 42 minutes.

Reflect and check Alternatively, convert the times to minutes, subtract, and convert back to hours and minutes: 20:14 = 1214 minutes, 13:32 = 812 minutes, 1214 − 812 = 402 minutes = 6 hours and 42 minutes.

Example 2 Determine the time 2 hours and 45 minutes after 10:25 a.m.

Create a strategy Break the addition into segments: minutes to the next hour, full hours, and remaining minutes. Check for a.m./p.m. transitions.

Apply the idea 10:25 a.m.

11:00 a.m.

+35 minutes

1:00 p.m.

+2 hours

From 10:25 a.m. to 11:00 a.m. is 60 − 25 = 35 minutes. Subtract 35 minutes from 45 minutes: 45 − 35 = 10 minutes remain Add 2 hours from 11:00 a.m. to 1:00 p.m. Add the remaining 10 minutes to get 1:10 p.m. The time is 1:10 p.m.

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Mathspace New South Wales – Year 11 Standard mathspace.co

1:10 p.m.

+10 minutes

Example 3 A movie starts at 11:50 a.m. and lasts 1 hour and 10 minutes. What time does it finish?

Create a strategy Add the duration to the start time, breaking into segments and checking for a.m./p.m. transitions.

Apply the idea From 11:50 a.m. to 12:00 p.m. is 10 minutes. Add the remaining 1 hour to 12:00 p.m. to get 1:00 p.m. Total = 1 hour + 10 minutes Sum the duration The movie finishes at 1:00 p.m.

Reflect and check Adding 1 hour and 10 minutes to 11:50 a.m. crosses midday, confirming 1:00 p.m.

Example 4 Determine the time 6 hours and 50 minutes before 9:10 a.m.

Create a strategy Subtract the duration from the given time, breaking into segments and checking for a.m./p.m. or midnight transitions.

Apply the idea 2:20 a.m.

3:00 a.m.

–40 minutes

9:00 a.m.

–6 hours

9:10 a.m.

–10 minutes

From 9:10 a.m. to 9:00 a.m. is 10 minutes. Subtract 10 minutes from 50 minutes: 50 − 10 = 40 minutes remain to subtract. Subtract 6 hours from 9:00 a.m. to get 3:00 a.m. Subtract the remaining 40 minutes to get 2:20 a.m. The time is 2:20 a.m.

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Tom plans to drive from Sydney to Canberra over two days with an overnight stay of 9 hours and rest breaks totaling 2 hours and 45 minutes. If he leaves on Friday at 8:30 a.m. and arrives on Saturday at 12:00 p.m., how much time does he spend driving?

A flight departs Newcastle at 14:50 and takes 2 hours and 55 minutes. What is the arrival time in 24-hour time?

Emma starts her workday at 8:45 a.m. and has two breaks of 20 minutes each, plus a lunch break of 50 minutes, before finishing at 4:55 p.m. How many hours and minutes does she work, excluding breaks?

A netball match starts at 3:15 p.m. and ends at 4:35 p.m. If a half-time break lasts 10 minutes, what is the duration of one half of the match?

A flight departs Melbourne at 13:20 and arrives in Perth at 16:50. Calculate the flight duration.

Extend your thinking 17

Sam works as a tutor. His weekly schedule is as follows: Day

Start time

End time

Break duration

Monday

9:30 a.m.

1:50 p.m.

30 minutes

Tuesday

10:15 a.m

2:45 p.m.

45 minutes

Wednesday

8:50 a.m.

1:20 p.m.

40 minutes

Thursday

11:00 a.m.

3:30 p.m.

50 minutes

Friday

9:45 a.m.

2:15 p.m.

35 minutes

Calculate the total number of hours and minutes Sam works each week, accounting for breaks. 18

A school excursion from Sydney to Blue Mountains departs at 7:30 a.m. and is scheduled to arrive at 11:00 a.m. The trip faces: • A departure delay of 20 minutes • A 15-minute break halfway • An additional 10-minute delay due to traffic

Calculate the new arrival time in 12-hour time.

If the group must arrive by 11:30 a.m., suggest one adjustment to meet this time.

A concert starts at 18:20 with shows of 1 hour and 25 minutes, a 10-minute break, and 1 hour and 30 minutes. The first show is delayed by 15 minutes. The concert must end by 21:30. Suggest one adjustment to meet this time.

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9.03

Elapsed time applications

After this lesson, you will be able to… • interpret information from timetables and schedules to determine relevant start and end times for journeys or activities. • apply various methods (counting on, subtraction, timeline) to accurately calculate elapsed time in practical contexts. • solve multi-step problems involving journey planning, including considerations for walking times, breaks, or connections. • calculate the total duration of activities composed of multiple segments or including breaks.

Elapsed time applications Calculating elapsed time, or the duration between two time points, is an essential skill for planning and managing a wide range of activities. This often involves interpreting start and end times, which may be presented in various schedules, timetables or charts. Key methods for calculating elapsed time include: • Counting on: Count the minutes from the start time to the next full hour, then count the number of full hours to the hour before the end time and finally add the remaining minutes to reach the end time. • Subtraction: Convert both times to 24-hour format if necessary. If the times are in hours and minutes, convert them entirely to minutes, subtract and then convert back to hours and minutes if needed. Alternatively, subtract the hours and minutes separately, borrowing 60 minutes for an hour if the start minutes are greater than the end minutes. • Using a timeline: Visually represent the time interval on a number line to help break down the calculation into manageable parts. Accurate calculations require understanding the relationship between units of time such as 1 hour = 60 minutes and proficiency in converting between 12-hour and 24-hour time formats. These calculations are frequently applied in contexts such as: • Journey planning: Determining travel durations, latest departure times or connection times using transportation timetables (e.g., bus or train schedules). • Activity scheduling: Calculating the duration of tasks, appointments or recreational activities and determining start or end times.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Exploration City Circular Quay to Parramatta (Limited Stops) Monday to Friday Circular Quay

14:38

14:50

15:00

15:10

Martin Place Station

14:44

14:56

15:06

15:16

Town Hall Station

14:50

15:02

15:12

15:22

Town Hall House

14:53

15:05

15:15

15:25

Victoria Rd at Hornsey St

14:58

15:10

15:20

15:30

Victoria Rd before Darling St

15:00

15:12

15:22

15:32

Victoria Rd after Lyons Rd

15:06

15:18

15:28

15:38

Riverside Girls High School

15:27

Trim Place

15:14

15:26

15:31

15:36

15:46

Blaxland Rd before Hatton St

15:24

15:36

15:41

15:46

15:56

Blaxland Rd opp

15:25

15:37

15:42

15:47

15:57

15:52

15:57

16:08

Community Health Centre Victoria Rd at Gaza Rd

15:35

15:47

The image displays a section of a bus schedule. Consider the following questions about calculating time intervals using such information: 1. A bus service departs Circular Quay at 15:10 and is scheduled to arrive at Parramatta station at 16:05. Calculate the total scheduled travel time for this journey. 2. The same bus service arrives at Town Hall at 15:32 according to the schedule. Calculate the time taken to travel from Circular Quay (departing 15:10) to Town Hall. 3. Discuss two different methods that could be used to calculate the duration of an activity that starts at 9:50 a.m. and finishes at 1:15 p.m. on the same day.

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Example 1 The provided train schedule shows a service departing Circular Quay at 1:23 p.m. and arriving at Central at 1:30 p.m. Station

Time (p.m.)

Museum

1.11

1:15

1:17

1:21

St. James

1:13

1:19

1:23

Circular Quay

1:17

1:20

1:23

1:27

Wynyard

1:19

1:22

1:29

Town Hall

1:22

1:25

1:32

Central

1:26

1:29

1:30

1:26

1:30

1:26

1:32 1:36

1:31

1:38 1:41

1:36

1:37

1:45

Calculate the journey time from Circular Quay to Central for this service.

Create a strategy To calculate the elapsed time, subtract the start time from the end time. Both times are in the afternoon (p.m.) and on the same day.

Apply the idea Departure time from Circular Quay: 1:23 p.m. Arrival time at Central: 1:30 p.m. Journey time = 1:30 p.m. − 1:23 p.m.

= 7 minutes

Subtract start time from end time Evaluate

The journey time is 7 minutes.

Example 2 Kate travels to work by bus. She lives a 10-minute walk from Merry Oaks bus stop and her workplace is a 5-minute walk from Highlands Drive. She must be at work by 8:00 a.m. The bus schedule is provided. Bus stop Pine St

Time (p.m.) 4:33

Westbrook Dr

7:30

8:32

9:31

5:38

7:39

8:38

7:42

8:47

9:43

8:50

9:40

Merry Oaks

4:40

5:41

6:36

Carlton Rd

4:47

5:42

6:43

Highlands Dr

4:51

5:48

6:51

7:48

8:51

9:48

6:17

7:12

8:18

9:13

10:18

West St

634

5:36

Mathspace New South Wales – Year 11 Standard mathspace.co

a What is the latest time Kate can leave home to arrive on time for work?

Create a strategy Work backwards from the required arrival time at work. First, calculate the latest arrival time at Highlands Drive bus stop. Then, identify the latest bus she can catch from Merry Oaks to meet this arrival. Finally, calculate her departure time from home.

Apply the idea Required arrival time at work: 8:00 a.m. Walking time from Highlands Drive to work: 5 minutes. Latest arrival at Highlands Drive = 8:00 a.m. − 5 minutes Subtract walking time from work arrival time = 7:55 a.m.

Evaluate

Consulting the schedule, the latest bus arriving at Highlands Drive by 7:55 a.m. is the one arriving at 7:48 a.m. This bus departs Merry Oaks at 7:42 a.m. Walking time from home to Merry Oaks bus stop is 10 minutes. Latest departure from home = 7:42 a.m. − 10 minutes Subtract walking time from bus departure time = 7:32 a.m.

Evaluate

The latest time Kate can leave home is 7:32 a.m.

b If she misses this bus (departing Merry Oaks at 7:42 a.m.), by how much time will she be late for work?

Create a strategy Identify the next available bus from Merry Oaks. Calculate her new arrival time at work using this bus. Then, determine the difference between this new arrival time and her required work start time of 8:00 a.m

9.03 Elapsed time applications mathspace.co

635

Apply the idea Bus stop Pine St

Time (a.m.) 4:33

Westbrook Dr

5:36

7:30

8:32

9:31

5:38

7:39

8:38

7:42

8:47

9:43

8:50

9:40

Merry Oaks

4:40

5:41

6:36

Carlton Rd

4:47

5:42

6:43

Highlands Dr

4:51

5:48

6:51

7:48

8:51

9:48

6:17

7:12

8:18

9:13

10:18

West St

If Kate misses the 7:42 a.m. bus from Merry Oaks, the next bus departs at 8:47 a.m. This bus arrives at Highlands Drive at 8:51 a.m. New work arrival time = 8:51 a.m. + 5 minutes = 8:56 a.m.

Add walking time from Highlands Drive Evaluate

Required work start time: 8:00 a.m. Work time delay = 8:56 a.m. − 8:00 a.m. = 56 minutes

alculate difference between new arrival C and required start Evaluate

Kate will be 56 minutes late for work.

Example 3 Liam allocates time to study mathematics on three different occasions during a week: • Monday: 4:30 p.m. to 6:15 p.m. • Wednesday: 5:00 p.m. to 6:45 p.m. • Friday: 3:45 p.m. to 5:05 p.m. a Calculate the duration of Liam’s study session on Monday.

Create a strategy Subtract the start time from the end time for Monday’s session. The “counting on” method can be used.

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Apply the idea Monday start time: 4:30 p.m. Monday end time: 6:15 p.m. Using the counting on method: • From 4:30 p.m. to 5:00 p.m. is 30 minutes. • From 5:00 p.m. to 6:00 p.m. is 1 hour. • From 6:00 p.m. to 6:15 p.m. is 15 minutes. Monday study duration = 1 hour + 30 minutes + 15 minutes Add the time intervals = 1 hour + 45 minutes

Evaluate

Liam studied for 1 hour and 45 minutes on Monday.

b Calculate the total time Liam spent studying mathematics during the week.

Create a strategy Calculate the duration for Wednesday’s and Friday’s sessions, then add all three durations together.

Apply the idea Wednesday session: 5:00 p.m. to 6:45 p.m. Wednesday duration = 6:45 p.m. − 5:00 p.m. Subtract start time from end time = 1 hour 45 minutes

Evaluate

Friday session: 3:45 p.m. to 5:05 p.m. Using the counting on method for Friday: • From 3:45 p.m. to 4:00 p.m. is 15 minutes. • From 4:00 p.m. to 5:00 p.m. is 1 hour. • From 5:00 p.m. to 5:05 p.m. is 5 minutes. Friday study duration = 1 hour + 15 minutes + 5 minutes Sum the time intervals = 1 hour + 20 minutes Total study time = Monday duration + Wednesday duration + Friday duration = (1h 45 min) + (1h 45 min) + (1h 20 min) Hours = 1 + 1 + 1

Add the hours

= 3 hours Minutes = 45 + 45 + 20

Add the minutes

= 110 minutes 110 minutes = 1 hour and 50 minutes Total study time = 3 hours + 1 hour 50 minutes = 4 hours 50 minutes

Since 60 minutes = 1 hour dd total hours and converted A minutes Evaluate

Liam spent a total of 4 hours and 50 minutes studying mathematics during the week.

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Example 4 Mia wants to watch a movie that starts at 10:50 a.m. and has a running time of 2 hours and 25 minutes. There are no advertisements before the movie. a Calculate the time the movie will finish.

Create a strategy Add the movie’s running time to its start time using a step-by-step addition method.

Apply the idea Start time: 10:50 a.m. Running time: 2 hours 25 minutes Add the minutes first: Total time = 10:50 a.m. + 25 minutes

Add minutes to start time

= 10:50 a.m. + 10 minutes + 15 minutes

reak down minutes to reach the B next hour

= 11:00 a.m. + 15 minutes

Add remaining minutes

= 11:15 a.m.

Evaluate

Now add the hours to 11:15 a.m.: Finish time = 11:15 a.m. + 2 hours = 1:15 p.m.

Add hours to the intermediate time Evaluate

The movie will finish at 1:15 p.m. on the same day.

b If Mia wants to catch a bus that leaves 15 minutes after the movie finishes, what time does her bus leave?

Create a strategy Add 15 minutes to the movie’s finish time.

Apply the idea Movie finish time: 1:15 p.m. Bus departure time = 1:15 p.m. + 15 minutes Add waiting time to movie finish time = 1:30 p.m. Mia’s bus leaves at 1:30 p.m.

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Evaluate

A Sydney school timetable is shown. Time slot

Monday

Tuesday

Wednesday

Thursday

Friday

8:40 a.m. – 9:50 a.m.

Maths

History

English

Science

Geography

9:50 a.m. – 11:00 a.m.

RECESS

11:00 a.m. – 12:10 p.m.

Science

English

Maths

P.E.

History

12:10 p.m. – 1:20 p.m.

Geography

P.E.

Science

Maths

English

1:20 p.m. – 2:30 p.m.

LUNCH

2:30 p.m. – 3:40 p.m.

P.E.

Maths

History

Geography

Science

Calculate the duration of the Maths lesson on Monday.

If a student arrives at school at 10:50 a.m. on Thursday, how long must they wait for their P.E. lesson to begin at 11:00 a.m.?

Practice Use this table to answer questions 5 and 6. Station

Ex 1

640

Time (a.m.)

Penrith

6:05

6:15

6:25

6:35

Blacktown

6:20

6:30

6:40

6:50

Parramatta

6:30

6:40

6:50

7:00

Strathfield

6:40

6:50

7:00

7:10

Central

6:50

7:00

7:10

7:20

Consider the Sydney Trains timetable for Penrith to Central: a

Calculate the total scheduled travel time from Penrith to Central for the train departing Penrith at 6:25 a.m.

Calculate the journey time from Parramatta to Central on the train that departs Parramatta at 6:50 a.m.

Alex arrives at Penrith station at 06:28. Based on the timetable, calculate how long he will have to wait at Penrith station to get in the first train he can catch to Central.

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 2

A bus timetable for Wollongong to Kiama is shown: Bus stop

Time (a.m.)

Wollongong Station

7:10

7:25

7:40

7:55

Fairy Meadow

7:20

7:35

7:50

8:05

Corrimal

7:25

7:40

7:55

8:10

Thirroul

7:40

7:55

8:10

8:25

Kiama Station

8:00

8:15

8:30

8:45

Chloe must be at work in Kiama by 9:00 a.m. It takes her 6 minutes to walk from Kiama Station to her work. What is the latest bus she can catch from Wollongong Station to arrive on time, and what is its scheduled arrival time in Kiama?

If Chloe’s home is a 10-minute walk from Wollongong Station, what is the latest time she can leave home to catch the bus identified in part (a)?

Consider the Sydney Ferries timetable from Circular Quay to Watsons Bay: Watsons Bay to Pyrmont Bay via Circular Quay and Barangar

Monday to Friday Day Restriction Wastons Bay

14:49

15:19

15:49

16:19

Rose Bay

14:59

15:29

15:59

16:29

16:49

17:09

Circular Quay

ARR

15:12

15:42

16:12

16:42

17:02

17:22

Circular Quay

DEP

15:17

15:47

16:17

16:47

17:07

17:27

Milsons Point

15:22

15:52

16:22

16:52

17:12

17:32

McMahons Point

15:25

15:55

16:25

16:55

17:15

17:35

Balmain East

15:30

16:00

16:30

17:00

17:20

17:40

Barangaroo

ARR

15:36

16:06

16:36

17:06

17:26

17:46

Barangaroo

DEP

15:40

16:10

16:40

17:10

17:30

17:50

Pyrmont Bay

15:44

16:14

16:44

17:14

17:34

17:54

Calculate the journey duration from Circular Quay to Watsons Bay on the service departing Circular Quay at 09:15. (Assume Watsons Bay arrival for this service is 09:37 based on the full timetable context).

If a passenger arrives at Circular Quay at 09:12, what is the time they will have to wait for the next ferry shown on this part of the timetable (departing at 09:15)?

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A Sydney Metro timetable from Tallawong to Chatswood is shown: Station

Ex 3

Time (a.m.)

Tallawong

7:00

7:06

7:12

Rouse Hill

7:05

7:11

7:17

Kellyville

7:10

7:16

7:22

Castle Hill

7:15

7:21

7:27

Chatswood

7:40

7:46

7:52

Calculate the waiting time between consecutive services departing Kellyville (e.g., between the first and second service shown, and between the second and third).

How long is the journey from Rouse Hill to Chatswood on the service that departs Rouse Hill at 7:11 a.m.?

Sarah has the following study plan for an afternoon: • English: 2:30 p.m. to 3:45 p.m. • Break: 3:45 p.m. to 4:00 p.m. • Mathematics: 4:00 p.m. to 5:30 p.m.

Ex 4

Calculate the duration of Sarah’s English study session.

What is the total time Sarah spends studying (excluding her break)?

If Sarah starts her Mathematics study 10 minutes late but still studies for the planned Mathematics duration, what time will she finish studying Mathematics?

A film festival is screening three short films back-to-back. There is a 5-minute break between each film. The durations of the films are: Film A: 25 minutes, Film B: 40 minutes, Film C: 30 minutes. Film A starts at 14:00. a

At what time will Film B start?

At what time will Film C finish?

A timetable from Gosford to Hornsby is shown: Station

Time (a.m.)

Gosford

5:50

6:05

6:20

Woy Woy

6:00

6:15

6:30

Hawkesbury River

6:10

6:25

6:40

Berowra

6:25

6:40

6:55

Hornsby

6:40

6:55

7:10

Calculate the journey duration from Woy Woy to Hornsby on the train that departs Woy Woy at 6:15 a.m.

If a passenger arrives at Gosford station at 06:10 a.m. and takes the earliest possible train to Hornsby shown on this timetable, what is their total journey time on that specific train service from Gosford to Hornsby?

An overnight train departs Sydney at 22:35 on Tuesday and arrives at its destination in Melbourne at 07:50 on Wednesday morning. Calculate the total journey time.

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A roast chicken needs to cook for 1 hour and 40 minutes. If it is placed in the oven at 5:50 p.m., at what time will it be ready?

Extend your thinking 15

Sophie needs to travel from Strathfield to Manly. She will take a train from Strathfield to Circular Quay and then a ferry from Circular Quay to Manly. The walk between the train station at Circular Quay and the ferry terminal takes 6 minutes. Use the timetables provided. Train timetable: Strathfield to Circular Quay (all times a.m.) Time (a.m.) Strathfield (departure)

8:00

8:10

8:20

Central (arrival)

8:10

8:25

8:35

Circular Quay (arrival)

8:20

8:30

8:40

Ferry timetable: Circular Quay to Manly (all times a.m.) Time (a.m.)

Circular Quay (arrival)

8:30

8:45

9:00

Manly (arrival)

8:55

9:10

9:25

If Sophie wants to arrive at Manly by 9:15 a.m., what is the latest train she can catch from Strathfield? Show calculations to support your answer.

Calculate Sophie’s total travel time from departing Strathfield to arriving at Manly if she takes the train identified in part (a).

Ethan is planning an evening out. He wants to attend a local community concert and have dinner at a nearby restaurant. • The walk from the concert hall to the restaurant (or vice versa) takes 8 minutes. • He needs to allow 15 minutes to find his seat and settle in before the concert starts. • Dinner bookings are available at 17:30 or 19:00. A dinner slot lasts for 1 hour. • The concert has three possible start times: 17:30, 18:15, or 19:00. The concert itself lasts for 1 hour and 30 minutes (including intermission/applause).

If Ethan decides to have dinner first, booking the 17:30 slot, what is the earliest concert start time he can attend afterwards?

If Ethan decides to attend the concert first, and it starts at 18:15, can he make a 19:00 dinner booking? Justify your answer with calculations of relevant times.

A painter starts a job at 08:30. They take a 45-minute lunch break starting at 12:15. They also take two 15-minute tea breaks, one at 10:00 and another at 14:30. If they finish the job at 16:45, calculate the actual time spent painting.

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A commuter is comparing three different transport options to get from Bondi Junction to North Sydney for work. • Option 1: Direct train from Bondi Junction departing at 8:10 a.m., arriving at North Sydney at 8:35 a.m. • Option 2: Bus from Bondi Junction to Town Hall, then train from Town Hall to North Sydney. • The bus departs from Bondi Junction at 8:05 a.m. and takes 25 minutes to reach Town Hall. • There is a 7-minute walk/wait time at Town Hall for the train. • The train journey from Town Hall to North Sydney takes 10 minutes. • Option 3: Go to Edgecliff station to catch a train to Circular Quay, ride a ferry from Circular Quay to Milsons Point, then walk to North Sydney office. • After departing at 8:00 a.m., walking from Bondi Junction to Edgecliff station takes 12 minutes. • Train from Edgecliff to Circular Quay departs at 8:15 a.m. and arrives at 8:25 a.m. • Ferry from Circular Quay to Milsons Point departs at 8:30 a.m. and arrive at 8:35 a.m. • Walking from Milsons Point to North Sydney office takes 5 minutes. Calculate the total travel time from Bondi Junction to arrival at the North Sydney office/ station for each option. Which option has the shortest total travel time?

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Exploration Consider a map of your local area. How might a grid of latitude and longitude differ from a street map’s coordinate system (e.g., letters and numbers)? Discuss why Earth’s spherical shape requires a different approach.

Example 1 A world map shows lines of longitude and latitude with several marked locations.

750 600

900

600

300 150 1500 00

750 450

B 1200

150 300

900

600

300 F

300 K

600

900

1200

1500

300 150 1800 00 150 300

450

450 60

900

a Which point is located on the Greenwich Meridian?

Create a strategy The Greenwich Meridian is the line of longitude at 0°.

Apply the idea Point A is on the vertical line marked 0°.

b Which location has coordinates 30°S, 60°W?

Create a strategy Locate 30° S by finding the Equator and moving 30° south. Locate 60°W by finding the Prime Meridian and moving 60° west.

Apply the idea The location at 30° S, 60° W is point F.

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c Find the coordinates of point I.

Create a strategy Determine the latitude by checking if point I is north or south of the Equator and by how many degrees. Determine the longitude by checking if it is east or west of the Greenwich Meridian and by how many degrees.

Apply the idea Point I is on the 60° latitude line north of the Equator, so its latitude is 60°N. It is on the 120° longitude line east of the Prime Meridian, so its longitude is 120°E. Thus, the coordinates are (60°N, 120°E).

d How many degrees south is point C from point B?

Create a strategy Calculate the difference in latitude between points B and C, noting their positions relative to the Equator.

Apply the idea Point B is at 15°N, and point C is at 15°S.

150 N

Jump 150 00 Jump 150 more

C 15 S 0

Difference = 15° + 15°

= 30°

Sum the distances from the Equator Evaluate

Point C is 30° south of point B.

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647

Time difference Time differences between locations arise from their longitudinal positions. Earth’s 360° circumference is divided into 24 hours, so each 15° of longitude corresponds to a 1-hour time difference. Locations east of a reference point are ahead in time, while those west are behind. For example, a location at 30°E is 2 hours ahead of one at 0°.

Example 4 A world map shows lines of longitude and latitude with marked locations.

75 60

900

45 300 150 1500 1200 900 00 150 D 300 450 600 750 900

B 60

300

600

750

600 450 300 0 900 1200 1500 1800 15 0 0 150 H C 300 450 0 60 750 900

a How many hours is point C ahead or behind point F?

Create a strategy Calculate the longitude difference, note the direction (east or west), and divide by 15° to find the time difference.

Apply the idea Point C is at 90°E, and point F is at 60°W. The longitude difference is 90° + 60° = 150°.

Time difference =

= 10

Divide longitude difference by 15° Evaluate

Since point C is east of point F, it is 10 hours ahead.

b How many hours is point F ahead or behind point J?

Apply the idea Point F and point J are both at 60°W, so the longitude difference is 0°.

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Name the point in the diagram that has the coordinates (40° S, 13° E): 130 E

600 E

530 E O C

Equator

400

150 S E

F G

400 S

Practice Ex 1

The world map shows the lines of longitude and latitude. Several locations are marked on the map: 900

900

600 300

1500 1200 900

600

300 300

300

600

900

150 300

450 60

J 750

150 00

1200 1500 1800

450 600

652

450

600

450 15

750

900

Name the point that is located on the Greenwich Meridian.

Find the point which has coordinates of (60° S, 60° W).

Find the coordinates of point I.

How many degrees north is point I from point B?

How many degrees east is point I from point F?

Identify the point that has the same latitude as point G.

Identify the point that has the same longitude as point D.

How many hours ahead or behind is point I from point F?

Mathspace New South Wales – Year 11 Standard mathspace.co

900

Ex 4

A world map shows lines of longitude and latitude with marked locations.

How many hours is point D ahead or behind point H?

How many hours is point A ahead or behind point B?

The table shows the latitude and longitude of several cities: List the cities that are in the northern hemisphere.

The table shows the latitude and longitude of several cities. Determine which of these cities is closest to the:

Line of latitude at 40° N

Line of latitude at 10° S

The table shows the latitude and longitude of several cities. Determine which of these cities is closest to the:

654

Line of longitude at 80° E

Line of longitude at 175° W

City

Latitude

Stavenegr

59°N

6°E

Melo

32°S

54°W

Uxbridge

52°N

0°W

Puerto Williams

55°S

68°W

El Tigre

9°N

64°W

Quissico

25°S

35°E

City

Longitude

Latitude

Longitude

Zagreb

46°N

16°E

Comallo

41°S

70°W

Eden

37°N

80°W

San José

10°N

84°W

Nagele

5°N

40°E

Lobamba

26°S

31°E

Latitude

Longitude

Bangkok

14°N

101°E

Paris

49°N

2°E

Wellington

41°S

175°E

Suva

18°S

178°E

Perm

58°N

56°E

Salem

37°N

80°W

City

Identify the coordinates of Town A given that: a

Town A is 8° south of Town B which has coordinates of (20° S, 155° E).

Town A is 6° west of Town B which has coordinates of (21° N, 140° E).

Town A is 7° south and 10° west of Town B which has coordinates of (25° N, 154° E).

Mathspace New South Wales – Year 11 Standard mathspace.co

The location of City A is (35°N, 173°W) and the location of City B is (44°N, 160°W): a

Calculate the difference in latitude between the pair of cities.

Which city is south of the other?

Consider the city of Alofi at latitude 19° S and longitude 170° W. Determine the latitude and longitude of:

15° north of Alofi

15° south of Alofi

15° east of Alofi

15° west of Alofi

Consider the city of Springbok at latitude 30°S and longitude 18°E. Determine the latitude and longitude of:

39° north of Springbok

39° south of Springbok

39° east of Springbok

39° west of Springbok

Patricia is going to travel to various destinations, and begins her journey at point D shown on the map: 900

900 75

750

600 45

450 C

300 150 0

600

1500 1200

900 600

300

600

900

300 150

1200 1500 1800

150

300

450

450 600

600 750

900

Identify the coordinates of her starting position, D.

Her change in latitude and longitude after each part of her journey is represented in the table. Complete the table with the coordinates of her starting and finishing positions after each part of her journey:

Part of journey

Starting position

Change in latitude

Change in longitude

Finishing position

(45° N, 75° E)

9° N

7° W

⬚

12° S

16° E

⬚ ⬚

Determine the difference in longitude and latitude between her finishing position at the end of the second part of the journey, and point A.

9.04 Latitude and longitude mathspace.co

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Extend your thinking The expression “travel to the ends of the Earth” suggests travelling to the farthest possible places.

Find the coordinates of the point on Earth that is the greatest possible distance from the point with coordinates (67° S, 99° E). A traveller starts their journey in a city located at (55° S, 104° E) at 12:00 p.m. (noon) local time. They take a direct flight to their destination at (55° N, 96° W), which takes 22 hours.

What is the local time at their destination (in 24 hour time) upon arrival, rounded to the nearest minute?

9.05 International time zones and time differences After this lesson, you will be able to… • define Coordinated Universal Time (UTC) and time zone • interpret UTC offsets to determine if a location’s time is ahead of or behind UTC+0 • calculate the time difference between two locations given their UTC offsets • determine the local time in one location when given the local time and UTC offset of another location, and the UTC offset of the first location • recognise the significance of the International Date Line

Coordinated Universal Time (UTC) The primary time standard used across the world. It is based on International Atomic Time, which is measured by atomic clocks, with leap seconds added occasionally to keep it close to the earth’s rotation. For every 15° longitude east or west of the Prime Meridian (known as 0° longitude), the time changes by 1 hour. Time zones The 24 divisions of the globe, where each change of 15 degrees longitude corresponds to one hour. As time zones often align with national or regional boundaries this is an approximate relationship.

The world is divided into time zones. The time zones are set to match daylight hours resulting from the rotation of the earth but can also be influenced by the politics of individual countries.

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Coordinated Universal Time (UTC) is the standard for comparing times globally. Greenwich, England, at 0° longitude, is designated UTC+0. Other locations have a UTC offset indicating hours ahead (+) or behind (−) Greenwich. Time zones east of UTC+0, like Sydney at UTC+10 are ahead in time, while those west, like Mexico at UTC−6 are behind.

Exploration Study the following world map. Pay attention to the time indicated at various locations around the world. 1. What do you notice about the times on the clocks? How do they compare to each other? 2. Why is the time different in each location? 3. How is the time in Greenwich, England (UTC) related to the time in the other locations? Greenland 6:00 AM Greenwich 8:00 AM Mexico 2:00 AM

India 1:30 PM Sydney 6:00 PM

+13+14 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 UTC+1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12

The International Date Line is an imaginary line that runs from the North Pole to the South Pole and is roughly at 180° longitude. It has an odd shape as it tries to avoid cutting through countries. UTC offsets can be used to calculate time differences. For example, if it is 2:00 p.m. on Sunday in UTC−11, then to find the time and day in UTC+12, add 11 + 12 = 23 hours to get 1:00 p.m. on Monday. Some time zones include fractional offsets (e.g., UTC+5.5 for Sri Lanka) due to regional preferences or daylight saving.

9.05 International time zones and time differences mathspace.co

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9.05 Practice questions What do you remember? 1

Describe these times as a number of hours ahead or behind UTC+0: a

Cities at UTC+1

Cities at UTC−9

Cities at UTC−3.5

Island at UTC+

Consider the map shown:

Vancouver (DST)

Lisbon (DST) Timbuktu

Moscow

Riyadh

Lima

Manila

Mumbai Perth

Santiago

Auckland (DST)

+13+14 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 UTC +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 Are these statements true or false?

Perth is 3 hours and 30 minutes ahead of Mumbai.

Santiago and Lima are both 5 hours behind of Timbuktu.

Riyadh and Moscow are both 5 hours behind of Manila.

If it’s 9:00 a.m. in Lisbon currently, then it is also 9:00 a.m. at Timbuktu.

Currently, the time zone in Auckland is UTC+13.

If it’s 5:45 p.m. in Vancouver without ADST, then it’s 4:45 p.m. with ADST.

Define these terms related to time zones: a

Coordinated Universal Time (UTC)

Using the table, identify the UTC offset for each city: City

660

International Date Line

UTC

Vancouver

−8

New Delhi

+5.5

Rome

Tokyo

Vancouver

New Delhi

Rome

Tokyo

Mathspace New South Wales – Year 11 Standard mathspace.co

Which city will have a time of 11:00 p.m. when it is 4:00 p.m. UTC+0? • Cairo (UTC+2) • Singapore (UTC+7) • Rio de Janeiro (UTC−3)

Ex 2

Consider the tables of time zones: UTC

UTC

Los Angeles

−8

Baku

Tijuana

−8

Dubai

Vancouver

−8

Yerevan

Chicago

−6

Afghanistan

+4.5

Mexico City

−6

Sri Lanka

+5.5

San Salvador

−6

Hong Kong

Havana

−5

Perth

New York

−5

Shanghai

Toronto

−5

Singapore

Athens

Seoul

Cairo

Tokyo

Kiev

Auckland

+12

How many hours is Toronto behind Perth?

How many hours is Afghanistan behind Shanghai?

When it is 11:00 a.m. in Chicago, what time is it in Dubai?

When it is 9:00 a.m. in Mexico City, what time is it in Sri Lanka?

Determine the time and day in Tokyo when it is 13:15 on a Thursday in Tijuana.

This table shows the time of several locations relative to UTC+0. UTC

662

City

Location

−5

New York, Cuba, Peru

−4

Chile, Barbados, Brazil (west), Bolivia

−3

Argentina, Brazil (east), Uruguay, Greenland

−2

South Sandwich Islands

−1

Azores

UK, Ireland, Iceland, Portugal, Ghana, Liberia, Mali

Algeria, Angola, Chad, Bosnia, Croatia

Finland, Greece, Lebanon, Egypt, South Africa

Russia (west), Saudi Arabia, Kenya, Madagascar, Iraq

+3.5

Iran

Mauritius, United Arab Emirates, Armenia

+4.5

Afghanistan

Pakistan, Maldives, Kazakhstan

+5.5

India, Sri Lanka

Mathspace New South Wales – Year 11 Standard mathspace.co

Consider the locations Uruguay and Algeria: i

Using the table, which location is ahead of the other in time?

Calculate the time difference between Uruguay and Algeria.

iii

When it is 21:25 on the 13th day of the month in Uruguay, what time and day is it in Algeria?

Consider the locations South Sandwich Islands and Brazil (west): i

Using the table, which location is ahead of the other in time?

Calculate the time difference between South Sandwich Islands and Brazil (west).

iii

When it is 6:00 p.m. on the 17th day of the month in South Sandwich Islands, what time is it in Brazil (west)?

Consider the locations Kenya and Mauritius: i

Using the table, which location is ahead of the other in time?

Calculate the time difference between Kenya and Mauritius.

iii

When it is 17:25 on the 15th day of the month in Kenya, what time and day is it in Mauritius?

Han, who is in Melbourne (UTC+10), has an interview over Skype with the human resources manager of an American trading firm at 4:00 p.m. on Sunday, Memphis (UTC−6) time: a

Calculate the time difference between the two cities.

What time and day will the interview fall on in Melbourne?

Three scientists collaborating on the same project all need to get together for weekly meetings. Each scientist is in a different location. Buzz is located in Rome (UTC+1). Han islocated in Cairo (UTC+2). Beth is located in San Francisco (UTC−8): a

Which scientist will have the earliest local time during the meeting?

Which of the scientists will have the latest local time during the meeting?

Beth suggests 8:00 p.m. in San Francisco. Determine the time it would be in the other two cities:

Buzz suggests 8:00 a.m. in Rome. Determine the time it would be in the other two cities:

i i 12

Rome Cairo

ii ii

Cairo San Francisco

Sophia, Quentin and Harry work for the same company from offices in three different cities. The cities and time zones of each person is given in the table: They want to organise an hour long online meeting within everyone’s working hours. a

Person

City

Time zone

Sophia

Los Angeles

UTC-8

Quentin Chicago

UTC−6

Harry

UTC+10

Melbourne

A person can attend the meeting if it starts after 8:00 and ends before 18:00 in their local time. Which of these times should Harry choose to make sure that Sophia and Quentin can both attend? The options are given in Harry’s time zone. A

16:00

13:00

9:00

6:00

Each person only works Monday to Friday. Which days could Harry choose to have the meeting in his time zone?

9.05 International time zones and time differences mathspace.co

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The table shows the current time in major capital cities beginning with the letter M: a

When it is 15:10 Minsk time, determine the time in Madrid. Give your answer in 24-hour time.

When it is 11:20 a.m. Montevideo time, determine the time in Minsk. City

Time

Madrid

Thu 1:58 a.m.

Managua

Wed 5:58 p.m.

Manila

Thu 7:58 a.m.

Melbourne

Thu 9:58 a.m.

Mexico City

Wed 6:58 p.m.

Miami

Wed 7:58 p.m.

Minneapolis

Wed 6:58 p.m.

Minsk

Thu 2:58 a.m.

Montevideo

Wed 8:58 p.m.

Montreal

Wed 7:58 p.m.

Moscow

Thu 3:58 a.m.

Mumbai

Thu 5:58 a.m.

A plane travels from El Paso (UTC−7) at 10:00 a.m. on Friday on a 13-hour flight to Brazilia (UTC−3): a

Calculate the time difference between the two cities.

What time is it in Brazilia when it is 10:00 a.m. in El Paso?

What time and day in Brazilia will the plane arrive?

When the time is 5:23 p.m. in Honolulu, it is 10:23 p.m. in Miami. You take 19 hours to travel from Honolulu to Miami. If you leave at 4:50 p.m. on Sunday, determine: a

The local time that you arrive in Miami

The day you arrive in Miami

Tina looks at her flight details. Her plane leaves at 6:00 a.m. on Wednesday and arrives ather destination at 8:00 p.m. on Tuesday (local time). The flight will take 12 hours. Calculate the time difference between her origin and destination.

The table displays the departure and flight times for three Air China flights: Flight number

664

Airport code

Departure time

Airport code

Flight time

CA994

San Francisco UTC-8 (SFO)

1:13 p.m.

Beijing UTC+8 (PEK)

12 hours and 31 minutes

CA927

Shanghai Pudong 5:56 p.m. UTC+8 (PVG)

Beijing UTC+8 (PEK)

8 hours and 31 minutes

CA950

Beijing UTC+8 (PEK)

San Francisco UTC+8 (SFO)

14 hours and 37 minutes

7:21 p.m.

Determine the time CA994 arrives in Beijing local time.

Determine the time CA927 arrives in Beijing local time.

Determine the time CA950 arrives in San Francisco local time.

Mathspace New South Wales – Year 11 Standard mathspace.co

Luke wants to travel from Moscow to Beijing by train. In order to do that he has to board three trains. These trains are shown in the table: Train

Departure time

Station

Arrival time

Moscow UTC+3

8:56 p.m Monday

Novosibirsk UTC+6

2:29 p.m. Wednesday

Novosibirsk UTC+6

1:25 p.m. Thursday

Ulaanbaatar UTC+8

9:47 p.m. Friday

Ulaanbaatar

6:46 a.m. Saturday

Beijing UTC+8

2:27 p.m.

Station

UTC+8

How long does the first train take?

How long does the third train take?

Sunday

How long does the second train take?

A flight departs from Dubai (UTC+4) at 15:30 on Monday and takes 8 hours to arrive in London (UTC+0). a

Calculate the time difference between Dubai and London.

What time and day will the flight arrive in London?

A global team has members in Tokyo (UTC+9), New York (UTC−5), and Cairo (UTC+2). They need to schedule a meeting that starts between 9:00 a.m. and 5:00 p.m. local time for all members. a

What is the time difference between Tokyo and New York?

Determine a suitable start time in Cairo that works for all members.

Extend your thinking 21

Explain these terms: a

Why would you ‘gain time’ if you travelled from New Zealand to the UK?

Why would you ‘lose time’ if you travelled from Paris to Melbourne?

Consider the table of time zones: City

UTC

Vancouver

−8

City

UTC

Auckland

+12

Kiki works in Auckland for a multi-national company. She works until lunchtime on Friday and then catches an afternoon flight to Vancouver, the flight taking hours. Explain how she can work on Friday in the Vancouver office.

Explain why this cannot happen when Kiki flies from Vancouver to Auckland.

9.05 International time zones and time differences mathspace.co

665

The two Diomede islands are only 4 km apart, but travelling this distance from one to the other means crossing the International Date Line: a

UTC+12 Big Diomede

At 3:00 p.m. on Wednesday, Hannah rows from Big Diomede to Little Diomede in minutes.

Little Diomede

What day will it be on Little Diomede when Hannah arrives? b

Hannah spends about an hour on Little Diomede before rowing back, with the return journey taking about the same amount of time as before. What day will it be on Big Diomede when she returns?

UTC–8

International Date Line

A traveller departs from Apia, Samoa (UTC+13) at 10:00 a.m. on Monday, flies to Honolulu (UTC−10) in 5 hours, stays for 2 hours, then flies to Anchorage (UTC−9) in 6 hours. a

What time and day does the traveller arrive in Honolulu?

What time and day does the traveller arrive in Anchorage?

Explain how crossing the International Date Line affects the traveller’s itinerary.

Some regions, like Nepal, use a UTC+5.75 offset. Explain why a country might choose a fractional offset like 15 minutes instead of a whole hour, and provide an example of how this affects time calculations between Nepal and a city with a standard offset, such as Tokyo (UTC+9).

Two students calculate the time in Seoul when it is 2:00 p.m. in Vancouver: Student A: = UTC+1, so 2:00 p.m. +1 hour = 2:00 p.m. Student B: UTC+9 − UTC−8 = 17 hours, so 2:00 p.m. + 17 hours = 7:00 a.m. next day Identify the error in each student’s work and provide the correct time.

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Mathspace New South Wales – Year 11 Standard mathspace.co

9.06 Australian time zones and daylight savings After this lesson, you will be able to… • identify Australia’s main standard time zones (AWST, ACST, AEST) and their UTC offsets. • explain Daylight Saving Time (DST) and identify which Australian states/ territories observe it, creating AEDT and ACDT. • calculate the time in different Australian locations, making necessary allowances for standard time or daylight saving time. • solve practical problems involving time differences and elapsed time within Australia, accounting for daylight saving.

Australian time zones and daylight savings Daylight saving A system of setting clocks 1 hour ahead of the standard UTC in spring, to give more hours of daylight to the working day in summer months. In Australia, this practice is observed in some states from the first Sunday in October to the first Sunday in April. Australia is divided into three time zones as shown: • Australian Eastern Standard Time (AEST) • Australian Central Standard Time (ACST) • Australian Western Standard Time (AWST) AWST (UTC+8)

ACST (UTC+9.5)

AEST (UTC+10)

2 h behind

0.5 h behind

Compared to AEST

NT QLD

WA SA

NSW VIC TAS

9.06 Australian time zones and daylight savings mathspace.co

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Australia also uses Daylight Saving Time (ADST) in the summer months. From October to April, some of the Australian states and territories turn their clocks forward one hour. There are two different ADST zones as shown in the map: AWST (UTC+8)

ACST (UTC+9.5)

AEST (UTC+10)

ACDT (UTC+10.5)

AEDT (UTC+11)

2 h behind

0.5 h behind

Compared to AEST

0.5 h ahead

1 h ahead

NT WA

QLD

NSW

VIC

TAS

NO DST

DST

Example 1 When the time is 14:15 in Perth what is the time in: AWST (UTC+8)

ACST (UTC+9.5)

2 h behind

0.5 h behind

NT WA

AEST (UTC+10) Compared to AEST

1 h ahead

QLD

SA NSW

VIC

TAS

Mathspace New South Wales – Year 11 Standard mathspace.co

0.5 h ahead

NSW

668

AEDT (UTC+11)

QLD

NO DST

ACDT (UTC+10.5)

TAS DST

9.06 Practice questions What do you remember? 1

Which Australian time zone is closest to the UTC time?

What is the UTC offset for Australian Central Daylight Time (ACDT)?

Do these Australian states or territories observe Daylight Saving Time (ADST)?

Victoria

Western Australia

New South Wales

Northern Territory

Queensland

South Australia

Tasmania

Australian Capital Territory

If it is 4:00 a.m. in Australian Eastern Standard Time (AEST), what time is it in daylight savings time (AEDT)?

Practice 5

Which location will be the first to celebrate New Year’s Eve? • Canberra (UTC+10) • Darwin (UTC+9) • Perth (UTC+8)

Convert these Daylight Savings Time in Sydney to Eastern Standard Time: a

670

8:00 a.m. AEDT

2:49 a.m. AEDT

11:50 p.m. AEDT

Convert these Eastern Standard Time in Sydney to Daylight Savings Time: a

16:14 AEDT

14:35 AEST

4:00 a.m. AEST

9:45 a.m. AEST

5:13 p.m. AEST

If it is 4:00 p.m. Daylight Savings Time (DST) in South Australia, determine the time in: a

New South Wales

Western Australia

Queensland

Northern Territory

Mathspace New South Wales – Year 11 Standard mathspace.co

The map shows the time zones across the Australian states, use it to answer questions 9, 10 and 11. AWST (UTC+8)

ACST (UTC+9.5)

2 h behind

0.5 h behind

NT WA

AEST (UTC+10) Compared to AEST

ACDT (UTC+10.5)

AEDT (UTC+11)

0.5 h ahead

1 h ahead

QLD

NSW VIC

Ex 1

Sydney during No ADST

Sydney during ADST

Use the time zone map to answer the following: a

If it is 8:16 p.m. AEST, determine the following: i

The time in ACST

The time in AWST

If it is 7:27 p.m. in ACST, determine the time in AWST.

If it is 12:10 a.m. AEST, determine the following, giving your answers in 24-hour time: i

d 11

TAS DST

When the time is 19:20 in Perth determine the time in: a

NSW VIC

TAS NO DST

QLD

The time in ACST

The time in AWST

If it is 1:16 p.m. in ACST, determine the time in AWST.

Use the time zone map to answer the following: a

A flight leaves Hobart Airport, TAS at 10:55 a.m. during ADST. It takes 4 hours and 30 minutes to reach Perth airport, WA. Determine the time the flight arrives in WA local time.

A flight leaves Alice Springs Airport, NT at 5:15 a.m. during ADST. It takes 3 hours and 10 minutes to reach Hobart airport, TAS. Determine the time the flight arrives in TAS local time.

If an aeroplane leaves Broome Airport in Western Australia at 7:30 p.m. and arrives at Albury (operating on daylight savings time) Airport in Victoria at 3:52 a.m., determine its flight time.

If an aeroplane leaves Albury Airport in Victoria at 7:14 p.m. and arrives at Adelaide Airport in South Australia (operating on daylight savings time) at 9:10 p.m., determine its flight time.

If an aeroplane leaves Adelaide Airport in South Australia at 9:53 a.m. and arrives at Broome Airport in Western Australia at 3:41 p.m., determine its flight time.

9.06 Australian time zones and daylight savings mathspace.co

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Use this time zone map to answer questions 12 and 13. AWST (UTC+8)

ACST (UTC+9.5)

AEST (UTC+10)

2 h behind

0.5 h behind

Compared to AEST

NT WA

QLD

SA NSW VIC TAS

Luke and Maria board a train at 7:58 p.m. on Saturday from Sydney, New South Wales towards Adelaide, South Australia. a

If the trip to Adelaide took 20 hours and 22 minutes, determine the time they arrived in Adelaide local time.

Luke boards a train from Adelaide, South Australia at 5:45 p.m. on Sunday to Darwin, Northern Territory. If his trip took 28 hours and 36 minutes, determine the time he arrived in Darwin local time.

Maria boards a train in Adelaide, South Australia at 4:55 p.m. Sunday towards Perth, Western Australia. If the trip took her 19 hours and 20 minutes, determine the time she arrived in Perth local time.

Determine the time in Perth when it is 6:42 p.m. in Hobart.

Complete the table to show the equivalent time in each state: Western Australia (WA)

South Australia (SA)

Queensland (QLD)

6:15

⬚

Complete the table to show the equivalent time in each time zone: AWST

ACST

AEST

⬚

5:25

⬚

5:11 ⬚ 672

11:45

10:15

5:25 c

⬚

Mathspace New South Wales – Year 11 Standard mathspace.co

⬚ 7:17

⬚

12:25 ⬚

Tina looks at her flight details. Her plane leaves at 6:00 a.m. on Wednesday and arrives at her destination at 8:00 p.m. on Tuesday (local time). The flight will take 12 hours. Calculate the time difference between her origin and destination. Use this time zone map to answer questions 15 and 16: AWST (UTC+8)

ACST (UTC+9.5)

AEST (UTC+10)

2 h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

Consider the map that shows the time zones across the Australian states: a

An aeroplane leaves Perth airport in Western Australia at 9:16 p.m. and takes 4 hours and 22 minutes to reach Brisbane airport, Queensland. What time will it arrive in Brisbane local time?

An aeroplane leaves Brisbane airport in Queensland at 3:24 a.m. and takes 6 hours and 34 minutes to arrive at Darwin airport in Northern Territory. What time will it arrive in Darwin local time?

An aeroplane leaves Darwin airport in the Northern Territory at 10:48 p.m. and takes 6 hours and 28 minutes to arrive at Perth airport in Western Australia. What time will it arrive in Perth local time?

Consider the map that shows the time zones across the Australian states: a

Han and Amelia board a train at 8:52 p.m. on Thursday in Sydney, New South Wales travelling to Adelaide, South Australia. If they arrived in Adelaide, South Australia on Friday at 6:48 p.m., determine the duration of their train trip.

Harry boards a train in Adelaide, South Australia at 8:43 p.m. on Friday travelling to Darwin, in the Northern Territory. He arrives at 8:11 p.m. Saturday. Determine the duration of his trip.

Amelia boards a train in Adelaide, South Australia at 7:53 p.m. on Friday travelling to Perth, Western Australia. She arrives at 10:54 a.m. Determine the duration of her train trip.

9.06 Australian time zones and daylight savings mathspace.co

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9 Chapter review 1

Convert 4.25 hours to minutes. a

255 minutes

265 minutes

425 minutes

A train departs at 09:35 and arrives at 11:15 on the same day. How long was the journey? a

1 hour 30 minutes

1 hour 40 minutes

1 hour 50 minutes

2 hours 20 minutes

If City A is at UTC+3 and City B is at UTC−4, what is the time difference between them? a

240 minutes

1 hour

4 hours

7 hours

−1 hour

Convert: a

Quarter past nine in the morning to b Half past three in the morning to 12-hour 12-hour time time

Ten minutes to six in the morning to d Twenty minutes past midnight to 24-hour 24-hour time time

7:40 a.m. to 24-hour time

4:15 p.m. to 24-hour time

10:00 a.m. to 24-hour time

1:45 p.m. to 24-hour time

18:55 to 12-hour time

00:35 to 12-hour time

11:25 to 12-hour time

14:10 to 12-hour time

A movie starts at 6:50 p.m. and lasts for 2 hours and 35 minutes. What time does it end?

Consider the following train timetable extract for services from Town A to Town E: Town A

Town B

Town C

Town D

Town E

08:10

08:35

08:50

09:10

09:25

08:40

09:05

09:20

09:40

09:55

09:10

09:35

09:50

10:10

10:25

Calculate the total scheduled travel time from Town A to Town E for the train departing Town A at 08:40.

If a passenger arrives at Town B at 09:00, how long must they wait for the next service to Town E shown on this timetable?

Calculate the difference in longitude between City X at (30°N, 15°W) and City Y at (25°S, 65°E). Then, determine the time difference in hours and state which city is ahead.

Sarah, who is in London (UTC+0), needs to join a video call with a colleague in New York (UTC−5) at 10:00 a.m. New York time on a Tuesday: a

Calculate the time difference between London and New York.

What time and day will the call be for Sarah in London?

Chapter 9 review mathspace.co

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Australia has several time zones: • Australian Eastern Standard Time (AEST) is UTC+10. • Australian Central Standard Time (ACST) is UTC+9.5. • Australian Western Standard Time (AWST) is UTC+8.

If it is 10:30 a.m. AEST, what time is it in ACST?

If it is 2:15 p.m. AWST, what time is it in AEST?

Use the provided map of Australian time zones. AWST (UTC+8)

ACST (UTC+9.5)

AEST (UTC+10)

ACDT (UTC+10.5)

AEDT (UTC+11)

2 h behind

0.5 h behind

Compared to AEST

0.5 h ahead

1 h ahead

NT QLD

QLD

WA SA

SA NSW

NSW

VIC

TAS NO DST

TAS DST

When it is 14:30 in Brisbane (QLD) on a day when Daylight Saving Time is active in other states:

676

What time is it in Sydney (NSW)?

What time is it in Adelaide (SA)?

What time is it in Perth (WA)?

The first digital clock shows 11:17 p.m. The second digital clock shows 3:08 a.m.:

A bus departs at the time shown on the first clock on Monday evening. Write this time in 24-hour format.

The bus arrives at the time shown on the second clock the next morning (Tuesday). Write this time in 24-hour format.

Calculate the duration of the bus journey.

Mathspace New South Wales – Year 11 Standard mathspace.co

Using the provided map of Australia, estimate the latitude and longitude of the following cities: 1100 E

1150 E 1200 E

150 S

1250 E

1400 E 1450 E 1500 E 1550 E 100 S

Darwin

150 S

Broome

200 S

Davenport Brisbane

250 S

Menzies 300 S

300 S

Perth

Canberra Sydney Adelaide

350 S

350 S 1250 E 1300 E 1350 E 1400 E a

Perth

Broome

Davenport

Sydney

A documentary starts at 19:30 and is scheduled to end at 21:45. There is a single 10-minute advertising break during the documentary. Calculate the actual running time of the documentary itself, excluding the break.

Michael needs to travel from Newtown to Bondi Beach. He will take a train from Newtown to Bondi Junction and then a bus from Bondi Junction to Bondi Beach. The walk and wait time between the train arrival at Bondi Junction and the bus departure is 8 minutes. Train Timetable: Newtown to Bondi Junction: Newtown (departure)

Central (arrival)

Bondi Junction (arrival)

1:05 p.m.

1:15 p.m.

1:25 p.m.

1:20 p.m.

1:30 p.m.

1:40 p.m.

1:35 p.m.

1:45 p.m.

1:55 p.m.

Bus Timetable: Bondi Junction to Bondi Beach: Bondi Junction (departure)

Bondi Beach (arrival)

1:30 p.m.

1:45 p.m.

2:00 p.m.

2:15 p.m.

If Michael wants to arrive at Bondi Beach by 2:05 p.m., what is the latest train he can catch from Newtown? Show your calculations.

Calculate Michael’s total travel time from departing Newtown to arriving at Bondi Beach if he takes the train identified in part (a).

Chapter 9 review mathspace.co

677

A flight departs from City P (Longitude 30°E) at 10:00 local time on Tuesday. It flies to City Q (Longitude 75°W). The flight duration is 14 hours. a

Calculate the longitude difference between City P and City Q.

Calculate the time difference between City P and City Q, and state which city is ahead.

Determine the local time and day of arrival in City Q.

Two islands, East Island (UTC+13) and West Island (UTC−11), are very close geographically but are on opposite sides of the International Date Line. A boat trip from East Island to West Island takes 30 minutes. If a boat departs East Island at 2:00 p.m. on a Wednesday:

What is the time difference between East Island and West Island?

What will be the local time and day on West Island when the boat arrives? Explain your reasoning, considering the International Date Line.

During a period when Daylight Saving Time (DST) is active, a flight departs from Adelaide (SA, normally UTC+9.5, with DST UTC+10.5) at 8:00 p.m. Adelaide time on a Saturday. The flight to Darwin (NT, UTC+9.5, no DST) takes 3 hours and 30 minutes. Explain whether the arrival in Darwin will be on Saturday or Sunday, and at what local time. Show your calculations.

The world map shows lines of longitude and latitude. Several locations are marked. 900 75

900

750 I

600 450 300 150 0

450 300

B 150

600

150

120

300

600

900

00 150

300

450

450 600

600

J 750

678

150

1500 1800

1200

750 90

900

What are the approximate coordinates of Point E?

How many degrees of longitude separate Point A and Point D?

If it is 10:00 a.m. at Point F, what would be the approximate time at Point I, assuming 15° longitude equals 1 hour time difference? State if Point I is ahead or behind Point F.

Mathspace New South Wales – Year 11 Standard mathspace.co

Big ideas Measures of centre and spread enable effective data summarisation and analysis by quantifying typical values and variability, facilitating informed interpretations in real-world contexts.

10 Measures of centre and spread Chapter outline 10.01 Measures of centre 10.02 Measures of spread Investigation: Statistical reports in the media 10.03 Compare datasets Investigation: Spreadsheets and centre and spread 10.04 Quartiles and interquartile range Chapter 10 review

682 694 706 722 739

Median Median The value in a set of ordered data that divides the data into 2 parts. It is frequently called the ‘middle value’.

is the number of scores

For an odd number of scores, the median is the middle score. For an even number, it is the average of the two middle scores.

Exploration Consider a dataset: 9, 3, 11, 5, 7. Order the scores and determine the median. Then, replace 11 with 100 and recalculate the median. Compare the change in the median to the change in the mean (from the previous exploration). Discuss why the median is less affected by extreme values. 1. What is the median of the original dataset? 2. How does the median change after replacing 11 with 100? 3. Why might the median be preferred over the mean for data with extreme values?

Example 2 Determine the median using the histogram:

Frequency

20 15 10 5

Score

Create a strategy Sum the frequencies to determine n, calculate the median position, and identify the score range containing the median.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 3

A basketball player scored 22, 28, 25 points in three games. To achieve a mean of 27 points over four games, calculate the points needed in the fourth game.

Six fruit stalls at a weekend market recorded their total kilograms of fruit sold on a particular day: • Stall A: 8 kg • Stall B: 10 kg

Which sales ﬁgure (in kilograms) occurs most often?

What is the mode of this dataset?

Identify the mode(s).

Classify the data as uniform, unimodal, bimodal, or multimodal.

8, 18, 5, 2, 2, 10, 8, 5, 14, 14, 8, 8, 10, 18, 14, 5

5, 9, 2, 5, 7, 9, 3, 2, 5, 9, 2

2, 2, 6, 8, 8, 8, 8, 12, 14, 14, 14, 14, 18, 18

31, 45, 22, 19, 50

7, 11, 7, 15, 11, 15, 7, 11, 15

• Stall E: 8 kg • Stall F : 15 kg

Determine the following for each of the given datasets:

Score

Frequency

A class of 20 students took a short test with a total of 24 marks. Scores are grouped and displayed in the histogram: Histogram of test scores 7 6

Frequency

Ex 4

• Stall C: 8 kg • Stall D: 12 kg

5 4 3 2 1 0

0-4

5-9 10-14 15-19 20-24

Score interval

690

Identify the modal class from the histogram.

Is this grouped distribution unimodal or bimodal?

How might knowing the modal class help the teacher?

Mathspace New South Wales – Year 11 Standard mathspace.co

One hundred students were asked to note the number of hours they study in the week prior to an examination. The frequency table shows the results:

0 ≤ h < 10

10 ≤ h < 20

Identify the modal class for the number of hours spent studying.

20 ≤ h < 30

Classify the distribution based on the modal class(es) as uniform, unimodal, bimodal, or multimodal.

30 ≤ h < 40

40 ≤ h < 50

50 ≤ h < 60

60 ≤ h < 70

70 ≤ h < 80

80 ≤ h < 90

90 ≤ h ≤ 100

Hours h

Explain brieﬂy how knowing the modal class might be useful for student support services.

The bar graphs show the number of books borrowed per day from a school library over two different weeks: Bar graph B - books borrowed

Frequency

Bar graph A - books borrowed

5 4 3

Number of Students

Number of books

Identify the mode for each graph.

Which graph shows a higher frequency for its mode?

Classify the distribution as uniform, unimodal, bimodal, or multimodal for both graphs.

For the number of homework tasks completed by 24 students in a week, as shown in the dataset: 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7 a

Express the dataset above as a frequency distribution table.

Which number(s) appear most often? What are the mode(s)?

Is the data distribution uniform, unimodal, bimodal, or multimodal?

10.01 Measures of centre mathspace.co

691

For each of the given real-life scenarios, determine whether the mode would be less useful, useful, or most useful as a measure of centre to report. Brieﬂy justify your choice for each scenario: a

The most common shoe size requested by customers at a footwear shop.

The typical time it takes for students to travel from home to school for planning bus routes.

The representative house price in a large suburb reported in the news.

Determining the most popular subject choice among Year 11 students based on enrolment numbers.

Summarising the ratings ( for example, 1 to 5 stars) given to a new movie by 100 reviewers.

Extend your thinking 25

The dot plot shows the number of hours spent studying by 10 students: a

Calculate the median.

Calculate the mean.

Explain why there is a signiﬁcant difference between your answers to parts (a) and (b).

Determine an extra value that, when added to the dataset, makes the median and mean equal. Justify your answer.

Hours spent studying by students

The frequency table shows pulse rates from a ﬁtness study: Pulse Rate

Class Centre (x)

Frequency ( f )

f×x

50–59

⬚

60–69 70–79 80–89

0 1 2 3 4 5 6 7 8 9 10 11 12

⬚ ⬚

⬚

Complete the table.

Calculate the median pulse rate.

Calculate the mean pulse rate, rounded to two decimal places.

On a particular day, 20 ﬂights were delayed at an airport. The dot plot shows delay times in hours: a

Calculate the median delay time in minutes.

Calculate the percentage of ﬂights delayed longer than the median time.

If a flight delayed over 60 minutes incurs a $4000 fine, calculate the total fines.

0.5

1.0

1.5

2.0

Hours late

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Mathspace New South Wales – Year 11 Standard mathspace.co

2.5

The frequency table shows luggage weights (kg) at an airport check-in: a

Complete the table.

Calculate the median weight.

Calculate the mean weight, rounded to two decimal places.

Adding items to a 20 kg bag affects which measure more signiﬁcantly, mean or median? Explain.

Weight (x)

Frequency ( f )

f (x)

⬚

Total

⬚

⬚ ⬚

⬚

A dataset of 10 values has a sum of 100. The initial dataset is: 10, 10, 10, 10, 10, 10, 10, 10, 10, 10 The mode is currently 10. Propose one way to adjust some values, without changing the total sum of 100, so that the value 12 becomes the new unique mode. Show the new dataset.

Create a dataset of exactly 8 values that is clearly bimodal, with the modes being 5 and 10, each appearing exactly 3 times, and the dataset having a mean of 6.5.

Consider a grouped frequency distribution representing the heights (in cm) of students, with 5 class intervals of equal width. The class intervals 160 cm ≤ h < 165 cm and 170 cm ≤ h < 175 cm each have the highest frequency, 15 students. No other class interval has a frequency greater than 10:

Based only on the grouped data, how would you classify this distribution (unimodal, bimodal, etc.)?

Does this guarantee that if we looked at the original ungrouped height data, the distribution would still be bimodal? Explain why or why not.

A researcher analyses a large dataset and obtains these results: • Exactly one score, A, appears 20 times. • Exactly one other score, B, appears 18 times. • All other distinct scores appear 10 or fewer times each. Can this dataset be classiﬁed as unimodal, bimodal, or multimodal? Justify your answer.

A student analysed the dataset shown, representing the number of siblings for 12 students: 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5 The student stated: “The scores 2 and 3 appear most often. Since there is more than one mode, the data is multimodal.” Identify any error(s) in the student’s statement or reasoning, and provide the correct analysis including classiﬁcation.

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10.02 Measures of spread After this lesson, you will be able to… • identify and describe the range and standard deviation as measures of spread. • calculate the range of a dataset presented in a list, table, or graph. • calculate the population standard deviation for datasets using a scientific calculator. • estimate the population standard deviation for grouped data by using class centres. • interpret the standard deviation to describe the consistency or variability of data.

Range Range (of a dataset) The difference between the highest and lowest values in a dataset. Measure of spread In statistics, different methods of calculating the variability of a set. The most commonly used measures of spread are the range, interquartile range, and standard deviation.

The range is the simplest measure of spread in a quantitative dataset, calculated as the difference between the maximum and minimum values in a dataset. Range = Maximum – Minimum Maximum : is the highest value in the dataset Minimum : is the lowest value in the dataset

Example 1 Determine the range of the scores: 10, 19, 19, 7, 20, 14, 2, 11

Create a strategy Subtract the lowest score (2) from the highest score (20).

Apply the idea Range = Maximum – Minimum

694

Write the formula

= 20 – 2

Substitute the values

= 18

Evaluate

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Example 4 The table shows the number of goals scored by a football team per game. Score (x)

Frequency ( f )

a In how many games were 0 goals scored?

Create a strategy

Apply the idea

Check the frequency for a score of 0.

The frequency is 3 games.

b Determine the median number of goals scored, rounded to one decimal place if necessary.

Create a strategy

Apply the idea

For an even number of observations, average the two middle scores.

The median is 3.5.

c Calculate the mean number of goals scored per game, rounded to two decimal places.

Create a strategy Sum the product of each score and its frequency, then divide by the total frequency:

Apply the idea Write the formula

Substitute the values

Simplify

Evaluate

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d Calculate the population standard deviation, rounded to two decimal places.

Create a strategy

Apply the idea

Enter the scores and frequencies into the calculator’s statistics mode and select σn.

1. Enter each score and its frequency in statistics mode. 2. Press the σn button. The population standard deviation is σn = 1.75.

Example 5 Fill in the table and answer the questions: a Complete the table. Class

Class Centre

Frequency ( f )

f×x

1−9

⬚

10 − 18 19 − 27 28 − 36 37 − 45 Totals

⬚

8 ⬚

Create a strategy

⬚ ⬚ ⬚

⬚

Calculate the class centre as the average of the class bounds. Compute f × x by multiplying the class centre by the frequency.

Apply the idea For class 1 − 9:

Continue for the following classes to complete the table: Class

Class Centre

Frequency ( f )

f×x

1−9

10 − 18

19 − 27

28 − 36

192

37 − 45

328

736

Total

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10.02 Practice questions What do you remember? 1

Are the following statements always, sometimes or never true? a

If all values in a dataset were increased by the same amount, the standard deviation would also increase by that amount.

If the range of a dataset is large, the standard deviation will be large.

If a single data value that is signiﬁcantly smaller than the other data values is added to a dataset, it will not affect the standard deviation.

The standard deviation of a dataset will always be positive.

Which of the following describes a dataset with a small standard deviation? A

The data points are spread far apart from the mean.

The data points are clustered closely around the mean.

The dataset has a large range.

The data points have a high maximum value.

Practice Ex 1

Calculate the range for each set of scores: a

10, 7, 2, 14, 13, 15, 11, 4

15, −2, −8, 8, 15, 6, −16, 15

−0.5537, 1.7444, −0.3381, 0.7200, −0.3381, 1.0435

Consider the data provided in the table: a

Determine the range of the scores.

Determine the mode.

Score

Frequency

A group of students had a range in marks of 14 and the lowest score was 9. What was the highest score in the group?

The range of a set of scores is 8 and the highest score is 19. What is the lowest score in the set?

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Determine the range for the data represented by these dot plots, rounded to two decimal places: a

A 9

Company X

Company Y

Order the following graphs, from the lowest standard deviation, to the highest standard deviation: A

100

Frequency

80 60 40 20 0

20 30 40 50 60 70 80 90 100 110 120

Length (cm)

100 80 60 40 20 0

20 30 40 50 60 70 80 90 100 110 120

Length (cm)

700

Company X and Company Y both have the same mean salary of $72 000. Company Y’s standard deviation is greater than Company X’s. Which company likely has a larger income gap between the highest and lowest-paid employees?

Frequency

Ex 2

Mathspace New South Wales – Year 11 Standard mathspace.co

100

Frequency

80 60 40 20 0

20 30 40 50 60 70 80 90 100 110 120

Length (cm)

100

Frequency

80 60 40 20 0

10 20 30 40 50 60 70 80 90 100 110 120

Length (cm) Ex 3

Use technology to calculate the population standard deviation σn of each set of scores, rounded to two decimal places. a

8, 20, 16, 9, 9, 15, 5, 17, 19, 6

−17, 2, −6, 9, −17, −9, 3, 8, 5

81, 90, 90, 88, 73, 80, 86, 87, 75, 82, 70, 81, 71, 81, 79, 81, 80, 86, 88, 79

20, 44, 27, 25, 21, 28, 41, 24, 27, 39, 35, 43, 30, 17, 40

3, 14, 11, 17, 3, 18, 15, 6, 17, 15

For the data: 2, 4, 6, 8, 12 a

Calculate the population standard deviation σn, rounded to two decimal places.

Calculate the population standard deviation if 8 is added to each data value. How is the standard deviation affected?

Calculate the population standard deviation if all data values are multiplied by 2. How is the standard deviation affected?

The number of lollies in each of 10 packets is: 10, 11, 12, 13, 14, 16, 17, 18, 19, 22 a

Use your calculator to calculate the population standard deviation σn, rounded to two decimal places.

If an 11th packet is found to have only 5 lollies, how might this affect the standard deviation? Explain.

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The table shows the results of a survey measuring how much time students spend on social media in a week, in minutes. Class

Class Centre

Frequency ( f )

f×x

40 ≤ x < 45

42.5

45 ≤ x < 50

47.5

⬚

50 ≤ x < 55

52.5

55 ≤ x < 60

57.5

60 ≤ x < 65

62.5

65 ≤ x < 70

67.5

70 ≤ x < 75

72.5

75 ≤ x < 80

77.5

⬚

Total

⬚ ⬚ ⬚ ⬚ ⬚

Complete the table by calculating f × x for each row.

Estimate the mean time spent using the class centres, rounded to two decimal places.

Estimate the population standard deviation using class centres, rounded to two decimal places.

Would the standard deviation of the ungrouped data be higher or lower than this grouped estimate?

For each histogram: i

Determine the range of the dataset.

Calculate the mean of the dataset using the centres, rounded to two decimal places.

iii

Use technology to calculate the population standard deviation σn of the dataset, rounded to two decimal places.

Frequency

⬚

Frequency

Ex 5

62 63 64 65

Scores

42 43 44 45 46

Scores

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Extend your thinking 18

The scores obtained by two classes are given: • Red Class: 55, 57, 49, 58, 68, 57, 60, 53, 56, 51 • Blue Class: 53, 57, 62, 51, 56, 62, 58, 55, 58, 51

Which class performed better on average? Use statistical calculations to justify your answer.

Which class produced more consistent results? Use statistical calculations to justify your answer.

Han, a cricketer, has achieved scores of 52, 20, 68, 70 and 150 in his ﬁrst ﬁve innings this season. In his sixth innings, he scores 0. a

Describe how his season batting average changed from before to after the sixth inning.

Describe how his population standard deviation σn changed from before to after the sixth inning.

Describe how his median score changed from before to after the sixth inning.

Describe how his range changed from before to after the sixth inning.

The table shows the heart rate data of a group of people after exercise:

Height of step

Stepping rate

Heart rate

Short step

Slow

Short step

Slow

Short step

Medium

106

Short step

Medium

105

Short step

Fast

124

Short step

Fast

128

Tall step

Slow

100

Tall step

Slow

Tall step

Medium

125

Tall step

Medium

129

Tall step

Fast

132

Tall step

Fast

127

Complete the table. Round all values to one decimal place. Height of step Short step Tall step

Data

Slow

Medium

Fast

Avg. heart rate

90.0

Standard deviation of heart rate

1.0

⬚

Avg. heart rate

⬚

Standard deviation of heart rate

704

⬚

⬚ ⬚

⬚

Which of the combinations of step height and stepping rate generated the higher heart rate?

Which combination of step height and stepping rate showed the least variability?

Mathspace New South Wales – Year 11 Standard mathspace.co

The trial times (in seconds) of a female athlete for the 100 m sprint are recorded: 11.0, 10.5, 12.2, 11.2, 11.6, 11.7, 10.8, 12.1, 11.0, 10.9 a

Use technology to calculate the population standard deviation σn of her times, rounded to two decimal places.

On her next attempt, she manages to run 100 m in 10.1 seconds. Would this time increase or decrease these statistics? i

Mean

Population standard deviation

Investigation: Statistical reports in the media Investigate online

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Did you know?

Ice cream shops use measures of centre and spread to understand customer preferences! For example, the mean and median show the typical number of scoops sold for each flavour, while the range and standard deviation reveal how much sales vary between popular and less popular flavours. These measures help shop owners keep the right flavours in stock, reduce waste, and satisfy every sweet tooth! They also make it easier to plan for busy seasons, like summer, when ice cream sales skyrocket.

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10.03 Compare datasets After this lesson, you will be able to… • calculate measures of centre and spread for two or more datasets. • compare datasets using measures of centre (mean, median) and spread (range, standard deviation). • draw conclusions and make judgements based on the comparison of summary statistics. • examine the merits of different measures of centre and spread for describing a dataset.

Compare datasets Dataset A collection of numbers or values relating to a particular subject. Datasets are normally presented in tables or represented by graphs. For example, the test scores of each student in a particular class.

Comparing datasets involves identifying similarities and/or differences in key characteristics such as measures of centre and measures of spread: • Spread: Is one dataset more dispersed than the other? • Skew: Is one dataset more symmetrical? • Centre: Are there signiﬁcant differences in mean, median, or mode?

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Exploration Consider the histograms showing the height of students in two basketball teams, one representing Year 12 students and the other Year 8 students. Team A

Frequency

10 8 6 4 2 0

155

160

165

170

175

180

185

190

195

200

Height (cm)

Team B

Frequency

10 8 6 4 2 0

135 140 145 150 155 160 165

Height (cm)

1. Are there the same number of students in each team? Does it matter? 2. What are the similarities and differences in terms of measures of spread, centre, and shape of data? 3. Which team likely corresponds to the Year 12 team, and which to the Year 8 team?

Comparing datasets enables conclusions about the data. For example, a score of 50% in a geography test and 60% in a history test suggests better performance in history. However, if the geography class averaged 40% and the history class 80%, the geography score is relatively higher, indicating better performance in geography.

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b Calculate the mean of the scores of Student B.

Create a strategy Use the formula: Mean =

Apply the idea Write the formula

Substitute the values

Evaluate the addition

Evaluate the division

c Which student performed better overall on their tests?

Create a strategy Compare the students by using the mean values calculated in parts (a) and (b).

Apply the idea Student A has a higher mean, 85.2, and performed slightly better than Student B.

d What is the highest score overall? Which student (A or B) obtained that score?

Create a strategy Refer to the score columns and determine the highest score. Then, identify the student that obtained that score by referring to the column heading.

Apply the idea Based on the table, the highest score is 98, which was obtained by Student A.

e What is the lowest score overall? Which student (A or B) obtained that score?

Create a strategy Refer to the score columns and determine the lowest score. Then, identify the student that obtained that score by referring to the column heading.

Apply the idea Based on the table, the lowest score is 72, which was obtained by Student B.

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Example 3 The column graphs show the season results of two soccer groups, Group A and Group B, and the number of games (frequency) in which they scored a certain number of goals (scores). Group A 5

Frequency

4 3 2 1 0

Score - Group A Group B 5

Frequency

4 3 2 1 0

Score - Group B a What is the mode for Group A?

Create a strategy Examine the column graph for Group A and assess which score occurred the most.

Apply the idea Based on the column graph for Group A, the score that occurred the most is 3.

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Comparisons of statistical measures Measures of centre and spread provide distinct insights into datasets. Measures of centre (mean, median, mode) summarise where most data lies. Beneﬁts

Drawbacks

Mean

Includes all data in calculation, widely used

Heavily impacted by outliers

Median

Indicates the middle, not impacted by outliers

Does not include all data values

Mode

Quick to identify, shows most frequent value(s)

Not necessarily central, excludes some data

Measures of spread (range, standard deviation, interquartile range) describe variability and consistency. • Range • Interquartile range (IQR)

• Standard deviation

Outliers can skew data, affecting certain measures of centre and spread.

Interactive exploration Discover this concept in action online

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Range and standard deviation are affected by outliers, as the mean, used in standard deviation calculations, is outlier-sensitive. Beneﬁts

Drawbacks

Range

Easy to calculate, shows data extremes

Heavily impacted by outliers

Standard deviation

Measures distance from mean, widely used in statistics

Outlier-sensitive, best calculated with technology

Example 4 Two machines, A and B, produce chocolate bars with the mean and standard deviation of the weight of the bars shown: Machine

Mean (g)

Standard deviation (g)

1.5

0.65

a What does a comparison of the mean of the two machines indicate?

Create a strategy Compare the means from the table.

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Apply the idea Machine B has a greater mean weight than Machine A, indicating that Machine B generally produces heavier chocolate bars.

b What does a comparison of the standard deviation of the two machines indicate?

Create a strategy Compare the standard deviations from the table.

Apply the idea Machine B has a smaller standard deviation than Machine A, indicating that Machine B produces chocolate bars with more consistent weight.

Example 5 Han, a cricketer, has made scores of 52, 20, 68, 70, 150 in all his innings this season. In his next innings, he scores no runs. a What is the change in his season batting average before and after the sixth inning?

Create a strategy Subtract the old batting mean from the new batting mean. To calculate the mean, use the formula: Mean =

Apply the idea The old list is: 52, 20, 68, 70, 150. The new list is: 52, 20, 68, 70, 150, 0. Write the formula

Substitute the values

Evaluate Write the formula

Substitute the values

Evaluate

Change in mean = New Mean − Old Mean

Write the formula

= 60 − 72

Substitute the values

= −12

Evaluate

The batting average dropped by 12.

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10.03 Practice questions What do you remember? 1

What are three statistical measures that can be used to compare datasets?

Can two datasets have the same mean but different medians?

Which dataset has the lowest score? Graph 1

Graph 2 5

9 8

Frequency

7 5 4 3 2 1 0

3 2 1

Score

Score Is each statement true or false? a

The standard deviation is a measure of centre.

The dataset with a smaller spread is more consistent.

The dataset with a smaller spread is more variable.

Practice 5

The mean income of people in Canada is $43 000. This is the same as the mean income of people in Germany. The standard deviation of Canada is greater than the standard deviation of Germany. In which country is there likely to be the greatest difference between the incomes of the rich and poor?

Two cricketers compare the mean and standard deviation of their runs made per match. They conclude that Ivan is a more consistent batter but Bianca generally scores more runs per match: a

Who has a higher mean?

Who has a lower standard deviation?

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Ex 3

The graphs show the performance of two hockey teams, Team X and Team Y, in various games, along with the number of goals they scored in each game: Team Y 6

Frequency

Team X 5

3 2 1 0

4 3 2 1

Score

Determine the mode for Team X.

Determine the mode for Team Y.

Determine the range for Team X.

Determine the range for Team Y.

Which team scored the lowest total number of goals during the season?

Which team has the most varied results?

Use the summary statistics provided to determine the dataset with the higher: i

Measure of deviation

Mean Median Range

Dataset A

Dataset B

Measure of spread Mean

Mode

Standard deviation

Dataset A

7.2

6.5

3.46

Dataset B

7.2

3.23

In a Mathematics test out of 100, the students in two classes scored the results displayed: • Class A: 72, 66, 92, 76, 77, 63, 67, 91, 66, 77, 66, 73, 80, 68, 68 • Class B: 84, 70, 53, 82, 65, 85, 81, 87, 64, 81, 86, 79, 69, 82, 77 a

Determine the median mark for: i

b Ex 4

Class A

Class B

Using the medians, which class did better on the test?

Two machines, P and Q, produce cookies with the mean and standard deviation of the weight of the cookies shown: Machine

Mean (g)

Standard deviation (g)

2.1

0.8

What does a comparison of the mean of the two machines indicate?

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Ex 5

Emma, a hockey player, has scored 4, 2, 6, 3, 5 goals in her ﬁrst ﬁve matches this season. In her sixth match, she scores no goals: a

What is the change in her season goal average before and after the sixth match? Round your answer to two decimal places.

What is the change in her standard deviation before and after the sixth match? Round your answer to two decimal places.

The monthly rainfall for two cities across a year is displayed: • City A: 122, 120, 121, 67, 79, 24, 8, 46, 27, 67, 80, 128 • City B: 51, 66, 33, 78, 79, 92, 78, 99, 41, 60, 26, 56 a

Determine the range in monthly rainfall for: i

b 16

City A

City B

Which city had more consistent rainfall throughout the year?

Harry is tracking the growth of two different types of sunﬂower plants. He records the number of sunﬂowers picked daily from each plant over 10 days: • Plant A: 15, 7, 8, 10, 9, 15, 6, 6, 12, 15 • Plant B: 11, 10, 9, 9, 12, 10, 11, 10, 9, 10 Compare the daily growth of Plant A and Plant B, referencing their centre, spread, and skew.

The pulse rates of two groups are given: • Group 1: 82, 85, 88, 65, 73, 89, 79, 90, 76, 68, 88, 65, 63, 62, 88, 82 • Group 2: 75, 88, 74, 73, 80, 76, 67, 81, 71, 83, 89, 62, 63, 80, 71, 78 Compare the pulse rates of Group 1 and Group 2, referencing their centre, spread, and skew.

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Luke, a cricketer, has made scores of 51, 25, 99, 35 and 90 in his ﬁrst ﬁve innings this season. In his sixth innings, he scores a duck (0). Describe how this new score affected: a

His season batting average

The standard deviation of his scores

His median score

The range of his scores

Seven millionaires with an average net wealth of $41 million with a standard deviation of $8 million are having a party. Suddenly Jaden, who has a net wealth estimated to be $34 billion, walks into the room: a

Calculate the new average net wealth in the room. Round your answer to the nearest million.

Will the new standard deviation be higher, lower or unchanged from before?

Will the mode be higher, lower or unchanged from before if at least two of the millionaires have the same net wealth?

Will the range be higher, lower or unchanged from before?

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Extend your thinking 20

The weights of two groups of turtles are measured, in kilograms, to determine whether they might belong to the same species. The results are presented: • Group 1: 56, 60, 58, 59, 62, 55, 57, 61, 56, 59 • Group 2: 62, 64, 60, 65, 66, 62, 61, 64, 67, 63 Do you think the two groups of turtles are from the same species? Explain your answer using relevant statistical measures.

Two Biology classes, each with 15 students, sit a 10-question multiple choice test, each with four possible answers (only one of which is correct). Their class results, out of 10, are: • Class 1: 3, 2, 1, 1, 2, 5, 2, 2, 2, 3, 2, 1, 5, 5, 4 • Class 2: 10, 6, 9, 8, 10, 9, 7, 7, 6, 8, 8, 10, 8, 9, 8 a

Complete the table, rounding your answers to one decimal place when necessary:

Class 1 Class 2 b 22

Mean

Median

Mode

Range

⬚

Which class was more likely to have studied for their test? Explain your answer.

The scores obtained by two classes are given: • Red class: 55, 57, 49, 58, 68, 57, 60, 53, 56, 51 • Blue class: 53, 59, 52, 49, 59, 49, 58, 57, 48, 54

Which class performed better? Explain your reasoning.

Which class produced more consistent results? Explain your reasoning.

A pharmaceutical company is interested in comparing the effectiveness of two different painkillers. After a 2-week trial, the pain levels (on a scale of 1 to 10) of the participants using the two different medications were recorded and found to be as follows: Reformulated painkiller

Original formulation

Calculate the mean and standard deviation of the reformulated painkiller.

Calculate the mean and standard deviation of the original formulation painkiller.

Using these results, compare the effectiveness of the different painkillers on the pain of the participants.

Investigation: Spreadsheets and centre and spread Investigate online

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10.04 Quartiles and interquartile range After this lesson, you will be able to… • determine the five-number summary (minimum, Q1, median, Q3, maximum) for a dataset. • calculate the interquartile range (IQR) of a dataset. • compare and contrast the use of range and IQR as measures of spread. • interpret the meaning of quartiles and the IQR in the context of a dataset. • estimate quartiles and the IQR from a cumulative frequency graph.

Quartiles The range, a measure of spread, is calculated as the difference between the maximum and minimum values in a dataset but does not indicate the spread within these values. The median, a measure of centre, identiﬁes the middle value of a dataset. For a dataset with n items, the median is the

value.

Quartiles The values that divide an ordered dataset into 4 (approximately) equal parts. There are 3 quartiles. The ﬁrst, the lower quartile (Q1), divides off (approximately) the lower 25% of data values. The second quartile (Q2) is the median. The third quartile, the upper quartile (Q3), divides off (approximately) the upper 25% of data values.

First quarter

Second quarter

Third quarter Q2

Q1 2,

Minimum

722

Fourth quarter

10,

Median

Mathspace New South Wales – Year 11 Standard mathspace.co

Q3 11,

13,

16,

18,

22,

Maximum

Minimum

Lowest value

Lower quartile, Q1

At most 25% of the data is below this value

Median, Q2

50% of the data lies on either side of this value

Upper quartile, Q3

At most 25% of the data is above this value

Maximum

Highest value

Consider the given data: 1

Ensure the dataset is ordered before ﬁnding quartiles or the median.

Median 1

Locate the median between the 4th and 5th scores.

With four scores in each half, split each half to ﬁnd the quartiles. Q1 1

The ﬁrst quartile, Q1, is between the 2nd and 3rd scores. The third quartile, Q3, is between the 6th and 7th scores.

Median 12

For 9 scores, the 5th term is the median, with four terms on either side. Thus, Q1 is between the 2nd and 3rd scores, and Q3 is between the 6th and 7th scores. Q1 8

Median 10

13 14 18

22 25

For 10 scores, the median is between the 5th and 6th scores, with 5 scores on either side. Thus, Q1 is the 3rd term, and Q3 is the 8th term. Q1 12

Median 19

Q3 22 22 28 30

Each quartile represents 25% of the dataset. For a dataset with n items: Q1 is the

value

Median is the

value

Q3 is the

value

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Example 1 Here are Ray’s scores from his last 13 rounds of golf played: 66, 66, 68, 68, 70, 78, 80, 84, 106, 116, 126, 130, 132 a What is his median?

Create a strategy Use the formula:

where n is the total number of scores, to ﬁnd the position of the median.

Apply the idea There are n = 13 scores in the list. Write the formula

Substitute n = 13

Evaluate

Choose the 7th score

b What is the lower quartile?

Create a strategy Find the median of the lower half of the scores excluding the median.

Apply the idea The lower half of the scores are: 66, 66, 68, 68, 70, 78. Average the middle scores

Evaluate

c What is the upper quartile?

Create a strategy Find the median of the upper half of the scores excluding the median.

Apply the idea The upper half of the scores are: 84, 106, 116, 126, 130, 132. Average the middle scores

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Evaluate

b Find the median.

Create a strategy Use the formula:

where n is the total number of scores, to ﬁnd the position of the median.

Apply the idea There are 18 scores in the list. Write the formula

Substitute n = 18

Evaluate

Therefore the median is between the 9th and 10th score. Using the frequency table we can see that the 9th score is 38 and the 10th score is 38. Average the middle scores Evaluate

c Find the lower quartile of the set of scores.

Create a strategy Find the median of the lower half of the scores.

Apply the idea There are 9 scores in the lower half excluding the median. The lower quartile will be the 5th score from the frequency table. Lower quartile = 18

d Find the upper quartile of the set of scores.

Create a strategy Find the median of the upper half of the scores.

Apply the idea There are 9 scores in the upper half excluding the median. The upper quartile will be the 5th in the upper half, which is the 9 + 5 = 14th score overall. Upper quartile = 50

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e Find the interquartile range.

Create a strategy Use the interquartile range formula: IQR = Q3 − Q1

Apply the idea IQR = Q3 − Q1

Write the formula

= 50 – 18

Substitute the quartiles

= 32

Evaluate

Example 3 Answer the questions using the dot plot shown:

a Find the total number of scores.

Create a strategy Count the number of dots in the dot plot.

Apply the idea Number of scores = 15 Count the dots

b Find the median.

Create a strategy Use the formula:

where n is the total number of scores, to ﬁnd the position of the median.

Apply the idea Write the formula

Substitute n = 15

Evaluate

The 8th score in the dot plot is on 15. Median = 15

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Quartiles and cumulative frequency graphs Quartiles can be determined using a cumulative frequency histogram, which plots the cumulative frequency—the running total of frequencies—against scores. To ﬁnd the ﬁrst quartile (Q1), median, or third quartile (Q3), identify the 25%, 50%, or 75% points of the total cumulative frequency and locate the corresponding scores on the histogram.

Example 4 Consider this frequency table: Score

Frequency

Using the cumulative frequency histogram to ﬁnd the IQR and median. a Find the cumulative frequencies.

Create a strategy Add the frequencies together and create a new column for cumulative frequency.

Apply the idea Score

Frequency

Cumulative Frequency

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b Draw the cumulative frequency histogram.

Create a strategy Mark out the axes with score on x-axis and cumulative frequency on y-axis, then draw the bar and add in the polygon starting from bottom left.

Apply the idea Cumulative histogram and polygon 35

Cumulative frequency

30 25 20 15 10 5 0

Scores

c Find the 25%, 50% and 75% points on the cumulative frequency graph.

Create a strategy Take total frequency, the last entry for cumulative frequency and multiply by 0.25, 0.50 and 0.75 to ﬁnd the Q1, median and Q3 points. Then draw horizontal dotted lines to see where these points hit the bars.

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10.04 Practice questions What do you remember? 1

Find the median for each dataset of scores: a

2, 10, 28, 35, 50

8, 9, 11, 12, 13, 15

41, 42, 43, 44, 49, 50, 51, 53, 58

3, 4, 6, 7, 7, 8, 12, 14

What is the formula for calculating the interquartile range (IQR) of a dataset?

Are these statements true or false?

The ﬁrst quartile Q1 is the median of the entire dataset.

The median is always the

The interquartile range (IQR) is affected by outliers in the dataset.

The third quartile Q3 divides off approximately the upper 25% of data values.

value in an ordered dataset with n items.

Match the term with the deﬁnition. i

Highest value

50% of the data lies on either side of this value

iii

At most 25% of the data is above this value

The lowest value

At most 25% of the data is below this value

Minimum

Lower quartile, Q1

Median

Upper quartile, Q3

Maximum

Practice Ex 1

The dataset shows Rachelle’s scores from her last 13 rounds of golf played: 56, 58, 60, 62, 64, 72, 74, 78, 98, 108, 120, 124, 128

Find her median score.

Find the lower quartile score.

Find the upper quartile score.

Find the interquartile range.

The dataset shows Luke’s scores from his last 17 exams: 42, 46, 48, 51, 52, 54, 56, 68, 72, 76, 78, 82, 85, 86, 88, 92, 96

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Find his median score.

Find the lower quartile score.

Find the upper quartile score.

Find the interquartile range.

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For each dataset, the median is shown. Find the upper and lower quartiles. a

0, 1, 4, 6, 8, 9, 11 Median

22, 24, 25, 29, 32, 34 Median: 27

5, 11, 19, 20, 22, 24, 26, 27, 27

14, 15, 16, 16, 17, 18, 19, 20, 24, 25, 30, 36

Median Median: 18.5 8

What is the interquartile range for this dataset? Q1 = 3

Q3 = 14

1, 2, 3, 5, 5, 7, 11, 14, 15, 16 Minimum

Ex 2

Ex 3

Median = 6 Maximum

A sample of matchboxes was selected for quality control and the number of matches in each box recorded in a given frequency table.

Matches

Frequency

How many matchboxes were sampled?

Find the range of the data.

Find the interquartile range.

For this dot plot:

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

Find the range.

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Ex 4

For each set of scores: i

Sort the scores in ascending order.

Find the number of scores.

iii

Find the median.

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

40, 39, 15, 17, 10, 6, 24

−4, −6, −1, 7, 9, 7, 9

8, 20, 19, 4, 15, 14, 10

42, 28, 22, 40, 20, 54, 32, 43

84, 85, 79, 71, 69, 88, 82, 78

102, 115, 110, 113, 100

228, 205, 198, 202, 207, 197

19.5, 29.6, 39.1, 22.4, 15.8, 46.2, 35.4

Consider this frequency table showing the luggage weights, in kilograms, of 30 passengers:

Weight

Frequency

Find the cumulative frequencies.

Draw the cumulative frequency histogram.

Find the 25%, 50%, and 75% points on the cumulative frequency graph.

Find the median, Q1, Q3, and the IQR using the cumulative frequency histogram.

A group of students were asked how many phone calls they had made the previous day. The information was collected in a given frequency table.

Phone calls

Frequency

How many students were surveyed?

Find the range of the data.

Find the interquartile range by drawing the cumulative frequency histogram.

The dot plot shows the ages of customers in a mobile phone store in one day: a

Find the lower quartile.

Find the upper quartile.

Find the interquartile range. 18 19 20 21 22 23 24 25 26 27 28

Age of customers

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Extend your thinking 21

For a week, the airline Fly Air decided to keep track of how many minutes behind schedule each ﬂight departed. 100 results are shown in the dot plot. a

Find the median. What is the meaning of the median in this scenario?

Find the lower quartile.

Find the upper quartile.

Find the interquartile range.

If a ﬂight is delayed for 10 minutes or more, the airline incurs a fee. Based on the dot plot, how many ﬂights did the airline incurred a fee for?

What percentage of ﬂights did the airline incur a fee for?

A rival airline, Fly High, had a median delay time during the same week of 45 minutes. How many delayed ﬂights did Fly Air have that were longer than 45 minutes?

What percentage of Fly Air’s ﬂights had delay times that were longer than Fly High’s median delay time?

0 5 10 15 20 25 30 35 40 45 50 55 60

minutes

To gain a place in the main race of a car rally, teams must compete in a qualifying round. The median time in the qualifying round determines the cut off time to make it through to the main race. Here are some results from the qualifying round: • 75% of teams finished in 159 minutes or less. • 25% of teams finished in 132 minutes or less. • 25% of teams finished between with a time between 132 and 142 minutes. a

Find the cut-off time required in the qualifying round to make it through to the main race.

Find the interquartile range in the qualifying round.

In the qualifying round, the ground was wet, while in the main race, the ground was dry. To make the times more comparable, the ﬁnishing time of each team from the qualifying round is reduced by 5 minutes. Find the new median time from the qualifying round.

The bar graph shows the marks (out of 10) that students received on a spelling test: a

Find the lower quartile.

Find the upper quartile. Find the interquartile range.

c d

Explain the effect scores less than 7 have on the summary statistics.

Frequency

16 12 8 4 0

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Mark

These datasets each contain 10 scores: Set 1: 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 Set 2: 10, 12, 14, 16, 18, 40, 42, 44, 46, 48 Set 3: 10, 42, 44, 46, 48, 50, 52, 54, 56, 58

Which dataset has the largest interquartile range and what does this tell you about the spread of data in this dataset?

Identify if each set is symmetrical, skewed-left or skewed-right and justify your response.

There is a test to measure the Emotional Quotient (EQ) of an individual. Here are the EQ results for two groups of 21 people listed in ascending order: Group 1: 90, 90, 91, 92, 93, 94, 95, 95, 95, 97, 99, 100, 108, 114, 116, 116, 117, 118, 118, 122, 129 Group 2: 85, 85, 85, 86, 86, 88, 90, 95, 97, 98, 100, 100, 100, 100, 102, 105, 105, 105, 110, 110, 115

Calculate the range, interquartile range, median, and mean for both datasets.

Compare the mean of both datasets. What does this tell you about the EQ scores of the groups?

Compare the spread of both datasets. What does this tell you about the variability in EQ scores within the groups?

The dot plot shows the number of pets students have, for 25 students.

Number of pets a

Find the lower quartile.

Find the upper quartile.

Find the percentage of students who have more pets than the upper quartile. Explain why it is not exactly 25%.

Is the percentage of data values between the upper and lower quartiles always exactly 50%? Explain why or why not, including an example to justify your conclusion.

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The daily sales (in hundreds of dollars) for two shops, Shop A and Shop B, over 8 days are: • Shop A: 20, 10, 12, 15, 14, 20, 8, 9 • Shop B: 15, 14, 13, 14, 16, 14, 15, 16 Compare the daily sales of Shop A and Shop B, referencing their centre, spread, and skew.

A gamer scores 100, 50, 150, 70, and 130 points in their ﬁrst ﬁve rounds. In the sixth round, they score 10 points. Describe how this new score affected: a

Their season points average.

The population standard deviation of their scores.

Their median score.

The range of their scores.

A horticulturalist is testing two different fertilisers on plant growth. After a trial period, the growth (in cm) of 10 plants for each fertiliser was recorded: Fertiliser X (cm)

Fertiliser Y (cm)

Calculate the mean and population standard deviation of growth for Fertiliser X, rounded to two decimal places.

Calculate the mean and population standard deviation of growth for Fertiliser Y, rounded to two decimal places.

Using these results, compare the effectiveness of the different fertilisers on plant growth.

The dataset shows the ages (in years) of 11 historical artefacts discovered at a site: 105, 110, 112, 118, 120, 125, 130, 135, 140, 142, 150 Determine:

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The median age

The lower quartile (Q1) age

The upper quartile (Q3) age

The interquartile range (IQR)

A survey recorded the number of apps on 50 students’ phones. The data is in the frequency table: a

How many students were surveyed?

Determine the range of the number of apps.

Calculate the interquartile range (IQR).

Mathspace New South Wales – Year 11 Standard mathspace.co

Number of Apps

Frequency

Big ideas • Box plots provide a visual framework for summarising and comparing data distributions, highlighting central tendencies, spreads, and shapes. • Identifying outliers, clusters, and gaps in datasets reveals underlying patterns, variability, and anomalies critical for accurate data interpretation.

11 Box plots, clusters and outliers Chapter outline 11.01 11.02 11.03 11.04 11.05

Five-number summaries and box plots Parallel box plots Histograms, dot plots and box plots Outliers Identify clusters and gaps Chapter 11 review

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11.01 Five-number summaries and box plots After this lesson, you will be able to… • determine the five-number summary from a set of numerical data or graphical representation • determine the interquartile range (IQR) of datasets • compare and contrast the use of range and IQR as measures of spread • represent numerical datasets using a box plot to display a five-number summary, with and without using digital tools • interpret box plots to draw conclusions and make inferences about a dataset

Five-number summary Five (5)-number summary A method for summarising a dataset using 5 statistics: the minimum value, the lower quartile, the median, the upper quartile and the maximum value.

Quartiles were introduced to measure the spread of a dataset. Unlike the range, which uses only the maximum and minimum, quartiles provide insight into the internal distribution. The five-number summary combines quartiles with the dataset’s extremes to summarise its distribution. It consists of: • Minimum: The smallest value • Q1 (Lower quartile): The median of the lower half • Median: The middle value • Q3 (Upper quartile): The median of the upper half • Maximum: The largest value Q1

Min 25%

Median 25%

50%

25%

Max 25%

50%

These values divide the dataset into four equal parts, each containing 25% of the scores.

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Example 1 The points scored by a basketball team in their previous season’s games are: 77, 97, 96, 89, 52, 99, 58, 69, 96, 59, 96, 55, 80, 52, 68 a Sort the data in ascending order.

Create a strategy

Apply the idea

Arrange the scores from smallest to largest.

52, 52, 55, 58, 59, 68, 69, 77, 80, 89, 96, 96, 96, 97, 99

b Calculate the maximum value.

Create a strategy

Apply the idea

Select the largest score.

Maximum = 99

c Calculate the minimum value.

Create a strategy

Apply the idea

Select the smallest score.

Minimum = 52

d Calculate the median value.

Create a strategy

Apply the idea

The median is the middle score, or the average of the two middle scores.

With 15 scores, the median is the 8th score. Median = 77

e Calculate the lower quartile.

Create a strategy

Apply the idea

Calculate the median of the scores below the median.

Lower half: 52, 52, 55, 58, 59, 68, 69 Q1 = 58

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The five-number summary was used to describe a dataset’s spread. Box plots (or box-and-whisker plots) visually represent this summary. Minimum Lower quartile Median Upper quartile Maximum

17 52 69 87 100 Maximum Value

Median Minimum value

Upper Quartile

Lower Quartile

100

The box shows the middle 50% of scores, with its length as the interquartile range (Q3 − Q1 ). Whiskers extend to the minimum and maximum, unless outliers are present, which are plotted as points. Minimum

Lower quartile Q1

~25%

Outlier

Median Q2

~25%

Upper quartile Q3 ~25%

Maximum

~25%

Interquartile range

Similar to other data displays, the shape can describe based on the distribution of the dataset.

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Symmetrical

A symmetrical dataset is distributed around the centre with a similar frequency on the left and right. 0

20 25 30 35

Left skew

A left-skewed dataset is where the majority of the data points have higher values, with some data points at lower values. It is sometimes called a negative skew. 0

20 25 30 35

Right skew

A right-skewed dataset is where the majority of the data points have lower values, with some data points at higher values. It is sometimes called a positive skew. 0

20 25 30 35

Uniform

A uniform dataset is evenly distributed across all values.

20 25 30 35

When describing distributions, for skewed distributions, use the median and interquartile range as they are less impacted by outliers. For symmetrical or uniform distributions, use the mean and range since they take the values of all data points into account.

Interactive exploration Discover this concept in action online

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mathspace.co

Example 2 Create a box plot for the data in the table. Minimum

Lower quartile

Median

Upper quartile

Maximum

Create a strategy Use the five-number summary to construct the box plot.

Apply the idea

10 20 30 40 50 60 70 80 90 100

Example 3 Complete the table using the box plot.

Minimum Maximum Outliers Range IQR 0

8 12 16 20 24 28 32 36 40 44

⬚ ⬚ ⬚ ⬚ ⬚

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Create a strategy Use the illustration shown to determine the values for the table. Minimum

Lower quartile Median Q1 Q2 ~25%

~25%

Outlier

Upper quartile Maximum Q3 ~25%

~25%

Interquartile range

The range is the difference between the minimum and maximum. For the outliers, determine the points beyond the whiskers. For the IQR, measure the distance from the lower to the upper quartile, or the length of the box.

Apply the idea Completed table: Minimum Maximum Outliers Range IQR

1 43 1, 41, 43 42 7

Example 4 Exam scores are shown in the stem-and-leaf plot. Exam scores 6

1779

0023455

0135 Key 6 | 1 = 61

a Construct the five-number summary.

Create a strategy

Apply the idea

Identify minimum, maximum, median, and quartiles.

Five-number summary:

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Minimum Lower quartile Median Upper quartile Maximum

61 69 73 80 85

b Construct a box plot.

Create a strategy Use the five-number summary in part (a) as the data.

Apply the idea Exam scores

85 90

c Describe the shape of the box plot.

Apply the idea The box plot shows slight positive skew.

Example 5 Mathematics test results for two classes are shown. Mr. Smith’s class

Ms. Johnson’s class

100

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a Calculate the range of Mr. Smith’s class.

Create a strategy Use the whiskers of the box plot to calculate the range.

Apply the idea Range = 95 − 70 = 25

Write the equation Evaluate

Range is 25.

b Calculate the interquartile range of Ms. Johnson’s class.

Create a strategy Subtract Q1 from Q3.

Apply the idea IQR = Q3 − Q1 Write the formula = 85 − 70 Substitute the values = 15

Evaluate

c Compare the medians of the two classes.

Create a strategy Subtract Ms. Johnson’s median from Mr. Smith’s median.

Apply the idea Difference = 85 − 80 Write the equation =5

Evaluate

Mr. Smith’s median is 5 marks higher.

d Which class performed better? Explain.

Create a strategy Compare measures of central tendency and spread.

Apply the idea Mr. Smith’s class has higher minimum, median, and maximum scores and a smaller range, indicating more consistent and better performance.

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For each five-number summary, determine the: i

Range

Interquartile range

Minimum 26

Minimum

Lower quartile

Median

Upper quartile

Maximum

Is each statement true or false for the dataset shown in the box plot? a

The maximum is 16.

The minimum is 8.

The median is 11.

There is an error as the median is not exactly in the middle of Q1 and Q3.

The interquartile range is 3.

The value 1 is an outlier.

10 12 14 16 18 20

For the box plot shown: a

What percentage of scores lie between each of these values? i

Q1 and Q3

Minimum and Q1

iii

Median and maximum

Q1 and maximum

Minimum and Q3

In which quartile(s) is the data the least spread out?

In which quartile(s) is the data the most spread out?

Practice Ex 1

The times (in minutes) taken by a group of students to complete a maths test are: 45, 62, 38, 50, 71, 42, 65, 53, 47, 60, 39, 68, 55, 44, 57

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Sort the data in ascending order.

Calculate the: i

Maximum value

Minimum value

iii

Median value

Lower quartile

Upper quartile

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For the box plot shown, determine the: a

Minimum

Maximum

Range

Median

Interquartile range

Construct a five-number summary for each box plot: Scores a b

8 10 12 14 16 18 20

Scores

8 10 12 14 16 18 20

Scores

38 40 42 44 46 48 50 52 54 56 58 60

8 10 12 14 16 18 20

Construct a box plot to represent the information in the tables: a

Ex 3

Scores

0 Ex 2

Minimum 5

Minimum

Lower quartile

Median

Upper quartile

Maximum

For the box plot shown, determine the: ⬚ Minimum Maximum Outliers Range Interquartile range

⬚ ⬚ ⬚ ⬚

4 8 12 16 20 24 28 32 36 40 44

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A mathematics test is given to two classes. The marks out of 20 received by students in each class are represented in the box plots. a

Complete the table: Class 10P

Class 10Q

⬚

Median

⬚

Lower quartile Upper quartile

Interquartile range

0 2 4 6 8 10 12 14 16 18 2022 Class 10Q

⬚

Outliers

Ex 5

⬚

Range

Class 10P

⬚

Which class tended to score better marks?

Would the range or interquartile range be better for describing the spread of Class 10Q?

0 2 4 6 8 10 12 14 16 18 2022

The box plot shows the age at which a group of people obtained their driving licences: a

What is the oldest age?

What is the youngest age?

What percentage of people were aged from 18 to 22?

The middle 50% of responders were within how many years of one another?

In which quartile are the ages least spread out?

The bottom 50% of responders were within how many years of one another?

Age

The box plots represent the daily sales made by Carl and Angelina over the course of one month. Angelina’s sales

Carl’s sales

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Calculate the range for Angelina’s sales.

Calculate the interquartile range of Carl’s sales.

Compare the medians of the two classes.

Which salesperson had a more successful sales month? Explain your answer.

Create a box plot for the data shown in each graph. Monthly salaries of employees Number of employees

Scores of students

30 25 20 15 10 5 0

20 25 30 35 40 45 50

100

Salaries (in thousands) Debbie’s blog visitors Number of days

10 9 8 7 6 5 4 3 2 1 0

Number of pages read

10 0

50 100 150 200

Number of visitors

Extend your thinking 17

Ten participants had their pulse measured before and after exercise with results shown in different box plots. Compare the pulse rates before and after exercise for these participants. Pulse before exercise

Pulse after exercise

45 50 55 60 65 70 75 80 85 90

80 85 90 95 100 105 110 115 120 125

How does a box plot help us to understand the centre and spread within a dataset?

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The test scores of 11 students in science and music are listed: • Science: 52, 59, 87, 66, 74, 76, 73, 63, 91, 86, 82 • Music: 87, 63, 60, 43, 74, 82, 66, 61, 88, 60, 84

Construct a box plot for the science scores.

Construct a box plot for the music scores.

In which subject did students perform better overall? Explain your answer with reference to measures of central tendency and spread.

These box plots shows the number of points scored by two basketball teams in each of their matches. Gamma Geckos

26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70

Delta Dragons

30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68

Compare the median score of the Gamma Geckos and the Delta Dragons.

Compare the range of scores for the Gamma Geckos and the Delta Dragons.

Compare the interquartile range for the Gamma Geckos and the Delta Dragons.

If the two teams were to play against each other in the next game, determine which team you expect to win. Justify your answer.

The box plots summarise results from a medical study. The treatment group received an experimental drug to relieve cold symptoms, and the control group received a placebo. The box plots show the number of days each group continued to report symptoms: Control group

Treatment group

-2 0 2 4 6 8 10 12 14 16 18

Describe the shape of the data from the control group.

Describe the shape of the data from the treatment group.

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A box plot was created for a dataset. The value 32 was a typo and should have been 23. a

Does a new box plot need to be redrawn?

Would fixing the typo have any impact on the range and the IQR? 10

14 18 22 26 30 34 38

The marks in an end-of-year exam for a class of students are shown:

52, 56, 59, 64, 66, 77, 78, 80, 80, 80, 81, 84, 86, 90, 95, 96 a

Construct a box plot for the data.

Calculate the interquartile range.

What percentage of marks lie in the range to 85 to 96?

Which values do the lowest 75% of scores lie between?

If the score 91 was added to the dataset, what percentage of the marks are below Q3? Why isn’t this 75%?

11.02

Parallel box plots

After this lesson, you will be able to… • compare and contrast the measures of centre, spread and shape using parallel box plots • interpret parallel box plots to draw conclusions and make inferences when comparing datasets • identify and quantify differences in medians between datasets represented by parallel box plots • compare the ranges and interquartile ranges (IQRs) to assess relative spread or consistency between datasets • describe the relative skewness or symmetry of datasets when viewed as parallel box plots

Parallel box plots A box plot displays the five-number summary. The range and interquartile range measure data spread, visible on a box plot. Parallel box plots compare two or more datasets by displaying their box plots on the same scale, ensuring clear labels for each.

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Under 30s

Over 30s

18 20 22 24 26 28 30

Seconds As shown, the box plots compare task completion times in seconds, highlighting differences in spread and medians. Key comparisons include: • Spread: Compare ranges and interquartile ranges • Skew: Assess symmetry or skewness of each dataset • Medians: Identify differences in central tendency

Example 1 The box plots show the lifespan of light bulbs (in thousands of hours) from two manufacturers. Manufacturer A

Manufacturer B

Thousands of hours a Complete the table using the box plots, converting values to hours by multiplying by 1000.

Median Lower quartile Upper quartile Range Interquartile range

Manufacturer A

Manufacturer B

⬚

⬚ ⬚ ⬚

⬚ ⬚

⬚

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Create a strategy Extract medians, quartiles, ranges (Max − Min), and IQRs (Q3 − Q1) from the box plots, then multiply by 1000.

Apply the idea Manufacturer A

Manufacturer B

Median

4 × 1000 = 4000

5 × 1000 = 5000

Lower quartile

2.5 × 1000 = 2500

3.5 × 1000 = 3500

Upper quartile

4.5 × 1000 = 4500

6 × 1000 = 6000

Range

(5 − 1) × 1000 = 4000

(8 − 1.5) × 1000 = 6500

(4.5 − 2.5) × 1000 = 2000

(6 − 3.5) × 1000 = 2500

Interquartile range

b Which manufacturer produces light bulbs with the best lifespan?

Create a strategy Compare the medians.

Apply the idea Manufacturer B’s median is 5000 hours, higher than Manufacturer A’s 4000 hours, indicating better lifespan.

Reflect and check In fact, the best lightbulb produced by Manufacturer A has a lifespan of 5000 hours, which is the same as the median of Manufacturer B. This means that about half of the lightbulbs produced by Manufacturer B have a greater lifespan than all the lightbulbs produced by Manufacturer A.

Example 2 The box plots show goals scored by two football players per season. Sophile

Holly

15 Goals

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a Who scored the most goals in a season?

Apply the idea By looking at the endpoints of the right whiskers, Sophie scored 18 goals, while Holly scored 19 goals. So Holly score the most goals in a season. b How many more goals did Holly score in her best season compared to Sophie in her best season?

Apply the idea Difference = 19 − 18 Subtract Sophie’s maximum from Holly’s =1

Evaluate

Holly scored 1 more goal. c What is the difference between the median number of goals scored in a season by each player?

Apply the idea Sophie’s median is 11; Holly’s is 10. Difference = 11 − 10 Subtract Holly’s median from Sophie’s =1

Evaluate

Sophie’s median is 1 goal higher. d What is the difference between the interquartile range for both players?

Apply the idea Use the formula: IQR = Q3 − Q1 Sophie’s IQR = 14 − 7 =7 Holly’s IQR = 15 − 6 =9 Difference = 9 − 7 =2

Substitute the quartiles Evaluate Substitute the quartiles Evaluate Subtract Sophie’s IQR from Holly’s Evaluate

Holly’s IQR is 2 goals larger.

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Example 3 The box plots show jump distances (in centimetres) by two high jumpers. 60

120

130

150

John

110

120

110

120

Bill

80 cm

100

130

140

150

a Who has a higher median jump?

Create a strategy

Apply the idea

Compare the median lines.

John has a higher median jump of 120 cm compared to Bill’s 110 cm.

b Who made the highest jump?

Create a strategy

Apply the idea

Compare the maximum values.

John’s highest jump was 150 cm and Bill’s was 120 cm. John made the highest jump.

c Who made the lowest jump?

Create a strategy

Apply the idea

Compare the minimum values.

Both John and Bill had a lowest jump of 60 cm.

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d Which score can be removed from John’s jump distance dataset so that the mean, median, and mode remain unchanged?

Create a strategy To keep the mean, median, and mode unchanged, remove a score close to the mean that does not affect the middle value(s) or the most frequent score. Estimate John’s dataset from the box plot, calculate the mean, median, and mode, and test removing a score.

Apply the idea John’s box plot has this five-number summary: Assume a dataset: [60, 70, 80, 100, 110, 120, 120, 130, 140, 150] • Min = 60 • Q1 = 80 • Median = 120 • Q3 = 130 • Max = 150 Sum the scores and divideby 10 Evaluate the addition

Evaluate

To determine the median, the order is [60, 70, 80, 100, 110, 120, 120, 130, 140, 150], this shows that the middle values (5th, 6th) are 110, 120, so median =

The mode is 120 (twice). Remove a score of 110: Add the remaining scores then divide by 9

Evaluate the addition

Evaluate

With the new median, the 5th value is 120, mode is 120 and the mean is 107.78 ≈ 108. The median (120 is close to 115), and mode are nearly unchanged, so a score of 110 can be removed.

Reflect and check The mean shifted slightly from 108 to 107.78 due to the small dataset, but it’s close enough to be considered unchanged, as in similar examples. The median moved from 115 to 120, which aligns with the box plot’s median. Removing a score like 120 would change the mode and median significantly, and removing 80 would shift the mean to The score 110 maintains the measures most effectively.

but alter the median.

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Practice Ex 1

A class took an english test and a mathematics test. Both tests had a maximum possible mark of 20. The parallel box plots show the results of the tests: English

Mathematics

0 a

Complete the table using the two box plots. English

Mathematics

Median Lower quartile Upper quartile Range Interquartile range b Ex 2

In which test did the class tend to score better marks?

The parallel box plots show the prices, in dollars, of the items on the menu of an upmarket restaurant and the menu of a fast food restaurant. Upmarket restaurant

Fast Food

Price

770

Which restaurant has the higher median price for the items they sell?

What is the difference between the median prices of the items sold by each restaurant?

Which restaurant has a greater price range for the items on their menu?

What is the price difference between the most expensive items sold by each restaurant?

What amount of the cheapest item at the fast food restaurant could be bought for the same price as the most expensive item at the upmarket restaurant?

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Two small businesses, Café A and Café B, track their daily energy usage (in kilowatt-hours, kWh) over 30 days to monitor sustainability in their NSW community. The parallel box plots show their daily energy usage: Café A a The sustainability target is 18 kWh per day. By how much does Café B’s median energy usage exceed this target? b

Ex 3

What percentage of Café A’s daily energy usage is greater than its lower quartile?

Café B

How much larger is Café B’s interquartile range compared to Café A’s?

Considering the data for Café A, would a daily energy usage of 45 kWh be considered an unusually high value? Justify your answer by referring to the maximum value on the box plot.

20 25 30 35 40 45 50

The parallel box plots show the distances, in centimetres, jumped by two high jumpers. Matt

Paul

140

150

160

170

180

Distance (cm)

Ex 4

Who had a higher median jump?

Who made the highest jump?

Who made the lowest jump?

The box plots show the monthly profits (in thousands of dollars) of two derivatives traders over a year. a

Who made a higher median monthly profit?

Whose profits had a higher interquartile range?

Whose profits had a higher range?

How much more did Ned make in his most profitable month than Tobias did in his most profitable month?

Ned

Tobias

10 15 20 25 30 35 40 45 50 55

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A builder can choose between two different types of brick that are coloured red or yellow. The parallel box plots shows the results of tests on the strength of the bricks: Red

Yellow

Using the box plot, explain why a builder might to prefer to use the red bricks.

Using the box plot, explain why a builder might to prefer to use yellow bricks.

The parallel box plots shows the data collected by the manufacturers on the life-span of light bulbs, measured in thousands of hours: Manufacture A

Manufacture B

Thousands of hours a

Complete the following table. Write each answer in terms of hours. Manufacturer A

Manufacturer B

⬚

Median Lower quartile Upper quartile Range Interquartile range b

772

⬚ ⬚

⬚

⬚ ⬚ ⬚

Hence, which manufacturer produces light bulbs with the best lifespan? Explain your answer.

Mathspace New South Wales – Year 11 Standard mathspace.co

A mathematics test is given to two classes. The marks out of 20 received by students in each class are represented in the box plots to the right: Class 9P a For class 9P, find:

The median

The lower quartile

iii

The upper quartile

The range

The interquartile range

For class 9Q, find: i

The median

The lower quartile

iii

The upper quartile

The range

The interquartile range

0 2 4 6 8 10 12 14 16 18 20 22 Class 9Q

Which class tended to score better marks? Explain your answer. 0 2 4 6 8 10 12 14 16 18 20 22

The box plots shown represent the daily sales made by Carl and Angelina over the course of one month: a

What is the range in Angelina’s sales?

What is the range in Carl’s sales?

By how much did Carl’s median sales exceed Angelina’s?

Considering the middle 50% of sales for both sales people, whose sales were more consistent?

Which salesperson had a more successful sales month? Angelina’s sales

10 20 30 40 50 60 70 0

Carl’s sales

10 20 30 40 50 60 70

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At every training session of the season, a cyclist measured her pulse rate before a sprint and after a sprint. The before and after rates, measured in beats per minute (bpm), recorded throughout the season are presented in the box plots: a

How much greater was her median pulse rate after the sprint than before the sprint?

Determine the interquartile range of her pulse rate before the sprint.

Determine the interquartile range of her pulse rate after the sprint.

Determine the range of her pulse rate before the sprint.

Determine the range of her pulse rate after the sprint.

Are her pulse rate readings more consistent before or after the sprint?

In the last session of the season, the cyclist recorded her highest pulse rate of the season both before and after the sprint. By how much did her pulse rate increase during this particular training session? Pulse rate before (bpm)

Pulse rate after (bpm)

60 70 80 90 100 110 120 130 140 60 70 80 90 100 110 120 130 140 14

Eileen competed in the high beam gymnastics event at both the 2006 and 2010 Olympics. Her judges’ scores in both years are presented in the parallel box plots. 2010 Games 2006 Games x 5

Judges Scores (/10)

774

What was the difference between the minimum scores she was awarded?

What was the difference between the maximum scores she was awarded?

One particular judge at the 2010 games gave Eileen score of 8.3. In which quartile of her 2006 scores would this lie?

In which year did the judges score Eileen most consistently?

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Extend your thinking 15

Two bookstores recorded the selling price of all their books. The results are presented in the parallel box plots. Bookstore A prices ($)

Bookstore B prices ($) x 50

100

110

120

130

140

150

Which bookstore had the more consistent prices? Explain your answer.

Comparing the most expensive books in each store, how much more expensive is the one in store B?

Is each statement true or false? i

25% of the books in Bookstore B are at least as expensive than the most expensive book in Bookstore A.

25% of the books in Bookstore B are cheaper than the cheapest book in Bookstore A.

Cooper and Marion are racing go-karts. The times (in seconds) for the 12 laps of their qualifying race are shown: • Cooper: 58.9, 46.5, 52.6, 66.6, 58.4, 53.1, 45.0, 52.1, 52.4, 52.7, 44.8, 51.7 • Marion: 47.8, 54.6, 68.5, 68.0, 62.8, 57.2, 54.8, 63.4, 58.1, 64.3, 66.2, 47.1 a

Construct the five-number summary for each set.

Create a parallel box plot of the two sets of times with the outlier displayed separately.

If pole position (the best starting spot for the final race) is awarded to the racer with the fastest qualifying lap time, which racer will earn it?

Does spinning out on a lap, causing a high outlier, impact the selection for pole position? Explain your answer.

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A cinema is showing three films, labelled A, B and C. The ages of people watching each of the films are illustrated in the parallel box plots: A

Which film do you think has an adults only rating, restricting it to viewers 18 years of age and older? Explain your answer.

Which film would you recommend for a group of 15 year olds to watch? Explain your answer.

Which film would you recommend to a family of two parents in their 40’s and two teenagers? Explain your answer.

Did you know?

Box plots help us see how data groups together — just like these beetles clustering on a flower! Each cluster shows where most data points sit, while any insect wandering off on its own would be an outlier. In statistics, box plots make it easy to spot patterns, compare groups, and notice when something doesn’t quite fit — just like in nature, those outliers can reveal something special about the whole picture. 776

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11.03 Histograms, dot plots and box plots After this lesson, you will be able to… • identify and compare the features highlighted by histograms, dot plots, and box plots for the same dataset • determine quartiles from datasets displayed in histograms and dot plots, often using cumulative frequency • construct a box plot from the five-number summary derived from a histogram or dot plot • recognise and describe consistent data shapes (symmetrical, skewed) across different graphical displays • interpret box plots, histograms, and dot plots to draw conclusions about a dataset

Different graphs of the same data Measures of centre (mean, median, mode) summarise a dataset with a single value, indicating typical performance. Measures of spread (range, interquartile range) describe data variability, with lower spread indicating greater consistency. Different displays highlight specific features: • Frequency displays (histograms, dot plots, stem-and-leaf plots) show shape, frequencies, mode, and outliers. • Box plots show shape, range, interquartile range, median, and outliers. Data shape (symmetrical, positively skewed, negatively skewed) remains consistent across displays.

Frequency

Symmetrical data: 10 9 8 7 6 5 4 3 2 1 0

8 9 10 11 12 13 14 15 16 17 18

Box plot of symmetric data

Score Histogram of approximately symmetric data

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Example 1 Match the box plot to the correct histogram.

2 3

2 3 4 5 6 7 8 9 10

8 9 10

2 3 4 5 6 7 8 9 10

4 5 6 7

2 3 4 5 6 7 8 9 10

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The histogram’s minimum is 0, maximum is 5. The five-number summary is (0, 1, 2, 3, 5). The box plot derived from the five-number summary:

0.5

1.5

2.5

3.5

4.5

For histograms with class intervals, use class midpoints for the five-number summary. Cumulative frequency histogram 20 17.5

Frequency

15 12.5 10 7.5 5 2.5 0

Scores So this time, the Q1, median, and Q3 will just be the class centres of the 25%, 50% and 75% points, giving 1.5, 2.5 and 3.5. The min and max would be the class centres of the min and max classes, 0.5 and 5.5. The box plot is just formed using these values:

0.5

1.5

2.5

3.5

4.5

5.5

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Example 2 A histogram shows the test scores (out of 10) for 24 students, using exact integer scores: 6

Frequency

5 4 3 2 1 0

Scores a Convert the histogram into a cumulative frequency table.

Create a strategy Add each frequency to the previous cumulative total.

Apply the idea

782

Score (x)

Frequency

Cumulative frequency

3+4=7

7 + 6 = 13

13 + 5 = 18

18 + 3 = 21

21 + 3 = 24

Mathspace New South Wales – Year 11 Standard mathspace.co

Draw box plots from dot plots A dot plot displays frequency with dots, similar to a histogram. To create a box plot, use a cumulative frequency table to find the five-number summary.

Dot plot showing frequency distribution of scores. Construct a cumulative frequency table by summing frequencies for each score. For a total frequency of 30: Score (x)

Frequency

Cumulative frequency

2+3=5

5+1=6

6+3=9

9 + 4 = 13

13 + 3 = 16

16 + 2 = 18

18 + 4 = 22

22 + 8 = 30

Use the table to find the five-number summary: • Q1 (25%): 30 × 0.25 = 7.5, at score 7 • Median (50%): 30 × 0.5 = 15, at score 9 • Q3 (75%): 30 × 0.75 = 22.5, at score 12 The minimum is 4, maximum is 12. The five-number summary is (4, 7, 9, 12, 12).

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 Box plot derived from the five-number summary. For small datasets (around 15 points), counting dots in ascending order to find quartiles may be faster. 784

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Example 3 A dot plot shows the number of books read by 15 students in a month:

Number of books read a Calculate the five-number summary.

Create a strategy Construct a cumulative frequency table and find Q1 at 25%, median at 50%, and Q3 at 75%.

Apply the idea Total frequency: 15 Minimum: 0 Maximum: 5 Score (x)

Frequency

Cumulative frequency

2+3=5

5+4=9

9 + 3 = 12

12 + 2 = 14

14 + 1 = 15

• Q1 : 15 × 0.25 = 3.75, at score 1 • Median: 15 × 0.5 = 7.5, at score 2 • Q3: 15 × 0.75 = 11.25, at score 3 Five-number summary: (0, 1, 2, 3, 5).

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Each pair displays the same dataset. What is the skew for each pair of histograms and box plots? a

10 9 8 7 6 5 4 3 2 1 0

10 20 30 40 50 60 70 80 90 100

20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 90 100

10 9 8 7 6 5 4 3 2 1 0

20 30 40 50 60 70 80 90

10 9 8 7 6 5 4 3 2 1 0

2 3 4 5 6 7 8 9 10

Which key features can be read off each display? a

Histogram

Box plot

Dot plot

Cumulative frequency histogram and polygon

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Practice Ex 1

Match the histograms to the corresponding box plots. Histogram A

Box plot 1

10 9 8 7 6 5 4 3 2

10 20 30 40 50 60 70 80 90

1 0 10 20 30 40 50 60 70 80 90 100

Histogram B 10 9 8 7 6 5 4 3 2 1 0

Box plot 2

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

Histogram C 10 9 8 7 6 5 4 3 2 1 0

Box plot 3

10 20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 90 100

Histogram D 10 9 8 7 6 5 4 3 2 1 0

788

Box plot 4

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 100

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Ex 2

A histogram shows the number of hours 20 students spent on homework last week, with exact integer hours (no class widths): Hours student spent studying 5

Frequency

4 3 2 1 0

Ex 3

Hours

Convert this histogram into a cumulative frequency (CF) table.

Draw the corresponding box plot.

What is the interquartile range (IQR)?

A plant nursery records the heights (in cm), of 30 saplings, grouped in this table: Height (cm)

Frequency

10–15

15–20

20–25

25–30

30–35

Create a cumulative frequency graph.

Calculate or estimate Q1, median, and Q3. Use class midpoints for the approximate locations.

Sketch the box plot using these approximate quartile values, plus the minimum and maximum (class midpoints).

A dot plot shows the daily number of cups of coffee consumed by a small group of 12 coworkers:

Daily cups of coffee consumed List all values in ascending order. a

Identify the five-number summary (minimum, Q1, median, Q3 and maximum).

Draw a box plot of the distribution.

Comment on whether the data appear symmetric, skewed, or neither.

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A histogram summarises the number of books read by 30 students in one month: 10

Frequency

8 6 4 2 0

Number of books read

Construct an approximate box plot and state the five-number summary.

Report the approximate IQR and describe the skewness of the distribution.

A teacher devises a reward system where students are placed into four tiers (Bronze, Silver, Gold, Platinum) based on quartile cutoffs. Provide the approximate cutoff values for each tier based on the box plot.

The residents of two blocks of townhouses were asked the number of pets they own. The frequency of various responses are presented in these dot plots. a

Is pet ownership a little lower or higher in Block A than Block B?

In Block A, how many pets do most households have?

In Block B, how many pets do most households have?

What is the shape of the data for Block A?

Calculate the range of the number of pets in Block A.

Which block has more variability in the number of pets?

Do either sets of scores have an outlier?

2 3 Block A

2 3 4 Block B

A teacher recorded the test scores (out of 20) for 25 students. The data are presented in two ways:

3 4 5 6 7 8 9 10 11 12 13 14 15 16

Dot plot data

Frequency

Histogram data 8 7 6 5 4 3 2 1 0

0 5 8 11 14 17

Marked groups

790

Draw a box plot for the dot plot and grouped histogram.

Compare the two five-number summaries and briefly comment on any differences.

The school awards extra credit to students whose scores are above the third quartile. Based on your box plots, what is the cutoff score for extra credit?

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Extend your thinking 11

Imagine a histogram with only three very wide classes (for example, 0–9, 100–199, 200–299), and the total number of data points is 30. a

How would you estimate Q1, median, and Q3 if each bar is quite tall or wide?

Discuss how reliable your quartile estimates are in this scenario.

Suggest one or two strategies for improving accuracy without collecting new data.

Imagine a box plot of weekly workout times (in minutes) for 12 individuals. Next month, each person adds 10 extra minutes to their weekly workout time. a

Describe precisely how the five-number summary changes when a constant is added to every data value.

How does this shift impact the box plot’s shape and the measures of spread (range and IQR)?

If a gym policy requires at least 60 minutes/week, how does the shift change the proportion of people meeting that requirement?

You have two groups (Group 1 has 10 data points, Group 2 has 30). Both are summarised: • Group 1 five-number summary: (4, 5, 7, 8, 12) • Group 2 five-number summary: (4, 5, 7, 8, 12) At first glance, they look identical.

Why might interpreting these box plots as “the same distribution” be misleading?

Discuss how the smaller sample size (Group 1) impacts the reliability of quartiles and the median.

Suggest at least one other piece of information needed to decide if these two groups are truly similar.

A large dataset is grouped into intervals for a histogram, but you only know each interval’s midpoint and frequency—not the exact bounds or distribution within each class. a

Propose a systematic method to approximate the quartiles using those midpoints.

Discuss two factors that can make these quartile estimates inaccurate or biased.

If you realised some intervals were much wider than others, how might that distort the final box plot?

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11.04 Outliers After this lesson, you will be able to… • define an outlier and identify potential outliers by observation • apply the Q1 – 1.5 × IQR and Q3 + 1.5 × IQR rule to formally identify outliers • calculate the fences (lower and upper bounds) for outlier detection • explain the impact of outliers on measures of centre (mean, median) and measures of spread (range, IQR) • compare summary statistics of a dataset with and without an outlier

Outlier A data value that appears to stand out from the other members of the dataset by being unusually high or low. An outlier is a value that deviates markedly from other data points, potentially due to errors or exceptional events. Outliers can affect measures like the mean, median, mode and range. Some outliers can be identified using general observations. However, this method is not accurate and is subjective. Therefore, a formal calculation for outliers provides an accurate method of determining which values are outliers in a set of data. To determine if a data value is an outlier, a rule that involves the lower and upper quartiles and interquartile range (IQR) is used. A data point is classified as an outlier if it falls below the lower fence or above the upper fence. These fences, also referred to as the lower bound and upper bound, respectively, are calculated using the following formulas: Lower fence = Q1 − 1.5 × IQR Q1 is the lower quartile (first quartile) IQR is the interquartile range, Q3 − Q1 Upper fence = Q3 + 1.5 × IQR Q3 is the upper quartile (third quartile) IQR is the interquartile range

Example 1 Identify the outlier(s) in the dataset {63, 67, 71, 76, 111}.

Create a strategy

Apply the idea

Organise the data in ascending order. Identify value(s) significantly greater or smaller than others.

The dataset in ascending order is: 63, 67, 71, 76, 111

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Mathspace New South Wales – Year 11 Standard mathspace.co

Outlier = 111

Example 2 Consider the dot plot shown.

5 6 Score

a Determine the median, lower quartile, and upper quartile scores.

Create a strategy

Apply the idea

The median is the middle score.

Median = 3

The lower quartile is halfway between the lowest score and the median, and the upper quartile is halfway between the median and the highest score.

Lower quartile = 2 Upper quartile = 4

If values lie between two scores, take the average. b Calculate the interquartile range.

Create a strategy

Apply the idea

The interquartile range is the difference between the upper and lower quartile values.

IQR = Q3 − Q1

Write the formula

=4−2

Substitute the values

Evaluate

c Calculate 1.5 × IQR, where IQR is the interquartile range.

Create a strategy

Apply the idea

The interquartile range is 2.

1.5 × IQR = 1.5 × 2

Substitute IQR = 2

Evaluate

d An outlier is a score more than 1.5 × IQR above or below the upper or lower quartile, respectively. Determine the outlier.

Create a strategy A score 1.5 × IQR above the upper quartile is 7. A score 1.5 × IQR below the lower quartile is −1.

Apply the idea Outlier = 9

Identify the score outside these values.

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Example 3 Consider the dataset: 9, 5, 3, 2, 6, 1 a Complete the five-number summary for this dataset. Minimum Lower quartile Median Upper quartile Maximum

⬚ ⬚ ⬚ ⬚ ⬚

Create a strategy

Apply the idea

Sort the values in ascending order.

Dataset in ascending order:

The minimum is the first value, and the maximum is the last.

1, 2, 3, 5, 6, 9

For an even number of scores, the median is the average of the two middle scores. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.

Minimum

Lower quartile

Median

Upper quartile

Maximum

b Calculate the interquartile range.

Create a strategy

Apply the idea

The interquartile range is the difference between the upper and lower quartiles.

IQR = Q3 − Q1 =6−2

Substitute the values

Upper quartile is 6, lower quartile is 2.

Evaluate

Write the formula

c Calculate the lower fence.

Create a strategy Use the lower fence formula: Q1 − 1.5 × IQR

Apply the idea Lower fence = Q1 − 1.5 × IQR

794

Write the formula

= 2 − 1.5 × 4

Substitute Q1 = 2 and IQR = 4

=2−6

Evaluate the product

=−4

Evaluate

Mathspace New South Wales – Year 11 Standard mathspace.co

Interactive exploration Discover this concept in action online

mathspace.co

Example 4 Consider the dataset: 36, 36, 70, 45, 52, 48, 36, 43, 44, 51 a Calculate the mean, rounding to one decimal place.

Create a strategy The mean is the average of the scores.

Apply the idea Add the dataset values then divide by 10

Evaluate the addition

Evaluate

b Calculate the median.

Create a strategy The median is the middle score after ordering the dataset.

Apply the idea Dataset in ascending order: 36, 36, 36, 43, 44, 45, 48, 51, 52, 70 With 10 scores, the median is the average of the 5th and 6th scores:

c Calculate the mode.

Create a strategy

Apply the idea

Identify the most frequent score.

The mode is 36.

d Calculate the range.

Create a strategy

Apply the idea

The range is the difference between the highest and lowest scores.

The range is 70 − 36 = 34.

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Mathspace New South Wales – Year 11 Standard mathspace.co

e Remove the outlier and recalculate the mean, rounding to one decimal place.

Create a strategy Identify and remove the score significantly different from others.

Apply the idea Remove 70, as it is much higher than other scores. Calculate the mean with 9 scores: Add the dataset values then divide by 9

Evaluate and round

Reflect and check This mean is lower than the original 46.1.

With the outlier removed, recalculate the median.

Create a strategy

Apply the idea

The median is the 5th score in the ordered set.

The median is 44, slightly lower than 44.5.

g With the outlier removed, recalculate the mode.

Create a strategy

Apply the idea

Identify the most frequent score.

The mode remains 36.

h With the outlier removed, recalculate the range.

Create a strategy Calculate the difference between the new highest and lowest scores.

Apply the idea The range is 52 − 36 = 16, smaller than 34.

Reflect and check The range is significantly reduced without the outlier.

11.04 Outliers mathspace.co

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Example 5 Consider the dataset: 53, 46, 25, 50, 30, 30, 40, 30, 47, 109 a Complete the summary statistics table. Mean Median Mode Range

⬚ ⬚ ⬚ ⬚

Create a strategy

Apply the idea

Mean: Sum of values divided by the number of values

Sorted dataset: 25, 30, 30, 30, 40, 46, 47, 50, 53, 109

Median (even number of values): Mean of the two middle values

Mean

Mode: Most frequent value

Median

Range: Difference between highest and lowest values

Mode

Range

b Identify the outlier.

Create a strategy

Apply the idea Outlier = 109

An outlier is significantly different from other values. Sorted dataset: 25, 30, 30, 30, 40, 46, 47, 50, 53, 109 Check if 25 or 109 is far from other values.

c Complete the summary statistics table after removing the outlier. Mean Median Mode Range

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Mathspace New South Wales – Year 11 Standard mathspace.co

⬚ ⬚ ⬚ ⬚

11.04 Practice questions What do you remember? 1

When an outlier is removed from a dataset, describe the effect on the: a

Mode

Range

For each of these scenarios, an outlier was removed. Was the outlier smaller or larger than the values that remain? a

The mean decreased after the outlier was removed.

The mean increased after the outlier was removed.

The median decreased after the outlier was removed.

The median increased after the outlier was removed.

Consider the given frequency table: a

What is the mode?

Which weight is the outlier?

If the outlier is removed, what is the new mode?

Compare the mode before and after removing the outlier.

Weight in kilograms

Frequency

Practice Ex 1

Ex 2

Identify the outlier(s) in the dataset {82, 85, 88, 92, 145}. a

Sort the dataset in ascending order.

Identify the outlier(s) by observation.

Identify possible outliers in each of these datasets by calculating the upper and lower fences: a

73, 77, 81, 86, 131

7, 25, 28, 35, 42

69, 79, 86, 72, 86, 77, 73, 82, 81, 76, 83, 47, 87, 70, 80, 85

58, 63, 58, 59, 64, 68, 68, 30, 73, 25, 72, 61, 65, 69, 75, 72

VO2 Max is a measure of how efficiently your body uses oxygen during exercise. The more physically fit you are, the higher your VO2 Max. Here are some people’s results when their VO2 Max was measured: 46, 27, 32, 46, 30, 25, 41, 24, 26, 29, 21, 21, 26, 47, 21, 30, 41, 26, 28, 26, 76 a

800

Sort the values into ascending order.

Determine the median VO2 Max.

Determine the upper quartile value.

Determine the lower quartile value.

Identify any outliers by calculating upper and lower fences.

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 3

For each of these sets of data: i

Construct the five-number summary.

Calculate the interquartile range.

iii

Calculate the value of the lower fence.

Calculate the value of the upper fence.

Would the value −5 be considered an outlier?

Would the value 16 be considered an outlier?

9, 5, 3, 2, 6, 1

3, 10, 9, 2, 7, 5, 6

12, 5, 11, 1, 9, 8, 5, 6

The five numbers 16, 16, 17, 24 and 17 have a mean of 18 and a median of 17. Describe the effect on the mean and median if a new number that is larger than 24 is added.

Carl has been recording his spelling test scores for the past semester. His scores were: 14, 16, 2, 15, 15, 16, 15

Ex 4

Ex 5

Calculate the median of Carl’s scores.

Calculate the mean of Carl’s scores, rounded to two decimal places.

Which measure of centre most accurately describes the centre of this dataset? Explain your answer.

For each of these sets of data: i

Determine the mean, median, mode and range. Round your answers to two decimal places where necessary.

Which data value is an outlier?

iii

Remove the outlier from the set and recalculate the values found in part (i).

Describe how each of the four statistics changed after removing the outlier.

27, 50, 24, 37, 47, 41, 27, 126, 44, 27

4.7, 2.8, 1.9, 0.9, 0.9, 2.2, 2.2, 1.2, 1.5, 0.9

4700, 4700, 4700, 4500, 5300, 4900, 5200, 4800, 1500, 5100

A coffee shop records the number of customers served per hour over 10 hours: 45, 50, 55, 60, 65, 70, 75, 80, 85, 150 a

Complete the summary statistics table, rounded to one decimal place where necessary.

Identify the outlier by observation.

Median

Complete the summary statistics table after removing the outlier. Round the mean to one decimal place.

Mode

Let A be the original dataset and B be the dataset without the outlier. Compare the statistics using >,<, or =.

Mean

Range

With outlier

>, <, or =

Without outlier

Mean

Median

⬚

Mode

Range

⬚

11.04 Outliers mathspace.co

⬚ ⬚ ⬚ ⬚

801

The selling prices of six houses are: $467 000, $413 000, $410 000, $456 000, $487 000, $929 000

What is the median selling price? Round your answer to the nearest thousand dollars.

What is the mean selling price? Round your answer to the nearest thousand dollars.

Which measure of centre most accurately represents the centre of the data?

Which of the selling prices appears to be an outlier?

Recalculate the mean selling price, without the outlier.

A group in a study volunteer to take a test that assesses their reaction time. The participants clicked a button as soon as they heard a sound which was played at random intervals. The reaction time, in milliseconds, of each participant is: 220, 280, 210, 220, 215, 180, 185, 190, 190, 195, 150, 190, 195, 195

Identify any outliers by observation.

Which of these statements is the most likely explanation for the outlier? A

The participant was not focused.

The sound took longer to play.

The participant missed the button.

The timer malfunctioned.

The number of three-pointers scored in different basketball games by a single team are shown in the table.

Three-pointers

Frequency

Describe the impact on each measure if the outlier is removed:

Mean

Median

Mode

Range

A teacher records the time (in minutes) taken by students to complete a mathematics quiz. The times are: 15, 18, 20, 22, 25, 28, 30, 35, 60 a

Calculate the mean time, rounded to one decimal place.

Calculate the median time.

Identify any outliers by observation.

Recalculate the mean time without the outlier, rounded to one decimal place.

The weights (in kilograms) of packages delivered by a courier service are: 2.5, 3.0, 3.2, 3.5, 4.0, 4.2, 4.5, 10.0

802

Construct the five-number summary.

Calculate the interquartile range.

Identify any outliers using the fence method.

Mathspace New South Wales – Year 11 Standard mathspace.co

Extend your thinking 17

The data point 5 is below the lower fence and is considered an outlier. The interquartile range is 12. Determine the smallest integer value of the lower quartile.

A journalist wanted to report on road speed cameras being used as revenue raisers. She obtained data that showed the number of times 20 speed cameras issued a fine to motorists in one month. The results were: 101, 102, 115, 115, 121, 124, 127, 128, 130, 130, 143, 143, 146, 162, 162, 163, 178, 183, 194, 977 The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which measure of centre should she use in her article? Explain your answer.

The selling prices of artworks sold at an auction are given: $18 000, $11 000, $17 000, $20 000, $18 000, $16 000, $15 000, $218 000 Which measure of centre best identifies the typical selling price of recently sold artwork? Explain your answer.

Emily is conducting a chemistry experiment. She repeats the experiment 15 times and records the outcomes. She calculates the mean of her results and notes it down. Unfortunately, before she can write up the report, she misplaces her results sheet, and one of the data values is no longer readable. Determine the missing data value and identify if it is an outlier if the mean is 2.5, and the 14 legible data values are: 2.4, 2.5, 2.6, 2.7, 2.4, 2.5, 2.8, 2.3, 2.5, 2.6, 2.4, 2.5, 2.6, 2.7

A medical study recorded the recovery times (in days) for two groups of patients following a new surgical procedure. The recovery times are: Dataset 1: 12, 14, 17, 19, 21, 20, 18, 15, 14, 18, 19, 20, 25, 30, 22 Dataset 2: 10, 13, 18, 20, 22, 19, 21, 16, 13, 17, 18, 19, 24, 29, 23 a

Identify any outliers in dataset 1 using the fence method.

Use the fence method to determine the outliers in dataset 2.

Comment on the answers you received for parts (a) and (b).

11.04 Outliers mathspace.co

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11.05 Identify clusters and gaps After this lesson, you will be able to… • identify clusters in a dataset from graphical displays and describe their location • identify gaps in a dataset from graphical displays and consider their potential causes • relate the presence of outliers to the formation of gaps in a distribution • describe the overall shape of a data distribution (symmetrical, skewed, unimodal, bimodal) considering clusters and gaps • explain the occurrence of clusters, gaps, and outliers in the context of given data

Symmetrical distribution When the 2 sides of the distribution are a mirror image of each other. A normal distribution is a true symmetric distribution of observed values. Data distributions can be symmetrical or skewed, affecting how clusters and gaps appear. Symmetrical data has a bell-shaped curve, with the mean, median, and mode roughly equal.

Symmetrical distributions, like daily temperatures in Sydney, have approximately 50% of values above and below the mean.

Positively skewed data has a longer right tail, with the mean greater than the median.

Positively skewed data, such as rainfall amounts, concentrates on lower values with a few high outliers.

Negatively skewed data has a longer left tail, with the mean less than the median. There are a few features that are related to negatively skewed data: The mean is usually less than the median. The median is usually less than the mode.

804

Mathspace New South Wales – Year 11 Standard mathspace.co

Interactive exploration Discover this concept in action online

This is the general shape of a frequency histogram with a symmetrical shape.

30 25

Percentage

mathspace.co

Even though the data is not exactly symmetrical, the dark line shows that a symmetrical curve can roughly be drawn over the histogram following the data.

20 15 10

To describe shape we do not need the data to be exactly the same, we are looking to describe what it is most like.

5 75

95 100 105 110

Frequency

Weight

This is the general shape shown over a histogram of positively skewed data.

Frequency

Score

This is the general shape shown over a histogram of negatively skewed data.

Score

11.05 Identify clusters and gaps mathspace.co

805

Example 1 Identify whether the distribution of daily maximum temperatures in Sydney is symmetrical, positively skewed, or negatively skewed. 10 9 8

Frequency

7 6 5 4 3 2 1 0

Score

Create a strategy

Apply the idea

Examine the histogram’s shape, focusing on the symmetry and tail lengths.

The histogram shows a bell-shaped curve with roughly equal frequencies on both sides of the centre. The distribution is symmetrical.

Example 2 A group of 15 students records their daily travel times (in minutes) to school. The stem-and-leaf plot shows these times. Identify whether the distribution is symmetrical, positively skewed, or negatively skewed. Daily travel times 1

0235568

025

Key 1 | 5 = 15 minutes

806

Mathspace New South Wales – Year 11 Standard mathspace.co

Clusters Cluster A group of data points that are close together or have similar values. A cluster is a group of data points concentrated within a narrow range, visible as peaks in histograms or dense regions in box plots. Clusters indicate common values in the dataset.

Frequency

A bimodal histogram, like cricket scores from two matches, shows two clusters. More than 2 peaks indicate a multimodal distribution.

Score

Example 3 The percentage of faulty computer chips in 42 batches were recorded in the histogram. 14 12

Frequency

10 8 6 4 2 0

Faulty chips (%)

808

Mathspace New South Wales – Year 11 Standard mathspace.co

Upper bound = Q2 + 1.5 × IQR Q1

is the third quartile

IQR

is the interquartile range

Gaps indicate unusual data patterns, requiring investigation into their causes, such as errors or exceptional events.

Exploration Analysis of monthly rainfall in Sydney shows most months recording 50 to 150 mm, but one month at 500 mm. Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

110

130

500

100

140

120

Rainfall (mm)

1. Looking at the June rainfall figure of 500 mm, what different types of reasons could explain why this one month is so different from the others? 2. If you were to plot all twelve monthly rainfall values, how would the 500 mm value for June visually create a space or ‘gap’ between it and the cluster of more typical rainfall amounts? 3. When analysing this rainfall dataset, why is it important to notice this 500 mm outlier and the gap it creates from the more common values? How might it affect your understanding of Sydney’s typical rainfall?

Histogram with gap 5

Frequency

4 3 2 1 0

Score There is a huge gap in this histogram between 4 and 10, and the data clusters around the peak at 3. This outlier of 10 creates a positive skew graph and gives a potentially misleading representation of the data. It’s up to the data analyst to figure out why this happened and if they need to either ignore the outlier or replace the outlier with another measure such as the median.

11.05 Identify clusters and gaps mathspace.co

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a Determine the five-number summary, mean, and identify outliers.

Create a strategy Calculate cumulative frequencies and score times frequency to calculate the mean and quartiles. Use IQR to identify outliers.

Apply the idea Score

Frequency

Cumulative frequency

Score × frequency

120

240

400

250

180

100

200

Total frequency is 38. Add Score × frequency then divide by the total frequency

Evaluate the addition

Round to two decimal places

For Q1, 0.25 × 38 = 9.5, so Q1 = 20. For median, 0.5 × 38 = 19, so Median = 40. For Q3, 0.75 × 38 = 28.5, so Q3 = 50. The interquartile range is IQR = 50 − 20 = 30. The lower bound is 20 − 1.5 × 30 = −25. The upper bound is 50 + 1.5 × 30 = 95. The score 100 is an outlier. Five-number summary: (10, 20, 40, 50, 60), with mean of 37.63, and outlier of 100.

11.05 Identify clusters and gaps mathspace.co

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11.05 Practice questions What do you remember? 1

Describe a characteristic of a symmetrical distribution.

Define positive skew and negative skew in the context of a dataset’s distribution.

How can clusters be identified within a dataset?

A set of data is strongly positively skewed. If the median is 60, is the mean less than, equal to or greater than 60?

Practice Identify if the dataset is positively skewed, negatively skewed, or symmetrical.

Frequency

Ex 1

10 9 8 7 6 5 4 3 2 1 0

10 14

25 28

15 20 25 30 35 40 45 50

Class

22 26 30 34 38 42 46

Class

34 37 40 43 46 49 52 55

Class 11.05 Identify clusters and gaps mathspace.co

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Frequency

Score

Frequency

Score

15 14 13 12 11 10

Frequency

9 8 7 6 5 4 3 2 1 0

Score

816

Mathspace New South Wales – Year 11 Standard mathspace.co

Frequency

Score

Frequency

20 15 10 5 0 1

Score Ex 2

The stem-and-leaf plot shows the ages of a group of people on a bus. a

Ages

Determine the: i

Median age

Difference between the lowest age and the median

011225

iii

Difference between the highest age and the median

479

The mean age, rounded to two decimal places

Is the data positively or negatively skewed?

01234566

Key 1 | 4 = 14 years old

11.05 Identify clusters and gaps mathspace.co

817

The stem-and-leaf plot shows the number of hours worked per week by a group of people. a

Is there any clustering of data? If yes, in what interval?

Determine the mode(s).

No. of hours 0

1 2

0366

145667

04679

Key 5 | 2 = 52 hours 8

The percentage of faulty mobile phones in 118 batches were recorded. a

Is the data unimodal, bimodal, or multimodal but not bimodal?

Determine the modal class(es) of the data.

20 18 16 14

Frequency

Ex 3

Determine the mode of the data.

12 10 8 6 4 2 0

Faulty mobile phones (%)

For the given column graph: a

Is there any clustering of data?

Where does the clustering occur: 1 to 4, 5 to 7, or 8 to 11?

Determine the mode.

Describe the shape of the distribution of the data. 45 40 35

Frequency

Ex 4

30 25 20 15 10 5 0

Score

818

Mathspace New South Wales – Year 11 Standard mathspace.co

9 10

The following histogram represent students’ heights in centimetres: a

Does the histogram represent grouped data or individual scores?

Estimate the value of the mean, rounded to one decimal place.

Describe the shape of the distribution.

Frequency

105 115 125 135 145 155 165 175

Score

Ex 5

The scores of 20 students in a mathematics test are recorded in the frequency table: Score

Frequency

Determine the five-number summary, mean, and identify any outliers.

Draw a box plot for this data and comment on any clusters or gaps.

If the outlier(s) are removed, calculate the new mean and discuss the impact of the removal on the dataset’s overall interpretation.

Using an example, explain the impact of positive skewness on mean, median and mode of a dataset.

The table shows the number of crime novels in a bookshop for different price ranges. Price of crime novel, to the nearest $5 Frequency

5 3

10 9

Plot this data as a column graph.

Describe the shape of the data in the column graph.

15 19

20 6

25 19

30 9

35 3

11.05 Identify clusters and gaps mathspace.co

819

Consider the stem plot: Leaf

Are there any outliers? If so, state the value.

Is there any clustering of data? If so, in what interval?

c d

What is the mode?

Describe the shape of the data.

133789

135888

5 6 7 8 9

Key 2 | 3 = 23 45

Consider the data shown in the histogram: a

Are there any outliers? If so, what is the value?

Is there any clustering of data? If so, in what interval?

What is the mode?

Describe the shape of the distribution of the data.

Frequency

30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 10 11

Score 16

If a set of data is strongly positively skewed and the median is 70, what can we conclude about the mean?

Identify whether these statements are true or false:

If two sets of data have the same median then the datasets must be the same.

If two sets of data have different modes then the highest values cannot be the same.

Two sets of data have the same highest and lowest values. This means they must have the same median.

Two sets of data that have the same highest and lowest values must have the same mean.

Which of these dot plots has the highest median? A

820

15 20 Score

Mathspace New South Wales – Year 11 Standard mathspace.co

15 Score

Consider the dot plot given:

Which score can be removed so that the mean, median and mode remain unchanged?

250

300

350

400

Consider the histogram:

Frequency

200

180 160 140 120 100 80 60 40 20 0

Score Determine the measure of centre that would be most appropriate to use to represent the data in this display. Justify with reasoning.

Extend your thinking 21

A teacher wants to compare the test scores of two different classes. One class has a symmetrical distribution, while the other has a positively skewed distribution. How might the teacher interpret and compare the performance of the two classes based on these distributions?

The six numbers 6, 2, 7, 18, 17 and an unknown number x have a median of 8.5. Calculate the value of x.

VO2Max is a measure of how efficiently your body uses oxygen during exercise. The more physically fit you are, the higher your VO2Max. Here are some people’s results, listed in ascending order, when their VO2Max was measured: 21, 21, 23, 25, 26, 27, 28, 29, 29, 29, 30, 30, 32, 38, 38, 42, 43, 44, 48, 50, 76 a

Determine the median.

Determine the upper quartile.

Determine the lower quartile.

Consider the box plot for this dataset: Are the results positively or negatively skewed?

Determine the value of the outlier.

An average untrained healthy person has a VO2Max between 30 and 40. What can we say about the majority of this group of people?

11.05 Identify clusters and gaps mathspace.co

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The price of petrol at a petrol station was recorded each day for two weeks. The results are presented in the table: Monday

Tuesday Wednesday Thursday

Friday

Saturday

Sunday

Week 1

$1.70

$1.50

$1.62

$1.46

$1.49

$1.46

$1.55

Week 2

$1.25

$1.36

$1.25

$1.21

$1.20

$3.30

The mean petrol price across the 14 days of records is $1.54 per litre. For which week is this mean a better indication of the price of petrol? Explain your answer using appropriate data display. 25

822

A regional weather service recorded a sample of monthly rainfall (in mm) over 24 months with this frequency distribution: Rainfall (mm)

Frequency

100

120

Compute the five number summary, mean, and standard deviation. Identify outlier(s) using the 1.5 × IQR rule.

Produce a histogram to illustrate the distribution; comment on clusters and gaps.

Determine the new five-number summary, mean, and standard deviation: i

Removing the outlier(s)

Replacing the outlier(s) with the median

Create parallel box plots for the original data, the outlier-removed data, the outlierreplaced data

Analyse how these adjustments affect the standard deviation and discuss which method better preserves the data’s variability.

Discuss the implications of adjusting outliers in the context of rainfall measurements for water resource management.

Mathspace New South Wales – Year 11 Standard mathspace.co

A dataset has the following five-number summary: Minimum = 15, Q1 = 22, Median = 30, Q3 = 38, Maximum = 45. a

Calculate the range and interquartile range (IQR).

Construct a box plot to represent this data.

The following dataset represents the number of hours students spent studying for an exam: 5, 7, 8, 9, 9, 10, 11, 12, 18, 20

Calculate the first quartile (Q1), third quartile (Q3), and the interquartile range (IQR).

Determine the lower and upper fences and identify any outliers in the dataset.

The histogram shows the number of hours students in a class spent on homework in a week. Hours students spent studying 5

Frequency

4 3 2 1 0

Hours

Determine the five-number summary from the histogram data.

Construct the box plot for this data.

The dataset shows the number of rainy days per month in a town over a year: 3, 5, 6, 6, 7, 8, 9, 10, 11, 12, 14, 25

Calculate the mean and median number of rainy days, including the value 25.

If the value 25 is identified as an outlier and removed, recalculate the mean and median.

The following data represents the number of goals scored by a school’s soccer team in 12 matches: 0, 1, 1, 1, 2, 2, 2, 3, 5, 5, 6, 9

824

Identify any clusters of data scores.

Identify any significant gaps in the data scores.

Mathspace New South Wales – Year 11 Standard mathspace.co

Big ideas Linear relationships enable effective modelling and analysis of real-world scenarios by defining constant rates of change and using algebraic and graphical representations to predict outcomes.

12 Linear relationships Chapter outline 12.01 12.02 12.03 12.04

Straight line graphs Gradient and intercept Gradient-intercept form Modelling linear relationships Investigation: Spreadsheets and linear relationships 12.05 Direct variation Chapter 12 review

830 841 853 864 887 896

12.01

Straight line graphs

After this lesson, you will be able to… • identify a linear relationship and its representation as a straight-line graph • construct a table of values from a linear equation • plot points on a Cartesian plane to construct a straight-line graph • determine if a graph or set of points represents a linear relationship • identify the y-intercept of a line from its graph

Linear relationships Linear relationships Two variables x and y are in a linear relationship (or form a linear function) if they are connected by an equation of the form y = mx + c. Graphically, m is the gradient and c is the intercept with the vertical axis of the corresponding linear graph. An equation representing a linear relationship will always form a straight line when plotted on the Cartesian plane. This type of relationship generally connects an independent variable (x) with a dependent variable (y), such that y’s value is a function of x. For example, the equation y = 2x + 1 describes a linear relationship. Substituting values of x into the equation gives corresponding y values, which, when plotted, form a straight line. Linear relationships can model real-world scenarios, such as: • Cost of hiring a taxi based on distance • Converting between temperature scales

• Monthly savings over time

Interactive exploration Discover this concept in action online

mathspace.co

Point and graph plots Graphing a linear equation involves creating a table of values, plotting points on the Cartesian plane, and connecting them with a straight line. Graphing software and calculators can also directly plot linear equations. For example, entering y = −4x + 1 into such a tool will display its line. A table of values lists pairs of x and y values. Each pair (x, y) represents a point, located by finding x on the x-axis and y on the y-axis. For example, the point (1, 3) is plotted by locating x = 1 and y = 3, then marking their intersection.

Interactive exploration Discover this concept in action online

830

Mathspace New South Wales – Year 11 Standard mathspace.co

mathspace.co

Example 1 For the equation y = 3x − 2, complete the table of values. x

−1

⬚

Create a strategy Substitute each x value into y = 3x − 2 to calculate the corresponding y value.

Apply the idea For x = −1: y = 3x − 2

Write the equation

= 3 × (−1) − 2

Substitute x = −1

= −3 −2

Simplify

= −5

Evaluate

For x = 0: y = 3x − 2

Write the equation

=3×0−2

Substitute x = 0

=0−2

Simplify

= −2

Evaluate

For x = 1: y = 3x − 2

Write the equation

= 3×1−2

Substitute x = 1

=3−2

Simplify

Evaluate

For x = 2: y = 3x − 2

Write the equation

= 3×2−2

Substitute x = 2

=6−2

Simplify

Evaluate

The completed table is: x

−1

−5

−2

12.01 Straight line graphs mathspace.co

831

Example 2 Consider the equation y = −4x + 1. a Complete the table of values. x

−1

⬚

Create a strategy Substitute each x value into y = −4x + 1 to calculate the corresponding value.

Apply the idea For x = −1: y = −4x + 1

Write the equation

= −4 × (−1) + 1 Substitute x = −1 =4+1

Simplify

Evaluate

For x = 0: y = −4x + 1

Write the equation

= −4 × 0 + 1

Substitute x = 0

=0+1

Simplify

Evaluate

For x = 1: y = −4x + 1

Write the equation

= −4 × 1 + 1

Substitute x = 1

= −4 + 1

Simplify

= −3

Evaluate

For x = 2: y = −4x + 1

Write the equation

= −4 × 2 + 1

Substitute x = 2

= −8 + 1

Simplify

= −7

Evaluate

The completed table is:

832

−1

−3

−7

Mathspace New South Wales – Year 11 Standard mathspace.co

b Plot the points from the table.

Create a strategy For each point (x, y), locate x on the x-axis and on the y-axis, then mark their intersection.

Apply the idea Plot (−1, 5) by finding x = −1 and y = 5. 5

4 3 2 1 –3

–2

–1

–2 –3 –4 –5 –6 –7

Plot the remaining points. 5

4 3 2 1 –3

–2

–1

–2 –3 –4 –5 –6 –7

c Draw the graph of y = −4x + 1.

Create a strategy Draw a straight line through the plotted points, extending across the Cartesian plane.

12.01 Straight line graphs mathspace.co

833

Example 3 For each set of points plotted on the Cartesian plane, determine if the graph formed by these points is linear.

a 8 6 4 2

x –3

–2

–1

Create a strategy Connect the points with a straight line and check if all points lie on it.

Apply the idea Draw a line through the points (−1, 1), (0, 3), (1, 5), and (2, 7).

y 8 6 4 2

x –3

–2

–1

All points lie on the straight line, confirming the graph is linear.

12.01 Straight line graphs mathspace.co

835

b 8 6 4 2

x –3

–2

–1

Create a strategy Connect the points and check if they form a straight line.

Apply the idea Attempt to draw a straight line through (−1, 1), (0, 0), (1, 1), and (2, 4).

y 8 6 4 2

x –3

–2

–1

The points do not lie on a straight line, indicating the graph is non-linear.

Reflect and check Calculate the rate of change between pairs of points: from (−1, 1) to (0, 0), the change is From (1, 1) to (2, 4), it is

836

. The varying rates confirm non-linearity.

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 2

For each equation: i

Complete the table of values.

Plot the points and sketch the graph.

y = 5x − 2

y = −3x + 6

x −1 0 1 2 y c

⬚

y = x + 4

⬚

For each equation:

⬚

Complete the table of values.

Plot the points and sketch the graph. b

⬚

−2

−1

⬚

−1

⬚

−5

⬚

For each graph shown, state the coordinates where the function crosses the y-axis: y a b y 4

1 –4 –3 –2 –1

838

x −2 0 2 4 y

y = −3x − 1

x −4 0 4 8 y

−1

y = −x − 2

x −2 −1 0 1 y

–1

x 1

1 –4 –3 –2 –1 –1

–2

–3

–4

Mathspace New South Wales – Year 11 Standard mathspace.co

x 1

2 1

–5 –4 –3 –2 –1 –1

2 3 4 5

–2

Determine if the points form a linear graph: a

–3

1 –3 –2 –1 –1

–1

x 1

–2

–3

(−3, −1), (−1, 1), (1, 3), (3, 5)

(−2, 3), (−1, 1), (0, 0), (1, 2)

(−2, 4), (0, 2), (2, 0), (4, 0)

A phone plan’s total cost (y) in terms of gigabytes used (x) is given by y = 2x + 15: a

b 12

–2

Determine if the points form a linear graph: a (−2, 0), (0, 2), (2, 4), (4, 6) c

–4

–5

–3

–4

–2

–3

Ex 3

–4 –3 –2 –1 –1

Complete the table of values for x = 0, 1, 2, 3. x

⬚

Plot the points from your table. Does the graph appear to be linear?

A savings plan’s total savings (y) in terms of the number of months (x) is given by y = 3x + 10: a

Complete the table of values for x = 0, 1, 2, 3. x

⬚

Plot the points from your table. Does the graph appear to be linear?

12.01 Straight line graphs mathspace.co

839

A student completes a table for y = 4x − 3 but makes an error: x

−1

−7

−1

Identify and correct the error.

Extend your thinking 14

A water tank’s level (y litres) is recorded at different times (x hours). The recorded data points are (0, 50), (1, 45), (2, 40), and (3, 35): a

Plot the points and determine if the graph is linear.

Explain why this relationship should be linear.

Explain why plotting only two points may not be sufficient to confirm a linear graph, and suggest how many points are ideal.

A student completes a table for

but makes errors, which are then used to plot

points and sketch a graph. The table and graph are: x

−2

Identify and correct the errors in the table.

Plot the corrected points and sketch the correct graph.

y 8

6 4 2

x –3 –2 –1

840

Mathspace New South Wales – Year 11 Standard mathspace.co

12.02

Gradient and intercept

After this lesson, you will be able to… • calculate the gradient of a line from a graph or two given points • interpret the meaning of positive, negative, zero, and undefined gradients • identify the y-intercept of a line from its graph • determine the y-intercept of a line algebraically, given its gradient and a point • construct the graph of a line given its gradient and y-intercept

Gradient Gradient The slope of a line. It is calculated as the gradient of a line segment it contains. If A(x1, y1) and B(x2, y2) are 2 distinct points on a line, the gradient of the line (or line segment AB) is given by

The gradient of a straight line tells us how steep it is and which way it slopes. The letter m is used for gradient. To calculate it, find the ratio of the vertical change (the ‘rise’) to the horizontal change (the ‘run’) between any two different points on the line.

rise is the vertical change in y-coordinates between two points run is the horizontal change in x-coordinates between two points For any two points with coordinates (x1, y1) and (x2, y2) on the line, the gradient is calculated using the formula:

is the gradient of the line

y2 − y1 is the change in y-coordinates (rise) x2 − x1 is the change in x-coordinates (run)

12.02 Gradient and intercept mathspace.co

841

Every straight line has a constant gradient, except for vertical lines, whose gradient is undefined. y

1 –4

–3

–2

–1

–4

–3

–2

–1

–2

–3

–4

Positive gradient (m > 0)

–2

–1

1 –3

Negative gradient (m < 0)

–4

–1

x 1

–4 –3 –2 –1

–1

–2

–3

–4

Zero gradient (m = 0)

x 1

Undefined gradient

Interactive exploration Discover this concept in action online

842

Mathspace New South Wales – Year 11 Standard mathspace.co

mathspace.co

Example 1 Determine the gradient of the lines shown. a

y 4 3 2 1

x –6

–4

–2

–1

Create a strategy Identify the coordinates of two distinct points on the line and use the formula

Apply the idea From the graph, two points on the line are (−1, 2) and (3, 4). Write the formula Substitute (x1, y1) = (−1, 2) and (x2, y2) = (3, 4) Simplify the numerator and denominator Simplify the fraction The gradient is .

12.02 Gradient and intercept mathspace.co

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b 8 6 4 2

x 2

Create a strategy Identify the coordinates of two distinct points on the line and use the formula

Apply the idea From the graph, two points on the line are (1, 5) and (5, 2). Write the formula Substitute (x1, y1) = (1, 5) and (x2, y2) = (5, 2) Simplify the numerator and denominator The gradient is

Example 2 Consider the points A(−1, 3), B(5, −1), and C(2, 1). a Determine the gradient of the line passing through points A, B, and C.

Create a strategy Use the gradient formula to confirm consistency.

844

with any two points on the line, then verify with another pair

Mathspace New South Wales – Year 11 Standard mathspace.co

Apply the idea Let (x1, y1) = (−1, 3) (point A) and (x2, y2) = (5, −1) (point B). Write the formula Substitute the coordinates of A and B Simplify the numerator and denominator Simplify the fraction Verify using points A(−1, 3) and C(2, 1). Substitute the coordinates of A and C Simplify the numerator and denominator Simplify the fraction The gradient of the line through points A, B, and C is

b Construct the graph of the line passing through points A, B and C.

Create a strategy Plot the points A(−1, 3), B(5, −1), and C(2, 1) on the Cartesian plane, then draw a straight line through them.

Apply the idea y 4 A 3 2 C

x –1

1 –1

5 B

–2

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Apply the idea For the gradient of the line: Write the formula Substitute (x1, y1) = (−5, 0) and (x2, y2) = (0, 3) Evaluate The graph shows that the line crosses y-axis at (0, 3), so the y-intercept is c = 3.

Reflect and check Confirm the gradient by calculating it with another pair of points on the line, such as (−10, −3) and (−5, 0). Write the formula Substitute coordinates Evaluate The gradient is , matching the previous result.

Example 4 Consider the points (2, 1) and (4, 4) on a line. a Determine the gradient.

Create a strategy Use the gradient formula

with the given points.

Apply the idea Write the formula Substitute (x1, y1) = (2, 1) and (x2, y2) = (4, 4) Evaluate The gradient is .

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b Determine the y-intercept.

Create a strategy Use the gradient (m) and the coordinates of one of the points (x, y) in the equation y = mx + c. Then, solve for c.

Apply the idea The gradient is

. Use the point (2, 1), so x = 2, y = 1. y = mx + c

Write the formula x=2y=1

Substitute 1=3+c

Evaluate the multiplication

c = −2

Subtract 3 from both sides making c the subject

The y-intercept is −2.

c Construct the graph of the line using the given points. Label the y-intercept on your graph.

Create a strategy Plot the given points (2, 1) and (4, 4) on the Cartesian plane, draw a straight line through them, and identify and label the point where the line crosses the y-axis.

Apply the idea y 5

(4,4)

4 3 2

(2,1)

1 1

–1

–1 –2

(0,–2)

–3

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Mathspace New South Wales – Year 11 Standard mathspace.co

x 4

Practice Ex 1

For each graph or set of points, determine the gradient: a

1 –5 –4 –3 –2 –1 –1

x 1

2 3 4 5

–2

–3

–4

–5

2 1

x 1

850

2 3 4 5

–3 –4

–3

–5

2 1

x 1

–2

2 3 4 5

Ex 2

2 3 4 5

–5 –4 –3 –2 –1 –1

–2

–4 –3 –2 –1 –1

–5 –4 –3 –2 –1 –1

–2

–5 –4 –3 –2 –1 –1

–4 –3 –2 –1 –1

–2

–3

–4

x 1

Calculate the gradient of the lines passing through the points: a

(−3, 2), (1, 6) and (0, 5)

(0, 5), (4, −3) and (6, −7)

(−2, 1), (3, −4) and (5, −6)

(2, 3), (−1, 3) and (8, 3)

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 3

For each set of points or line shown: i

Determine the gradient.

Determine the y-intercept. y

x 1

–5 –4 –3 –2 –1 –1

2 3 4 5

–1

–2

–1

x 1

–3

–4 –3 –2 –1

2 3 4 5

–2

Ex 4

x 1

–5 –4 –3 –2 –1 –1

–2

–4

–5

–2

–6

–3

–7

–4

–8

Given points A(−2, 5) and B(1, −1): a

Calculate the gradient of the line AB.

Determine the y-intercept of the line AB.

Sketch the graph of the line AB, labelling the y-intercept and points A and B.

For each linear equation: i

Determine the gradient..

Determine the y-intercept.

iii

Sketch the graph, labelling the y-intercept.

y = -2x + 5

y = 3x - 2

A line has a y-intercept of (0, −3) and passes through (2, 5): a

Determine its gradient.

Sketch the graph, labelling the y-intercept and the point (2, 5).

12.02 Gradient and intercept mathspace.co

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A hiker climbs a hill that rises 120 metres vertically over a horizontal distance of 300 metres. Calculate the gradient of the hill’s slope.

Extend your thinking 11

A line has a gradient of

and passes through (3, 5):

Determine its y-intercept.

Sketch the graph, labelling the y-intercept.

The cost of renting a bicycle is $10 plus $3 per hour, where y is the total cost in dollars and x is the number of hours: a

Identify the gradient and explain its meaning in this context.

Identify the y-intercept and explain its meaning.

A ramp rises 1.8 metres over a horizontal distance of 2.4 metres: a

Calculate its gradient and explain what it means in this context.

Find the y-intercept and explain what it means in this context.

Harry must find the gradient and the y-intercept. His answers are:

Gradient:

y-intercept: (0, −3)

–1

However, he only got 1 out of 2 parts correct.

–2

–8

–6

–4

–2

x 2

–3 –4 –5 –6 –7

852

Identify which parts of Harry’s answers are incorrect and justify why.

Calculate the correct answers.

A student calculates the gradient of a line through points (−6, 4) and (3, −11) as : a

Explain the student’s error.

Calculate the correct gradient and determine the y-intercept of this line.

Mathspace New South Wales – Year 11 Standard mathspace.co

12.03

Gradient-intercept form

After this lesson, you will be able to… • identify the gradient and y-intercept from an equation in the form y = mx + c • write the equation of a line in gradient-intercept form given its key features • determine the equation of a line from its graph • sketch the graph of a line from its equation in gradient-intercept form • recognise and graph horizontal (y = c) and vertical (x = b) lines

Write the equation in gradient-intercept form The gradient-intercept form of the equation of a straight line is a way to express the relationship between the x-coordinates and y-coordinates of any point on the line. This form directly shows the line’s gradient and where it crosses the y-axis.

y = mx + c y

is the y-coordinate of any point on the line

m is the gradient of the line, which measures its steepness x

is the x-coordinate of any point on the line

c is the y -intercept, which is the y-coordinate of the point where the line crosses the y-axis To write the equation of a non-vertical line in this form, substitute the known gradient (m) and the known y-intercept (c) into the general equation y = mx + c. If the gradient and y-intercept are not directly given but can be determined from a graph, first calculate the gradient using two points on the line and identify the y-intercept by observing where the line crosses the y-axis.

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12.03 Gradient-intercept form mathspace.co

853

Example 1 Write the equation of a line with gradient

and y-intercept −2.

Create a strategy Substitute the given gradient m and y-intercept c into the gradient-intercept form y = mx + c.

Apply the idea y = mx + c

Write the gradient-intercept form Substitute

and c = −2

Simplify

Example 2 Write the equation of the line shown in each graph: a

y 4 3 2 1 –4

–3

–2

–1

x 1

–1 –2 –3 –4

Create a strategy Identify the y-intercept (c) from the graph, the coordinates of two distinct points on the line to calculate the gradient (m) using the formula . Then, substitute m and c into the gradient-intercept form y = mx + c.

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Mathspace New South Wales – Year 11 Standard mathspace.co

y 4 3 2 1 –4

–3

–2

–1

x 1

–1 –2 –3 –4

Apply the idea From the graph, the line crosses the y-axis at (0, −1). Therefore, the y-intercept c = −1. Two distinct points on the line are (0, −1) and (3, −4). Write the gradient formula Substitute (x1, y1) = (0, −1) and (x2, y2) = (3, −4) Evaluate the subtraction = −1 Simplify The gradient is m = −1. Now substitute m =−1 and c = −1 into y = mx + c. y = mx + c Write the gradient-intercept form y = (−1) x + (−1) Substitute m = −1 and c = −1 y = −x −1 Simplify The equation of the line is y = −x − 1.

Reflect and check To verify the equation, substitute the coordinates of the second point, (3, −4), into the equation y = −x − 1. y = −x − 1 −4 = −3 − 1 −4 = −3 − 1 −4 = −4

Write the equation Substitute x = 3 and y = −4 Evaluate Compare the left-hand side and the right-hand side

Both sides of the equation are equal; therefore, the equation is correct.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Apply the idea The equation is y = 3. This is a horizontal line where every point has a y-coordinate of 3. The y-intercept is 3 (the point (0,3)). The gradient is 0. Plot the point (0, 3). Since the gradient is 0, another point could be (1, 3), (2, 3), etc. Draw a horizontal line through these points. y 4

y=3

(0, 3)

(2, 3)

2 1

x –3

–2

–1

d x = −2

Create a strategy Recognise that this equation is in the form x = b, which represents a vertical line. Identify the x-intercept. All points on this line will have this x-coordinate. The gradient is undefined.

Apply the idea The equation is x = −2.

x = –2

This is a vertical line where every point has an x-coordinate of −2. The x-intercept is −2 (the point (−2, 0)).

3 (–2, 2) 2

Plot the point (−2, 0). Since it’s a vertical line, another point could be (−2, 1), (−2, 2), etc. Draw a vertical line through these points.

1 –4

–3

(–2, 0) –2 –1

1 –1

–2

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Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 2

Determine the equations of the following lines in gradient-intercept form: a

b 4

x 2

–4 –3 –2 –1

–1

–2

–3

–4

y 4

3 2

1 2

–4 –3 –2 –1 –1

x 2

Ex 3

862

Sketch the following lines on a Cartesian plane: a

The line y = −x + 3.

A line with gradient 2 and y-intercept −3.

The line x = −2.

A line with gradient

The line

The line y = −2x − 1.

A line with gradient

The line y = 3.

and y-intercept 1.

. and y-intercept 4.

Mathspace New South Wales – Year 11 Standard mathspace.co

–2 –3 –4

Consider the line plotted: a b

By how much does the y-value change as the x-value increases by 1?

Determine the gradient of the line. What does this value represent in terms of the change in the y-value for each unit increase in the x-value?

6 4 2

x –1

For each pair of points: i

Calculate the gradient of the line passing through them.

Write the equation of the line in gradient-intercept form.

(2, 5) and (4, 9)

(−1, 3) and (1, −1)

(0, −2) and (3, 4)

(−2, −3) and (2, 5)

A plumber charges a call-out fee of $75 and then $50 per hour for labour. a

Write an equation for the total cost C (in dollars) in terms of the number of hours h worked, in the form C = mh + b.

Determine the total cost if the plumber works for 3.5 hours.

Extend your thinking 11

Convert the following equations to gradient-intercept form: a

2x − 4y = 8

3x + 6y = 12

3x − 10y = −2

16x + 12y = 10

3y = 12x − 15

−9x + 9y = 27

Consider the lines with the following equations: • Line A: 5x + 3y + 5 = 0 • Line B: 7x + 6y − 3 = 0

Express Line A in the form y = mx + c.

Identify which line is steeper, A or B.

A straight line has gradient −1 and goes through the points (0, 2) and (a, −6): a

Write the equation of the line in the form y = mx + c.

Determine the value of a.

Two lines have equations y = 4x + 1 and y = 4x − 5: a

Describe the relationship between these lines.

Explain why they never intersect.

A student claims the line through (0, 3) and (2, 7) has equation y = 2x + 1: a

Identify the error.

Write the correct equation. 12.03 Gradient-intercept form mathspace.co

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12.04 Modelling linear relationships After this lesson, you will be able to… • model a practical situation using a linear function • interpret the gradient and vertical intercept of a linear model in a real-world context • use a linear model to make predictions through interpolation and extrapolation • identify and describe the limitations of a linear model

Model linear relationships Model A mathematical, conceptual or physical representation that describes, simplifies, clarifies or provides an explanation of the structure, workings or relationships within an object, system or idea. Models can provide a means of testing and predicting behaviour within limited conditions. Models may be physical or exist in digital form.

Many real-world situations can be described or approximated using a linear relationship. This process is called linear modelling. A linear model uses the familiar equation of a straight line, often written in the gradient-intercept form: y = mx + c. In these real-world models, variables often represent quantities like time, distance, or cost, and may be denoted by letters other than x and y. The gradient (m) indicates the rate of change, and the vertical intercept (c) shows an initial value or fixed component. For instance, a delivery service charging according to the model C = 2d + 5 (where C is cost and d is distance in kilometres) has a rate of change of $2 per kilometre and a vertical intercept of $5 (a base fee).

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Unlike mathematical graphs that extend indefinitely, many real-world linear models are restricted to values that make sense in context. For instance, quantities like distance, volume or time cannot be negative, so their graphs typically appear in the first quadrant. However, some physical quantities, such as temperature, can take negative values. When analysing a linear model, always consider the appropriate range of values for the situation.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Example 1 A phone plan has a monthly fee of $25 plus $0.20 per minute of calls. a Model the total cost C in dollars as a function of call time t in minutes.

Create a strategy Use the fixed monthly fee as the vertical intercept (c) and the cost per minute as the gradient (m) to form the linear model C = mt + c.

Apply the idea The fixed monthly fee is the vertical intercept, so c = 25. The cost per minute is the rate of change or gradient, so m = 0.20. C = mt + c

Write the general linear model form

C = 0.20t + 25

Substitute m = 0.20 and c = 25

The model is C = 0.20t + 25.

b What does the gradient represent in this context?

Create a strategy Identify the rate of change from the equation and interpret it in terms of cost and time.

Apply the idea The gradient is 0.20. It represents the cost increasing by $0.20 for each additional minute of calls. This is the rate of change of the cost.

c What does the vertical intercept represent in this context?

Create a strategy Identify the initial value from the equation and interpret it in the context of the phone plan.

Apply the idea The vertical intercept is 25. It represents the fixed monthly fee of $25, which is the cost even if no calls are made (t = 0).

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d Sketch the graph of the model.

Create a strategy Plot the vertical intercept (0,25). Use the gradient

to find another point (for

example, for every 5 minutes, the cost increases by $1). Draw the line, considering that time t cannot be negative.

Apply the idea The vertical intercept is at (0, 25). The gradient is 0.20, which means for every 50 minutes increase in time, the cost increases by 0.20 × 50 = $10. So, another point could be (50, 25 + 10) = (50, 35). Draw a line through (0, 25) and (50, 35), starting from t = 0. Cost (C dollars) 40

(50,35) 30

(0,25) 20

Time (t minutes) 0

100

Example 2 A car’s fuel consumption is modelled by F = −0.1d + 50, where F is fuel in litres and d is distance in kilometres. a Estimate the fuel after 100 kilometres.

Create a strategy Substitute d = 100 into the model.

Apply the idea F = −0.1d + 50 = −0.1(100) + 50

Substitute d = 100

= −10 + 50

Evaluate the product

= 40

Evaluate

After 100 kilometres, 40 litres of fuel remain.

866

Write the given model

Mathspace New South Wales – Year 11 Standard mathspace.co

b What is the maximum distance the car can travel on one full tank of fuel?

Create a strategy Substitute F = 0 into the equation.

Apply the idea A negative fuel amount is impossible. The model is only valid until the fuel runs out. The car runs out of fuel when F = 0. F = −0.1d + 50

Write the equation

0 = −0.1d + 50

Substitute F = 0

0.1d = 50

Add 0.1d to both sides

d = 500

Divide both sides by 0.1

So, the model is not valid for d > 500 km.

c Describe a limitation of the model.

Create a strategy Identify a real-world constraint that the mathematical model does not account for.

Apply the idea A significant limitation is that the model predicts negative fuel for distances greater than 500 kilometres (when F would become 0). Fuel cannot be negative, so the model is only valid for 0 ≤ d ≤ 500.

Example 3 A carpenter charges a call-out fee of $150 plus $45 per hour. a Determine an equation to represent the total amount charged, C, by the carpenter as a function of the number of hours worked, h.

Create a strategy The call-out fee is the vertical intercept (c or b in C = mh + b). The hourly charge is the gradient (m). Use these to form the linear model.

Apply the idea The vertical intercept (call-out fee) is c = 150. The gradient (hourly rate) is m = 45. C = mh + c

Write the general linear model form

C = 45h + 150

Substitute m = 45 and c = 150

The equation is C = 45h + 150.

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Interpolation and extrapolation Lines of best fit can be drawn by hand, but with the use of technology, a regression line can be determined, providing the best summary of the data set. The least-squares regression line is a line of fit that is calculated using technology.

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Once the equation of a line of best fit is obtained, it can be used to make predictions. The following steps can be taken: 1. Substitute the x-value into the equation of the line to calculate the predicted y-value. 2. Evaluate whether the prediction is reliable. Predictions can also be made using the graph of the line of best fit. Starting from the x-axis, move upward to the line of fit and then across to the y-axis to find the predicted value. S

Ice cream sales

400

300

200

100

20 30 Temperature (0C)

These terms describe the range in within which predictions are made: Interpolation Making predictions between known data values. For example, working between 2 known points on a graph to predict a value in between these points. Extrapolation Occurs when the fitted model, such as a line of best fit, is used to make predictions using values that are outside the range of the original data upon which the fitted model was based.

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Extrapolations

Interpolations

The reliability of predictions depends on the correlation strength, whether the data is interpolated or extrapolated, and the number of points in the data set. • Strong correlation + interpolation = reliable prediction • Moderate/weak correlation + interpolation = less reliable prediction • Extrapolation = generally unreliable prediction • More data points increase reliability

Example 4 The equation y = −8.71x + 6.79 is a line of best fit for a bivariate data set. When x = 3.49 predict the value of y.

Create a strategy Substitute the value of x into the equation for the line of best fit.

Apply the idea y = −8.71x + 6.79

Write the equation

= −8.71 × 3.49 + 6.79

Substitute x = 3.49

= −23.6079

Evaluate

Example 5 Chirping crickets can be an excellent indication of how hot or cool it is outside. Different species of crickets have different chirping rates, but for a particular species the following data was recorded:

870

Temperature (°C)

Number of chirps per minute

115

150

176

Mathspace New South Wales – Year 11 Standard mathspace.co

Chirps per minute

220 200 180 160 140 120 100 80 60 40 20

Temperature 0C 0

8 10 12 14 16 18 20 22 24 26 28 30 32

a According to the graph, what is the temperature when the crickets make 140 chirps each minute?

Create a strategy Move from the frequency of chirps to the temperature to find the x-value for a given y-value. Start with y = 140 and find the x-value.

Apply the idea A line can be drawn horizontally from the y-value, 140, to the line of the relationship, then another line vertically down to the x-axis. Chirps per minute 220 200 180 160 140 120 100 80 60 40 20

Temperature 0C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 The answer is 20°C.

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b According to the graph, how many chirps per minute will the crickets make if the temperature is 27°C?

Create a strategy Find the value of y when x = 27.

Apply the idea Draw a line from x-value, 27, vertically to the line of the relationship, then another line horizontally left to the y-axis. Chirps per minute 220 200 180 160 140 120 100 80 60 40 20

Temperature 0C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

The answer is 210 chirps per minute.

c According to the graph, how many chirps are the crickets making each minute if the temperature is 19°C?

Create a strategy Find the value of y when x = 19.

Apply the idea Draw a line from x-value, 19, vertically to the line of the relationship, then another line horizontally left to the y-axis.

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Mathspace New South Wales – Year 11 Standard mathspace.co

Chirps per minute 220 200 180 160 140 120 100 80 60 40 20

Temperature 0C 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

The answer is 130 chirps per minute.

d Which prediction is more reliable, part (b) or (c)?

Create a strategy Remember that for reliability of predictions: • Strong correlation + interpolation = reliable prediction • Moderate/weak correlation + interpolation = less reliable prediction • Extrapolation = generally unreliable prediction • More data points increase reliability

Apply the idea On the initial scatterplot, the minimum number of chirps per minute is 77 and the maximum number of chirps per minute is 176. This means that part (b) uses extrapolation and part (c) uses interpolation. Part (c) is more reliable than part (b).

Reflect and check There are only four points, so none of the predictions are incredibly reliable. However, the correlation is strong, so that improves the reliability somewhat.

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Is each statement true or false? a

The line of best fit can only be used for interpolation, not extrapolation.

Extrapolation using the line of best fit is generally more reliable than interpolation.

Extrapolation can lead to inaccurate predictions if the underlying trend does not continue beyond the range of the known data points.

Scatterplots may not accurately represent the relationship between variables if there are outliers, non-linear relationships, or if the data is too sparse.

It is not important to consider data validity and limitations when using interpolation or extrapolation.

Considering data validity and limitations helps ensure that the predictions made are accurate and reliable, and that they do not lead to incorrect conclusions or decisions based on the data.

Does each graph show interpolation or extrapolation? a

10 20 30 40 50 60 70 80 90

20 10

x 0

10 20 30 40 50 60 70 80 90

x 0

10 20 30 40 50 60 70 80 90

Find the y-value for the given line of best fit and x-value: a

y = 0.45x + 7.1, at x = 3

y = 1.75x − 3.2, at x = −2.4

y = −2x − 7, at x = 5

12.04 Modelling linear relationships mathspace.co

875

Practice 8

Ex 1

The amount of medication M (in milligrams) in a patient’s body gradually decreases over time t (in hours) according to the equation M = 1050 − 15t: a

After 61 hours, how many milligrams of medication are left in the body?

Calculate the number of hours it will take for the medication to be completely removed from the body.

A streaming service charges a base fee of $30 plus $0.15 per movie watched. a

Suggest a model for the total cost C in dollars as a function of movies watched m.

What does the gradient represent?

What does the vertical intercept represent?

Sketch the graph of the model in a Cartesian plane.

A racing car starts the race with 250 litres of fuel. From there, it uses fuel at a rate of 5 litres per minute: a

Ex 2

876

Complete the table of values: Number of minutes passed, x

Amount of fuel left in tank, y

⬚

Determine an equation relating the number of minutes passed, x, and the amount of fuel left in the tank, y.

Describe how the amount of fuel in the car is changing over time.

For which values of x does the graph accurately model the amount of fuel left in the tank?

A battery’s charge is modelled by B = −0.2t + 80, where B is charge in percent and t is time in hours: a

Estimate the charge after 50 hours.

Predict the charge after 500 hours.

Describe a limitation of the model.

Calculate the charge after 100 hours.

Determine when the battery is fully depleted.

Mathspace New South Wales – Year 11 Standard mathspace.co

Consider the pattern of blue boxes:

Complete the table of values: Number of columns (c)

Number of blue boxes (b)

⬚

Determine an equation that describes the relationship between the number of blue boxes, b, and the number of columns, c.

Calculate the number of blue boxes if this pattern were to continue for 38 columns.

If there were 45 blue boxes, how many columns would there be?

After Mae starts running, her heart rate in beats per minute increases at a constant rate as shown in the table: Number of minutes passed, t

Heart rate, H

Determine Mae’s heart rate after 12 minutes.

Calculate the change in heart rate per minute.

Determine an equation that describes the relationship between the number of minutes passed, t, and Mae’s heart rate, H.

Explain what the gradient represents in this context.

The height (H in cm) of a young plant after w weeks is modelled by the equation H = 0.75w + 4: a

Calculate the height of the plant after 6 weeks.

Determine how many weeks it will take for the plant to reach a height of 13 cm.

The number of fish in a river is approximated over a five-year period. The results are shown in the table: Time in years (t)

Number of fish (F)

4800

4600

4400

4200

4000

3800

Calculate the gradient of the line.

Explain the meaning of the gradient in this context.

Determine the value of F when the line crosses the vertical axis.

Determine an equation for the line using the given values.

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The graph shows the conversion between temperatures in Celsius and Fahrenheit:

100

Use the graph to convert 10°C into Fahrenheit.

0°C is 32°F. For every 1°C increase, by how much does the Fahrenheit temperature increase?

80 60 50

Would 80°F be above or below normal body temperature (approximately 37°C)?

Determine the rule for conversion between Celsius (°C) and Fahrenheit (°F).

30 10

0 17

Ex 3

15 20 25 30 35 40

The volume of water (V in litres) in a tank draining over time (t in minutes) is given by V = −12t + 300: a

What is the initial volume of water in the tank?

Calculate when the tank will be empty.

Describe one limitation of this linear model if considering very large values of t (for example, significantly after the tank is empty).

A freelance graphic designer charges a $75 consultation fee for new projects and then $50 per hour for design work: a

Write a linear equation for the total cost (C) for h hours of design work.

Calculate the total cost for a project that requires 4.5 hours of design work.

If a client’s budget is $325, how many hours of design work can they afford?

Are these predictions an extrapolation or an interpolation? a

878

A prediction for the y-value when x = 5 is made from the data set: x

A prediction for the y-value when x = 33 is made from the data set: x

102

112

A prediction for the x-value when y = 95.69 is made from the data set: x

94.4

96.4

94.4

97.8

94.9

95.9

94.4

Mathspace New South Wales – Year 11 Standard mathspace.co

Ex 4

A line of best fit for this data set is y = 1.26x − 57.01 When x = 3.49, predict the value of y. Is this an example of interpolation or extrapolation? Round your answer to two decimal places:

51.2

25.4

38.9

58.6

38.2

60.8

26.3

28.5

5.4

Scientists conducted a study to see people’s reaction times after having different amounts of sleep. The results are recorded in the table: Number of hours sleep (x)

1.1

1.5

2.1

2.5

3.5

Reaction time in seconds (y)

4.66

4.1

4.66

3.7

3.6

3.4

The data has been graphed along with a line of best fit found by eye. a

Predict the reaction time for someone who has slept for 5 hours.

Predict the number of hours someone sleeps if they have a reaction time of 4 seconds.

Reaction time 5 4 3 2 1

Number of hours sleep 0

The number of fish in a river is measured over a five year period: Time in years (t)

Number of fish (F)

1903

1994

1995

1602

1695

1311

The data has been graphed along with a line of best fit found by eye. a

Predict the number of years until there are no fish left in the river.

Predict the number of fish remaining in the river after 7 years.

According to the line of best fit, how many years are there until there are 900 fish left in the river?

2000 1800 1600 1400 1200 1000 800 600 400 200

t 2 4 6 8 10 12 14 16 18 20

12.04 Modelling linear relationships mathspace.co

879

Ex 5

The number of people at the beach can be affected by the temperature. After observing a particular beach, the data was recorded in a table: Temperature (°C)

Number of people at the beach

120

180

230

Sketch a line of best fit by eye.

Approximately, how many people are at the beach when the temperature is 20°C?

200

Approximately, what is the temperature when there are 200 people at the beach?

150

Estimate how many people are at the beach when the temperature is 19°C?

100

Is the prediction for 30°C or 19°C more reliable?

c d e

225

People

175 125 75 25

Temperature 0C 12 14 16 18 20 22 24 26 28

One litre of gas is raised to various temperatures and its pressure measured is shown in a table: Temperature (K)

300

302

304

308

310

312

314

316

318

Pressure (Pa)

2400

2416

2434

2462

2478

2496

2512

2526

2546

The data has been graphed along with a line of best fit.

880

Use the line of best fit to predict the pressure when the temperature is 306 K.

Is the prediction in part (a) an example of interpolation or extrapolation?

Is the prediction in part (a) reliable?

Is it reasonable to use the line of best fit to predict pressure within each of these range of temperatures? i

300 ≤ Temp ≤ 320

300 ≤ Temp ≤ 600

iii

0 ≤ Temp ≤ 320

280 ≤ Temp ≤ 340

Mathspace New South Wales – Year 11 Standard mathspace.co

2580 Pressure (Pa) 2560 2540 2520 2500 2480 2460 2440 2420 2400

Temperature (K) 296 300 304 308 312 316

An ice cream shop records the number of ice creams sold and the maximum temperature of each day for two weeks: Temp (°C)

Sales

The data has been graphed along with a line of best fit: a

Predict the number of ice creams sold if the maximum temperature for a day is 38°C.

Is it reasonable to use the line of best fit to predict ice cream sales within each of these range of temperatures? ii

20 ≤ Temp ≤ 40

iii

20 ≤ Temp ≤ 60

0 ≤ Temp ≤ 60

90 70

Predict the maximum temperature if 10 ice creams were sold.

0 ≤ Temp ≤ 40

Sales

100

50 40 30 20 10

Temp (0C) 22 24 26 28 30 32 34 36 38 40

A plane’s altitude (A) is measured at several times (t) during its descent: Time (t seconds)

200

400

1700

Altitude (A metres)

9000

7815

7092

593

The data has been graphed along with a line of best fit drawn 9000 by eye. a

Predict the altitude of the plane 100 seconds into the descent.

8000

Predict the altitude of the plane 500 seconds into the descent.

6000

For how many seconds has the plane been descending when it is at an altitude of 7500 metres?

4000

How many seconds did the plane take to descend to the ground?

7000

5000

3000 2000 1000 t 0

200 400 600 800 1000 1200 1400 1600

12.04 Modelling linear relationships mathspace.co

881

A car company looked at the relationship between how much it had spent on advertising (A) and the amount of sales (S) each month over several months. The data has been plotted on a graph and a line of best fit drawn: a

Two points on the line are (3200, 300) and (5600, 450). Find the gradient of the line of best fit.

S 800 700

The line of best fit can be written in the form S = mA + c, where m is the gradient, c is the vertical intercept, S is sales in thousands of dollars, and A is the amount spent advertising costs.

600 500 400 300

Find the value of c, the vertical intercept of the line. d

200

Use the line of best fit to estimate the number of sales next month if $4800 is to be spent on advertising.

100 A 0

1000 2000 3000 4000 5000 6000 7000 8000

The depth a diver, x, has descended below the surface of the water is plotted against her lung capacity, y: y 105 100 95 90 85 80

x 1

882

Does the line of best fit have a positive or negative gradient?

Find the gradient of the line.

Find the equation of the line of best fit.

Use the line of best fit to estimate the lung capacity, y, at a depth of 4 metres.

Mathspace New South Wales – Year 11 Standard mathspace.co

Extend your thinking 29

A mobile phone salesman earned $600 in a particular week during which he sold 26 phones, and $540 in another week during which he sold 20 phones: a

Determine a linear equation to represent the weekly earnings of the salesman, E, as a function of the number of phones sold, n.

Determine how much the salesman will earn in a week during which he sells 36 phones.

Gardening Company A charges a fixed call-out fee plus a certain amount for every hour of work performed on any particular garden. One customer pays $380 for 4 hours of work. Another customer pays $490 for 6 hours of work: a

Using a linear equation as a model, determine an expression for the amount of money, P, the company charges in terms of t, the number of hours worked.

How much would the company charge for a job involving 7 hours of work?

Gardening Company B charges a fixed fee of $130 for every job, plus $65 for every hour of labour. David usually takes 5.5 hours to complete his garden work but injured his hand and is unable to. Which company should he call to work on his garden?

Explain a limitation of Company A’s pricing model.

A cafe notices that for every 5°C increase in temperature outside, they sell 20 fewer cups of coffee a day. On a certain day when the temperature was 15°C, they sold 250 cups of coffee: a

If the temperature outside is 25°C, predict how many cups of coffee the cafe will sell.

Determine the temperature at which the cafe will sell no coffee.

The cafe decides to implement a marketing strategy that increases their cups of coffee sales by two cups per degree Celsius increase in temperature. On a 15°C day, they sell 275 cups of coffee. Determine a new equation for the model.

Explain why the new model may not hold for extreme temperatures.

The population of City A can be modelled by the equation P = 80000 + 1500t, where t is the number of years since 2020. The population of City B is modelled by P = 60000 + 2000t: a

In how many years will the populations of the two cities be equal?

If the population growth of City A continues at this rate, in what year will its population reach 150000?

12.04 Modelling linear relationships mathspace.co

883

Several cars underwent a brake test and y their age, x, in years was measured 70 against their stopping distance, y, in metres. The scatterplot shows the 60 results and a line of best fit of y = 3x + 30 that approximates the 50 positive correlation: a

Use the line of best fit to estimate 40 the stopping distance of a car that is 7.5 years old. 30

Is the predicted value in part (a) reliable? Explain your answer.

20 10

x 0

9 10 11 12

Are these predictions examples of interpolations or extrapolations? Justify your answer. a

Using the provided table that shows a car’s fuel efficiency at different speeds, a prediction for the fuel efficiency at 70km/h is made from the data set.

Using the table of a company’s annual revenue over the past 5 years, a prediction for the next year is made from the data set.

Using this scatterplot that shows a student’s test scores at different hours of study, a prediction for the test score at 15 hours of study is made from the data set.

Speed (km/h)

Fuel efficiency (L/100 km)

6.5

5.8

5.2

4.9

5.4

100

6.1

Year

Revenue ($ million)

2015

2016

2017

2018

2019

22 Test score

90 80 70 60 50 40 30 20 10

Hours of study 0

884

Mathspace New South Wales – Year 11 Standard mathspace.co

10 12 14 16 18

Discuss the limitations of using interpolation and extrapolation for making predictions in the context of climate change.

The distance in kilometres, x, of several locations from the equator and their temperature, y in °C, is measured and graphed on a scatterplot: y

50 45 40 35 30 25 20 15 10 5

x 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Find the equation of the line of best fit shown.

If your distance from the equator is 4000 km, estimate the temperature, y, using the equation in the previous part.

Is the predicted value in part (b) reliable? Explain your answer.

The fleet manager for the Australian Automotive Association wants to estimate how car maintenance costs, C (in hundreds of dollars), are related to the distance, K (in thousands of kilometres), driven each year. The data collected and the line of best fit are shown on a scatterplot: a

The equation of the least-squares regression C line for the graph is C = 9.9K + 146.2 550 Interpret what the coefficient of K represents in this context.

Explain what the constant term in the equation represents in this context.

Is this interpretation of the constant term reasonable given the context? Explain your answer.

500 450 400 350 300

Use the least-squares equation C = 9.9K + 146.2 to predict the annual maintenance cost for a car that is driven 36 000 km per year.

200

Is this predicted value reliable? Explain your answer.

100

Is there any reason to believe that there is a causal relationship between distance driven and maintenance costs? Explain your answer.

250

150

K 0

12.04 Modelling linear relationships mathspace.co

885

Based on the given scatterplot and line of best fit, are either interpolation or extrapolation reliable? Explain your answer. y 9 8 7 6 5 4 3 2 1

x 0

Investigation: Spreadsheets and linear relationships Investigate online

mathspace.co

Did you know?

Spreadsheets are powerful tools for modelling data and visualising trends! By organising numbers into tables and charts, they help users analyse information, spot patterns, and make informed decisions. 886

Mathspace New South Wales – Year 11 Standard mathspace.co

12.05

Direct variation

After this lesson, you will be able to… • identify and describe a direct variation relationship • represent direct variation using the equation y = kx • determine the constant of variation (k) from given information • use the direct variation equation to solve problems • recognise and interpret the graph of a direct variation as a line through the origin

Direct variation A proportional relationship where one quantity directly varies with respect to a change in another quantity. This implies that if there is an increase (or decrease) in one quantity then the other quantity will experience a proportionate increase (or decrease). Constant of variation In a direct variation relationship y = kx, the non-zero constant k is the constant of variation. It represents the rate of change in this context. Direct variation describes a relationship where one variable is directly proportional to another. If a variable y is directly proportional to a variable x, an increase in x results in a proportional increase in y, and a decrease in x results in a proportional decrease in y. This is denoted as y ∝ x, where ∝ indicates direct proportionality. The relationship is expressed as an equation:

y = kx y

is the dependent variable

is the independent variable

i s the constant of variation. This is a non-zero constant representing the gradient of the line and the rate of change of y with respect to x.

The equation y = kx produces a straight-line graph passing through the origin (0, 0). The gradient of the line equals k, the constant of variation, which also represents the rate of change in these relationships.

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Exploration Consider a scenario where the cost of fuel varies directly with the volume purchased. For example, suppose 20 litres of fuel cost $40. 1. If the relationship between cost (C) and volume (V) is a direct variation, what is the general form of the equation connecting them? 2. Using the given information, determine the constant of variation (k). What does this constant represent in the context of fuel cost? 3. Sketch a graph of cost (vertical axis) versus volume (horizontal axis) for this relationship. What are the key features of this graph (for example, where does it start, what is its shape)? 4. How does the gradient of your sketched graph relate to the constant of variation you determined? Explain its meaning in this scenario.

Problems involving direct variation require determining the constant of variation (k) using a known pair of values (x, y). Once k is found, the equation y = kx can be used to find unknown values of x or y. This method is used in contexts where one quantity varies directly with another, for example, earnings proportional to hours worked or distance proportional to time at a constant speed. A graph representing direct variation, will always be a straight line that passes through the origin (0, 0). In other words, its vertical intercept will always be zero. The gradient of the line will be equal to the constant of variation. The diagram shows a linear graph, where variable y is directly proportional to variable x. y 4

x 0

888

Mathspace New South Wales – Year 11 Standard mathspace.co

Example 1 The distance D (in kilometres) travelled by a cyclist is directly proportional to the time T (in hours). If the cyclist travels 45 km in 3 hours, determine the constant of variation and represent the relationship as an equation and a graph for 0 ≤ T ≤ 5.

Create a strategy Express the relationship as D = kT. Use the given values to determine k, write the equation, and sketch the graph. The constant of variation k will represent the cyclist’s speed (rate of change).

Apply the idea D = kT Write the equation 45 = k × 3

Substitute D = 45 and T = 3

k = 15

Divide both sides by 3 making k the subject

The constant of variation is k = 15. This means the cyclist travels at a rate of 15 km/h. The equation is D = 15T. The graph of D = 15T is a straight line through the origin (0, 0) with gradient 15.

D (km) 70 60 50

(3, 45) 40 30 20 10

T (hours) 0

12.05 Direct variation mathspace.co

889

Example 2 The cost C (in dollars) of hiring a tutor is given by the formula C = 50T + 20, where T is the time in hours. Determine whether this relationship represents a direct variation between cost and time

Create a strategy A direct variation has the form y = kx (or C = kT in this context), which means its graph must pass through the origin (0, 0). Check if C = 0 when T = 0.

Apply the idea C = 50T + 20 Write the equation = 50 × 0 + 20

Substitute T = 0

= 20

Evaluate

Since C = 20 when T = 0, the graph does not pass through the origin (0, 0). Therefore, the relationship is not a direct variation. It is a general linear relationship with a vertical intercept of 20.

Example 3 The cost C (in dollars) of repairing a bicycle is directly proportional to the time T (in hours) spent on it. A repair taking 3 hours costs $255. a Determine the cost of a repair that takes 2.5 hours.

Create a strategy Express the relationship as C = kT. Determine k using the given values, then determine the cost for T = 2.5.

Apply the idea Determine the constant of variation k. C = kT Write the equation 255 = k × 3 k = 85

Substitute C = 255 and T = 3 Divide both sides by 3 making k the subject

The equation is C = 85T. Here, the constant of variation, $85 per hour, is the rate of change of cost with respect to time. Now, determine the cost for T = 2.5 hours: C = 85T Write the equation = 85 × 2.5

Substitute T = 2.5

= 212.50

Evaluate

The cost is $212.50.

890

Mathspace New South Wales – Year 11 Standard mathspace.co

b Determine the time required for a repair costing $357.

Create a strategy Substitute C = 357 into the equation C = 85T.

Apply the idea C = kT Write the equation 357 = 85T T= 4.2

Substitute C = 357 Divide both sides by 85 making T the subject

The time required is 4.2 hours.

Example 4 The earnings E (in dollars) of a worker are directly proportional to the hours H worked. If the worker earns $180 for 6 hours, determine the hours worked to earn $270 and sketch the graph for 0 ≤ H ≤ 10 to verify.

Create a strategy Express the relationship as E = kH. Determine k using the given values, then determine the hours for E = 270, and sketch the graph.

Apply the idea Determine the constant of variation k: E = kH Write the equation 180 = k × 6 k = 30

Substitute E = 180 and H = 6 Divide both sides by 6 making k the subject

Determine the hours worked to earn $270: E = kH Write the equation 270 = 30H H=9

Substitute E = 270 and k = 30 Divide both sides by 30 making H the subject

The worker earns $270 in 9 hours. The graph of E = 30H is a straight line through (0, 0) with gradient 30.

300

E (dollars) (9, 270)

250 200 150

(6, 180)

100 50

H (hours) 0 1 2 3 4 5 6 7 8 9

12.05 Direct variation mathspace.co

891

Ex 3

Ex 4

The force F (in newtons) applied by a spring is directly proportional to its extension E (in metres). If a force of 50 N corresponds to an extension of 0.2 m, calculate the force for an extension of 0.5 m.

The cost C (in dollars) of repairing a car is directly proportional to the time T (in hours) spent. A repair taking 4 hours costs $280:

Calculate the cost for a repair taking 1.5 hours.

Calculate the time for a repair costing $490.

The earnings E (in dollars) of a barista are directly proportional to the hours H worked. If $225 is earned in 5 hours: a

Calculate the hours needed to earn $405.

Sketch the graph of the relationship for 0 ≤ H ≤ 10.

A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth of the diver over 4 minutes: Number of minutes passed, x Depth of diver in metres, y

0 0

1 1.4

2 2.8

3 4.2

4 5.6

Calculate the increase in depth each minute.

Determine a linear equation for the relationship between the number of minutes passed, x, and the depth, y, of the diver.

Calculate the depth of the diver after 6 minutes.

Calculate how long the diver takes to reach 12.6 metres beneath the surface.

The graph shows the amount of Euros that can be bought with Australian Dollars on a particular date:

How many Euros can 20 AUD buy?

How much Australian currency is required to buy 6 Euros?

Calculate the number of Euros that $1 AUD buys.

Write the rule for conversion between AUD (A) and Euros (E).

Euros

6 4 2

AUD 2

8 10 12 14 16 18 20

Beth’s income is based solely on the number of hours she works, and she is paid a fixed hourly wage. She earns $25 per hour. Let I represent Beth’s income after working h hours: a

Determine an equation relating h and I.

Calculate Beth’s income when she works 25 hours.

Calculate the number of hours that Beth must work to earn $125.

Explain the meaning of the gradient in this context.

12.05 Direct variation mathspace.co

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The cost, C, for a business to operate, can be expressed in terms of h, the total number of hours it has operated for. The cost is $120 an hour: a

Sketch a graph that displays the cost against time.

Determine the gradient of the line.

Determine an equation relating h and C.

Calculate the total cost for the business to operate for 28 hours.

The cost C (in dollars) of electricity is directly proportional to the energy used E (in kilowatt-hours). The table shows some values: Energy (kWh), E

Cost ($), C

2.5

⬚

Determine the missing cost for 30 kWh.

Write the equation relating C and E.

The speed S (in km/h) of a conveyor belt is directly proportional to the number of items N processed per minute. If 60 items per minute correspond to a speed of 12 km/h, calculate the speed for 100 items per minute.

The number of calories C burned is directly proportional to the time T (in minutes) spent exercising. If 240 calories are burned in 30 minutes: a

Determine the constant of variation.

Calculate the calories burned in 45 minutes.

The distance D (in metres) a car travels is directly proportional to the time T (in seconds). If D = 25T, sketch the graph for 0 ≤ T ≤ 10.

The cost C (in dollars) of printing posters is directly proportional to the number of posters N printed. If 50 posters cost $200: a

Determine the constant of variation.

Calculate the cost for 75 posters.

What does the constant of variation represent in this context?

Sketch the graph of the relationship for 0 ≤ N ≤ 100.

Extend your thinking 22

The cost C (in dollars) of data usage is directly proportional to the data D (in GB). A user pays $15 for 5 GB. If another plan charges $22.50 for 7.5 GB. which plan is more costeffective for 10 GB?

894

Mathspace New South Wales – Year 11 Standard mathspace.co

The power P (in watts) of an electrical circuit is directly proportional to the current I (in amperes). The table shows some values: Current (A), I

Power (W), P

⬚

Determine the missing power for 8 A.

Determine the constant of variation.

A bakery’s revenue R (in dollars) from selling cupcakes is directly proportional to the number of cupcakes N sold. If 100 cupcakes generate $250: a

Determine the constant of variation.

Calculate the number of cupcakes sold for $400 revenue.

Explain why this model may not hold if the bakery offers discounts for bulk purchases.

Sketch the graph of the relationship for 0 ≤ N ≤ 200.

Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body. Consider the given graph and table, which show the amount of medication in the bloodstream for the liquid and capsule form of the medication: Liquid form

Capsule form Time (mins), t

Amount in blood (mgs), A

24.6

42.3

77.7

10 5

t 0

In which form is the medication absorbed more rapidly?

Explain your reasoning.

12.05 Direct variation mathspace.co

895

12 Chapter review 1

The graph shows a straight line. What is the y-intercept of this line? A

−1

y 4 3 2 1 –4

–3

–2

–1

–1 –2 –3 –4

What is the gradient of the line with the equation y = −4x +7? A

−4

Which of these equations represents a line with a gradient of A

and a y-intercept of −5?

y = 2x − 5

For the graph shown: i

Identify the y-intercept.

Determine the gradient of the line. y 4 3 2 1 –4

–3

–2

–1 –1 –2 –3 –4

896

Mathspace New South Wales – Year 11 Standard mathspace.co

x 1

For the equation y = 3x − 4: i

Complete the table of values.

Plot the points and sketch the graph. x

−1

⬚

A small business’s profit (P in dollars) is recorded at different months (m). The recorded data points are (0, 100), (1, 140), (2, 180), and (3, 220): a

Plot the points and determine if the graph is linear.

Explain why this relationship might be linear in its early stages.

A student completes a table for

but makes errors, which are then used to plot

points and sketch a graph. The table and graph are: x

−3

y 7 6 5 4 3 2 1

x –3 –2 –1

Identify and correct the errors in the table.

Plot the corrected points and sketch the correct graph.

Calculate the gradient of the line passing through the points (−2, 1), (2, 7), and (0, 4).

Given points P(−1, 6) and Q(2, 0):

Calculate the gradient of the line PQ.

Determine the y-intercept of the line PQ.

Write the equation of the line PQ in the form y = mx + c.

Sketch the graph of the line PQ, labelling the y-intercept and points P and Q.

For each equation: a

Calculate its gradient (as a positive value representing the steepness) and explain what it means in this context.

Write the equation of the line representing the slope’s height, assuming the top of the slope starts at y = 100 when x = 0 and descends from there. Chapter 12 review mathspace.co

897

Samira must find the gradient, the y-intercept, and write the equation of the graph shown. y 6

Her answers are: i

Gradient:

y-intercept: (0, 6)

iii

Equation:

3 2

However, she only got 1 out of 3 parts correct.

1 –7 –6 –5 –4 –3 –2

x 1

–1 –1

Identify which parts of Samira’s answers are incorrect and justify why.

Calculate the correct answers.

Determine the equation of the following lines in gradient-intercept form: a

1 –4 –3 –2 –1

–1

x 1

–4 –3 –2 –1

–2

–3

–4

–4 –3 –2 –1

–1

–2

–3

–4

Mathspace New South Wales – Year 11 Standard mathspace.co

x 1

–1

–2

898

x 1

For each equation: i

Determine the value of the gradient, m.

Determine the y-intercept, c.

y = −3x + 5

4x − 6y + 18 = 0

e 14

2y = 8x − 10

6x − 3y = 2x + 5y − 16

A straight line has gradient −2 and goes through the points (0, 5) and (b,−3): a

Write the equation of the line in the form y = mx + c.

Determine the value of b.

Consider the line y = −3x + 6 graphed on a Cartesian plane: a

If you start at the y-intercept and move 2 units right, what is the resulting y-coordinate? Show your reasoning.

y 7 6

Interpret the gradient’s effect on the line’s direction and steepness compared to a line with gradient −1.

5 4 3 2 1

x –1

–1

A gym membership charges a sign-up fee of $50 plus $0.20 per visit: a

Suggest a model for the total cost C in dollars as a function of visits v.

What does the gradient represent?

What does the vertical intercept represent?

Sketch the graph of the model in a Cartesian plane for up to 100 visits.

A phone’s battery life is modelled by L = −0.25h + 90, where L is battery life in percent and h is hours of use: a

Estimate the battery life after 40 hours of use.

Predict the battery life after 400 hours of use.

Describe a limitation of the model.

Calculate the battery life after 120 hours of use.

Determine when the battery is fully depleted.

Chapter 12 review mathspace.co

899

A submarine starts at the surface and descends. The table shows its depth over 4 minutes: Time (minutes), m

Depth (metres), D

2.5

5.0

7.5

10.0

Calculate the increase in depth each minute.

Determine a linear equation for the relationship between time m and depth D.

Calculate the depth of the submarine after 7 minutes.

Calculate how long it takes the submarine to reach 22.5 metres.

A delivery driver earned $550 in a week making 30 deliveries, and $490 in another week making 24 deliveries: a

Determine a linear equation to represent the weekly earnings of the driver, E, as a function of the number of deliveries made, d.

Determine how much the driver will earn in a week making 40 deliveries.

Plumbing Service X charges a fixed call-out fee plus an hourly rate. One client pays $320 for 3 hours of work. Another client pays $470 for 5 hours of work: a

Using a linear equation as a model, determine an expression for the amount of money, P, Plumbing Service X charges in terms of t, the number of hours worked.

How much would Service X charge for a job involving 4.5 hours of work?

Plumbing Service Y charges a fixed fee of $150 for every job, plus $60 for every hour of labour. Sarah needs 2.5 hours of plumbing work. Which service should she call?

Explain a limitation of Service X’s pricing model for very short jobs like 30 minutes.

A line of best fit has been drawn to A approximate the relationship between sea temperature, T in °C, and the area 800 of healthy coral, A in hectares, in a 700 particular location. Two particular points, (2, 700) and (23, 175), lie on the line: 600 a

Find the equation of the line of best fit.

Using the line of best fit, find the area A of coral expected when the sea temperature is 26°C.

Is the predicted value in part (b) reliable? Explain your answer.

500 400 300 200 100 T 0

900

Mathspace New South Wales – Year 11 Standard mathspace.co

The table shows data on the number of kilograms of litter collected each week in a national park x weeks after the park managers started an anti-littering campaign: Weeks (x) Kilograms of litter collected (y)

1 3.9

2 3.7

3 3.5

4 3.1

5 3.1

6 2.9

7 2.7

Construct a scatterplot for this data.

Sketch a line of best fit on your scatterplot.

Find the equation for your line of best fit.

Use your equation of the line of best fit to find the number of kilograms of litter collected 8 weeks after the start of the anti-littering campaign.

Is the predicted value in part (d) reliable? Explain your answer.

The cost C (in dollars) of buying bananas is directly proportional to the weight W (in kilograms). If 3kg costs $10.50, calculate the cost of 5kg.

The cost C (in dollars) of hiring a typist is directly proportional to the time T (in hours) spent. A job taking 5 hours costs $325: a

Calculate the cost for a job taking 2.5 hours.

Calculate the time for a job costing $195.

The cost C (in dollars) of internet data is directly proportional to the data D (in GB). Plan Alpha offers $20 for 8GB. Plan Beta offers $27 for 9GB. Which plan is more cost-effective if you need 12GB?

A florist’s revenue R (in dollars) from selling bouquets is directly proportional to the number of bouquets N sold. If 80 bouquets generate $2800: a

Determine the constant of variation.

Calculate the number of bouquets sold for $1750 revenue.

Explain why this model may not hold if the florist has to discard unsold flowers at the end of the day.

Sketch the graph of the relationship for 0 ≤ N ≤ 100.

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Big ideas Financial calculations and budgeting enable informed purchasing decisions by modelling costs, including taxes, loans, and ongoing expenses, to optimise personal financial planning.

13 Purchases and budgets Chapter outline 13.01 13.02 13.03 13.04 13.05

Percentage increase and decrease Profit and loss Purchase options Purchase a car On-road and running costs of a car Investigation: Getting on the road Investigation: Choose the right car Investigation: Car costs with spreadsheets 13.06 Household bills 13.07 Prepare a personal budget Investigation: Create a budget Chapter 13 review

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13.01 Percentage increase and decrease After this lesson, you will be able to… • apply percentage increase and decrease to find a new value. • calculate the final value after repeated percentage changes. • calculate the Goods and Services Tax (GST) for an item. • determine the pre-GST price of an item from a GST-inclusive price. • solve a variety of practical problems involving percentage change and GST.

Increase or decrease by percentage Percentage decrease The reduction in a value compared to its initial value, expressed as a percentage.

Percentage increase The growth in a value compared to its initial value, expressed as a percentage.

A percentage represents parts out of 100. For example, a score of 75 out of 100 is 75%, which is equivalent to

or 0.75.

Percentages are used to adjust amounts by increasing or decreasing them. An increase might apply to prices during a sale markup, while a decrease could reflect a discount. These calculations are useful for determining costs, savings, or profits. To increase an amount by y%, two methods can be used: • Calculate y% of the original amount by multiplying by • Multiply the original amount by

, and add it to the original amount.

in a single step.

To decrease an amount by y%, two methods can be used: • Calculate y% of the original amount and subtract it from the original amount. • Multiply the original amount by in a single step.

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Example 1 Increase 1300 by 40% using these steps: a Find 40% of 1300.

Create a strategy Turn 40% into a fraction and multiply by 1300.

Apply the idea

= 520

Evaluate

b Add the extra amount to the original to get the new total.

Create a strategy Add the extra amount to the original amount.

Apply the idea Original amount + Extra = 1300 + 520 = 1820

Substitute the values Evaluate

Example 2 Decrease 200 by 25% using these steps: a Find 25% of 200.

Create a strategy Turn 25% into a fraction and multiply by 200.

Apply the idea

= 50

Evaluate

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Repeated percentage change Repeated percentage changes happen when an amount is adjusted by percentages multiple times in a row. Each change uses the result of the previous one as the new starting amount. To calculate successive changes, apply each percentage one after another. For an increase by y1%, multiply by

. For a decrease by y2%, multiply by

. The order of changes matters

because each percentage is based on the current amount. This is useful for real-life situations, like stacking discounts during a sale or adjusting prices over time.

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Example 3 A pair of shoes priced at $70 is on sale for 30% off, followed by an additional 10% off. Calculate the final sale price.

Create a strategy Apply each percentage decrease sequentially, using the result of the first discount as the starting amount for the second.

Apply the idea Price after first discount = 70% of $70

Decrease $70 by 30%, equivalent to finding 70% Convert 70% to

= 49

Evaluate

Price after second discount = 90% of $49 Decrease $49 by 10%, equivalent to finding 90% Convert 90% to = 44.1

Evaluate

The final sale price is $44.10.

Example 4 A business purchases an item for $500 and increases the price by 20% for retail. Due to low demand, the retail price is then reduced by 15%. Calculate the final selling price.

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Create a strategy Apply the percentage increase followed by the percentage decrease sequentially.

Apply the idea Retail price = 120% of $500

Increase $500 by 20%, equivalent to finding 120% Convert 120% to

= 600 Final selling price = 85% of $600

Evaluate Decrease $600 by 15%, equivalent to finding 85% Convert 85% to

= 510

Evaluate

The final sale price is $510.

Example 5 An investment of $2500 increases in value by 8% in the first year, and then its new value increases by a further 12% in the second year. Calculate the single overall percentage increase over the two years.

Create a strategy First, calculate the final value of the investment after both increases. Then, determine the total increase in dollars. Finally, express this total increase as a percentage of the original investment amount.

Apply the idea Final value = $2500 × (1 + 0.08) × (1 + 0.12) Calculate the final value using single-step multipliers = $2500 × 1.08 × 1.12

Evaluate the factors inside the brackets

= $3024

Evaluate to find the final value

Total increase = $3024 − $2500 = $524

Calculate the total increase in dollars Evaluate the subtraction Express the increase as a percentage of the original value

Overall percentage increase = 0.2096 × 100%

Evaluate the fraction

= 20.96%

Evaluate the final percentage

The overall percentage increase is 20.96%. Note that this is not the same as simply adding the percentages (8% + 12% = 20%).

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What percentage should a quantity be multiplied by in order to increase the quantity by: a 17% b 5% c 40% d 9.4% e

33.51%

a b c

Find 12% of 1800 and subtract your answer from the original amount of 1800. Calculate 88% of 1800. Are your answers to part (a) and part (b) the same? Explain why this is.

What percentage should a quantity be multiplied by in order to decrease the quantity by: a 10% b 57% c 5% d 75% e

20.7%

27.275%

15.7%

Calculate the 10% GST applicable on the following original prices: a $80 b $37 c $95

112%

0.8%

$59

Calculate the sales price, including GST, for an item with an original price of $20.

Practice Ex 1

Increase 2500 by 20% using these steps: a Find 20% of 2500. b Add the amount to the original to get the new total.

Ex 2

Decrease 400 by 15% using these steps: a Find 15% of 400. b Subtract the amount from the original to get the new total.

Increase: a 700 by 10%. c e

$7 by 106%. 260 L by 0.8%.

45 cm by 71%.

22.6 kg by

A bag of potatoes weighs 50 kg. When more potatoes are added, the weight of the bag increases by 35%. Find the new weight of the bag of potatoes.

A basket of goods was valued at $43.70 in January 2011. The inflation rate for the year was 8%. What is the expected cost of the basket of goods in January 2012?

A bag of rice weighs 100 kg. When some rice is poured out, the weight of the bag decreases by 30%. Find the new weight of the bag of rice.

A tennis racket marked at a price of $90 is advertised to be selling at 45% off the marked price. Find the discounted price.

What does 10% off the price of a car mean?

Steph is going to buy a dress that is marked as 75% off. The original price was $36: a What is the value of the discount? b What is the price that Steph will pay for the dress? 13.01 Percentage increase and decrease mathspace.co

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Duncan is going to buy a hat that is marked as 25% off. The original price was $30: a Find the discount amount. b Find the discounted price that Duncan will pay for the hat.

Ex 3

A laptop priced at $120 is on sale for 20% off, followed by an additional 15% off. Calculate the final sale price.

Ex 4

A retailer buys a gadget for $300 and increases the price by 15% for sale. Due to a promotion, the sale price is then reduced by 10%. Calculate the final selling price.

Ex 5

An investment of $3500 grows by 7.5% in the first year, and then its new value increases by an additional 12.5% in the second year. Calculate the single overall percentage increase over the two years, rounded to two decimal places.

A dining table selling for $1000 is discounted at 5% and then again at 6%: a What is the price of the dining table after the first discount is applied? b What is the price of the dining table after the second discount is applied? c The percentage discount is changed by adding the previous percentages (5% + 6%) to get 11%. Find the new discounted price. d Is a single discount of 11% equal to successive discounts of 5% and 6%? e What is the single discount rate that is equivalent to successive discounts of 5% and 6%? Give your answer as a percentage to one decimal place.

The price of a heater selling for $234 is initially discounted by 14% and later marked up by 14%. What is the final sales price?

The price of a phone was increased by 40% and then again by 40%: a What was the overall percentage increase? b Is the overall percentage increase when the price is increased by 40% twice equal to 2 × 40%?

The price of a dress was reduced by 30% and then again by 30%: a What was the overall percentage decrease? b Is the overall percentage decrease when the price is reduced by 30% twice equal to 2 × 30%?

Find the overall percentage increase of the following: a The price of a shirt was increased by 97% and then again by 97%. b The price of a dress was reduced by 20% and then increased by 44%. c The price of a dress was increased by 19% and then reduced by 14%.

The price of a phone was reduced by 25% and then again by 25%. Find the overall percentage decrease.

The Run4Fun charity race is increasing in popularity. A year ago, 60 000 people registered to run and this number is expected to increase by 6% this year and then by another 8% next year. How many people are expected to run next year?

Sarah receives a bill of $45 for a service fee and $15 for materials from a technician. If GST is applied to the service fee only, calculate the total bill.

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Ex 6

John received a bill of $68 for consultation and $10 for herbal supplements from a medical professional. If GST is only applied to the consultation fee, what is the total bill that John must pay?

Calculate the GST on an item with a sales price (including GST) of: a $40 b $83.90

The marked price of a particular grinder sold in a store is $350. The store offers a 12% discount, followed by an additional 7% discount for cash payment.

Extend your thinking 31

After discounting, a clothes dryer sold for $855. Calculate the original price if the discount was: a 20% b 20% and 16% successively

Which offer gives the greater discount? a • Offer 1: A single discount of 13% b

• Offer 2: Successive discounts of 6% and 11% • Offer 1: A single discount of 16% • Offer 2: Successive discounts of 5% and 11%

A PC originally sells at $4700: a Calculate its sale price after successive discounts of 6% and 9%. b Calculate its sale price after successive discounts of 9% and 6%. c Does the order of discounting affect the sale price? d Calculate its sale price after successive discounts of 8%, 7% and 5%.

Bob has invested heavily in a certain company. One day, it was announced that their factory was damaged by a fire and its share price fell by 40%. However, when it was announced the next day that the company had insurance and wouldn’t be too badly affected, the shares recovered by 40%: a Express the share price after the original announcement as a percentage of the previous day’s price. b Express the share price after the insurance announcement as a percentage of the price after the original fire announcement. c Calculate the final share price as a percentage of the price before either announcement. d Hence, evaluate the percentage change over the two days. e Is this change an overall increase or decrease?

Calculate the original price of the item, before the application of GST for each of the following sales price: a $99 b $56.45 c $937.70

Fruits and vegetables do not incur GST of 10%. If the total shopping bill is $109, and fruits and vegetables amount to $62, what is the cost of the other items prior to adding GST?

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13.02 Profit and loss After this lesson, you will be able to… • calculate profit or loss from given cost and selling prices. • determine the selling price or cost price given profit or loss information. • calculate profit or loss as a percentage of the cost price. • solve multi-step practical problems involving profit and loss.

Profit and loss Profit The excess of total revenue above the total cost of selling goods or services. Loss The difference between the total cost and the total revenue of producing and selling goods or services where the cost is greater than the revenue. Profit occurs when the revenue from selling an item exceeds the cost of acquiring it, while a loss results when the cost exceeds the revenue. These concepts are critical in business to assess financial performance. The cost price is the amount paid to acquire an item, and the selling price is the amount for which it is sold. The marked price is the labelled price, which may differ from the selling price due to discounts. Revenue refers to the total income earned, typically the selling price multiplied by the quantity sold. Profit is calculated as the selling price minus the cost price, expressed as Selling price − Cost price. A loss occurs when this difference is negative, and the absolute value represents the loss. The break-even point is reached when the selling price equals the cost price, resulting in neither profit nor loss. Percentage profit or loss is expressed relative to the cost price, calculated as

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Example 1 Calculate the selling price of an item with a cost price of $333 and a profit of $269.

Create a strategy Add the profit to the cost price to find the selling price.

Apply the idea Selling price = Cost price + Profit Write the formula = 333 + 269

Substitute the values

= $602

Evaluate

The selling price is $602.

Example 2 Xavier bought a property for $371 000. In the first year, it increased in value by 12%, but in the second year, it decreased in value by 6%. If running costs during the two years amounted to $1500, calculate Xavier’s profit or loss at the end of two years, rounded to the nearest cent. a Calculate the amount earned from the changing value of the property, rounded to the nearest cent.

Create a strategy Determine the property’s value after the first year’s increase and second year’s decrease, then subtract the original cost to find the amount earned.

Apply the idea Value after first year = 112% of $371 000

Increase by 12%, equivalent to 112% Convert 112% to

= $415 520 Value after second year = 94% of $415 520

Evaluate Decrease by 6%, equivalent to 94% Convert 94% to

= $390 588.80

Evaluate

Amount earned = $390 588.80 − $371 000 Subtract original cost = $19 588.80

Evaluate

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Calculate the profit (or loss) when: a The selling price is $344 and the cost price is $288. b The selling price is $154 and the cost price is $363. c The money received is $5526 and expenses are $2396. d The money received is $3836 and expenses are $5640. e The selling price is $343 and the cost price is $134. f The selling price is $1697 and the cost price is $2853. g The money received is $470.78 and expenses are $333.27. h The money received is $4451.88 and expenses are $4520.75.

Calculate the cost price when: a The selling price is $357 and the profit is $199. b The selling price is $2983 and the loss is $2386. c The selling price is $258 and the profit is $139. d The selling price is $549.62 and the profit is $525.67. e The selling price is $2050 and the loss is $1427.

Calculate the expenses when: a The money received is $388 and the profit is $192. b The money received is $8567 and the loss is $1798. c The money received is $244 and the profit is $235. d The money received is $3915 and the loss is $1848. e The money received is $883.21 and the loss is $410.79.

Calculate the total revenue if: a The expenses are $322 and the profit is $437. b The expenses are $2973 and the loss is $2052. c The expenses are $241 and the profit is $228. d The expenses are $586.88 and the profit is $635.01. e The expenses are $3842 and the loss is $3082.

Every item in the store is sold at 8% more than its cost. If a product was bought from the manufacturer for $360, calculate: a The sale price b The profit

Every item in the store is sold at 9% more than its cost. If a product is sold for $110, calculate: a The cost price b The profit

A game retailer sells new games at a markup of 33% above the cost price and old games at a markup of 18%. Calculate the total amount paid if Sally bought the new Super Zora 5 game which has a cost price of $90, and the old Super Zora 3 game which has a cost price of $31.

Maria bought a car for $5400 and sold it two years later, making a loss of 15%. How much did she sell it for?

Amelia bought a truck for $17 300 and sold it four years later, making a loss of 25%. How much did she sell it for?

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Ex 2

James bought 2 pens for $9.20 each. He then sold one of the pens for $10.71 and the other for $5.92. a What was the total cost of the two pens? b How much did he receive for the two pens? c Hence, find James’s profit (or loss).

Buzz bought a mobile phone for $170 and sold it 7 months later, making a loss of 30% on the purchase price. Find the selling price.

Sophie purchased a motorcycle for $12 500 and sold it three years later, incurring a loss of 20%. Calculate the selling price.

Liam bought a house for $425 000. In the first year, its value increased by 10%, but in the second year, it decreased by 5%. If maintenance costs over the two years totaled $2000, calculate Liam’s profit or loss at the end of two years, rounded to the nearest cent. a Calculate the amount earned from the changing value of the house, rounded to the nearest cent. b Calculate Liam’s total profit or loss, rounded to the nearest cent.

In its opening month, the local pizza store sold 161 large pizzas at $15 each and 152 small pizzas at $5 each. The store’s expenses for the month included electricity costs of $29, water costs of $20, rent of $284, wages of $1164 and ingredient costs of $253. Calculate: a The total revenue b The total expenses c The profit (or loss) for the month

Extend your thinking 21

A manufacturer is selling its old factory machinery for $34 320, which represents a loss of 12%. At what price did the manufacturer purchase the machinery?

Calculate the percentage profit if an item is sold for: a 3.5 times its cost price b 1.5 times its cost price

A retailer purchased a refrigerator for $270 and sold it for $310 three years later. Calculate the average annual profit as a percentage of the cost price, expressing your answer as a percentage rounded to two decimal places.

Liz bought a chest of drawers for $230 and sold it two years later for $179. a Calculate her loss in dollars. b Calculate her percentage loss, correct to two decimal places.

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13.03

Purchase options

After this lesson, you will be able to… • calculate costs associated with credit card use, including interest and minimum payments. • determine the deposit, balance, and repayment schedule for a lay-by purchase. • analyse the costs of buy now, pay later schemes, including late fees. • compare the total cost of different purchasing methods for a given item.

Credit cards Credit card A method for borrowing funds to purchase goods and services. Banks or financial institutions agree to loan an amount of money to an individual up to a pre-approved limit, and the individual agrees to pay back the amount plus any interest and fees that are charged. Loan An amount of money borrowed from a bank or other financial institution. Balance The difference between the total debit entries and the total credit entries of an account during a financial period. Interest Money paid or received in return for using or lending money. Interest rate A percentage at which interest is charged or paid. Repayment A set amount, paid at regular intervals over the period of a loan, to pay off a loan. Minimum payment The lowest amount of money required for a borrower to pay off their credit card balance each month to remain in good standing with the credit card company. Debit cards withdraw directly from a bank account and can only be used to pay if there is enough money in the account, as the money is withdrawn immediately. Debit cards can be used in person and online in some cases. Credit cards allow people to buy goods and services that may be too expensive to buy in a one-off purchase otherwise or for purchases that can’t be done with cash or debit such as purchases online or from another country. Also, credit cards can have rewards programs such as cash back or travel points that make purchasing using the credit card beneficial. 13.03 Purchase options mathspace.co

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Credit cards are a form of loan, in the form of a continuous line of credit, where consumers are allowed to have a continuing balance of debt if they cannot repay the full balance. However, credit card interest rates are often very high and if the full balance is not paid off, the customer can see their owing balance rise very quickly. The interest rate is normally written as a rate per annum however, the interest may actually be calculated daily, weekly, biweekly, or monthly. The periodic or nominal interest rate is calculated by dividing the annual interest rate by the number of periods per year.

A = P (1 + r)n A is the balance with interest P is the closing balance r is the periodic interest rate

n is the number of periods per year

Example 1 The opening balance on a credit card is $1600, and purchases of $631 and repayments of $419 are made during the month. a If the credit card company requires a minimum payment of 8% of the closing balance, find the minimum payment required to be rounded to the nearest cent.

Create a strategy Determine the closing balance and multiply by the minimum payment rate of 8% or 0.08 as decimal.

Apply the idea Determine the closing balance by adding the opening balance and purchases and subtracting any payments made. Closing balance = 1600 + 631 − 419 = 1812 Determine the minimum payment by multiplying the closing balance by the minimum payment rate. Minimum payment = 1812 × 0.08 = $144.96

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Multiply closing balance by 8% Evaluate

Lay-by is a method of payment, where an initial deposit is made to secure an item for purchase and the balance is paid off at a later date, usually after a number of repayments. The initial deposit may be a fixed amount or a percentage of the total price, and sometimes includes a small administrative fee. Once the total amount is paid, the item can be taken home. If payments are made on time, then there is no additional cost beyond the administrative fee. However, if payments are not made on time or the customer changes their mind, then lay-by can be much more expensive than paying cash or using a debit card because of cancellation fees.

Example 2 Maria purchased on lay-by an item valued at $180. If she paid a deposit of 19% and paid the balance off with fortnightly payments of $26, how many fortnights are there until she receives ownership of the item, rounded to the nearest integer?

Create a strategy First, find the amount owing after the deposit is subtracted, and then divide it by the amount of each fortnightly payment.

Apply the idea Find the amount owing: Amount owing = Item value - deposit

Write the formula

= 180 − (180 × 0.19)

Substitute the values

= 180 − 34.20

Evaluate the multiplication

= $145.80

Evaluate

Find the number of fortnights: No. of fortnights

Divide $145.80 by $26 =6

Evaluate and round

Rounding up to the nearest integer, Maria will own the item after 6 fortnights.

Reflect and check The owing balance could also be determined by calculating 180 × 0.81 because if 19% was paid in the deposit, then 81% is left to pay. Amount owing = 180 × 0.81 Write the expression = 145.80

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Evaluate

Example 3 Andre buys a new TV for $1500 with a buy now-pay later plan. The plan involves five equal payments and no administrative fees or interest. However, if he misses a payment, an initial fee of $15 is charged. If the payment is not received within a fortnight, another fee of $5 will also apply. He pays the first two payments on time but is three weeks late on the third payment and one week late on the fourth payment. He pays the last payment on time. a Determine the total amount of fees Andre had to pay.

Create a strategy There are no fees on three of the payments. There are two fees on one payment and one fee on the other payment. These fees all need to be added up.

Apply the idea Fees on third payment = Initial fee + More than a fortnight fee = 15 + 5

Substitute the values

= 20

Evaluate

Fees on fourth payment = Initial fee = 15 Total fees = Fees on third payment + Fees on fourth payment

Write the formula Substitute the value Write the formula

= 20 + 15

Substitute the values

= 35

Evaluate

The total amount of fees is $35.

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b Determine the percentage of the original cost he paid in fees, rounded to one decimal place.

Create a strategy To find the percentage, the fees need to be divided by the original cost then multiplied by 100%

Apply the idea Percentage

Write the formula Substitute the values = 2.3%

Evaluate and round

The fees are 2.3% of the original cost.

Reflect and check 2.3% is fairly minimal compared to other options like credit cards which often charge around 20% if payments are late.

Example 4 A store offered an interest-free period for 23 months on all purchases. Valentina purchased a $322 treadmill by paying an initial $27 deposit followed by 8 monthly instalments. She was also charged an account-keeping fee of $3 per month. a Determine the amount of each instalment to the nearest cent.

Create a strategy First, find the amount owing and divide it by the number of instalments, and then add the monthly account-keeping fee.

Apply the idea Find the amount owing: Amount owing = Item value - deposit

Write the formula

= 322−27

Substitute the values

= $295

Evaluate

Find the size of each instalment: Size of each instalment

Divide $295 by 8 instalments = 36.88 + 3 Evaluate the division = $39.88

Evaluate

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a b

xplain the main difference between a credit card and a debit card in terms of funding E the purchase. Identify one advantage of using a credit card over a debit card. I dentify the condition under which no additional fees are incurred in an interest-free buy now-pay later plan. Describe what happens to a credit card balance if the minimum payment is not met.

Practice 5

Ex 1

6 The opening balance on a credit card is $2000, and purchases of $500 and repayments of $350 are made during the month. a If the credit card company requires a minimum payment of 8% of the closing balance, determine the minimum payment required. b If the minimum payment is made, find the owing balance after one month of interest at 18% p.a., rounded to the nearest cent. 7

Ex 2

Calculate the closing balance on a credit card with opening balance $1068, if purchases to the amount of $802 and repayments to the amount of $643 are made during the month.

In the past month, Tracy made purchases to the amount of $482 and repayments to the amount of $114 on her new credit card, which charges interest at 3% p.a. compounded monthly. a Determine the closing balance before interest. b Determine the closing balance after interest. c If the credit card company requires a minimum payment of 5% of the closing balance, what is the minimum payment Tracy will have to pay? d If Tracy only pays the minimum amount, what is her balance owing?

8 Han purchased an item on lay-by that was valued at $340. If he paid a deposit of $5 and paid the balance off with weekly payments of $30, after how many whole weeks will he receive ownership of the item? 9

These people purchased different items on lay-by. Determine how much more they still need to pay for the item: a Maya’s item was valued at $165. She paid a deposit of $16.00. b Darwin’s item was valued at $418. He paid a deposit of 8%. c James’ item was valued at $110. The deposit required was 5% and the lay-by fee was $17.

Amelia purchased an item on lay-by that was valued at $110. If she paid a deposit of 22% and paid the balance off with monthly payments of $38, after how many full months will she receive ownership of the item?

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These people purchased different items on lay-by. Determine the size of their regular payments given that the item is purchased on lay-by: a Valentina’s item was valued at $200. She paid a deposit of 15% and paid the balance off with regular weekly payments over 14 weeks. b Dave’s item was valued at $160. He paid a deposit of $30 and paid the balance off with regular monthly payments over 3 months.

Ex 3

12 Samantha buys a new laptop for $2000 with a buy now-pay later plan. The plan involves six equal payments and no administrative fees or interest. However, if she misses a payment, an initial fee of $20 is charged. If the payment is not received within two weeks, another fee of $7 will also apply. She makes the first three payments on time, but is two weeks late on the fourth payment and one week late on the fifth payment. She makes the last payment on time. a Determine the total amount of fees Samantha has to pay. b Determine the percentage of the original cost she paid in fees.

Ex 4

A gym offered a membership plan with a one-time joining fee of $100 and a monthly fee of $20 for 12 months. Sheryll joined the gym by paying the joining fee and plans to pay the monthly fee for the entire year. a Determine the average monthly cost throughout the year, including the joining fee. b Determine the total cost of the membership for one year.

Mae purchased a $4084 sofa on a deferred payment plan. She paid a $622 deposit, followed by nothing for the first year and then 33 fortnightly instalments of $134. a How much did Mae pay for the sofa in total? b How much interest did she pay? c What was the annual rate of interest charged for buying on a deferred payment plan, rounded to two decimal places?

15 Dan purchased a desk, valued at $878, on interest-free terms for 21 months, with an initial $80 deposit and 9 monthly payments. A penalty fee of 10% of the balance owing is charged for instalments that are not paid on time. a Determine the monthly instalment. b Determine the penalty fee due if Dan is late in paying the third monthly instalment. 16

The sale price of a bike is $800. Vincent chooses to purchase the item on a credit card, with 16 days interest-free. His card has an interest rate of 14% per annum, compounded daily. a Vincent pays $483 on his payment due date. How much does he still owe? b Vincent pays the remaining balance 22 days later. How much does he pay overall for the item? c How much more does Vincent pay by using his credit card instead of cash?

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Extend your thinking 17

Beth purchased a $3200 second-hand car on deferred payment terms, and was given the choice of two payment plans: • Plan 1: 7% deposit with monthly instalments of $104 over 3 years. • Plan 2: $362 deposit with 77 weekly instalments of $63. Which plan is most affordable for Beth? Support your answer with appropriate calculations.

Alexia is interested in buying a new computer valued at $1818.09. She doesn’t have the money to pay up front, but the store has an option to pay by hire purchase, with 12 equal monthly payments. a Alexia can only afford a monthly payment of $165. If the store offers a simple interest rate of 15% p.a., is this rate feasible for Alexia given her budget? Calculate the monthly payment and determine if it fits within her budget. b If Alexia proceeds with the hire purchase at the 15% interest rate, how much total interest does she pay compared to paying with cash?

Irene wants to buy a second-hand car for $1685 and is deciding between using her credit card or taking out a loan. The interest rates and additional fees for a credit card are shown in the table: a If Irene decides to apply for a credit card and pay with that, how much will she spend on fees in the first year?

Credit card interest rates and fees Application Fee

$200

Annual Fee

$80

Interest (p.a.)

17.75%

Compounding interval

Daily

Irene pays $825 during each monthly payment interval (or the remaining balance if less than $825) using a credit card with daily compounding at 17.75% p.a. A record of the payments and owing balance has been started in this table: Balance (before interest)

Balance (after interest)

Payment

Balance (owing)

March

$1685.00

$1710.59

$825.00

$885.59

April

$885.59

May b c d

Complete the table. ow much does Irene pay using a credit card, including fees? H If Irene uses a credit card with 40 interest-free days and pays her balance within the interest-free period, how much does she pay including fees?

The sale price of a television is $581. Noah chooses to purchase the television on a credit card, with 7 days interest-free. His card has an interest rate of 26.75% p.a., compounded daily. a Noah pays $208 on the day of purchase. How much does he still owe? b Noah wants to avoid paying more than $610. How many full days, n, from the date of purchase does Noah have before the total costs exceed $610?

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13.04

Purchase a car

After this lesson, you will be able to… • calculate simple interest and repayment amounts for a car loan. • calculate stamp duty for a vehicle using a set of rules or a table. • identify various on-road costs such as registration and insurance. • determine the total upfront cost of purchasing a vehicle. • solve problems comparing different car purchase scenarios.

Car purchase Simple interest The interest accumulated when the interest payment in each period is a fixed percentage of the principal (the initial lump sum of money). The simple interest formula is given by I = PRn, where I is the interest earned, P is the principal value invested, R is the rate of interest and n is the number of time periods over which the interest is applied. Purchasing a car involves multiple costs critical for budgeting. These include the purchase price, insurance, registration and loan repayments if financed. The purchase price is the car’s cost, paid outright or financed via a loan. A loan involves borrowing a principal and repaying it with interest over a term. Car loans often use simple interest, calculated as:

I = PRn I interest P principal R interest rate per period (as a decimal) n number of periods Total loan amount is P + I where the repayment amount is total loan amount divided by the number of repayment periods. Online loan calculators provide precise repayments but simple interest gives reasonable estimates. Insurance includes compulsory third-party (CTP) insurance (≈$500 annually, required for registration), optional comprehensive insurance ($800 to $2500) and non-compulsory third-party insurance ($200 to $400). Registration in NSW costs $350 to $800 annually.

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Example 1 Marie takes out a car loan for $14 000 at a fixed simple interest rate of 10.5% per annum over 5 years. a Calculate the total interest paid.

Create a strategy Use the simple interest formula I = PRn, where P is the principal, R is the interest rate per year, and n is the number of years.

Apply the idea I = PRn

Write the formula

= $14 000 × 0.105 × 5

Substitute P =$14 000, R = 10.5% = 0.105 and n = 5

= $7350

Evaluate

b Calculate the total loan amount paid.

Create a strategy Add the principal and interest to find the total loan amount.

Apply the idea Total loan amount = P + I

Sum principal and interest

= $14 000 + $7350 Substitute P = $14 000, and I = $7350 = $21 350

Evaluate

c Calculate the monthly repayment amount rounded to two decimal places.

Create a strategy Divide the total loan amount by the number of monthly repayments over 5 years.

Apply the idea Years to months = 5 × 12 = 60 months Monthly repayment

Convert years to months Evaluate Write the formula Substitute the values

= $355.83

Evaluate and round

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d Calculate the fortnightly repayment amount.

Create a strategy Divide the total loan amount by the number of fortnightly repayments over 5 years. 1 year is equivalent to 26 fortnights.

Apply the idea Years to fortnights = 5 × 26 = 130 fortnights Fortnightly repayment

Convert years to fortnights Evaluate Write the formula Substitute the values

= $164.23

Evaluate and round

Example 2 John purchases a car with a market value of $30 000. He pays $500 for CTP insurance, $1000 for comprehensive insurance and $400 for registration. The stamp duty is $900. Calculate the total upfront cost of purchasing the car.

Create a strategy Sum the purchase price, CTP insurance, comprehensive insurance, registration and stamp duty.

Apply the idea Total cost = $30 000 + $500 + $1000 + $400 + $900 Add all cost components = $32 800

Evaluate

The total upfront cost is $32 800.

Reflect and check Verify by listing costs: purchase price ($30 000), CTP ($500), comprehensive ($1000), registration ($400), stamp duty ($900). Summing confirms $32 800.

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b $59 000

Create a strategy Apply $1350 plus $5 per $100 or part thereof above $45 000.

Apply the idea Excess value = $59 000 − $45 000 = $14 000 Number of $100 units

Calculate value above $45 000 Evaluate Divide excess by $100

= 140

Evaluate

Stamp duty on excess = 140 × $5 = $700

Multiply by rate per unit Evaluate

Total stamp duty = $1350 + $700 = $2050

Add flat rate and excess duty Evaluate

Example 4 In another region, vehicle stamp duty is 2% of the purchase price up to $30 000, plus 14% for every dollar between $30 000 and $40 000, plus 5% for every dollar over $40 000. a Complete the table of stamp duty values: Vehicle value Stamp duty ($)

$0 ⬚

$30 000 600

$45 000 ⬚

$60 000 ⬚

$75 000 3750

Create a strategy Calculate stamp duty for each vehicle cost based on the given rates, applying each percentage to the appropriate value range.

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Apply the idea For $0: Stamp duty for $0 =2% of $0

Apply 2% to zero

=$0

Evaluate

For $45 000: Stamp duty for $30 000 = $600

Given value

Excess for $40 000 = $40 000 − $30 000 = $10 000

Calculate value above $30 000 Evaluate

Stamp duty on excess = 0.14 × 10000

Apply 14% rate

= $1400

Evaluate

Excess for over $40 000 = $45 000 − $40 000 = $5000

Calculate value above $40 000 Evaluate

Stamp duty for over $40 000 = 0.05 × 5000

Apply 5% rate

= $250

Evaluate

Stamp duty = $600 + $1400 + $2250 = $2250

Sum both parts Evaluate

For $60 000: Excess 1 = $40 000 − $30 000 = $10 000

Calculate value between $30 000 and $40 000 Evaluate

Excess 2 = $60 000 − $40 000 = $20 000

Calculate value above $40 000 Evaluate

Stamp duty on $30 000 = $600

Use previous value

Stamp duty on $10 000 = 0.14 × 10 000

Apply 14% rate

= $1 400

Evaluate

Stamp duty on $20 000 = 0.05 × 20 000

Apply 5% rate

= $1000

Evaluate

Stamp duty for $60 000 = $600 + $1400 + $1000 = $3000 Vehicle value Stamp duty ($)

$0 ⬚

$30 000 600

Sum both parts Evaluate

$45 000 2250

$60 000 3000

$75 000 3750

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b The piecewise graph shows the stamp duty for cars from $0 to $100 000:

Stamp duty ($1000s)

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

Vehicle cost ($1000s) 0

10 20 30 40 50 60 70 80 90 100

Use the graph to determine, rounded to the nearest $100: • The stamp duty on a $80 000 car. • The cost of a car with stamp duty of $5000.

Create a strategy Read the graph, noting axes in thousands, to find the stamp duty at $80 000 and the vehicle cost at $5000 stamp duty, rounding to the nearest $100.

Apply the idea Stamp duty at x = 80 ≈ 4 (in $1000s) = $4000

Read y-value at x = 80 Convert to dollars

For a $80 000 car: $4000 Vehicle cost at y = 5 ≈ 100 (in $1000s) = $100 000 For stamp duty of $5000: $100 000

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Read x-value at y = 5 Convert to dollars

Ex 4

In NSW, car stamp duty is calculated as $3 per $100 or part thereof for vehicles valued at $44 999 or less, and a flat $1350 plus $5 per $100 or part thereof above $45 000 for vehicles valued at $45 000 or more. a Complete the table of stamp duty values: Vehicle value ($) Stamp duty ($) b

30 000 900

50 000

70 000

90 000 3600

The piecewise graph shows the stamp duty for cars from $0 to $100 000: Stamp duty ($1000s) 4 3.5 3 2.5 2 1.5 1 0.5

Vehicle cost ($1000s) 0

10 20 30 40 50 60 70 80 90 100

Use the graph to determine, rounded to the nearest $100: i

The stamp duty on a $85 000 car.

The cost of a car with stamp duty of $2100.

Calculate the monthly repayments for a $10 000 car loan, given that the fixed simple interest rate is 8.6% p.a. and the loan period is 2 years.

For a loan of $11 000, with a fixed simple interest rate of 9.3% p.a. and a loan period of 10 years, find the cost of one repayment instalment if it is paid: a Weekly b Monthly c Quarterly

If Han takes out a car loan worth $25 000 with a fixed, simple interest rate of 4.8% p.a. and wants to make monthly repayments over a loan period of 6 years, calculate: a The monthly interest rate as a percentage b The amount of interest he pays c The total amount he pays

If stamp duty on a vehicle is $80 for the first $5000 of the purchase price plus 5% for every dollar over $5000, calculate the stamp duty payable on a vehicle with a purchase price of: a $9000 b $17 000 c $78 750

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Ian wishes to purchase a car valued at $50 000. He has $25 000 in savings and intends to take out a car loan to cover the remainder of the cost. After researching, he finds a lender that offers a fixed simple interest rate of 7.5% p.a. and a loan period of 4 years. If Ian wants to pay off the loan in monthly instalments, calculate: a The principal amount of the loan b The total amount of interest on the loan c The total loan amount to be repaid d His monthly repayments e The stamp duty payable on the car. Use the following NSW rates: • For vehicles valued at $44 999 or less: $3 per $100 or part thereof. • For vehicles valued at $45 000 or more: $1350 plus $5 per $100 or part thereof for the value above $45 000.

Stamp duty charges based on the value of a car are listed in this table: Car value $800 or less $800 to $30 000 $30 000 to $35 000 $35 000 or more

Stamp duty $40 5% $1500 + 12% of amount over $30 000 6%

Calculate the stamp duty on a car valued at: a $756 b $29 900 e $32 150 16

$106 500

$1160

Stamp duty in one state is based on the market value of the vehicle, regardless of the price paid. The rates are: • 2.5% of the market value, for vehicles up to $52 009, or • $1301 plus 5% of every dollar over $52 009 Calculate the stamp duty, correct to the nearest dollar, on a vehicle with a market value of: a $52 100 b $18 000 c $111 800

Extend your thinking 17

A loan of $20 000 has an interest rate of 12.4% p.a. for a period of 5 years. a Is the total interest paid lower when you make fortnightly, monthly, or weekly repayments? Explain your reasoning. b If the period of the loan is reduced from 5 years to 1 year, is the total interest paid unchanged, less, or more? c If the original loan amount is tripled to $60 000 but the interest rate and period remain constant at 12.4% p.a. and 5 years respectively, what happens to the weekly repayments?

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A buyer takes out a loan for 80% of a $45 000 car at 7% p.a. simple interest over 4 years. The NSW stamp duty is $3 per $100 or part thereof for vehicles valued at $44 999 or less, and $1350 plus $5 per $100 or part thereof above $45 000. CTP insurance costs $500. a Calculate the total cost, including loan principal, interest, stamp duty, and CTP insurance. b How much would be saved on interest if the loan period is reduced to 3 years?

Two cars are financed: • Car A ($35 000, loan of $20 000 at 6.5% p.a. for 5 years) • Car B ($60 000, loan of $40 000 at 8% p.a. for 3 years) NSW stamp duty applies: $3 per $100 or part thereof up to $44 999; $1350 plus $5 per $100 or part thereof above $45 000. a Calculate the total cost (loan principal, interest, stamp duty) for each car. b Which car has the lower monthly repayment, and by how much?

A buyer compares two loan options for a $50 000 car: • Option A (loan of $40 000 at 6% p.a. over 5 years) • Option B (loan of $30 000 at 8% p.a. over 4 years) NSW stamp duty is $3 per $100 or part thereof up to $44 999; $1350 plus $5 per $100 or part thereof above $45 000. Registration costs $400. a Determine which option results in a lower total cost (loan principal, interest, stamp duty, registration). b Calculate the difference in monthly repayments between the two options.

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13.05 On-road and running costs of a car After this lesson, you will be able to… • calculate the total annual cost of different types of car insurance. • determine the total annual registration cost for a vehicle. • identify and calculate other running costs such as maintenance and depreciation. • calculate the total ongoing costs of owning and operating a vehicle.

On-road and running costs of a car Depreciation A decrease in value due to wear and tear, decay, decline in price etc. On-road and running costs of a vehicle encompass expenses required to operate and maintain a car after purchase, critical for financial planning. These include insurance, registration, dealer delivery charges and for certain vehicles, luxury car tax. Insurance protects against financial loss from accidents, covering injury, death, or property damage. Four main types exist: • Compulsory Third Party (CTP): Mandatory in NSW, covers injury or death to others, costing approximately $450 to $550 annually. • Third Party Property: Covers damage to others’ property, typically $200 to $400 per year. • Third Party, Fire and Theft: Adds coverage for fire or theft of the insured vehicle, around $300 to $600 annually. • Comprehensive: Covers damage to the insured vehicle, others’ property, fire, and theft, costing $800 to $2500 per year. Premiums, the annual cost of insurance, vary by driver age, driving history, car type, and location, and may be paid annually, monthly, or fortnightly. An excess is a fee paid when making an at-fault claim, as specified in the insurance policy. A no-claim bonus reduces premiums for claim-free years. Registration ensures a vehicle is roadworthy and identifiable, mandatory in NSW for all vehicles, including cars, caravans, trailers, and motorcycles. Annual costs include a motor vehicle tax based on tare mass (e.g. $295 for up to 975 kg, $487–$804 for 1505 to 2504 kg as of January 1, 2025) and a registration fee of $79. Additional fees may apply, such as vehicle safety checks (pink slips), transfer fees, or renewal fees for used cars. Dealer delivery charges cover preparation costs, such as documentation, safety checks, or cleaning, and are often negotiable. Luxury car tax, a compulsory government tax, applies to vehicles over $73 073 (or $81 050 for fuel-efficient models in 2024-25), calculated at 33% of the value above the threshold.

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Example 1 Lisa buys a second-hand car for $12 580. Her Compulsory Third Party insurance is $455 per year, and her Comprehensive insurance premium is $42 per fortnight. She was involved in an accident causing $3750 damage to another car and $2390 to her own, with a Comprehensive policy excess of 7.5% of repair costs if at fault. Calculate: a Calculate the annual insurance cost.

Create a strategy Multiply the fortnightly comprehensive premium by the number of fortnights in a year and add the CTP premium.

Apply the idea Annual comprehensive cost = 26 × $42

Calculate fortnightly premiums for 26 fortnights

= $1092

Evaluate

Total insurance cost = $455 + $1092 Add CTP premium = $1547

Evaluate

b Calculate the excess paid for the accident.

Create a strategy Sum the repair costs for both cars and calculate 7.5% of the total.

Apply the idea Total repair costs = $3750 + $2390 Add damage to both cars = $6140

Evaluate

Excess = 0.075 × 6140 = $460.50

Calculate 7.5% of total Evaluate

c Calculate the insurance company’s payout for repair costs.

Create a strategy Subtract the excess from the total repair costs.

Apply the idea Payout = $6140 − $460.50 = $5679.50

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Example 2 Xavier wanted to know how much of his savings were spent on his car. His yearly sources of running costs are given as shown. Assume that there are 52 weeks in a year. Description Fuel CTP insurance and comprehensive car insurance Price of servicing Cost of tyres NRMA gold membership Tolls and parking fees Parking fines

Cost $3769 $1393 $348 for the first service and $369 for the second service. $201 each $150 $720 $320

a What is his weekly running costs rounded to the nearest cent?

Create a strategy Add all sources of running costs throughout the year then divide by 52.

Apply the idea For the total yearly running cost: Total yearly running cost = 3769 + 1393 + 348 + 369 + 4 × 201 + 150 + 720 + 320 Add the costs = $7873

Evaluate

For the weekly running cost: Weekly running costs =

Write the formula

Substitute the values

= $151.40

Evaluate

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13.05 Practice questions What do you remember? 1

Determine whether each statement about vehicle running costs is true or false: a

Compulsory Third Party (CTP) insurance is mandatory in NSW and covers damage to your own vehicle.

Comprehensive insurance typically costs more than Third Party Property insurance.

Registration fees include a motor vehicle tax based on the car’s tare mass.

Luxury car tax applies to all vehicles regardless of their value.

Match each type of car insurance to its primary coverage: a Compulsory Third Party (CTP) i Covers damage to others’ property b Third Party Property ii Covers injury or death to others c Comprehensive iii Covers damage to own vehicle, others’ property, fire, and theft

What is the approximate annual motor vehicle tax for a car with a tare mass of 1500 kg, based on NSW rates?

Practice Ex 1

Raj purchases a used car for $15 000. His Compulsory Third Party insurance costs $475 per year, and his comprehensive insurance premium is $48 per fortnight. He causes an accident resulting in $4500 damage to another vehicle and $2800 to his own, with a comprehensive policy excess of 6% of repair costs if at fault. Calculate: a The annual insurance cost b The excess paid for the accident c The insurance company’s payout for repair costs

Ex 2

Sophie tracks her annual car running costs as shown in this list: Description Fuel CTP and comprehensive insurance Price of servicing Cost of tyres NRMA membership Tolls and parking fees Parking fines a b

Cost $4120 $1450 $325 for the first service and $395 for the second service. $190 each, replaced 4 times $130 $680 $280

alculate her weekly running costs, rounded to the nearest cent. Assume 52 weeks in a C year. If depreciation of $6000 is included as a running cost, calculate the new weekly running cost rounded to the nearest cent. 13.05 On-road and running costs of a car mathspace.co

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Calculate the annual cost of Compulsory Third Party (CTP) insurance priced at $465 and Third Party Property insurance priced at $25 per month.

A car with a tare mass of 1100 kg has a motor vehicle tax of $279 and a registration fee of $66. Calculate the total annual registration cost.

Emma pays $38 per fortnight for comprehensive insurance and $480 annually for CTP insurance. Calculate her total annual insurance cost.

A vehicle with a tare mass of 1800 kg has a motor vehicle tax of $466. If the registration fee is $66 and a safety check costs $42, calculate the total annual registration cost.

A luxury car is valued at $72 000. Calculate the luxury car tax, given the threshold is $66 331 and the rate is 33% on the amount above the threshold.

James has a Third Party, Fire and Theft insurance policy with a premium of $15 per fortnight. If he makes an at-fault claim for $4200 in damage to another vehicle, and his policy has an excess of $300, calculate: a

His annual insurance premium

The insurance company’s payout for the claim

Maria’s car requires two services per year, costing $320 and $410. If she replaces four tyres at $180 each, calculate her total annual maintenance cost.

A dealer delivery charge for a new car includes $250 for documentation, $150 for safety checks, and $100 for cleaning. Calculate the total dealer delivery charge.

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Tom’s comprehensive insurance policy has a premium of $1200 per year and a no-claim bonus that reduces the premium by 10% after a claim-free year. If he remains claim-free for one year, calculate his new annual premium.

A car with a tare mass of 1400 kg incurs a motor vehicle tax of $364. If the registration fee is $66 and a transfer fee of $34 applies for a used car, calculate the total registration cost for the year.

Extend your thinking 16

A luxury car valued at $85 000 incurs a luxury car tax with a threshold of $66 331 and a rate of 33% on the amount above the threshold. If the dealer delivery charge is $600 and the car’s comprehensive insurance costs $1400 annually, calculate the total first-year cost of these expenses.

A car owner pays $455 for CTP insurance, $900 for comprehensive insurance, and $430 for registration (including $66 fee). If fuel costs $80 per week and servicing is $700 annually, calculate the percentage of the total annual running cost attributed to insurance.

A vehicle owner has a choice between Third Party Property insurance at $350 per year or Comprehensive insurance at $1100 per year. If an at-fault accident causes $5000 damage to another vehicle and $3000 to their own, with a comprehensive excess of $400, compare the total cost (premium plus claim costs) for each insurance type.

Investigation: Getting on the road Investigate online

mathspace.co

Investigation: Choose the right car Investigate online

mathspace.co

Investigation: Car costs with spreadsheets Investigate online

mathspace.co

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13.06

Household bills

After this lesson, you will be able to… • calculate utility usage from previous and current meter readings. • interpret information presented on a sample household utility bill. • calculate the cost of usage based on different tariff structures. • calculate a total bill amount by combining usage and supply charges. • solve a variety of practical problems involving household utility costs.

Measure water and energy usage Household water, electricity and gas usage is measured to calculate costs, essential for managing bills. Meters track consumption over a billing period, typically three months, for quarterly bills. Separate meters record water (in kilolitres, kL), electricity (in kilowatt-hours, kWh), and gas (in megajoules, MJ). Kilowatt-hours (kWh) measures kilowatts for how many hours used, and megajoule (MJ) means 1 million joules. Usage is determined by subtracting the previous meter reading from the current reading: Usage = Current reading − Previous reading Readings are actual (taken on-site), estimated if access is restricted, or provided by smart meters for real-time accuracy, indicated on bills as A, E, or S respectively.

Example 1 Examine this quarterly water bill and answer the questions: Last bill $259.84

Payments $259.84

Balance $0.00

This bill $203.38

Amount due $203.38

Please pay by: 12/02/2019 Account for residential property

I sample Ave Sampletown

Water meter details

meter reading period: 20 Oct 15 - 22 Jan 16

Meter no. ABCD 1234

Last reading 1405

This reading 1425

Consumption (kL) 20

Total water used in 95 days was 20 kilolitres Fixed charges - GST free

20 Oct 15 - 22 Jan 16

Water service

$32.53

Wastewater (sewerage) service

$125.33

Usage charges - GST free Water

20 kL at $2.2760 per kL

$45.52 Total Amount Due: $203.38

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a What is the total amount due?

Create a strategy Typical location of the total amount due is at the top or bottom of the bill.

Apply the idea Total amount due is $203.38.

b What was the amount due on the previous water bill?

Create a strategy Look for the amount of the last bill.

Apply the idea Amount due on the previous water bill is $259.84.

c How much water was consumed during this quarterly billing period?

Create a strategy This is located under ‘consumption (kL)’.

Apply the idea 20kL

Reflect and check It can also be found by calculating the difference between the water meter readings.

d What is the water usage charge?

Create a strategy This is located under ‘usage charges’.

Apply the idea $45.52

Reflect and check It is calculated by multiplying the amount of water consumed in kL by the price of $2.2760 per kL.

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Household bills calculations Calculating household bills involves applying tariffs to water, electricity and gas usage, ensuring accurate cost management. Water costs typically include a fixed supply charge and a usage charge. Electricity and gas bills combine usage costs (based on tariffs) with supply charges. Bills reflect actual or estimated usage, with credits reducing costs.

Example 3 A household’s electricity meter shows 10 500 kWh (previous) and 11 000 kWh (current), with a tariff of 25 cents/kWh and a supply charge of $0.90 per day for a 90-day quarter. Calculate the electricity bill. a Calculate the usage.

Create a strategy Subtract the previous reading from the current reading.

Apply the idea Usage = 11 000−10 500 = 500 kWh

Subtract readings Evaluate

b Calculate the bill.

Create a strategy Multiply usage by the tariff (in dollars), add the supply charge for 90 days.

Apply the idea Tariff = = $0.25 per kWh Usage cost = 500 × $0.25 = $125 Supply cost = 90 × $0.90

Evaluate Multiply by usage Evaluate Calculate for 90 days

= $81

Evaluate

Total bill = $125 + $81

Sum costs

= $206

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Mathspace New South Wales – Year 11 Standard mathspace.co

Evaluate

Practice Ex 1

Consider this quarterly water bill:

Last bill $272.15

Payments $272.15

Balance $0.00

This bill $218.76

Amount due $218.76

Please pay by: 30/05/25 Account for residential property

I sample Ave Sampletown

Water meter details meter reading period: 20 Feb 25 - 20 May 25 Meter no. ABCD 1234

This reading 1530

Last reading 1508

Consumption (kL) 22

Total water used in 92 days was 22 kilolitres Fixed charges - GST free

20 Feb 25 - 20 May 25

Water service

$34.80

Wastewater (sewerage) service

$130.16

Usage charges - GST free Water

22 kL at $2.3950 per kL

$52.69 Total Amount Due: $218.76

What is the total amount due?

What was the amount due on the previous water bill?

How much water was consumed during this quarterly billing period?

What is the water usage charge?

What is the price of water per kilolitre?

How many litres of water were used on average per day to the nearest litre?

If the price of water rises by 10% next quarter, calculate the new price per kilolitre.

Ex 2

A household uses 720 kWh of electricity at a fixed tariff of 28 cents/kWh. Calculate the cost.

Ex 3

A household’s electricity meter shows 9800 kWh (previous) and 10 300 kWh (current), with a tariff of 22 cents/kWh and a supply charge of $0.95 per day for a 92-day quarter. Calculate: a

The usage for the billing period.

The total electricity bill.

An electricity meter’s previous reading was 5432 kWh and the current reading is 5768 kWh. Calculate the usage for the billing period.

A water company charges $2.10 per kilolitre. If a household uses 489 kL of water, calculate the usage cost.

An electricity company charges $0.185 per kWh. Calculate the cost for a household using 1757 kWh.

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An electricity company charges $0.235 per kWh. How much more would a household expect their electricity bill to cost in Spring than in Autumn, based on the data shown? Usage per day (kWh)

Usage for time period (kWh)

Summer

16.4

1495

Autumn

17.4

1589

Winter

19.5

1776

Spring

19.3

1757

Year

18.1

6617

The table shows Uther’s average water use for one day: a

How much water is Uther using per day?

How much water does Uther use in a week?

If water costs $2.70 per kilolitre, how much would Uther have to pay per day?

Usage

Litres per day

Dishwasher

Washing Machine

Gardening

Toilet

Shower

A water-saving shower head uses 60% of the water an ordinary shower head does. a

If Sharon uses 5.8 kL of water to shower each month, how many litres would she save with a water-saving shower head?

If water costs $1.90 per kilolitre, how much money would Sharon save?

The pie graph shows the annual water usage in kilolitres for a household: a

Calculate the amount of water that was used (in kilolitres).

If water costs $1.10 per kilolitre, how much does this household have to pay?

85 KL shower

81 KL laundry

36 KL car

67 KL graden

Consider the following households and the approximate gas usage: • Households with 1 − 2 residents use on average 3100 MJ of gas per quarter. • Households with 3 − 4 residents use on average 5400 MJ of gas per quarter. • Households with 5 or more residents use on average 6400 MJ of gas per quarter. If gas is charged at $0.02751 per MJ for the first 1220 MJ, and $0.02283 per MJ for the remaining, calculate the quarterly bill for a household with 3 to 4 residents.

13.06 Household bills mathspace.co

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Sophia’s gas usage over the past year for each month is given in the table: a

Month

Usage

March 2019

2430 MJ

February 2019

2436 MJ

January 2019

2351 MJ

December 2018

2447 MJ

November 2018

2354 MJ

October 2018

2444 MJ

September 2018

2353 MJ

August 2018

2448 MJ

July 2018

2434 MJ

June 2018

2361 MJ

May 2018

2362 MJ

April 2018

2444 MJ

Predict Sophia’s energy usage for the next month by averaging her monthly usage, rounded to the nearest megajoule. With a fixed usage cost of $0.028 per MJ and a supply charge of $18 per month, calculate the cost of Sophia’s expected gas bill for the next month.

The table shows details from an electricity bill of a small business: Usages and supply charges

Units

Price

Amount

Peak

5498.662 kWh

$0.3312 per kWh

$1821.16

Off-peak

4339.136 kWh

$0.1755 per kWh

28 days

$1.8275 per day

Supply charge

Total Amount Due:

Calculate the amount for the off-peak electricity usage.

Calculate the supply charge for the 28 days of the billing period.

Calculate the total amount due.

Consider Holly’s internet bill and billing history: How much internet are you using? Bill period: 31 Aug to 30 Sep 2018 (31 Days?) Bill history

Balance brought forward: $0.00 +

New charges: $54.50

956

Previous balance $29.30 We received: $29.30

$ 60

Account activity

Amount due: $54.50 Nov 17

Jan Mar May 18 18 18 Monthly charges

Jul 18

Mathspace New South Wales – Year 11 Standard mathspace.co

Sep 18

Due date: 22 Oct

Using the current bill and bill history, calculate Holly’s predicted weekly expenditure on internet, assuming 52 weeks in a year.

Month

Bill charge

August

$29.30

July

$37.50

June

$30.00

May

$45.00

April

$23.20

March

$33.60

February

$47.10

January

$38.30

December

$25.10

November

$18.70

October

$30.00

The following information is taken from Angela’s gas bill: Your gas supply detais Supply address

21 Bill Drive KENSINGTON NSW 2033

Supply period

31 Oct 2018 to 30 Nov 2018 (31 days)

DPI:

9992341551

Energy Plan:

Flexible

Meter no: CQ654321 Read date

Read type

Start read

End read

Heating Value

Conversation factor

31 Oct

Estimate

6012.19

6103.01

137.76000

1.030500

Usage MJ

Your next meter read is due between 29 Dec 18 and 5 Jan 19. Please ensure easy access to your meter on these days. a

Determine the amount of gas, in cubic metres, consumed during the billing period.

Determine the amount of gas, in megajoules, consumed during the billing period, rounded to the nearest megajoule.

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Consider the following gas bill: How much energy are you using?

Account activity

Bill period: 31 Aug to 30 Sep 2018 (31 Days?)

Previous balance $143.68

$ 8

20 Sep Payment: $143.68 CR

160

Balance brought forward: $0.00

120

Sep 17

Nov 17

Jan 18

Mar 18

May 18

Jul 18

Sep 18

New charges: $96.24 = Amount due: $96.24

Due date: 22 Oct Average daily cost

Average daily use

What is the total gas usage in megajoules (MJ) for this billing period?

What was the average daily gas usage (in MJ) for this billing period?

Extend your thinking 21

A NSW electricity company offers a Green Power plan with 100% renewable energy. Bill pays a flat rate of $0.3445 per kWh. Switching to Green Power increases the rate by 5 cents/kWh. a

Bill’s last bill showed an average daily consumption of 8.5 kWh. Calculate his average daily cost on the current plan.

Calculate the cost of his average weekly consumption if he switches to Green Power.

How much extra would Bill pay weekly on Green Power?

Consider Sally’s water bill: Water meter details

meter reading period: 21 Oct 18 - 21 Jan 19

Meter no.

This reading

Last reading

Consumption (kL)

AK32457

1695

1678

Total water used in 93 days was 17 kilolitres Fixed charges - GST fees 21 Oct 18 - 21 Jan 19 Water service

$25.63

Wastewater (sewerage) service

$152.28

Usage charges - GST free Water

17 kL at $2.2760 per kL

$38.69

Total Amount Due: $216.60

958

Calculate the percentage of the total amount due attributed to supply charges, rounded to the nearest percent.

If the price of water rises by 12 cents per kL next quarter and Sally’s usage remains the same, calculate the extra cost.

Mathspace New South Wales – Year 11 Standard mathspace.co

The table shows monthly charges for Deborah’s natural gas account:

Deborah uses 5000 MJ in January. Find the total monthly cost.

In February, Deborah uses 13 500 MJ more than January. Calculate her total monthly cost increase.

Calculate the average charge per MJ in February.

Amount used

Charges

First 5000 MJ

$0.0192/MJ

Next 13 500 MJ

$0.0155/MJ

Supply Fee

$80.00

Sophia is on a fixed rate electricity plan at 31.41 cents/kWh and considers a flexible plan with time-of-use rates. She uses 9.5 kWh daily. Flexible plan Charge type

Fixed rate plan Rate

Charge type

Rate

Peak usage

58.79 cents per kWh

Usage

31.41 cents per kWh

Shoulder usage

26.72 cents per kWh

Supply

89.76 cents per day

Off-peak usage

16.26 cents per kWh

Supply

89.76 cents per day

Calculate her total electricity costs for a 92-day period on the fixed rate plan.

Sophia estimates her daily usage on the flexible plan is 2.5 kWh during peak times, 5 kWh during shoulder times, and 2 kWh during off-peak times. Calculate her costs for a 92-day period on the flexible plan.

13.07 Prepare a personal budget After this lesson, you will be able to… • prepare a personal budget for a given income, considering fixed and discretionary spending. • calculate total income, total expenses, and the resulting savings or deficit. • convert expenses from various timeframes into a consistent unit for a budget. • analyse a budget to make informed financial decisions.

Prepare a personal budget Budget A tool for managing money and achieving financial goals. It is a list of all the expected income and costs for a certain period of time. A budget aims to balance the income and expenses, so that the expenses are less than or equal to the income.

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Discretionary spending A type of spending that is not essential or mandatory. It includes expenses such as entertainment, clothes, and gifts that can be varied based on choice and preference. Fixed spending A type of spending that is essential or mandatory. It includes expenses like rent or board that cannot be varied or changed.

A personal budget plans income and expenses to manage finances, balance spending and achieve savings goals. Key components include: • Income: Money earned, typically weekly, fortnightly or monthly. • Expenses: Money spent, divided into: • Fixed spending: Regular, constant payments (for example rent, insurance). • Discretionary spending: Variable, optional costs (for example entertainment, clothing). • Savings: Income minus expenses, adjustable by reducing discretionary spending. Budgets, often weekly or monthly, remain flexible to adapt to changes, typically managed via spreadsheets.

Example 1 Amerie pays rent of $207 per week and budgets $611 per quarter for electricity and water. Calculate the weekly amount to cover these expenses.

Create a strategy Divide quarterly costs by weeks per quarter, add to weekly rent.

Apply the idea Weekly utility cost = = $47 Total weekly cost = $207 + $47 = $254

Divide quarterly cost by 13 weeks Evaluate Add rent Evaluate

Example 2 Luke’s weekly expenses for three weeks are $225, $218 and $208. To maintain an average of $221 over four weeks, calculate the maximum expense for the fourth week.

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Match each definition to its budget term: a

Money earned, typically weekly or monthly

Discretionary spending

Regular, constant payments (e.g., rent)

Fixed expenses

Variable, optional costs (e.g. entertainment)

iii

Income

Income minus expenses

Savings

Classify each expense as fixed or discretionary: a

Monthly internet bill

Dining out

Car insurance

Movie tickets

Identify whether each is a main component of a personal budget: a

Income

Expenses

Investments

Savings

Practice 5

Last month, Neil saved $235, which was 15% of his income. Calculate his income last month.

Sally earns $610 per week. Her weekly work-related expenses are $185 on travel and $86 on lunch. Calculate the percentage of her income spent on these expenses, rounded to two decimal places.

The table shows Fiona’s weekly budget: a

Find the value of: i

Income Wages Umpiring

$80 $25

Total

Fiona saves any money not spent. How much does she save each week?

Expenses Phone $8.50 Bus ticket $13 B Entertainment Food $45.25 Total $91.15

Amelia earns a weekly salary of $723 and receives other income of $126. Her weekly expenses are: rent $199, groceries $51, petrol $33, electricity $33, telephone $23, magazines $13, clothes $62. Calculate: a

Her weekly income

Her weekly expenses

Her weekly savings

Her annual savings

James’s monthly expenses include: rent, food, cable TV, travel, insurance, clothing, repairs. Determine whether the following expenses are fixed:

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Cable TV

Travel

Repairs

Rent

Mathspace New South Wales – Year 11 Standard mathspace.co

Insurance

Clothing

Ex 1

Ex 2

Ex 3

Zane pays rent of $239 per week and budgets $533 per quarter for electricity and water. How much should she put aside weekly for these expenses?

Sally earns a weekly salary of $1195 and other income of $181. Her weekly expenses are listed. Calculate: a

The percentage of income saved, correct to two decimal places.

• •

Rent $122 Groceries $97

The number of full weeks needed to save for a holiday costing $3800.

• •

Petrol $60 Electricity $46

•

Telephone $28

•

Magazines $10

•

Clothes $79

Emma’s weekly expenses vary. In the first 3 weeks, she spends $312, $298, and $285. To maintain an average weekly expense of $300 over four weeks, calculate her maximum expense in the fourth week.

Amelia earns a gross annual salary of $34 500. After paying 24% income tax and $19 650 in annual expenses, how much can she set aside monthly for starting a business?

Liam’s expenses are 25% of his $1080 weekly income. If his expenses increase by 8%, what percentage of his income will expenses now represent?

Tina budgets for car-related costs: • Petrol: $50 per week • Insurance: $110 per month • License renewal: $40 per annum • Servicing: $310 every six months • Green Slip: $750 per annum • Registration: $600 per annum a

Calculate the annual cost of: i

Petrol

Insurance

iii

License

Servicing

If Tina’s annual income is $59 000 and she can spend 19% on car costs, calculate her maximum monthly car loan repayment.

Buzz’s income increases from $327 by $6.54, and his expenses increase from $114.45 by 4%: a

Calculate the percentage of income currently consumed by expenses.

Calculate the percentage of income consumed by expenses after the increases, to two decimal places.

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Tracey’s bank statement for January: Date 01/01/15 06/01/15 10/01/15 17/01/15 20/01/15 31/01/15

Item Opening balance Groceries Salary Train ticket Socks Magazine

Credit

Debit

Balance $55.40 $ $144.50 $ $123.10 $

$40.00 $ $18.70 $ $6.15

Calculate the new bank balance at the end of January.

Extend your thinking 18

Income Earnings

Liz’s weekly budget is shown: a

Calculate Liz’s weekly expenses.

Calculate Liz’s weekly savings.

Calculate the percentage of her income spent on mortgage payments, rounded to two decimal places.

$147 $23 $64 $22 $20 $64 $46 $56

Marge’s weekly budget spreadsheet includes income and expenses: A 1 2 3 4

B Personal budget (weekly) Income Earnings

$950.00

5 6 7 8 9 10 11 12 13 14 15

964

$505

Expenses Mortgage Electricity Food Council Rates Insurance Water Clothing Entertainment

Total

$950.00

Savings

$350.00

What formula did Marge use in cell F13?

What formula did Marge use in cell C15?

Mathspace New South Wales – Year 11 Standard mathspace.co

Expenses Rent Electricity and gas Food Internet Gym membership Public transport Phone Clothing Entertainment Total

$272.00 $51.00 $135.00 $29.00 $22.00 $76.00 $50.00 $80.00 $85.00 $600.00

Valerie’s weekly budget spreadsheet plans savings for a holiday: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

B Personal budget (weekly)

Income Earnings

$650.00

Total

$650.00

Savings

$140.00

Expenses Rent Electricity and gas Food Internet Fitness club Public transport Phone Clothing Subscriptions Total

$180.00 $54.00 $100.00 $16.00 $25.00 $25.00 $10.00 $85.00 $15.00 $510.00

Valerie allocates $125 weekly savings for a holiday and $15 for emergencies. Calculate her holiday savings after one year (52 weeks).

Calculate her emergency fund after one year at $15 weekly.

If Valerie buys a new washing machine for $555 using her emergency fund, how much remains in the fund?

Investigation: Create a budget Investigate online

mathspace.co

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13 Chapter review 1

A concert ticket was priced at $80. The price is increased by 25% due to high demand. What is the new price of the ticket? a

$100

$105

$60

A florist buys roses for $2.50 each and sells them for $5.00 each. What is the profit made on each rose? a

$90

$1.50

$2.00

$2.50

$7.50

Mia takes out a car loan for $22 000 at a fixed simple interest rate of 8.5% per annum over 5 years. Calculate the total interest paid. a

$8500

$9350

$11 870

$12 000

A mountain bike is marked at a price of $650. It is advertised as selling at 20% off the marked price. Find the discounted price.

Emily hired a plumber. The plumber charged $180 for labour and $95 for parts. GST (10%) is applied to the labour cost only. Calculate the total bill.

After a clearance discount of 30%, a sofa was sold for $560. What was the original price of the sofa?

A retailer sells items at 15% more than their cost. If an item cost the retailer $250, calculate: a

The sale price.

The profit made.

A bakery sold 200 croissants at $3.50 each and 150 muffins at $2.75 each in one day. The bakery’s expenses for the day (ingredients, electricity, wages) totalled $650. Calculate: a

The total revenue for the day.

The total profit (or loss) for the day.

A shop is selling a discontinued laptop for $760, which represents a loss of 15% on the original cost price. What was the original cost price of the laptop?

Liam purchased a new washing machine valued at $750 on lay-by. He paid a deposit of $50 and then weekly payments of $70. After how many whole weeks will he receive ownership of the item?

Sarah buys a new refrigerator for $1500 using a “buy now, pay later” plan. The plan involves twelve equal monthly payments with no administrative fees or interest. However, if she misses a payment, an initial fee of $15 is charged. If the payment is not received within two weeks, another fee of $5 will also apply.

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Sarah makes the first six payments on time, but is three weeks late on the seventh payment and one week late on the eighth payment. She makes the remaining payments on time.

Determine the total amount of fees Sarah had to pay.

Determine the percentage of the original cost she paid in fees, rounded to one decimal place.

Chloe wants to buy a new laptop valued at $2200. The store offers a hire purchase option with 12 equal monthly payments. a

Chloe can afford a monthly payment of $200. If the store offers a simple interest rate of 12% p.a., calculate the required monthly payment and determine if this interest rate is feasible for Chloe’s budget.

If Chloe proceeds with the hire purchase at the 12% interest rate, how much total interest does she pay compared to paying with cash?

The sale price of a new camera is $950. Michael chooses to purchase it using his credit card, which has 20 interest-free days. His card has an interest rate of 18.5% per annum, compounded daily. a

Michael pays $400 on his payment due date (within the interest-free period). How much does he still owe?

Michael pays the remaining balance 30 days after the initial payment due date. How much does he pay for the camera overall (including the initial $400 payment)? Round your final answer to the nearest cent.

How much more does Michael pay by using his credit card and not paying the full amount within the interest-free period, compared to paying the $950 in cash initially?

In NSW, car stamp duty is calculated as $3 per $100 or part thereof for vehicles valued at $44 999 or less, and a flat $1350 plus $5 per $100 or part thereof above $45 000 for vehicles valued at $45 000 or more. Find the stamp duty levied on a car with a market value of: a

$28 000

$52 000

$39 500

$59 750

Two friends are buying cars: • Car X: Valued at $28 000. Loan of $18 000 at 7.2% p.a. simple interest for 4 years. • Car Y: Valued at $48 000. Loan of $35 000 at 8.5% p.a. simple interest for 5 years. NSW stamp duty applies: $3 per $100 or part thereof up to $44 999; $1350 plus $5 per $100 or part thereof above $45 000. a

Calculate the total cost for each car.

Which car has the lower monthly repayment, and by how much?

Ahmed pays $42 per fortnight for comprehensive car insurance and $520 annually for CTP insurance. Calculate his total annual insurance cost.

A car with a tare mass of 1650 kg has an annual motor vehicle tax of $430. If the registration fee is $70 and a safety inspection costs $45, calculate the total annual registration cost.

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A luxury car is valued at $92 000. The luxury car tax threshold is $75 526 with a tax rate of 33% on the amount above this threshold. The dealer delivery charge is $750 and the car’s comprehensive insurance costs $1600 annually. a

Calculate the luxury car tax.

Calculate the total first-year cost of the additional expenses.

A car owner pays $480 for CTP insurance, $1050 for comprehensive insurance, and $495 for registration (which includes a $70 registration fee component). Fuel costs an average of $75 per week, and annual servicing is $650. Calculate the percentage of the total annual running costs that is attributed to insurance (CTP and comprehensive combined). Round your answer to one decimal place.

A household’s electricity meter showed a previous reading of 7650 kWh and a current reading of 8230 kWh. The tariff is 25 cents/kWh, and there’s a supply charge of $1.05 per day for a 91-day quarter. a

Calculate the electricity usage for the billing period.

Calculate the total electricity bill for the quarter.

The table shows Ethan’s average daily water use: Usage

Litres per day

Dishwasher

Washing Machine

Gardening

Toilet

Shower

How much water (in Litres) is Ethan using per day?

How much water (in kL) does Ethan use in a 90-day billing period?

If water costs $2.50 per kilolitre, how much would Ethan have to pay for his water usage in this 90-day period?

An electricity company charges for gas based on the following quarterly tariff: • First 1500 MJ: $0.0315 per MJ • Remaining MJ: $0.0255 per MJ • Quarterly supply charge: $75.00 A household uses 4800 MJ of gas in a quarter. Calculate their total gas bill for that quarter.

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Olivia is on a fixed rate electricity plan at 33.50 cents/kWh and has a daily supply charge of 95.00 cents. She is considering switching to a flexible plan with the following time-of-use rates and the same daily supply charge: Flexible plan

Fixed rate plan Charge type

Rate

Usage

33.50 cents per kWh

Supply

95.00 cents per day

Charge type

Rate

Peak usage

60.50 cents per kWh

Shoulder usage

28.00 cents per kWh

Off-peak usage

17.50 cents per kWh

Supply

95.00 cents per day

She uses an average of 10.5 kWh of electricity daily.

Calculate her total electricity costs for a 90-day period on her current fixed-rate plan.

Olivia estimates her daily usage on the flexible plan would be: 3 kWh during peak, 4 kWh during shoulder, and 3.5 kWh during off-peak. Calculate her estimated total electricity costs for a 90-day period if she switched to the flexible plan.

Based on these calculations, would switching to the flexible plan save Olivia money?

Michael earns a weekly salary of $850 and receives other income of $150. His weekly expenses are: • Rent $250 • Groceries $70 • Transport $40 • Utilities $50 • Phone $25 • Entertainment $60 Calculate:

His total weekly income.

His total weekly expenses.

His weekly savings.

His annual savings.

Maria is creating an annual budget. Her known annual expenses are: • Rent: $1500 per month • Utilities (Electricity & Gas): $240 per quarter • Groceries: $90 per week • Car Insurance: $1200 per year • Public Transport: $60 per month • Health Insurance: $180 per month Assume 52 weeks per year, 12 months per year, 4 quarters per year: a

Calculate the total annual cost for each of these expense categories.

If Maria’s net annual income is $75 000, and she wants to allocate 15% of her net income to savings, how much can she allocate to other discretionary spending per year after accounting for the listed expenses and her savings goal?

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The table shows Sam’s proposed monthly budget: Income Salary (after tax) Investment Income

4200 150

Total

Expenses Rent Utilities Groceries Transport Insurance Discretionary Total

1500 200 450 180 120 X Z

Sam wants to save at least 20% of his total monthly income.

Calculate Y, Sam’s total monthly income.

Calculate the minimum amount Sam wants to save per month.

If Sam achieves his savings goal, what is the maximum value for X?

Calculate Z, Sam’s total monthly expenses, using the maximum value of X from part (c).

Marge’s weekly budget is shown in a spreadsheet extract: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

970

B Personal budget (weekly)

Income Earnings

1050.00

Total

1050.00

Savings

⬚

Expenses Rent Electricity and gas Food Internet Gym membership Public transport Phone Clothing Entertainment Total

What formula could Marge use in cell F13 to calculate total expenses?

What formula could Marge use in cell C15 to calculate her savings?

Calculate the values for F13 (total expenses) and C15 (savings).

Mathspace New South Wales – Year 11 Standard mathspace.co

320.00 55.00 140.00 25.00 20.00 70.00 45.00 75.00 90.00 ⬚

Answers

7 a 8c2

c 0

1.01 Simplify algebraic expressions

f 18ab

h 3

What do you remember?

k 9c

m 3a3

n 3u6 v5

o 15x5 y4

p 7a6 b5

b x and y

1 a 6 c 5

d −1

e −9

f −3y and 5y

2 a True

b False

c True

e False

f False

g False

d False

Practice 3 a 13x

b 3y

c 9c

d 9c + 2d

e −2x

f 5x + 5y

g 5x + 7y + z

h 4x + 7 − 2y

i 8x + 5

j 5ab + 10cd

k x2 + 11x + 10

l 3ab + 5bc

m x2 − 16

n 7a2 b + 4ab2 2

o 4xy + 2x y − 2xy − 7 p x2 − 4x 4 a 45u

b 28r

c 6xy

d 24rs

e 15cd

f −25ef

3 5

g 15u v

h 60pq

i 30xy

j 36x2 y2 z2

k −54a3

l 20a2 b2

m 20a7 b5

n −18w4 x5 y5 14 7

p −15e f

o 42c d

c 3

d 4

e −4

i 3r

j −2

5 a

9 a 1.4x3 y + 1.8x2 y2

6 a 15

g 35 fg

c 7.2x2 y2

e 1.2a4 b + a4 b2

f x2 y

10 a 1.5x4 y2 + 1.5x3 y3

c −7.44a5 b3

11 a 18x2 y2

b 2x2 y2

d 4x2 y + x3 y2

e 5.4w2 z2 − w3 z2

12 6b + 2c cm 13 10x − 2y cm 14 The equation 3a × 2a = 5a2 simplifies to 6a2 = 5a2. Subtracting 5a2 from both sides gives a2 = 0, which means a = 0. This is the only value for which the equation holds. 15 Multiplication first gives 7a + 10a = 17a. Addition first gives 12a × 2 = 24a, which is incorrect. The order of operations ensures consistent correct results. 16 a

11a + 3b 4a + b a + 2b

4a + 3b

5x − 2y

Mathspace New South Wales – Year 11 Standard mathspace.co

7a + 2b

3a − b

3x + 2y

972

c 3y

Extend your thinking

7 8

b 2a3 b

8 a 2x

2x − 4y

3x − y

−x − 3y

17 36a2 cm2

12 a b 3xy2

18 a xy

−3

−1

−3

−5

−7

−9

−11

19 39 − 4x cm b

1.02 Substitution What do you remember? 1 Order of operations ensures accurate evaluation; in 3x + 4y2, substitute values, compute y2 first, multiply by 4, then add 3x to avoid errors. 2 Substituting into an expression evaluates its value, while substituting into an equation verifies if the value satisfies the equality. 3 Units ensure the result is meaningful; for A = l × w, length and width units (e.g., cm) yield area in cm2. 4 The order of operations ensures consistent and correct evaluation. For 2 + 3 × 42, the correct order is: exponents first, 42 = 16, then multiplication, 3 × 16 = 48, then addition, 2 + 48 = 50. The incorrect order is left to right: (2 + 3) × 42 = 5 × 16 = 80. The difference (50 vs. 80) shows that order of operations prevents errors. 5 a 32

b 17

c −2

d 32

e −442

Substitute x = 3

=6+5

Evaluate the multiplication

= 11

Evaluate

= RHS 2

b x – 4 = 32 − 4

Substitute x = 3

=9−4

Evaluate the power

= 5

Evaluate

= RHS 14 F = 59°F 15 D = 1.1 g/cm3 16 T = 38 b −2

17 a 4

c 28

d 18

18 a −105

b −84

19 5x + 7 = 5 × (−2) + 7

Substitute x = −2

= −10 + 7

Evaluate the multiplication

= −3

Evaluate

= RHS

20 I = $420

Practice 6 a 36

13 a 2x + 5 = 2 × 3 + 5

b 30

c 16

d 16

f −28

g 123

h 28

7 a A = 42 cm2 c A = 200 cm2

b d A = 45 cm2

21 S = 382 cm2 22 S = 435 23 a E = 12 edges 24 C = 30°C 25 F = 7.47 Newtons

8 a P = 36 cm

b A = 36 cm2

c A = 6 cm2

d P = 16 cm

Extend your thinking

b A = 12.6 cm2

Verifying:

e P = 30 cm 9 a A = 24 cm2 c

d A = 20 mm2

10 a V = 105 cm3

b V = 63.96 cm3

c V = 5 cm3

d V = 3000 mm3

b E = 12 edges

b F = 8.58 Newtons

26 x = 5 3x – 5 = 3 × 5 − 5

Substitute x = 5

= 15 − 5

Evaluate the multiplication

= 10

Evaluate

= RHS

11 y = −15

Answers mathspace.co

973

27 a i y = 9.8 m

13 a = −10

ii y = 0 m

iii y = −196 m b At t = 2 s, since y = 9.8 m is positive. 28 a R = 35.35 m c R = 55.23 m

b R = 24.52 m d R = 6.14 m

1.03 Evaluate the subject of a formula What do you remember?

b An operation that reverses another operation to isolate a pronumeral (e.g., addition reverses subtraction) c The pronumeral being solved for, typically isolated on one side of the equation d Replacing pronumerals in a formula with given numerical values 2 a Divide both sides by l b Divide both sides by 2, then subtract a c Multiply both sides by 2, then divide by b d Multiply both sides by t e Divide both sides by I f Divide both sides by 2π g Multiply both sides by h2 h Subtract at from both sides

c A = 30 mm2 e E = 100 J

15 a di = 15 cm b The rearranged formula is Substituting the values, f = 10 cm. b TMars ≈ 3.24 s

1.04 Speed, distance and time What do you remember? 1 a ii

b i

2 a 33.33 m/s

b 16.67 m/s

c 25 m/s

d 8.33 m/s

3 Add the speeds of the two vehicles to ﬁnd their relative speed, used to calculate the time or distance until they meet, as it represents how quickly the gap between them closes. 4 Higher speed increases both reaction distance (as s × t grows with s) and braking distance (as ks2 grows quadratically with s); longer reaction time increases reaction distance (as s × t grows with t).

5 a 54.2 km/h

b 60 km/h

b V = 24 V

c 60 km/h

d 90 km/h

d s = 60 km/h

e 15 km/h

f 80 km/h

f S = 94 cm

6 a 267.75 m

b 348 km

4 b = 10 m

c 45 km

d 240 m

5 h = 12 mm

e 132 km

f 135 km

6 a u = 33

Practice

Practice 3 a A = 28 cm

b anew = 7.20

14 a b = 21.60

16 a L ≈ 0.99 m

1 a A letter representing a variable or unknown value in a formula

Extend your thinking

b m = 55.04 kg

7 a 3h

b 4h

c 3h

d 3h

c r = 5 cm

d d = 150 km

e 1.25 h

7 a r = 7.5 cm

b A = 176.6 cm2

8 a 140 km/h

b 2.5 hours

8 a a = −0.02

b s = 49

9 a 38.44 m

b 70 m

9 a v = 8.94 m/s

b v = 6.32 m/s

10 The ﬁrst motorcycle stops at 16.72 m, while the second at 46.17 m. The second motorcycle’s higher speed and longer reaction time result in a greater stopping distance.

10 V = 9822 11 h = 10 mm 12 a = 4

11 156.6 km 12 5 h

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f 2.5 h

16 a Male: 0.061, Female: 0.098

Extend your thinking

b Females have a lower water content, increasing BAC.

13 390 km 14 Distance: 280.0 km, Speed: 46.7 km/h 15 200 m

17 3 18 3:20 a.m.

16 222 km 17 The truck will not stop in time, missing by 5 m.

1.05 BAC (Blood Alcohol Content)

1.06 Medication doses What do you remember? 1 a Age range: 1 to 2 years. Primary variable: Age in months.

What do you remember? 1 The rate at which the average person metabolises alcohol, approximately 0.015 g /100 mL per hour. 2 The density of pure alcohol in grams per millilitre. 3 Females generally have higher body fat and lower water content, leading to higher BAC for the same alcohol intake. 4 0.00

b Age range: 1 to 12 years. Primary variable: Age in years. c Age range: 2 to 17 years. Primary variable: Weight in kilograms. 2 a 150 b Age in years plus 12 c 70 3 Clark’s formula Practice

5 Food intake, which can slow alcohol absorption and reduce BAC compared to the formula’s prediction.

5 227 mg

Practice

6 300 mg

6 a 1.0

b 7.4

7 a 0.070

4 22 mg

c 0.9

7 35 mg

b 4.67 hours

8 a 200 mg

b 179 mg

9 153 mg

8 0.044 9 a 2.00 hours

b 6.67 hours

10 a 24 mg

b 67 mg

10 a 1.4

b 7.5

11 a 114 mg

b 257 mg

11 a 0.017

b 0.095

Extend your thinking

12 a 4.0

b 0.018

12 a 19 mg

13 a 23.67

b 0.42

b 18 mg

14 Standard BAC: 0.041, Adjusted BAC: 0.028 Extend your thinking 15 Factors like lack of food, carbonated drinks, or poor health can increase alcohol absorption, leading to higher BAC than predicted.

c Fried’s formula uses a ﬁxed denominator (150) based on months, while Young’s uses a variable denominator (age +12) based on years, leading to slight variations. 13 a 110 mg

b 1100 mg

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14 a Clark’s: 46 mg, Young’s: 59 mg b Clark’s uses mass, which may be more accurate if the child’s weight deviates from average, while Young’s uses age, which is simpler but less precise for atypical sizes. 15 a 549 mg b 2 tablets c The calculated 549 mg is not exactly divisible by 300 mg tablets; 2 tablets (600 mg) is chosen as the nearest practical dose, which is slightly higher but assumed to be within safe limits.

Extend your thinking 8 1.80 s 9 a $600 b Second scenario: $480. The impact of each variable depends on the context. In the change from the second scenario to the ﬁrst, the time (T ) has the largest relative change (a 50% increase). The combined effect of the changes results in the interest increasing from $480 to $600. 10 a 7.24 m3

b 7238 kg

1.07 Other formulas

1.08 Change subject of a formula

What do you remember?

1 a 4

b 3

c 4

2 a I = interest in dollars, P = principal in dollars, R = rate in decimals, T = time in years b BMI = body mass index in kg/m², m = mass in kilograms, h = height in metres c P = final population, P0 = initial population, r = growth rate per year, t = time in years Practice 3 a $300

b 2 years

c $6000

d 0.025

e $270

f 1 year

g $6000

h 0.02

4 a 26.23 kg/m2

b 1.71 m

c 54.45 kg

d 29.30 kg/m2

e 1.65 m

f 40.50 kg

g 24.91 kg/m2

h 1.68 m

i 60.69 kg 5 a 929.5

b 1097

6 a 0.524 m3

b 2.00 m

7 a 20 N

b 10 kg

c 5 m/s2

d 15 N

e 20 kg

f 4 m/s2

1 a y

b p

c z

d k

2 a Subtract b from both sides. b Divide both sides by b. c Multiply both sides by b. d Subtract c from both sides. Practice 3 a

e x=k−n

4 a

5 a

8y = x + 2z

8y − 2z = x b

Given Subtraction property of equality

x = 8y − 2z Solution Given

Multiplication property of equality

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Write the formula

Subtract 2l from both sides

Divide both sides by 2

Make w the subject

Substitute P = 20 and l = 7

1.09 Write and solve equations What do you remember? 1 a i Drink cost ii 5, 12 iii 2

Evaluate the multiplication

Evaluate the division

Verify: P = 2 × 7 + 2 × 3 = 14 + 6 = 20 cm, matches given perimeter. 7 x = 100 items

b i Time ii 30, 50 iii 10 c i Number of oranges ii 12, 30 iii 2 2 A

8 x = ±1 m 9 x = 60 items

3 a +

b ×

c −

d =

4 a ii

b iii

c v

d iv

10 x = ±1 m Practice Extend your thinking

5 a n + 7 = 12

11 b = 12 cm

n = 5 m = 3 months

The positive value ensures a valid length for the table’s lower base, suitable for a stable trapezium shape.

t = 6 tubes

12 T = 6016.85 K

m = 20 meals

This calculation is critical in industrial gas storage to ensure safe operating conditions, as extreme temperatures could affect tank integrity.

c 15 + 2t = 27

b 25 + 10m = 55 d 2w + 2(w + 5) = 26 w = 4 cm

6 40 + 7m = 180 7 c + 11 = 3c − 5 c = 8 crates 8 4c − 10 = 50 c = 15 cups

13 x = 14 hours This time helps plan production schedules, ensuring eﬃciency and resource allocation.

9 3n − 4 = 17 n = 7

14 Ruﬁno has an error in his work.

10 20 + 0.5m = 35

The equation for KE as given

m = 30 minutes

Multiplying both sides by 2

Now he should have divided by v2

x = 5

Make m the subject

12 15b + 30 = 90 b = 4 books 13 s + 15 = 3s − 10 s = 12.5 km/h

Answers mathspace.co

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7 a 35

Extend your thinking

b 28

e −47

14 15 000 + 200u = 300u

f −27 2

−3

−7

−11

−15

9 a

p = 4 packages The solution ensures accurate delivery cost calculations for planning. 16 100 + 30 ×

= 340

h 31 d 35 km2

Selling 150 units covers costs, enabling ﬁnancial planning for proﬁtability.

Correct equation: 20 + 3p = 32

g 20 c 9 cm

8 a 40 cm

d 9

b 72 m

u = 150 units

15 The student’s equation is incorrect. The phrase “plus $3 per package” means +3p, not +3( p + 1), which implies an extra package.

c 25

10 a 68°F

b 95°F

c −5°C

d 100°C

11 a 55.23 m

b 33.06 m

c 79.53 m

d 12.41 m

12 15 kg

c = 12 classes

13 5 cm

The solution helps the gym verify billing accuracy for discounted classes.

14 a 12.0 kg

b 15.0 m/s

17 15k − (200 + 5k) = 400

15 a $3000

b 4 years

16 a 50.2 m

b 97.1 m

k = 60 cakes Selling 60 cakes achieves the target proﬁt, aiding inventory and sales planning. + 5.5 = 10

n = 6.75 Veriﬁcation:

× 6.75 + 5.5 = 4.5 + 5.5 = 10,

conﬁrming the solution ﬁts the problem.

Chapter 1 review 1 C

17 a 150 km/h

3 C 4 a 11a + 3b

b y + 9y + 20

c 30mn

d 12x3 y7

h 10pq

b 0.059

19 a 4.6

b 0.047

20 3 : 30 a.m. 21 a 83 mg

b 71 mg

22 a 115 mg

b 374 mg

23 a 1541

b 1967 3

j 3x y

k a

26 a

27 a

5 12x cm 6 20w2 m2 b 8 m/s2

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Mathspace New South Wales – Year 11 Standard mathspace.co

b 0.38 kg

25 a

4 2

i 2

c 210 km

18 a 0.045

24 a 2.14 m

2 B

b 3 hours

28 Error in Step 1: Dividing KE by 2 is incorrect; multiply both sides by 2 to eliminate .

9 a Survey; faster and cheaper, a random sample can estimate preferences reliably.

Final formula: 29 a x = 24

b m = 5 months

c r = 6 roses

d w=9m

b 672 students

11 a Census

b 15x

c 12 000 + 15x

d 75x = 12 000 + 15x

e 200 tickets

f 250 tickets

b Survey; only a sample is tested, not the entire population. Extend your thinking

2.01 Census or survey

12 No; it only includes cafeteria users, not a random sample, so it may be biased.

What do you remember? 1 a True

b False

c True

d False

2 a ii

b iii

c iv

d i

3 B

13 a Census: Advantage - complete accuracy, Disadvantage - time-consuming. Survey: Advantage - faster, Disadvantage - potential sampling error. b Survey; large population makes census too costly, a random sample provides a good estimate.

4 B 5 B

14 a 0.315

Practice 6 a Census

b Yes, census; small size makes collecting all data practical and accurate. 10 a Survey

30 a = 8 apples 31 a 75x

8 Census. With only 18 employees, conducting a census is highly manageable and ensures every individual’s concerns and preferences regarding a signiﬁcant change like health insurance are heard. This provides complete data, avoids potential dissatisfaction from unheard employees, and is more suitable than a survey which might miss crucial perspectives in such a small, impactful decision.

b Survey

c Census

d Survey

e Census

f Survey

g Census

h Survey

i Census

j Survey

k Census

l Survey

m Census

n Survey

7 Contacting all 6000 members for a census would be time-consuming and potentially costly (even if digital, managing responses is effortful). A survey of 300 members, if randomly selected, is much faster, more cost-effective, and can still provide a reliable estimate of overall member interest in the new organic section, making it more suitable than a census.

b Census is 0.32; close match and large sample size suggest reliability. 15 Three reasons why a survey may provide more useful results than a census: 1. Faster data collection: Surveys sample a subset, enabling quicker results, especially for large populations. 2. Lower cost: Surveys require fewer resources, making them more feasible for budgetconstrained studies. 3. Timely insights: Surveys provide estimates rapidly, allowing for more current data compared to a time-consuming census. 16 Second; larger sample size reduces sampling error, making it more reliable.

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b Survey question: “Do you support the popular policies of the current mayor?” Sample: Voters surveyed at a political rally.

2.02 Samples and populations What do you remember? 1 a False

b True

c True

d False

2 a All members of a speciﬁc group of interest. b A subset of the population used to estimate characteristics. c A sample that mirrors the population’s characteristics. 3 A census involves collecting data from an entire population, while taking a sample means collecting data from a group or subset from within the population.

c Survey question: “How often do you walk your dog to the local park?” Sample: Pet owners surveyed at a dog training class. 10 a Biased b Biased c Suﬃcient, as random selection increases the likelihood of a representative sample. d Suﬃcient, as a random sample of 2500 likely captures the diversity of 25 000 residents. 11 a All products in the factory.

4 Answers vary, but may include: • Use random sampling: Ensure that the sample is representative of the entire population by selecting participants randomly. • Design neutral survey questions: Avoid leading or loaded questions that suggest a particular answer. Questions should be worded neutrally to avoid influencing the respondent’s answers. • Ensure anonymity and confidentiality: Allow respondents to remain anonymous and assure them that their responses will be confidential.

b 44% c No 12 a All grapes in the vineyard. b 72% c No 13 a Fair c Biased b Loaded question c Leading question d Leading question b Yes

c No

16 a Emotional language

5 a Population: all library visitors; Sample: the 200 visitors surveyed. b Population: all 800 gym members; Sample: the 50 members surveyed. c Population: all classmates; Sample: the 25 classmates polled. 6 a No

b Yes

c No

7 a Yes

b Yes

c No

8 a Yes

b No

c Yes

9 a Survey question: “Do you agree that our morning coffee blend is the best in town?” Sample: Customers surveyed during morning rush hour.

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d Biased

14 a Leading question

15 a Yes Practice

b Fair

Mathspace New South Wales – Year 11 Standard mathspace.co

b Question is unclear: how large is “large”? c This question is too personal. Individuals might not be comfortable discussing this openly. 17 Advantage: Easy and cost-effective. Disadvantage: Excludes pet owners who don’t visit the store, causing bias. Extend your thinking 18 No, a sample of 50 is too small to reﬂect the diversity of 25 000 members, risking bias. 19 a Use a random number generator to select phone numbers from a directory. b Excludes people unavailable from 3 : 00 p.m. to 6 : 00 p.m., causing bias.

20 a 375 000 b No, as 800 participants may not reflect the diversity of 2 500 000 residents in age or health. c Irrelevant, as household size does not affect side effects, unlike age or health. 21 a First sample: 3333 fish; Second sample: 3571 fish. b Random sampling varies the proportion of tagged fish. 22 a Biased, because the sample size is small and the sample is not representative of the population. b Biased, because the group of people being asked are already users of public transport and so will benefit if the government spends more money on the public transport. c Biased, because the question is leading and contains negative information about shark nets. 23 a The question uses emotive or leading language by including the Prime Minister’s beliefs. Question should be: “Do you think taxes are too high, too low, or just right?” b The question asks more than one question and makes a false assumption that a person has one of the accounts. Question should be: “Do you have a Snapchat account?” and “Do you have a Twitter account?” c The question makes a false assumption that the person has a sibling and that the sibling can drive. Question should be: “Do you have a sibling? If yes, how often do they drive you to school?” d The question uses emotive or leading language as it suggests social media is time-wasting. Question should be: “How much time do you spend on social media each day?” e The question makes a false assumption that a person uses shoe polish. Question should be: “Do you use shoe polish? If yes, what brand do you use most often?”

24 a Country b No, because the question uses emotive or leading language and suggests that country music is cool. It also asks two questions rather than one, as it’s possible to like both types of music (or neither).

2.03 Sampling techniques What do you remember? 1 a Each individual in the population has an equal chance of being selected, typically using a random number generator or similar tool. b Individuals are selected at regular intervals from a list of the population, starting from a randomly chosen point. c The population is divided into subgroups (strata) based on a characteristic, and a random sample is taken from each stratum proportional to its size. d Individuals volunteer to participate, often through open invitations like online surveys, without random selection. 2 a Reduces bias by giving every individual an equal chance of selection. b Simple to implement with a list and ﬁxed interval. c Ensures proportional representation of key subgroups. d Easy and cost-effective to collect responses. 3 a True

b False

e False

f True

c False

d True

4 a self-selected sampling b convenience sampling c simple random sampling Practice 5 a The target population is all students at Bluewater High School. The sampling frame is the school’s enrolment database. b 150 students c Non-response occurs when selected students do not participate in the survey, here 150 − 120 = 30 students.

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6 a Convenience sampling

Systematic sampling: List all 6000 residents and calculate the interval: 6000 ÷ 300 = 20. Select a random starting point between 1 and 20, then select every 20th resident. Advantage: Simple to implement with a list. Disadvantage: Risks bias if the list has a hidden pattern.

b Self-selected sampling c Systematic sampling d Quota sampling 7 a All residents in the council area b All students in the teacher’s class c All members of the sports club d All patients who used the hospital’s emergency services 8 a 30 students

b 80%

9 a Every 30th tablet

b Every 50th driver

10 a Students numbered 5, 12, 19, 27 b Students numbered 3, 15, 28, 42 11 a Junior years: 120 students; Senior years: 120 students b Age 18–35: 160 residents; Age 36–50: 120 residents; Age 51+: 120 residents 12 a i Self-selected sampling

ii 40%

b i Convenience sampling

ii 85%

13 a i Simple random sampling

ii 100%

b i Systematic sampling

ii 94%

14 a Not a sample, as in-person attendees are unlikely to be remote workers. b A sample, as all are remote workers, matching the target population. c Not a sample, as the meeting may include non-remote workers. d Not a sample, as the full list includes non-remote workers. 15 a 40 patrons

b 80%

Extend your thinking 16 Stratiﬁed sampling: Divide residents into strata (e.g., by age: 40% aged 18 − 35, 35% aged 36 − 55, 25% aged 56 and above). Then, select a proportional sample from each stratum. For a sample of 300, this would be 0.40 × 300 = 120 from the 18 − 35 group, 0.35 × 300 = 105 from the 36 − 55 group, and 0.25 × 300 = 75 from the 56 and above group. Advantage: Ensures representation of all age groups. Disadvantage: Requires detailed demographic data, increasing complexity.

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17 Systematic sampling: Low bias if the list is random, easy to implement with a fixed interval, but may miss subgroups if the list is ordered. Self-selected sampling: High bias due to self-selection, very easy to implement (e.g., online surveys), poor representation as only motivated individuals respond. Random sampling: Low bias with equal selection chance, moderately easy with random number tools, good representation but may miss small subgroups. Stratified sampling: Very low bias with proportional strata, complex due to demographic data needs, excellent representation of subgroups. 18 a Year 7: 50 students; Year 8: 50 students; Year 9: 40 students; Years 10–12: 60 students b Stratified sampling ensures proportional representation of each year level, reducing bias and improving accuracy for diverse study habits. Self-selected sampling risks bias, as only motivated students may respond, skewing results. 19 a Surgical: 80 patients; Medical: 60 patients; Outpatient: 60 patients b Response rate: 80%. Non-response may introduce bias if non-respondents (e.g., dissatisfied patients) differ systematically from respondents, skewing satisfaction results. 20 Stratified sampling: Divide residents by age groups, randomly select proportional samples. Justified for ensuring diverse age representation. Bias risk: Requires accurate demographic data; errors in strata may skew results. Volunteer sampling: Post an online survey for open responses. Justified for ease and cost. Bias risk: Self-selection favours motivated or opinionated residents, reducing representativeness.

2.04 Survey design

10 a How often do you engage in physical activity?

What do you remember?

b Which of these income ranges best describes your household?

1 a A question with a fixed set of response options. b A question allowing respondents to answer in their own words. c A question with predefined options and an additional choice for respondents to specify their own answer. 2 a A question that prompts or encourages a specific response. b A question that is unclear or open to multiple interpretations. c A bias where the order of response options influences the answers given. 3 B, C Practice 4 a Partially closed

b Open

c Closed 5 a Biased; How satisﬁed are you with our hospital services? b Fair c Biased; What are your thoughts on the school’s canteen services? 6 a Ambiguous due to vague term “often.” Improved: How many times have you visited a doctor in the past 12 months? b Ambiguous due to subjective term “good.” Improved: How satisﬁed are you with the program’s quality? c Ambiguous due to broad term “travel.” Improved: How many times per year do you travel interstate? 7 C 8 a Is the school library spacious? Is the school library well-resourced? b Are the park facilities clean? Are the park facilities safe? 9 a Do you think it’s necessary to upgrade the school facilities? b Do you always attend community events?

c To what extent do you agree or disagree with participating in community volunteering? 11 Placing the opinion question first may influence students’ responses. The structure should be improved by asking factual or behavioural questions before moving on to opinion-based ones. Extend your thinking 12 Importance: Pretesting ensures questions are clear, unbiased, and relevant, improving survey reliability. Steps: 1. Administer the survey to a small, diverse group of NSW residents. 2. Collect feedback on question clarity and response options. Issue: Pretesting could identify leading questions (e.g., “Don’t you agree recycling is important?”), allowing revisions to remove bias. 13 Fault: Order bias – activities listed first (train, bus) may be chosen more due to their position. Address: Randomise the order of activities for each respondent. 14 Misleading due to a small, non-representative sample (one school, only 60 responses), which may not reflect all NSW students. Improvement: Use stratified random sampling to survey students from multiple NSW schools, ensuring a larger, diverse sample. 15 This could introduce visual design bias, where the more prominent option attracts more responses regardless of true preference. To improve fairness and reliability, all response options should be presented with equal visual weight—consistent font, size, colour, and layout. 16 This strategy may compromise voluntary participation by pressuring respondents to answer sensitive questions to receive a reward. To improve ethical integrity, the prize draw should not be tied to answering optional questions, and those questions should be clearly marked as voluntary.

c Do you support the new policy?

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2.05 Privacy, bias and ethics

b The sample is from self-selecting participants.

What do you remember?

c The sample is too small.

1 A sampling error is the variation between the true population value (parameter) and the sample estimate (statistic), often due to randomness or poor sampling methods. 2 a True

b False

c True

d False

3 a Cherry-picking data b Strategic choice of measures of centre and spread c Exaggeration d Confusing correlation and causation e Comparing dissimilar datasets g Lack of context b i

c iii

d ii

Practice

b Measurement error c Measurement error 6 a Over-generalisation and selection bias b Exaggeration c Strategic choice of measures of centre and spread d Cherry-picking data e Comparing dissimilar datasets c Yes

d No

8 1. Unrepresentative sample: The sample only includes Year 10 students, who may have different screen time habits than other year levels. 2. Bias: The survey measured screen time, not opinions on whether it was excessive. Sarah’s claim of “excessive screen time” reﬂects her personal bias. 3. Procedural issues: Details about the survey’s administration, such as timing or response conditions, are missing, which could affect reliability. 9 a The sample does not represent the population.

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c Disregarding cultural diversity. d Publishing identiﬁable participant data. 11 a The negative phrasing may confuse respondents, leading to inaccurate responses. b The question is personal and may lead to social desirability bias, as respondents may underreport consumption.

12 a The timing of the survey after a promotional event may bias responses due to heightened positive sentiment.

13 a The sample does not represent the population.

d Sampling error

b Yes

b Collecting sensitive data without consent.

b Measurement error: Responses may reﬂect temporary excitement from the event rather than typical satisfaction, skewing results.

5 a Sampling error

7 a No

10 a Using leading or biased questions.

c The question is leading, implying the eco-friendly choice is superior.

f Misleading graphs 4 a iv

d The sample is not random.

Mathspace New South Wales – Year 11 Standard mathspace.co

b The environmental fair likely attracts eco-conscious individuals, overestimating support for the recycling program among all residents. 14 Fault: The question assumes all respondents prioritise longevity, ignoring cultural or personal values that may define health differently. Improvement: Rewrite as, “What practices do you follow to maintain your health?” to respect diverse health perspectives. Extend your thinking 15 a Yes b Exaggeration due to a small base value, making the increase seem more significant. 16 a Strategic choice of measures, using the mean influenced by outliers. b The mean overstates typical calorie burn, as the median better represents most users’ experience.

17 Issues:

9 a All t-shirts produced by the factory.

• Non-response bias: Only 55% responded, so the claim does not represent all users. • Ethical issue: Claiming 100% satisfaction is deceptive and lacks transparency. • Bias: The survey may exclude users with accessibility issues or language barriers. Improvement: Provide the survey in multiple formats (e.g., online, phone, translated versions) to include diverse users.

Chapter 2 review 1 C

b 92% c Sample proportion (92%) is below the aim (95%). It suggests the aim may not be met, but further sampling is needed to conﬁrm. 10 a Yes. Use a user database or broader sampling frame (e.g., electoral roll, random digit dialing) to randomly select and invite participants to a secure poll. b Self-selected sample. Only engaged readers participate, likely unrepresentative of all citizens, causing selection bias. 11 a 540 000

2 C 3 C 4 a Survey

b Census

c Survey

d Census

5 A survey of 500 voters is faster and more cost-effective than a census of 25 000, which requires significant resources (printing, postage, data entry). A random sample can provide a reliable estimate of community opinion efficiently. 6 a 0.430 b Census: 0.45. Survey: 0.430. The survey is close to the census, and 5000 hectares is a large sample, suggesting reliability despite minor sampling variation. 7 a Census: Advantage - complete, accurate data. Disadvantage - costly, time-consuming. Survey: Advantage - faster, cost-effective. Disadvantage - estimates with potential sampling error. b Survey. A census for 800 000 is too expensive and slow. A well-designed survey with a large, representative sample provides timely, accurate insights efficiently. 8 a Population: All students using university food services. Sample: 150 students questioned. b Population: 1500 electric cars in the batch. Sample: 75 cars tested. c Population: All festival attendees. Sample: 40 interviewees.

b No. Volunteer beta testers, likely tech-savvy and from Sydney, may not reflect the diverse Australian user base (e.g., internet speeds, devices). c Relevant. Usability issues may vary by OS. A sample skewed to one OS may miss issues on others, reducing representativeness. 12 a Convenience sampling b Self-selected sampling c Systematic sampling d Stratified sampling 13 a Every 50th subscriber b Every 30th loaf 14 Strategy 1: Simple random sampling. Justification: Each resident has an equal chance of selection. Bias: Non-response bias; non-responders may have different opinions. Strategy 2: Convenience and self-selected sampling. Justification: Targets market-goers who choose to participate. Bias: Selection bias; market attendees may not represent all residents. 15 a Arts: 120; Science: 100; Engineering: 80; Business: 100. b Stratified sampling ensures proportional representation of each faculty, critical if career aspirations vary by faculty. Simple random sampling risks uneven faculty representation. 16 a Closed b Open c Partially closed

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17 a Ambiguous: “Regularly” is vague. Improved: How many days per week do you engage in at least 30 minutes of moderate physical activity? b Ambiguous: “Satisfactory” is subjective. Improved: On a scale of 1 (Very Dissatisfied) to 5 (Very Satisfied), rate the park’s cleanliness. c Ambiguous: Unclear time frame. Improved: On a typical weekday, how many hours do you use a computer for leisure? 18 Importance: Pretesting ensures questions are clear, unambiguous, and not leading, improving data reliability. Steps: 1. Administer the survey to 10 − 20 similar residents. 2. Collect feedback on confusing or sensitive questions. Issue: Leading question, e.g., “Don’t you agree the library’s hours are fantastic?” Pretesting could reword it to “How satisfied are you with library hours?” 19 Fault: Order bias (primacy effect). Early options (Weekly, Fortnightly) may be over-selected. Fix: Randomise option order electronically or use multiple versions for paper surveys. 20 a 0.416

b 250

21 a 24

c 0.400 b 0.629

22 a Around 0.48 to 0.62, clustered near 0.55.

d Cultural insensitivity. 26 Issues: Misleading claim (exaggerates satisfaction), non-response bias (70% non-responders may differ), lack of transparency about response rate. Improvement: Offer the survey in multiple languages and ensure accessibility (e.g., mobile-friendly, screen reader-compatible) to engage diverse subscribers. 27 a Using mean inﬂated by outliers instead of median. b The mean (20 hours) overstates typical usage compared to the median (8 hours), giving a false impression of platform engagement.

3.01 Classify data What do you remember? 1 a False

b True

2 a Numerical c Continuous

c True

d False

b Nominal d Context-dependent

3 a Produces categories, e.g., types of fruit b Produces countable whole numbers, e.g., number of books c Produces categories with a natural order, e.g., class ranks

b Decreases spread, making proportions narrower around 0.55.

d Produces measurable quantities with decimal values, e.g., weight of a person

c 0.55

e Produces data type depending on survey question phrasing

d Yes. 60% is close to 55% and within expected sampling variability for size 200. 23 a Sampling error (selection bias). b Measurement error (response bias). c Measurement error (systematic error). d Sampling error (convenience sample). 24 a Exaggeration (small base inﬂates percentage). b Over-generalization, selection bias. c Cherry-picking data. d Confusing correlation and causation. 25 a Lack of informed consent. b Leading questions.

986

c Failure to ensure anonymity.

Mathspace New South Wales – Year 11 Standard mathspace.co

Practice 4 a Numerical

b Categorical

c Categorical

d Categorical

e Categorical

f Numerical

g Numerical

h Categorical

i Categorical

j Numerical

k Categorical 5 a Nominal

b Nominal

c Ordinal

d Nominal

e Ordinal

f Ordinal

g Ordinal

h Nominal

Disneyland visitors

160

Number of people (in hundred thousand)

159 158

b 23.81%

16 a India

b $2.47

c $113.22

157 156

d 9.5 GB

17 a Bus

155 154

b 17 604 000 trips

c 17.5%

153 152 151 150

15 a 2100 movies

2008

2009

2010

2011

2012

Year

9 a Non-smoker

b Approximately 63%

c Approximately 27% 10 a 17 males 11 a

d 69.0%

18 a 4500 visits

b 2500 visitors

c 0.375

d 2900 visitors

19 a 0 – 10

b 11 − 20

c 26.9%

d 41 − 50

Extend your thinking

b 10 females

c 29 males

Red

Brown

Blonde

Black

Straight

Curly

Local shops

Online marketplaces

Men

Women

20 a Large sedan

b $55.40

c $86.50

d $9.40

21 a 35%

b Rome and Shanghai

c Dubai

d 65%

22 a Cinemas

SecondMalls hand markets

Comedy Action Drama Horror Women

Men

12 a 100

b 44

13 a 43 girls

b 10 c The local art gallery and the annual theatre production. 23 a Metropolitan b Off path (on curve) c Intersection, rear end, pedestrian and side swipe d i False

ii False

iii True

24 a Under 25 years old, Male b $286

c 44% b 25 students

c Science

c $1298 d i No

ii No

iii Yes

iv No

3.03 Sector and line graphs

14 a Swimming carnival results

What do you remember?

100 90

1 A pie graph is a circular chart divided into sectors to represent proportions of a whole, and the sum of the central angles is 360°.

80 70

Points

2 a True

50 40 30 20 10 0

Yellow

b Green

988

Blue

Red

Green

c Red

Mathspace New South Wales – Year 11 Standard mathspace.co

b True

c True

d False

3 A divided bar graph is a graph where the bar represents the whole dataset, divided into segments proportional to each category’s size. Choosing a bar length as a multiple of 5 or 10 simpliﬁes calculations for segment lengths.

Practice

4 a Science

How long homesickness lasted

b 5%

10%

d 45

25% 5 a Purple

b 12 children

20%

c 30 children 15%

6 a Pizza

30%

b Nuggets and noodles 7 a Snakis

b Wrinkles

c Hot Cheezos 8 a i $60

d 224 students

ii $120

iii $360

1−2 weeks

1−2 months

3−4 months

15 a 0.5 m

9 a 144°

b 360°

c 54°

10 a 25%

b 41%

c 19%

Genre

Frequency

Rock

b 7.5 m

c 1.5 m

e The second year

f 10 m

16 a $1.50 Fraction Angle

d $2.75

e September and April f $1375

100°

17 a

Pop

80°

Pareto hotness scale

Soundtracks

40°

Number of customers

RnB

140°

Total

180

360°

g $625

RnB Pop

Soundtracks

b Monday and Tuesday c 8°C warmer

d 4 customers

e 13 customers

f 69 customers b 65 mm d 30 millimetres

e 65 millimetres

Rock

13 a Sunday

c 3 customers

18 a 70 mm

b 50

, , , , ,

c Berlin

12 a 2013

d 3.5 m

b December and June

c $750

3−4 weeks

iv $300

b $1800

11 a

Less than a week

c 25

19 a

110 Rainfall (mm) 100 90 80 70 60 50 40 30 20 10

Month

0 1 2 3 4 5 6 7 8 9 10 11 12

b Rainfall is highest in January, then drops sharply to April. It remains low from April to September, with September having the least rain. Rainfall rises steadily from October to December.

Answers mathspace.co

989

popular and prove to be excellent choices for after school programs, catering to a broad interest base and promoting physical and creative development.

Number of vegetables

9 8 7 6 5 4 3 2 1

There is an increasing trend in homework as students advance through school levels, with it becoming the most dominant activity in high school. This suggests the potential for introducing structured after school homework sessions at the high school level, which, if successful, could also be considered for middle schools to support academic achievement.

Day 0

21 a 1800 books

b 180 000

22 500 000 000 23 a 11.7 10.8 9.9 9 8.1 7.2 6.3 5.4 4.5 3.6 2.7 1.8 0.9

25 Answers may vary. Possible answer: I agree with DeShaun.

UV index

Larger sample sizes typically decrease sampling error, making generalisations more reliable. Lucille collected her data from her classmates, so her sample size is much smaller than the 10 000 people represented in DeShauns graph. 1 mark for 1 sampling feature (sampling error)

Time 600 800 1000 1200 1400 1600 1800

b 1400

c Between 1000 and 1600

Extend your thinking 24 Answers may vary. Possible answer: Across all school levels, working for a family business remains the least popular activity, highlighting minimal involvement in such activities amongst school-aged children. This trend suggests that after school programs would benefit from focusing on other areas. There is a noticeable decrease in TV watching as students progress from elementary to high school, reflecting changing interests, capabilities, and responsibilities as students age. Given that watching TV does not benefit the developmental and educational needs of students, it would not be appropriate as an after school program. A good proportion of students participate in sports teams and artistic activities across all educational stages. These activities remain

Lucille’s sample is likely to be less diverse, since her classmates all attend the same school and are likely to be from the same community. Consequently, the results of this graph may reﬂect localised or limited preferences and may not accurately capture broader trends across a more diverse population. 1 mark for 1 sampling feature (diversity) Answers may vary. Possible answer: I disagree with DeShaun. Lucille collected the data herself, whereas DeShaun used data from an external source. There are no details provided about the procedure used to collect the data. Who conducted the survey, what sampling technique was used, how was the data collected, did the survey use biased questioning? 2 mark for multiple sampling features all linked to procedure. 26

20 Water 18 16 14 12 10 8 6 4 2 0

990

Mathspace New South Wales – Year 11 Standard mathspace.co

Time 3

The separate stem-and-leaf plots show Soil Type A’s distribution is more even, while Soil Type B has clusters at lower values (e.g., two 4s).

Practice 4 a

Conclusion: Soil Type A is slightly more effective for seed germination, with a higher mean (10.47 vs. 9.67), median (11 vs. 10), and total seeds germinated (157 vs. 145). Both soil types have similar variability (range = 12, IQR = 7) and show increasing trends, but Soil Type A performs better in early and middle stages. The difference of 12 seeds may not be signiﬁcant enough to strongly prefer Soil Type A unless other factors (e.g., cost) are considered.

Frequency

b 16 + 18 + 16 + 22 + 10 + 10 = 92 5

23 a 56%

Score

Class

Frequency

30–39

d 46 books

40–49

e The mode is 39 books. In the context of the reading challenge, this means that the most common number of books read by children across both the summer and winter challenges was 39. Three children (two in summer and one in winter) read exactly 39 books, more than any other number. Since 39 is less than the challenge goal of 47 books, it indicates that the most frequent reading achievement fell short of the target, suggesting that 39 books was a typical effort level for participants, but the goal may have been ambitious for many children.

50–59

60–69

70–79

80–89

b 44% c Summer challenge

6 a

Mass (g)

Frequency

401

402

403

404

405

406

407

408

409

410

3.05 Histograms and grouped frequency tables What do you remember? 1 a Ungrouped

b Grouped

c Ungrouped

d Grouped

2 a False

b False

c True

d True b

3 The class centre is the midpoint of a class interval, calculated as: Frequency

4 3 2 1 0

401

402

403

404

405

406

407

408

409

410

Mass (g)

Answers mathspace.co

995

7 a 22.1

b 19–27

8 a 9.3

b 1–5

Score (x)

Frequency ( f )

10–19

20–29

30–39

40–49

50–59

Score (x)

Frequency ( f )

30–39

40–49

50–59

60–69

70–79

80–89

Frequency

10 a

20 18 16 14 12 10 8 6 4 2 0

Extend your thinking

c Yes d No. The table shows ranges of lengths and not individual leaves. We do not know if any of the leaves in the range 0 ≤ x < 20 are less than 5 mm. 14 a Sylvia’s is more useful, as the class intervals are smaller so we can describe the shape of the data with more accuracy and identify trends in more detail. b No, Lee is not correct as the histogram shows that 51 patients lived between 21–30 years. In fact, by using Salvia’s histogram, we can see that 0 patients lived 30 years after being diagnosed. 15 a

b 2–3 coffees 11 a 140.9 cents

Frequency

80 100

b 11 days

b i No 16 a

ii Yes

iii No

Battery life (minutes)

iv No

Frequency

400 ≤ x < 500

500 ≤ x < 600

600 ≤ x < 700

700 ≤ x < 800

800 ≤ x < 900

900 ≤ x < 1000

1000 ≤ x < 1100

1100 ≤ x < 1200

1200 ≤ x < 1300

Total

117

2 1 0

120.9 125.9 130.9 135.9 140.9

Price (in cents per litre) d 140.9 < p ≤ 145.9

996

20 18 16 14 12 10 8 6 4 2 0

Commute time (minutes)

Number of coffee purchases

b 60 mm ≤ x < 80 mm

13 a 35

Frequency

9 a

12 a 25 students b 13 students c 16%

Mathspace New South Wales – Year 11 Standard mathspace.co

b 89.7%

c 9.4%

3.06 Cumulative frequency tables and graphs

6 a

What do you remember? 1 A cumulative frequency polygon, or ogive, is a line graph connecting the cumulative frequencies at the upper right endpoint of each class interval. 2 a False

b True

c False

d True

3 In a cumulative frequency histogram, bars always increase in height from left to right, as each represents the running total of frequencies, unlike a regular frequency histogram where bar heights vary based on individual class frequencies.

Practice 5 a

Frequency (f )

Cumulative frequency (cf )

20 − 24

25 − 29

30 − 34

35 − 39

40 − 44

45 − 49

50 − 54

Cumulative frequency ( cf )

4 Cumulative frequency is the running total of frequencies, calculated by adding each frequency to the previous cumulative total. The ﬁnal value represents the total frequency of the dataset.

Score (x)

80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

Usage Level

Trips (x)

Frequency (f )

Cumulative Frequency (cf )

Minimal

0≤x<5

Low

5 ≤ x < 10

Moderate

10 ≤ x < 15

7 a 26

Frequent

15 ≤ x < 20

8 a

Heavy

20 ≤ x < 25

105

c 75 d 34 b 6

c 18

Score (x) Frequency ( f )

Cumulative frequency (cf )

c 5

d Approximately three-quarters of the adults made fewer than 17.75 trips.

b 105

Score (x)

Answers mathspace.co

997

Extend your thinking

Cumulative frequency ( cf )

60 54 48 42 36 30 24 18 12 6 0

13 a 2 b 12 c

Score (x)

c 60

Cumulative frequency (cf )

Score (x)

Frequency (f )

1−4

5−8

9 − 12

13 − 16

17 − 20

b 27

d 7 14 a

c 4

e 11

f 18

Number of Number of sightings locations ( f )

Cumulative frequency (cf )

Score

Frequency

11 a 14 12

b 2

c 0

d 71.4%

Frequency

b 19 locations

c 12 locations

15 a

50 100 150 200 250 300 350

Mass (g)

998

Frequency ( f )

d 14.5

9 a

Score (x)

Mathspace New South Wales – Year 11 Standard mathspace.co

Time recorded

Frequency

Cumulative frequency

16:00 ≤ x < 16:10

16:10 ≤ x < 16:20

16:20 ≤ x < 16:30

16:30 ≤ x < 16:40

16:40 ≤ x < 16:50

16:50 ≤ x < 17:00

b 25

c 16

d 48%

Cumulative Frequency

16 a

22 20 18 16 14 12 10 8 6 4 2 0

6 Positively skewed 7 Negatively skewed 8 Positively skewed 9 In a positively skewed distribution, the long right tail increases the mean, making it greater than the median, which is greater than the mode. 145 150 155 160 165

Height (cm)

170

b 16 students

c 6 students

d 2 students

e 153

3.07 Shape of distribution What do you remember? 1 Possible answers: • The data is evenly distributed, with the left side (less than the mean) mirroring the right side (greater than the mean). • The left side of the distribution (less than the mean) mirrors the right side (greater than the mean). • The mean, median, and mode are very similar. 2 Positive skew refers to a distribution where the tail is on the right side of the distribution. Negative skew refers to a distribution where the tail is on the left side of the distribution. 3 Approximately 50% of the data lies above the mean, and 50% lies below the mean. 4 The median is greater than the mean. Practice 5 a Negatively skewed b Positively skewed c Symmetric d Positively skewed e Positively skewed f Negatively skewed g Positively skewed h Symmetrical

10 The median ($190) better represents the typical bill because, in a positively skewed distribution, the mean is inﬂated by high values in the right tail, while the median is less affected by extremes. 11 Nearly symmetric. The distribution of scores is highly balanced around the central class (the 70s). The frequency of scores in the stems below and above the centre is almost identical. While the mean is slightly lower than the median, suggesting a very slight negative skew, the overall shape is best described as nearly symmetric. Extend your thinking 12 Positively skewed. The majority of data is concentrated at lower times (10–15 minutes), with a longer right tail toward higher times (up to 25 minutes). 13 The heavy right skew in the sales data suggests that most salespeople have low sales numbers, while a few have very high sales numbers. 14 The symmetric distribution indicates consistent performance, while the positively skewed distribution suggests most students scored lower, with a few high achievers. The teacher should use measures of centre and spread for a more accurate comparison. 15 The distribution is positively skewed, with most wait times short (2–12 minutes) and a long right tail (up to 75 minutes). The median should be used to report the typical wait time, as it is less affected by extreme high values. The positive skew suggests that most customers are served quickly, but a few experience signiﬁcantly longer waits, possibly due to occasional delays or complex requests.

Answers mathspace.co

999

3.08 Misleading graphs and appropriate displays

d Pie graph. It summarises large categorical data into proportions.

What do you remember?

e Dot plot. It displays frequency for a small categorical dataset.

1 a False. Bar graphs are better for comparing categories, not continuous trends. b True. Pie graphs effectively display parts of a whole.

f H istogram. It shows frequency distribution for continuous data. g Line graph. It shows trends and patterns of step counts over weeks (time series data).

c True. Dot plots clearly show individual data points for small datasets.

8 a No. Dot plots are unsuitable for large categorical datasets.

d False. Stem plots are better for small ranges to avoid clutter.

b Yes. Bar graphs compare categorical frequencies effectively.

e True. Both can show categorical data, but dot plots show counts, while pie graphs show proportions.

c No. Picture graphs are less precise for numerical data.

g False. Dot plots can clearly display outliers for small datasets. 2 A misleading graph distorts data, leading to incorrect conclusions. Features: manipulated scales (e.g., non-zero y-axis), omitted data (e.g., missing outliers), inappropriate graph types (e.g., pie graph for non-proportional data). 3 a i

b iv

c ii

d iii

4 Advantage: Visually compares proportions. Limitation: Cluttered with many categories. 5 A manipulated scale distorts data by altering axis increments or starting points. Example: A y-axis starting at 50 instead of 0 exaggerates differences. Practice

d Yes. Pie graphs show proportions of categorical data. 9 a Answers may vary. Possible answer is any a column graph where the y-axis is not starting at 0. Graph 1 50

No. of days with freezing temperatures

f F alse. Histograms show frequency in intervals, not individual values.

b Line graph. It displays continuous changes over time. c Dot plot. It shows individual responses for a small dataset.

1000 Mathspace New South Wales – Year 11 Standard mathspace.co

47 46 45 44 43 2017

2018

b Start the y-axis at 0 to provide accurate context. Graph 2 50

No. of days with freezing temperatures

7 a Pie graph. It shows proportions of categorical data effectively.

6 a The vertical axis lacks a scale. b Without a vertical scale, sales values are unclear, making it impossible to gauge the trend’s signiﬁcance. A truncated axis may exaggerate differences, leading to false conclusions about sales growth.

45 30 25 20 15 10 5 0

2017

2018

12 The frequency still sums to 28 according to the histogram and a 10% jump from 90% to 100% exaggerates the peak, leading to misinterpretation of occupancy distribution. 13 The scale skips from 16 to 18, omitting 16 to 18, minimising the high score at 19 and suggesting less variability. 14 Compressed spacing exaggerates trends, making changes seem dramatic, while expanded spacing flattens trends, downplaying variations, both potentially leading to misinterpretation. 15 A wide scale compresses data, minimising changes, while a narrow scale exaggerates fluctuations, leading to misinterpretation of trend significance. b Both graphs clearly show Debate as the most popular activity (highest bar, highest point). 17 a The sum of the percentages exceeds 100%, not accurately showing proportions. b Bar graph allows direct comparison of vote counts, making differences clearer than proportional segments. Presidential run 2012 75

Supports (in %)

Number of books Deana read

9 8 7 6 5 4 3 2 1 0

Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec. Months

b Answers may vary. Possible answer: A line graph is best to show trends in reading over time, clearly displaying ﬂuctuations and patterns. 20 A side-by-side bar graph is best for comparing team scores across matches, clearly showing differences. 6 5 4

Team 1 Team 2

3 2 1

16 a Column graph

C Match

Extend your thinking 21 The graph lacks a vertical scale, and the Argentinian column is disproportionately tall, exaggerating similarity. Actual data shows a signiﬁcant gap, making the claim misleading. 22 The claim is misleading. The box plot shows only 25% of apartments are below $630 000, with a median of $687 500, indicating most are above. The plot obscures individual prices, limiting precise conclusions.

60 35 30

23 a Bar graph. It clearly shows Constantina’s lead, but may exaggerate differences. Pie graph obscures exact vote counts.

15 0

19 a A line graph:

Number of books

11 Segment sizes misrepresent viewer counts. Soccer (180 viewers) appears smaller than football (120 viewers), distorting proportions and leading to incorrect conclusions.

18 The line of best ﬁt is too steep, exaggerating the relationship between usage and satisfaction, leading to overestimation of the trend’s strength.

Scores

10 The histogram misrepresents the data by starting its horizontal axis at 8000, which excludes the significant outliers of 2000 and 4000 steps. This makes the daily counts appear higher and more consistent than they actually are.

Huckabee

Romney Candidates

Palin

b Pie graph. It minimises vote differences, but lacks precise counts, potentially misleading viewers about closeness.

Answers 1001 mathspace.co

Using a pie graph:

24 Using a line graph:

Total annual sales distribution of bakery items

Monthly sales of cookies over time Number of cookies sold

200 175 150 125 100

23.6%

25.6%

29.8%

21.0%

Bread

Cakes

Pastries

75 50 25 0

Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec. Months

• The line graph highlights the trend in cookie sales over the year. • You can see ﬂuctuations and identify months with higher or lower sales.

• The pie graph shows the proportion of total annual sales each item contributes.

Using a column graph:

• Cakes have the smallest segment(21.0%), revealing their lesser contribution to total sales.

Number of bakery items sold

• For example, sales peak in December (175 cookies) and drop to their lowest in February (120 cookies), showing a general upward trend over the year.

160

Sales of different bakery items in January

• This visualisation helps in understanding the overall sales distribution and identifying the most and least popular items over the year.

150 140 130 120

Chapter 3 review

110

1 C

100 90 80

• Bread is the largest segment, indicating it has the highest sales proportion (29.8%).

2 B Cookies

Cakes Bread Bakery items

Pastries

• The column graph provides a snapshot of sales for each item in January. • You can easily compare the sales of different items. • Bread had the highest sales (150 units), while Cakes had the lowest (90 units), indicating the relative popularity of each item in January.

3 B 4 a Numerical

b Categorical

c Numerical

d Categorical

e Numerical 5 a Answer may vary. Possible answer: How often do you exercise per week? • (A) Rarely or never • (B) 1 − 2 times • (C) 3 − 4 times • (D) 5 or more times b Answer may vary. Possible answer: On average, how many minutes do you spend exercising per session? 6 a 12 females b 15 males c Practice papers

1002 Mathspace New South Wales – Year 11 Standard mathspace.co

c 5 hours

50 45 40 35 30 25 20 15 10 5 0

d 57.5%

Visitors (thousands)

7 a

15 a

Spring

Summer Autumn Season

b Summer

ii $1125

iii $675

iv $450 b 25°C

c May and September d 15°C 10 a 20 students

b 28 points

c 42 points

d 30 marks

11 a 175 cm

b 198 cm

c Rockets 12 a

d 1 cm

Age Group

Frequency

10 − 19

20 − 29

30 − 39

40 − 49

50 − 59

60 − 69

Hours (x) Frequency ( f )

5− < 10

10− < 15

15− < 20

20− < 25

75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

c 70 calls

d 53 calls

16 Positively skewed 17 Median (410 ms). In positively skewed data, the mean is inﬂated by large values, making the median a better measure of centre.

b 25 − 29 minutes

14 a

0− < 5

Call duration (mins.)

b 55 members 13 a ≈ 26.2 minutes

Cumulative Frequency (cf )

Cumulative frequency

b $590 9 a July

Frequency (f )

Winter

c 18 000 visitors

8 a i $1350

Call Duration (mins)

Cumulative Frequency (cf )

0−4

5−9

10 − 14

15 − 19

20 − 24

18 a Histogram. Suitable for continuous data with a large dataset. b Pie graph. Shows proportions of categorical data effectively. c Dot plot. Ideal for discrete data with a small dataset, showing individual values. 19 a The vertical axis lacks a clear numerical scale, potentially exaggerating profit changes. b An unclear scale may exaggerate small profit increases, misleading viewers about financial growth.

b 40 students

Answers 1003 mathspace.co

4.01 Multiplication and division by powers of 10

Extend your thinking 13 1.5 × 10−4 mg 14 The diameter is scaled by 102.

What do you remember?

15 The number is 4.5. Then, dividing it by 102 gives 0.045.

1 a i

b ii

2 a Divide by 100

b Multiply by 1000

c Divide by 100

d Multiply by 100

16 420 mm

3 a n=2

b n=3

4.02 Units of length and area

c n = −2

d n = −4 What do you remember?

Practice

1 a m

4 a 10

b 1000

c 100

d 10 000

e 10

f 1000

g 100

h 10 000

5 a 60

d 510 000

e 430

f 45 000

g 1000

h 950 000

3 a 1000 m = 1 km

b 100 cm = 1 m

c 1 cm = 10 mm

d 1000 mm = 1 m

e 1 mm2 = 0.000 000 000 001 km2 f 1 km2 = 10 000 000 000 cm2

b 4

c 6.4

d 7.8

Practice

e 27

f 92

g 7

h 5

4 a 30 mm

7 a 8.2

b 370

c 48

d 12 500

e 0.6

f 9400

g 130

h 680

b 0.1794

c 0.000 003 7

d 0.096

e 0.0619

f 0.024 05

g 0.000 008 1

h 0.000 734

9 a 1000

b 10 000

c 100

d 10 000

e 10 000

f 1000

10 a 10

b 100

c 1000

d 10 000

e 100

f 10 000

g 1000

h 10 000

11 a 7200

b 0.031

c 7

d 9.8

e 560

f 3.4

12 a 45 000

d mm

b mm2, cm2, m2, km2

6 a 8

8 a 0.528

c cm

2 a There are 10 mm in 1 cm, 100 cm in 1 m, and 1000 m in 1 km.

b 800

c 12 000

b km

b 0.0072

d 8m

e 6000 m

f 9 km

g 4.5 cm

h 72 mm

i 1250 cm

j 3.5 m

k 2800 m

l 15 km

5 a 24 000 mm

b 3.484 m

c 600 000 cm

d 8.668 km

e 15 000 000 mm

f 7.5 m

g 3200 mm

h 0.125 km

6 a The desk is 19 cm longer than the rug. b 5.34 m

c 1.1 m

d The bridge is 300 m longer. 7 a 30 000 cm2 2

b 560 000 m2

c 60 000 m

d 2 ha

e 0.045 m2

f 12 000 cm2

g 8 ha

h 0.015 km2

8 8.41 m2 9 76 000 m2

c 81 000 000

d 930

e 0.0015

f 7000

10 16.5 m2

g 6.2

h 1 200 000

11 0.025 km2

1004 Mathspace New South Wales – Year 11 Standard mathspace.co

b 6 cm

c 600 cm

2 a mL

Extend your thinking 12 a Name

Height Height (metres) (kilometres)

One World Trade Center

541.3

0.5413

Shanghai Tower

632

0.632

Petronas Twin Towers

451.9

0.4519

Zifeng Tower

450

0.45

Burj Khalifa

828

0.828

b Zifeng Tower, Petronas Twin Towers, One World Trade Center, Shanghai Tower, Burj Khalifa 13 Area has two dimensions, length and width, and 100 × 100 = 10 000.

3 a Capacity

b Volume

c Volume

d Capacity

d L

4 1 L = 1000 mL 5 a 1 km = 1000 m, therefore to convert from km3 to m3 multiply by 10003 which is 1 000 000 000 m. b 1 m = 100 cm, therefore to convert from m3 to cm3 multiply by 1003 which is 1 000 000 cm. c 1 cm = 10 mm, therefore to convert from cm3 to mm3 multiply by 103 which is 1000 mm. 6 Divide by 1003 = 1 000 000. Practice 7 1000 mm3 8 a 3500 mm3 c 4 000 000 000 mm3

15 16 years Farm

c kL

b 4 200 000 000 mm3

14 666.67 years

16 a

b L

d 19 000 mm3 Length, m Width, m

Area, m2

9 a 5.36 L

d 1670 L

b 5L

200

100

20 000

c 9000 L

350

7350

10 a 18 620 L

b 2000 L

c 7.85 L

d 0.8 L

100

2400

400

16 000

100

10 000

11 a 68 L

b 52 000 mL

c 88 kL

d 63 000 L

e 1 mL

f 1 cm3

b Farm 1 and Farm 4

g 36 mL

h 720 cm3

c Farm 5

i 1 kL

j 1 m3

k 83 kL

l 270 m3

m 400 mL

n 0.0038 L

o 1670 L

p 0.0846 kL

17 a 72 200 m2

b 18.05 acres

18 4 m from each long side and 2 m from each short side. 19 203 years

b 1 kL

c 1 000 000 mL

20 195 m2

13 a 144 m3

21 46 400 chickens 22 a 136 ha

b 1 360 000 m

What do you remember? b m3

b 144 kL

14 4.3 m3 2

4.03 Units of volume and capacity

1 a m3

12 a 1000 L

c cm3

d m3

15 a 100 cm3 b v = lwh = 10 × 5 × 2 = 100 cm3 c Answers may vary: length = 5 cm, width = 5 cm and height = 4 cm

Answers 1005 mathspace.co

16 a e.g. multiply the volume by 1 000 000 to convert to cm3. 1 cm3 = 1 mL so we now have the capacity in mL. Divide by 1000 to convert to L. 3

b 1000 a L

5 200 000 mg 6 4000 mg 7 2 kg and 468 g 8 500 g

c a3 kL

9 5.5 kg

d 1 m3 = 1 kL

10 $168 000

17 Method 1: 40 × 60 × 150 = 360 000 mm3 Method 2: 4 × 6 × 15 = 360 cm3 3

11 290 g 3

360 cm = 360 × 10 mm = 360 000 mm 3

12 a 6400 kg

18 1 L is equal to 1000 cm , and to convert the tank’s volume to litres, we need to divide the volume in cubic centimetres by 1000. So, 120 000 cm3 ÷ 1000 = 120 L.

15 21.8 kg

19 4.8 L

16 5.43 kg

20 a 1600 cm

b 40 cm

b 10 400 kg

13 1700 g 14 5.2 kg

17 400 g

Extend your thinking

21 150 cups

18 32 loads

22 400 23 a 36.469 m

19 0.138 g 3

c 34 015 L

b 36 469 L d 1718 L

e 20 24 7

20 10 750 g 21 a 0.12 m3

b 942 kg

c 9 beams

d 40 500

4.04 Units of mass

4.05 Scientific notation and significant figures

What do you remember?

1 a 6000 mg c 470 mg 2 a 0.087 g

b 6 mg

1 Measurement

Preﬁx

Scientiﬁc Notation

0.000 000 007 2 m

nano-

7.2 × 100 nm

4500 g

kilo-

4.5 × 100 kg

0.065 s

milli-

6.5 × 101 ms

d 72 668 mg b 0.155 g

c 8.505 g

d 52.005 g

3 a 0.023 kg

b 0.0565 kg

c 2.452 kg

d 0.014 526 kg

e 0.152 620 kg

f 1.265 253 kg

12 000 000 bytes

mega-

1.2 × 101 MB

g 3200 kg

h 54 000 kg

0.000 000 45 m

micro-

4.5 × 10−1 μm

9 800 000 000 bytes giga-

9.8 × 100 GB

15 000 000 000 000 bytes

1.5 × 101 TB

Practice 4 a 6251 g

b 12 000 g

c 0.946 kg

d 11 000 kg

e 3 kg

f 5000 kg

g 9272 kg

h 589 kg

i 1 tonne

j 8.5 tonnes

1006 Mathspace New South Wales – Year 11 Standard mathspace.co

tera-

2 a True

b False

c False

3 a 3

b 2

c 5

d 4

4 a 5.7 × 10−8 s −3

c 8.2 × 10

b 3.46 × 106 W 8

d 9.877 × 10 bytes

Practice

b 1.6 × 10−7

18 a 6 × 1011 3

d 3 × 102

c 3 × 10

e 4.5 × 104

f 2.1 × 1010

h 4.4 × 105

g 2.4 × 10

5 a 2.49

b 20 000 000

c 41 300

d 0.009

Extend your thinking

e 897 000

f 50 300

19 a 2.107 × 1024 atoms

g 3014

h 8 000 000

b 9g

i 13 008 000

j 0.836

c 1.495 × 10−23 g

k 7 910 000

l 0.007 14 −3

b 0.005 = 5 × 10

6 a 300 = 3 × 10 c

−5

= 4 × 10

d 1 × 10–2 × 2 × 10−5 = 2 × 10−7 7 a i Yes

ii No

iii No

iv Yes

–4

20 7 × 108 m 21 At any particular time, approximately 2.32 × 105 aircraft are registered as being in ﬂight over 7.772 × 1010 square metres of air space. 22 999 23 1.4130 × 109 days

b 4.5 × 10

b 2.004

24 1.25 × 1014

c 12 010

d 0.000 456 1

25 1.89 × 10−8 m3

9 a 3 × 102

b 4.5 × 104

8 a 340 200

−2

c 7 × 10

d 6.23 × 10−3

e 2 × 103

f 2 × 10−3

g 1 × 10−4

h 8.84 × 105 7

26 a 3.2 × 100 body lengths per second b 5797.89 days

Chapter 4 review

i 8.4626 × 10

j 6.14 × 100

k 3.47 × 10−4

l 3 × 10−5

m 7 × 10−2

n 7.64 × 10−1

2 C

o 2 × 106

p 1 × 10−6

3 B

10 a i 7.27 × 10−7

ii 2.22 × 10−7

ii 1.25 × 10

−6

c i 9.37 × 10−4

ii 6 × 10−9

b i 1.25 × 10 d i 8.31 × 10

−9

ii 5.13 × 10

1 C

4 a 0.617

b 0.2385

c 0.000 004 2

d 0.081

e 0.456

f 0.0073

g 0.1284

h 0.0905

11 a 3.844 × 10−3

b 1.300 × 10−2

−15

d 3.342 × 102

c 10 000

d 10 000

12 a i 3

ii 422 m3

e 100

f 10 000

b i 2

ii 2.5 mL

g 10

h 10 000

c 1.120 × 10

13 a 1

b 1.09

14 a 1.332 × 109 km3 c 1.332 × 10−18 Tm3

b 1.332 × 1021 L d 1.332 × 1012 GL b 138 000 000

15 a B 16 a 4 × 10

c 1.2

17 135 000 kg

b 1.5 × 10

c 375

5 a 10

b 100

6 a 18 000 mm

b 4.25 m

c 400 000 cm

d 0.75 km

e 950 cm

f 1.2 m 2

7 a 50 000 cm

b 750 000 m2

c 80 000 m2

d 3.5 ha

e 25 000 cm2

f 1.2 ha

8 15.5 m

Answers 1007 mathspace.co

9 a 4200 mm3

b 2 500 000 000 mm3 3

c 500 000 000 mm

d 23 000 mm

e 1800 mm3

f 750 000 000 mm3

10 a 75 L

b 48 000 L

c 52 mL

d 65 kL

e 12.5 L

f 33 mL

11 a 7152 g

b 0.820 kg

c 15 000 g

d 7.2 t

e 0.950 kg

f 3800 kg

b 64 c No, the triangle is not a right-angled triangle because a2 + b2 ≠ c2.

14 a 451 000 c 15 000

e t = 16

f u = 10

6 12.65 m

b 3.01

7 a 17.0 cm

b 13.4 mm

c 35.0 m

d 15.8 cm

8 a 15.2 cm

b 15.6 mm

b 0.058

16 16 people

9 200 m

17 7.6 × 10 s 18 1.25 g 19 a

c r = 52

d 276 000 d 0.000 322

15 a 5.8

5 a

b 50 900

c 0.007

Practice 4 a 65

12 6 kg 13 a 3.18

3 The triangle is a right-angled triangle since 92 + 122 = 152

Plot

Length (m)

Width (m)

Area (m2)

150

12 000

250

7500

8100

10 a Yes

b No

c No

d Yes

e Yes

f No

g Yes

h No

11 a 17 cm

12 5.69 m 13 a y = 5.7 m

b y = 1.2 cm

Extend your thinking 14 Yes, Pythagoras’ theorem still holds after enlargement by factor of 4.

b Plots A and C 20 15 187 trees

15 a Yes

21 180 glasses

16 Yes, point B is the midpoint of Y Z because Y B = BZ

22 0.3 g

17 a $1260

23 2500

d No

b $5040

19 Hiker B walks further by approximately 207 m

b 18.0 g/mol −23

c 2.99 × 10

5.02 Perimeter

5.01 Pythagoras’ theorem

What do you remember?

What do you remember? 2

c Yes

18 205 cm

24 a 1.51 × 1024 molecules

b No

1 a 4 metres = 400 centimetres 2

1 a r =p +q

b p =r −q

2 a a = 15

b a = 25

c a = 12

d a = 9.22

1008 Mathspace New South Wales – Year 11 Standard mathspace.co

b 12 centimetres = 120 millimetres c 6 kilometres = 6000 metres d 3.2 metres = 3200 millimetres

2 Multiply the length of one side by ﬁve.

4 a False

• 5 cm by 19 cm

b C = 2π y

3 a d = 2y b True

c False

• 4 cm by 20 cm

d True

• 6 cm by 18 cm • 7 cm by 17 cm • 8 cm by 16 cm

Practice 5 a Square or rhombus

b Triangle

c Square or rhombus

d Decagon

6 a 32 cm

b 24 cm

e 32 cm

f 42 cm

c 30 cm

7 a 46 m

b 44 m

8 a 16 units

b 22 units

• 9 cm by 15 cm • 10 cm by 14 cm • 11 cm by 13 cm d 44 cm

9 57 cm

b Arc length = π x

c P = π x + 2x

23 The triangle’s perimeter is 12 × 3 = 36 m. If the square’s side length is s and the triangle’s side length is t, then 4s = 3t.

b 49.01 m

b True. An octagon has 8 sides, so each side is = 7 cm.

14 6.4 cm 15 a 36 cm

b 60 mm

c 10 cm or 100 mm

Number of posts is

17 13 m 18 a i 25.13 cm

ii 57.13 cm

b i 12.88 m

ii 29.28 m

c i 14.92 cm

ii 33.92 cm

d i 18.85km

ii 42.85 km b 4π cm

20 Angle of the sector is θ = 90°. Perimeter is 28.57 cm (rounded). 21 a 55.70 cm

b 31.68 cm

c 190.90 mm

d 33.48 m

e 33.96 cm

f 20.47 m

Extend your thinking 22 The possible dimensions of the rectangle are: • 1 cm by 23 cm • 2 cm by 22 cm • 3 cm by 21 cm

c True. The circumference is 2π r, which is always greater than r since 2π > 1. 25 The perimeter is 2 × (25 + 10) = 70 m.

16 29 cm

19 a

= 9 m.

24 a True. The square’s perimeter is 4 × 8 = 32 cm. The rectangle’s perimeter is 2 × (10 + 6) = 32 cm. They are equal.

12 6x + 16 13 a 16π cm

To ﬁnd the dimensions, divide the perimeter by 2 (which is 24), and ﬁnd pairs of whole numbers that add up to 24.

The square’s side length is

10 84 cm 11 a d = 2x

• 12 cm by 12 cm

= 14.

5.03 Perimeter of composite shapes What do you remember? 1 B 2 Breaking down a composite shape into its simple shapes allows you to identify and measure each exterior side individually, making it easier to calculate the total perimeter by summing the lengths of all outer boundaries. 3 a True. A composite shape is deﬁned as a shape made up of two or more simple shapes that are touching or connected to form a single shape. b True. Composite shapes can be made up of any simple shapes, including circles, combined with other shapes like triangles or rectangles.

Answers 1009 mathspace.co

c False. A composite shape requires two or more simple shapes combined; a single shape is not considered composite.

to form a composite shape with the same perimeter as a hexagon made up of six equilateral triangles.

d False. For a shape to be composite, the simple shapes must be touching or connected; separated shapes do not form a single composite shape.

c True. A composite shape with a hole in the middle has more sides than a solid shape with the same outer dimensions because the perimeter of the hole is also added. Therefore, its perimeter will be greater.

Practice

15 42 m

4 a i Five rectangles

16 123 m

ii 148 cm b i Two rhombuses

ii 78 cm c i One square and four semicircles ii 113.10 cm

5 52 mm 6 a 62 m

b 84 m

7 68 m 8 a 58 cm

b 44 mm

c 65 cm

d 68 cm

e 86 mm

f 164 m

9 182.80 units

18 a No, because the radius is counted an extra four times. b 8π + 48 cm 19 715 m

10 53.13 m 11 a 13 m

b 60 m

12 a 45.7 cm

b 50.1 cm

c 50.3 cm

d 37.7 cm

e 37.7 m

f 392.8 units

g 108.5 cm

h 260.5 cm

i 120.7 cm

j 220.8 cm

13 He has included all four sides of the rectangle, but one of the sides of 20 cm is inside the shape. He has worked out the circumference of an entire circle of diameter 20 cm, but he only has a semicircle. 14 a False. The perimeter of a composite shape is the sum of the lengths of its sides, which may not include shared sides or internal sides. Therefore, it is not always equal to the sum of the perimeters of its simple shapes. b False. It is possible for two composite shapes with the same perimeter to have a different number of component shapes. For example, a rectangle and a square could be combined

1010 Mathspace New South Wales – Year 11 Standard mathspace.co

20 a y = 81.39

b 372.79 m

c $7843.46 21 a x = 6.71

b y = 9.85

c 32.56 m

d $1172.16

22 a x = 11.2

b 3.2 L

c $172.80 23 $3868.56 24 a 300.00 m b 305.78 m c 96.33 m d The starting points are staggered to ensure a fair race where all athletes run the same distance. Since Lane 2 has a longer perimeter, its starting line is placed further ahead to compensate for the extra distance the athlete must travel. 25 Approximately 80 m 26 a 74 m

b $3478

27 a 42.1 m

b $1347.20

Extend your thinking

a=8m

28 a 780 m

b 21.84 km

29 a 337 m

b 11.80 km

30 a l = 19.65 m

b 85.30 m

h=6m b=4m

c $1647

A = 36 m2

31 a 3530.44 cm

b 14

5.04 Area

h = 10 cm

What do you remember? 1 a False

b True

c False

A = 120 cm2

2 a i Incorrect

ii Exact

10 a 21.0 cm2

b 4.2 cm2

b i Correct

ii Approximate

c i Incorrect

ii Approximate

d i Correct

ii Exact

11 80 mm2

b 0.4 cm

12 a 225π cm2

b 900π mm2

c 121π cm2

d 72.25π m2

3 a 500 cm c 120 000 m2 4 a

d 4 ha c

b 16π mm2

c 49π mm2

d 75π cm2

Practice 6 a 33 cm2

b 225 mm2 2

d 65.5 cm2

c 20.0 cm

13 a 380.1 cm2

b 615.8 mm2

d 254.5 m2

b 12 mm

c 25 cm

c 530.9 cm 14 452.39 cm2

5 a 50π cm2

b = 12 cm

d False

e True

c 225 cm

d 77 m2

e 196 cm2

f 1386 cm2

g 63 m2

h 432 cm2

i 120 m

j 110 cm

k 364 mm2

l 55 cm2

m 120 mm2

n 101.5 mm2

Base

Height

Area

7 mm

4 mm

28 mm2

12 cm

8 cm

96 cm2

10 cm

6 cm

60 cm2

36 m2

15 a 5 cm 16 a i

ii 81π cm2

b i

ii 50π cm2

17 a 319.1 m2

d 6m

b 158.4 mm2 2

c 108.6 cm

d 102.1 cm2

18 6.25 cm2 19 50π cm2 20 102.63 cm2 21 a 48°

b 60 cm2

22 25π cm2 23 40 cm2 24 a 61.8 m

b 188.3 m

25 a 117 600 m2

b 11.76 ha

Answers 1011 mathspace.co

Extend your thinking

35 1206.4 m2

26 The circumference of a circle is C = 2 × π × r, so we can divide the circumference by 2 × π to

36 9.8 cm2

ﬁnd the radius,

b $426

38 33.5 cm2

Now, we can put the radius into the area formula A = π r2 to get A = π × 102 = 100π. We can leave this as an exact value or round to get the approximate area. 27 a θ = 36° 2

b 31 m

c No, the perimeter of the sector from part (a) is required to calculate the area. Knowing only the radius is not suﬃcient. 28 a 418.9 m2 b The area of the crop ﬁeld watered is directly proportional to the rotation angle. A larger angle means a larger portion of the circle is covered, increasing the area. For example, doubling the angle would double the area watered, assuming the sprinkler radius stays the same. 29 a 223.4 m2 b The area of a sector is directly proportional to its angle. Since the distance to the fence (radius) is constant, doubling the beam angle will double the illuminated area. The new area would be 2 × 223.4 = 446.8 m2. 30 a

37 a 71 mm2

5.05 Area of composite shapes What do you remember? 1 Addition method: Divide the composite shape into basic shapes, calculate each area, and sum them. Subtraction method: Calculate the area of a larger enclosing shape and subtract the areas of smaller shapes. 2 Total area = Area of rectangle + Area of semicircle 3 Rectangles, triangles, circles, trapeziums, parallelograms, or sectors. 4 False. The addition method sums the areas of component shapes, but if shapes overlap, the overlap area is counted twice and must be subtracted. The subtraction method may also be used, where the area of a larger shape is reduced by removing smaller areas. Practice 5 a i One trapezium and two semicircles ii 161.11 cm2 b i Two trapezia

Farm

Length (m)

Width (m)

Area(m2)

250

20 000

300

6000

120

3000

350

17 500

c 100.53 cm

150

12 000

8 15 254.87 cm2

b Farm 1, Farm 4, Farm 5 c Farm 2, Farm 3

ii 144 cm2 6 a 132.7 cm2

b 98.2 cm2

d 589.1 cm2

c 339.3 cm

7 a 47.12 cm2

9 a 201.1 cm2

e 1.17 ha

10 12.86 m2 b 5 plants

32 30 m

33 a 120 cm2 34 138.6 m

b 15 tiles

1012 Mathspace New South Wales – Year 11 Standard mathspace.co

d 110.47 cm2

Extend your thinking

d None 31 a 6 m2

b 84.82 cm2 2

b A = π (R2 − r2)

5.06 Surface area

16 a 36π cm2

What do you remember? 1 a 6

b 5

c 6

d 8

2 a False

b True

c True

d False

b A = π x2

3 a C = 2π x 4 a True

b False

c False

5 a True

b False

c False

d 1115.56π cm2

e 144π cm2

f 676π mm2

17 a 60 499.670 m2

b 8527.599 cm2

18 a 225 cm2

b 15.00 cm

19 804.25 cm d True

b 144π cm2

Extend your thinking 20 509.7 m2

Practice 6 a

8 cm

Substituting into the

8 cm

area of a trapezium Substituting into the area of a rectangle Factorising out the common term, l

Sum of all 6 sides

b 384cm

7 a 96 cm2

b 1176 cm2

c 328 m2

d 340 cm2

e 448 m2

f 724 mm2

8 a 288 cm2

b 920 cm2

c 1520 cm

d 552 cm

What do you remember?

11 a 412.69 cm2

b 216 m2

12 a 301.59 m2

b 251.33 m2

13 a i 75.40 cm2

ii 100.53 cm2

b i 75.40 m2

ii 131.95 m2

c i 201.06 m2

ii 301.59 m2

ii 353.43 m2

14 a 565.49 cm2

b 188.50 m2

1 A composite solid is a three-dimensional shape formed by combining two or more basic solids. An example is a storage tank made of a cylinder with a spherical cap. 2 Identify all exposed faces of the composite solid, calculate the surface area of each exposed face using the appropriate formula, and sum them, excluding any shared surfaces where solids are joined. 3 a Two hemispheres 2

c 314.16 cm

d 15 891.77 cm

e 9424.78 cm2

f 66 826.58 cm2

15 a 201.06 cm

c 1661.90 cm2

b 201.06 m

Simplify

5.07 Surface area of composite solids

10 26 480 cm2

b 5.64 cm

9 1088 cm

Substitute

22 a TSA = 4π r2 + 6π rh

d i 226.19 m

b Two rectangular prisms c A trapezoidal prism and a triangular prism d A cylindrical hole in a rectangular prism

d 15 446.63 mm2

Answers 1013 mathspace.co

5.08 Volume and capacity

Practice 2

4 a i 121π mm

ii 616π mm

iii 242π mm2

What do you remember? 1 a Square, 15 cm

b 3075.62 mm2

b Triangle, 12 cm

5 867.08 cm2

c Parallelogram, 4 cm

6 365.45 cm2 7 a 120 cm

d Trapezium, 3 cm

b 228 cm

c 6500.78 mm2 8 a 1413.72 cm2

Volume

d 854.51 cm2 b 2780.31 cm2

c 4194.03 cm2 9 a • Three square faces with side length of 12 cm.

1 cm

Capacity 1 mL

75 cm

75 mL 3

500 cm

500 mL

• Three faces which are squares of side length 12 cm, with quarter-circles of radius 6 cm cut out of them.

1000 L

3 a True

b True

• One-eighth of a sphere with radius 6 cm.

4 Circle

c True

d False

b 835.73 cm2 10 a 1658.76 cm2

b 131.95 cm2

c 1507.96 cm2

d 3298.67 cm2

Extend your thinking

Practice 5 169 646 L 6 a 225 cm3

b 300 m3

11 Calculate the total surface area of the rectangular prism:

7 a 452.4 cm3

b 18 095.6 cm3

8 a 169.6 cm3

b 294.5 cm3

2 × 7 × 6 + 2 × 6 × 8 + 2 × 7 × 8 = 292 cm2

9 a 560 mL

b 504 mL

Subtract the triangle at each end of the triangular hole:

10 482 549 L

292 – 2 ×

× 3 × 3 = 283 cm2

Add in the three rectangular faces that are part of the triangular prism going through the shape. Determine the slant length of the triangle using the Pythagoras’s theorem:

11 210 000 cm3 12 2010.6 cm3 13 a 133.33 cm3

b 336 cm3

14 a 5625π cm3

b 3000π mm3

15 a 37.7 cm3

b 837.8 cm3

16 a 75π cm3

b 84π cm3

17 a 23.07 cm

b 3479 cm3

c 3L 18 a 523.60 cm3

19 462 mL

283 + 2 × 2

×8+3×8= 2

cm = 361 cm (nearest whole number)

20 a

b 4500π mm3 b 2 094 395.102 cm3

12 a 40.79 m2

b $8728.08

21 a 1072.330 units3

13 a 16.3 cm2

b 614 complete nuts

22 30 m3

1014 Mathspace New South Wales – Year 11 Standard mathspace.co

23 0.6 L

Practice

24 6 cm

4 a 1583.4 cm3

b 792 cm3

c 1233.3 cm Extend your thinking 25 No, a cube and a rectangular prism with the same volume do not necessarily have the same surface area. This is because the surface area depends on the dimensions (length, width, height) of the prism, not just the volume. A cube has equal dimensions, while a rectangular prism can have different dimensions, leading to different surface areas. For example, a cube with sides of length 4 units has a volume of 4 × 4 × 4 = 64 cubic units, and a surface area of 6 × (4 × 4) = 96 square units. A rectangular prism with dimensions 2 unit by 2 unit by 16 units has the same volume (2 × 2 × 16 = 64 cubic units) but its surface area is 2 × (2 × 2 + 2 × 16 + 2 × 16) = 136 square units.

5 1288 cm3 6 576 cm3 7 a 64π cm3

b 216π cm3

c 477.5 cm3 8 659 734 L 9 768 cm3 10 a 980π cm3

b 3015.9 cm3

11 646 mL 12 515 681 L 13 251 327 L 14 a 54π m3

b 6π m3

c 135 717 L 15 8378 L

26 33 cm 27 6 m

Extend your thinking

28 2.5 cm

16 1320 m3

29 22 619.47 L

17 a 49 m3

30 75 000 cm3

18 a Alice: 15.71 m

Bob: 47.12 m3

31 375π cm3

b Alice pays: $1906.31

32 691.6 L

Bob pays: 5718.92

33 $1500

5.10 Trapezoidal rule

5.09 Volume of composite solids What do you remember?

b Calculate the volume of the half-cylinder and the rectangular prism

using

using V = l × w × h, then add them together. 2 Calculate the volume of the triangular prism

(V = π × 52 × 12), then add them together. c False

d False

b True

2 B 3 The trapezoidal rule approximates the area by averaging the lengths of the parallel sides and multiplying by the perpendicular distance between them. 4 a ii

and the cylinder

b True

What do you remember? 1 a False

1 a A half-cylinder and a rectangular prism

3 a True

b 31 416 L 3

b i

c iii

Practice 5 a 232.5 cm2

b 748 cm2

6 36 m2

Answers 1015 mathspace.co

7 432 m2

Extend your thinking

8 a i 51 754 m2 ii 95 824 m

iii 147 578 m

iv Less than Explanation: The trapezoidal rule approximates the area by assuming straight boundaries between measurement points. For the ﬁrst piece of land, the creek boundary is concave (curving inward), which means the actual area under the curve is less than the area of the trapeziums. b i 142 284 m2 ii 184 702 m2 iii 326 986 m

24 a 18.6 m2

b 100.44 m3

25 a 342 m2

b 423 m2

c The two-application approximation is more accurate because it divides the shape into smaller trapeziums, reducing the error caused by assuming straight lines between measurement points. With only one application, the larger trapezium over-simpliﬁes the shape’s curvature, leading to a less precise estimate. 26 a 19.8 m2

b 5 816 448 m3

27 a 9770.5 cm2

b 159 cm

iv Greater than. Explanation: The trapezoidal rule assumes straight boundaries, and the creek boundary here is convex (curving outward). This means the actual area under the curve exceeds the area of the trapeziums. 9 26.8 m2

Chapter 5 review 1 13 m 2 12 m 3 60 cm2 4 40 m2 5 28 cm

10 126 m2

6 7.1 cm

11 3.12 ha

7 28.3 cm2 2

12 1 707 750 m

8 30.5 cm

13 a 232 484 m2

b 8183 m3

14 a 354 036 m

b 5488 m

b 479.54 m

16 a 14 513 m2

b 2003 m3

15 a 14 104 m

9 30 m3 10 18 m2 3

c 2 003 000 L 17 a 893.5 m2

b 37.7 m3

c 37 700 L 18 a 8204.6 m2

b 7384.14 m3

c 7 384 140 L 19 a 10.5 m2

b 84 m3

20 a 33.6 m2

b 67 kL

21 a 39 m2

b $159.55

22 118.3 m3 3

23 31.16 m

1016 Mathspace New South Wales – Year 11 Standard mathspace.co

11 12 cm2 12 26 m 13 96 cm2 14 a 14.1 cm

b 16.6 mm c 25 m

d 4.7 cm

15 a $2250.00

b $5625.00

16 a 54.87 cm

b 17.72 cm

c 80.55 cm

d 32.48 cm

17 80 18 The perimeter is 2(l + w) = 56 cm, so l + w = 28 cm. List pairs of whole numbers (l, w) where l ≥ w and l + w = 28: (14, 14), (15, 13), (16, 12), (17, 11), (18, 10), (19, 9), (20, 8), (21, 7), (22, 6), (23, 5), (24, 4), (25, 3), (26, 2), (27, 1).

Method: Divide perimeter by 2 to ﬁnd sum of length and width, then ﬁnd all whole number pairs summing to 28. 19 a 24.70 m

b 121.70 m

20 a 8.06 m

b 11.66 m

c 38.72 m

b 2850 m3

34 a 27.22 m

38 a 53 200 m2 b 69 900 m2

b 36.7 m

c 123 100 m2 d Less, as the river’s inward curve reduces the area compared to the trapezoidal approximation.

24 a One rectangle and two semicircles 2

b 112.27 cm

25 a 452.4 cm2

d 1078.26 cm3

37 25 m2

b 45π cm2

23 a 1060.3 m

c 2400 cm3

36 2565.6 cm3

21 50.85 mm

b 532 cm3

35 96 000 cm3

d $1548.80

22 a 72°

33 a 405 cm3

b $339.30

6.01 Salaries and wages

What do you remember? 1 52 weeks 2 a 52

b 26

c 12

3 Pay per period = 4 a True

b False

c True

Practice 5 $7523.50 27 a 726 cm2

b 526 m2

c 1008 cm2

d 552 cm2

e 490.09 cm2

f 8746.02 cm2

6 a $16 640

b $36 400

7 a $399

b $1411.50

c $366

h 1182.36 cm

8 a 7.5 hours

c $1005

g 145.27 m

28 a 314.16 cm

b 471.24 cm

29 Surface area is the sum of two pentagonal bases and ﬁve rectangular lateral faces. Bases: 2Ab. Lateral faces form a rectangle (unrolled) with width Pb = 5s and height L, so lateral area is Pb × L = 5sL. Total surface area: 2Ab + 5sL. b 135.34 cm2

30 a 10.77 cm 2

c 100.53 cm

d 236 cm

b $201

9 $14 694 10 $3758.08 11 a $711.54 c $3083.33

b $1423.08 d $18.72

12 a $144

b $90

c $207

d $288

e $387

31 a 16.77 cm

b 1628.1 cm2

13 a 41 hours

b $47 per hour

32 a 17.09 cm

b 1180.2 cm2

14 a $1026

b $2052

c $53 352

d $4446

Answers 1017 mathspace.co

15 a $414 c Valentina

b $548.90

3 2 times the normal hourly rate

d $134.90

4 a False

b True

c False

d False

16 $738.34 Practice

17 $3201

5 a $180 per hour

18 $27.76 19 a $65 709

b $240 per hour

c $1440

Extend your thinking

6 a i $39.00

ii $52.00

b i $35.40

ii $47.20

c i $40.80

ii $54.40

b $1263.63

20 Offer 1, because his annual salary will be $49 710, which is higher than Offer 2 at $32 298.24 and Offer 3 at $48 000.

7 a i $159.00

ii $212.00

b i $126.00

ii $168.00

c i $213.75

ii $285.00

d i $294

ii $392

8 a 12 hours

b 8 hours

21 Offer 2, because the weekly pay for Offer 2 is higher than Offer 1 at $915 and Offer 3 at $904. 22 a i $40.79 per hour

ii $5200

b i $41.25 per hour

ii $5200

23 Weekly earnings with bonus distributed: $1026.92. Original monthly pay without bonus: $4200. Weekly is higher when converted to monthly: $1026.92 × 52 ÷ 12 ≈ $4451.67 vs $4200.

c 7 hours 9 a 48 equivalent hours at normal pay b 47 equivalent hours at normal pay c 52 equivalent hours at normal pay d 30.5 equivalent hours at normal pay 10 a $1064

b $1092

c $1050

6.02 Overtime

11 a 23

What do you remember?

b 9

12 a $28.42 per hour

1 38 hours

c $56.84 per hour

2 1.5 times the normal hourly rate

c 5

d $1139.25

b $42.63 per hour d $1421

e $73 892 13

TIMESHEET

Name: Avril Smith

Day

Start

Finish

Total hours

Normal hours

Time and a half hours

Double hours

Earnings ($)

Tue

11:00 a.m.

5:00 p.m.

143.34

Wed

11:00 a.m.

8:00 p.m.

262.79

Thu

11:00 a.m.

6:00 p.m.

179.18

Fri

11:00 a.m.

8:00 p.m.

262.79

Sat

1:00 p.m.

5:00 p.m.

191.12

1039.22

Total

1018 Mathspace New South Wales – Year 11 Standard mathspace.co

TIMESHEET

Name: Maria

Day

Start

Finish

Total hours

Normal hours

Time and a half hours

Double time hours

Earnings

Mon

8:30 a.m.

3:30 p.m.

$157.50

Tue

9:00 a.m.

5:00 p.m.

$191.25

Wed

8:00 a.m.

4:00 p.m.

$191.25

Thu

9:00 a.m.

7:00 p.m.

1.5

$275.63

Sat

10:00 a.m.

2:00 p.m.

$180.00 Total

TIMESHEET

$995.63

Name: David

Day

Start

Finish

Total hours

Normal hours

Time and a half hours

Double time hours

Earnings

Mon

7:00 a.m.

4:00 p.m.

$188.10

Tue

8:00 a.m.

6:00 p.m.

$217.80

Wed

7:00 a.m.

5:00 p.m.

$217.80

Thu

8:00 a.m.

4:00 p.m.

$158.40

Sat

9:00 a.m.

1:00 p.m.

$158.40 Total

16 a 19 hours b 12 hours and 40 minutes

$940.50

6.03 Commission, piecework and royalties

c 9 hours and 30 minutes 17 a 22.5 hours

b 15 hours

b 7 hours Extend your thinking 18 $33.55 per hour 19 a $18.57

b $16.55

c $16.11 20 4.61 hours 21 Annual earnings = [(a × h) + (b × h × 2)] × 50 or simpliﬁed Annual earnings = 50h(a + 2b)

What do you remember? 1 Commission is a percentage of sales earned by employees like salespeople, often with a retainer for stability. Piecework is a ﬁxed payment per unit of work completed, with no hourly wage or beneﬁts. Royalties are percentage-based payments to creators based on revenue from sales or usage of their work, providing ongoing income. 2 a False

b True

c False

d False

3 a Commission

b Piecework

c Royalties

d Commission

e Piecework

f Royalties

g Commission

h Piecework

4 a $50

b $75

c $96

d $60

5 $1800

Answers 1019 mathspace.co

Practice 6 a $240

b $182

c $672

d $750

7 a 29

b 40

c 84

d 167

8 a $1.40

b $5250

9 a $120

b 300 loaves

c $40 per loaf 10 a $25.60

d $300 b $1536

Offer B provides a higher total income. Furthermore, Offer B is the safer choice because its income relies more heavily on the guaranteed retainer and less on commission, making it less risky if she does not meet her sales targets. 30 $110 000

c 19 200 kg 11 $0.62

6.04 Government allowances

12 a $19.24/hour

b 5.06 kg

13 a $750

b $1150

14 $1190 15 a $5820

b $12 760

16 a $135

b $385

17 $1800

What do you remember? 1 a To support students aged 25 or older studying full-time or undertaking apprenticeships. b To assist young people aged 16-24 with study, training, or job-seeking expenses. c To provide income support for people aged 67 or older who meet income and asset tests.

18 12% 19 a $6000

b 3000 prints

20 a $10 000

b $561 000

21 a i $294

ii 3%

b i $390

ii 5%

22 a $1914

• Offer B: Her predicted annual income would be $61 600 (from $52 000 in retainer plus $9600 in commission).

b $520

Extend your thinking 23 a $2000

b $2250

24 a $1200

b $4100

d To help unemployed individuals aged 22 or over, or those temporarily unable to work/ study, with living costs. 2 a 25

b 16

c 67

d 22

3 a No

b Yes

c No

d Yes

4 a Yes

b No

c Yes

d No

e Yes

f No

g Yes

h Yes

5 a $410.30

b $663.30

c $756.90

d $1149.00

c 1200 artworks 25 a $1700

b $3400

26 $19 200 27 $240 28 $24 000 to $38 400 29 Elna should choose Offer B. Based on her expected annual sales of $192 000, the total annual earnings for each offer are: • Offer A: Her predicted annual income would be $45 200 (from $26 000 in retainer plus $19 200 in commission).

1020 Mathspace New South Wales – Year 11 Standard mathspace.co

Practice 6 a $221 7 a i $446.00

b $415

c $391 ii $11 596.00

b i $425.10

ii $11 052.60

c i $511.10

ii $13 288.60

8 a i $836.60

ii 21 751.60

b i $663.30

ii $17 245.80

c i $718.10

ii 18 670.60

9 $663.30 10 a $0.00 c $1134.90

b $1051.30 d $1595.60

11 $9286.64

6 a $1400

b $5600

12 $6345.56

c $980

d $6580

13 a $692.10

b $404.30

c $339.70

7 $606.48 8 $752.50

14 a $1732.20

b $1577.50

9 $2585.00

15 a i $944.00

ii $921.00

10 $3863.40

iii $921.00 b i $739.70

ii $1366.20

iii $739.70 c i $2298.00

ii $2298.00

iii $2298.00 16 a $397.30

b $410.30

17 a $190.08

b $14 950

18 a i $1149.00

ii $70.00

iii $1079.00 ii $8.25

iii $857.85 c i $1149.00

12 a $1055.39

ii $20.00

iii $1129.00

16 $66 270.25 17 a $635.60

b $69.60

18 $2637.14 Extend your thinking 19 a $483.81

b $588.81

c $80 752.62 b $64 152.12

c Salary A

19 53 hours

21 Salary increase

20 $444.00 21 a $160.98

b $2259.62

15 $777

20 a $852.12 Extend your thinking

b $3166.16

13 $734.38 14 a $336.54

c $209.66

b i $866.10

11 $2232.50

22 $37.55 b 1 USD = 2.08 AUD

22 a $480 b Reduction: $0, Austudy received: $663.30

Chapter 6 review 1 B 2 B

c $1143.30

6.05 Annual leave loading

3 C 4 $539.00

What do you remember?

5 a 7.75 hours

1 A percentage of normal wage as a bonus on annual leave. 2 $4765.00 3 $2502.36 4 $1019.20 Practice 5 a $665 c $1120

b $875

b $182.13

c $910.63 6 a $60 547.50

b $1164.38

7 i $39.23 per hour

ii $5460

8 a $937.50

b $987.50

c $1125

d $1062.50

9 a $1681.50

b $87 438.00

10 a $135.00

b 300 keychains

c $8.00 per keychain

d $427.50

Answers 1021 mathspace.co

11 a $500

b $850

8 $65 020

12 a $1425

b $1725

9 a $665.60

b $42 665.60

10 a $91 872.00

b $1872.00

13 $28 157.89 14 $449.50

Extend your thinking

15 $21 751.60

11 a $300.00

16 $733.10 17 a $1004.00

12 a Not deductible: Commuting expenses are private, not work-related.

b $1041.00

b Deductible: Travel for work purposes is an allowable deduction.

c $1004.00 18 43 hours 19 a $1320

b $5280

c $924

d $6204

c Deductible: Work-related books fall under tools and equipment expenses. 13 $124 761

20 $649.25

14 $800.00

21 $2937.50

15 a $82 500

22 a $500.77

b $549.23

c $1501

c $79 050.00

16 a $900.00

23 $24.43

c $2600.00

7.01 Allowable deductions

7.02 Tax tables

What do you remember?

1 a An allowable deduction is an expense recognised by the ATO that can be subtracted from gross income to reduce taxable income. b Examples include work-related travel costs and union fees. c Taxable income = Gross income − Allowable deductions d Commuting costs are considered private expenses, not work-related. 2 a Yes

b No

c Yes

d No

3 Gross income is total earnings before deductions, while taxable income is the amount left after subtracting allowable deductions. 4 $48 000

7 $48 150

1022 Mathspace New South Wales – Year 11 Standard mathspace.co

b $69 600.00 d $5300.00

c Tax is applied to all income, increasing the amount payable. 2 a 0 − $18 200

b $18 201 − $45 000

c $45 001 − $135 000 d $190 001 and over 3 Taxable income is gross income minus allowable deductions.

Practice b $60 084

d $5001

b A person may get a refund if they have paid more tax than required based on their actual income, or a bill if they have underpaid due to changes in income or deductions.

4 $0 5 $58 340

b At least $2501

1 a Tax tables are used to calculate the amount of tax to be withheld from income by matching taxable income to the correct bracket and applying the formula speciﬁed for that bracket.

Practice

6 a $416

b $25.00

5 $11 278.00 6 $64 040.00

7 a $55 840.00

b $7540.00

6 $29.00

8 $57 668.00

7 $706.00

9 $20 064.70

8 a $2740.25

10 $35 235.90

9 $3067–$3069

11 $55 217.75

10 $96

12 $45 335.47

11 a $252.14

b $1422.00

b $307.00 Extend your thinking 13 $64 040.00 14 a Stewie multiplied by 0.30 correctly but wrote the result as $7000 instead of $6000 b $10 288.00

c No, she does not claim the tax-free threshold as the calculated amount in part (b) matches the provided tax withheld by her employer. 12 $845.00 13 $131.00

15 a $5788.00

14 a $962.30

b $1047.86

b $6500.00 c The proposal does not beneﬁt individuals with an income of $50 000.00, as the tax increases. 16 a $14 788.00

Extend your thinking 15 a $843.00 c $1686.00 16 a $707.00

b $15 450.00 c The original system beneﬁts Alex, as he pays $662 less than under the ﬂat rate.

c $136.00 17 a $842.00 c $6152.00–$6154.00 18 a $846.00

What do you remember?

b It deducts income tax from pay each period to avoid a large year-end tax bill.

7.04 Medicare levy

c It reduces tax withheld by exempting the ﬁrst $18 200 of annual income if claimed.

What do you remember?

d For secondary jobs, to ensure tax is withheld from all earnings. c False

d False

3 Allowable deductions reduce the gross pay to calculate taxable income for a pay period. The lower the taxable income, the less PAYG tax is withheld, as tax withholding is based on the taxable income amount. 4 They match different pay periods: weekly, fortnightly, or monthly. Practice

b $846.00

c $3384.00

1 a Pay-As-You-Go

b True

b $843.00

b $3076.00–$3077.00

7.03 PAYG tax

2 a False

b $843.00

1 To help fund Medicare, Australia’s public healthcare system. 2 2% 3 Medicare levy = 0.02 × T 4 Taxable income is gross income minus allowable deductions. Practice 5 $734.00 6 $1100.00 7 a $568.00

b $668.00

5 $237.00

Answers 1023 mathspace.co

8 The withheld amount is correct, as 61 200 × 0.02 = 1224, which matches $1224.

9 $45 932.00 10 a $744.00

9 The withheld amount is correct, as 42 500 × 0.02 = 850, which matches $850.

Extend your thinking

10 a $2140.00

11 $69 000.00

b $0

11 $709.30, which is 2% of their taxable income.

12 $143 511.76

12 The withheld amount of $1810 is not correct. The correct amount should be $90 000 × 0.02 = $1800.

13 a $7020.00

Extend your thinking 13 Zara’s correct taxable income is $100 000, and the withholding was incorrect by $100. 14 a $1368.00

b $1573.20

c 25.90% 15 $143 461.54 16 Ava’s levy is $1000, Ben’s is $1500. The ratio of levies is 1000 : 1500 = 2 : 3, and the ratio of incomes is 50 000 : 75 000 = 2 : 3, so they are equal. 17 a $1600.00 b Charlotte likely used a rate of 2.25% instead of 2%, as 80 000 × 0.0225 = 1800.

7.05 Net earnings

b $42 666.00

14 Deductions reduce taxable income, lowering income tax and the Medicare levy, which increases net earnings since less is subtracted from gross income. 15 a $60 212.00 b $56 812.00 c 2.45% d The increase in deductions reduces taxable income, but income tax and Medicare levy are calculated at ﬁxed rates (30% tax plus 2% levy in this bracket), so only a portion of the deduction increase directly affects net earnings. 16 a $3400.00 b The difference becomes $1800, reducing the gap by $1600. 17 a $36 712.00

b $105 209.02

18 a No, her calculation is incorrect. The correct net earnings should be $52 052.00.

What do you remember? 1 The amount an individual retains after subtracting allowable deductions, income tax, and the Medicare levy from gross income. 2 Taxable income is gross income minus allowable deductions, used to calculate income tax and the Medicare levy. 3 Net earnings = Gross income − Allowable deductions − Income tax − Medicare levy 4 2%

b Mia likely forgot to include the Medicare levy ($1260) and made an additional error. Correct income tax is $9688, so

65 000 − 2000 − 9688 = $53 312

Subtracting the levy gives 53 312 − 1260 = $52 052, not $53 312.

7.06 Yearly tax liability What do you remember? 1 The total amount of income tax and Medicare levy an individual must pay based on their taxable income for the ﬁnancial year.

Practice 5 $37 352.00

2 Total tax liability = Income tax + Medicare levy

6 $46 272.00 7 a $48 800.00

b $33 416.00

b $42 396.00

8 $46 714

1024 Mathspace New South Wales – Year 11 Standard mathspace.co

3 It compares PAYG withholdings to total tax liability; if withholdings exceed liability, a refund is owed; if less, additional tax is payable.

4 a A refund of $500.00.

c $16 388.00

b Additional tax of $1700.00. c A refund of $1200.00.

d An additional tax of $2388.00.

5 A refund of $272.00.

19 PAYG withholdings are estimates based on standard rates and may not account for speciﬁc deductions, variable income, or precise tax brackets, leading to over- or under-withholding reconciled at tax return time.

6 A refund of $12.00.

20 a $21 828.00

d Additional tax of $500.00. Practice

7 a The income tax is correct, as 4288 matches the tax for $45 000 in the $18 201–$45 000 bracket: 0.16 × (45 000 − 18 200) = 4288. b The Medicare levy is correct, as 45 000 × 0.02 = 900. Total tax liability is correct, as 4288 + 900 = 5188.

b Additional tax of $1828.00. c The employer likely used a tax rate or withholding table without accounting for deductions, overestimating the tax needed for a taxable income of $97 000.00.

Chapter 7 review

8 Additional tax of $1348.00.

1 C

9 Additional tax of $2388.00.

2 D

10 a • Taxable income: $89 500.00 • Income tax: $17 638.00 • Medicare levy: $1790.00 b An additional tax of $10 228.00. 11 a $3200 b Ben owes more additional tax ($1988) compared to Amelia’s ($1212 refund). 12 a $1600.00 b Ethan owes less additional tax of $988.00 compared to Kim’s $1588.00. 13 a $1000.00

b $460.00

3 B 4 a $450 5 $1501 6 a $47 500

b $5038

7 a $3488

b $3924

c More tax by $436. 8 $210 9 a $3075.40 b $1826

14 PAYG withholdings of $8700 match 15% of gross income, as 0.15 × 58 000 = 8700.

10 a $918

15 The refund is correct, as total tax liability is $4648 ($3808 income tax + $840 Medicare levy), and 4800 − 4648 = 152.

11 a $650

Extend your thinking 16 a $14 788

b $7150

17 a A refund of $12.00.

b $69 750

b $919

c $1837 b $790

12 a $1450 b $1624 c Old total tax: $4288 + 0.30 × ($72 500 − $45 000) + $1450 = $13 988. New taxable income: $72 500 × 1.12 = $81 200.

b Each $1000.00 in deductions decreases total tax liability by $320.00.

New total tax: $4288 + 0.30 × ($81 200 − $45 000) + $1624 = $16 772.

c A deduction of at least $1525.00 is needed to achieve a refund of at least $500.00.

Percentage increase:

18 a $17 188.00 b An additional tax of $3188.00.

× 100 ≈ 19.90%. 13 $49 264

Answers 1025 mathspace.co

14 $50 484

Practice

15 a $72 500

b $59 312

6 a 6 vertices and 9 edges b Directed

16 Brian is entitled to a refund of $148. 17 a • Taxable Income: $94 750 • Income Tax: $19 213 • Medicare Levy: $1895 b Laura is entitled to a refund of $1108. 18 a $14 500

7 a 6 vertices

b 9 edges

c Vertex P

d S and U

e Undirected

f 5

8 a 5 vertices and 3 edges b One edge

b $74 100

c $7900

9 a Yes

b No

c Yes

d No

19 a $1100

b $72 400

10 a Undirected

b Undirected

c $2400

d $5500

c Directed

d Directed

e Undirected

f Undirected

g Directed

h Directed

20 a $68 292 b $69 892 c 2.34% d Additional deductions reduce taxable income, saving tax at the marginal rate plus Medicare levy (32%). This saving increases net earnings, but only by a fraction of the deduction amount, as tax is a percentage of the deduction.

e True

f False

2 a 3 edges c C 3 a Non-valid

b i 7

ii 63

12 a 0

b 2

c 0

d 2

13 a i A = 1, B = 1, C = 2, D = 1, E = 1 ii Not connected b i A = 2, B = 2, C = 1, D = 3 c i A = 3, B = 2, C = 4, D = 2, E = 3 ii Connected

What do you remember? b False

ii 33

ii Connected

8.01 Introduction to networks

1 a True

11 a i 9

c False

d True

d i A = 0, B = 1, C = 2, X = 1 ii Not connected

b A, C, D

Extend your thinking

d D

14 Jonah has counted the intersection of edges as cutting the edge into two pieces which is not correct. The number of edges is 4.

b Valid

c Non-valid

d Valid

e Valid

f Valid

g Valid

h Valid

15 Yes, it is possible if you connect the two odd vertices, A and E, all vertices are even.

4 a Edges = 3, Vertices = 4

16 a Undirected

b Directed

b Edges = 5, Vertices = 5

c Directed

d Directed

c Edges = 4, Vertices = 5

e Undirected

f Undirected

d Edges = 3, Vertices = 4 e Edges = 4, Vertices = 5 f Edges = 5, Vertices = 6 g Edges = 5, Vertices = 6 h Edges = 5, Vertices = 6 5 a ii

b i

c iii

d iv

1026 Mathspace New South Wales – Year 11 Standard mathspace.co

17 a 3

b 53

c 100

18 a 1

b 20

c No

19 The sum of the degrees is 2 + 2 + 3 + 1 = 8. The number of edges is 4. Double 4 is 8 therefore the sum of the degrees is equal to double the number of edges.

This happens because each edge has two ends which are counted when ﬁnding the degree of a vertex.

8 a Aaron has worked with Rochelle and Bianca. • Rochelle and Bianca

8.02 Network representations

9 a Teribithia

b • Aaron and Mario b Oz

c A directed edge from Oz to Teribithia What do you remember? 1 Complete as the graph is simple and each vertex is joined to every other vertex. 2 a i The ﬁbre optic cables ii The cities iii V = 7

b Complete as the graph is simple and each vertex is joined to every other vertex.

iv E = 11 v Yes

c Since a loop refers to an edge that connects a vertex to itself, having one means the currency will be converted to itself. So, there are no loops because a currency doesn’t need to convert into itself. There would be no conversion fee for staying in the same currency.

vi No b i The wires ii The electrical components iii V = 8 iv E = 9 v No vi No

11 a 4 minutes

b 8 minutes

c 38 minutes

3 B

12 a 124 likes

Practice

b 117 likes

c This graph is directed and weighted.

4 a Not simple

b Simple

c Not simple 5

10 a Based on the graph, the edge with the highest value represents the highest conversion fee. The value of 7 between New Zealand Dollars and Euros is the largest so, the conversion between New Zealand Dollars and Euros has the highest fee percentage.

d Not simple

Temora

Boorowa Harden

Cootamundra Junee

Yass

6 a Yes. James and Jenny’s vertices are connected by an edge which indicates that they played together. b James, Jenny, and Belinda 7 a Yes. Xanthe and Tina’s vertices are connected by an edge which indicates that they have worked together. • Tina and Derek • Xanthe and Xavier

b No

c Yes

d No

Extend your thinking

Young

b • Tina and Xavier

13 a Yes

14 No, it cannot be redrawn as a simple graph. A simple graph is undirected with no loops or multiple edges. If we changed edges from directed to undirected or removed edges or loops to make it simple, then it would be a different network. So, it is not possible without changing the network. 15 No. If there is only one vertex, then the only edge would be a loop which is not allowed for simple graphs. 16 There are four simple graphs with three vertices.

We can rotate and move around the vertices, but they would represent the same information.

Answers 1027 mathspace.co

17 a It is an app where people are friends. Otherwise, it would need to be a directed graph.

b It would not be possible to make this a non-simple graph. To do so, you would need to be a friend with yourself to make a loop or be a friend with a person multiple times, neither of which are possible. c It would be possible to make a non-simple graph using a different app with a followerfollowing set-up as it would be a directed graph. 18

Foxes

Hawks

Squirrels

Rabbits

Plants c

Killer Whale Tuna

8.03 Weighted graphs

What do you remember?

Mackerel Zooplankton

b A and D; B and C.

1 a 12 c 4

19 a

d The connection between A and B.

Airport

Expo center

Hillsboro

2 a vertices

Beaverton

Gateway

Rose quarter

Pioneer square

b Airport - Gateway - Rose Quarter - Pioneer Square - PSU 20 a

Kingfisher Frog

Small fish

b True. A symmetrical table represents an undirected network where the weight is the same in both directions. c False. The weight of a path is the sum of the weights of the edges in that path. d False. A dash typically means there is no direct edge connecting those two vertices. Practice

Water beetle

4 a 3 hours Snail

Tadpole

d connection

3 a False. A weight can represent any numerical value, such as time, cost, or capacity.

Clackmas Gresham town center

PSU

b weights

c undirected

b 26 hours

5 a

F 14

Algae H

6 G 12 10 J

1028 Mathspace New South Wales – Year 11 Standard mathspace.co

b F and H, with a travel time of 14 minutes.

c I and J have no direct path to F. 6 a 6 b 10

c 13

R 32

150

d No. The total weight is decided by the edges. The travel to vertices in part (b) is longer than (c) and (b) since they had a smaller weight. 7

41 9

90 120

100

4 110

200

2 A

5 3

T 8 a

9 a

11 6

3570

Lookout Waterfall Cave Ridge

Lookout

−

Waterfall Cave

Base Camp

−

Ridge

−

Base Camp

−

N b 410

65 1240 535

965

Waterfall

325

3 P

F 230

Lookout

475

Cave

Ridge 7

Base camp

Answers 1029 mathspace.co

10 a

−

L b

−

15 a 21 b 19 c 20 d Direct path A – C Extend your thinking 16

−

− A

−

7 5

12 D

Eagles

Hawks

17 a

A 15 8

9 Lions

Tigers

12 a $2050

b $2000

13 a 27 L/min

b 43 L/min

1030 Mathspace New South Wales – Year 11 Standard mathspace.co

C b No. The direct path from A to E has a weight limit of 8 tonnes, which is less than the truck’s weight of 10 tonnes.

c A valid path is A-B-C-E. The weight limits for this path are 12, 11, and 14 tonnes respectively, all of which are greater than or equal to 10 tonnes. 18 a

(50, 1)

8.04 Spanning trees What do you remember? 1 a A connected graph with no cycles (or a connected graph in which every edge is a bridge).

(90, 3)

b A subnetwork which is a tree that connects every vertex of a graph.

(60, 2)

c A spanning tree in a weighted graph that has the minimum total weighting possible.

Q (120, 4)

R (40, 1)

2 a False

b True

3 a No

c True

d True

b Yes

4 Answers may vary. Possible answers: S b The cheapest path is P-R-S with a total cost of $130. The fastest paths are P-R-S and P-Q-R-S, both taking 4 hours. The paths are not the same, as the set of fastest paths includes more options than the single cheapest path.

b 76 minutes c Yes. The path A – E – D has a weight of 28 minutes. This is much faster than the peak hour path A – B – C – D, which takes 76 minutes. 21 a Yes. The total distance is 12 + 8 + 18 = 38 km, which is less than 50 km. b Yes. The total distance is 10 + 15 + 18 = 43 km. c No. The total path length is 60 km, which exceeds the van’s 50 km range.

B C

b Path A – C – E has a battery usage of 30% and takes 30 minutes.

20 a 43 minutes

19 a Path A – C – E has a time of 30 minutes and uses 30% battery.

c The path A-C-D has a battery usage of 22% each way, for a total of 44%. The path A-B-D uses 40% each way, for a total of 80%. The path A-C-D is faster and uses significantly less battery, making it the better choice.

E S3

Practice 5 a No, because it is disconnected and has a cycle. b No, because it has a cycle. c Yes d No, because it has a cycle. e Yes f No, because it has a loop.

d Path: Start → W2 → W4 → W5. The total distance is 37 km, which is within the range.

Answers 1031 mathspace.co

Extend your thinking

6 Lisa

Alex

Sandy

Kate

Amy

14 a Answers will vary. One possible spanning tree includes edges A-D, A-B, B-C. Another includes edges D-C, C-B, B-A. B

Robert

Sarah

Joseph

Matt

Chris

12 14

Barbara

Tom

7 Mabel

Unknown Unknown

Rebecca Patrick

April

Todd

Mitchell Juliet

Daisy

Leonard Connor

10 12

Unknown Mark

Richard

8 The minimum spanning tree includes edges A-E(3), E-C(4), C-B(2), B-D(6). The total weight is 3 + 4 + 2 + 6 = 15.

9 The minimum spanning tree includes edges R-T(3), Q-R(5), T-U(6), S-T(7), P-Q(10). The total weight is 3 + 5 + 6 + 7 + 10 = 31.

b For the tree A-D, A-B, B-C: Weight = 30. For the tree D-C, C-B, B-A: Weight = 37. (Answers depend on part (a)).

10 The minimum spanning tree includes edges B-E(2), E-F(3), B-C(4), A-D(5), D-E(7). The total length is 2 + 3 + 4 + 5 + 7 = 21.

c No. The spanning tree with edges A-D, A-B, and B-C has the lowest weight of 30. The minimum cost spanning tree uses edges A-D, A-B, and B-C for a total cost of $30 000.

11 The minimum spanning tree includes edges A-B(2), B-C(3), D-E(3), Router-A(4), C-D(5). Total weight is 2 + 3 + 3 + 4 + 5 = 17. The minimum cost is $1700. 12 The minimum spanning tree includes edges P-S(7), O-P(8), S-T(9), H-P(12). Total weight is 7 + 8 + 9 + 1 2 = 36. The minimum cost is $36 million. 13 The minimum spanning tree includes edges Berries-Fruit(8), Roses-Veggies(10), VeggiesHerbs(12), Herbs-Berries(15), Pump-Roses(20). Total cost is 8 + 10 + 12 + 15 + 20 = 65.

1032 Mathspace New South Wales – Year 11 Standard mathspace.co

15 a Using all routes would create multiple cycles in the network. A spanning tree connects all vertices with the minimum number of edges (n − 1) and no cycles, ensuring there are no redundant, costly connections. b The spanning tree with the lowest cost includes edges B-E (2), C-D (3), A-B (4), and B-C (5). The total cost is 2 + 3 + 4 + 5 = 14, which is $14 million. 16 a 7

b 5

17 The minimum spanning tree includes the edges A-B(3), C-D(4), B-C(5), and D-E(5). The total weight is 3 + 4 + 5 + 5 = 17.

18 Consider the existing connections A-F and C-D as two super-vertices. The problem is to connect these two groups and the remaining vertices (B, E) with minimum cost. Using Prim’s algorithm starting from the A-F group, the cheapest edge out is E-F(6). Now we have A-F-E. Cheapest edge out is E-B(5). Now we have A-F-E-B. Cheapest edge out is D-E(7) which connects the C-D group. All vertices are connected. The new walkways are E-F, E-B, and D-E. Minimum additional cost is 6 + 5 + 7 = 18 thousand dollars, so $18 000.

7 F

8 2

P S

5 U

4 2

4 6

b 24 km

5 D

C B

3 7

4 R

6 a The minimum spanning tree consists of the edges: CD, BE, AB, FG, DF, and BC. A

b 19 hours

What do you remember?

5 a Example answer:

8.05 Prim’s algorithm

1 a

Practice

c $48 000

7 a 4

c • AB with weight 6

• AF with weight 1

500

• FC with weight 7

720

740

• CD with weight 6

• DE with weight 5

520

Note: If there are multiple edges with the same minimum weight, other orders might be possible depending on tie-breaking, but this is one valid sequence starting from A and always picking the globally smallest available edge connected to the current tree without forming a cycle.

C 640

720 E

560

c $12 650 000

2 11 3 Prim’s algorithm starts by selecting any vertex and grows the tree from there. Kruskal’s algorithm starts by selecting the edge with the lowest weight anywhere in the graph. 4 a True

b False

c True

d True

Answers 1033 mathspace.co

8 a

12 a 5

F 4

B 2 F

c $2295

9 F

17 11

c 21 million dollars

E 16

15 H

100

15 a

160 B

E 2

134

10 4

6 D

117

126

11 K

111

D 15

10 26

10 21

C I

17 A

b 27 m

17 C

1034 Mathspace New South Wales – Year 11 Standard mathspace.co

31 18

H A

b 120 m

c $5850

16 a

19 The graph representation would show 5 vertices (A, B, C, D, E ) with edges and weights as listed.

D E H

One possible MST (using Prim’s starting at A, for example): B–C(90 m), C–D(100 m), A–B(120 m), D–E(130 m). Total length: = 440 m.

13 13

200

150

11 G

b 83 m

c $5810

17 a

D 8 15

13 17 J

b 117 m

180

100

20 a The graph representation would show 5 vertices (P, Q, R, S, T ) with edges and weights as listed in the table. Q 40

50 I

18 a Alex’s order (Prim’s from A): (A, B), (B, C), (C, D), (D, E), (E, F ). Total: 19 km. b Ben’s order (Prim’s from F ): (F, E), (E, D), (D, C), (C, B), (B, A). Total: 19 km. c Yes, MSTs identical. Trails: {(A, B, 4), (B, C, 3), (C, D, 5), (D, E, 2), (E, F, 5)}. This graph has a unique MST, which Prim’s ﬁnds regardless of start. Not always guaranteed; multiple MSTs can exist, leading to different results based on start/ties. d No, D–F (6 km) not included. From A, after A, B, C, D, E connected (cost 14 km), to connect F, options are A − F (10), C − F (9), D − F (6), E − F (5). Prim’s selects E − F (5) (cheapest). MST remains 19 km. e Max length for C–F is 5 km. When A, B, C, D, E are connected (cost 14 km), edges to F are A − F (10), C − F (x), E − F (5). For C − F (x) to be chosen, x ≤ 5 km (to be ≤ E − F (5)). If C − F = 5 km, it’s an alternative MST path.

80 60

c $359 500

Extend your thinking

130 E

C 19

220

A 22

120

b 180 000

8.06 Shortest path What do you remember? 1 The shortest path between two vertices is the path that has the minimum total weight, found by summing the weights of all edges along the path. 2 The shortest path is the route with the mathematically smallest total weight. The best path is the most suitable route based on other factors like traﬃc, tolls, scenery, or accessibility, and it might not be the shortest one. 3 a False

b True

c False

d True

4 The shortest path might have heavy traﬃc, low speed limits, poor road conditions, or tolls, making a longer path a better option in terms of time or cost.

Answers 1035 mathspace.co

Practice

5 A, B, D, C, E 6 a

G 10

b The shortest path from hub S to hub A is S, C, A, with a total shipping time of 9 hours.

1 B 2 C 3 A

9 A, C, B, E, F, H, J. Weight is 58.

4 a Vertices: 5, Edges: 4

10 D, C, E, G, F, I. Weight is 48.

b Vertices: 6, Edges: 5 c Vertices: 5, Edges: 5

11 a The shortest path is Central-Uptown-Museum with a time of 30 minutes. b The best path is Central-Market-UptownMuseum with a time of 43 minutes.

d Vertices: 6, Edges: 6 5 a Degrees: A = 2, B = 2, C = 1, D = 3. Connected b Degrees: U = 4, Y = 4, A = 4, E = 4, N = 4. Connected.

12 D, C, A, E, B with a total weight of 21. 13 B, F, D, C, E, A. Weight is 39.

6 a 3

14 B, C, D, A, E, F, G. Weight is 50.

b 53 km

Extend your thinking A 6 W

Shortest route: S-T-X-W-V-U Total distance: 26 km

1036 Mathspace New South Wales – Year 11 Standard mathspace.co

22 12 G

6 E 16

3 C

7 5

4 V

Chapter 8 review

8 A, D, F, G, H. Weight is 57.

7 C, A, B, F, I. Weight is 65.

Quickest route: D, E, C, G, A, F, B

5 5

8 a Yes. Their vertices are connected by an edge, which indicates they have worked together. b David and Grace; Fiona and Ben. 9 a Location B b Location A

C D 11

−

12 a 5

b 6

Lookout 5 Start

3 Waterfall 6

Summit

c 17

13 The shortest path is A-D-C, with a weight of 25 minutes. 14 a It is a tree. b It is a tree. c Not a tree; it is disconnected. d Not a tree; it is disconnected and has a cycle. 15 a 9 edges.

b 6 edges.

16 The minimum cost is $135 million. 17 The shortest path is A-B-F-I, with a total time of 53 minutes.

24 The van cannot take the route through B because the Depot-B bridge has a height limit of 4.0 m. The only alternative is the Depot-A-C path, which is valid. However, both roads from C, C-D (4.0 m) and C-B (3.8 m), are too low. Since all routes from the Depot to D include at least one impassable bridge, no valid path exists. 25 a The minimum cost is $1900. b The new MST uses is S-T-U-P-Q-R. The cost is $2200. 26

Museum

18 a The shortest path is A-C-E with a time of 40 minutes.

b The best path is A-B-D-E with a time of 65 minutes. It includes one scenic route (D-E).

Gallery

12 8

20 18

19 Path: A-E-C-D-F-B. Total time: 39 minutes.

Park

20 a Undirected b Directed c Directed d Undirected (usually, as ﬂights go both ways) 21 The student has incorrectly counted the intersection point of two edges as a vertex. The network has 4 vertices and 6 edges.

Tower

Wharf

Minimum total time is 35 minutes. 27 a Path A-C-E-G-F-J. Weight = 41 metres. b Path B-F-G-H. Weight = 35 metres.

Answers 1037 mathspace.co

28 a The shortest path is A-B-E with cost $400.

Extend your thinking

b The best path for them is still A-B-E with a cost of $400.

17 4 hours 15 minutes

29 Yes. If there’s a tie for the lowest weight edge, the choice can be arbitrary. Prim’s algorithm builds from one point, so its choice might be limited to edges connected to its current tree. Kruskal’s algorithm can pick any of the tied-weight edges from anywhere in the graph. These different choices at a tie-break point can lead to two different, but equally minimal, spanning trees.

18 315 seconds 19 2 hours 30 minutes. Using 24-hour time (11:50 to 14:20) avoids confusion across midday, simplifying subtraction. 20 460 seconds 21 a 1 hour 35 minutes

b 2 hours 25 minutes

c 3 hours 5 minutes

d 7 hours 5 minutes

9.01 Units of time

9.02 Time intervals

What do you remember?

1 60 minutes

1 a True

b True

2 20:30

2 60

3 3:45 p.m.

3 1 hour and 15 minutes

4 9 minutes

4 a 3 hours and 45 minutes

c False

b 3 hours and 15 minutes Practice Practice

5 225 minutes 6 a 240 minutes

5 3 hours and 45 minutes

b 48 hours

c 18 000 seconds

d 4320 minutes

6 12:45 p.m.

e 420 minutes

f 86 400 seconds

7 1:30 p.m.

7 a 30 minutes

b 3 hours

8 6 hours and 20 minutes

c 60 minutes

d 2 hours

e 15 minutes

f 5 hours

9 6:30 a.m.

8 a 11:30

b 18:45

c 00:10

d 14:25

9 a 08:30

b 15:45

c 00:21

d 12:15

e 06:20

f 22:50

10 13:40 11 11:45 a.m. 12 15 hours and 45 minutes

10 a 22:17

b 00:08

13 17:45

11 a 7:45 a.m.

b 3:30 p.m.

14 6 hours and 40 minutes

c 12:00 a.m.

d 12:00 p.m.

15 35 minutes

e 9:15 a.m.

f 11:50 p.m.

16 3 hours and 30 minutes

12 2 hours 30 minutes 13 35 minutes 14 2 hours 30 minutes 15 4 hours 5 minutes 16 2 hours 35 minutes

1038 Mathspace New South Wales – Year 11 Standard mathspace.co

Extend your thinking 17 19 hours 18 a 11:45 a.m. b Answers may vary. A possible adjustment is to cancel the 15-minute break. This would

save the required 15 minutes, resulting in an arrival time of 11:30 a.m. 19 Shorten the second show by 10 minutes to 1 hour and 20 minutes, so the concert ends at 21:30.

16 a 19:00 (Dinner 17:30 − 18:30. Arrives concert hall 18:30 + 8 mins = 18:38. Ready for concert 18:38 + 15 mins = 18:53. So, the 19:00 concert is the earliest.)

What do you remember? 1 C b True

c False

d True

3 a 2 hours and 25 minutes b 2 hours and 55 minutes c 4 hours and 25 minutes d 2 hours and 20 minutes 4 a 1 hour and 10 minutes b 10 minutes Practice 5 a 45 minutes

b 20 minutes

6 7 minutes (to catch the 6:35 a.m. train) 7 a The 7:55 a.m. bus from Wollongong (arrives Kiama at 8:45 a.m.) b 7:45 a.m. 8 a 22 minutes

15 a The 8:10 a.m. train from Strathﬁeld. (Arrives C.Q. at 8:30. Walk 6 mins, ready for ferry at 8:36. Catches 8:45 ferry, arrives Manly 9:10 a.m.) b 1 hour (or 60 minutes)

9.03 Elapsed time applications

2 a False

Extend your thinking

b 3 minutes

9 a 6 minutes (between 7:10 and 7:16, and between 7:16 and 7:22) b 35 minutes 10 a 1 hour and 15 minutes b 2 hours and 45 minutes c 5:40 p.m. 11 a 14:30 b 15:45 12 a 40 minutes b 50 minutes (departs Gosford 6:20 a.m., arrives Hornsby 7:10 a.m.)

b No. Concert (18:15 for 1h 30m) ﬁnishes at 19:45. Earliest arrival at restaurant is 19:45 + 8 mins = 19:53. This is too late for a 19:00 dinner booking. 17 Total time span: 16:45 − 08:30 = 8 hours 15 minutes. Total break time: 15 mins + 45 mins + 15 mins = 75 minutes = 1 hour 15 minutes. Painting time: 8h 15m − 1h 15m = 7 hours 18 Option 1: Total time 25 minutes. Option 2: Bus journey (25 min) 8:05 → 8:30. Wait (7 min) to 8:37. Train (10 min) → 8:47. Total time 25 + 7 + 10 = 42 minutes. Option 3: Walk (12 min) 8:00 → 8:12. Train to C.Q. (8:15 → 8:25, journey 10 min, wait 3 min). Ferry (8:30 → 8:35, journey 5 min, wait 5 min). Walk to oﬃce (5 min). Total time 12 + 3 + 10 + 5 + 5 + 5 = 40 minutes. Option 1 has the shortest total travel time (25 minutes).

9.04 Latitude and longitude What do you remember? 1 a Latitude is a measure of how far a position is north or south of the Equator. b Longitude is a measure of how far a position is east or west of the Greenwich Meridian. 2 a ( 35° South, 147° East) b (28° South, 124° East)

13 9 hours and 15 minutes

c (13° South, 129° East)

14 7:30 p.m.

d (19° South, 122° East) 3 Point G

Answers 1039 mathspace.co

Practice

Extend your thinking

4 a Point A

b Point J

18 67°N, 81°W

c (60° N, 120° E)

d 45°

e 180°

f Point I

19 20:40

g Point G

h 12 hours ahead

5 a Nelson

b Port Douglas

6 a Landmass

b Water

c Landmass

d Water

What do you remember? 1 a Cities are 1 hour ahead of UTC+0 b Cities are 9 hours behind of UTC+0

7 a South, 7.4 hours behind

c Cities are 3 hours and 30 minutes behind of UTC+0

b South, 0.33 hours ahead c North, 0.73 hours ahead

d The island is 12 hours and 45 minutes ahead of UTC+0

d North, 0.47 hours behind 8 94°, 6.27 hours, B ahead 9 a 17 hours behind

b 2 hours ahead

10 Stavenegr, Uxbridge, El Tigre 11 a Eden

b Nagele

12 a Bangkok

b Suva

13 a (28° S, 155° E)

b (21° N, 134° E)

2 a True

b True

e True

f False

b City A

15 a Latitude: 4°S

b Latitude: 34°S

4 a UTC-8

b UTC+5.5

c UTC+1

d UTC+9

Practice

Longitude: 170°W Longitude: 170°W

5 a Dobo and Tokyo c 5:00 a.m.

d Latitude: 19°S

Longitude: 155°W Longitude: 175°E

6 a 10 hours

b Latitude: 69°S

c 1:00 p.m.

Longitude: 18°E Longitude: 18°E

7 Singapore

c Latitude: 30°S

d Latitude: 30°S

Longitude: 57°E Longitude: 21°W 17 a (45° N, 75° E) b Part of Starting journey position

d True

b An imaginary line near 180° longitude where the date changes by one day when crossed.

14 a 9°

16 a Latitude: 9°N

c True

3 a The standard time reference at UTC+0, based in Greenwich, England.

c (18° N, 144° E)

c Latitude: 19°S

9.05 International time zones and time differences

8 a 13 c 9:00 p.m.

b 10 hours d 9:00 p.m. b 07:00

b 3.5 d 8:30 p.m.

e 06:15 on Friday Change Change Finishing in in position latitude longitude

(45° N, 75° E)

9° N

7° W

(54° N, 68° E)

12° S

16° E

(42° N, 84° E)

c Difference in latitude: 18°, difference in longitude: 39°.

1040 Mathspace New South Wales – Year 11 Standard mathspace.co

9 a i Algeria ii 4 hours iii 01:25, on the 14th b i South Sandwich Islands ii 2 hours iii 4:00 p.m. c i Mauritius ii 1 hour iii 18:25 on the 15th

10 a 16 hours

b 8:00 a.m. on Monday

23 a Tuesday

b Wednesday

11 a Beth

24 a 4:00 p.m. on Sunday

b Han

b 1:00 a.m. on Monday

c i 5:00 a.m.

ii 6:00 a.m.

d i 9:00 a.m.

ii 11:00 p.m.

12 a C b Tuesday, Wednesday, Thursday, or Friday. 13 a 14:10

b 5:20 p.m.

14 a 4 hours

b 2:00 p.m.

c 3:00 a.m. on Saturday 15 a 4:50 p.m.

b Monday

16 22 hours 17 a 5:44 p.m.

b 2:27 a.m.

c 5:58 p.m. 18 a 38 hours and 33 minutes b 30 hours and 22 minutes c 31 hours and 41 minutes 19 a 4 hours b 19:30 on Monday 20 a No suitable start time in Cairo meets the 9:00 a.m. to 5:00 p.m. requirement for all members. b 3:00 p.m. Extend your thinking 21 a The UK is 12 hours behind New Zealand, so if you have, for example, a 22-hour flight the local time when you arrive is only 10 hours later than your local time when you left. b There is a 9-hour difference (depending on the time of year) between Paris and Melbourne, so if you have a 20-hour flight your local time when you arrive in Melbourne will be 29 hours later than the local time when you left Paris. 22 a There is a 20-hour time difference between the two cities, but the flight is only 13 hours, so the local time when Kiki arrives is 7 hours earlier than the local time when she left. b We would need to add the 20-hour time difference to the 13-hour flight, so local time on arrival will be 33 hours later than local time on departure.

c The traveller departs Apia on Monday at 10:00 a.m. (Apia time). After a 5-hour flight and crossing the International Date Line from west to east, they arrive in Honolulu on Sunday at 4:00 p.m. (Honolulu time). This means they arrive on the day before they departed due to the IDL crossing. After a 2-hour stay, they depart Honolulu on Sunday at 6:00 p.m. (Honolulu time). After a 6-hour flight to Anchorage, they arrive on Monday at 1:00 a.m. (Anchorage time). 25 Fractional offsets align local time with geographical or cultural preferences, such as Nepal’s offset to be slightly ahead of India. The time difference between Nepal and Tokyo is 9 − 5.75 = 3.25 hours, or 3 hours and 15 minutes. For example, if it’s 12:00 p.m. in Nepal, it’s 3:15 p.m. in Tokyo. 26 Student A incorrectly divided UTC offsets instead of subtracting them. Student B correctly calculated the 17-hour difference but should clarify the day change. The correct time is 7:00 a.m. the next day in Seoul, as 9 − (−8) = 17 hours ahead.

9.06 Australian time zones and daylight savings What do you remember? 1 Australian Western Standard Time (AWST) 2 UTC+10.5 3 a Yes

b No

c Yes

d No

e No

f Yes

g Yes

h Yes

4 5:00 a.m. Practice 5 Canberra 6 a 15:14 AEST c 1:49 a.m. AEST 7 a 15:35 AEDT c 10:45 a.m. AEDT

b 7:00 a.m. AEST d 10:50 p.m. AEST b 5:00 a.m. AEDT d 6:13 p.m. AEDT

Answers 1041 mathspace.co

8 a 4:30 p.m.

b 1:30 p.m.

c 3:30 p.m.

d 3:00 p.m.

9 a 21:20

b 22:20

10 a i 7:46 p.m.

ii 6:16 p.m.

the 3-hour time difference gives a Perth arrival time of 11:30 p.m. on Thursday. However, Brisbane (AEST) does not use daylight saving and is only 2 hours ahead of Perth. The same ﬂight from Brisbane would also land at 2:30 a.m. Friday in Brisbane time. Subtracting the smaller 2-hour time difference gives a Perth arrival time of 12:30 a.m. on Friday. Therefore, the arrival would not be on the same day.

b 5:57 p.m. c i 23:40

ii 22:10

d 11:46 a.m. 11 a 12:25 p.m. b 9:55 a.m.

18 During Daylight Saving Time, Sydney is UTC + 11 and Perth is UTC + 8, creating a 3-hour difference. A ﬂight leaving at 6:00 a.m. AEDT is equivalent to 3:00 a.m. AWST. Adding the 5-hour ﬂight duration results in an arrival time of 8:00 a.m. AWST.

c 5 hours and 22 minutes d 2 hours and 26 minutes e 7 hours and 18 minutes 12 a 3:50 p.m.

b 10:21 p.m.

c 10:45 a.m. 13 a 4:42 p.m. b

Western Australia (WA)

South Australia (SA)

Queensland (QLD)

6:15

7:45

8:15

9:45

11:15

11:45

8:45

10:15

10:45

5:25

6:55

7:25

19 Since both Melbourne and Sydney are in the same time zone (AEDT) during Daylight Saving Time, the train’s arrival time is simply 3 hours after departure, making it 1:00 a.m. the next day.

Chapter 9 review 1 B 2 B 3 C

4 a 9:15 a.m.

b 3:30 a.m.

c 05:50

d 00:20

e 07:40

f 16:15

7:11

g 10:00

h 13:45

12:25

i 6:55 p.m.

j 12:35 a.m.

7:47

k 11:25 a.m.

l 2:10 p.m.

AWST

ACST

AEST

3:55

5:25

5:55

5:11

6:41

10:25

11:55

5:47

7:17

5 9 : 25 p.m. 14 22 hours 15 a 3:38 a.m.

6 a 1 hour 15 minutes b 9:28 a.m.

c 3:46 a.m.

b 5 minutes 7 Longitude difference: 80°.

b 23 hours 28 minutes

Time difference: 5 hours and 20 minutes. City Y is ahead.

c 16 hours 31 minutes

8 a 5 hours

16 a 22 hours 26 minutes

Extend your thinking 17 During daylight saving, Melbourne (AEDT) is 3 hours ahead of Perth (AWST). A 5.5-hour ﬂight departing at 9:00 p.m. Thursday would land at 2:30 a.m. Friday in Melbourne time. Subtracting

1042 Mathspace New South Wales – Year 11 Standard mathspace.co

b 3:00 p.m., Tuesday 9 a 10:00 a.m.

b 4:15 p.m.

10 a 15:30 (DST: UTC + 11) b 15:00 (DST: UTC + 10.5) c 12:30 (UTC + 8, no DST)

11 a 23:17

3 a 26th

b 03:08

b Between 54th and 55th

c 3 hour 51 minutes 12 a (31°S, 116°E)

b (17°S, 122°E)

c (21°S, 133°E)

d (34°S, 151°E)

13 2 hours 5 minutes 14 a Latest train: 1:05 p.m. (arrives Bondi Junction at 1:25 p.m., meaning he is ready for a bus by 1:33 p.m., and will arrive at Bondi Beach at 2:00 p.m.). b 55 minutes

4 a ii

b v

e vi

f i

d iv

5 a The modal class is the class interval in grouped data that has the highest frequency. b For ungrouped data, the mode is the speciﬁc value that appears most often. In grouped data, the modal class is the interval with the highest frequency, not a single value. Practice

15 a 105° b 7 hours, City P is ahead c 5:00 p.m., Tuesday 16 a 24 hours b 2:30 p.m., Tuesday. The time difference is 24 hours, and crossing the International Date Line from east to west subtracts one day. 17 Arrival: 10:30 p.m., Saturday. Time difference is 1 hour (Adelaide DST UTC + 10.5, Darwin UTC + 9.5).

6 a 10.4

b 16.96

c 9

7 a Yes

b Yes

c Yes

8 a 43.4

b 45.6

c 44.5

9 a 85.6 b 83.4 c Student A performed better with a mean of 85.6 compared to Student B’s 83.4. 10 a 330 000 ≤ C < 340 000 b $335, 000

Flight departs 8:00 p.m. for 3.5 hours, so arrival is 11:30 p.m. Adelaide time, or 10:30 p.m. Darwin time.

12 30

18 a (75° E, 45° N)

13 36

b 120°

11 a 17

b 28.5

14 a 17

c 10:00 p.m., Point I is ahead

c 56.75

d 43.45

b 34

15 300

19 a 8 hours, Hobart is ahead

16 75

b 8:00 a.m. (next day)

17 33

20 a 15:25 b 16:35 c First available ferry: 16:55, arrives Circular Quay at 17:02

18 a The sales ﬁgure 8 kg occurs most often (3 times). b The mode is 8 kg. 19 a i The mode is 8.

10.01 Measures of centre

ii The data is unimodal. b i The modes are 8 and 14.

What do you remember? 1 The mean is the average, calculated as the sum of all scores divided by the number of scores, acting as the balance point of the data. 2 The median is the middle value in an ordered dataset, found using the position is the number of scores.

c iii

, where n

ii The data is bimodal. c i The mode is 26. ii The data is unimodal. d i The modes are 2, 5, and 9. ii The data is multimodal. e i There is no mode. ii The data is uniform.

Answers 1043 mathspace.co

f i The modes are 7, 11, and 15. ii The data is uniform. 20 a The modal class is 10 − 14. b The grouped distribution is unimodal. c The modal class (10 − 14) shows the most common score range, helping the teacher identify typical performance and adjust teaching or assessments. 21 a The modal class is 40 ≤ h < 50.

c The dataset is positively skewed, with larger values increasing the mean, while the median remains at the central value. d It is not possible. When a new value x is added, the new median becomes 4. For the which solves to x = −5. Since hours cannot be negative, no such value exists. 26 a

b The grouped distribution is unimodal. c The modal class (40 ≤ h < 50) shows the most common study range, aiding in tailoring support resources. 22 a Graph A: Mode = 2. Graph B: Mode = 2. b Graph A has a higher frequency (8 vs. 7). c Both graphs are unimodal. 23 a

Tasks

Frequency

Pulse Rate

Class Frequency ( f ) Centre (x) 54.5

545

60–69

64.5

967.5

70–79

74.5

1490

80–89

84.5

1014

b 71.25

c 70.46

27 a 90 minutes

b 25%

c $44 000 28 a

Frequency ( f )

f (x)

150

240

340

450

285

b All numbers (2, 3, 4, 5, 6, 7) are modes.

200

c The data distribution is uniform.

Total

1665

b Less useful. Mean or median better represent typical travel time. c Less useful. Median is better for house prices due to outliers. d Most useful. The mode identiﬁes the subject with the highest enrolment. e Useful. The mode shows the most common rating.

f×x

50–59

Weight (x)

24 a Most useful. The mode identiﬁes the most frequently requested size for stock management.

= 4,

mean to also be 4, the equation is

b 18 kg c 17.53 kg d The mean is more affected, as it accounts for all values, while the median depends only on the middle position. 29 Dataset: {12, 12, 12, 12, 12, 5, 5, 10, 10, 10}. Mode is 12. 30 Dataset: {3, 4, 5, 5, 5, 10, 10, 10}. Modes are 5 and 10, mean is 6.5. 31 a The grouped distribution is bimodal.

Extend your thinking 25 a 4 hours b 4.9 hours

1044 Mathspace New South Wales – Year 11 Standard mathspace.co

b No, the ungrouped data may be unimodal as grouping can mask speciﬁc height frequencies.

32 The dataset is unimodal, as A has the highest frequency.

16 a

33 Error: The student incorrectly used “multimodal” instead of “bimodal” for two modes. Correct analysis: Modes are 2 and 3; the dataset is bimodal.

10.02 Measures of spread What do you remember? 1 a Never true

b Sometimes true

c Never true

d Sometimes true

2 B

7 a 5

170

45 ≤ x < 50

47.5

522.5

50 ≤ x < 55 52.5

840

55 ≤ x < 60 57.5

977.5

60 ≤ x < 65 62.5

437.5

65 ≤ x < 70

67.5

810

70 ≤ x < 75

72.5

797.5

75 ≤ x < 80

77.5

387.5

4942.5

b 72

d Higher Standard Deviation

f×x

17 a i 5

ii 62.62

iii 1.35

b i 6

ii 42.78

iii 1.61

Extend your thinking

b 4

18 a The Red Class, because their mean was 56.4, higher than the Blue Class’s mean of 56.3.

8 B 9 D-A-C-B b 9.57

c 5.81

d 8.63

e 5.54 11 a 3.44 b The population standard deviation remains at 3.44. c The population standard deviation is doubled to 6.88. 12 a 3.36 b Adding an 11th packet containing only 5 lollies, which is signiﬁcantly less than the mean of 15.2, will increase the spread of the data, leading to a higher standard deviation. Therefore, the standard deviation would increase. b 3.94 d Standard deviation

14 a i 60

ii 18.54

b i 82

ii 25.37

15 1.58

42.5

c 9.76

6 11

c 1.61

40 ≤ x < 45

c 2.2981

5 23

13 a 4

Frequency (f )

b 59.55 minutes b 31

4 a 5

10 a 5.30

Class Centre

Totals

Practice 3 a 13

Class

b The Blue Class, because their population standard deviation was 3.74, lower than the Red Class’s 4.98. 19 a Decreased by 12

b Increased by 4.57

c Decreased by 8

d Increased by 20

20 a

Height of step

Data

Slow

Medium Fast

Short step

Avg. heart rate

90.0

105.5

126.0

Standard deviation of heart rate

1.0

0.5

2.0

Avg. heart rate

98.0

127.0

129.5

Standard deviation of heart rate

2.0

2.5

Tall step

b A tall step at a fast stepping rate, c A short step at a medium stepping rate.

Answers 1045 mathspace.co

21 a 0.54 b i Decrease

ii Increase

14 a The goal average dropped by approximately 0.67.

10.03 Compare datasets

b 0.56.

What do you remember?

15 a i 120

1 Compare datasets using: • Measures of centre (mean, median and mode) • Measure of spread (range) • Shape of the data 2 Yes 3 Graph 1 4 a False

b True

c False

Practice

6 a Bianca

b Ivan

7 a 21 minutes

b 25.7 minutes

c Primary students ii False

ii 73

b City B 16 On average, Plant A shows slightly higher growth (Mean 10.3 vs 10.1). However, Plant B has a higher median (10 vs 9.05) and is signiﬁcantly more consistent, with a much smaller range (3 vs 9). The positive skew in Plant A’s data indicates occasional days of very high growth, while Plant B’s symmetrical distribution reﬂects stable, predictable performance. 17 For Group 1, the mean is 77.69 bpm, median is 80.5 bpm, range is 28 bpm, and the dataset is negatively skewed (mean < median).

5 Canada

8 a i False

produces cookies with more consistent weight.

iii True

b Yes c Yes 9 a 85.4 b 84.4 c Student X has a higher mean, 85.4, and performed better than Student Y.

For Group 2, the mean is 75.69 bpm, median is 75.5 bpm, range is 27 bpm, and the dataset is slightly positively skewed (mean > median). Group 1 has a higher centre (mean and median), indicating higher average pulse rates. Group 1 also has a slightly larger spread (range), suggesting less consistency. The negative skew in Group 1 indicates a lower pulse rates, while Group 2’s slight positive skew suggests a few higher pulse rates.

d The highest score is 95, obtained by Student Y.

18 a Decreased by 10

b Increased by 5.50

c Decreased by 8

d Increased by 25

e The lowest score is 71, obtained by Student Y.

19 a $4286 million

b Higher

c Unchanged

d Higher

10 a 4

b 3

c 5

d 6

Extend your thinking

e Team X

f Team Y

11 a i Dataset B

ii Dataset B

b i Dataset A

ii Dataset A

20 Looking at the data, although the ranges are identical (both are 7 kg). The mean for Group 1 is 58.3 kg. The mean for Group 2 is 63.4 kg. The mean values are quite different. Group 2 has a noticeably higher average weight than Group 1. This difference in mean weights might suggest that these two groups of turtles are not of the same species. However, it’s important to note that the determination of species is based on multiple factors, not just weight. Additional data and biological information would be needed to conclusively determine species.

12 a i 72

ii 81

b Class B 13 a Machine Q has a greater mean weight than Machine P, indicating that Machine Q generally produces heavier cookies. b Machine Q has a smaller standard deviation than Machine P, indicating that Machine Q

1046 Mathspace New South Wales – Year 11 Standard mathspace.co

21 a

Mean

Median

Mode

Range

Class 1

2.7

Class 2

8.2

b Class 2, because their mean, median and mode were higher. 22 a The mean for the Red Class is 56.40, and the mean for the Blue Class is 53.8. This shows Red Class performed better because their mean is higher. b If we look at the standard deviations, Red class has σ ≈ 5.25 and Blue class has σ ≈ 4.29, so Blue class is more consistent because their standard deviation is lower.

7 a Q1 = 1, Q3 = 9 c Q1 = 15, Q3 = 26.5

d Q1 = 16, Q3 = 24.5

8 11 9 a 51

b 7

c 1

10 a 5

b 16

c 11

d 17

11 a i 6, 10, 15, 17, 24, 39, 40 ii 7 iii 17 iv 10 v 39 vi 29 b i −6, −4, −1, 7, 7, 9, 9 ii 7

23 a

= 3.2, σ = 2.10

iii 7

= 3.9, σ = 2.73

iv −4

c Both painkillers have similar mean effectiveness with 3.2 and 3.9, yet the original painkiller’s higher standard deviation indicates a wider spread in pain levels among users, suggesting less consistent results. The reformulated painkiller appears more reliable due to its smaller standard deviation which shows consistency.

b Q1 = 24, Q3 = 32

v 9 vi 13 c i 4, 8, 10, 14, 15, 19, 20 ii 7 iii 14 iv 8 v 19 vi 11

10.04 Quartiles and interquartile range What do you remember? 1 a 28

b 11.5

iii 36 c 49

d 7

b True

iv 25 v 42.5

2 IQR = Q3 − Q1 3 a False

d i 20, 22, 28, 32, 40, 42, 43, 54 ii 8

c False

d True

4 a iv: Minimum: Lowest value

vi 17.5 e i 69, 71, 78, 79, 82, 84, 85, 88 ii 8

b v: Lower quartile, Q1: At most 25% of the data is below this value

iii 80.5

c ii: Median: 50% of the data lies on either side of this value

v 84.5

d iii: Upper quartile, Q3: At most 25% of the data is above this value e i: Maximum: Highest value

iv 74.5 vi 10 f i 100, 102, 110, 113, 115 ii 5 iii 110 iv 101

Practice 5 a 74

b 61

c 114

d 53

6 a 72

b 51.5

c 85.5

d 34

v 114 vi 13

Answers 1047 mathspace.co

g i 197, 198, 202, 205, 207, 228

Cumulative frequency

ii 6 iii 203.5 iv 198 v 207 vi 9 h i 15.8, 19.5, 22.4, 29.6, 35.4, 39.1, 46.2 ii 7

35 29

10 0

iii 29.6

45 47

Number of phone calls

iv 19.5 v 39.1

Weight

Frequency

Cumulative Frequency

The interquartile range (IQR) is the difference between the upper quartile (Q3) and the lower quartile (Q1). From the cumulative frequency graph, Q1 = 12th value is at 1 phone call and Q3 = 36th value is at 5 phone calls. Thus, the IQR is 5 − 1 = 4.

14 a 19.5

15 a 72 hours

c 13

vi 19.6 12 a

16 a

Cumulative frequency

Totals

14 3 15 16 17 18 19 20 21

Weight (kg)

b 85 hours

Freq ( f )

c 7

Score (x)

b 30

b 26.5

b 7

c 6

d 8

17 a 1

b 3

c 2

18 a 75

b 8

c 3

19 a 79

b 64

c 84

e 2

d 20%

e 20% c 25% = 7.5 50% = 15 75% = 22.5 d The median is 18 kg, Q1 is 16 kg, Q3 is 19 kg, and the IQR is 3. 13 a 47 b 7

20 a 74

b 79.5

c 24

d 35

e 16.5

f 18.5

g • Centre: The Sapphire Sharks have a higher median score (79.5 vs. 74), suggesting they generally performed better. • Spread: The Sapphire Sharks have a larger range (35 vs. 24) and a larger IQR (18.5 vs. 16.5), indicating their scores are

1048 Mathspace New South Wales – Year 11 Standard mathspace.co

more spread out and less consistent than the Emerald Eagles.

third quartile and the maximum, indicating a few lower values causing a left skew.

• Skew: For the Sapphire Sharks, the mean (79.33) is slightly less than the median (79.5), suggesting a slight left skew. For the Emerald Eagles, the mean (75.22) is slightly greater than the median (74), suggesting a slight right skew.

25 a For Group 1, the range is 39, the interquartile range is 23(116.5 − 93.5), the median is 99 and the mean is approximately 104.24.

Extend your thinking 21 a

minutes. This means that about half of the

ﬂights had a delay less than

minutes and

half had a delay of more than

minutes.

b 10 minutes

c 45 minutes

d 35

e 83

f 83%

g 22

h 22% 22 a 142 minutes

b 27

c 137 minutes 23 a 8 marks b 9 marks c 1 d In this dataset, the scores less than 7, while constituting the minimum values, don’t significantly impact the mean and median due to their low frequency. However, they do increase the range and could potentially affect the first quartile by shifting it lower. 24 a Set 2 has the largest interquartile range. This indicates that the middle 50% of data in Set 2 is spread over a larger range of values compared to the other two datasets. b Set 1 appears to be symmetric as the differences between corresponding quartiles are nearly equal, indicating a symmetrical distribution. Set 2 is skewed right because the difference between the third quartile and the maximum is much larger than the difference between the minimum and the first quartile, indicating a few higher values causing a right skew. Set 3 is skewed left as the difference between the minimum and the first quartile is much larger than the difference between the

For Group 2, the range is 30, the interquartile range is 18(105 − 87), the median is 100 and the mean is approximately 97.48. b The mean of Group 1 is higher than that of Group 2, which suggests that Group 1, on average, scored higher on the EQ test. c The spread of Group 1 is larger than that of Group 2, indicating that there is a greater variability in EQ scores within Group 1 than within Group 2. 26 a 2.5 b 8 c There are only 2 students who have more than 8 pets (the upper quartile), which is equal to 8% of the total number of students (2 out of 25). It is not exactly 25% because the quartiles are determined by dividing the data into 4 equal parts, but this does not guarantee that the percentage of values above or below a quartile will always be 25%. 27 Yes, the percentage of data values between the upper and lower quartiles is always exactly 50%. This is because the lower quartile (Q1) represents the 25th percentile of the data and the upper quartile (Q3) represents the 75th percentile of the data. The difference between the 75th and 25th percentiles is 50% of the data. For example, consider the dataset: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The lower quartile is 3 (25th percentile) and the upper quartile is 8 (75th percentile). The data values between Q1 and Q3 are 4, 5, 6 and 7, which represent 50% of the total dataset.

Chapter 10 review 1 B 2 D 3 B 4 a 5

b 19

c 107

d 0.3

5 a 7

b 110

c 21

d 15

Answers 1049 mathspace.co

6 a $102 000

b $98 000

c $100 000 7 $62 8 a 35 ≤ H < 40 hours

b 36.6 hours

9 a 9.78

b 5.13

c 5.56

d 6.95

10 a 29

b 8.90

11 a 4.24

c Fertiliser Y resulted in a higher average growth (Mean Y = 16 cm vs Mean X = 13 cm). Fertiliser Y also produced more consistent growth, as shown by its smaller standard deviation ( σ Y ≈ 1.73 cm compared to σ X ≈ 2.57 cm). Therefore, Fertiliser Y appears to be more effective and reliable. 18 a 125 years

b 112 years

c 140 years

d 28 years

b The population standard deviation remains at 4.24. It is unaffected.

19 a 50

b 6

c 3

20 a 1

b 2.5

c 1.5

c The population standard deviation is tripled to 12.73.

21 a 88 marks

12 a 19 kg

b 23.5 kg

c Group B 13 a i 11

ii 7

b Club B (smaller range indicates more consistency). 14 a Battery Y has a greater mean life than Battery X, indicating that Battery Y generally lasts longer. b Battery Y has a smaller standard deviation than Battery X, indicating that Battery Y provides a more consistent battery life. 15 Shop A: Mean = $1350, Median = $1300, Range = $1200. Its graph will be slightly positively skewed (Mean > Median). Shop B: Mean = $1462.50, Median = $1450, Range = $300. Its graph will be slightly positively skewed (Mean > Median). Comparison: Shop B has a higher centre (both mean and median sales are higher), indicating generally better sales performance. Shop B also has a much smaller spread (range), suggesting more consistent daily sales compared to Shop A, whose sales are more variable. Both shops show a slight positive skew.

c Decreased by 15 points.

b 23 marks

c 80 marks 22 a i 75

ii 77

b i 25

ii 38

c i 17.5

ii 21

d Class B has a slightly higher median score than Class A. Class B also has a larger range and a larger IQR, indicating a wider spread of scores and less consistency compared to Class A. Both distributions appear reasonably symmetric, though Class B’s spread is wider overall.

11.01 Five-number summaries and box plots What do you remember? 1 a

16 a Decreased by 15 points. b Increased by approximately 10.64.

d 4

d Increased by 40 points. 17 a Mean X = 13.00 cm, σ X ≈ 2.57 cm b Mean Y = 16.00 cm, σ Y ≈ 1.73 cm

1050 Mathspace New South Wales – Year 11 Standard mathspace.co

2 a i 19

ii 11

b i 34

ii 15

3 a True

b False

e True

f True

4 a i 50%

ii 25%

8 a

c True

d False

iii 50%

iv 75%

Data

v 75% b Second 0

c First and third

10 20 30 40 50 60 70 80 90 100

Practice

Data

5 a 38, 39, 42, 44, 45, 47, 50, 53, 55, 57, 60, 62, 65, 68, 71 b i 71

ii 38

iii 53

iv 44

b 18

c 15

d 10

v 62 6 a 3 e 7 7 a

Minimum

Lower quartile

Median

Minimum

Maximum

Upper quartile

Outliers

2, 3, 40

Maximum

Range

Interquartile range

Minimum

Lower quartile

Median

Upper quartile

Maximum

Minimum

Lower quartile

Median

Upper quartile

Maximum

Minimum

Lower quartile

Median

51.5

Upper quartile

Maximum

10 a i

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

10 12 14 16 18 20 22 24 26 28 30

iii Left skewed

Answers 1051 mathspace.co

b i

Minimum

Lower quartile

Median

31.5

Upper quartile

36.5

Maximum

Data

iii Somewhat symmetrical, with slight right skew e i

20 22 24 26 28 30 32 34 36 38 40 42

Minimum

Lower quartile

Median

50.5

Upper quartile

Maximum

iii Symmetrical c i

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

40 42 44 46 48 50 52 54 56 58 60

iii Somewhat symmetrical, with slight left skew f i

Minimum

Lower quartile

Median

Upper quartile

Maximum

iii Somewhat symmetrical, with slight right skew d i

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

1052 Mathspace New South Wales – Year 11 Standard mathspace.co

iii Symmetrical g i

Minimum

Lower quartile

Median

Upper quartile

Maximum

Data

iii Somewhat symmetrical, with slight right skew h i

Minimum

Lower quartile

Median

Upper quartile

Maximum

iii Right skew j i

Minimum

Lower quartile

Median

15.5

Upper quartile

Maximum

Data

iii Symmetrical

Data

11 a The cyclist’s heart rates varied by 36 beats per minutes (bpm). b The middle half of the cyclist’s heart rates varied by 19 beats per minute. 12 a i D ataset A - skewed left, Dataset B - symmetrical

iii Somewhat symmetrical, with slight right skew

ii Dataset B iii Dataset B b i D ataset A - skewed right, Dataset B - skewed left ii Dataset B

i i

Minimum

Lower quartile

Median

Upper quartile

Maximum

iii Dataset A

Answers 1053 mathspace.co

13 a

Class 10P

Class 10Q

Median

Lower quartile

Upper quartile

Range

Interquartile range

Outliers

None

d 7 f 5

17 The typical pulse rates increases from about 68 to about 108 and are spread out over a larger range after exercise.

b 17 years

e 1st

Extend your thinking

c Interquartile range c 25%

Number of pages read

b Class 10P 14 a 31 years

15 a 49 b 19 c 12 d It’s Carl, because he has the higher minimum, median, and maximum scores and a smaller interquartile range, indicating more consistent and better sales.

18 Answers may vary. One possible explanation: A box plot visually displays the median, providing a measure of the central value of the dataset. The interquartile range, represented by the box, indicates the variability within the central half of the data, offering insight into the spread of data. Lastly, the ‘whiskers’ of the box plot highlight the overall range of data, contributing to our understanding of the data’s distribution. 19 a

Scores

16 a Monthly salaries (in thousand dollars)

b Scores of students

100

Scores

100

c Debbie’s blog visitors

−50

100

150

200

250

1054 Mathspace New South Wales – Year 11 Standard mathspace.co

c Science. The highest score is the maximum value of the data. Science has a highest score of 91, while music has a highest score of 88. Science has a minimum score of 52, while music has a minimum score of 43. Science has a median on 74 and Music has a median score of 66.

20 a The median score of the Gamma Geckos is 5 points higher than the median of the Delta Dragons. b The range of the Delta Dragons is 12 points wider than the Gamma Geckos. c The interquartile range of the Delta Dragons is 19 points wider than the interquartile range of the Gamma Geckos. d Answers can support either team, but should use summary statistics to justify the answer. For example: • We can expect the Gamma Geckos to win since the median number of points scored is 5 points higher than the median number of points scored by the Delta Dragons. • We can expect the Delta Dragons to win since in about 25% of games they scored more points than the highest number of points scored by the Gamma Geckos. 21 a Negatively skewed b Positively skewed c Yes. The group that received treatment had a lower median and most people had symptoms for less than 9 days, whereas the control group had half the people having symptoms for more than 9 days. 22 a Yes, correcting the typo from 32 to 23 affects the maximum value and potentially changes any identiﬁed outliers. A new box plot is necessary to accurately represent the corrected data. b Fixing the typo changes the range from 21 to 14, resulting in a decrease of 7. However, since the positions of Q1 and Q3 remain unchanged, there is no impact on the interquartile range (IQR).

11.02 Parallel box plots What do you remember? 1 Minimum value, Lower quartile, Median, Upper quartile, Maximum value 2 To compare the ﬁve-number summary and also the range and interquartile range visually. 3 The two measures of spread are the range and the interquartile range (IQR). The range is represented by the total distance from the tip of the leftmost whisker (the minimum value) to the tip of the rightmost whisker (the maximum value). The interquartile range is represented by the length of the box itself. Practice 4 a

English

Mathematics

Median

Lower quartile

Upper quartile

Range

Interquartile range

b Mathematics 5 a Upmarket restaurant b $22 c Upmarket restaurant d $36 e 26 items 6 a 7 kWh b 75% c 4 kWh d Yes, this would be an unusually high value. The maximum daily energy usage shown on the box plot for Café A is 30 kWh, and 45 kWh is significantly higher than this.

23 a

50 55 60 65 70 75 80 85 90 95 100

b 20 c 25% d 52 and 85 e There are now 17 scores, so there are 13 values below Q3 and

= 76.5%. At least

75% of the data lies below Q3.

7 a Matt

b Matt

c Matt

8 a Tobias

b Ned

c Ned

d $5000

9 a Red bricks have a higher median and higher maximum value. b Yellow bricks have a higher minimum and a smaller range so their strength is more consistent.

Answers 1055 mathspace.co

10 a

Manufacturer Manufacturer A B Median

4000

5000

Lower Quartile

2500

3500

4500

6000

Range

4000

6500

Interquartile Range

2000

2500

b Manufacturer B, because 50% of their light bulbs last longer than all of Manuafacturer A’s light bulbs. ii 10

iii 18

iv 13

ii 11

iii 14

iv 11

v 8 b i 13 v 3 c Class 9P. The top 25% of their marks are between 18 and 20, while the top 25% for 9Q is between 14 and 20. 12 a 50

b 50

c 12

d Carl

e Carl

b 12 bpm

c 12 bpm

d 52 bpm

e 56 bpm

f Before

g 16 bpm 14 a 0.5

b 0.75

c 3rd

d 2006

Extend your thinking 15 a Bookstore A, it has a lower range and interquartile range. b $10 c i True 16 a

ii True

c Cooper d No, a high outlier does not change what the minimum score is, hence would not alter the selection for pole position. 17 a Film C, because all the ages are 18 or older. b Film B, because the median age is 16. c Film A, the ages include teenagers and adults with a median of 38.

11.03 Histograms, dot plots and box plots What do you remember? 1 a The histogram

b The box plot

c The box plot

d The histogram

2 a Negatively skewed

b Symmetrical

3 a • • • •

Frequencies for each group Mode Shape Outliers

b • • • •

Quartiles Median Shape Outliers

• Range and interquartile range c • • • •

Frequencies for each group Mode Shape Outliers

Cooper

Marion

Minimum

44.8

47.1

49.1

54.7

Median

52.5

60.45

55.75

65.25

4 • Histogram A/ Box plot 2

68.5

• Histogram B/ Box plot 4

Maximum

66.6

d • Total • Rate of change • Outliers Practice

• Histogram C/ Box plot 1 • Histogram D/ Box plot 3

1056 Mathspace New South Wales – Year 11 Standard mathspace.co

Time (seconds)

c Symmetrical

13 a 12 bpm

Cooper

Marion 40

Upper Quartile

11 a i 13

5 a Hours (x)

Frequency

Cumulative frequency

2+3=5

5+4=9

9 + 5 = 14

14 + 4 = 18

18 + 2 = 20

7 a (0, 1, 2, 3, 5) b

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

c Slightly right-skewed or roughly mildly skewed to the right. 8 a

0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

c IQR = 2.5

9 10 11 12 13 14 15 16

Five-number summary:

Minimum

3.5

Median

6.5

Maximum

6 a

Cumulative histogram of sapling heights

Frequency

b IQR: 10 − 3.5 = 6.5

Skewness: Moderately right-skewed

5 0

c • Bronze: Up to 3.5 books 10

Height bins b • Q1 = 17.5

• Silver: 3.5 to 6.5 books • Gold: 6.5 to 10 books • Platinum: Above 10 books

• Median = 22.5

9 a Lower

• Q3 = 27.5 c

b 0

c 2

d Positively skewed

e 4

f Block B

g No 10 a Dot plot box plot: 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Answers 1057 mathspace.co

Histogram box plot:

c If the threshold is 60, more people might now surpass it after adding 10 minutes (or some borderline individuals cross that cutoff), thus increasing the proportion who meet the requirement. 10

b The two five-number summaries are notably different, indicating that the dot plot and histogram represent separate datasets. The dot plot’s summary (Min=3, Q1 = 7, Median = 10, Q3 = 12.5, Max = 16) has a lower median and is less spread out than the histogram’s estimated summary (Min = 0, Q1 = 9.5, Median = 12.5, Q3 = 15.5, Max = 17). These significant discrepancies in the measures of centre and spread confirm that the two graphs do not represent the same set of scores. c Approximately 12.5 (dot plot), 15.5 (histogram). Extend your thinking 11 a • With only three broad classes, you can do a rough cumulative frequency approach. If, for instance, 10 data points fall in the first class, 15 in the second, and 5 in the third, you can see where each quartile might fall. • But you’ll only know that Q1, median, and Q3 are somewhere inside one of those classes. b Not very reliable—wide intervals cause large uncertainty because you do not know the distribution within each interval. The quartile might be near the lower edge or upper edge of the interval. c • Refine the class intervals: If possible, break each large class into smaller ranges to get more detail. • Use additional summary statistics: The mean or a stem-and-leaf/dot-plot if partial data is available. • Ask for raw data or more finely binned data to reduce interval width. 12 a Adding a constant c to all data adds c to each minute, Q1, median, Q3. b The box plot slides to the right but keeps the same overall shape. Range and IQR stay unchanged since they depend on differences, not absolute values.

13 a Identical five-number summaries do not guarantee that the underlying shape or frequency distribution is identical. One is a small set of 10 points, the other is 30 points. b With fewer data points, the quartiles and median can change dramatically if a single score differs. They are less stable in small samples. c A histogram or dot plot would show if the data are truly distributed similarly. Also, knowledge of sample size or any repeated values can clarify differences. 14 a Use each midpoint as a representative value for all points in that class. Calculate a cumulative frequency and then locate the 25%, 50% and 75% positions. b Potential inaccuracies: • Actual data may cluster near one edge of an interval, making a single midpoint misleading. • Widely varying interval widths (some classes might span 5 units, others 10) can shift quartile locations significantly. c Wider intervals can lump dissimilar data together, causing overestimation or underestimation of quartiles. The resulting box plot might not reflect the true spread or centre accurately.

11.04 Outliers What do you remember? 1 a The mode usually does not change, but it can change if the outlier is also the mode. b The range always decreases. 2 a Larger

b Smaller c Larger

3 a 18 kg

b 14 kg

c 18 kg

d The mode are the same before and after removing the outlier. Practice 4 a 82, 85, 88, 92, 145

1058 Mathspace New South Wales – Year 11 Standard mathspace.co

d Smaller

b 145

5 a 131

b 7

c 47

d 25, 30

6 a 21, 21, 21, 24, 25, 26, 26, 26, 26, 27, 28, 29, 30, 30, 32, 41, 41, 46, 46, 47, 76 b 28

c 41

7 a i

d 25.5

Minimum

Median

Maximum

ii 4

iii −4

iv 12

v Yes

ii 126 iii

e 76

Mean

Median

Mode

Range

iv The mean, median and range decreased, but the mode stayed the same. b i

Mean

1.92

Median

1.7

Mode

0.9

Range

3.8

Mean

1.61

vi Yes b i

Median

1.5

Mode

0.9

Maximum

Range

1.9

ii 6

iii −6

iv 18

v No

vi No c i

ii 4.7

Minimum

iii

iv The mean, median and range decreased, but the mode stayed the same. c i

Mean

4540

Minimum

Median

4750

Mode

4700

Median

Range

3800

Maximum

ii 1500 iii

Mean

4877.8

Median

4800

vi No

Mode

4700

8 The mean will be higher, but the median will stay the same.

Range

800

ii 5

iii −2.5

iv 17.5

v Yes

9 a 15

b 13.29

c The median, because it is not affected by the outlier. 10 a i

Mean

Median

Mode

Range

102

iv The mean and median increased, the range decreased, but the mode stayed the same. 11 a

Mean

73.5

Median

67.5

Mode

None

Range

105

Answers 1059 mathspace.co

b 150 c

Mean

65.0

Median

Mode

None

Range

19 Median, because it is the least affected by the outlier. 20 The missing value 2.0 is an outlier as it is outside the IQR-based outlier boundaries [2.1, 2.9]. 21 a No outliers

With outlier

>, < or =

Without outlier

Mean

Median

Mode

Range

12 a $462 000

b $527 000

c Median

d $929 000

e $446 600 13 a 280

b A

b The median will be unchanged since the middle of a large group of data points is mostly unchanged when extreme values are removed. c The mode of the dataset is unchanged. d The range will decrease since the largest data value is being removed.

16 a

b 25

c 60

c Both datasets are found to have no outliers when analysed using the 1.5 × IQR fence method. This is a reasonable outcome, as the two datasets exhibit very similar statistical properties. Both have a median of 19 days and an interquartile range (IQR) of 6 days. The fences calculated for Dataset 1 ([6, 30]) and Dataset 2 ([7, 31]) are also very similar. This indicates that while the datasets contain different individual recovery times, the overall distribution and spread of the central 50% of the data are almost identical, and neither contains values extreme enough to be classified as statistical outliers.

11.05 Identify clusters and gaps

14 a The mean will be lower since the outlier is larger than the rest of the values.

15 a 28.1

b No outliers

d 24.1

What do you remember? 1 Possible answers: • The data is evenly distributed around the mean. It means that the left side of the distribution (less than the mean) mirrors the right side (greater than the mean). • The left side of the distribution (less than the mean) mirrors the right side (greater than the mean). • The mean, median and mode are very similar.

Minimum

2.5

3.1

Median

3.75

4.35

Maximum

10.0

b 1.25

2 Positive skew refers to a distribution where the tail is on the right side of the distribution. Negative skew refers to a distribution where the tail is on the left side of the distribution.

c 10.0

Extend your thinking 17 n = 24 18 She should use the mean since it is the higher measure of average because it is affected by the outlier.

1060 Mathspace New South Wales – Year 11 Standard mathspace.co

3 Clusters within a dataset can be identiﬁed by looking at areas where data points are concentrated. These are typically groups of data points that are relatively close together compared to the rest of the dataset. 4 Greater than 60 Practice 5 a Negatively skewed c Symmetrical

b Positively skewed d Positively skewed

e Positively skewed

f Negatively skewed

g Positively skewed

h Symmetrical ii 11 years

iii 30 years

iv 23.32 years

Novel genres 20

Frequency

6 a i 21 years

13 a

b Positively skewed 7 a Yes. The data is clustered mostly in the 20s to 30s (around 26 – 37 hours). There is also a smaller group in the 40s to 50s, but less tightly clustered. b 26 and 36

15 10 5 0

c 3

d Negatively skewed

15 a No

c 4.5% and 8.5% 9 a Yes

b Yes, 30s - 40s

c 48

b 4%–5% and 8%–9%

b Symmetrical 14 a Yes, 92

8 a Bimodal

Price

b Yes, 1 − 4 b 1 to 4

c 3

d Positively skewed

10 a Grouped data

16 The mean is greater than 70.

b 151.5 cm

17 a False

c Negatively skewed

b False

c False

d False

18 A

11 a Mean: 71.25 Five-number summary:(40, 65, 70, 77.5, 90)

19 300

Outlier: 40

20 Median, because the data is negatively skewed.

b Extend your thinking

Mathematics test scores

100

c The original mean was 71.25. After removing the outlier of 40, the new mean increased to approximately 72.89. This demonstrates that the low outlier was pulling the original mean down. By removing it, the mean becomes a more accurate measure of the central tendency for the main body of students’ scores. The removal of the outlier provides a clearer picture of the typical performance, which is concentrated between 60 and 90 and is slightly positively skewed. 12 In a positively skewed distribution, the mean is typically greater than the median, which is greater than the mode. This is because the mean is affected by the long tail of high values, pulling it further to the right.

21 The teacher might consider that the symmetrical distribution indicates a more consistent performance across students, while the positively skewed distribution suggests that most students scored lower, with a few high achievers. The teacher should compare medians for typical performance and use IQR to assess spread. 22 x = 10 23 a 30 b 42.5 c 26.5 d Positively skewed e 76 f T he majority of this group of people are likely to do none to moderate amounts of exercise.

Answers 1061 mathspace.co

24 Week 1, because there are no outliers. Should draw parallel box plots to compare.

d Original rainfall (mm)

Week 1 Petrol Price ($) Outlier-removed rainfall (mm) Outlier-replaced rainfall (mm)

1.0

1.2

1.4

1.6

1.8

.20

Week 2 Petrol Price ($)

1.0

1.2

1.4

1.6

1.8

2.0

25 a Original Data: • Five number summary: [30, 40, 45, 54.5, 120] mm • Mean: Approximately 52.75 mm • Standard deviation: Approximately 24.44 mm • Outliers: The 100 mm and 120 mm readings are identiﬁed as outliers. b 5

Frequency

4 3 2

90 100 110 120 130 140 150

e Removing outliers reduces standard deviation (24.44 to 8.40) significantly, reflecting less variability. Replacing outliers yields a similar standard deviation (7.86) but slightly higher mean (44.21 vs. 44.10). Removing outliers better preserves typical variability, as replacement slightly inflates the mean. This comes from comparing standard deviations and means. However, in real life, for water resource management, extreme rainfall values (even if rare) are critical for flood and drought planning. This means we should still use the original dataset. f A lthough adjusting outliers (by removal or replacement) offers a clearer picture of typical conditions, maintaining awareness of outliers (extreme rainfall) is essential for proper water resource management. Hence, reporting both typical and extreme values is advisable.

1 0

Chapter 11 review 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120

Rainfall (mm)

Clustered around the peak of 40. Huge gap between 55 and 100, small gap between 100 and 120. c i • Five number summary: [30, 37.5, 45, 54, 55] mm • Mean: Approximately 44.10 mm • Standard deviation: Approximately 8.40 mm ii • Five number summary: [30, 40, 45, 54, 55] mm • Mean: Approximately 44.21 mm • Standard deviation: Approximately 7.86 mm

1 C 2 B 3 B 4 a 11.2, 11.5, 11.8, 11.9, 12.2, 12.5, 12.7, 12.9, 13.1 b

Minimum

11.2

Lower quartile

11.65

Median

12.2

Upper quartile

12.8

Maximum

13.1

5 a Minimum = 5°C, Maximum = 25°C b Median = 14°C, Q1 = 10°C, Q3 = 20°C

1062 Mathspace New South Wales – Year 11 Standard mathspace.co

6 a Range = 30, IQR = 16 b

12 a Mean increases because the low outlier pulls the average down. Median may increase slightly or remain unchanged, as it is less sensitive to outliers

Data Values

b Range decreases as the minimum value becomes higher. IQR is minimally affected, as it focuses on the middle 50% of data

b IQR, because it measures the central 50% and is less inﬂuenced by skewness and outliers than the range

7 a Q1 = 8, Q3 = 12, IQR = 4 b Lower Fence = 2, Upper Fence = 18, Outliers: 20 8 a

Minimum

Lower quartile

1.5

Median

Upper quartile

Maximum

13 a Median, because it is less affected by the higher ages in the skewed tail, better representing the typical age

14 a Histogram 1 matches Box Plot A, as both show positive skew b Histogram 2 matches Box Plot B, as both are relatively symmetrical 15 a

Homework Hours

Minimum

Lower quartile

16.5

Median

Upper quartile

27.5

Maximum

Minimum

Lower quartile

Median

Upper quartile

Maximum

9 a Mean = 9.67, Median = 8.5 b Mean = 8.27, Median = 8 10 a Cluster around 0 − 3 goals, and another smaller one around 5 − 6 goals b Gap between 3 and 5, and a gap between 6 and 9 11 a Class Alpha b Class Beta c Class Alpha’s middle 50% is slightly more consistent than Class Beta’s, as its IQR is smaller

16 a 50% b The data has the same maximum spread in three sections: from Q1 to the Median, from the Median to Q3, and from Q3 to the Maximum. c 8 hours d More than 75% 17 a Median = 22 minutes, Mean = 23.8 minutes b Positively skewed

d Class Beta, 5

Answers 1063 mathspace.co

12.01 Straight line graphs

b i

What do you remember? 1 A linear relationship is represented by an equation where y depends on x, such as y = mx + c, and its graph forms a straight line on the Cartesian plane.

−1

y 10 8 6

2 Locate x on the x-axis, then move vertically to y on the y-axis and mark the intersection.

4 2

3 Plot the points and check if they lie on a straight line when connected, indicating a constant rate of change. c i

Practice 4 a

x −2

−1

−2

−1

−2

−1

−3

−1

−2

−3

−2

−1

−3

−5

y 6

d i

5 a The graph is linear because it forms a straight line.

−1

−2

−1

−2

−3

y x

b The graph is not linear because it forms a curve, indicating a non-constant rate of change. 6 a i

−3

−2

−1

1 −1

−1

−2

−7

−2

−3

ii 8

7 a i

6 4 2 −3 −2 −1

−2

x 1

−4 −6

1064 Mathspace New South Wales – Year 11 Standard mathspace.co

−4

y 1

−6 −4 −2 −1

2 4 6 8 10

−2 1

−3 x

−4 −2

b i

−5

c (0, −3)

−1

8 a (0, 2)

−1

−4

−7

9 a The points form a linear graph because they lie on a straight line when connected.

y 2 x −2

−1

−4

b The points do not form a linear graph because they do not lie on a straight line when connected.

−6

−2

d (0, 4)

10 a The points form a linear graph because they lie on a straight line when connected.

−2

c The points form a linear graph because they lie on a straight line when connected. d The points do not form a linear graph because they do not lie on a straight line when connected.

11 a

5 4 3

y 25

x −2

d i

b (0, −1)

c i

−4

−5

−1

−3

−5

10 x 1

The graph appears to be linear as the points lie on a straight line. 12 a

Answers 1065 mathspace.co

y 10

8 15

4 2 x

5 −3 −2 −1

x −1

2 3 4 5

The corrected graph is linear.

The graph appears to be linear as the points lie on a straight line.

12.02 Gradient and intercept

13 The error is at x = 0, where y should be −3, not −1. Correct table:

What do you remember?

−1

−7

−3

direction, calculated as m = points (x1, y1) and (x2, y2).

y 50

3 A horizontal line has a gradient of 0. A vertical line has an undeﬁned gradient.

40 30

Practice

4 a −2

d 0 x 1

The points form a linear graph. b The relationship should be linear if the water level changes at a constant rate (e.g., drains or ﬁlls at a steady pace). 15 Two points will always form a straight line. Plotting at least three points is ideal, as if the third point also lies on the line connecting the ﬁrst two, it provides stronger evidence of linearity. If it doesn’t, the relationship is not linear.

−2

1066 Mathspace New South Wales – Year 11 Standard mathspace.co

c 1

e −1.5 or

f 1.5 or

b −2

5 a 1

c −1

6 a i m=1

d 0

ii c = 2

b i m = −3

ii c =

c i m=

ii c = −3

d i m=

ii c = −8

7 a Gradient: −2

b y-intercept: 1

5 4 3

16 a The error for x = −2 is y = 7; it should be y = 9. The error for x = 2 is y = 5; it should be y = 3. Corrected table:

using two

2 The y-intercept is the y-coordinate of the point where a line crosses the y-axis (where x = 0). For an equation y = mx + c, the y-intercept is c, occurring at the point (0, c).

Extend your thinking 14 a

1 The gradient measures a line’s steepness and

2 1 (0, 1) −2

−1 −1

ii y-intercept: −3

8 a i Gradient: iii

9 a Gradient: 4 b

y 1 −4

(2, 5)

4 x

−2

−1

x −3 −2 −1

−2

−3 (0, −3)

−2

−4

(0, −3)

−5

b i Gradient: −2

ii y-intercept: 5

iii

y 6

or 0.4

Extend your thinking (0, 5)

11 a y-intercept: 1

y 5

x −3 −2 −1

2 1 (0, 1)

ii y-intercept: −2

c i Gradient: 3 iii

−3 −2 −1

−1

3 2 1 −2

−1

x 1

−1

12 a Gradient: 3. It represents the cost per hour of renting the bicycle ($3/hour). b y-intercept: 10 (Point (0, 10)). It represents the ﬁxed initial cost ($10) of renting the bicycle before any hours are used.

−2 (0, −2) −3 −4

13 a Gradient: ii y-intercept: 0

d i Gradient: iii

b The y-intercept is 0. This means the ramp starts at ground level (a height of 0 metres at a horizontal distance of 0 metres).

y 2 1

(0, 0) −4 −3 −2 −1 −1 −2

or 0.75. The gradient means the

ramp rises 0.75 metres vertically for every 1 metre horizontally (or 3 m rise for 4 m run).

2 3 4

14 a i The calculated gradient is incorrect. Harry calculated the gradient as

, likely by

computing run/rise instead of rise/run. Using points (−7, 0) and (0, −3), the correct gradient is m =

ii The calculated y-intercept is correct, as the line crosses the y-axis at (0, −3).

Answers 1067 mathspace.co

b i Gradient: ii y-intercept: −3

3 2

15 a The student likely made a sign error in the numerator or denominator, or incorrectly subtracted. Correct calculation: =

−4 −3 −2 −1 −1

2 3 4

−2

−3

, y-intercept: −6

b Gradient:

−4

12.03 Gradient-intercept form

What do you remember?

2 1

b y-intercept

1 a Gradient

b True

c False

−4

4 a c y=5 5 a Equal

3 2

−2 −3 −4

b y = −2x – 1 d

−4 −3 −2 −1 −1

d Equal

6 a y=x+2

b y = −x + 2

b Not equal

c Not equal

7 a

−2 −3

d True

Practice

−4 −3 −2 −1 −1

2 This equation is in gradient-intercept form because it is written as y = mx + c. The gradient m = −2 and the y-intercept c = 5. 3 a True

y 4

3 2

1 −4 −3 −2 −1 −1

x 1

2 3 4

−2

−6 −5 −4 −3 −2 −1

−3

−1

−4

1068 Mathspace New South Wales – Year 11 Standard mathspace.co

1 2 3 4 5 6

13 a y = −x + 2

2 1

−4 −3 −2 −1 −1

−2

b Because they have the same gradient, the lines increase or decrease at the same rate. Since they start at different y-intercepts, the vertical distance between them is always constant, and they will never intersect.

−4

3 2 1

−4 −3 −2 −1 −1

15 a The gradient is m =

= 2, but substituting

x = 0, y = 3 into y = 2x + c gives c = 3, not 1.

2 3 4

b y = 2x + 3

−2 −3

12.04 Modelling linear relationships

−4

b 8 14 a They are parallel because they have the same gradient m = 4 but different y-intercepts.

2 3 4

−3

b Line A

12 a

What do you remember?

y 4

1 a 15

b 30

2 a Rate of change

b Initial value

c Time cannot be negative in most contexts.

d Temperature

1 −4 −3 −2 −1 −1

x 1

2 3 4

3 a 50 L/min

b 200 L

c Increasing

d Rate of ﬁlling

4 a $3.00 b $2.20 per kilometre

8 a −4 b The gradient is −4. This means the y-value decreases by 4 for every 1-unit increase in the x-value. 9 a i m=2

ii y = 2x + 1

b i m = −2

ii y = −2x + 1

c i m=2

ii y = 2x − 2

d i m=2

ii y = 2x + 1

5 a False

b False

e False

f True

c True

d True

6 a Interpolation

b Extrapolation

c Extrapolation

d Interpolation

7 a y = 8.45

b y = −17

c y = −7.4 Practice

10 a C = 50h + 75 b $250 Extend your thinking

8 a 135 mg

b 70 hours

9 a C = 0.15m + 30

b Cost per movie

c Base fee 11 a

c y = 4x − 5

d y=x+3

Answers 1069 mathspace.co

b The ﬁsh population is decreasing by 200 ﬁsh each year.

C 50

c F = 4800

(100, 45)

d F = −200t + 4800

30 (0, 30)

16 a 50°F

20 10 0

Number of minutes passed, x

b 25 minutes

120

c The model would predict negative volume, which is physically impossible. 18 a C = 50h + 75

b $300

c 5 hours 19 a Interpolation

b Extrapolation

c Interpolation

Amount of fuel 250 225 200 left in tank, y

175

150

b y = 250 − 5x c The amount of fuel in the car is decreasing at a constant rate of 5 litres per minute.

20 y = −52.61, Extrapolation 21 a 3 seconds

b 2.5 hours

22 a 20 years

b 1300 ﬁsh

c 11 years 23 a

d 0 ≤ x ≤ 50 11 a 70 percent b −20 percent, which is unrealistic c The model predicts negative charge beyond 400 hours, which is impossible for a battery. d 60 percent e 400 hours 12 a

d F = 1.8C + 32

17 a 300 L

10 a

b 1.8°F

c Below

225 People 200 175 150 125 100 75 50 25

Temperature °C

12 14 16 18 20 22 24 26 28

Number of columns (c) Number of blue boxes (b)

1 1

2 3

3 5

5 9

10 20 19 39

b 150 people

c 23°C

d 140 people

e 19°C

24 a 2448 Pa

b b = 2c − 1

b Interpolation

c 75 boxes

c Yes

d 23 columns

d i Yes

iii No

iv No

ii Yes

iii No

iv No

25 a 90

13 a 85 beats/minute b 3 beats/minute

b 22°C

c H = 3t + 49

c i No

d Rate of heart rate increase per minute 14 a 8.5 cm

ii No

b 12 weeks

15 a −200

26 a 8500 m c 300 seconds 27 a c $400 000

1070 Mathspace New South Wales – Year 11 Standard mathspace.co

b 6500 m d 1800 seconds b c = 100

28 a Negative gradient

b −2.5

c y = −2.5x + 105

d y = 95

Extend your thinking 29 a E = 10n + 340

b $700

37 a For each 1000 km increase in distance driven, the least-square line predicts that the maintenance costs increase by $990. b A car that is not driven at all is predicted to cost $14 620 to maintain.

b $545

c No, because for a car that hasn’t been driven, it should have little to no cost to maintain.

c Company A

d $50 260

d The high ﬁxed fee of $160 may make the company uncompetitive for jobs that take a very short amount of time.

e No. The prediction is likely unreliable. The value K = 36 is outside the range of the provided data (which goes up to K = 35), making this an extrapolation. While it is close to the data range, reliability decreases with extrapolation.

30 a P = 55t + 160

31 a 210 cups of coffee b 77.5° C c C = −2t + 305 d Zero sales at high temperatures is unrealistic; demand may stabilise. 32 a 40 years b In the year 2066 33 a 52.5 m b Yes, because the prediction is determined using interpolation.

f Y es, a car that is driven more will have more wear and tear and would be likely to have greater maintenance costs. 38 Neither interpolation, nor extrapolation would be overly reliable. This dataset is more non-linear, so a non-linear model would be more appropriate for making predictions.

12.05 Direct variation

34 a Interpolation. The speed of 70 km/h falls within the range of known data points, which is from 40 km/h to 100 km/h.

What do you remember?

b Extrapolation. The prediction for the revenue for the next year is beyond the range of known data points (2015 to 2019).

2 Constant of variation

c Extrapolation. The number of hours of study at 15 hours does not fall within the range of known data points, which spans from 2 to 14 hours. 35 Interpolation and extrapolation can be limited by the accuracy and validity of the data, as well as the assumption that the relationship between variables remains constant. In the context of climate change, factors such as human intervention, natural disasters, and technological advancements can inﬂuence the relationship between variables, making predictions less reliable. 36 a y = −0.005x + 49

b y = 29°C

1 y = kx

b k

3 a (0, 0)

4 No, because it does not pass through (0, 0) due to the constant term +5. Practice 5 a k = 12, D = 12T b 60

50 40 30

(2, 24)

20 10

c Yes, because it is using interpolation.

T 1

6 $21

Answers 1071 mathspace.co

7 No, because when T = 0, C = 10, so the graph does not pass through (0, 0).

8 192 pages

200

9 15 minutes

150

10 125 N

100

11 a $105

b 7 hours

12 a 9 hours b

450 400 350 300 250 200 150 100 50

b $300 c Cost per poster d

(5, 225)

400

350 300 250

H 2

200

13 a 1.4 m/min

b y = 1.4x

c 8.4 m

d 9 min

14 a 12 Euros

b 10 AUD

c 0.60 Euros 15 a I = 25h

4500 4000 3500 3000 2500 2000 1500 1000 500

d E=

d Hourly wage

N 20

Extend your thinking 22 Both plans cost $30 for 10 GB; they are equally cost-effective. 23 a 40 W

b 5

24 a 2.5 b 160 cupcakes

Hours 5

10 15 20 25 30 35

b 120 c C = 120h d $3360 17 a 7.5

100

Cost

(50, 200)

150 50

b $625

c 5 hours

b C = 0.25E

18 20 km/h 19 a 8

21 a 4

16 a

(10, 250)

250

b 360 calories

c Discounts mean that the price per cupcake is no longer constant. Therefore, the relationship between revenue and the number of cupcakes would no longer be a direct variation. d

500 R 450 400 350 300 250 200 150 100 50 0

1072 Mathspace New South Wales – Year 11 Standard mathspace.co

(100, 250)

N 50

100

150

25 a In capsule form

y 7

b The capsule has a higher rate of absorption (5.9 mg/min) compared to the liquid form (4 mg/min).

6 5 4

Chapter 12 review

1 B

2 C

3 B 4 i −1 5 i

ii 3 −1

−7

−4

−1

6 a

8 9 a −2 P 6

x 1

−1 −2 −3 −4 −5 −6 −7

c y = −2x + 4

b 4

2 1

1 2 3 4 5 6

The corrected graph is linear.

4 (0, 4) 3 2 1

−1

P (profit $)

10 a Gradient:

or 0.625. The gradient means the

slope descends 0.625 metres vertically for every 1 metre horizontally.

200

150

100

m (months) 1

The points form a linear graph. b The relationship might be linear if the proﬁt increases by a consistent amount each month, for example, due to steady sales growth or ﬁxed incremental cost savings. 7 a The error for x = −3 is y = 6; it should be y = 7. The error for x = 3 is y = 4; it should be y = 3. Corrected table:

−3 −2 −1

−3

11 a Parts (i) and (iii) (gradient and equation) are incorrect. Samira calculated run/rise instead of rise/run. The y-intercept (part ii) is correct. Using points (−5, 2) and (0, 6), the correct = .

gradient is m = b Gradient: Equation:

. y-intercept: 6 (or (0, 6)). .

12 a y = −2x + 3

b y = −2x + 2

c y=x−2

d y=x−1

13 a i −3

ii 5

b i 4

ii −5

c i

ii 3

Answers 1073 mathspace.co

d i

ii 10

e i

f i

ii 2

c Sarah should choose Service X, as it is cheaper by $17.50. d The high call-out fee of $95 makes very short jobs disproportionately expensive compared to competitors with lower call-out fees. 21 a A = −25T + 750

14 a y = −2x + 5

b A = 100 hectares

b 4 15 a The y-intercept is (0, 6). The gradient −3 means a rise of −3 per 1 unit run. For 2 units right, the rise is −3 × 2 = −6. Starting at y = 6, 6 − 6 = 0. The y-coordinate is 0.

c No, because it is using extrapolation. 22 a

b The gradient −3 indicates a decreasing line, falling 3 units vertically for every 1 unit horizontally. A line with gradient −1 also slopes downward but falls 1 unit vertically for every 1 unit horizontally. The magnitude ∣−3∣ = 3 indicates y = −3x + 6 is steeper than a line with gradient −1 (magnitude ∣−1∣ = 1). 16 a C = 0.20v + 50

Weeks 0 1 2 3 4 5 6 7 8 9

Litter collected (kg) 5

C (cost $)

(100, 70)

60 50

c Sign-up fee 70

b Example answer:

b Cost per visit d

Litter collected (kg) 5

4 3

(0, 50)

Weeks

0 1 2 3 4 5 6 7 8 9

v (visits) 20

d Approximately 2.4 kg

17 a 80% b −10%, which is unrealistic c The model predicts negative battery life beyond 360 hours, which is impossible. d 60% e 360 hours 18 a 2.5 m/min c 17.5 m

c Example answer: y = −0.2x + 4

b D = 2.5m d 9 min

19 a E = 10d + 250 b $650 20 a P = 75t + 95 b $432.50

1074 Mathspace New South Wales – Year 11 Standard mathspace.co

e No, because it is using extrapolation. 23 $17.50 24 a $162.50

b 3 hours

25 Plan Alpha: $30. Plan Beta: $36. Plan Alpha is more cost-effective. 26 a 35 b 50 bouquets c Discarding unsold ﬂowers reduces effective revenue per bouquet sold, as it introduces losses not accounted for in the direct variation model.

d 3500

11 $47.20

R (revenue $)

3000

12 70 kg

(80, 2800)

13 $49.50

2500 2000

14 You pay 10% less for the car.

1500 1000 500 0

N (bouquets) 20

15 a $27

b $9

16 a $7.50

b $22.50

17 $81.60

13.01 Percentage increase and decrease

18 $310.50 19 20.94%

What do you remember?

20 a $950

1 a 1950

b $893

c $890

d No

e 10.7%

b 1950 c Yes, because ﬁnding 130% of 1500 is the same as ﬁnding 30% + 100% of 1500, which is the same as adding 1500 and 30% of 1500. 2 a 117%

b 105%

c 140%

d 109.4%

e 133.51%

f 127.275%

h 212%

21 $229.41 22 a 96%

b No

23 a 51%

b No

24 a 288.09%

b 15.2%

c 2.34% 25 43.75% 26 68 688 people

3 a 1584

27 $64.50

b 1584 c Yes. Decreasing an amount by 12% means that 100% − 12% = 88% of the original amount remains. This is mathematically equivalent to calculating 88% of the original amount directly. 4 a 90% e 79.3% 5 a $8

b 43%

c 95%

d 25%

f 84.3%

h 99.2%

b $3.70

c $9.50

d $5.90

6 $22.00

b 3000

8 a 60

b 340 b 76.95 cm d 26.8375 kg

b $7.63

30 $286.44 Extend your thinking 31 a $1068.75

b $1272.32

32 a Offer 2

b Offer 1

33 a $4020.38

b $4020.38

c No

d $3820.25

c 84%

7 a 500

c $14.42

29 a $3.64

34 a 60%

Practice

9 a 770

28 $84.80

b 140% d (−16) %

e Decrease 35 a $90

b $51.32

c $852.45 36 $42.73

e 262.08 L 10 67.5 kg

Answers 1075 mathspace.co

16 a $18.40

13.02 Profit and loss

b $16.63

c Loss of $1.77 What do you remember?

17 $119

1 Revenue is the total income generated from normal business operations, such as the sale of items. The break-even point is when the selling price equals the cost price, resulting in neither proﬁt nor loss. 2 Proﬁt is calculated as Selling price − Cost price. It occurs when the selling price exceeds the cost price. 3 Percentage Loss = × 100 4 Cost price is the amount paid to acquire an item. Marked price is the labelled price of the item, which may differ from the selling price due to discounts.

7 a Proﬁt of $56

b $17 125

20 a $3175

b $1750

c Proﬁt of $1425 Extend your thinking 21 $39 000 22 a 250% 23 4.94% 24 a $51

b $503

r is the periodic interest rate

d $158.05

and n is the number of periods.

b Loss of $209

2 a True

d Loss of $1804

e False

e Proﬁt of $209

f Loss of $1156

g Proﬁt of $137.51

h Loss of $68.87

8 a $158

b $5369

c $119

d $23.95

e $3477 9 a $196

b $10 365

c $9

d $5763

e $1294 c $469

b 22.17%

13.03 Purchase options

c Proﬁt of $3130

10 a $759

b 50%

1 The formula is A = P (1 + r)n, where A is the balance with interest, P is the closing balance,

Practice

c $5635

19 a $19 125

What do you remember?

5 False

6 a $5601

18 $10 000

b $921 d $1221.89

b False

c True

b $28.80

12 a $100.92

b $9.08

13 $156.28 14 $4590 15 $12 975

1076 Mathspace New South Wales – Year 11 Standard mathspace.co

d True

3 a A credit card uses borrowed money from the bank, while a debit card withdraws directly from the user’s own bank account. b Credit cards can offer rewards programs, such as cash back or travel points. 4 a No additional fees are incurred if all payments are made on time. b If the minimum payment is not met, interest charges and potential penalties may be applied, increasing the balance and risking the account’s good standing.

e $760 11 a $388.80

Practice 5 $1227 6 a $172

b $2007.67

7 a $368

b $368.92

c $18.45

d $350.47

8 12 weeks

9 a $149.00

b $384.56

c $121.50

10 3 months

Practice 4 a $4815

b $19 815

5 a $6624

b $24 624

11 a $12.14

b $43.33

12 a $47

b 2.4%

13 a $28.33

b $340

6 $280.00

14 a $5044

b $960

7 $27 950

c i $513.00

ii $236.77

c 12.22%

8 a $660

b $1500

15 a $88.67

b $62.07

c $1850

d $3200

16 a $317

b $802.69

9 a

c $2.69 Extend your thinking

Vehicle value ($)

30 000

50 000

70 000

90 000

Stamp duty ($)

900

1600

2600

3600

17 Plan 1 18 a Since $174.23 exceeds $165, the 15% interest rate is not feasible for Alexia’s budget. b $272.71 19 a $280 b

b i $3400

ii $60 000

10 $488.33 11 a $40.83

b $176.92

c $530.75

12 a 0.4%

b $7200

c $32 200

13 a $280

b $680

c $3767.50

Balance (before interest)

Balance (after interest)

Payment

Balance (owing)

14 a $25 000

b $7500

March

$1685.00

$1710.59

$825.00

$885.59

c $32 500

d $677.08

April

$885.59

$898.51

$825.00

$73.51

e $1600

May

$73.51

$74.62

$0.00

c $2004.62 20 a $373

d $1965.00 b 110 days

15 a $40

b $1495

c $6390

d $58

e $1758 16 a $1306

b $450

c $4291

13.04 Purchase a car Extend your thinking What do you remember? 1 I = PRn, where I is the interest, P is the principal, R is the interest rate per period (as a decimal), and n is the number of periods. 2 For vehicles valued at $44 999 or less, stamp duty is $3 per $100 or part thereof. For vehicles valued at $45 000 or more, it is $1350 plus $5 per $100 or part thereof above $45 000. 3 CTP insurance is mandatory for vehicle registration, costing ≈ $500 annually, and covers third-party injury. Comprehensive insurance is optional, costing $800 to $2500, and covers damage to the car and other property.

17 a With simple interest loans where interest is not compounded, the frequency of repayments does not affect the total interest paid, as the interest is calculated on the principal amount for the entire period. However, if repayments reduce the principal immediately, then more frequent repayments reduce the principal faster, leading to lower interest overall. Therefore, making weekly repayments would result in the lowest total interest paid. b Less c They are also tripled. 18 a $47 930

b $2520

Answers 1077 mathspace.co

19 a Car A: $27 550, Car B: $51 700 b Car A, by $936.11 20 a Option B, with a total cost of $41 600 compared to Option A’s $54 000.

3 a No

b Yes

c No

4 a C

b D

c B

d E

e A Practice

b $41.67

13.05 On-road and running costs of a car What do you remember? 1 a False

b True

c True

2 a ii

b i

c iii

d False

5 a $218.76

b $272.15

c 22 kL

d $52.69

e $2.3950 per kL

f 239 L

g $2.6345 per kL 6 $201.60 7 a 500 kWh

3 $466

b $197.40

8 336 kWh Practice

9 $1026.90

4 a $1723

b $438

10 $325.05

c $6862 5 a $156.54

11 $39.48

b $271.92

12 a 162 L

6 $765

b 1134 L

c $0.44

7 $345 8 $1468 9 $574

13 a 2320 L

b $4.41

14 a 269 kL

b $295.90

15 $128.99

10 $1870.77. 11 a $390

b $3900

12 $1450

16 a 2405 MJ

b $85.34

17 a $761.52

b $51.17

c $2633.85

13 $500

18 $7.93

14 $1080

19 a 90.82 m3

b 3534 MJ

15 $464

20 a 2480 MJ

b 85 MJ per day

Extend your thinking

16 $8160.77

21 a $2.93

17 20.39%

c $2.98

18 Third Party Property: $3350; Comprehensive: $1500

b $23.47

22 a 82%

b $2.04

23 a $176.00

b $209.25

c $0.02

13.06 Household bills

24 a $357.10

b $370.63

What do you remember? 1 a True

b False

2 a kL (kilolitres)

c True

d False

b kWh (kilowatt-hours)

c MJ (megajoules)

1078 Mathspace New South Wales – Year 11 Standard mathspace.co

13.07 Prepare a personal budget What do you remember? 1 a True

b True

c False

d iv

7 a $287.50

b $37.50

3 a Fixed

b Discretionary

8 a $1112.50

b Profit of $462.50

c Fixed

d Discretionary

9 $894.12

c No

10 10 weeks

2 a iii

4 a Yes

b ii

b Yes

c i

d Yes

Practice

11 a $35

5 $1566.67

12 a Monthly payment = $205.33. Not feasible as it exceeds $200.

6 44.43%

b 2.3%

b $264

7 a i $105

ii $24.40

13 a $550

b $13.85

b $958.43

c $8.43

8 a $849

b $414

c $435

d $22 620

9 a Yes

b No

c Yes

d No

e No

f Yes

b $1700

c $1185

d $2090

15 a Car X: Total cost = $24 024.

10 $280 11 a 67.88%

14 a $840

Car Y: Total cost = $51 375. b 5 weeks

b Car X is lower by $348.25.

12 $305

16 $1612

13 $547.50

17 $545

14 27.00%

18 a $5436.42

b $7786.42

15 a i $2600

ii $1320

19 23.3%

iii $40

iv $620

20 a 580 kWh

b $240.55

21 a 170 L

b 15.3 kL

b $440.00 16 a 35%

b 35.69%

c $38.25

17 $116.95

22 $206.40

Extend your thinking

23 a $402.08

18 a $442

b $63

c 29.11% 19 a = SUM(F4:F12)

b = C13 − F13

20 a $6500

b $780

c $225

Chapter 13 review 1 B

b $404.78 c No, the flexible plan is more expensive by $2.70. 24 a $1000 c $505

b $495 d $26 260

25 a Rent: $18 000. Utilities: $960. Groceries: $4680. Car Insurance: $1200. Public Transport: $720. Health Insurance: $2160. b $36 030

2 C

26 a $4350

b $870

3 B

c $1030

d $3480

4 $520 5 $293 6 $800

27 a = SUM(F4:F12) b = C13-F13 c F13 = $840.00, C15 = $210.00

Answers 1079 mathspace.co