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© 2023 Mathspace Pty Ltd Copyright Notice This Work is copyright. All rights are reserved. Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL). Reproduction and communication for other purposes Except as permitted under the Act (for example, a fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher. For permission to use material from this text or product, please email hello@mathspace.com.au For our full digital offering, visit mathspace.co Mathspace Pty Ltd Level 11, 477 Pitt St Sydney 2000 The Victorian Curriculum F-10 content elements are © VCAA, reproduced by permission. The VCAA does not endorse or make any warranties regarding this resource. The Victorian Curriculum F-10 and related content can be accessed directly at the VCAA website. Title: Mathspace Victorian Curriculum - Year 8 - 2024 edition Print Run: 1 ISBN: 978-1-963022-43-8 Editors and lead authors: Erin Gallagher, Julia Reid, Sarah Butterworth, Anjelica Sieger, Stephen George, Dot Yung Further reviewers: Melissa Archey, Warren Bardsley, Lianna Beeching, Kathryn Bennett, Benjamin Bui, Eric Chan, Cory Duker, Bahija Elmir, Wayne Gillie, Jeremy Hunter, Shahad Odeh, Helen Soar, Patricia Toutzaris, Paul Unkles, Jie Yang Writing and development team: Adam Humphries, Neil Lopez, Ma. Lourdes Ona, Bernadette Barillo, Sheramie Castillo, Abegail Fontanilla, Shaira Llanita, Lesley Lopez, Kerby Mendoza, Regine Jara, Albert Jara, Rolly Apdo, Michaela Belen, Valerie Baja, Marielle Belen, Allen Calabia, Adrian Doctolero, Dhave Fernandez, Janine Francisco, Rose Francisco, Rona Josue, Crystel Lontoc Images and Design team: Alex Duckmanton, Chastine Marquez, Keith Gimeno

Contents 1

Real numbers 1.01

Add and subtract integers

1.02

Multiply and divide integers

1.03

Four operations with integers

1.04

Negative fractions

1.05

Negative decimals

1.06

Rational and irrational numbers

1.07

Recurring and terminating decimals

Investigation: Golden ratio

Index laws

Chapter 1 review

1.08

Percentages 64 2.01

Percentage, decimal and fraction conversion

2.02

Find the whole

2.03

Percentage increase and decrease

Investigation: Percentages

2.04

Profit and loss

2.05

Consumer percentages

Chapter 2 review

106

Ratios and rates

110

3.01

Ratios

112

3.02

Maps and scale

123

3.03

Rates

131

3.04

Financial rates

138

3.05

Best buys

148

Chapter 3 review

154

Algorithms and coding

160

4.01

Algorithms in code

162

4.02

Loops

170

4.03

Debugging

180

4.04

Algorithms for divisibility

189

Chapter 4 review

195 Contents mathspace.co

iii

Time 202 5.01

Elapsed time

204

5.02

12 and 24-hour time

209

5.03

Time zones and the date line

213

Investigation: Plan with time zones

227

Chapter 5 review

231

Algebra 234 6.01

Substitution

236

6.02

Add and subtract algebraic terms

241

6.03

Multiply and divide algebraic terms

246

6.04

Simplify algebraic expressions

252

6.05

The distributive law

256

6.06

Numeric factors

261

Chapter 6 Review

266

Linear relationships

270

7.01

Solve equations

272

7.02

Tables of values

278

7.03

Linear graphs

288

Investigation: Linear solutions

302

7.04

Multi-step equations

304

7.05

Intersecting lines

312

7.06

Inequalities

325

Investigation: Experiment with linear inequalities

341

Chapter 7 review

344

Geometry 350 8.01

Angles and triangles

352

8.02

Transformations and congruence

362

8.03

Congruent triangles

374

8.04

Transformations and similarity

385

Mathspace Victorian Curriculum – Year 8 mathspace.co

Similar triangles

395

Investigation: Determine congruence or similarity

407

Properties of quadrilaterals

411

Investigation: 3D coordinate systems

424

Chapter 8 review

429

Pythagoras’ theorem

436

Squares and square roots

438

Investigation: Proof of Pythagoras’ theorem

444

9.02

Pythagoras’ theorem

449

9.03

Find the hypotenuse

456

9.04

Find a short side

461

9.05

Applications of Pythagoras’ theorem

468

Chapter 9 review

477

8.05 8.06

9.01

Measurement 480 10.01

Perimeter

482

10.02

Circumference of a circle

491

10.03

Perimeter of composite shapes

499

10.04

Estimation and units of area

507

10.05

Area of rectangles, triangles and parallelograms

515

10.06

Area of composite shapes

523

10.07

Area of trapeziums, kites, and rhombuses

530

Investigation: Area of circles

544

10.08

Area of a circle

548

10.09

More composite areas

554

10.10

Units of volume and capacity

561

10.11

Volume of right prisms

569

Chapter 10 review

579

Probability 586 11.01

Calculate probability

588

11.02

Intersection and union

601

11.03

Venn diagrams

615

Contents mathspace.co

11.04

Two-way tables

627

11.05

Mutually exclusive events

637

Investigation: Probability simulations

647

Chapter 11 review

652

Statistics 658 12.01

Measures of centre and spread

660

Investigation: Random sampling

678

12.02

Sampling methods

680

12.03

Bias

689

12.04

Compare samples to populations

698

Investigation: Plan and conduct statistical investigations

710

Chapter 12 review

712

Answers

716

Mathspace Victorian Curriculum – Year 8 mathspace.co

"Pure mathematics is, in its way, the poetry of logical ideas." Albert Einstein

Big ideas • Understanding integers and operations with integers develops logical reasoning, problemsolving skills, and mathematical abstraction. • Real numbers are either rational or irrational. • Index laws provide essential rules for simplifying mathematical calculations.

Curriculum Outcomes VC2M8N01 Recognise irrational numbers in applied contexts, including π and numbers that develop from the square root of positive real numbers that are not perfect squares, and recognise that irrational numbers cannot develop from the division of integer values by natural numbers VC2M8N02 Establish and apply the exponent laws with positive integer exponents and the zero exponent, using exponent notation with numbers VC2M8N03 Convert between fractions and terminating or recurring decimals, using digital tools as appropriate VC2M8N04 Use the 4 operations with integers and with rational numbers, choosing and using efficient mental and written strategies, and digital tools where appropriate, and making estimates for these computations VC2M8N06 Use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts involving profit and loss; formulate problems, choosing efficient mental and written calculation strategies and using digital tools where appropriate; interpret and communicate solutions in terms of the context, reviewing the appropriateness of the model

1 Real numbers Chapter outline 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

Add and subtract integers Multiply and divide integers Four operations with integers Negative fractions Negative decimals Rational and irrational numbers Recurring and terminating decimals Investigation: Golden ratio Index laws Chapter 1 review

4 12 16 21 29 36 41 48 52 60

The "Golden Ratio" is a special number that shows up everywhere from architecture to nature! If a staircase spirals a certain way, it might follow the Golden Ratio.

1.01 Add and subtract integers After this lesson, you will be able to… • perform addition and subtraction of integers using a number line. • understand combining adjacent signs. • add and subtract integers by combining adjacent signs.

Addition and subtraction of integers Interactive exploration Discover this concept in action online

mathspace.co

Recall these directions for moving along a number line when adding or subtracting integers: • Adding a positive integer means to move right • Subtracting a positive integer means to move left • Adding a negative integer means to move left • Subtracting a negative integer means to move right When dealing with integers and addition, the order in which we add does not matter, as addition is commutative. This means that for any integers a and b, a+b=b+a However, when dealing with subtraction, the order does matter, as subtraction is not commutative. So, for subtraction of integers, a − b ≠ b − a, unless a = b When working with a mix of addition and subtraction, however, we can rearrange the terms strategically so that the sums and differences of integers are correct. For instance, we may write 12 + (−8) as −8 + 12 and further, we may write −8 + 12 as 12 − 8 The negative or positive sign on the left of the integer is its sign. The sign must stay with the integer when rearranging addition and subtraction problems.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 1 Evaluate −7 + 15

Create a strategy Plot the necessary points on the number line to solve the expression.

Apply the idea We are starting at −7 and we want to add 15 to this. Adding a positive integer means we move to the right. We need to move 15 units right so we end at 8:

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

−7 + 15 = 8

9 10

Evaluate

Reﬂect and check Strategically, we may rearrange the problem to make it a subtraction problem, so −7 + 15 = 15 − 7 = 8

Example 2 Evaluate −4 − 3 − 7

Create a strategy Plot the necessary points on the number line to solve the expression.

Apply the idea We are starting at −4 and we want to subtract 3 and then 7 from this. Subtracting a positive integer means to move left on the number line, so we will move 3 spaces left to get −7. Then we move another 7 spaces left to ﬁnally get −14:

−15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2

−4 − 3 − 7 = −7 − 7 = −14

−1

Subtract 3 from −4 Evaluate

1.01 Add and subtract integers mathspace.co

Idea summary To add or subtract using a number line: • • • •

Adding a positive integer means to move right Subtracting a positive integer means to move left Adding a negative integer means to move left Subtracting a negative integer means to move right

The negative or positive sign on the left of the integer is its sign. The sign must stay with the integer when rearranging addition and subtraction problems.

Combine adjacent signs When we add a negative number, such as 2 + (−3), it’s actually the same as 2 − 3. When we write the problem with one sign instead of two it is called combining adjacent signs. For a number line with the positive direction to the right, we ﬁnd the following results: Adding a positive integer means we move right, and simplify the problem to addition Adding a negative integer means we move left, and simplify the problem to subtraction Subtracting a positive integer means we move left, and simplify the problem to subtraction Subtracting a negative integer means we move right, and simplify the problem to addition

3 + (+5) = 3 + 5 3 + (−5) = 3 − 5 3 − (+5) = 3 − 5 3 − (−5) = 3 + 5

Adjacent signs have different meanings. The ﬁrst tells us the operation (either addition or subtraction), while the second tells us the sign of the integer (either positive or negative). To combine adjacent signs: • When the adjacent signs are the same, the problem is addition. • When the adjacent signs are different, the problem is subtraction.

Example 3 Find the value of 9 + (−8).

Create a strategy

Apply the idea

Combine the adjacent signs to simplify the expression.

The adjacent signs are different, so we have the subtraction problem 9 + (−8) = 9 − 8. Counting down 8 units from 9 gives 1.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 4 Find the value of −8 + 2 − (−7).

Create a strategy When adjacent signs are the same, the result is to add. This means that −(−7) is the same as + 7

Apply the idea We can combine the adjacent signs by writing: −8 + 2 − (−7) = −8 + 2 + 7

Combine adjacent signs

= −6 + 7

Perform −8 + 2 ﬁrst

Evaluate

Example 5 Find the value of −12 + 16 + (−18).

Create a strategy Combine adjacent signs and simplify the expression.

Apply the idea −12 + 16 + (−18) = −12 + 16 − 18

Combine adjacent signs

= 4 − 18

Perform −12 + 16 ﬁrst

= −14

Evaluate

Idea summary To combine adjacent signs: • When the adjacent signs are the same, the problem is addition. For example, 3 − (−4) = 3 + 4 • When the adjacent signs are different, the problem is subtraction. For example, 3 − (+4) = 3 − 4

1.01 Add and subtract integers mathspace.co

1.01 Practice questions Understanding 1

Explorer

Adventurer

1, 3

In which direction on the number line do you move when you: a

Add a positive number, for example −2 + 5?

Subtract a positive number, for example −4 − (+3)?

Add a negative number, for example 2 + (−6)?

Subtract a negative number, for example 5 − (−3)?

What is the difference between subtracting a negative number and adding a positive number?

Consider the number line: −15

−10

−5

• Move 7 units to the right • Move 14 units to the left

Write the steps in part (a) as an equation involving integers.

Fluency 4

Find the position of the plotted integer after these moves: • Move 6 units to the right • Move 2 units to the left

4, 5a–f, 6, 7a–f, 8a–e

4f–h, 5e–l, 6e–h, 7a–f, 8a–h 4h, 5e–l, 6h, 7g–l, 8i–p

Use the plotted point on the number line to add the integers: a

2+8 −5

−10

−5

−10

−5

−10

−5

−8 + 8 −15

−7 + 4 −15

−5 + 9 −15

2+7+3 −5

Trailblazer 2, 3

1, 3

Mathspace Victorian Curriculum – Year 8 mathspace.co

−7 + 9 + 5 −10

−5

−14 + 6 + 11 −15

Ex 1

−9 + 9 + 3 −10

−10

−5

Evaluate: a

16 + 6

−7 + 14

−11 + 0

−10 + 10

−13 + 8

−40 + 102

−25 + 15

−56 + 56

−19 + 10 + 5

−34 + 20 + 34

−22 + 0 + 22

−98 + 28 + 84

Use the plotted point on the number line to subtract the integers: a

7 − 10 −10

−5

−15

−10

−5

−10

−5

−5 + 8 − 1 −15

−6 − 6 −20

4−4 −10

−7 − 2 −10

−5

15 − 11 − 5 −10

1.01 Add and subtract integers mathspace.co

11 − 16 + 3 −10

8 Ex 3, 4, 5

−20

−15

−10

−5

−6 − 13 − 5 −25

Ex 2

−5

Evaluate: a

10 − 7

15 − 15

−30 − 55

−12 − 9 − 2

31 − 16 − 7

−4 − 2 − 34

8 − 11

−11 − 11

40 − 63

−15 − 4 + 22

0 − 22 + 38

18 − 38 + 44

Rewrite the expression by combining adjacent signs and then evaluate: a

7 + (−5)

4 − (−1)

−10 − (−10)

25 − (−26)

21 − 29 + (−9)

−7 + (−8)

−3 − (−9)

−3 ÷ (−9) + (−4)

m −20 + 35 + (−19)

Reasoning

9, 10

15 + 12 + (−18)

−12 − 5 − (−10)

−13 + 43 + (−60)

13 + (−11)

−50 − (−42)

−7 − (−4) − (−8)

35 − 72 − (−45)

10, 11

11–13

James calculates −3 − (−3) = 3 + 3 = 6. Explain the error he has made and correct the answer.

If the sum of two integers is negative, can both integers be positive? Explain your answer.

Given −5 + a = b, calculate the smallest value of a so that b is not negative.

Given −3 − c = d, calculate the largest value of c so that d is not negative.

Given m > n, determine whether the following will be negative or positive: a

m+n

m−n

Problem-solving 14–16, 20

n−m

14, 16, 18–20, 22

m − (−n)

14, 17–22

Find two numbers that have a sum of 6 and a difference of −10.

Vanessa gets in a lift on floor 34. She descends 32 floors, then goes up 6 floors before exiting the lift. What floor is she now on?

One night in Cologne the temperature fell to −15 °C. By lunchtime the next day, the temperature was forecast to be 9 °C. Find the expected rise in temperature.

Dry ice can be used to keep things cold. Dry ice turns into a gas at −78 °C, and can be as cold as −100 °C. What is the directed number that represents the dry ice cooling from −78 °C to −100 °C?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Three contestants are heading into the ﬁnal round of a game show. Joy has 75 points and Margaret has −8 points. If Emily has 19 more points than Joy, calculate how many points separate Margaret and Emily.

A submarine is 298 m below sea level. It rises 58 m, then dives 45 m. Find the distance of the submarine below sea level.

A company had a profit of $1540 in the first quarter and a loss of $650 in the second quarter. In the third quarter, they made a profit of $1400 and then incurred a loss of $440 in the fourth quarter. What is the total profit or loss for the year?

Barrels of crude oil used to make petrol have an average capacity of 158 L. Ten barrels are inspected by the quality assurance department. The results of the inspection are recorded in the table: Barrel Amount below or above the average capacity (L) a

−24

−32

+22

+15

−10

−17

−35

+12

Find the actual volume of: i

Barrel 8

Barrel 1

Calculate the total capacity above or below the average capacity that is contained in the 10 barrels.

Each letter has been assigned an integer. A word must be formed using these letters and each letter can only be used once. The table shows the assignment of the integers: Letter Integer

−8

−6

−4

The word score is the sum of the integers assigned to its letters. a

Calculate the score for the word STEAM. b

Calculate the score for the word PRAISE.

Write a word with a score of 2.

Write a word with a score of 4.

1.01 Add and subtract integers mathspace.co

1.02 Multiply and divide integers After this lesson, you will be able to… • multiply and divide integers.

Multiplication and division of integers Interactive exploration Discover this concept in action online

mathspace.co

We can use these rules to determine the sign of the answer when we multiply or divide two integers: When we multiply or divide two integers with the same sign, their product or quotient will be positive. When we multiply or divide integers with different signs, their product or quotient will be negative.

−2 × −3 = 6 and −8 ÷ −4 = 2 −2 × 3 = −6 and 8 ÷ −4 = −2

As each pair of negative signs make a positive when multiplying and dividing integers, we can expand on the previous rules as follows: If the total number of negative signs is odd, the answer will be negative. For this example there are 3 negative signs, so the answer will be negative: If the total number of negative signs is even, the answer will be positive. For this example there are 4 negative signs so the answer will be positive: Using these two rules helps with more complicated multiplication and division. If we know whether the answer is positive or negative, based on the number of negative signs, we can focus on the numbers, complete the evaluation and add the appropriate sign at the end.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 1 Evaluate 5 × (−9)

Create a strategy We are multiplying integers with different signs, so the product will be negative.

Apply the idea 5 × (−9) = −45

Perform the multiplication

Example 2 Evaluate (−6)3

Create a strategy Write the multiplication in expanded form and complete one multiplication at a time.

Apply the idea (−6)3 = −6 × (−6) × (−6)

Write in expanded form

= 36 × (−6)

Multiply −6 by −6

= −216

Multiply 36 by −6

Example 3 Evaluate −20 ÷ 4

Create a strategy

Apply the idea

We are dividing a negative integer by a positive integer, so the answer will be negative.

−20 ÷ 4 = −5

Perform the division

Idea summary • When we multiply or divide integers with the same sign, their product or quotient will be positive. For example, −2 × −3 = 6 and −8 ÷ −4 = 2 • When we multiply or divide integers with different signs, their product or quotient will be negative. For example, −2 × 3 = −6 and 8 ÷ −4 = −2

1.02 Multiply and divide integers mathspace.co

1.02 Practice questions Understanding 1

The product of a positive number with a negative number is negative.

The product of an even number of negative numbers is negative.

When dividing two negative integers, the result is always negative.

The quotient of a positive integer divided by a negative integer is positive.

When multiplying two integers, the order in which they are multiplied does not affect the product.

When dividing an integer by 1, the result is always equal to the original integer.

Determine whether the expressions will be positive or negative:

(−2432) ÷ 32

(−6) × 4

3 × (−2) × 5

(−42) ÷ 7

(−924) ÷ 28

(−36) × (−9)

1587 ÷ (−23)

(−7614) ÷ (−81)

6 × (−4)

(−3) × (−2) × (−5)

(−6) × (−4)

(−3) × 2 × (−5)

(−3) × (−2) × 5

42 ÷ (−7)

(−42) ÷ (−7)

924 ÷ (−28)

(−924) ÷ (−28)

8g–s, 10, 12e–n, 14, 16c–f

8h–t, 10, 12f–p, 14, 16e–h

6 × (−6)

(−12) × (−10)

(−9) × 0

(−4) × (−4) × (−4)

(−4) × (−11)

(−4)2

(−12)2

15 × (−3)

(−9) × 1 (−3) × 4 × (−3)

(−11) × 4

(−4) × (−4)

4 × (−7) × 5

−(−12)2

−(−8)3

(−5)3

(−6) × (−8) × (−3)

−4 × 4 × 5

(−8) × (−5) × (−3) × 6

Complete the equations: a

⬚ × (−5) = 55

−6 × ⬚ = −66

⬚ × 8 = −56

7 × ⬚ = −84

(−11) × r = −66

n × (−3) = −36

7 × q = −28

t × 7 = −42

Solve: a

Evaluate:

(−855) ÷ (−15)

8a–m, 9, 11, 12a–i, 13, 15, 16a–d

m (−4)3

(−8) × (−45)

Consider the equation 924 ÷ 28 = 33, and evaluate:

Fluency

Ex 1, 2

Consider the equation 42 ÷ 7 = 6, and evaluate:

75 × (−7)

Consider the equation 3 × 2 × 5 = 30, and evaluate:

a 7

Consider the equation 6 × 4 = 24, and evaluate:

a 6

(−2) × 5

Determine whether the expressions will be positive or negative:

a 5

Trailblazer 1, 5, 7

a 4

Adventurer 1–3, 5, 7

Are the statements true or false?

a 3

Explorer 2–4, 6

Mathspace Victorian Curriculum – Year 8 mathspace.co

Ex 3

Find the value of n in the statements: a

A number n is multiplied with −3 to give 21.

A number n is multiplied with −10 to give −40.

Evaluate: a

(−2) ÷ (−2)

12 ÷ (−4)

4 ÷ (−1)

0 ÷ (−36)

(−30) ÷ 6

21 ÷ (−7)

m (−72) ÷ (−9)

(−22) ÷ 11

⬚ ÷ 5 = −7

⬚ ÷ 5 = −11

(−72) ÷ ⬚ = 9

(−108) ÷ n = −12

p ÷ (−8) = 11

(−48) ÷ t = 6

k ÷ 8 = −6

Find the value of n in the statements: a

⬚ ÷ (−6) = −9

Solve: a

(−50) ÷ (−10)

Complete the equations: a

(−2) ÷ 1

A number n, when divided by 3, gives −9. b

A number n when divided by −5 gives −4.

Evaluate: a

(−4) × 3 ÷ (−2)

5 × (−3) × (−7)

10 × 8 ÷ (−5)

(−6) × (−4) × (−3)

(−14) × (−4) ÷ 7

8 × (−15) ÷ 4

(18 ÷ (−3)) ÷ (−2)

(−126) ÷ ((−6) × (−3))

Reasoning

18, 19

17, 20, 21

Explain why there is no solution to the equation x2 = −1.

Is it possible for the product of two integers to be zero? Explain your answer.

Evaluate −4 × 8 × (−2).

Evaluate −4 × (−2) × 8.

Explain why the answers to parts (a) and (b) are the same.

Is the answer to (9 × −6) × 3 and 9 × (−6 × 3) the same? Why?

Complete the table: Product −1 × −1 −1 × −1 × −1 −1 × −1 × −1 × −1 −1 × −1 × −1 × −1 × −1 −1 × −1 × −1 × −1 × −1 × −1

Number of negative terms 2 3

Result 1 −1

Explain how the number of negative terms can be used to determine whether the result is a negative or positive number. 1.02 Multiply and divide integers mathspace.co

Problem-solving 22, 25

22, 24

23, 24, 26

Find two integers that have a product of −35 and a sum of −2.

Consider the Cartesian plane. Which quadrant(s) must the point (x, y) lie in if:

xy > 0

x÷y<0

y Quadrant 2 Quadrant 1

xy < 0 0

In golf your score for each hole is determined by comparing it to the par (or expected) number of shots for that hole. Taking one less shot than expected will give a score of −1, one more shot will give +1, and so on. Ivan played 18 holes of golf, and the frequency of each score he obtained is recorded in the table: a

Find Ivan’s total score for his 6 scores of +2.

Find Ivan’s total score for his 5 scores of −1.

Find Ivan’s total score for the 18 holes.

Score −1 Par +1 +2

Frequency 5 3 4 6

Victoria earns $1100 a month, and deposits it into her bank account. She also withdraws $200 a month to pay for rent. a

How much, in total, does Victoria deposit into her bank account in one full year?

How much, in total, does Victoria withdraw from her bank account in one full year?

Find the total change in her bank account over the year. An increase will be represented by a positive number and a decrease will be represented by a negative number.

Find x rounded to two decimal places, if the mean of −5, 6, −2x and −10 is −7.

1.03 Four operations with integers After this lesson, you will be able to… • use the order of operations to evaluate expressions involving integers.

Mixed operations with integers When we work with integers, we can perform the same four operations that we use with whole numbers: addition, subtraction, multiplication, and division.

Quadrant 3 Quadrant 4

Mathspace Victorian Curriculum – Year 8 mathspace.co

The main difference when working with integers is that we have to take into account the signs of the numbers. To solve problems with mixed operations, we need to follow the order of operations: 1. Evaluate operations inside grouping symbols such as brackets. 2. Evaluate any powers. 3. Evaluate any multiplication or division, reading from left to right. 4. Evaluate any addition or subtraction, reading from left to right. 5. Consider and check the sign of the ﬁnal answer, especially if negative values were involved in the calculation.

Example 1 Evaluate 5 + 9 × (−6)

Create a strategy We ﬁrst need to evaluate the multiplication, and then the addition.

Apply the idea 5 + 9 × (−6) = 5 + (−54)

Evaluate the multiplication

= 5 − 54

Combine adjacent signs

= −49

Evaluate

Example 2 Evaluate (7 − 8) × (−5)

Create a strategy We ﬁrst need to evaluate the operation in the brackets, and then the multiplication.

Apply the idea (7 − 8) × (−5) = (−1) × (−5) Evaluate 7 − 8 =5

Evaluate

Example 3 Evaluate 21 ÷ (−7) × 2

Create a strategy Evaluate the multiplication and division from left to right.

Apply the idea 21 ÷ (−7) × 2 = −3 × 2 = −6

Evaluate 21 ÷ (−7) Evaluate

1.03 Four operations with integers mathspace.co

Example 4 Evaluate −5 + 12 ÷ 22

Create a strategy

Apply the idea

Evaluate in order the power, the division and the addition.

−5 + 12 ÷ 22 = −5 + 12 ÷ 4 Evaluate 22 = −5 + 3

Evaluate the division

= −2

Evaluate the addition

Example 5 Evaluate ((36 − (10 + 10)) ÷ 2) + 14 × 6

Create a strategy We ﬁrst need to evaluate the operation in the innermost brackets, and the next resulting brackets, and then the division, the multiplication and the addition.

Apply the idea ((36 − (10 + 10)) ÷ 2) + 14 × 6 = ((36 − 20) ÷ 2) + 14 × 6

Evaluate 10 + 10

= (16 ÷ 2) + 14 × 6

Evaluate 36 − 20

= 8 + 14 × 6

Evaluate the division

= 8 + 84

Evaluate the multiplication

= 92

Evaluate the addition

Idea summary The order of operations is as follows: 1. Evaluate operations inside grouping symbols such as brackets. 2. Evaluate any powers. 3. Evaluate any multiplication or division, reading from left to right. 4. Evaluate any addition or subtraction, reading from left to right. 5. Consider and check the sign of the ﬁnal answer, especially if negative values were involved in the calculation. If the problem has brackets inside another set of brackets, simplify the inside brackets ﬁrst.

Mathspace Victorian Curriculum – Year 8 mathspace.co

1.03 Practice questions Understanding 1

When evaluating an expression with mixed operations, you should always perform addition and subtraction before multiplication and division.

If an expression contains only multiplication and division operations, you should evaluate them from left to right.

Two integers can be added in any order but not subtracted.

Two integers can be multiplied in any order and also divided in any order.

The expression 2 × (3 + 4) − 5 is equivalent to (2 × 3) + (2 × 4) − 5, due to the distributive property.

Which expression is equivalent to 5 − 3 + (−2)? 5 + (−3) + (−2)

−8 + 3

7 Ex 1, 2

−5 − 3 − 2

5+3−2

3−8

−(4 − 2) + 3

3−2−5

Which expression is equal to −3 − (2 + 5)? −3 − 2 − 5

−3 + 2 + 5

List the operations in the correct order for the expression 5 − 2 × 3 + 1.

Fluency 6

Which expression is not equivalent to −(4 × 2) + 3?

A 5

Trailblazer 1, 3–5

A 4

Adventurer 1, 3–5

Determine whether each statement is true or false:

A 3

Explorer 1a–c, 2, 3, 5

6a–e, 7a–f, 9a–f, 10a–c, 11b

6d–h, 7c–h, 9c–h, 10bce, 6d–h, 7e–j, 8, 9e–j, 11ab 10c–f, 11

Evaluate: a

−5 + 4 + 3

−3 + 5 − 7

6 − (−3) + 7

−2 − 1 − 4

1 − (3 + (−7))

1 − (4 − (−5))

((−8) + 2) − 5

(4 − (−2)) − 6

(−4 + 2) × 6

−3 − 6 × (−4)

(8 − 9) × (−8)

6 × (−5) − 4

−8 × (−6 − 4)

Evaluate: a

7 + 8 × (−7)

−5 × (−7) + 8

6 × (−4 + (−7))

−6 × (−4) × (−2)

(2 × 27) − 3 × 5 − 8

Complete the equation: 2 + ⬚ × 5 = −18

Evaluate: a

−7 − ((−8) ÷ 2)

(−39 + 9) ÷ (−5)

−9 + 28 ÷ 4

(21 − 6) ÷ (−3)

48 ÷ (−6) + (−4)

14 ÷ (−5 + 3)

63 ÷ (−9) − (−8)

(−54) ÷ (−4 − 2)

(−36) ÷ (−6) ÷ (−3) j

15 ÷ (40 ÷ (−8))

1.03 Four operations with integers mathspace.co

10 Ex 3, 4

Evaluate: a e

Ex 5

−8 + (24 ÷ 4)2 2

(8 − 10 ÷ 2)

−6 + 16 ÷ 22

(−84) ÷ (2 × 7)

24 ÷ (−8) × 2

(−56 × 5) ÷ (−7)

Evaluate: a

((−43 + 8) ÷ (−5))2

16 × (26 ÷ (9 + 4))

(48 − ((−9) × 6) ÷ 3) + 16

((88 − (8 + 28)) ÷ 4) + 21 × 7

Reasoning 12

12, 13

13, 14

14, 15

Explain why the statement is incorrect, and ﬁnd the correct answer: 6 − (−60) ÷ (−3) = −22

Are the results of (−5 + 3) × (−4) and −5 × (−4) + 3 × (−4) the same? Why?

An expression involving mixed operations with integers is shown, along with a series of working steps to solve the expression. However, a mistake was made during the working steps. Identify where the mistake occurred and correct the mistake. Expression: (−3) + 10 × 2 − (−5) 1

(−3) + 10 × 2 − (−5) = 7 × 2 − (−5)

= 14 − (−5)

= 14 + 5

= 19

During a math competition, the participants were given this expression to evaluate: 7 − 4 × (3 − 9) ÷ 2 + 5 × 2 a

The judges received several different answers from the participants, including 1, −21, and 29. Which one of these is the correct answer?

Using only brackets, change the expression in two ways so that the other two answers would be correct.

Problem-solving 17–19 16

16, 18, 19, 21

16, 20–22

Three people on a farm are given ﬁve minutes to pick as many apricots as they can. They get to keep all the ripe apricots they pick, as well as an equal split of 30 ripe apricots between the three of them (which were picked earlier in the day). If Justin (one of the three people) picks 33 apricots, 15 of which are unripe, how many apricots will Justin get to keep in total?

Each week Lisa gets paid an allowance based on her age and how many chores she did that week. The allowance consists of her age, $2 for every chore she completes in a week, and is decreased by $4 for each chore she forgets to complete. If Lisa is 11 years old, completes 4 chores in a week, but forgets to complete a chore, how much will her allowance be that week?

Mathspace Victorian Curriculum – Year 8 mathspace.co

he temperature in Paris is 13 °C less than double the temperature in London. If it is 8 °C T in London then what is the temperature in Paris?

The temperature in Sydney is 21 °C more than triple the temperature in New York. If it is −4 °C in New York then what is the temperature in Sydney?

Fred busks with 2 friends whilst selling CD copies of his own music. They make a total of $36 from busking and decide to split it evenly. Fred makes an additional amount by selling 9 CDs at $15 each. If Fred pays a permit of $31, what is the total amount he earns?

Laura and a group of 11 friends are ordering some pizza. They think that they will eat two pizzas for every three people. Pizzas cost $14 each and there is a $8 delivery fee. If they split the total cost evenly between each person, how much will Laura need to pay? At the start of the week, Quiana had $131 in her bank account. On Monday, she withdrew $43. On Wednesday, she deposited $11, and on Thursday, she deposited three times as much as she deposited on Wednesday. Finally, she withdrew $25 on Friday.

Find Quiana’s ﬁnal bank balance at the end of the week. 22

Starting from his house, Jerry walked 1 m/s due south for 30 minutes, then 2 m/s due north for 15 minutes and then due south again 2 m/s for 45 minutes. Find Jerry’s ﬁnal distance from his house.

1.04 Negative fractions After this lesson, you will be able to… • add negative fractions. • subtract negative fractions.

• multiply negative fractions. • divide negative fractions.

Addition and subtraction with negative fractions When adding and subtracting negative fractions, we are combining the ideas from adding and subtracting fractions with adding and subtracting negative integers. We use the following skills to help us add and subtract negative fractions: • Add or subtract the numerators of fractions with common denominators.

• Convert fractions with uncommon denominators to fractions with common denominators using the lowest common denominator.

1.04 Negative fractions mathspace.co

• Rearrange addition and subtraction problems with negative fractions by keeping the sign on the left of the fraction with the fraction when you move it.

• When there are adjacent signs, two adjacent signs that are the same become addition and two adjacent signs that are different become subtraction.

Simplify fraction answers where possible and pay close attention to the signs of the values.

Example 1 Evaluate

Create a strategy Find the lowest common multiple (LCM) of the two denominators.

Apply the idea The LCM of 8 and 7 is 56. So we need to multiply both parts of

by 7, and multiply both parts of Multiply

and

by 8.

Evaluate the multiplication Rearrange the new fractions Evaluate

Idea summary Adding and subtracting negative fractions combines the ideas from adding and subtracting fractions with adding and subtracting negative integers.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Multiplication and division with negative fractions If we are multiplying The visual of a circle with

, then we are calculating parts shaded, then

of .

of each part shaded reveals that

To multiply two fractions, we multiply the numerators and the denominators separately, then simplify the answer. To divide two fractions, we multiply the ﬁrst fraction by the reciprocal of the second fraction. To calculate

: Multiply by the reciprocal

Multiply the numerators and denominators

Evaluate the multiplication

Simplify the fraction

When negative fractions are involved, we use the same rules for the sign of the answer as when working with multiplication and division of integers: • When we multiply or divide fractions with the same sign, their product or quotient will be positive. • When we multiply or divide fractions with different signs, their product or quotient will be negative. When negative fractions are involved, we use the same rules for the order of operations as when working with integers: 1. Evaluate operations inside grouping symbols such as brackets. 2. Evaluate any powers. 3. Evaluate any multiplication or division, reading from left to right. 4. Evaluate any addition or subtraction, reading from left to right. 5. Consider and check the sign of the ﬁnal answer, especially if negative values were involved in the calculation.

1.04 Negative fractions mathspace.co

Example 2 Evaluate

Create a strategy Multiply numerators and denominators separately. The product of two negative integers is a positive integer.

Apply the idea Multiply numerators and denominators

Evaluate

Reﬂect and check It is useful to know whether to expect a positive or negative answer before evaluating, as this can help us know if we have made a mistake if we end up with an answer that is not the expected sign.

Example 3 Evaluate

Create a strategy Divide fractions by multiplying the ﬁrst fraction by the reciprocal of the second fraction. The product or quotient of two integers with different signs is a negative integer.

Apply the idea Multiply by the reciprocal

Multiply numerators and denominators

Evaluate

Simplify

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 4 Evaluate

Create a strategy Convert the mixed numbers to improper fractions.

Apply the idea Convert the mixed numbers to improper fractions

Multiply by the reciprocal

Multiply the numerators and denominators

Evaluate the multiplications

Convert the improper fraction to a mixed number

Simplify the fraction

Example 5 Evaluate

Create a strategy We ﬁrst need to evaluate the operation in the brackets, and then the multiplication. Rewrite the two fractions inside the brackets with a common denominator.

Apply the idea For

, the LCM of 9 and 6 is 18. Rewrite

and

with same denominator

Evaluate the multiplication inside the bracket

Evaluate the subtraction

Evaluate

Simplify

1.04 Negative fractions mathspace.co

Idea summary To multiply two fractions, we multiply the numerators and the denominators separately, then simplify the answer. To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. When we multiply or divide fractions with the same sign, their product or quotient will be positive. When we multiply or divide fractions with different signs, their product or quotient will be negative. We use the same order of operations rules when working with fractions as when working with integers.

1.04 Practice questions Understanding

Explorer 1–4, 5a–e

Adventurer 2–4, 5b–f

Trailblazer 2, 3, 5c–h

True or false: Fractions can be added in any order without changing the result.

Which equation is true? A

In the expression

Which operation requires common denominators? Select all that apply. A

, which expression is the correct ﬁrst step? B

A 4

Addition

Subtraction

Multiplication

Which operation should be performed ﬁrst? a

Mathspace Victorian Curriculum – Year 8 mathspace.co

Division

Fluency Ex 1

Ex 2

Ex 3

Ex 4

Ex 5

6a–e, 7a–e, 8a–f, 9a–d, 10a–d, 11a–c

6c–g, 7b–f, 8c–h, 9e–h, 10d–g, 11c–f

6h–l, 7g–l, 8e–l, 9h–l, 10e–l, 11d–h

Evaluate the sum and write your answer as a simpliﬁed fraction. a

Evaluate the difference and write your answer as a simpliﬁed fraction. a

Evaluate the product and write your answer as a simpliﬁed fraction. a

Find the quotient and write your answer as a simpliﬁed fraction. a

Evaluate and write your answer as a mixed number in simplest form. a

Evaluate:

1.04 Negative fractions mathspace.co

Reasoning 12

12, 13

13, 14

13, 15

Wade and Avery want to evaluate the expression:

Wade wants to follow the order of operations and add

and

add fractions in any order, so she wants to add

ﬁrst.

and

ﬁrst. Avery says you can

Whose method will be more eﬃcient? Explain your reasoning.

If they do not make any mistakes in their calculations, will Wade and Avery get the same result? Justify your answer.

To simplify the expression

Then he noticed that

, Weston began by rewriting the expression as

is bigger than , so he switched the order of the fractions:

He evaluated the expression to get an answer of

Identify and explain Weston’s mistake. 14

Paul has a part-time job at the local news agency. He spends and

of his pay on comic books

on biscuits. Explain how to calculate the fraction of his pay Paul has left over.

In the staff room, there is

of a cake left over from a meeting. If 20 members of staff would

all like a piece, what fraction of the original cake will they each receive? Explain how you reached your answer.

Problem-solving 17, 18 16

At the end of last month, a dam was

17, 18

16–18

full. Over the current month,

was used up by the town. Meanwhile, rain ﬁlled

of the dam’s capacity

of the dam’s capacity. What fraction of the

dam is full at end of this month? 17

There is

of a birthday cake left. If Kirk eats

of the remaining cake, what fraction of the

original cake did he eat? 18

Moscow’s coldest maximum temperature was

and warmest maximum was

Calculate the average maximum temperature as a fraction.

Mathspace Victorian Curriculum – Year 8 mathspace.co

1.05 Negative decimals After this lesson, you will be able to… • add negative decimals. • subtract negative decimals. • multiply negative decimals. • divide negative decimals.

Addition and subtraction with negative decimals To add and subtract negative decimals, we combine the ideas from adding and subtracting decimals with adding and subtracting negative integers. These strategies can be used to add and subtract decimal numbers: • To add or subtract decimals, we can use the same vertical algorithm that we use with whole numbers, making sure the decimal points line up. • If there are negative decimal numbers, ﬁrst simplify any adjacent signs and then rearrange the problem so that it becomes an addition or subtraction problem. For example, the problem −5.4 + 9.2, is rearranged to 9.2 − 5.4:

Another example, to calculate −4.3 − (−6.7), we simplify the adjacent signs to −4.3 + 6.7, then rearrange to 6.7 − 4.3:

• To add two negative decimal numbers, add the two numbers and then make the result negative. For example, −3.45 + (−4.82) becomes −3.45 − 4.82 which becomes − (3.45 + 4.82). • In order to write the decimals in the correct order it helps to consider the sign of the answer. For example, 8.05 − 9.60 will have a negative answer because the bigger number is negative. It is the difference between 9.60 and 8.05. The problem becomes −(9.60 − 8.05).

Example 1 Evaluate: −8.2 − 1.1

Create a strategy Add the two numbers and then make the result negative.

1.05 Negative decimals mathspace.co

Apply the idea Rewrite the expression as addition

Evaluate the addition

Make the result negative

Example 2 Evaluate: 2.5 + (−8.1)

Create a strategy Simplify the adjacent signs first then consider the sign of the answer. The answer will be a negative number that is the difference between the numbers.

Apply the idea Combine adjacent signs

Rewrite the expression as subtraction

Evaluate the subtraction

Make the result negative

Example 3 Evaluate: −9.3 − (−2.2)

Create a strategy • Simplify adjacent signs and then rearrange the problem so that it becomes an addition or subtraction problem. • Consider the sign of the answer. The answer will be a negative number that is the difference between the numbers.

Apply the idea Simplify the adjacent sign

Rewrite the expression as subtraction

Evaluate the subtraction

Make the result negative

Mathspace Victorian Curriculum – Year 8 mathspace.co

Idea summary These strategies can be used to add and subtract decimal numbers: • We can use the same vertical algorithm that we use with whole numbers, making sure the decimal points line up. • First simplify any adjacent signs and then rearrange the problem so that it becomes an addition or subtraction problem. For example, the problem −5.4 + 9.2, is rearranged to 9.2 − 5.4. • To add two negative decimal numbers, add the two numbers and then make the result negative. For example, −3.45 + (−4.82) becomes −(3.45 + 4.82). • To write the decimals in the correct order, it helps to consider the sign of the answer. For example, 8.05 − 9.60 will have a negative answer that is the difference between 9.60 and 8.05. The problem becomes −(9.60 − 8.05).

Multiplication and division with negative decimals Just like when we multiplied and divided integers, we need to pay attention to the signs of the decimals when multiplying and dividing. • If both decimals have the same sign, the answer will be positive. • If the two decimals have different signs, the answer will be negative. For example, to calculate −3.6 × (−0.8): 1. Find the sign of the product. We can see we have (−) × (−) = (+), so our answer will be positive. 2. Using the vertical algorithm, ignore the decimal points and multiply as if they are whole numbers.

3. To place the decimal point, we count the total number of decimal places in the original numbers. In this case, since the question has two decimal places, we insert the decimal point into our answer so that it also has two decimal places to get: −3.6 × (−0.8) = 2.88 To divide any number by a decimal, we multiply both numbers by the same power of 10 to make the divisor a whole number. For example, to calculate −9.3 ÷ 0.3: 1. Find the sign of the answer. We can see we have (−) ÷ (+) = (−), so our answer will be negative. 2. Multiply both numbers by 10 so that we are dividing by 3 and not 0.3. The problem becomes: −93 ÷ 3. 3. Calculate the division and write the answer with the correct sign: −93 ÷ 3 = −31 therefore −9.3 ÷ 0.3 = −31

1.05 Negative decimals mathspace.co

Example 4 Evaluate the product 7.4 × (−4.1).

Create a strategy

Apply the idea

Multiply the numbers together as whole numbers and place the decimal point accordingly. The product of a positive integer and a negative integer is a negative integer.

Ignore the decimal point for the meantime and multiply the numbers together as whole numbers. 7 4 7 + 2 9 6 3 0 3 ×

4 1 4 0 4

The result is 3034. The original question has 2 decimal places altogether. So we insert the decimal point into this answer so that it also has 2 decimal places. 7.4 × (−4.1) = −30.34

Example 5 Evaluate the quotient 7.36 ÷ (−0.08)

Create a strategy Multiply both numbers by a power of 10 to make the divisor a whole number then perform the division. The quotient of a negative and a positive integer is a negative integer.

Apply the idea 7.36 ÷ (−0.08) = (7.36 × 100) ÷ (−0.08 × 100)

Multiply both decimals by 100

= 736 ÷ (−8)

Evaluate multiplication

= −92

Evaluate division

Reflect and check The solution to the original division is equivalent to 736 ÷ −8. Let’s consider the proportionality between the two expressions: 7.36 ÷ −0.08 is proportional to 736 ÷ −8. The reason they are proportional is that −7.36 and −736 differ by a factor of 100 (as −7.36 × 100 = −736), and 0.08 and 8 also differ by a factor of 100 (as 0.08 × 100 = 8). Since both the numerators and denominators differ by the same factor, the two fractions are proportional. This confirms that the final result of 7.36 ÷ (−0.08) is −92, which is consistent with our calculation of −736 ÷ 8.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Idea summary To multiply or divide two decimals, first find the sign of the answer. • If the two signs are the same the answer is positive. • If the two signs are different the answer is negative. To multiply two decimal numbers, ignore the decimal points and multiply the two numbers. The number of decimal places in the answer is equal to the total number of decimal places in the numbers being multiplied. To divide decimal numbers, first multiply both numbers so that the divisor is a whole number. Then perform the division as for whole numbers.

1.05 Practice questions Understanding 1

Explorer 2, 3

Adventurer 2, 3

Trailblazer 1, 2, 4

Determine whether the statements are true or false. a

When dividing a decimal by 1, the result remains unchanged.

When you multiply two positive decimals, the result is always positive.

When you multiply a negative decimal by a positive decimal, the result is always positive.

When you multiply two decimal numbers with one decimal place each, the number of decimal places in the answer is also 1.

When you divide a decimal by a decimal you can multiply both numbers by a power of ten to make the calculation easier.

Identify the correct sequence of operations when solving the expression: 3 − (7.54) × 4 + 0.5

Subtraction, multiplication, addition

Addition, subtraction, multiplication

Multiplication, subtraction, addition

Multiplication, addition, subtraction

Which strategy is most eﬃcient for dividing a decimal by 1000? A

Shift all digits three places to the right.

Shift all digits three places to the left.

Use long division.

Convert the decimals to fractions

Determine whether the calculations can be evaluated: a

1.05 Negative decimals mathspace.co

5a–f, 6a–f, 7a–f, 8a–d, 9, 10

Fluency

5 Ex 1, 3

Ex 2

Ex 4

Ex 5

5g–l, 6g–l, 7h–l, 8e–h, 9, 10

Evaluate: a

−8.2 − 1.1

9 − (−8.3)

−2.6 − 8.9

6.2 − (−1.5)

5.6 − (−1.8)

−2.4 − (−9.26)

−9.3 − (−2.2)

−1.6 − 3.04

−3.1 − (−3.9)

4.5 − (−6.93)

−2.3 − 5.26

−10.7 − (−1.45)

Evaluate: a

−6.7 + 10.5

−2.5 + (−3.5)

0.3 + (−0.7)

−1.9 + 2.8

−4.2 + (−5.6)

−3.6 + 2.4

−9.1 + (−7.3)

2.8 + (−4.9)

−1.2 + 0.5

3.9 + (−6.2)

−7.4 + (−9.7)

5.5 + (−3.2)

Find the product: a

8 × (−2.4)

(−7.6) × 7.8

(−0.9) × 0.6

6.4 × (−7.8)

(−3.9) × (−9.8)

(−8) × (−7.2)

(−4.3) × 9.7

1.99 × (−8)

(−7.26) × 9

7.4 × (−4.1)

(−2.77) × (−2)

(−4.9) × (−7.3)

Find the quotient: a

9.2 ÷ (−2)

(−29.4) ÷ 3

(−0.35) ÷ 0.7

(−0.27) ÷ (−0.9)

0.81 ÷ (−0.9)

4.16 ÷ (−0.08)

(−8.28) ÷ (−0.09)

(−61.4) ÷ (−15.35)

(−6.4)2

(3.7)3

(−6.4)3

−2.5 + 3.5 × 2.0 ÷ (−1)

−5.5 − 3.5 × 2.0 ÷ (−1.4)

−1.5 − 0.5 × (−3.0) + (−2.5) ÷ 0.5

−7.4 + 9.4 × 0.8 ÷ (−1.6)

12, 13

12, 14

Calculate: a

5e–j, 6d–i, 7e–j, 8c–f, 9, 10

−(3.7)2

Evaluate:

Reasoning

11, 12

Yuri knows that −25 × 36 is equal to −900. Explain how Yuri can ﬁnd the answer to −0.25 × 0.36 without using a calculator.

Zoe used a calculator to evaluate 2.086 × −1.47 but forgot to enter the negative sign and the decimal points. The answer she got was 306 642. What should the answer have been? Why?

Consider 13.8 + (−14.52) − 22.03 Mia and James evaluate the expression in different ways. Mia’s working: 13.8 + (−14.52) − 22.03 = −0.72 − 22.03 = −22.75 James’ working: 13.8 + (−14.52) − 22.03 = 13.8 − 7.51 = 6.29 Who do you think is correct? Explain your answer.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Consider the expression: −3.83 − (−11.63) = d ÷ 4.95 Describe the steps you could take to ﬁnd the value of d.

Problem-solving 15–17, 19 15

15–18, 20

16, 18–21

Consider the expression: 9 + (−2.4). a

Evaluate 9 + (−2.4).

Which of these situations could be calculated using 9 + (−2.4)? A

Rose has $9 in her account. If she spends $2.40, how much does Rose have left in her account?

Lily has $9 in her account. If she spends $2.40, how overdrawn is Lily?

Ben has $9 in his account. If he deposits $2.40 into his account, how much does Ben have?

Consider the table that shows the temperature in different locations: a

Find the change in temperature from London to Sitka, Alaska.

If the temperature change from one of these locations to another was 9.3 °C, what was the journey?

Cairo is 40.9 °C warmer than Sitka, Alaska. Find the temperature in Cairo.

London, England 12.1 °C Sitka, Alaska, USA −20.3 °C Dudinka, Russia −29.6 °C Vancouver, Canada −10.5 °C

Peggy owes $98.96 in total to 4 friends. a

Assuming she owes the same amount to each friend, how much does she owe one of them?

If Peggy pays just 3 of her friends back, what will be the resulting change in her bank account?

Jenny takes out a loan of $2200. She pays back $42.60 each month without interest. If she has made 5 repayments so far, how much does Jenny still owe?

At midnight, the temperature in a certain city is 6.71 °C. Each hour, the temperature decreases by 0.87°. Find the temperature after 5 hours.

A diver is 3.5 m below the surface of the water. He descends 1.47 m/s for 3 seconds, then rises 0.46 m/s for 6 seconds. Note that surface level is 0 m, and 1 m below is represented by −1 m.

Find the depth of the diver after 9 seconds.

The diver is running out of oxygen and needs to get to a depth of 0.35 m below the surface within 4 seconds. How many metres does he need to rise each second, assuming he will rise at a constant speed?

Debbie is considering installing solar panels for her home. In direct sunlight, the solar panels generate 8.71 kWh of electricity each hour, and on average her house gets 7 hours of sunlight a day. Any unused power is returned to the main grid and you receive 33 cents per kWh. Over a 24-hour period, a household’s usage averages out to 1.87 kWh each hour. How much will you expect to make each day by selling back electricity if you install the solar panels?

1.05 Negative decimals mathspace.co

1.06 Rational and irrational numbers After this lesson, you will be able to… • classify numbers as rational or irrational.

Rational and irrational numbers We have compared, ordered, and solved problems with numbers. These numbers are all called real numbers. Most of the numbers that we know are called rational numbers. Real numbers also consist of irrational numbers. A rational number is a real number which can be written as a fraction where both the numerator and denominator are integers, and the denominator is never zero.

Real Numbers Rational numbers Integers Whole numbers ... −4 −3 −2 −1 0 1 2 3 4 ...

An irrational number is also a real number, but it cannot be written as a fraction of two integers. You might be familiar with one irrational number already: π.

Irrational numbers

A repeating decimal or a recurring decimal is a decimal number that has a pattern of digits that repeats endlessly after the decimal point. For example, 6.333..., 0.181 818 ..., 0.245 245 245... All repeating decimals can be written as fractions. A common example you might already be familiar with is

which is 0.333...

A terminating decimal is a decimal number that has a finite number of digits after the decimal point, meaning it eventually comes to an end. For example, 1.25 All terminating decimals can be written as fractions, and so they are rational: 1.25 =

, 0.673 =

, 0.3 =

Rational numbers include integers, fractions and repeating and terminating decimals. Some examples of rational numbers are:

Evaluate these on your calculator. Note the decimal expansions for rational numbers are repeating or terminating.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Some decimals are non-repeating and non-terminating and cannot be written as fractions. This makes them irrational numbers. Irrational numbers are non-repeating and non-terminating. Some examples of irrational numbers are:

The square root of a number that is not a perfect square cannot be written as a fraction so is always an irrational number. These numbers are also called surds.

Interactive exploration Discover this concept in action online

mathspace.co

Example 1 Is

rational or irrational?

Create a strategy Check whether the number can be written as a fraction.

Apply the idea 35 is not a perfect square, so

will not equal a whole number.

Using a calculator, we can get = 5.916 079 783 099 616… The decimals are non-terminating and non-recurring. This means that fraction. So

cannot be written as a

is irrational.

Example 2 Given the irrational numbers

a Plot the irrational numbers on the following number line: 0

1.06 Rational and irrational numbers mathspace.co

Create a strategy Approximate the values of each irrational number to make it easier to plot them on the number line.

Apply the idea Calculate the approximate decimal values for each irrational number using a calculator: ≈ 7.07, π ≈ 3.14, ≈ 5.39, and ≈ 9.90.

≈ 1.41,

Now that we have approximate decimal values for each number, we can plot them on the number line. ≈ 1.41 0

π ≈ 3.14

≈ 5.39

≈ 7.07 7

≈ 9.90 9

b Arrange the irrational numbers in descending order.

Create a strategy

Apply the idea

In part (a) we plotted all the numbers on a number line, we can use the number line to order them from largest to smallest.

Based on the positions of the numbers on the number line, the order is:

Idea summary A rational number is a number which can be written as a fraction where both the numerator and denominator are integers and the denominator is not equal to zero. An irrational number is a number which cannot be written as a fraction of two integers. Rational numbers may be repeating or terminating decimals. Some examples of rational numbers are: 1 2 45, , , 81, 0.34, 1.45 3 7 Irrational numbers are non-repeating and non-terminating decimals. Some examples of irrational numbers are:

ππ, 7, 15, 20 The square root of a number that is not a perfect square is always an irrational number. These numbers are also called surds.

Mathspace Victorian Curriculum – Year 8 mathspace.co

1.06 Practice questions Understanding

Explorer 1, 3, 4

Adventurer 1, 3, 4

Trailblazer 2–4

How do we know a number is a rational number?

Christa decides to express her favourite rational number as a decimal. Will the decimal be repeating, terminating or neither?

Select the irrational number from the set of numbers:

Select the rational number from the set of square roots:

Fluency

5 Ex 1

5i–x, 6c–h, 7h–p, 8, 9e–n

c 1.618 03…

4.435

0.693 147…

m −0.7

−4

912

5d–o, 6b–e, 7e–m, 8, 9c–j

Identify the numbers as rational or irrational:

5a–l, 6a–d, 7a–i, 8, 9a–h

For each square or cube root: i

Evaluate, with answer rounded to two decimal places.

Determine whether the value is an exact value or an approximation.

Evaluate and round your answer to two decimal places when applicable. a

1.06 Rational and irrational numbers mathspace.co

8 Ex 2

Consider the irrational numbers a

Plot the irrational numbers on the number line: 0

b 9

Write the numbers in descending order.

Maddie knows that when you add or subtract a rational number and an irrational number, the result is an irrational number. For each expression: i

Identify whether the two numbers are rational or irrational numbers.

Find the value of the expression, rounded to two decimal places.

iii

Is the result a rational or an irrational number?

Reasoning

10, 11, 13, 14

12–14

12, 13, 15

Explain why the decimal 0.345 is a rational number.

Akin states that 24.737 337 333… is a rational number because it has a repeating pattern. Is Akin correct? Explain your answer.

Are all integers rational numbers? Explain your answer.

Why is π an irrational number?

Can a number be both rational and irrational? Explain your answer.

Every whole number is also a rational number but not every rational number is a whole number. Explain why this statement is true.

Problem-solving 16

16, 18

16–18

A square has an area of 10 square metres. Calculate the side length of the square. Is the side length rational or irrational?

Some sets of numbers are: A

Rational numbers

Integers

Whole numbers

Positive real numbers

All real numbers

Match the set of numbers most appropriate to describe these situations:

The populations of towns.

The times of the runners (in seconds) in a 100 m sprint.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Distance between Jupiter and Venus.

The position of a submarine relative to sea level. (Note that height above sea level is expressed as a positive quantity.)

The goal difference of a hockey team at the end of a season. (Note that the goal difference is the number of goals the team scored in the season minus the number of goals that were scored against them.)

The cost (in dollars) of sending a text message.

The circumference of a circle is given by 2π r, where r is the radius. Determine whether each statement is true: a

The circumference is an irrational number if the radius is rational.

The circumference is a rational number if the radius is equal to π.

The circumference is a rational number if the radius is irrational.

The circumference is a rational number if the radius is equal to

where a is an integer.

1.07 Recurring and terminating decimals After this lesson, you will be able to… • recognise and represent recurring and terminating decimals. • convert fractions to decimals.

Recurring and terminating decimals We now know that rational numbers may be further described as recurring or terminating, and that irrational numbers are both non-recurring and non-terminating. A repeating decimal or a recurring decimal is a decimal number that has a pattern of digits that repeats endlessly after the decimal point. A terminating decimal is a decimal number that has a ﬁnite number of digits after the decimal point, meaning it eventually comes to an end. Recurring decimals are written with a line above the repeating pattern, and sometimes with a dot above the number if it is the only repeating value:

Fractions are rational numbers. Therefore, the decimal expansion of a fraction will either be terminating or recurring. We can tell if a fraction’s decimal expansion will be terminating if its denominator, when simpliﬁed, has only prime factors of 2 and/or 5. Otherwise, the fraction’s decimal expansion must be recurring.

1.07 Recurring and terminating decimals mathspace.co

Interactive exploration Discover this concept in action online

mathspace.co

Example 1 Write the following recurring decimals with the appropriate notation using a line to indicate the recurring digits: a 0.777 77…

Create a strategy

Apply the idea

Identify the repeating value.

0.777 77… has one repeating value which is 7. So we write a line above 7: 0.777 77… =

b 0.456 565 6…

Create a strategy

Apply the idea

Identify the repeating value(s).

0.456 565 6… has two repeating values which is 56. So we write a line above 56: 0.456 565 6… =

Idea summary Recurring decimals are written with a line above the repeating pattern, and sometimes with a dot above the number if it is the only repeating value: 0.012 121 212... = 0.012 . 0.333... = 0.3 or 0.3 The decimal expansion of a fraction will either be terminating or recurring. We can tell if a fraction’s decimal expansion will be terminating if its denominator, when simpliﬁed, has only prime factors of 2 and/or 5. Otherwise, the fraction’s decimal expansion must be recurring.

Convert fractions to decimals Consider one third. This can be written as a fraction , and as a decimal that repeats forever as 0.333 333… If we were given

and asked to express it as a decimal, we could enter it into our calculator

and get 0.777 777 777. Our calculator eventually runs out of space, but we know that we have a recurring-decimal

and that the sevens in 0.777 777 777… go on forever.

Mathspace Victorian Curriculum – Year 8 mathspace.co

To convert from a fraction to a decimal without a calculator, we can rewrite the fraction as division. For example, is 3 divided by 7. Then we can use short division, adding extra zeros as required to the numerator.

Example 2 Write the fraction

as a recurring decimal.

Create a strategy Use short division, and add extra zeros as required to the numerator.

Apply the idea 9 7.000

Set up the short division including some trailing zeros.

0.7 7 9 7.000

How many multiples of 9 are in 7? 0. So we place a 0 above the 7 and place the decimal point in position. How many multiples of 9 are in 70? Because 9 × 7 = 63, there are 7 multiples. So we place a 7 above the first zero and carry the remaining 7 (70 − 63 = 7) to the next place value.

0.7 7 77 9 7.000

This is the same calculation as the previous step.

0.7 7 7 77 9 7.000

Again, this is the same calculation. So we place a 7 above the third zero. We can now see that this pattern will continue forever so we label this as a recurring decimal.

So we place a 7 above the second zero and carry the remaining 7 to the next place value.

Idea summary To convert from a fraction to a decimal without a calculator, we can rewrite the fraction as a division expression. For example, 3 is the same as 3 ÷ 7 7 Then we use short division, adding extra zeros as required to the numerator.

1.07 Recurring and terminating decimals mathspace.co

1.07 Practice questions Understanding

Ex 1

Explorer 1, 2a–d, 3, 4a–d

Adventurer 1, 2c–f, 3, 4c–f

Trailblazer 1, 2e–h, 3, 4e–h

Describe these decimals as terminating or recurring:

Write these recurring decimals with the appropriate notation using a line to indicate the recurring digits:

0.333…

5.3434…

0.6123123…

1.777…

2.34111…

9.999…

5.8787…

Round these decimals to two decimal places: a

0.66…

Write these fractions as decimals: a

Fluency 5

5a–d, 6a–d, 7a–h, 8a–d, 10, 12

5b–e, 6c–f, 7c–j, 8b–e, 10, 12

5c–f, 6e–h, 7e–l, 8e–h, 9, 11, 13

Complete the boxes with the missing numbers: a

Use short division to convert each fraction into a decimal rounded to two decimal places. Identify whether the decimal is terminating or non-terminating. a

2 4

Mathspace Victorian Curriculum – Year 8 mathspace.co

7 Ex 2

Write these fractions as recurring decimals: a

Write each decimal as a fully simpliﬁed fraction: a

0.7

0.09

0.907

5.8

7.19

1.0625

4.625

223.205

Express 2.9 ÷ 9 as: a

A recurring decimal, rounded to two decimal places.

A simpliﬁed fraction.

Write these recurring decimals as fractions: a

Write these mixed numbers as recurring decimals: a

The fraction as 1 ÷ 9.

b can also be written

Fraction

Recurring decimal

The quotient is 0.111… which we read as “zero point one recurring”. 0.111… can be written as

Rounded to two decimal places 0.11

Complete the table.

0.89 1

1.00

1.07 Recurring and terminating decimals mathspace.co

The fraction as 1 ÷ 7.

can also be written

Fraction

Recurring decimal

Rounded to four decimal places

The quotient is 0.142 857 142 857… when written as a decimal, which can be written as .

0.1429

Complete the table.

Reasoning

14, 15

15, 16

1.0000

15, 17

Is π considered a terminating or a non-terminating decimal? Explain your reasoning.

Investigate the decimal representation of these square roots:

Are the square roots of whole numbers ever recurring decimals? Explain your reasoning.

When are the square roots of whole numbers terminating decimals?

an irrational number? Explain your reasoning.

Describe the pattern you observe when the denominators of fractions are multiples of 3 . What can you say about the decimal representation of these fractions?

Consider these fractions:

Identify which of the fractions are terminating and which are recurring.

Write each denominator as a product of prime factors.

What are the only prime factors that must be present in the denominator of a fraction for it to have a terminating decimal representation?

Given the fraction

, without converting it to a decimal, determine if it will have a

terminating or recurring decimal representation. Justify your answer.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Problem-solving 18–20 18

A rectangle has a length of

20–22

metres and a width of

20–23

metres. What is the area of the

rectangle in square metres? Express your answer as a decimal. 19

A rope is cut into pieces, each measuring

metres in length. If the total length of the rope

is 6 metres, how many pieces can be cut? Express your answer as a decimal and determine whether it is terminating or recurring. 20

A rectangular garden has a length of

metres and a width of

metres. Calculate the

perimeter of the garden in metres. Is the resulting decimal terminating or recurring?

A container is ﬁlled with litres of water. If litres of water are poured out, what is the remaining amount of water in the container? Write your solution as both a terminating or recurring decimal.

Leanora has metres of ribbon. She wants to cut it into equal pieces, each measuring metres. How many pieces can she cut?

A local pizza restaurant wants to divide a pizza into equal slices, so that each slice represents a recurring decimal of the pizza. If the pizza must be divided into more than 2 slices but less than 20, what are the possible numbers of slices the pizza can be divided into? Explain your reasoning.

1.07 Recurring and terminating decimals mathspace.co

INVESTIGATION

Golden ratio Investigate the evidence of the golden ratio in nature and graphic design.

Objectives To explore the golden ratio

Materials • •

Calculator Pen or pencil

• •

Measuring tape Grid paper

The golden ratio The golden ratio can be found everywhere. It’s in architecture, art and even in nature. In this investigation, we will delve into the possible presence of the golden ratio in the proportions of the human body, bearing in mind that its occurrence is not always precise.

Measure the following lengths on your own body Total height head to toe Belly button to feet Waist to ﬂoor Top of head to waist Tip of nose to chin Middle of eye to the tip of nose Length of the lips Width of the nose

Mathspace Victorian Curriculum – Year 8 mathspace.co

Top of head to chin Top of head to the middle of eye Outside distance between the eyes Hairline to the pupil of eye Lips to the chin Wrist to elbow Tip of ﬁnger to wrist

Find the ratio of A : B for each of the 9 pairs of measurements. Each pair should result in a ratio very close to the golden ratio. For example, to create the ﬁrst ratio, write the ratio Total height head to toe : Belly button to feet using your measurement values. A Total height head to toe Waist to floor Tip of nose to chin Length of the lips Top of head to chin Top of head to the middle of the eye Outside distance between the eyes Tip of the nose to the chin Wrist to the middle of the elbow

1 2 3 4 5 6 7 8 9

B Belly button to feet Top of head to waist Middle of eye to the tip of nose Width of the nose Width of the head Pupil of the eye to lip Hairline to the pupil of the eye Lips to the chin Tip of finger to elbow

Convert each of the ratios you just found into a decimal. Use your calculator if necessary.

AB C

Compare all of the ratio decimals you just found. Calculate the average of these ratios. Use the average to estimate what the golden ratio is.

Once you have made your estimate, read about the golden ratio.

Investigate 1.

Is your estimation of the golden ratio from the activity a terminating decimal, recurring decimal, or neither? Explain.

How close was your average or your guess to the actual value of the golden ratio?

With a group, calculate the average of your ratio and their ratios. Is it closer to the actual value of the golden ratio? Is this new number calculated a terminating decimal, recurring decimal, or neither?

Investigation: Golden ratio mathspace.co

The Fibonacci sequence Given the following sequence, can you guess what number comes next? What is the pattern? 0, 1, 1, 2, 3, 5, 8, 13, 21, … The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, often associated with many natural patterns and structures. Phi Recall that the golden ratio, represented by the Greek letter Phi (Φ), is approximated to 1.618 033 988 75. Compute the ratio between each consecutive pair of Fibonacci numbers by dividing the larger number by the smaller one. 8÷5=⬚

13 ÷ 8 = ⬚

21 ÷ 13 ≈ ⬚

⬚ ÷ 21 ≈ ⬚ ⬚÷⬚≈⬚

What do you notice about the ratios as the Fibonacci numbers increase?

Now, we will draw the Fibonacci spiral. Begin with a sheet of grid paper. Draw a 1 × 1 square

Draw a 1 × 1 square adjacent to the side length of the ﬁrst

Draw a 2 × 2 square adjacent to the side lengths of the previous two squares

Mathspace Victorian Curriculum – Year 8 mathspace.co

Draw a 3 × 3 square adjacent to the side lengths of the previous two squares

Continue drawing squares with side lengths that follow the Fibonacci sequence of numbers so that they are adjacent to the side lengths of the previous squares. After drawing as many squares as you can ﬁt, sketch a spiral that connects the corners of these boxes. Once you have made your golden spiral, read about the golden ratio in design.

Investigate Once you have completed the activity, answer these questions. 1.

How does the golden ratio relate to the Fibonacci sequence?

Research and discuss how the golden ratio may be used in design with a partner.

Why are the ratios of successive numbers in the Fibonacci sequence close to, but never truly equal to, the golden ratio?

Discussion 1.

Research artwork that involves the golden ratio and share with a partner. How do artists use the golden ratio or the golden spiral in their art?

Where in nature can you think of the Fibonacci golden spiral appearing?

Investigation: Golden ratio mathspace.co

1.08 Index laws After this lesson, you will be able to… • apply index laws to simplify expressions.

Product of powers rule Index notation, also called exponential form allows us to write repeated multiplication in a simpler way. Recall that we say 103 has a base of 10 and an exponent or power of 3. Exponent/power 3

10 = 10 × 10 × 10 Multiplied 3 times Base

Consider the product 25 × 23. Notice that the terms share like bases of 2. Let’s think about what this would look like if we expanded the expression: 25 × 23 =2×2×2×2×2 × 2×2×2 We can see that there are eight 2’s being multiplied together, and notice that 8 is the sum of the powers in the original expression. We can write this in exponential form as 28, where 2 is the base and 8 is the power. So, in our example above,

25 × 23 = 25 + 3 = 28

We can avoid having to write each expression in expanded form by using the product of powers rule or the multiplication law. It states that when multiplying two powers with the same base, we add the powers. For any base number a, and any numbers m and n as powers, am × an = am + n. That is, when multiplying terms with a common base: • Keep the same base • Find the sum of the powers

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 1 Simplify 34 × 310

Create a strategy

Apply the idea 34 × 310 = 34 + 10

We can use the multiplication law: am × an = am + n.

Add the powers Simplify

Reflect and check The instruction to simplify is different to if we were asked to evaluate. Simplify means leave in index notation and not work out the actual value.

Idea summary The product of powers rule states that for any base number a, and any numbers m and n as powers, a m × an = am + n

Division of powers rule The method for dividing like bases raised to a power involves subtracting the powers, rather than adding them as in the case of multiplication. Say we wanted to ﬁnd the value of 27 ÷ 23. By evaluating each term in the quotient separately we would have: 27 ÷ 23 = 128 ÷ 8 = 16 = 24 Another way to look at this is that we could write out each term in expanded form and simplify it. Simplify using

We can avoid having to write each expression in expanded form by using the division of powers rule or the division law:

where a is any base number, and any numbers m and n as powers. That is, when dividing terms with a common base: • Keep the same base • Find the difference in the powers

1.08 Index laws mathspace.co

Example 2 Simplify 3730 ÷ 3718

Create a strategy

Apply the idea 3730 ÷ 3718 = 3730 − 18

We can use the division law:

= 37

Subtract the powers Simplify

Idea summary The division of powers rule states that for any base number a, and any numbers m and n as powers, am = am – n, an ≠ 0 an

Power of a power rule Let’s consider how we might rewrite expressions that have a power raised to another power. Consider the expression (52)3. What is the resulting power of base 5? To ﬁnd out, have a look at the expanded form of the expression: (52)3 = (52) × (52) × (52) = (5 × 5) × (5 × 5) × (5 × 5) =5×5×5×5×5×5 = 56 In the expanded form, we can see that we are multiplying 5 together six times. This result demonstrates the power to a power rule: (52)3 = 52 × 3 = 56 We can avoid having to write each expression in expanded form by using the power of a power rule. For any base number a, and any numbers m and n as power, (am)n = am × n That is, when simplifying a term with a power that itself has a power: • Keep the same base • Multiply the powers

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 3 Simplify (26)8

Create a strategy

Apply the idea

Apply the power of a power rule, which states that for any base number a, and any numbers m and n as powers,

(26)8 = 2(6 × 8) Apply the power of a power rule. = 248

Multiply the powers.

(am)n = am × n

Idea summary The power of a power rule states that for any base number a, and any numbers m and n as powers, (am)n = am × n

The power zero Interactive exploration Discover this concept in action online

mathspace.co

Let’s consider a whole number base, such as the number 4. When we follow the pattern of the equivalent values when 4 is raised to whole number powers, we get: 41 = 4

42 = 16,

43 = 64,

44 = 256

We can have a power of the value zero as well. Let’s consider what 40 might equal by working backwards. Since powers represent repeated multiplication, we may work backwards with division. We will divide by 4 and the power gets smaller. If 41 = 4, then:

40 = 41 ÷ 4 =4÷4 =1

We can extend the observation of the zero power to any other base. For any base number a, a0 = 1, a ≠ 0 This means that any number to the power of zero is equal to 1.

1.08 Index laws mathspace.co

Example 4 Evaluate 80

Create a strategy

Apply the idea

Use the rule a = 1 where a is the base.

a0 = 1

Write the rule

Substitute a = 8

8 =1

Idea summary For any base number a,

a0 = 1, a ≠ 0

This says that any number to the power of zero is equal to 1.

1.08 Practice questions Understanding 1

b c d

When you divide terms with the same bases you ⬚ the powers. Example: 312 ÷ 34 = 38

When you multiply the same bases you ⬚ the powers. Example: (73) (77) = 710 When you have a power to a power you ⬚ the powers. Example: (32)3 = 36 When you have a power of zero it is equal to ⬚. Example: 50 = 1

32 × 33 = (⬚ × ⬚) × (⬚ × ⬚ × ⬚) = 3⬚

b d

Consider the expression (23)2. a

b c

53 ÷ 53 = 5⬚ − ⬚ = 5⬚ = ⬚

Determine whether the expressions are equivalent to (23)2: 23 × 23

(2 × 2 × 2) × (2 × 2 × 2)

iii

(2 × 2 × 2)2

2×2×2×2×2×2×2

23 × 22

(2 × 2 × 2) × (2 × 2)

(23)2 = 23 × 2

Are the equations true or false? i

Trailblazer 1, 3, 4

Complete the expressions: a

Adventurer 1–3, 4a–d

Complete each statement: a

Explorer 1–3

(23)2 = 23 + 2

Complete the equation:

Mathspace Victorian Curriculum – Year 8 mathspace.co

(23)2 = 2⬚

Describe the expressions as positive, negative or zero: a

(−3)0

(−3)

Ex 3

Ex 4

−(−3)

5b–h, 6, 7bc, 8a–e, 9a–d, 10e–j, 11e–l, 12c–f, 13c–h, 14b–f

−37

−(−3)3

5e–j, 6, 7cd, 8c–g, 9e–h, 10g–l, 11i–p, 12, 13e–j, 14c–h

54 × 53

66 × 67

1744 × 1748

4 × 56 × 57

26 × 28 × 27 × 25

(−2)2 × 23 × (−2)4

−9 × (−2)2 × (−2)6

9 × (−10)4 × 108

Use your calculator to ﬁnd the product of the expressions: 34 × 33

92 × 95

55 × 33

6 × 52 × 53

613 ÷ 66

108 ÷ 106

1533 ÷ 1527

Simplify: a

a Ex 2

(−3)

−36

Simplify:

i 6

c 2

5a–g, 6, 7ab, 8a–c, 9a–d, 10a–f, 11a–h, 12a–c, 13a–e, 14a–c

Fluency

Ex 1

−30

38 ÷ 36

Simplify: a

Use your calculator to ﬁnd the quotient of the expressions: a

39 ÷ 34

47 ÷ 44

44 ÷ 24

48 ÷ 45

Simplify: a

(53)4

(66)3

(74)5

(95)8

(3 )

(10 )

(5 )

(47)4

(83)5

(122)5

(35)2

(8319)40

(−983)0

Evaluate: b

0.50

−100

720 + 23

7410

(−3)0

3 ×3

65 + 81

890 − (−95)0

70 + (8 × 7)0

−9 (12 × 15)0

25 × 110

(2 × 13)0

4 (2 × 13)0

1.08 Index laws mathspace.co

Complete the statements: a e

b f

119 = ⬚

5⬚ = 625

10⬚ = 1

28 = ⬚

9⬚ × 96 = 99

2⬚ = 64

9⬚ = 729

Find the value of the blank that will make the equations true. a e i

042 = ⬚ 35 × 3⬚ = 37

24 × 2⬚ = 27 12 ⬚

b f

108

(45 ) = 45

58 × 5⬚ = 512

910 × 9⬚ = 918 ⬚ 14

700

(37 ) = 37

(1111)⬚ = 1133

31⬚ × 3137 = 3156 (22⬚)8 = 2264

Find the value of the blank that will make the equations true. a e

5⬚ ÷ 55 = 57

68 ÷ 68 = 6⬚

Reasoning

b f

69 ÷ 6⬚ = 67

16, 18, 19

c g

1110 ÷ 11⬚ = 114 49 ÷ 4⬚ = 49

15, 17, 18

d h

31⬚ ÷ 3121 = 3113

15, 17, 20

Show that 64 ÷ 64 = 1 without using the division of powers law.

Show that 53 × 52 = 55 without using the product of powers law.

Explain why the expression 24 × 34 cannot be simpliﬁed using the product of powers rule.

Sam is asked to simplify the expression (43) × (42) × (45). He claims the result is 410. Is Sam’s answer correct? Justify your answer.

Jane simpliﬁes the expression Explain your reasoning.

A student simpliﬁes the expression 115 ÷ 1112 and gets 117 as the result. Explain the student’s mistake.

and gets 52 as the result. Is Jane’s solution correct?

Problem-solving 21, 22, 25

22–26

23–27

If (2m)n = 215 and m = 3, what is the value of n?

If 10m × 10n = 109, and m = 4, what is the value of n?

If 52 m ÷ 5m = 54, what is the value of m?

Sally is reading a 500 page book that she wants to finish in 5 days. She starts off by reading 2 pages on the first day, but quickly realises that she needs to read more to finish the book in time. She decides to square the number of pages she reads each day.

How many pages has she read after:

i 2 days ii Will Sally be able to ﬁnish the book in time?

Mathspace Victorian Curriculum – Year 8 mathspace.co

4 days

A square has side length 74 units. Write the area of the square in index notation.

A rectangular box has dimensions 22, 23, and 24 units. Write the volume of the box in index notation.

The number of students in a university grows according to the formula, S = 120 × 1.05t, where t is the number of years after 2015. a

How many students were there at the university in 2015?

How many students were there in 2020?

Did you know?

Real numbers also help you track how far you can drive on that tank of petrol. If your car averages 12.3 kilometres per litre, those 42.5 litres will give you approximately 522.75 kilometres of driving range. This kind of calculation is crucial for planning long trips or just knowing when you’ll need to refuel. Real numbers make it easy to manage fuel efficiency, costs, and distance, giving you a clear picture of your vehicle’s performance. From the pump to the road, real numbers keep everything in check!

1.08 Index laws mathspace.co

1 Chapter review 1

identify the irrational number from the set of numbers:

identify the rational number from the set of square roots:

Describe these decimals as terminating or recurring:

Evaluate:

8 × (−5)

3 × (−6) × 7

−5 × 3 × 4

−(−15)2

0 ÷ (−45)

(−33) ÷ 11

(−5) × 4 ÷ (−2)

(−18) × (−6) ÷ 9

(24 ÷ (−4)) ÷ (−3)

(−150) ÷ ((−5) × (−3))

Evaluate: a

8 − (−4) + 9

−3 − 2 − 5

2 − (5 − (−7))

((−10) + 3) − 7

9 + 7 × (−9)

−4 − 7 × (−5)

7 × (−6) − 5

−7 × (−5) × (−3)

−10 + (30 ÷ 5)2

−7 + 16 ÷ 23

(−63 × 6) ÷ (−9)

(−90) ÷ (3 × 5)

18 × (30 ÷ (12 + 3))

((98 − (10 + 72)) ÷ 4) + 25 × 8

m ((−39 + 9) ÷ (−6)) o 6

Evaluate each expression. Round your answer to three decimal places where necessary. a

8.5 + (−4.25) − 9.25

−3.15 − 5.6 + 2.25

2.1 − (7.35 − (−5.5))

((−10.75) − 7.85 + 3.65)

9.2 + (−9.1) × 7.3

−4.15 − (−5.3) × 7.85

1200.7 × (−0.06) − 5.55

−7.5 × (−3.3) × (−5.6)

−10.25 + (5.3 ÷ 30.6)2

−2007.5 + 23 ÷ 1600.3

(−0.06 × 6.35) ÷ (−9.75)

(−900.05) ÷ (5.25 × 3.75)

m ((−48.95 + 9.05) ÷ (−6.35))2

1.85 × ((1.25 + 0.35) ÷ 3.05)

((9.85 + (3.05 + 1.05)) ÷ 0.45) + 0.85

(−150 + (11 × 7) × 3) + 20

(5.45 − (0.75 × (−1.15)) ÷ 0.35) + 2.05

Mathspace Victorian Curriculum – Year 8 mathspace.co

Evaluate: a

Identify the numbers as rational or irrational: a

e i 9

−0.9

2.718 281 …

Evaluate and round to two decimal places when applicable. a

Consider the irrational numbers a

Plot the irrational numbers on the number line: 0

b 11

3.141 59 …

Arrange the irrational numbers in ascending order.

Write these fractions as recurring decimals: a

i 12

Write each decimal as a fully simpliﬁed fraction: a

0.5

0.04

0.123

3.7

8.47

2.0625

6.375

103.205

Write these mixed numbers as recurring decimals: a

Chapter 1 review mathspace.co

Write the expressions in simplest index form. a

77 × 78

6 × 35 × 36

415 ÷ 47

99 ÷ 96

i 15

Evaluate: a

0.70

(−850)0

80 + (9 × 8)0

36 × 120

(3 × 14)0

5 (3 × 14)0

(120)7

m (73)2

44 × 40

990 − (−100)0

h (34)2

(43)4

Find two integers that have a product of −56 and a sum of −1.

Gualberto invests $1500 a month in his stock portfolio. He also sells stocks worth $300 a month. a

How much, in total, does Gualberto invest in his stock portfolio in one full year?

How much, in total, does Gualberto sell from his stock portfolio in one full year?

Find the total change in his stock portfolio over the year. An increase will be represented by a positive number and a decrease will be represented by a negative number.

Wei and his two friends set up a stand to sell homemade dumplings during a local festival. They make a total of $1080 from the sales and decide to share it evenly. In addition to that, Wei makes an extra amount by selling 9 jars of his special sauce at $45 each. If Wei needs to pay a stall fee of $93, what is the total amount he earns?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Maria and her group of 11 friends are having a party and want to order tacos. They estimate that for every three people, they will need two orders of tacos. Each order of tacos costs $42 and there is a $24 delivery fee. If they split the total cost evenly between each person, how much will Maria need to pay?

At the beginning of the week, Takashi had $3923 in his bank account. On Monday, he withdrew $129. On Wednesday, he deposited $33, and on Thursday, he deposited three times as much as he deposited on Wednesday. Finally, he withdrew $75 on Friday. Find Takashi’s ﬁnal bank balance at the end of the week.

A square has an area of 29 square metres. Calculate the side length of the square. Is the side length rational or irrational?

A square patio has a side length of

metres. Calculate the area of the patio in square

metres. Is the resulting decimal terminating or recurring?

A bottle is ﬁlled with litres of juice. If litres of juice are consumed, what is the remaining amount of juice in the bottle? Write your solution as both a fraction and a recurring decimal.

If (3p)q = 318 and p = 2, what is the value of q?

The number of trees in a forest increases every year according to the formula, T = 200 × 1.07y, where y is the number of years after 2020.

How many trees were there in the forest in 2020?

How many trees will there be in 2025?

If the number of trees continues to increase at the same rate, how many trees will there be in the forest in 2030?

A cuboid box has dimensions 32, 33, and 34 units. Write the volume of the box in index notation. Chapter 1 review mathspace.co

Big ideas Percentages play an essential role in analysing and solving real-world problems.

Curriculum Outcomes VC2M8N05 Solve problems involving the use of percentages, including percentage increases and decreases and percentage error, with and without digital tools VC2M8N06 Use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts involving profit and loss; formulate problems, choosing efficient mental and written calculation strategies and using digital tools where appropriate; interpret and communicate solutions in terms of the context, reviewing the appropriateness of the model VC2M8A03 Use mathematical modelling to solve applied problems involving linear relations, including financial contexts involving profit and loss; formulate problems with linear functions, and choose a representation; interpret and communicate solutions in terms of the context, and review the appropriateness of the model

Percentages Chapter outline 2.01 2.02 2.03 2.04 2.05

Percentage, decimal and fraction conversion Find the whole Percentage increase and decrease Investigation: Percentages Profit and loss Consumer percentages Chapter 2 review

66 73 81 89 91 97 106

Let's watch some sport! If a stadium holds 20,000 people and it's 75% full, that means there are 15,000 people cheering. Aussie Aussie Aussie!

2.01 Percentage, decimal and fraction conversion After this lesson, you will be able to… • convert from a given fraction, decimal or percentage to another form. • compare and order rational numbers.

Convert between percentages and fractions We can convert between fractions and percentages by remembering what percentage means. Percentage means ‘per hundred’ and any percentage can be written as a fraction with the denominator 100. To convert any fraction to a percentage, we can do the following: 1. Change the denominator of the fraction to 100 if possible and write as a percent. For example: 2. If the denominator isn't easy to change to 100, such as , then convert the fraction to a decimal first by dividing the numerator by the denominator. Then multiply by 100 and attach the % symbol. For example,

= 3 divided by 7 = 0.4286 = 42.86%

To convert any percentage to a fraction: 1. Write the percentage as a fraction with a denominator of 100 and remove the % symbol. 2. If there is a decimal within the fraction like in

, then multiply both the numerator and

denominator by a power of 10 so that the numerator becomes an integer, like

3. Simplify the fraction.

Example 1 Write 55% as a fraction in its simplest form.

Create a strategy To write a percentage as a fraction, write it as a fraction with a denominator of 100 and simplify.

Apply the idea Write as a fraction Divide the numerator and denominator by 5

Mathspace Victorian Curriculum – Year 8 mathspace.co

Simplify the fraction

Example 2 Express the fraction

as a percentage.

Create a strategy To write a proper fraction as a percentage, we need to make the denominator equal to 100.

Apply the idea We can achieve this by multiplying both the numerator and denominator of the given fraction by the same number. In this case, we multiply by 4 to get: Multiply both numerator and denominator by 4

Evaluate

Write as a percentage

Example 3 Write

as a percentage.

Create a strategy

Apply the idea

To write a fraction as a percentage, convert to a decimal then multiply by 100.

First, simplify the fraction to its simplest form: 21 3 = 35 5 = 0.6

Remove the common factor of 7 Convert to a decimal

= 0.6 × 100% Multiply by 100% = 60%

Rewrite as a percentage

Example 4 Write

as a percentage.

Create a strategy To write a mixed number as a percentage, ﬁrst write it as an improper fraction and then make the denominator 100.

Apply the idea Write as an improper fraction

Change the denominator to 100

Write as a percentage

2.01 Percentage, decimal and fraction conversion mathspace.co

Idea summary To convert any fraction to a percentage, we can do the following: 1. Change the denominator of the fraction to 100 if possible and write as a percent. 7 14 = = 14% For example: 50 100 3 2. If the denominator isn't easy to change to 100, such as , then convert the fraction to a 7 decimal first by dividing the numerator by the denominator. Then multiply by 100 and attach the % symbol. 3 For example, = 3 divided by 7 = 0.4286 = 42.86% 7 To convert any percentage to a fraction: 1. Write the percentage as a fraction with a denominator of 100 and remove the % symbol. 2.5 , then multiply both the numerator and 2. If there is a decimal within the fraction like in 100 25 . denominator by a power of 10 so that the numerator becomes an integer, like 1000 3. Simplify the fraction.

Convert between percentages and decimals We can convert between decimals and percentages using the idea of the hundredths place value. Percent means ‘per hundred’ or divide by 100. We can convert any percentage to a decimal by dividing the percentage by 100 and removing the % sign. For example, 83% =

which is read as “83 hundredths”. This is 0.83 when written as a decimal.

To convert from a decimal to a percentage, multiply the decimal by 100 and add a % sign. For example, 0.567 = 0.567 × 100% = 56.7%

Example 5 Write 54% as a decimal.

Create a strategy To write a percentage as a decimal divide by 100.

Reﬂect and check

Apply the idea Divide by 100 Write as a decimal

Mathspace Victorian Curriculum – Year 8 mathspace.co

We wrote the percentage as a decimal representing the number of hundredths indicated by the percentage. Notice that converting the percentages to decimals had the same effect as decreasing the place value of the digits in the percentage by two places, then removing the % symbol. This is equivalent to dividing by 100 and removing the % symbol.

Example 6 Write 0.09 as a percentage.

Create a strategy To write a decimal as a percentage, multiply by 100 and add the percentage sign.

Apply the idea 0.09 = 0.09 × 100% = 9%

Multiply by 100 and add the percentage sign Evaluate

Example 7 Write 6.3 as a percentage.

Create a strategy To write a decimal as a percentage, multiply by 100 and add the percentage sign.

Apply the idea 6.3 = 6.3 × 100% = 630%

Multiply by 100 and add the percentage sign Evaluate

Idea summary To convert any percentage to a decimal, divide the percentage by 100 and remove the % sign. 83 × 0.83 For example, 83% = 100 To convert from a decimal to a percentage, multiply the decimal by 100 and add a % sign. For example, 0.567 = 0.567 × 100% = 56.7%

Compare fractions, decimals, percentages To compare fractions, decimals and percentages, they need to be in the same form. This means we must convert each number so that they are all of one type. Arranging numbers in ascending order means smallest to largest whereas descending order means largest to smallest.

Interactive exploration Discover this concept in action online

mathspace.co

2.01 Percentage, decimal and fraction conversion mathspace.co

Example 8 In this question, we will be working with the numbers , 60% and 0.3. a Convert

into a percentage. Do not round your answer.

Create a strategy Multiply

by 100 and add a % sign.

Apply the idea 1 1 = × 100% 4 4 1 100 = × % 4 1 100 = % 4 = 25%

Multiply by 100 and add the percentage sign Write as a fraction Multiply Evaluate the division

b Convert 0.3 into a percentage.

Create a strategy Multiply the decimal by 100 and add a % sign.

Apply the idea 0.3 = 0.3 × 100% = 30%

Multiply by 100 and add the percentage sign Evaluate

c Which of the following arranges , 60% and 0.3 from largest to smallest? A 60%, , 0.3

60%, 0.3,

, 60%, 0.3

0.3, , 60%

Create a strategy

Apply the idea

Compare the percentages.

The percentages in order are: 60%, 30%, 25% So the original numbers in order are: 60%, 0.3, The answer is option B.

Idea summary Ascending order means smallest to largest. Descending order means largest to smallest. To compare decimals, percentages and fractions, convert them to the same type then compare.

Mathspace Victorian Curriculum – Year 8 mathspace.co

2.01 Practice questions Understanding

Fill in the blanks to make the statements true: 29% is equivalent to 29 out of ⬚.

Ex 4

How many squares are shaded?

What percentage of the grid is shaded?

What fraction of the grid does this percentage represent?

4c–f, 5, 6a–f, 7–11, 13b, 16 4e–h, 5, 6c–h, 7–11, 13c, 15, 17

68%

13%

45%

275%

37%

94%

450%

2.5%

For each of these fractions: i

Write an equivalent fraction with a denominator of 100.

Write it as a percentage. b

Write these fractions as percentages: a

Write these fractions as percentages: b

Which number is the largest in each pair? a

Ex 5

4a–d, 5, 6a–d, 7–10, 12, 13a, 14, 16

Write these percentages as simpliﬁed fractions:

a 8

53% is equivalent to ⬚ out of 100.

a Ex 3

Use the grid to answer the following:

Fluency

Trailblazer 3

What does the % symbol mean when placed after a number?

Ex 2

Adventurer 1–3

Ex 1

Explorer 1–3

Write these percentages as a decimal: a

34%

60%

270%

303%

76%

500%

182%

6897%

2.01 Percentage, decimal and fraction conversion mathspace.co

10 Ex 6, 7

Write these decimals as percentages: a

0.24

1.71

13.04

1.1

0.857

0.041

Convert between percentages, fractions and decimals to complete the table: Fraction

Decimal

Percentage

1.21

Write these percentages as both a fraction and a decimal: a

Ex 8

80%

For these groups of numbers: i

Rewrite the group as percentages.

Order the original group from smallest to largest.

Sarah attended a workshop for 125 minutes but stepped out for a quarter of an hour phone call. What percentage of the workshop did she miss due to the phone call?

In a basketball game, Michael scored 28 points out of his team’s total of 112 points. Find the percentage of points scored by Michael in this game.

In a book collection, 12 out of the 40 books were sold at a garage sale. What percentage of the original number of books remain in the collection?

A town has 13 675 registered voters, and 8752 of them are registered with the Blue Party. What percentage of registered voters in the city are aﬃliated with the Blue Party?

Reasoning

18, 19

18–20

19–21

When we convert a decimal to a percentage we multiply by 100 and add the percentage sign. Explain why this changes the format of the number but not the size of the number.

Consider the fraction . a

Is there an equivalent fraction that has a denominator of 100, and a whole number as its numerator?

What is the closest whole percent to ?

Explain why converting a percentage to a decimal is the same as dividing by 100 and removing the % sign.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Suppose that p, q and r represent different digits from 1 to 9.

Convert 0.pqr to a percentage.

Convert 0.pqr to a fraction.

Convert p.qr% to a fraction.

Convert p.qr% to a decimal.

Convert

to a percentage.

Problem-solving 22, 23, 25

22, 24, 25, 27

to a percentage.

24–27

A student scores 24 out of 30 on a test. Determine the student’s score as a percentage, decimal, and fraction.

A glass is ﬁlled with water up to of its capacity. What is the percentage and decimal representation of the ﬁlled portion of the glass?

Opie’s soccer team has won 13 of its 15 games so far this season. There are 20 games in a season. a

What is the largest percentage of wins Opie’s team could achieve over the whole season?

What is the smallest percentage of wins Opie’s team could achieve over the whole season?

Josie scored 90% on a test. If the test was out of 40 marks, how many marks did she get?

Louis ate

of a birthday cake and then Beth ate 0.23 of the cake. What percentage of the

cake is left? 27

Alex is considering three different investments. • Investment A: interest rate • Investment B: interest rate 0.13 • Investment C: interest rate 15% Which investment should he choose, and why?

2.02

Find the whole

After this lesson, you will be able to… • use fractions to find the whole. • use the unitary method to find the whole.

The unitary method Percentages are helpful because they tell us how much of the whole we currently have. However, there could be situations when we know the percentage, but we do not know “the whole” or the original amount. 2.02 Find the whole mathspace.co

For example, 39 is 30% of some number. What is this number? An approach to solving this would be to ﬁnd 1% of the number ﬁrst. We know that 30% of the number is 39, so if we divide 39 by 30, we now know 1% of the number.

From here, we can ﬁnd the whole or any other percentage of the whole. The whole is 100% of the original number, so we can multiply 1% by 100:

To ﬁnd 4% of the original number, we multiply 1% by 4:

This is known as the unitary method.

Example 1 9% of a number is 72. a Complete the statement: Because 9% of the number is ⬚, we know that 1% of the number is ⬚.

Create a strategy

Use the ﬁrst step of the unitary method.

Apply the idea The question tells us that 9% of the unknown number is 72, so this is the number for the ﬁrst box. To ﬁnd 1%, we divide 72 by 9 to get 72 ∏ 9 = 8. So we get: Because 9% of the number is 72, we know that 1% of the number is 8. b What is 5% of the number?

Create a strategy To ﬁnd 5% of the number, we can use the answer we found in part (a) and multiply it by 5.

Apply the idea We know from part (a) that the value of 1% of the number is 8. To ﬁnd 5% of it, we have: 5% of the number = 8 × 5 Multiply 8 by 5 = 40 Evaluate Therefore, 5% of the number is 40.

Mathspace Victorian Curriculum – Year 8 mathspace.co

c What is the number?

Create a strategy To find 100% of the number, we multiply our result from part (a) by 100. If 1% = 8, then 100% = number.

Apply the idea Number = 8 × 100

Multiply 1% by 100 to get the whole

= 800

Evaluate

Idea summary We can find the whole by using the unitary method: 1. Divide the given amount by the percentage number. This finds 1% of the whole. 2. Multiply the result by the percentage you want to find. • Multiplying by 100 will give us the whole amount. • Multiplying by another number will give us a different percentage of the whole.

Finding the whole To find the whole amount, one method is to determine what number we can multiply the percent by to get 100%. For example, if 14 is 50% of a number, we can multiply by 2 to find 100% of the number. 50% of a number = 14 50% of a number × 2 = 14 × 2 100% of a number = 28 If the percent is not a factor of 100, we can first divide by a number that would result in a factor of 100, then multiply to get 100%. For example, if 150% of a number is 27, we can divide by 3 to get 50%, then multiply by 2 to get 100%. 150% of a number = 27 150% of a number ∏ 3 = 27 ∏ 3 50% of a number = 9 50% of a number × 2 = 9 × 2 100% of a number = 18 Another method is to use what we know about finding percentages of a quantity to set up an equation. For example, “20% of a number is 7” can be translated into an equation, and we can use algebra to solve the equation.

2.02 Find the whole mathspace.co

Example 2 20% of a number is equal to 7. What is this number?

Create a strategy Let x be the number and form an equation.

Apply the idea 20% × x = 7 20 1 20% = = 100 5 1 × x =7 5 1 5× × x =7×5 5 x = 35

Write the equation Replace the percentage with a fraction Rewrite 20% as

1 5

Multiply both sides by 5 Evaluate

The number is 35.

Reﬂect and check We can check our answer by thinking that: 20% of the number = 7

Given

20% of the number × 5 = 7 × 5 Multiply both sides by 5 100% of the number = 35 Evaluate This tells us that 35 is the whole.

Example 3 Find the whole amount if

1 of the amount is 30. 14

Create a strategy If we know 1 fourteenth, then 14 fourteenths, is a whole. If 1 fourteenth is 30, then 14 fourteenths will be 14 × 30.

Apply the idea 11 x = 30 14 x = 30 14 x = 30 × 14 x = 30 × 14 x = 420 x = 420

Mathspace Victorian Curriculum – Year 8 mathspace.co

Write the equation Multiply both sides by 14 Evaluate

Example 4 150% of a quantity is 135. a What is 10% of the quantity?

Create a strategy If we know 150%, we can find 10% by dividing the value by 15 because

150% = 10%. 15

Apply the idea We divide 150% by 15 to get 10% of the quantity: 10% of the quantity = = 9

Divide 135 by 15 Evaluate

Reﬂect and check We can also find the whole first by letting x be the quantity and forming an equation.

Write the equation Write the percentage as a decimal

Divide both sides by 1.50

Evaluate

Now we know the value of the quantity, we can find 10% of it: 10% × 90 = 9 b What is the whole quantity?

Create a strategy We can use the answer from part (a) and consider the percentage as 100%.

Apply the idea We need to make the percentage, 10%, as a whole. 10% of the quantity = 9 10% of the quantity × 10 = 9 × 10 100% of the quantity = 90

Use the answer from part (a) Multiply both sides by 10 Evaluate

So the whole quantity is 90.

2.02 Find the whole mathspace.co

Idea summary If the given percentage is a factor of 100%, we can ﬁnd the whole by multiplying by the value that would make 100%. If the given percentage is not a factor of 100%, we can ﬁrst divide by a number that would result in a factor of 100, then multiply to get 100%. Or, we can translate the problem into an equation, then use algebra to solve for the original number.

2.02 Practice questions Understanding 1

Explorer 1, 2a, 3, 4ac

Adventurer 1, 2a, 4ab

Trailblazer 1, 2b, 4bd

Choose the statement that is the same as 500% of a quantity: A

Five hundredths of that quantity

Five hundred times that quantity

Five times that quantity

Five hundred more than that quantity

Complete the table of conversions between fractions and percentages. Fully simplify each fraction. a

Fraction

Percentage

Fraction

Percentage

50%

4% 10%

For each percentage, ﬁll in the value we can multiply by to get 100%: a

5% × ⬚ = 100%

10% × ⬚ = 100%

25% × ⬚ = 100%

2% × ⬚ = 100%

What number do you divide by to make the factor of 100% in the bracket?

What number do you then multiply by to make 100%?

70% (10%)

For each percentage:

Fluency 5

75% (25%)

5a, 6a–d, 7a–d, 8a–d, 9a–e, 10ab, 11, 12, 13ab

1% of a quantity is 12. i

What is 10% of the quantity?

What is 100% of the quantity?

10% of a quantity is 50. i

What is 1% of the quantity?

What is 5% of the quantity?

iii

What is 100% of the quantity?

Mathspace Victorian Curriculum – Year 8 mathspace.co

120% (20%)

5b, 6c–f, 7c–f, 8c–f, 9b–f, 10–12, 13a–f

18% (2%)

5c, 6e–h, 7e–h, 8d–h, 9c–h, 10, 12, 13c–h

Ex 1

Ex 2

Ex 3

Ex 4

5% of a quantity is 30. i

What is 1% of the quantity?

What is 10% of the quantity?

iii

What is 100% of the quantity?

For each number: i

Find 1% of the number.

Find the whole number.

6% of the number is 36.

9% of a number is 63.

60% of a number is 480.

47% of a number is 188.

4% of a number is 236.

15% of a number is 7500.

12% of a number is 576.

8% of a number is 230.

Find the whole quantity.

For each number: i

Find 1% of the quantity.

5% of a quantity is $320.

11% of a quantity is $16.50.

30% of a quantity is 2700 mL.

54% of a quantity is 6210 mL.

14% of a quantity is 7280 g.

22% of a quantity is 88 000 g

34% of a quantity is 918 L.

78% of a quantity is 10 959 cm.

50% of the number is equal to 11.

Find the whole number if: a

10% of the number is equal to 9.

20% of the number is equal to 17.

5% of the number is equal to 12.

600% of the number is equal to 54.

25% of the number is equal to 12.

10% of the number is equal to 6.5.

300% of the number is equal to 36.

Find the whole number if: a

of the number is equal to 6.

of the number is equal to 20.

of the number is equal to 8.

of the number is equal to 14.

of the number is equal to 11.

of the number is equal to 82.

of the number is equal to 9.

of the number is equal to 10.

Find the whole quantity if: a

10% of the quantity is equal to 840 g.

50% of the quantity is equal to $0.75.

25% of the quantity is equal to $2.50.

200% of the quantity is equal to 120 mL.

For each quantity: i

Find 10% of the quantity.

Find the whole quantity.

120% of a quantity is 156.

240% of a quantity is 1464.

80% of a quantity is $160.

320% of a quantity is $13 280.

2.02 Find the whole mathspace.co

For each quantity: i

Find 25% of the quantity.

Find the whole quantity.

125% of a quantity is 30.

50% of a quantity is 85.

150% of a quantity is 75 000.

250% of a quantity is 1500.

For each of the following, ﬁnd the whole quantity: a

14% of a quantity is 637.

26% of a quantity is 10 400.

c e

32% of a quantity is $10 240.

120% of a quantity is $87.30.

80% of a quantity is 840 mL.

57% of a quantity is 4651.2 g.

64% of a quantity is $10 400.

92% of a quantity is 4738 m.

Reasoning 14

1% of the quantity

8100 × 27

100% of the quantity

Opie shares $10 000 between Jasper and Josie. He gives Jasper p% of the money. a

What percentage of the money does he give to Josie?

Write an expression for the amount of money Jasper receives.

Write an expression for the amount of money Josie receives.

Given that the money that Opie shared with Jasper and Josie was q% of an amount that he won, write an expression for the total amount.

Problem-solving 17–21, 24–25 17

15, 16

p% of a quantity is q kg. Write an expression for: a

14, 15

If $8100 represents 27% of an amount of money. Which of the following calculations would give the whole amount? A

14, 15

18, 21–24, 26, 28

19, 22–23, 25–30

Melted snow accounts for 4800 ML of a town’s water supply in a year. This represents 12% of its total supply. a

Find 1% of the yearly water supply.

Calculate the town’s whole yearly water supply.

A rock sample was taken from a dig site. 3.7 kg of the sample was pure gold. The team determined that the sample contained 25% of the total gold found at the site. Calculate how many kilograms of gold were at the dig site overall.

Kate spends 40% of her weekly income on rent. If her rent is $480 each week, ﬁnd her weekly income.

A shop has sold 150 units of a product, representing 25% of its stock. How many units were in stock originally?

Jack has completed 40 pages of a book, which is 80% of the book. How many pages are in the whole book?

Mathspace Victorian Curriculum – Year 8 mathspace.co

A total of 45 000 new farmers entered the workforce last year. This is 9% of all the farmers currently employed in the workforce. Find the number of farmers that are in the workforce altogether.

In a school, 80% of the students participate in extracurricular activities, and there are 960 students who take part in these activities. How many students are there in the school?

In a store sale, 600 electronic items were sold, representing 25% of the total items sold. Determine the total number of items sold during the sale.

In a lottery, only 4% of tickets won prizes. If there were ﬁve prizes, how many tickets were sold?

Tara knows that there are 423 radio towers in her city, and that this represents 9% of the total number of towers in the city. Find the total number of towers in Tara’s city.

Quentin knows that his house is 1.84 km along the road from its start, and that the distance from the start of the road to his house represents 23% of the road’s total length. Find the road’s total length.

A factory produces car seats for children. Last year 2400 car seats were identiﬁed as faulty by quality control experts and removed from the production line before sale. These faulty seats represented 3% of the total number of car seats produced by the factory that year. Find the total number of car seats produced by the factory.

On a recent survey, 22% of flight attendants for an airline said they prefer to have a long haul flight and 33% of flight attendants said they prefer to work multiple short haul flights. All other flight attendants said they prefer a return medium length flight. If 108 flight attendants prefer a return medium length flight, how many flight attendants are employed by the airline?

At a high school athletics carnival, the following numbers of students were present. The percentage stated represents the percentage of the year group present. • Year 7: 170 (85%) • Year 8: 152 (80%) • Year 9: 165 (75%)

• Year 10: 200 (80%) • Year 11: 102 (85%) • Year 12: 81 (90%)

How many students attend the school?

2.03 Percentage increase and decrease After this lesson, you will be able to… • apply the two-step method to find the result of a percentage increase or decrease. • apply percentage increase and decrease in practical problems and real-life scenarios. • calculate the percentage change, both increase and decrease, by comparing it to the original amount.

2.03 Percentage increase and decrease mathspace.co

Percentage increase or decrease using two-steps Percentage increase and percentage decrease are used to describe how much a value has changed in comparison to its original value. To ﬁnd the result of a percentage increase: 1. Find the percentage of the original amount. 2. Add the result to the original amount. For example, to increase 40 by 25%, we need to ﬁnd 25% of 40 and add it to the original. 25% of 40 = 10 10 + 40 = 50

Find the percentage of the original Add the result to the original

To ﬁnd the result of a percentage decrease: 1. Find the percentage of the original amount. 2. Subtract the result from the original amount. So if we wanted to decrease 40 by 25%, the working would be: 25% of 40 = 10

Find the percentage of the original

40 − 10 = 30

Subtract the result from the original

Example 1 Bob wants to decrease 110 by 60%, so he calculates 110 − (60% × 110). What was his result?

Create a strategy Convert the percentage to a fraction and calculate.

Apply the idea

Convert the percent

Evaluate the brackets

Evaluate the subtraction

Reﬂect and check Decreasing an amount by 60%, is the same as finding 40% of the amount. Check by finding 40% of 110.

40% x 110 = 0.4 x 110

Bob's calculation was correct.

Mathspace Victorian Curriculum – Year 8 mathspace.co

= 44

Idea summary To find the result of a percentage increase, we find the percentage of the original amount and add it to the original amount. To find the result of a percentage decrease, we find the percentage of the original amount and subtract it from the original amount.

Percentage increase or decrease using a single percentage Interactive exploration Discover this concept in action online

mathspace.co

Recall that the original amount is equal to 100% of itself. This fact is particularly useful if we want to ﬁnd the result of a percentage increase or decrease directly. To increase 40 by 25%, we add 25% of 40 to 100% of 40. This is the same as ﬁnding 125% of 40: 25% of 40 + 100% of 40 = 125% of 40

Add the percentages

= 1.25 × 40

Change percentage to decimal and multiply

= 50

Evaluate

To decrease 40 by 25%, we take 25% away from the original 100%, leaving only 75% of 40: 100% of 40 − 25% of 40 = 75% of 40

Subtract the percentages

= 0.75 × 40

Change percentage to decimal and multiply

= 30

Evaluate

Example 2 By what percentage should a quantity be multiplied by in order to increase it by 38%? A 138%

100.38%

62%

38%

Create a strategy

Apply the idea

To increase a quantity by 38%, we need to add 38% to the original 100%:

So, the percentage by which the quantity should be multiplied to increase it by 38% is 138%, which is option A.

100% + 38% = 138%

Example 3 Owen wants to decrease a quantity by 88%. By what percentage should they multiply the quantity? A 12%

99.12%

188%

88%

2.03 Percentage increase and decrease mathspace.co

Create a strategy

Apply the idea

The quantity starts at 100% of its value. To decrease this by 88%, we want to multiply it by a value that is 88% less than 100%.

To calculate 88% less than 100%, subtract 88% from 100%. So, the answer is A: 12%

Idea summary To increase a quantity by a percentage, ﬁrst add the percentage to 100% and then calculate the percentage of the quantity. To decrease a quantity by a percentage, ﬁrst subtract the percentage from 100% and then calculate the percentage of the quantity.

Calculate percentage change If we want to know the percentage by which an amount has increased or decreased, we can calculate the percentage change by comparing it to the original amount. If the new amount is larger than the original amount, the original amount has increased. To ﬁnd the percentage increase, we compare the amount of increase to the original amount using the formula:

For example, if a store sold 100 items last month and 120 items this month, the increase in items is 120 − 100 = 20 items. The percentage increase would be

If the new amount is smaller than the original amount, the original amount has decreased. To ﬁnd the percentage decrease, we compare the amount of decrease to the original amount with the formula:

For example, if a store sold 200 items last month and 120 items this month, the decrease in items is 200 − 120 = 80 items. The percentage decrease would be

Remember that when calculating percentage increase or decrease, the original amount is always the denominator.

Example 4 Find the percentage decrease needed to reduce 180 to get 117.

Mathspace Victorian Curriculum – Year 8 mathspace.co

a How much do we need to subtract from 180 to get 117?

Create a strategy We can calculate the amount by which we need to decrease 180 by ﬁnding the difference between 180 and 117.

Apply the idea Amount of decrease = 180 – 117

Subtract 117 from 180

= 63

Evaluate the subtraction

b What is the decrease as a percentage of 180?

Create a strategy We know from part (a) that the decrease amount is 63. We want to ﬁnd this as a percentage of 180.

Apply the idea Write 63 as a fraction of 180 Evaluate the fraction using division Convert the decimal to percentage

Idea summary To ﬁnd the increase as a percentage of the original quantity, write it as a fraction and multiply by 100%: Percentage increase =

Increase × 100% Original quantity

To ﬁnd the decrease as a percentage of the original quantity, write it as a fraction and multiply by 100%: Percentage decrease =

Decrease × 100% Original quantity

2.03 Practice questions Understanding 1

Adventurer 1–3

Trailblazer 1–3

Which of the following multipliers represent a percentage increase? Select all that apply. A

Explorer 1–3

103%

65%

1.09

0.99

Which of the following multipliers represent a percentage decrease? Select all that apply. A

84%

102%

1.4

0.79

2.03 Percentage increase and decrease mathspace.co

Here is a grid of 100 blue squares. Each small square represents 1% of the total: a

Choose the image that represents 3% more than the original grid: A

How many squares were added to the original grid to increase it by 3%?

Fill in the blanks to complete an expression that represents the 3% increase in squares: ⬚% + ⬚% = 103% = 1.03

Choose the image that represents 6% less than the original grid: A

Ex 2

How many squares were subtracted from the original grid to decrease it by 6%?

Fill in the blanks to complete an expression that represents the 6% decrease in squares: ⬚% − ⬚% = 94% = 0.94 4, 5, 6a–d, 7, 8a–c, 9abef, 10, 11a–d, 12, 15, 16a–d, 17, 18

4, 5, 6c–f, 7, 8b–d, 9c–f, 10, 11b–e, 13, 14, 16c–f, 18, 19

6e–h, 7, 8d–f, 9d–h, 10, 11c–f, 13, 14, 16c–h, 19, 20

Hiro wants to decrease 120 by 40%. a

Calculate 40% of 120.

Should Hiro add or subtract the amount from part (a) from 120? Explain your answer.

Decrease 120 by 40%.

Lucy wants to increase 70 by 60%. a

Calculate 60% of 70.

Should Lucy add or subtract the amount from part (a) from 70? Explain your answer.

Increase 70 by 60%.

Increase: a

80 by 5%

120 by 10%

72 by 40%

320 by 25%

$140 by 9%

$30 000 by 20%

80 km by 15%

400 mL by 35%

What single percentage should a quantity be multiplied by in order to increase the quantity by: a

Fluency

Ex 1

17%

Mathspace Victorian Curriculum – Year 8 mathspace.co

9.4%

212%

Ex 3

Evaluate the following by multiplying the original number by a single percentage: a

Increase 80 by 60%

Increase 400 by 45%

Increase 900 by 8%

Increase 80 by 40%

Increase 1200 by 45%

Increase 1200 by 8%

Decrease: a

70 by 8%

260 by 10%

250 by 20%

96 by 60%

$480 by 18%

$55 000 by 45%

200 km by 22%

650 mL by 30%

What single percentage should a quantity be multiplied by in order to decrease the quantity by: a

15 Ex 4

17%

40%

52%

Evaluate the following using a single percentage: a

Decrease 80 by 60%

Decrease 400 by 45%

Decrease 900 by 8%

Decrease 80 by 40%

Decrease 1200 by 45%

Decrease 1200 by 8%

A percentage increase was applied to the number 60 in order to get 108. a

Find how much was added to 60 to get 108.

Express the increase as a percentage of 60.

A percentage increase was applied to the number 180 in order to get 207. a

Find how much was added to 180 to get 207.

Express the increase as a percentage of 180.

A percentage decrease was applied to the number 90 in order to get 27. a

Find how much was subtracted from 90 to get 27.

Express the decrease as a percentage of 90.

A percentage decrease was applied to the number 200 in order to get 140. a

Find how much was subtracted from 200 to get 140.

Express the decrease as a percentage of 200.

For each of the following: i

State whether the value increased or decreased.

Calculate the percentage change rounded to two decimal places.

82 changed to 38

33 changed to 47

$63 changed to $98

122 minutes changed to 109 minutes

79 m changed to 157 m

141 g changed to 142 g

199 km changed to 92 km

135 students changed to 85 students

A bag of rice weighs 120 kg. The company who makes these bags decides to increase their weight by 45%. Find the new weight of the bag.

A discus thrower has a personal best of 60 m. In their most recent competition, they fell short of their personal best by 30%. Find the distance of their throw in this competition.

2.03 Percentage increase and decrease mathspace.co

Sharon’s daily pay was $150. When she began working shorter hours in order to study, her pay decreased to $90 per day. Express this decrease as a percentage of her original pay.

Maria bought a bicycle for $600. She later sold it for $180. What is the new value as a percentage of the original?

Reasoning 21

21–23

22–24

23–25

Abed wants to increase 1500 by 30%. a

Calculate 30% of 1500 and add the answer to 1500.

Calculate 130% of 1500.

Explain why the answers to (a) and (b) are the same.

Tracy wants to increase 80 by 30%. a

What single percentage should she multiply 80 by?

Explain how you got your percentage in part (a).

The car Derek wants to buy is advertised for $12 000. It is on sale with 20% off, but it has to have 10% GST added to it. Derek calculates −20% + 10% = −10% and thinks his discount will be 10% of $12 000. Is he correct? Explain your reasoning.

An amount is decreased by 30%, then the result is increased by 30%. Is the ﬁnal amount equal to the original amount? Explain your reasoning.

If a discount of a% is applied to an amount and then an increase of b% is applied, write an expression for the single multiplier that is applied to the amount.

Problem-solving 26

26, 27

26–30

During the editing phase when making a movie, the director ordered that the movie be cut down to 124 minutes, which was a 38% decrease from the original duration. What was the movie’s original duration length?

In training for her next marathon, Kathy increased her practice route by 11%, running a new route of 6660 m. What distance was Kathy running before she increased the distance of her practice route?

While packing for a trip, Ember’s suitcase was heavier than the maximum weight allowance of 23 kg. She removed several pairs of jeans and shoes, reducing the weight by 9.8%, making her suitcase exactly 23 kg. How heavy was her suitcase before removing the jeans and shoes? Round your answer to one decimal place.

The length of a rope is 120 metres. The length is increased by 15% and then the result is decreased by 15%. What is the new length of the rope, rounded to the nearest metre?

Kevin earns a salary of $1500 per week for a 45-hour week. His weekly salary is increased by 8% and his number of hours worked is decreased by 12%. Calculate his new hourly salary.

Mathspace Victorian Curriculum – Year 8 mathspace.co

INVESTIGATION

Percentages Investigate percentage changes in the price of market items using authentic data.

Objectives To model trends in market items with percentages To apply percent change to a real-life situation

Percent increase and decrease

In this activity, we will look at how the price of milk, bread, and eggs have changed over time.

Analyse the data and the line graph provided. Prices of market items in Australia Milk 2010 $2.09 2011 $1.58 2012 $1.58 2013 $1.52 2014 $1.48 2015 $1.47 2016 $1.41 2017 $1.43 2018 $1.43 2019 $1.62 2020 $1.63 2021 $1.70 2022 $2.08

Bread $3.42 $2.68 $2.90 $3.00 $2.85 $2.75 $2.62 $2.54 $2.50 $2.95 $2.81 $2.86 $3.28

Eggs $4.38 $4.06 $4.20 $4.27 $4.48 $4.67 $4.76 $4.42 $4.08 $4.48 $4.81 $4.94 $5.48

5.5

Price in AUD

Eggs

5 4.5 4 3.5

Bread

3 2.5 2

Milk

1.5 Year 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

Investigation: Percentages mathspace.co

Investigate Use the data and the line graph provided to answer the questions that follow. 1.

At first glance, which item do you think had the largest percentage decrease in 2011? Explain your reasoning.

Calculate the percentage decrease for milk, bread, and eggs in 2011. Which item saw the largest percentage decrease in 2011? Is this what you expected?

Which item saw the largest percentage increase in a single year?

What was the percentage change in the price of eggs from 2010 to 2022?

What was the percentage change in the price of bread from 2010 to 2022?

In which year did eggs have the largest percentage decrease?

In 2015, the price of milk decreased by $0.01. In 2020, the price of milk increased by $0.01. Calculate the percentage changes in the price of milk for 2015 and 2020. If the flat changes are the same, why are the percentage changes different? Which percentage change was larger? Why?

Is the percentage increase higher for bread in 2019 or eggs in 2022?

With a partner, discuss reasons that may have caused the prices in milk, bread, and eggs to change over time.

Discussion

If two items increased by the same amount of money, why might they have different percentage increase values? What determines how large or small the percentage increase will be? Why?

Would you expect the price of milk, bread, and eggs to increase or decrease after 2022? Explain your reasoning.

Research reasons why the prices of milk, bread, and eggs decreased in 2011 and increased in 2018 and 2022. Based on your research, what are factors that affect the price of milk, bread, and eggs?

If you were to compare the price change of a luxury item, like a high-end smartphone, over the same period, do you think the percentage change would be similar, higher, or lower? Why?

Mathspace Victorian Curriculum – Year 8 mathspace.co

2.04

Profit and loss

After this lesson, you will be able to… • solve word problems involving net change (profit or loss), cost and sale prices and percentage profit or loss.

Profit and loss The cost price of an item is how much money we paid for it. The sale price of an item is how much money we sell it for. We can ﬁnd the net change in money by subtracting the cost price from the sale price. Net change = Sale price − Cost price If the net change is positive, we gain money. The money gained is called a proﬁt. If the net change is negative, we lost money. The money lost is called a loss. If the net change is $0, we neither gained nor lost money. When this happens, we describe it as breaking even.

Example 1 After buying and then selling a bicycle, the net change in Laura’s money was − $44. Which of the following describes her change in money? A Proﬁt of $44

Loss of $44

Loss of − $44

Breaking Even

Create a strategy

Apply the idea

To describe the change look at the sign of the value.

If the net change in money is a negative value, it is a loss. We can describe her change in money as a loss of $44. The correct option is B.

Example 2 Sandy bought a stove for $238 and made a proﬁt of $39. What was the sale price of the item?

Create a strategy We can calculate sale price using the formula: Net change = Sale price − Cost price

2.04 Profit and loss mathspace.co

Apply the idea 39 = Sale price – 238

Substitute the values

39 + 238 = Sale price

Add 238 to both sides

$277 = Sale price

Evaluate

Idea summary The net change in money is the directed number indicating how much money was gained or lost after buying and selling an object. Net change = Sale price − Cost price A positive net change indicates a proﬁt. A negative net change indicates a loss. A net change of $0 means the seller broke even.

Percentage profit and loss Percentage proﬁt and loss describe how much a value has changed in comparison to its cost price. The percentage helps us understand the relative amount of money we made or lost when comparing the amount to what we paid originally. Percentage proﬁt can be calculated with the formula:

Percentage loss can be calculated with the formula:

Example 3 Charlie bought a cake for $220 and sold it for $209. Which of the following describe Charlie’s change in money after buying and selling the cake? Select all that apply. A Loss of 11%

Loss of $11

C Proﬁt of 5%

Proﬁt of $11

E Loss of 5%

Loss of $5

Create a strategy Use the formula: Net change = Sale price − Cost price If the answer is positive, it is a profit. If the answer is negative, it is a loss.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Apply the idea Since the cost price was greater than the sale price, the change in money will be a loss. Loss = Cost price − Sale price

Use the formula

= 220 – 209

Substitute the values

= $11

Simplify Find the percentage loss Substitute the values Evaluate Simplify

So he lost $11 which is equal to 5%. Options B and E are correct.

Idea summary Percentage profit and loss describe how much a value has changed in comparison to its cost price. To find the percentage loss or percentage profit, we use these formulas: Percentage profit =

profit × 100% cost price

Percentage loss =

loss × 100% cost price

2.04 Practice questions Understanding

1–3

Adventurer 3

Trailblazer 3

John bought a bike for $120 and sold it for $230. Is this a proﬁt or a loss?

Joanna bought a coffee machine for $520 and sold it for $130. Is this a proﬁt or a loss?

Describe what it means if the net change in money is: a

Positive

Fluency

Ex 1

Explorer

Negative

4, 5ace, 6ace, 7abd, 8a–c, 9acef, 10ace, 11, 13–16

4, 5ace, 6ace, 7abd, 8b–d, 9ce–g, 10ce–g, 11, 12, 14–16

4, 5bdf, 6bdf, 7cef, 8c–e, 9bdgh, 10bdgh, 12, 14, 16, 17

For each sale, describe the change in money as a proﬁt or a loss: a

After buying and then selling a bicycle, the net change in Sarah’s money was −$56.

After buying and then selling a scarf, the net change in Anjelica’s money was $5. 2.04 Profit and loss mathspace.co

9 Ex 2

After buying and then selling a chair, the net change in Derek’s money was −$8.

After buying and then selling a scarf, the net change in Bianca’s money was $0.

After buying and then selling a toaster, the net change in Edward’s money was $65.

Find the proﬁt on these items: a

Cost price: $65, sale price: $82

Cost price: $120, sale price: $186

Cost price: $1400, sale price: $2050

Cost price: $86.50, sale price: $93.50

Glen bought a trading card for $11 and sold it for $24.

Roxanne bought a dress for $712 and sold it for $829.

Find the loss on these items: a

Cost price: $70, sale price: $45

Cost price: $225, sale price: $138

Cost price: $5870, sale price: $4250

Cost price: $850, sale price: $375.50

Judy bought a basketball for $99 and sold it for $92.

Charlie bought a sculpture for $660 and sold it for $585.

Find the proﬁt or loss for each sale: a

The sale price is $540 and the cost price is $284.

The sale price is $2065 and the cost price is $3120.

The sale price is $340.50 and the cost price is $290.80.

The sale price is $1800 and the cost price is $3500.

The cost price is $276.90 and the sale price is $320.55.

The cost price is $2480 and the sale price is $1770.

For each sale, write the net change in money as a negative or positive integer: a

Valentina bought a pillow for $2 and sold it for $16.

Bill bought a chair for $17 and sold it for $6.

Hannah bought a tennis racquet for $82 and sold it for $169.

Aoife bought a car for $12 000 and sold it ﬁve years later for $4500.

Yan bought an apartment for $380 000 and sold it ten years later for $520 000.

Find the sale price of these items: a

Cost price: $58, proﬁt: $10

Cost price: $120, proﬁt: $32.50

Cost price: $80, loss: $27

Cost price: $250, loss: $135

Alex bought a stove for $129. She eventually sold the stove and made a proﬁt of $29.

William bought a wallet for $12. He eventually sold the wallet and made a proﬁt of $2.

Pauline bought a plane ticket for $469. She then had to sell the plane ticket and made a loss of $9.

Ryan bought a guitar for $176. He eventually sold the guitar and made a loss of $37.

Mathspace Victorian Curriculum – Year 8 mathspace.co

11 Ex 3

Find the cost price of these items: a

Sale price: $72, proﬁt: $15

Sale price: $10 000, proﬁt: $2200

Sale price: $520, loss: $210

Sale price: $780, loss: $360

Ben sold a toothbrush for $12 and made a proﬁt of $3.

Pauline sold a washing machine for $486 and made a loss of $7.

Sarah sold a laptop for $233 and made a proﬁt of $55.

David sold a pair of earrings for $161 and made a loss of $55.

Xander bought a bag for $140 and sold it for $133. Describe Xander’s loss as a dollar amount and a percentage loss, after buying and selling the bag.

Sean bought a spade for $36 and sold it for $45. Describe Sean’s proﬁt as a dollar amount and a percentage proﬁt, after buying and selling the spade.

Oliver bought a baseball bat for $130 and sold it for $143.

Calculate Oliver’s proﬁt.

Find the percentage proﬁt of the cost price of the baseball bat.

Bianca bought a microwave for $50 and sold it for $35. a

Calculate Bianca’s loss.

Find the percentage loss of the cost price of the microwave.

John buys a used laptop for $391 and resells it for $554. a

Calculate the proﬁt John made on the laptop.

Find this proﬁt as a percentage of the cost price.

Paul invests $9000 in the stock market only to see his shares plummet in value to $4509. a

Calculate the loss Paul made on the shares.

Find this loss as a percentage of the cost price.

Sally purchased a dress for $20 and auctioned it off on eBay for $41. Find: a

The proﬁt (or loss).

The proﬁt (or loss) as a percentage of the cost, rounded to two decimal places.

The proﬁt (or loss) as a percentage of the sale price to two decimal places.

Reasoning 18

19, 20

19, 21

18, 20, 21

The formula to calculate proﬁt or loss percentage is:

The formula to calculate percentage increase or decrease is:

How is the formula for calculating the proﬁt or loss percentage similar to the formula for calculating percent increase or decrease? How are they different? 2.04 Profit and loss mathspace.co

Juliette and Alice decide to try to earn money by purchasing clothes that look almost new from a resale shop, then selling them at a higher price on social media. After a few weeks, they realised they are not making a proﬁt. If the selling price is higher than the purchase price, what are some reasons that could explain why Juliette and Alice are not making a proﬁt?

Buzz bought a mobile phone for $170 and sold it 7 months later, making a loss of 30% on the purchase price. a

Buzz’s friend did the math in his head and ﬁgured Buzz must have sold the phone for $51. Explain the mistake Buzz’s friend made.

Find the selling price of the mobile phone.

Harper sold a laptop for $1200 that was purchased 12 months earlier, making a loss of 20% on the purchase price. a

Harper’s friend calculated that she must have bought the laptop for $1200 + 20% of $1200 = $1440. Explain the mistake Harper’s friend made.

Find the purchase price of the laptop.

Problem-solving 22, 23, 25–27 22

Calculate the total revenue of the store in the ﬁrst month.

Calculate the total expenses of the store in the ﬁrst month.

Calculate the net change in money by the store in the ﬁrst month.

Every item in store is sold at 8% above cost. If a product was bought from the manufacturer for $360, ﬁnd:

The profit

Find the proﬁt made on the bicycle.

Find the sale price of the bicycle.

Hermione bought a phone for $200 and sold it for a 16% loss. a

The sale price

Sarah bought a bicycle for $400 and sold it for an 18% proﬁt. a

22, 28–31

In its opening month, a local pizza store sold 190 large pizzas at $20 each and 200 small pizzas at $6 each. The store incurred electricity costs of $26, water costs of $10, rent of $408, wages of $1345 and ingredient costs of $210.

a 24

22, 24, 25, 27, 28

Find the loss made on the phone.

Find the sale price of the phone.

Noah sold a jumping castle for $700 which was a 12% proﬁt. a

Find the sale price of the jumping castle as a percentage of the cost price.

Find the cost price of the jumping castle.

James sold a jacket for $600 which was a 4% loss. a

Find the sale price of the jacket as a percentage of the cost price.

Find the cost price of the jacket.

Yvonne bought a car for $17 600 and sold it 2 years later, making a loss of 19%. Find the sale price of the car.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Tania bought a painting for $2000 and sold it two years later at an overall loss of 20%. At what price did she sell the painting?

David bought a skateboard for $120. He then sold the skateboard for $90 to a friend. What was the percentage loss on the original price of the skateboard?

Alex buys an antique vase for $300 and later sells it at an auction for $450. a

Did Alex make a proﬁt or loss and by how much?

Calculate the proﬁt or loss as a percentage of the cost price.

2.05 Consumer percentages After this lesson, you will be able to… • solve word problems involving discounts, mark-ups and GST.

Mark-ups and discounts When buying products, we often encounter a mark-up or a discount. • A mark-up is a percentage increase in the price of a product. • A discount is a percentage decrease in the price of a product. Since mark-ups and discounts are simply percentage increases and decreases, we can calculate them using the same methods. Recall that we can apply percentage increases and decreases in one step: • For a percentage increase, add the percent to 100% and multiply the result by the original price. • For a percentage decrease, subtract the percent from 100% and multiply the result by the original price.

Example 1 A pair of boots priced at $80 is discounted by 35%. What is the discounted price of the pair of boots?

Create a strategy To ﬁnd the discounted price, multiply the original price by (100 − 35) % = 65%.

Apply the idea Discount price = 65% × 80

Multiply by 65%

= 0.65 × 80

Convert the percentage to a decimal

= $52

Evaluate

2.05 Consumer percentages mathspace.co

Idea summary A mark-up is a percentage increase in the price of a product. A discount is a percentage decrease in the price of a product.

Multiple mark-ups and discounts Suppose a video game was marked up by 20%, later discounted by 25% and ﬁnally discounted again by 40%. If the original price of the video game was $50, what is its current price? • The video game starts at the price of $50. • After a mark-up of 20%, its price is $50 × 120% = $60. • After a discount of 25%, its price is $60 × 75% = $45. • After a ﬁnal discount 40%, its price is $45 × 60% = $27. However, this is a lot of steps. Notice that the only operation we used was multiplication. We can reach the same answer by applying the three percentage changes in one step instead of three steps. Current price = $50 × 120% × 75% × 60%

Example 2 A bladesmith began her career selling her swords for $1500. After a few months, she marked up the price by 16%. A few weeks after that, a video of her making the swords went viral, and she marked up the price by 11%. In a few days she is due to appear on a popular podcast, and in anticipation of the increased demand, she marks up the price by 13%. What is the price of the bladesmith’s swords after all the mark-ups?

Create a strategy Multiply the original price by the three percentage changes for each mark-up.

Apply the idea To mark-up the price by 16% then 11% then 13%, we need to multiply by 116%, 111% and 113%. Final price = 1500 × 116% × 111% × 113%

Multiply by the percentages

= 1500 × 1.16 × 1.11 × 1.13

Convert the percentages to decimals

= $2182.48

Evaluate

Idea summary We can ﬁnd the ﬁnal price after multiple mark-ups and discounts by multiplying the original price by all the percentage changes at once.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Goods and services tax (GST) Almost every item we buy or service we use has a compulsory mark-up applied to it called a goods and services tax (GST). The GST is a 10% price increase. The extra money we pay goes to the government, and they can spend it on essential things like schools and roads. Some items we buy are excluded from GST. The most common examples are fresh fruit and vegetables, as well as other basic foods such as bread, ﬂour, oil, and dairy products. We can calculate the total price, inclusive of GST, in the same way we calculate a 10% percentage increase: Price including GST = Pre-GST price × 110% To calculate a pre-GST price when given the total price, we divide by 110%: Pre-GST price = Price including GST ∏ 110%

Example 3 Quentin is buying items he needs for school. The prices marked for the items are all pre-GST. He buys a laptop for $1360, exercise books for $18, a pack of pens for $7, and a calculator for $38. Find the total price Quentin pays including GST.

Create a strategy Add the prices and increase the total by 10%.

Apply the idea Total price = 1360 + 18 + 7 + 38 = 1423 Total including GST = 110% × 1423

Add the prices Evaluate Multiply by 110%

= 1.10 × 1423

Convert the percentage to decimal

= $1565.30

Evaluate

Example 4 The price of a microwave is $247.50 including GST. Calculate the GST included in the price.

Create a strategy Recognise that the total price represents 110% of the pre-GST price. Then, calculate 10% of the pre-GST price to find the GST.

Apply the idea 110% of the Pre-GST price = 247.50 247.50 10% of the Pre-GST price = 11 = 22.50

$247.50 is 110% of the pre-GST price Divide $247.50 by 11 Evaluate

The GST is $22.50. 2.05 Consumer percentages mathspace.co

Example 5 A book’s price including GST is $55. Find the pre-GST price of the book.

Create a strategy To calculate a pre-GST price, divide the ﬁnal price by 110%.

Apply the idea Original price = Final price ∏ 110%

Divide by 110%

= 55 ∏ 110%

Substitute the value

= 55 ∏ 1.1

Convert the percentage to a decimal

= $50

Evaluate

Reﬂect and check We can solve the problem algebraically by letting the pre-GST price be x. Since GST is 10%, the after GST price is 100% + 10% = 110%. 110% × x = 55 Write the equation 1.10 × x = 55 Convert the percentage to a decimal 55 x= Divide both sides by 1.10 1.10 Evaluate = $50

Example 6 The price of a collectable coin was marked up by 15% and is now $42.32. Find the original price.

Create a strategy Let the original price be x. Since mark-up is 15%, the final price is (100 + 15)% = 115% of the original price.

Apply the idea 115% × x = 42.32 Write the equation 1.15 × x = 42.32 Convert the percentage to a decimal 42.32 x= Divide both sides by 1.15 1.15 Evaluate = $36.80

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Idea summary The GST is a 10% tax that increases the price of goods and services. We can calculate the total price, inclusive of GST, in the same way we calculate a 10% percentage increase: Price including GST = Pre-GST price × 110% To calculate a pre-GST price when given the total price, we divide by 110%: Pre-GST price = Price including GST ∏ 110%

2.05 Practice questions Understanding 1

b c d

Ex 1

Trailblazer 1–3

• Loss • Discount • Proﬁt An item is sold for more than it was bought for. • Mark-up A shopkeeper reduces the price of an item by 15% during a • GST sale. 10% is added to the price of goods and services as a tax that • Selling price • Cost price goes to the Government. A shopkeeper buys an item for $35 and adds 10% to the price.

Find the percentage increase in price after the mark-ups are applied: a

The new price of a wallet after a mark-up is calculated by multiplying the original price by 110%.

The new price of a hat after a mark-up is calculated by multiplying the original price by 121%.

Find the percentage decrease in price after the discounts are applied: a

The new price of a phone after a discount is calculated by multiplying the original price by 95%.

The new price of a shirt after a discount is calculated by multiplying the original price by 77%.

Fluency 4

Adventurer 1–3

Select the term that is best used to describe the situation: a

Explorer 1–3

4ab, 5ab, 8–11, 15–22

4cd, 5cd, 6–9, 12–15, 17–20, 22–23

Find the discounted price: a

A pair of boots priced at $80 is discounted by 25%.

A desk priced at $80 is discounted by 40%.

A board game priced at $25 is discounted by 30%.

A cricket bat originally marked at a price of $150 is advertised to be selling at 45% off this marked price.

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Find the marked-up price: a

A book priced at $90 is marked up by 40%.

A dress at $120 is marked up by 30%.

A brush set priced at $140 is marked up by 45%.

A t-shirt originally priced at $170 is marked up to sell for 45% more.

Find the original price of the items: a

The price of a basketball was marked up by 60% and is now $144.

The price of a singlet was marked up by 45% and is now $26.10.

The price of a backpack was discounted by 30% and is now $63.

The price of a golf club was discounted by 35% and is now $11.70.

Find the original price of the items: a

A shop owner decides to mark-up the price of a sled by 31% so that it now sells for exactly $60.

A bakery sells garlic twist loaves, baked fresh every day. At 4:30 p.m. just before closing, they discount any remaining loaves by 43% and sell them for $2 each.

A popular hotel increased its nightly rates from $277 to $349. a

Calculate how much more a single night’s stay costs after the price increase.

By what percentage did the nightly rate increase? Round your answer to two decimal places.

A holiday resort in Tasmania reduced its overnight rates from $313 to $239. a

Calculate how much less a single night's stay costs after the decrease in price.

By what percentage did the nightly rate decrease? Round your answer to two decimal places.

Find the ﬁnal price after applying both mark-ups: a

A mattress originally priced at $90 is marked up by 70%. A few months later, the mattress is marked up from its new price by a further 60%.

A world trip originally priced at $9000 is marked up by 17%. A few months later, the world trip is marked up from its new price by a further 49%.

Find the ﬁnal price after applying both discounts: a

A painting canvas originally priced at $80 is discounted by 40%. A few months later, the painting canvas was discounted from its new price by a further 20%.

A second-hand boat originally priced at $12 000 is discounted by 23%. A few months later, the boat was discounted from its new price by a further 6%.

Find the ﬁnal price after applying both the discount and the mark-up: a

A backpack originally priced at $90 is marked up by 40%. A few months later, the backpack was discounted from its new price by 10%.

A washing machine originally priced at $80 is discounted by 15%. A few months later, the washing machine was marked up from its new price by 45%.

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?

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The price of a heater selling for $234 is initially discounted by 14% and later marked up by 14%. Find the ﬁnal sale price of the heater.

A jeweller began her career selling her necklaces for $1700.

Ex 2

After a few months she marked up the price by 11%. A few weeks after that, a video of her making the necklaces went viral, and she marked up the price by 13%. In a few days, she is due to appear on a popular podcast and, in anticipation of the increased demand, she marks up the price by 16%. Find the price of the jeweller’s necklaces after all the mark-ups.

In order to keep up with competition, a local shoemaker is forced to discount the price of his signature boots by 23%. Noticing that he still isn’t selling enough, the shoemaker discounts his boots by a further 22% the next week, and then another 35% the week after that. The boots were originally priced at $110. Find the ﬁnal price of the shoemaker’s boots after all the discounts.

In response to the changing economy, the price of a car is marked up by 35%, then discounted by 13%, marked up again by 28% and then ﬁnally discounted again by 14%. If the car was originally priced at $33 000, ﬁnd the new price of the car.

Find the price of the items inclusive of GST:

Ex 3

Ex 4

20 Ex 5

A board game has a pre-GST price of $30.

A hat has a pre-GST price of $7.

A laptop has a pre-GST price of $1070.

The price of a television is $682 including GST. a

What percentage of the pre-GST price does the total price represent?

By what number should we divide the total price to find the 10% GST amount?

Calculate the amount of GST included in the total price.

If the total price of a different television is $660 including GST, calculate the GST amount.

Find the pre-GST price of the items: a

A textbook has a price including GST of $77.

A fridge has a price including GST of $737.

A jacket has a price including GST of $48.29.

Amelia is buying items she needs for school. The prices marked for the items are all pre-GST. She buys: a laptop for $1470, exercise books for $22, a pack of pens for $6, and a calculator for $42. Find the total price Amelia pays including GST.

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Ex 6

The price of a collectable stamp was marked up by 12.5% and is now $81.45. Find the original price.

Hannah is buying items she needs for school. The prices marked for the items all include GST. She buys: a laptop for $1716.00, exercise books for $18.70, a pack of pens for $7.70, and a calculator for $36.30. Find the total pre-GST price of the items.

Reasoning 24

24–26

24, 26, 27

24, 26, 28, 29

A laptop originally sells at $4700. a

Calculate its sale price after successive discounts of 6% and 9%.

Calculate its sale price after successive discounts of 9% and 6%.

Does the order of discounting affect the sale price? Explain your answer.

Calculate its sale price after successive discounts of 8%, 7% and 5%.

An artist paints 1 landscape and 1 portrait painting, pricing them both at $3200. • The price of the landscape painting is marked up once by 8% and later marked up by a further 23%. • The price of the portrait painting is marked up once by 23% and later marked up by a further 8%.

Find the ﬁnal price of the landscape painting after applying both mark-ups.

Find the ﬁnal price of the portrait painting after applying both mark-ups.

The artist notices that, after calculating all the mark-ups, the price of the landscape and portrait paintings are equal. Explain why the prices are equal.

A skipping rope priced at $10 is marked up by 40% and then later discounted by 40%. a

Calculate the new price of the skipping rope.

Explain why the new price of the skipping rope is not equal to the original price of the skipping rope.

A ﬂute priced at $600 is marked up by 10% and then marked up again by 40%. a

Instead of calculating 2 separate mark-ups, Elizabeth claims that she can calculate the ﬁnal price in a single step using the expression 600 × 150%, because 100 + 10 + 40 = 150. Is Elizabeth correct? Explain your answer.

Instead of calculating 2 separate mark-ups, Judy claims that she can calculate the ﬁnal price in a single step using the expression 600 × 110% × 140%. Is Judy correct? Explain your answer.

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Find the ﬁnal price of the ﬂute.

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A bookcase priced at $110 is discounted by 20% and then discounted again by 15%. a

Instead of calculating 2 separate discounts, Edward claims that he can calculate the ﬁnal price in a single step using the expression 110 × 80% × 85%. Is Edward correct? Explain your answer.

Instead of calculating 2 separate discounts, Carl claims that he can calculate the ﬁnal price in a single step using the expression 110 × 65%, because 100 − 20 − 15 = 65. Is Carl correct? Explain your answer.

c 29

Find the ﬁnal price of the bookcase.

A computer game priced at $80 is discounted by 15% and later marked up by 20%. a

Instead of calculating 1 discount and 1 mark-up, Tom claims that he can calculate the ﬁnal price in a single step using the expression 80 × 105%, because 100 − 15 + 20 = 105. Is Tom correct? Explain your answer.

Instead of calculating 1 discount and 1 mark-up, Luigi claims that he can calculate the ﬁnal price in a single step using the expression 80 × 85% × 120%. Is Luigi correct? Explain your answer.

Find the ﬁnal price of the computer game.

Problem-solving 32, 33

31–33

30, 31, 33, 34

An item costing $x is marked up by m% then marked down by n% then has 10% GST applied. Write an expression for the new value in terms of x.

An electronics store marks up all products by 35%. If a laptop has been marked up by $245:

What was the original cost of the laptop?

What was the selling price of the laptop?

Lucy wanted to buy a new refrigerator and decided to wait for the annual home appliance sale. The original price of the refrigerator was $1800. At the sale, the refrigerator’s price was reduced by 15%, and customers who bought energy-eﬃcient appliances received an extra 7% discount. a

If Lucy purchased the energy-eﬃcient refrigerator, how much did she pay for it?

What single percentage discount is equivalent to the two successive discounts?

Vanessa is looking to purchase a new sofa. She found that this year, it comes with 10% GST added and a 5% discount. But next year, 11% of GST is added with 7% discount. Which deal offers better value for Vanessa?

Sarah bought a pair of shoes for $79.99. The next week, the store had a 20% off sale, and she decided to buy another pair. a

How much did she save on the 2nd pair of shoes? Round your answer to the nearest cent.

How much did she pay for the 2nd pair of shoes? Round your answer to the nearest cent.

Find the GST that is included in the original price, assuming that GST is 10%. Round your answer to the nearest cent. 2.05 Consumer percentages mathspace.co

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2 Chapter review 1

Convert between percentages, fractions and decimals to complete the table:

Fraction

Decimal

Percentage

1.47 2

For these groups of numbers: i

Rewrite the group as percentages.

Order the original group from smallest to largest.

a 3

Jenny is examining three different savings accounts. • Savings Account X: interest rate • Savings Account Y: interest rate 0.18 • Savings Account Z: interest rate 20% Which savings account should she select, and why?

For each number: i

Find 1% of the quantity.

Find the whole quantity.

7% of a quantity is 77.

47% of a quantity is $188.

3% of a quantity is 177 mL.

54% of a quantity is 6210 mL.

34% of a quantity is 918 L.

78% of a quantity is 10 959 cm.

Find the whole quantity if: a

of the quantity is equal to $22.

of the quantity is equal to 9 kg.

of the quantity is equal to 40 mL.

of the quantity is equal to 18 m.

5% of the quantity is equal to $11.

10% of the quantity is equal to 840 g.

25% of the quantity is equal to $12.

800% of the quantity is equal to 96 mL.

According to a recent survey, 30% of teachers in a school said they prefer teaching Science and 40% of teachers said they prefer teaching mathematics. All other teachers said they prefer teaching Literature. If 90 teachers prefer teaching Literature, how many teachers are working in the school?

Laura knows that her farm is 3.5 km from the beginning of the road, and that the distance from the beginning of the road to her farm represents 35% of the road’s total length. Calculate the road’s total length.

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Increase: a

20 by 80%

200 by 55%

$600 by 8%

80 mL by 40%

Decrease: a

90 by 10%

150 by 20%

84 by 50%

360 by 25%

$160 by 15%

$40 000 by 30%

100 km by 20%

500 mL by 40%

Lucas wants to decrease 2400 by 20%. a

Calculate 20% of 2400 and subtract the answer from 2400.

Calculate 80% of 2400.

Explain why the answers to (a) and (b) are the same.

Michael’s monthly salary was $2500. When he decided to go part-time in order to start his own business, his salary decreased to $1500 per month. Express this decrease as a percentage of his original salary.

James can type 300 characters in a minute. After spraining his wrist, this decreased to 195 characters a minute. Express this decrease as a percentage of his original speed.

When reviewing a manuscript for a book, the editor recommended that the manuscript be shortened to 186 pages, which was a 15% decrease from the original length. What was the manuscript’s original page count?

When preparing for a large event, Nolan discovered that his cooler was ﬁlled above its maximum capacity of 40 kg. He removed several bottles and cans, reducing the weight by 12.5%, making his cooler exactly 40 kg. How heavy was the cooler before removing the bottles and cans? Round your answer to one decimal place.

The volume of a liquid in a container is 80 litres. The volume is increased by 20% and then the result is decreased by 20%. What is the new volume of the liquid, rounded to the nearest litre?

Calculate the proﬁt on these transactions:

Purchase price: $75, sale price: $95

Purchase price: $150, sale price: $200

Purchase price: $1600, sale price: $2100

Purchase price: $98.50, sale price: $108.50

Elliot bought a comic book for $15 and sold it for $30.

Daisy bought a handbag for $750 and sold it for $850.

Calculate the loss on these transactions: a

Purchase price: $80, sale price: $60

Purchase price: $240, sale price: $180

Purchase price: $6000, sale price: $4700

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Purchase price: $900, sale price: $410.50

Steven bought a baseball bat for $110 and sold it for $100.

Nina bought a painting for $700 and sold it for $625.

Determine the original price of these items: a

Selling price: $80, proﬁt: $20

Selling price: $12 000, proﬁt: $2400

Selling price: $550, loss: $220

Selling price: $800, loss: $370

Rachel sold a hairbrush for $15 and made a proﬁt of $4.

Edward sold a refrigerator for $500 and made a loss of $10.

Lisa sold a tablet for $250 and made a proﬁt of $60.

Kevin sold a necklace for $170 and made a loss of $60.

Jacob bought a vintage camera for $200 and sold it online for $350. Determine: a

The proﬁt or loss.

The proﬁt or loss as a percentage of the cost, rounded to two decimal places.

The proﬁt or loss as a percentage of the sale price, rounded to two decimal places.

In its ﬁrst quarter of operations, a local bakery sold 450 loaves of bread at $4 each and 600 muﬃns at $2 each. The bakery had expenses for electricity costs of $200, water costs of $50, rent of $1000, wages of $3000 and ingredient costs of $500. a

Calculate the total revenue of the bakery in the ﬁrst quarter.

Calculate the total expenses of the bakery in the ﬁrst quarter.

Calculate the net change in money by the bakery in the ﬁrst quarter.

Megan purchased a vintage lamp for $200. Later, she sold the lamp at a yard sale for $150. What was the percentage loss on the original price of the lamp?

Andy bought a vintage record player for $500 and sold it three years later at an overall loss of 15%. At what price did he sell the record player?

Alice bought a digital camera for $320 and sold it 6 months later, making a loss of 25% on the purchase price.

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Alice’s sister did the math in her head and ﬁgured Alice must have sold the camera for $80. Explain the mistake Alice’s sister made.

Find the selling price of the digital camera.

Find the original price of these items: a

The price of a laptop was marked up by 40% and is now $840.

The price of a pair of headphones was marked up by 25% and is now $75.

The price of a coat was discounted by 20% and is now $240.

The price of a bicycle was discounted by 15% and is now $212.50.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Find the marked-up price: a

A vinyl record priced at $60 is marked up by 50%.

A jacket at $80 is marked up by 25%.

A pair of sneakers priced at $120 is marked up by 20%.

A book originally priced at $50 is marked up to sell for 60% more.

Find the ﬁnal price after applying both mark-ups: a

A laptop originally priced at $800 is marked up by 20%. A few weeks later, the laptop is marked up from its new price by a further 15%.

A bicycle originally priced at $300 is marked up by 25%. A few days later, the bicycle is marked up from its new price by a further 10%.

A collector started selling vintage comic books for $500. After a famous movie was announced based on one of the comic books, he marked up the price by 25%. A few weeks after that, a major auction house showed interest in his collection and he marked up the price by 20%. Following this, a well-known collector offered to buy his entire collection, and he marked up the price by 15%. Find the price of the vintage comic books after all the mark-ups.

Peter is buying electronics for his new apartment. The prices are all pre-GST. He buys: a television for $800, a sound system for $250, a coffee machine for $120, and a gaming console for $300. Find the total price Peter pays including the 10% GST.

Liam purchased a computer for $999.99. During a weekend sale, the same store offered a 25% discount on the computer, and he decided to buy one more. a

How much did he save on the second computer? Round your answer to the nearest cent.

How much did he pay for the second computer? Round your answer to the nearest cent.

Assuming there is a value added tax (VAT) of 20%. Find the VAT included in the original price.

Jackson is considering buying a new car. This year, the car is being sold with a 12% goods and services tax and an 8% discount. However, next year, the goods and services tax will increase to 14% and the dealership will offer a 10% discount. Which deal would be better for Jackson?

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Big ideas Proportional relationships can be used to solve real-world problems.

Curriculum Outcomes VC2M8M05 Recognise and use rates to solve problems involving the comparison of 2 related quantities of different units of measure VC2M8M07 Use mathematical modelling to solve practical problems involving ratios and rates, including distance-time problems for travel at a constant speed and financial contexts; formulate problems; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model

Ratios and rates Chapter outline 3.01 3.02 3.03 3.04 3.05

Ratios Maps and scale Rates Financial rates Best buys Chapter 3 review

112 123 131 138 148 154

The speeds at which gears rotate are determined by the ratio of their sizes.

3.01

Ratios After this lesson, you will be able to… • find equivalent and simplified ratios. • find a quantity from a given share. • divide a quantity by a ratio using the unitary method or the fraction method. • describe a ratio between three quantities.

Equivalent and simplified ratios A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another. There is 1 blue dot for every 3 green dots. The ratio is 1 : 3. It is important that both quantities are in the same units. We can make equivalent ratios by multiplying or dividing each value in the ratio by the same number to preserve the proportions. For example: 1 : 3 = 1 × 2 : 3 × 2 Multiplying each value by 2 1 : 3 = 2 : 6

Evaluating

Examples of equivalent ratios to 1 : 3

2:6

3:9

4 : 12

A ratio is a simpliﬁed ratio if there is no equivalent ratio with smaller integer values. We simplify ratios by multiplying or dividing both sides of a ratio expression by a common factor. For example:

Dividing each value by 5 Fully simpliﬁed ratio

If parts of a ratio have different units, ﬁrst convert them to the same units before simplifying.

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Example 1 Write 30 : 70 as a fully simpliﬁed ratio.

Create a strategy Divide the ratio by the highest common factor.

Apply the idea

Reﬂect and check Divide by 10

Evaluate

So we see that 30 : 70 and 3 : 7 are equivalent ratios, since we can divide the ﬁrst ratio by 10 to get the second. Also, there are no common factors between 3 and 7, so we know this is fully simpliﬁed.

Example 2 Simplify the ratio

Create a strategy We can clear each fraction one at a time by multiplying by the denominators.

Apply the idea

Reﬂect and check

We can start by multiplying both sides by 7 and then by 5 to clear the fractions.

Another way to do this is to cross multiply and then simplify. Cross multiply

Multiply both sides by 7

Simplify

Multiply by 5

Simplify

Evaluate

Divide by common factor

Evaluate

We can simplify the ratio by ﬁnding the highest common factor (HCF) of 20 and 56. 20 : 56 = 5 : 14 Divide both numbers by their HCF of 4 So the simpliﬁed ratio is 5 : 14.

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Example 3 Simplify the ratio 5.4 : 0.75

Create a strategy We ﬁrst need to write an equivalent ratio of integers and then simplify fully.

Apply the idea To simplify the ratio 5.4 : 0.75, we can ﬁrst convert it to a ratio of integers by multiplying both numbers by 100: 5.4 : 0.75 = 5.4 × 100 : 0.75 × 100

Multiply both numbers by 100

= 540 : 75

Evaluate

We can simplify the ratio by ﬁnding the highest common factor (HCF) of 540 and 75. 540 : 75 = 12 : 5

Divide both numbers by their HCF of 15

Therefore, the simpliﬁed ratio of 5.4 : 0.75 is 12 : 5.

Example 4 Write 540 cents to $3.00 as a fully simpliﬁed ratio.

Create a strategy Convert both amounts to the same unit and simplify the ratio.

Apply the idea Convert dollars to cents: $3 = 3 × 100 cents

There are 100 cents per dollar

$3 = 300 cents This means our ratio is 540 : 300. We can simplify the ratio by ﬁnding the highest common factor (HCF) of 540 and 300. 540 : 300 = 9 : 5 So the simpliﬁed ratio is 9 : 5.

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Divide both numbers by their HCF of 60

Reﬂect and check When simplifying we can also divide by any common factor we can see until there are no more common factors. For example: Notice a common factor of 10 Simplify Notice a common factor of 2 Simplify Notice a common factor of 3 Simplify Dividing by 10, then 2, then 3 is the same as dividing by 60 since 10 × 2 × 3 = 60.

Idea summary Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio. A ratio is a simpliﬁed ratio if the values are integers and have no common factors other than 1.

Find a quantity from a share When we know the ratio of two quantities and the amount of one of the quantities, we can find the amount of the other quantity. For example, if we know that a recipe uses oats and flour in the ratio of 3 : 4 and that we need 180 g of oats, we can find how much flour we need. In this case, we know the amount of oats and we want to work out the amount of flour. The 3 in the ratio 3 : 4 refers to the oats and this is 180 g. 180 g

3 parts

4 parts

One part would therefore be 180 ÷ 3 = 60 g.

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Flour is then 4 parts, so 4 × 60 = 240 g of ﬂour should be used. 60 g

60 g

Oats

60 g

3 : 4 180 g : 240 g

60 g

Flour

Another way to think about this is using proportional thinking: If 3 parts is 180 grams, then 1 part is 60 grams. Therefore, 4 parts must be 240 grams. We can use visuals or we can work more algebraically using ratios.

Example 5 A salad dressing is supposed to have a 5 : 16 ratio of vinegar to oil. If there are 13 mL of vinegar, how many mL of oil should be added? Round your answer to the nearest whole mL.

Create a strategy We can use the idea that the ratios of the amounts must be the same as the ratios of the parts.

Apply the idea

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Reﬂect and check We could also use equivalent ratios: Vinegar : Oil = 5 : 16

Given

Vinegar : Oil

Multiply by

Vinegar : Oil

Simplify

to get 13 mL of vinegar

Evaluate the amount of oil to the nearest whole number

Idea summary Ratios describe a relationship between 2 or more quantities. Two ratios are equivalent if one is a multiple of the other. To solve ratio problems we reason about the proportions, often using equivalent ratios.

Divide a quantity by a ratio Interactive exploration Discover this concept in action online

mathspace.co

Consider the problem of sharing $200 in the ratio 2 : 3. This can be done using two main methods. Unitary method • Calculate the total number of parts (2 + 3 = 5 parts). • Calculate what one part is worth (1 part = 200 ∏ 5 = 40). • Multiply each number of the original ratio by the amount for one part to calculate each number in the new ratio (2 × 40 : 3 × 40 = $80 : $120). Fraction method • Calculate the total number of parts (2 + 3 = 5 parts). • Divide each side of the ratio by the total number of parts

• Each share is worth the corresponding fraction multiplied by the total amount . We can also use ratios to relate three quantities in the form a : b : c. All three quantities must be multiplied or divided by the same number when simplifying or ﬁnding equivalent ratios.

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Example 6 Three siblings ﬁnd 700 kg of treasure. They split it up in the ratio 1 : 4 : 9. The youngest sibling receives the smallest portion, and the eldest sibling receives the largest. a How much treasure will the youngest receive?

Create a strategy 1. Find the total number of parts. 2. Find one part of the treasure.

Apply the idea Find the total number of parts. Total number of parts = 1 + 4 + 9

= 14

Sum the parts Evaluate

If the treasure was divided into 14 parts, then there would be: • 1 part for the youngest sibling • 4 parts for the middle sibling • 9 parts for the eldest sibling Find one part of the treasure: Divide 700 kg by the total number of parts

Evaluate

The youngest sibling receives one part of the treasure which is equal to 50 kg.

b How much treasure will the eldest sibling receive?

Create a strategy The ratio is 1 : 4 : 9, so the eldest will earn 9 parts of treasure for every 1 part the youngest receives. We already know how much treasure the youngest sibling receives, so we just need to multiply it by the parts received by the eldest sibling.

Apply the idea Eldest sibling receives = 50 kg × 9 = 450 kg

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Multiply one part of the treasure by 9 Evaluate

c After some negotiation, the eldest sibling gives up one share of her treasure to her youngest sibling. What is the new ratio written in its simplest form?

Create a strategy The original ratio is 1 : 4 : 9, after the negotiation we need to change the parts of the youngest and eldest sibling.

Apply the idea Add 1 part to the youngest sibling and subtract 1 part from the eldest sibling Evaluate Simplify by dividing all parts by 2 Evaluate

Idea summary Unitary method • Calculate what one part is worth • Calculate what each share of the ratio is worth, by multiplying what one part is worth with each number in the ratio Fraction method • Divide each side of the ratio by the total number of parts • Each share is worth the corresponding fraction multiplied by the total amount We can use ratios to relate three quantities in the form a : b : c.

3.01 Practice questions Understanding

Explorer 1–3, 5

Adventurer 2–5

Trailblazer 2–5

In a snack mix there are 4 cashews to every 5 almonds. Write this as a ratio.

What number would you divide both sides by to simplify the ratio 15 : 18?

Write the ratios as simpliﬁed fractions. a

12 : 16

100 : 50

45 : 25

4:5

Write the fractions as simpliﬁed ratios. a

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Select the pair of equivalent ratios: 2 : 10, 6 : 12, 5 : 1, 1 : 5, 1 : 4

Fluency 6 Ex 1, 2, 3

300 : 350

21 : 12

10 : 24

48 : 135

0.3 : 0.45

0.24 : 0.56

0.012 : 0.028

0.9 : 7.2

12 : 45 : 18

0.035 : 0.063

1 : 0.7

m 0.75 : 2

0.21 : 0.56

0.1 : 0.05

0.9 : 1.35

3 : 2.4 : 0.45

⬚ : 30 = 4 : 15

Ex 5

9 : 8 = ⬚ : 32 7 : 4 = ⬚ : 22

120

f j

⬚ : 5 = 132 : 60 ⬚:3=5:2

7 : 9 = 88 : ⬚

c g k

8:⬚=1:5

⬚:6=7:8

0.4 : ⬚ = 10 : 28

h l

50 cents to $2.10

3.8 kg to 180 g

5 years to 33 months

624 hours to 71.5 days

9:4=⬚:5

1.8 : 3 = 27 : ⬚

A cake recipe calls for a 3 : 5 ratio of ﬂour to sugar. If you have 360 g of ﬂour, how much sugar is needed? Tom and Jack divide their earnings of $714 in the ratio 10 : 7. Find how much Tom receives.

Find how much Jack receives.

Dave and Luke bought a scratch ticket that cost $10. Dave contributed $8 and Luke’s contribution was $2. They won $30 000. They decide to share their winnings in the same ratio as they contributed. How much money should Dave receive?

How much money should Luke receive?

Quentin is making a scarf that uses two materials. He will have to use for every

a 12

⬚ : 2 = 48 : 16

Express as a simpliﬁed ratio:

a 11

Complete these equivalent ratios:

6h–t, 7f–l, 8–10, 12–15

Ex 4

6g–q, 7c–f, 8–11, 13, 15

Simplify the ratios:

6a–j, 7a–d, 8–11, 13, 15

of green material

of blue material.

Write the simpliﬁed ratio of green to blue material.

If he needs 15 m of green material, how many metres of blue material will he need?

There are two celebration dinners happening at Happy Mo’s Restaurant. Each has 6 men at the table. At Beth’s birthday table there is a ratio of men to women of 1 : 3. At Buzz’s table the ratio is 3 : 1. a

Find the number of women at Beth’s birthday table.

Find the number of women at Buzz’s table.

Find the ratio of the total number of men to the total number of women at the two tables.

Mathspace Victorian Curriculum – Year 8 mathspace.co

15 Ex 6

The movement of Earth and Venus orbiting the Sun is compared. It takes the 8-Earth years and 13-Venus years to go around the sun before they are both at their starting positions. a

Write the ratio of the number of Earth years to the number of Venus years.

One Earth year consists of approximately 365.25 Earth days. How many Earth days make up one Venus year? Round your answer to two decimal places.

Jonathan, John and Jefferson found 750 kg of treasure. They decided to split it up in the ratio 5 : 8 : 12. Jonathan being the richest of the three, receives the smallest portion, and Jefferson being the poorest receives the largest. a

How much treasure will Jonathan receive?

How much treasure will Jefferson receive?

After some discussion, Jonathan gives up four shares of his treasure equally to John and Jefferson. Find the new ratio.

Reasoning

16–18

17, 18

17–19

A bag contains 5 red marbles, 3 blue marbles, and 6 yellow marbles. Explain what the ratio 5 : 14 would represent.

The ratio of dogs to cats to rabbits in a pet store is 2 : 3 : 4. If there are 45 pets in the store, explain the process of ﬁnding the number of rabbits.

Can 20 marbles be divided into two groups in the ratio 1 : 5? Explain.

Abel’s soccer team won

of their games during a season.

Explain why his team cannot have played 26 games.

Assuming that all of their games were either won or lost, express their wins to losses as a ratio.

Problem-solving 20, 25–27

20, 25, 26, 28

20–24, 27–29

Daniel and Emma share a $6 bag of 180 jelly beans. Daniel paid $4 and Emma paid $2. How many jelly beans does each person get if they are shared in the same ratio as the amount they paid?

Mr. Johnson is a history teacher. His full-time yearly salary in 2024 was $100 000 to be more realistic for a full time teacher.

Midway through 2024, Mr. Johnson could only work Monday, Tuesday and Wednesday each week. Ms. Wier will now take the Thursday and Friday lessons. The total salary of $100 000 now needs to be split between Mr Johnson and Ms Wier. The ratio of Mr. Johnson’s pay to Ms. Wier pay will be 3 : 2. How much will Ms. Wier be paid?

When Mr. Johnson returned to working Monday to Friday in 2025, he was given a pay raise. The ratio of Mr. Johnson’s 2024 full-time salary of $100 000 to his new salary is 4 : 5. What is his new 2025 full-time salary?

If 8 workers can complete a landscaping project in 12 hours, how long will it take if only 5 workers were employed to do the job? Leave your answer in hours and minutes.

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Alice, Bob, and David, share $y dollars evenly among themselves. Three other friends, Emily, Frank, and Helen, share $y dollars in the ratio of 3 : 4 : 5, respectively. Using an equation, determine how much money Frank will receive if David receives $45 more than Emily.

A prize pool is to be divided among three contestants, Xena, Yeng, and Zion, in the ratio 3 : 4 : 5. Before the prize money is distributed, Zion is disqualiﬁed, and his share is divided among Xena and Yeng in the ratio 2 : 1. If after Zion’s share is divided, Xena has $6000 more than Yeng, use an equation with the total amount reflected as $12x to ﬁnd how much money there was all together.

All students in a school play sport on Friday afternoons. The ratio of students who play tennis, soccer and rugby is 3 : 8 : 11. a

If there are 55 students who play rugby, how many students play tennis?

How many students play sports on Friday afternoons?

Roxanne is mixing red, blue and yellow paint in a container. The tub of paint is ﬁlled to 300 mL. She remembers that she added 60 mL of red paint and that the ratio of red to blue paint is 4 : 7. a

How much blue paint did she use?

How much yellow paint did she use?

She wants to ﬁnd the full ratio of red to blue to yellow paint so she can mix it again. Write the new triple ratio.

There are three exoplanets orbiting the star Kepler-37. The planets are named Kepler-37b, Kepler-37c, and Kepler-37d. The number of times each exoplanet orbits the star before they are all at their starting positions is shown in the table: Planet

Kepler-37c

Kepler-37b

Kepler-37d

Number of orbits

Write the triple ratio of the number of orbits for Kepler-37b to Kepler-37c to Kepler-37d.

The entire cycle takes approximately 199 Earth days for all three exoplanets to return to their starting position. How many Earth days make up one orbit of Kepler-37d?

Hiro, Anna, Keia and Nico shared a bottle of soda. The ratio of the amount of soda Hiro, Anna, Keia and Nico received is 3.2 : 4.6 : 5.8 : 0.4 respectively. The size of the soda is 140 ml. How many millilitres of soda did Hiro drink?

Abby is a journalist and she spent a total of 45.25 hours researching, writing and editing a new report. She spent 22.36 hours researching and 10.28 hours writing.

122

How many hours did Abby spend editing the news report?

Find the ratio in which her time was divided between researching, writing and editing.

Mathspace Victorian Curriculum – Year 8 mathspace.co

3.02 Maps and scale After this lesson, you will be able to… • relate scale drawings to ratios. • find actual lengths and distances given the ratios in maps or drawings. • identify and interpret scale bars on drawings and maps.

Scale ratios Ratios are very useful in representing lengths or distances on scale drawings or maps. If the scale ratio is 1 : 1000 it means the length in the real world is 1000 times the length on the scale drawing or map. We say the scale factor is 1000. To convert a length on the drawing or map into a length in the real world we multiply by 1000. To convert a length in real life to a length drawn on a scaled drawing or map we divide by 1000.

Example 1 Bianca is looking over a map of her local area and notices that the scale of the map is given as 1 : 100 in the map legend. a Find the actual distance (in centimetres) between two points which are drawn 12 cm apart on that map.

Create a strategy

Apply the idea

Multiply the distance between the two points on the map by the scale of the map.

Distance = 12 × 100 = 1200

Evaluate the multiplication

So, the distance is 1200 cm. b Find the distance in metres between the two points from part (a).

Create a strategy Remember that there are 100 centimetres in 1 metre.

Apply the idea We found in part (a) that the two points are 1200 cm apart in the real world.

Divide this distance in centimetres by 100 to ﬁnd the distance in metres

Simplify to ﬁnd the distance in metres

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Example 2 The scale on a map of a garden is 1 : 2000. How far apart on the map should two fountains be drawn if the actual distance between the fountains is 100 metres? Express your answer in centimetres.

Create a strategy

Apply the idea

Divide the actual distance between the two fountains in centimetres by the scale of the map. Remember that 1 metre = 100 centimetres, therefore, 100 metres is 100 x 100 cm, or 10 000 cm.

Evaluate

Example 3 The following is a 1 : 66 000 scale drawing of the sailing route from the mainland to an island off the coast. The captain approximates the distance to be 10.3 cm on the map. What is the distance of the boat trip in kilometres?

Create a strategy Multiply by 66 000

Apply the idea We want the left side of the ratio to be 10.3 so we multiply the ratio by 10.3. 1 : 66 000 = 1 × 10.3 : 66 000 × 10.3 = 10.3 : 679 800

Multiply the ratio by 10.3 Evaluate

The distance of the boat trip is 679 800 cm. To convert this to kilometres we use the fact that 1 km = 100 000 cm.

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Divide by 100 000 Evaluate

Idea summary To convert a length on the scale drawing to a length in real life we multiply by the scale factor. To convert a length in real life to a length on a scale drawing or map we divide by the scale factor.

Scale bars Consider the scale drawing of a granny ﬂat: The scale bar on the drawing is 1 cm long which means 1 cm on the drawing is equal to 1 metre in real life. To write this as a ratio we convert to the same units. 1 metre is equal to 100 cm, therefore we say the diagram has a scale ratio of 1 : 100.

1 metre

The bed measures 2.5 cm on the scale drawing. This is 2.5 m in real life.

Example 4 The map designer for a new amusement park measures Park Map the main street to be 4 cm. The walk along the main street is known to be 120 m.

cm 1 0

Main Street

2 3 4 5 6 7 8 9

What ratio is the map using? Give your ﬁnal answer in the form, 1 : ⬚.

Create a strategy

Simplify the ratio by converting to the same units and dividing by the HCF.

Apply the idea Write the ratio Convert to centimetres Divide by 4 Evaluate

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Idea summary Scale can be indicated by a scale bar on a drawing or map. To ﬁnd the scale ratio, convert the lengths to the same units.

3.02 Practice questions Understanding 1

Explorer 1–4

Adventurer 1–4

Trailblazer 1–4

The scale bar on this map measures 2 cm. How many actual kilometres are represented by 2 cm on the map? Perth

Sydney

1000 km

If the scale ratio on a diagram is 1 : 400, is the real length smaller or larger than the diagram length?

If the scale ratio on a diagram is 25 : 1, is the real length smaller or larger than the diagram length?

A map has a scale of 1 : 30 000. If I measure 1 cm on the map, what is the actual distance in centimetres?

What is the actual distance in kilometres?

What distance on the map represents an actual distance of 6 km?

A diagram has a scale ratio of 1 : 20. a

What is the scale factor between the diagram measurements and real measurements?

How many times larger is the actual item than the diagram?

Fluency 5

126

5a–c, 6, 7, 9–13

5b–d, 6–13

Convert these descriptions to a scale ratio: a 3 cm represents an actual distance of 9 m. b 8 mm represents an actual distance of 40 m. c 60 mm represents an actual distance of 5400 m. d 5 cm represents an actual distance of 30 km. e 20 cm represents an actual distance of 16 km. f 98 mm represents an actual distance of 9.8 km.

Mathspace Victorian Curriculum – Year 8 mathspace.co

5c–f, 6–9, 11–14

Ex 1

Ex 2

For a map with a scale of 1 : 2000. a Find the actual distance (in centimetres) between two points that are drawn 19 cm apart on the map. b Find the distance between the two points in metres.

For a map with a scale of 1 : 500. a Find the actual distance (in millimetres) between two points that are drawn 12.5 mm apart on the map. b Find the distance between the two points in centimetres.

The scale on a map of a garden is 1 : 2500. How far apart on the map should two fountains be drawn if the actual distance between the fountains is 200 m? Express your answer in centimetres.

The scale on a map of a shearing shed is 1 : 300. How far apart on the map should two shearing stations be drawn if the actual distance between the stations is 6 m? Express your answer in centimetres.

The following is a 1 : 58 000 scale drawing of the sailing route from the mainland to an island off the coast. The captain approximates the distance to be 15.3 cm on the map. Estimate the distance in kilometres of the boat trip.

Ex 3

11 Ex 4

The map designer for a new amusement park measures the main street to be 4 cm. The walk along the main street is actually 280 m.

Park Map

Write down the scale ratio of the map. 2 cm 1 0

Main Street

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127

Estimate the distance from Perth to Brisbane using the map scale: 1000 km Australia

Brisbane

Perth

Find the distance between house A and the park.

Find the distance between house C and the park.

Find the length of the shorter side of the park.

Find the length of the longer side of the park.

Find the shortest distance travelled by driving from house A to house C along the roads.

Find the shortest distance travelled by ﬁrst driving from house A to house C, then driving to house B, travelling along the roads.

D 400 m

A map of New York is shown. The actual dimensions of one rectangular block are 230 m by 80 m.

Zara

7th Avenue

6th Avenue

Jason

5th Street 6th Street

Harper

Prospect Park West

A map of a town is drawn to scale as shown:

Fiske Place

7th Street Eli

128

8th Street

What is the shortest distance Zara will need to drive to reach the pick-up points, marked with an X?

Whose car is the closest to the pick-up location?

Mathspace Victorian Curriculum – Year 8 mathspace.co

15, 16

This is a map of where Adam and Carl live. The side of a small square on the map 1 represents 100 m.

in Way

Major S treet

6 7 8

4 5

Mounta

2 3

e Lan

What is the map reference of Carl’s house? Explain your answer.

Top

Adam is on the corner of Top Lane and Major Street (map reference E5). He walks 300 m along Top Lane to Carl’s house.

16, 17

Brick R o ad

Reasoning

Valley Avenue

Gree

n Cir

cuit

Vincent is comparing three parks from three different maps as shown. Which park has the largest area? Explain your answer. Lesse Park

Millenial Park

Tibsen Park

Scale: 1 : 2000

Scale: 1 : 5000

Scale: 1 : 7000

Is a square that measures 4 cm on a map with a scale of 1 : 200 the same size as a square that measures 4 mm on a map with a scale of 1 : 2000 in the real world?

A Boeing 747 is 71 m long. Sebastian represents a Boeing 747 on a drawing as 35.5 cm long. What is the scale of his drawing?

Mike has two maps that have 8 mm squares on them. The scales of the maps are 1 : 50 and 1 : 150. What is the ratio of the areas of the actual objects represented by the squares?

Could part (c) have been answered if only the two scales were known and not the size of the square? Explain with reference to two maps with scales 1 : a and 1 : b.

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Problem-solving 18–20

18–20

20–23

The scale on the map of a public garden is 1 : 5000. How far apart on the map should two ponds be if the actual distance between the ponds is 350 m? Give your answer in centimetres.

This is a 1 : 200 floor plan of a house. The homeowner wishes to add a dining room table, which is 175 cm long, placing it where the × is marked on the floor plan. Find the table’s length that should be drawn on the floor plan.

130

A plan of a building with ratio 1 : 90 is enlarged so its length and width are both doubled as shown: a

Find the new scale ratio for the larger plan.

The scale bar on the original plan represents 5 m. When the plan is doubled, the scale bar is doubled as well. Find the length that the new scale bar represents.

An oil tanker has a spill off the coast of Australia. An expert says that the oil could travel 1700 km in the ﬁrst day. Their map is shown sketched with circles with radiuses of 10 cm, 20 cm and 30 cm: a

Which locations will be affected in the ﬁrst day?

Location B is 26 cm from the oil spill on the map. Find the actual distance of this location from the oil spill.

Mathspace Victorian Curriculum – Year 8 mathspace.co

30 cm

20 cm

10 cm C

Spill

B A

Scale 1 : 8 000 000

A map is being drawn for a medium-sized rural town. The map will ﬁt on an A4 piece of paper which has the dimension of 21 cm by 29.7 cm. Write an appropriate scale for the map.

An architect is designing a house that is 21 m long and 15 m wide. a

She wants to print the plans for this house in a 84 cm by 60 cm section of a large piece of paper. Find the ratio she should use to ﬁt the plans on the paper.

After printing the plans, the architect measures the length of the hallway to be 24 cm on paper. She wants to order a rug that covers the full length of the hallway. How long should the rug be in metres?

3.03 Rates After this lesson, you will be able to… • solve problems involving rates. • convert units to the indicated unit rate.

Unit rates Relationships such as 60 km/hr and $1.50 per kg are called rates. They compare quantities measured in different units. 60 km/hr is read as 60 kilometres ‘per’ hour. Both of these examples are called unit rates because they measure per ‘one’ hour and per ‘one’ kilogram. To convert a rate to a unit rate, both quantities must be divided or multiplied by the same value. For example: 100 km per 2 hours is a rate. We divide both quantities by 2 to convert to the unit rate of 50 km per hour.

Example 1 Write 91 people per 7 buses as a unit rate.

Create a strategy 91 people per 7 buses needs to be written as a rate in the form of ⬚ people per 1 bus.

Apply the idea

91 people per 7 buses, can be written as: 917 7 91 people 91 people 91 Divide buses both by 7both parts by 7 = buses people= ÷7 people buses ÷ 7Divide both Divide parts by 7 parts 7 7 buses 7 7 13 people= 13 people Simplify = Simplify Simplify 1 bus 1 bus

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131

Idea summary A rate compares quantities measured with different units. For example: $6 per 3 kgs. A rate written per ‘one’ unit is called a unit rate. For example: $2 per kg. $2/kg is read as 2 dollars ‘per’ kilogram.

Apply rates Rates can be used to calculate how much one measurement changes based on the change in another. If a bike travels at 10 m/s then for every one second it travels 10 metres. This is a unit rate. We multiply a given number of seconds by 10 to ﬁnd the distance travelled in that time. When trying to calculate the quantity of something that is given as part of a non-unit rate, ﬁrst convert it to a unit rate then multiply by the number of units.

Example 2 On a road trip, Tracy drives with an average speed of 90 km/hr. How far does she travel in 8 hours?

Create a strategy

Apply the idea

If Tracy can travel 90 kilometres in 1 hour, then Tracy can travel 8 times as far in 8 hours.

Tracy travels at 90 km/h. This means that the car travels 90 kilometres per 1 hour. Distance = 90 × 8

Multiply 90 by 8

= 720 km

Idea summary To calculate the change in one value of a rate, ﬁrst convert it to a unit rate, then multiply by the number of units.

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Convert rates Interactive exploration Discover this concept in action online

mathspace.co

To convert rates from one set of units to another set of units, change one unit at a time by dividing or multiplying by the conversion rate. To change 12 km/min to m/sec ﬁrst change the kilometres to metres and then change the minutes to seconds. There are 1000 m in 1 km therefore change the kilometres to metres by multiplying by 1000 which gives: 12 × 1000 m/min = 12 000 m/min There are 60 seconds in a minute. Therefore, to calculate how many metres there are per second we divide the number of metres by 60 which gives:

Example 3 192 metres per 240 seconds a Express this is as a unit rate in terms of metres and seconds.

Create a strategy If we divide both numbers in the rate by 240, the rate is equivalent.

Apply the idea The original rate is equivalent to

metres per

seconds.

Divide 192 by 240 Simplify

3.03 Rates mathspace.co

133

b Express this as a unit rate in terms of metres and minutes.

Create a strategy We need to change the rate from metres per seconds to metres per minutes. There are 60 seconds to a minute. So we will need to multiply both parts by 60 to convert seconds to minutes.

Apply the idea

4 4 m / sec = × 60 metres / 1 × 60 minutes Convert Convert to 1 minute to 1 minute 44 4 5 5 / sec metres / 1 × 60 minutes Convert to 1 minute = × /60 m / sec =5 m metres 1 60 minutes Convert to 1 minute × 60 × 5 240 5 = metres / 1 minute Simplify 240 5 Simplify 240 Simplify = metres / 1 minute metres /51 minute Simplify = 48 metres 5 = Simplify 48 metres= 48 metres Simplify1 minute Simplify = Simplify 1 minute 1 minute 4 So, the rate m / sec is equivalent to 48 m/min. 5

Example 4 A worm takes 20 seconds to travel 25 cm. How long does it take the worm to travel a metre?

Create a strategy Remember that there are 100 cm in a metre. The worm will have to wriggle 25 cm four times to travel a metre.

Apply the idea If it takes the worm 20 seconds to cover 25 cm, it will take

= 4 times longer to wriggle one metre.

Write the conversion Simplify fractions Evaluate

Idea summary To convert rates from one set of units to another set of units, change one unit at a time by dividing or multiplying by the conversion rate.

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Mathspace Victorian Curriculum – Year 8 mathspace.co

3.03 Practice questions Understanding 1

Ex 1

Adventurer 1–3

Trailblazer 1–3

Select the rate in each set: i

3:4

15 cm

22%

$15/hr

iii

$1.50 per litre B

91 cm2

$22.50

112.5 cm

6 cm/min

25 : 35

125 cm

17 : 3

12 hrs

65 km per hr

10 days

The speed of a car is usually measured using the units km/hr. Write suitable units for each rate: a

The price of petrol

Typing speed

Heart rate

Casual pay rate

Which rates are examples of unit rates? A

$2 per min

5 men/3 days

95 km per hour

14 days in 2 weeks

Fluency 4

Explorer 1–3

4, 6–9, 12–16, 18, 21, 22b, 4, 5, 7, 9, 11–14, 16, 18, 21, 23–25 23, 25

4, 5, 7, 9, 10, 12, 13, 17, 19, 20, 22, 24, 25

Write the rate as a unit rate: a

1040 books per 13 libraries

85 people per 5 buses

234 kilometres per 4.5 hours

9 cakes per 72 guests

A plane burns 406 litres of fuel every 14 kilometres while in the air. Express the fuel consumption as a unit rate in litres per kilometre.

Tricia reads 20 pages of her book in 2 hours. Find her reading rate of pages per hour.

A brick paving manufacturer makes 165 brick pavers in 5 hours. At what rate are the pavers made per hour?

Justin earns $420 in 6 hours. Find Justin’s hourly rate.

Luke writes 900 words in 20 minutes. Find Luke’s writing speed in words per minute.

A tap ﬁlls up a 259 litre tub in 7 hours. Find the rate of water ﬂow of the tap in litres per hour.

Luke earns $328 in 8 hours. Find Luke’s hourly rate.

Two shops sell mangoes at the following prices: • 148 mangoes for $384.80 at Shop A • 252 mangoes for $453.60 at Shop B a

Find the cost of a single mango at Shop A.

Find the cost of a single mango at Shop B.

Which is the best place to buy mangoes, if you want the best price per item? 3.03 Rates mathspace.co

135

Neville is deciding between four cars to buy. He wants to buy the most fuel eﬃcient one. a

Complete the table: Car

b 14

Litres used

Distance travelled (km)

Magnum

13.405

38.3

Falcador

20.71

54.5

Canyonero

6.336

19.2

Civil

2.754

8.1

Which car is the most fuel eﬃcient?

A car travels 320 km in 4 hours. a

Complete the table of values: Time taken (hrs) Distance travelled (km)

Ex 2

4 320

Find the speed of the car in kilometres per hour.

A toy manufacturer makes 720 toys in 9 hours. Find the rate at which the toy manufacturer works.

Find the distance travelled by a car if it travels 60 km/hr for 9 hours.

A bus travels at a speed of 80 km/hr. How far does the bus travel in 5 hours?

Tina burns 640 calories by jogging 8 kilometres. Find the rate at which Tina burns calories per kilometre.

A 3-episode TV series had a total run time of 69 minutes. Find the run time for a single episode.

An animator uses 12 frames for every 0.5 seconds of an animation. Find the frame rate of the animation in frames per second (fps).

Consider the rate 180 metres per 240 seconds.

Ex 3

Ex 4

L/km

Express this as a unit rate in terms of metres and seconds.

Express this as a unit rate in terms of metres and minutes.

Convert the rates. Round your answer to two decimal places where necessary. a

6 L/h to mL/h

25 m/min to cm/sec

80 words/min to words/h

680 km/h to m/h

75 beats/min to beats/h

5.2 L/h to mL/min

500 mL/g to L/kg

800 cm/min to m/h

A cyclist travels 70 metres per 10 seconds. Find the speed of the cyclist in metres per minute.

Tina counts 26 heart beats over a 20 second interval. Find Tina’s heart rate in beats per minute (bpm).

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Patricia eats 7.2 litres of ice cream in 6 minutes in an ice cream eating contest. Patricia wants to ﬁnd her rate of ice cream consumption in millilitres per second. a

How many millilitres of ice cream did Patricia consume?

How many seconds did it take for Patricia to consume the ice cream?

Find her rate of consumption of ice cream in mL/s.

Reasoning

26, 27

27, 28

29, 30

Explain how to convert 120 km/hr into metres per minute.

Luke is a locksmith. On weekends, if he is on call, he gets paid $64.50 per hour plus a ﬂat rate call out fee of $75. On Saturday he has 3 jobs that will take 1.5, 0.25 and 1.25 hours respectively to complete. How much will he earn on Saturday? Explain your answer.

Lisa is trying to save money so she deposits $3.20 into her bank account for every 8 coffees she purchases. a

Describe the steps to determine how much Lisa deposits for each cup of coffee.

What is the deposit per cup as a unit rate?

A tap is dripping at a rate of 1000 mL/hr. A 5 L bucket is placed underneath it to catch the water. Will the bucket overﬂow in 5 hours? How do you know?

Avril is running from John in a game of tag. Avril runs 30 metres every 5 seconds and John runs 21 metres every 3 seconds. State whether John will catch Avril and explain your reasoning.

Problem-solving 32, 33, 36–38

32, 33, 36–38

31, 34, 35, 37, 39

Caitlin and Roslyn play cricket over the summer holidays. Caitlin has a rate of 24 runs per 2 games and Roslyn has a rate of 24 runs per 4 games. If there are 30 games in the season, how many runs did Caitlin and Roslyn score in total?

A worm takes 15 seconds to travel 20 cm.

How long does it take for the worm to travel 1 metre?

Find the rate of the worm’s time spent per metre distance travelled.

Adam really likes apples and eats them at a rate of 4 per day. a

How many apples does Adam eat in one week?

If Adam buys 44 apples how many days will this last him?

The cost per apple is $0.45. If Adam has $18, does he have enough to buy apples for 2 weeks?

The water level of a pool increases by 5 cm for every 10 litres of water added. After adding some water, the water level of the pool increased by 0.45 m. a

How much water was added to the pool?

If the cost of water is $0.91 for every 7 litres, how much did it cost to add the water to the pool?

If the pool is being ﬁlled by a hose that can deliver 15 litres of water every 4 minutes, how long did it take to add the water to the pool? 3.03 Rates mathspace.co

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In 2018, the population growth of Benin was 35 per 1000. This means that for every 1000 people, the number of people at the end of the year would be 1035. For a town with a population of 640, what would you expect the population to be at the end of the year? Explain your answer.

Tom is attending to a patient on the way to the emergency department. In the ambulance, he takes a quick measurement of the patient’s heart rate and ﬁnds it to be 36 beats per 20 seconds. An hour later, in the hospital, he takes a more detailed measurement and ﬁnds the patient’s heart rate to be 240 beats per 120 seconds. a

Calculate the heart rate in beats per minute (bpm) for the ﬁrst reading.

Calculate the heart rate in beats per minute (bpm) for the second reading.

Did the patient’s heart rate increase or decrease between these readings?

Peter runs daily and usually runs 19 km in 75 minutes. The last time he ran the 21 km Sydney half marathon, it took him 126 minutes.

Find his running rate in km/h during his daily run.

Find his running rate in km/h during the half marathon.

In which activity was he running faster on average?

Two shearers wanted to work out who was the faster shearer.

Jenny sheared 144 sheep over 6 days, and Sean sheared 115 sheep in 5 days.

Find Jenny’s rate per day in shearing sheep.

If Jenny continued at this rate, how many would she be able to shear in 25 days?

Find Sean’s rate per day in shearing sheep.

Who sheared sheep at a faster daily rate?

Jasper has a tank that holds 360kL of water. It is currently empty. He has two hoses that he can use to fill the tank. The first hose will fill the tank in 20 minutes if used alone and the second hose will fill the tank in 40 minutes if used alone. a

Find the rate at which the ﬁrst hose delivers water into the tank in kL/min.

Find the rate at which the second hose delivers water into the tank in kL/min.

How long would it take to ﬁll the tank from empty using both hoses?

3.04 Financial rates After this lesson, you will be able to… • solve problems involving financial rates such as simple interest, income tax, and exchange rates.

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Simple interest A loan is a service that allows you to borrow money and then repay the money over time. A percentage of the original amount borrowed is charged by the lender of the money to cover the administration of the loan. This extra amount is called interest. Interest can also be earned on money that is invested. The original amount invested grows by earning interest over time. Simple interest is calculated as a ﬂat percentage of the original amount borrowed or invested using: I = PRT Where I is the interest to be paid, P is the principal or original amount borrowed or invested, R is the rate of interest per year as a decimal and T is the amount of time the money is borrowed or invested in number of years. We can also calculate the principal, interest rate or time of the loan or investment given the other three variables by substituting the known values into the simple interest equation and then solving for the unknown. The term p.a. means per annum, which is per year. Most interest rates are quoted as a p.a. rate.

Interactive exploration Discover this concept in action online

mathspace.co

Example 1 What is the total interest to be paid on a 2-year $3000 loan at 17% p.a. ﬂat interest?

Create a strategy We can use the simple interest formula: I = PRT.

Apply the idea From the given, we have P = 3000, R = 17% = 0.17, and T = 2. I = PRT Write the formula = 3000 × 0.17 × 2

Substitute P, R, and T

= $1020 Evaluate

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Example 2 Sally made loan repayments totalling to $4320 on a loan of $4000 over 4 years. a Calculate I, the total interest charged.

Create a strategy Subtract the initial value of the investment from the ﬁnal value of the investment.

Apply the idea I = 4320 – 4000

Subtract the initial value from the ﬁnal value

= $320

Evaluate

b Calculate R, the annual interest rate.

Create a strategy We can use the simple interest formula: I = PRT.

Apply the idea Substitute I = 320, P = 4000, T = 4

Evaluate

Divide both sides by 16 000

Evaluate

Convert the rate into a percentage

Example 3 For a simple interest rate of 6% p.a., calculate the number of years T, needed for $1957 to be invested in order to earn $1174.20 in interest.

Create a strategy We can use the simple interest formula: I = PRT.

Apply the idea From the given, we have I = 1174.20, P = 1957, and R = 6% = 0.06.

Substitute I, P, and R Evaluate

Divide both sides by 117.42

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Evaluate

Idea summary We can calculate the simple interest of a loan or investment using the formula:

I = PRT I P R T

is the interest to be paid is the principal amount borrowed or invested is the rate of interest per year is the time in number of years

Taxation rates The total tax payable by most individuals will be made up of income tax and the Medicare levy, both of which are based on their taxable income for the ﬁnancial year. The correct amount of income tax for Australian residents, not including the Medicare levy, is determined from the following table: Resident tax rates (2024-2025) Taxable income 0 − $18 200 $18 201 − $45 000 $45 001 − $135 000 $135 001 − $190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

The use of this table is best explained using an example. To calculate the tax on a taxable income of $50 000 we complete the following steps. 1.

Determine the tax bracket the income falls into using the left-hand column of the table. $50 000 falls in the $45 001 − $135 000 bracket. Reading across the row we can see that the tax payable is: $4288 plus 30c for each $1 over $45 000

Calculate additional tax The $4288 covers tax for the ﬁrst $45 000 of the income. $50 000 is $5 000 over $45 000. On this $5 000 we pay 30c in each $1. This is calculated as: 0.30 × $5 000 = $1 500

Calculate total income tax Total income tax = $4288 + $1500 = $5788

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Calculate Medicare levy The Medicare levy is an additional tax that is paid to help fund Medicare, Australia’s public health care system. It is calculated as: Medicare levy = 2% of taxable income $50 000 × 0.02 = $1 000

Calculate total tax payable The total tax payable is found by adding together the income tax and the Medicare levy. Total tax payable = income tax + Medicare levy = $5788 + $1000 = $6788

An individual’s net income can be calculated by subtracting their total tax payable from their taxable income. net income = taxable income − total tax payable Therefore, the net income of someone who has a taxable income of $50 000 is: $50 000 − $6788 = $43 212

Example 4 Last ﬁnancial year, Roxanne earned $157 872. Using the income tax rates shown in the table, evaluate the following. Resident tax rates (2024-2025) Taxable income 0 − $18 200 $18 201 − $45 000 $45 001 − $135 000 $135 001 − $190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

a Calculate the income tax payable on Roxanne’s income.

Create a strategy Use the income bracket where Roxanne’s income lies, and use the necessary rule.

Apply the idea Income tax payable = 31 288 + (157 872 − 135 000) × 0.37

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Substitute the values

= 31 288 + 22 872 × 0.37

Evaluate the subtraction

= 31 288 + 8462.64

Evaluate the multiplication

= $39 750.64

Evaluate

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b Calculate the Medicare levy payable, given that it is 2% of taxable income. Round to the nearest cent.

Create a strategy Multiply Roxanne’s income by 2%.

Apply the idea Medicare levy payable = 157 872 × 0.02 = $3157.44

Substitute Income = 157 872 Evaluate

c Calculate the total amount Roxanne needs to pay in income tax and Medicare levy.

Create a strategy Add the income tax payable from part (a) and the Medicare levy payable from part (b).

Apply the idea Total amount payable = 39 750.64 + 3157.44 = $42 908.08

Add the payables altogether Evaluate

Idea summary Most working people in Australia pay income tax and the Medicare levy. Medicare levy = 2% of taxable income Total tax payable = income tax + Medicare levy Net income = taxable income − total tax payable

Exchange rates Different countries throughout the world have their own currency. In Australia, the currency is the Australian Dollar, which is represented by the three letter code AUD. If we know the value of 1 AUD in another currency, this is called the exchange rate. To convert AUD to another currency we multiply by the exchange rate. To convert another currency to AUD we divide by the exchange rate. If you are referring to the currency code, such as AUD, USD or NZD then there is no need to use the $, although it’s still used in some places to help reflect that it is a currency. Some countries do not call their currencies dollars, and have other names, such as Euro, Pounds or Yen. These currencies would not have a $ sign, they have their own symbols.

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Example 5 Assuming that 1 AUD can buy 0.90 USD, how many USD is equivalent to 79 AUD?

Create a strategy

Apply the idea

The question is asking:

79 AUD = 79 × 0.90 USD

if then

1 AUD = 0.90 USD

= 71.1 USD

Multiply 79 by 0.90 Evaluate

79 AUD = ⬚ USD

Multiply both parts of the equation 1 AUD = 0.90 USD by 79 AUD.

Example 6 Assuming that 1 AUD can buy 0.90 USD, how many AUD is equivalent to 29 USD?

Create a strategy

Apply the idea

Divide the amount of USD by the amount of USD that can be bought with 1 AUD.

Divide 29 by 0.90 Evaluate

Reflect and check If

1 AUD = 0.90 USD

then ⬚ AUD = 29 USD

To solve this we could also go via the unit rate for USD. If 1 AUD = 0.90 USD 1 AUD = 1 USD 0.90 Then 32.2 AUD = 29 USD

Divide both sides by 0.90 Multiply both sides by 29

Idea summary If we know the value of 1 Australian dollar in another currency this is called the exchange rate. To convert AUD to another currency we multiply by the exchange rate. To convert another currency to AUD we divide by the exchange rate.

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3.04 Practice questions Understanding 1

Explorer

Adventurer

1–4

Trailblazer 1–4

Using the simple interest formula I = PRT, what does I represent? A

The ﬁnal amount of a loan or investment after T years.

The interest on a loan or investment after T years.

What percentage of their income is the Medicare levy for most people?

What currencies are represented by these codes? a

AUD

Ex 1

6 Ex 2

7 Ex 3

USD

EUR

IDR

1 Australian dollar is equal to approximately 1.09 New Zealand dollars. a

To convert from AUD to NZD do you multiply or divide by the exchange rate?

To convert from NZD to AUD do you multiply or divide by the exchange rate?

Fluency 5

5, 6a–c, 7–9, 10a–c, 11–13

5, 6a–d, 7–9, 10b–d, 11–13

5, 6b–e, 7–9, 10b–f, 11–14

Calculate the simple interest for each scenario: a

An investment of $8840 at 6% p.a. for 2 years

A loan of $8000 at 8% p.a. for 6 years

An investment of $2540 at 9% p.a. for 2 years

An investment of $7000 at 7.2% for 9 years

Find the annual interest rate as a percentage to one decimal place for each scenario: a

The simple interest on an investment of $2650 over 3 years is $787.05.

The simple interest on a loan of $2830 over 10 years is $2603.60.

The simple interest on an investment of $3600 over 2 years is $504.00.

The simple interest on a loan of $7550 over 9 years is $6523.20.

The simple interest on a loan of $6600 over 33 months is $1252.35.

$552 is invested at 5% p.a simple interest. Dave wants to know the number of years it will take the investment to grow to $828.00. a

Calculate the interest that will be earned on the investment.

Calculate the number of years it will take the investment to grow to $828.00

Find the total amount to be paid on a 4-year, $90 000 loan at 12% p.a. ﬂat interest.

Isabelle is a working holiday maker. She has a job in Australia and her taxable income for the year is $47 600. Use the table to ﬁnd how much income tax she is required to pay: Working holiday makers income tax rates for 2024 to 2025 Taxable income $0 – $45 000 $45 001 – $135 000 $135 001 – $190 000 $190 001 and over

Tax on this income 15c for each $1 $6750 plus 30c for each $1 over $45 000 $33 750 plus 37c for each $1 over $135 000 $54 100 plus 45c for each $1 over $190 000 3.04 Financial rates mathspace.co

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The table shows the tax rates for 2024 to 2025: Resident tax rates (2024-2025) Taxable income 0 – $18 200 $18 201 – $45 000 $45 001 – $135 000 $135 001 – $190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

Calculate the income tax payable for these taxable incomes:

11 Ex 4

$8326

$36 810

$194 388

$225 810

$93 395

$146 664

Last ﬁnancial year, Roxanne’s taxable income was $197 955. Based on the Resident Tax Rate (2024-2025) table, calculate: a

The income tax payable on Roxanne’s income.

The Medicare levy payable, given that it is 2% of taxable income.

The total amount Roxanne needs to pay in income tax and Medicare levy.

Ex 5

Assuming that 1 AUD can buy 5.25 HKD (Hong Kong Dollars), how many HKD’s are equivalent to 36 AUD?

Assuming that 1 AUD can buy 5.25 HKD, how many AUD’s are equivalent to 57 HKD?

Ex 6

Assuming that 1 AUD can buy 0.67 USD, how many USD’s are equivalent to 88 AUD?

Reasoning

15–17

15, 17, 18

Luke’s investment of $3000 earned simple interest of 6% p.a. for the ﬁrst 8 years and 2% p.a. for the next 8 years. Luke says this is the same as earning 4% p.a. for 16 years. Is Luke correct? Show your working.

Most people pay income tax and the Medicare levy, which is charged at 2% of taxable income. Janet has a taxable income of $17 000 and says that her total payment to the tax department is 2% of her income. Is Janet correct? Explain your answer.

Suppose that 1 AUD is equivalent to 0.75 USD. James wants to exchange 1000 AUD to USD ready for his holiday to Disneyland. His calculation is: 1000 ÷ 0.75 = 1333.33 USD Is his calculation correct? Explain your answer.

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The table shows the tax rates for 2024 to 2025: Resident tax rates (2024-2025) Taxable income 0 – $18 200 $18 201 – $45 000 $45 001 – $135 000 $135 001 – $190 000 $190 001 and over

Tax on this income Nil 16c for each $1 over $18 200 $4288 plus 30c for each $1 over $45 000 $31 288 plus 37c for each $1 over $135 000 $51 638 plus 45c for each $1 over $190 000

A proposal suggested that to help those on low incomes the tax-free upper bound be increased from $18 200 to $20 000. Part of the proposal is that the tax paid by people earning more than $45 001 should remain the same. In order to achieve this it is proposed that those earning between $20 000 and $45 000 should pay 18c for each $1 instead of the current 16c for each $1. a

Will this proposal achieve its stated aim? Explain.

An alternative proposal is to raise the rate for those earning between $20 000 and $45 000 to pay 18c for each $1 over $20 000. What effect would this have on people earning more than $45 000?

If the 18c for each $1 over $20 000 is adopted, what change would it make to the instructions for ‘Tax on this income’ for those earning between $135 001 and $190 000?

Problem-solving 19–21 19

19–22

The graph shows the amount of simple interest charged each year by a particular bank, on some 9-year loans. a

Find the total amount of simple interest charged on a loan of $4000.

Calculate the simple interest rate per annum charged by the bank on these 9-year loans. Give your answer as a percentage.

19, 20, 23, 24 Interest 300 250 200 150 100 50 Principal 1000 2000 3000 4000

If $2196 is invested at 2% p.a. simple interest for 8 years, what simple interest rate would earn the same amount of interest in only 5 years? Give your answer as a percentage rounded to two decimal places.

Emma is a researcher with a taxable income of $97 589. Throughout the year, her employer has deducted $560 per week in pay-as-you-go (PAYG). a

Calculate the income tax payable using the Resident Tax Rate (2024-2025) table.

Calculate the Medicare levy, which is 2% of taxable income.

Since Emma’s employer has deducted more than is needed from her income, Emma is entitled to a refund for the year. Find the amount of this refund.

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Elias has just been hired in a new role where he will earn a fortnightly income of $5697 for an entire year. a

Calculate how much Elias earns in a year. Assume 26 fortnights in a year.

Calculate the income tax payable on Elias’s annual income using the Resident Tax Rate (2024-2025) table.

Elias also needs to pay $113.94 alongside his income tax. How much PAYG tax would his employer need to withhold from his fortnightly pay so that Elias’s total PAYG tax instalments for the year exactly cover his total tax payable?

Cinta has a gross weekly income of $1023. Each week, her employer withholds $193 in PAYG tax. At the end of the ﬁnancial year, Cinta has to submit a tax return. She has $842 in allowable tax deductions for the year. Assuming there are 52 weeks in a year, calculate:

Her taxable income.

Her income tax payable using the Resident Tax Rate (2024-2025) table.

How much Cinta pays towards the Medicare levy, which is 2% of her taxable income.

How much Cinta has already paid in PAYG tax instalments throughout the year.

How much Cinta is entitled to as a tax refund.

If 80 AUD is worth 56 USD, how much is 350 USD worth in AUD?

3.05 Best buys After this lesson, you will be able to… • calculate the unit price of items in order to compare their value. • determine best buy using unit price. • compute quantity per unit cost. • calculate the sale price using discounts. • determine best buy using discounts

Cost per unit If the two options we are choosing between involve different quantities, we can use what we know about rates to ﬁnd out how much a single unit costs. For example, if something costs $4 for 2 kg, the unit price would be $2 per kg.

Example 1 Isabelle is buying juice for her nephew’s birthday party. A 3.2 L bottle of apple juice costs $13.76. A 2.1 L bottle of orange juice costs $6.30.

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a How much does apple juice cost per litre?

Create a strategy Divide the cost of the bottle by the number of litres.

Apply the idea Apple juice price = 13.76 ÷ 3.2

Divide 13.76 by 3.2

= $4.30

Perform the division

b How much does orange juice cost per litre?

Create a strategy Divide the cost of the bottle by the number of litres.

Apply the idea Orange juice price = 6.30 ÷ 2.1 = $3.00

Divide 6.30 by 2.1 Perform the division

c Which juice is the best buy? A

Orange juice

Apple juice

Create a strategy

Apply the idea

Compare the answer in parts (a) and (b) and choose which juice is cheaper per litre.

The price of apple juice per litre is $4.30 while the price of orange juice per litre is $3.00. So the best buy is the orange juice. So the correct answer is option A.

Idea summary In choosing between two options, we can ﬁnd the cost per unit to determine which is the better buy.

Quantity for unit cost We can also compare prices by determining how much a single dollar buys. The idea is very similar to the previous one, but we now divide the quantity by the number of dollars. For example, if something costs $4 for 2 kg, the quantity per unit cost would be 0.5 kg for each $1.

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Example 2 A 1.35 kg packet of mince costs $15. How many grams of mince does each dollar buy?

Create a strategy To get the weight per dollar, divide the total weight of the mince by the number of dollars it costs.

Apply the idea $1 buys = 1.35 kg ÷ $15

Divide by 15

= 0.09 kg

Perform the division

= 0.09 × 1000 g

Convert to grams

= 90 g

Evaluate

Idea summary We can also compare prices by determining how much a single dollar buys. Notice that when using this method, the rate with the larger unit value represents the better value, as we get more for the same price.

Discounts Stores frequently offer a percentage discount on items which converts the original price into a sale price. This discount is often presented as “30% off” or “reduced by 25%”, and it is important to know how to calculate the new price. When a sale price has a percentage discount applied, we ﬁnd that percentage of the original price and subtract it to ﬁnd the sale price: Sale price = Original price − (Original price × Discount percent) We can take a shortcut by thinking about what happens after the discount is applied. If an item is reduced by 30%, then the sale price is 70% of the original price. Similarly, a 25% discount means the sale price is 75% of the original price. The remaining percent is 100 minus the discount.

Example 3 A pair of Bevi jeans normally costs $120, and the store is offering a 20% discount. A similar pair of Genie jeans normally costs $150, and the store is offering a 30% discount. Which pair of jeans is the best buy?

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Create a strategy Find the remaining percentages and multiply them by the original prices.

Apply the idea A 20% discount is the same as paying 80% of the original price, so we should ﬁnd 80% of $120. Similarly, a 30% discount is the same as paying 70% of the original price, so we should ﬁnd 70% of $150. For Bevi jeans: Sale price = 80% × 120

Find the remaining percent

= 0.80 × 120

Convert the percentage to a decimal

= $96

Evaluate

For Genie jeans: Sale price = 70% × 150 = 0.70 × 150

Find the remaining percent Convert the percentage to a decimal

= $105 $96 < $105 so the best buy is the pair of Bevi jeans.

Idea summary When a sale price has a percentage discount applied, we ﬁnd that percentage of the original price and subtract it to ﬁnd the sale price: Sale price = Original price − (Original price × Discount percent) We can also use a shortcut by multiplying the original price by the remaining percent which is 100% minus the discount percentage.

3.05 Practice questions Understanding

Explorer 1–3

Adventurer 1–3

A carton of 10 eggs cost $6.50. Calculate the cost of 1 egg in cents.

What does it mean for an item to be a better buy or the best buy?

Which is the better buy: 30 cents per apple or $1 buys 4 apples?

Trailblazer 1–3

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Fluency Ex 1

4–8, 10

4, 5, 7–10

Charma is buying juice for her nephew’s birthday party. • A 5.2 L bottle of apple juice costs $13.76 • A 2.1 L bottle of orange juice costs $6.30

Calculate the cost per litre of an apple juice.

Calculate the cost per litre of an orange juice.

Which juice is the best buy?

Sally is deciding which of two different cereals she should buy. • Choco Pops costs $0.84 for 200 g • Special Q costs $0.36 for 100 g

Calculate the cost per kilogram of Choco Pops.

Calculate the cost per kilogram of Special Q.

Which of the cereal brands is the best buy?

A supermarket sells two different brands of eggs, Happy Hens and Sunny Side Up. • Happy Hens eggs cost $6.00 for 12 eggs • Sunny Side Up eggs cost $1.50 for 5 eggs

8 Ex 2

Ex 3

Calculate the cost in cents per egg from Happy Hens.

Calculate the cost in cents per egg from Sunny Side Up.

Which brand is the better buy?

Iain feels like buying some ice-cream for himself and his friends. • A box of 8 Cornettos costs $18.00 • A box of 5 Magnums costs $7.25 a

Calculate the cost of a Cornetto.

Calculate the cost of a Magnum.

Which type of ice-cream is the better buy?

A hardware store sells nails in a box for $10. The box contains 200 nails. How many nails does each dollar buy?

At Emporio, durian is sold for 76 cents per kilogram. At Greenery, each 5.5 kg durian is sold for $2.97. Which store is the best buy?

A bookstore sells two different series of books, The Adventures Series and The Mystery Series. The Adventures Series costs $100 for 10 books and is currently being offered at a 25% discount. The Mystery Series costs $80 for 8 books and is currently being offered at a 20% discount.

152

Calculate the cost per book for The Adventures Series after the discount.

Calculate the cost per book for The Mystery Series after the discount.

Which series of books is the better buy?

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Reasoning

11, 12

12, 13

Fiona wants to buy some salmon from Whole-sail Seafood. She can purchase a 1.9 kg Atlantic salmon or a 2 kg Tasmanian salmon for $31.35 or $51.00, respectively. Explain how to ﬁnd which salmon is the better buy.

Consider each scenario: • A 200 m2 apartment is sold for $440 000 • An adjacent apartment that is 220 m2 in size is sold for $440 000 Without calculating the price per square metre of each apartment, explain how you know which apartment is the best value per square metre.

A 0.75 L bottle of milk costs $1.05. a

Calculate the number of litres for each dollar in two decimal places.

Explain why the answer to part (a) is not the unit price.

Problem-solving 14, 15 14

14, 15

14–16

Consider each scenario: • Han paid $124.62 for 67 L of petrol in Humbleton • Amelia paid $82.08 for 57 L of petrol in Dunkilderry a

Calculate the cost per litre of petrol in Humbleton.

Calculate the cost per litre of petrol in Dunkilderry.

In which suburb is petrol the better buy?

Han buys 30 L in Statesota, and this petrol station matched the best price between Humbleton and Dunkilderry. How much did he pay?

A local grocery store is offering two different deals on cereal. The ﬁrst deal is a buy one get one 50% off offer on a box of Bouncy Berries priced at $6. The second deal is a buy two get one free offer on a box of Krispy Krunchers priced at $5. Which deal offers the better value for money?

You want to buy 21 backpacks. Consider each scenario: • Barry’s Bargain Bins sells backpacks for $5.00 each, and if you buy 5 backpacks the 6th is free. • Lucie’s Low Price Supplies sells backpacks for $6.50 each, and if you buy 6 backpacks the 7th is free. a

How many free backpacks do you get at these stores? i

Barry’s Bargain Bins

Lucie’s Low Price Supplies

In which store would it be better to buy 21 backpacks?

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3 Chapter review 1

I f the scale ratio on a diagram is 1 : 500, is the real length smaller or larger than the diagram length?

If the scale ratio on a diagram is 30 : 1, is the real length smaller or larger than the diagram length?

Which is the better deal: 55 cents per pear or $1.50 gets you 3 pears?

Complete the equivalent ratios: a e

⬚ : 4 = 60 : 20 ⬚:2=6:5

b f

⬚ : 6 = 168 : 72

7:⬚=1:3

⬚:7=8:9

11 : 5 = ⬚ : 6

Express as a simpliﬁed ratio:

d h

75 cents to $2.40

4.6 kg to 200 g

7 years to 42 months

840 hours to 85 days

10 : 9 = ⬚ : 36 8 : 5 = ⬚ : 25

A cake recipe calls for a 2 : 4 ratio of ﬂour to sugar. If you have 400 g of ﬂour, how much sugar is needed?

Convert to a scale ratio: a

2 cm represents an actual distance of 8 m

25 cm represents an actual distance of 20 km

7 mm represents an actual distance of 35 m

70 mm represents an actual distance of 6300 m

6 cm represents an actual distance of 36 km

100 mm represents an actual distance of 10 km

The scale on a map of a botanical garden is 1 : 400. How far apart on the map should two particular trees be marked if the actual distance between the trees is 8 m? Express your answer in centimetres.

Shown here is a map of New York. The actual dimensions of one rectangular block are 200 m by 60 m.

Zara

7th Avenue

6th Avenue

Jason

5th Street 6th Street

Harper

Prospect Park West

Fiske Place

7th Street Eli

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8th Street

How many metres will Eli need to drive to reach the pick-up point, marked with an x?

Whose car is the farthest to the pick-up location?

Calculate the unit rate in each scenario: a

1200 pages printed by 15 printers

100 students in 5 classrooms

270 litres consumed over 5 days

10 loaves of bread for 80 people

A pump ﬁlls up a 300 litre swimming pool in 8 hours. Find the ﬂow rate of the pump in litres per hour.

Consider the rate 200 metres per 250 seconds.

Express this as a unit rate in terms of metres and seconds.

Express this as a unit rate in terms of metres and minutes.

Convert the rates. Round your answer to two decimal places when necessary. a

8 L/h to mL/h

90 words/min to words/h

720 km/h to m/h

85 beats/min to beats/h

6.0 L/h to mL/min

30 m/min to cm/sec

600 mL/g to L/kg

900 cm/min to m/h

Calculate the simple interest on: a

An investment of $9500 at 5% p.a. for 3 years

A loan of $6500 at 7% p.a. for 4 years

An investment of $3000 at 8% p.a. for 3 years

An investment of $8000 at 6.5% for 5 years

$620 is invested at 4% p.a simple interest. Emma wants to know the number of years it will take the investment to grow to $930. a

Calculate the interest that will be earned on the investment.

Calculate the number of years it will take the investment to grow to $930

Andrew is on a working holiday and has secured employment in Australia. His annual taxable income amounts to $58 400. Using the table provided, determine the total income tax he is obligated to pay: Table B: Income tax rates for working holiday makers for 2023 to 2024 Taxable income

Tax on this income

$0 - $45 000

15% on each $1 up to $45 000

$45 001 - $120 000

32.5% for each $1 over $45 000 to $120 000

$120 001 - $180 000

37% for each $1 over $120 000 to $180 000

$180 001 and over

45% for each $1 over $180 000

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James is deciding which of two different brands of pasta he should buy. • Twisty Pasta costs $1.20 for 250 g • Pasta O’s costs $0.50 for 100 g a

Calculate the cost per kilogram of Twisty Pasta.

Calculate the cost per kilogram of Pasta O’s.

Which of the pasta brands is the best buy?

At Fruitland, oranges are sold for 90 cents per kilogram. At VeggieMart, each 4.5 kg sack of oranges is sold for $3.60. Which store is the best buy?

Hiroshi, Mei, Ling, and Juan shared a bottle of tea. The ratio of the amount of tea Hiroshi, Mei, Ling, and Juan received is 2.8 : 3.6 : 4.2 : 0.6 respectively. The size of the tea is 550 ml. How many millilitres of tea did Hiroshi drink?

Eduardo is a writer and he spent a total of 50.25 hours researching, drafting and editing a new manuscript. He spent 24.50 hours researching and 12.25 hours drafting. a

How many hours did Eduardo spend editing the manuscript?

Find the ratio in which his time was divided between researching, drafting and editing.

The scale on the map of Lumina town is 1 : 4000. How far apart on the map should two coffee shops be if the actual distance between the coffee shops is 200 m? Give your answer in centimetres.

An engineer is planning a bridge that is 24 m long and 16 m wide. a

She wants to sketch the blueprints for this bridge on a 96 cm by 64 cm section of a large piece of paper. Find the ratio she should use to ﬁt the blueprints on the paper.

After sketching the blueprints, the engineer measures the length of the walkway to be 28 cm on paper. She wants to order steel beams that cover the full length of the walkway. How long should the steel beams be in metres?

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Daniel and Oliver play football over the summer holidays. Daniel has a rate of 28 goals per 3 matches and Oliver has a rate of 28 goals per 5 matches. If there are 35 matches in the season, how many goals did Daniel and Oliver score in total?

A snail crawls at a pace of 10 seconds for 5 cm.

How long does it take for the snail to crawl 1 metre?

Find the rate of the snail’s time spent per metre distance crawled.

Oliver has a large container that can store 500 kL of oil. The container is currently empty. He has two pumps that he can use to fill the container. The first pump can fill the container in 25 minutes if used alone and the second pump can fill the container in 30 minutes if used alone. a

Find the rate at which the ﬁrst pump delivers oil into the container in

Find the rate at which the second pump delivers oil into the container in

How long would it take to ﬁll the container from empty using both pumps?

The graph represents the annual simple interest accrued by a certain ﬁnancial institution on several 7-year loans. Interest ($) 350 300 250 200 150 100 50 Principal ($) 1000 2000 3000 4000 5000

Determine the total simple interest accrued on a loan amounting to $5000.

Find the annual simple interest rate levied by the institution on these 7-year loans. Present your answer in percentage form.

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Benjamin recently secured a job which pays him a monthly salary of $4890 for a full year. The table shows the tax rates for 2024 to 2025: Resident tax rates (2024-2025) 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

Determine Benjamin’s total annual earnings.

Compute the amount of income tax Benjamin has to pay on his annual earnings using the Resident Tax Rate (2024-2025) table.

In addition to income tax, Benjamin must also pay $97.60 per year. What would be the PAYG tax that his employer should withhold from his monthly paycheck so that Benjamin’s total PAYG tax instalments for the year accurately cover his total payable tax?

John receives a weekly salary of $1150. Each month, his employer withholds $880 monthly in PAYG tax. At the end of the ﬁscal year, John must ﬁle a tax return. He has $1024 in permissible tax deductions for the year which can be deducted from his total taxable income before tax is calculated. The table shows the tax rates for 2024 to 2025: Resident tax rates (2024-2025) 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

Considering there are 52 weeks in a year, compute:

158

His taxable income

His income tax due using the Resident Tax Rate (2024-2025) table

The amount John contributes towards the Medicare levy, which is 2% of his taxable income.

The total amount John has already paid in PAYG tax instalments during the year.

The amount John is entitled to as a tax refund.

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If 100 GBP (Great British Pound) is equivalent to 130 EUR (Euro), how much is 500 EUR worth in GBP?

Assuming that 1 AUD can buy 0.55 GBP, how many GBP’s are equivalent to 48 AUD?

Assuming that 1 AUD can buy 0.64 EUR, how many AUD's are equivalent to 72 EUR?

Assuming that 1 AUD can buy 1.07 NZD (New Zealand Dollar), how many NZD’s are equivalent to 95 AUD?

For the scenarios: • Jackson paid $158.42 for 82 L of diesel in Applewood • Lily paid $104.16 for 64 L of diesel in Stonebridge

Calculate the cost per litre of diesel in Applewood.

Calculate the cost per litre of diesel in Stonebridge.

In which town is diesel the better buy?

Jackson buys 35 L in Oakhaven, and this diesel station matched the best price between Applewood and Stonebridge. How much did he pay?

You plan to buy 25 notebooks. Consider each scenario: • Danny’s Discount Depot sells notebooks for $4.00 each, and if you buy 6 notebooks the 7th is free. • Caroline’s Cost Cutters sells notebooks for $5.00 each, and if you buy 7 notebooks the 8th is free. a

How many free notebooks do you get at these stores? i

b 34

Danny’s Discount Depot

Caroline’s Cost Cutters

In which store would it be better to buy 25 notebooks?

A nearby supermarket is presenting two different discounts on biscuits. The ﬁrst discount is a buy one get one 40% off deal on a packet of Jammy Dodgers priced at $8. The second discount is a buy three get one free offer on a packet of Crunchy Cookies priced at $6. Which discount offers the better value for money?

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Big ideas Algorithms provide structured methods to automate tasks, identify and fix errors, and outline clear steps for problem-solving.

Curriculum Outcomes VC2M8A04 Use algorithms and related testing procedures to identify and correct errors

4 Algorithms and coding Chapter outline 4.01 4.02 4.03 4.04

Algorithms in code Loops Debugging Algorithms for divisibility Chapter 4 review

162 170 180 189 195

Do you know why it’s called debugging? The first was an actual bug! A moth was trapped inside one of the earliest computers, causing it to malfunction.

4.01 Algorithms in code After this lesson, you will be able to… • recognise common notations in pseudocode such as INPUT, OUTPUT, SET, IF-THEN-ELSE. • translate a flow chart into pseudocode. • apply the IF-THEN-ELSE statement to make decisions in code. • interpret pseudocodes and understand what they represent.

Pseudocode and its common notations Before translating an algorithm into a specific programming language, or language that a computer can understand, the steps of the algorithm can be written as pseudocode. Pseudocode is a way of writing code that is human-friendly. With pseudocode, we don’t need to worry about writing an algorithm in the specific way the computer needs. Some common notations for pseudocode are: • INPUT - indicates a user will be inputting something. • OUTPUT or PRINT - indicates that an output will appear on the screen. • SET - assign an initial value to an expression. START

This algorithm (ﬂow chart) can be written in pseudocode as shown.

SET a = 3

SET b = 5 PRINT a + b

END

START

SET a = 3

SET b = 5

PRINT a + b

END

To choose between two options in code, we use the IF - THEN - ELSE statement.

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Example 1 Examine the given pseudocode. 1

START

SET x = 10

IF x IS AN EVEN NUMBER

4 THEN x = x + 1 5

END IF

PRINT x * 2

END

What is the printed number at the end if the initial value is a 23?

Create a strategy Create a flow chart from the code, then follow the flow chart using the initial value.

Apply the idea The flow chart shows to start with the initial value, 23.

START

Since 23 is not even, double it: 23 x 2 = 46

Start with the number ____

Add one

YES

Is the number even?

So the printed value is 46.

Double it

END

b 24?

Create a strategy Create a flow chart from the code, then follow the flow chart using the initial value.

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Apply the idea The flow chart shows to start with the initial value, 24.

START

Since 24 is even, add 1: 24 + 1 = 25

Start with the number ____

Then, double the sum: 25 × 2 = 50 So the printed value is 50.

Add one

YES

Is the number even?

Double it

END

Example 2 Refer to the rectangle displayed in the image. The rectangle’s dimensions are W metres for width and L metres for length.

W L

a What does the following pseudocode calculate? 1

START

SET W = 4

SET L = 7

SET P = 2*(W + L)

PRINT P metres

END

Please note: the symbol ∗ indicates multiplication. A Volume

Units

Area

Perimeter

Apply the idea The pseudocode PRINTs the value of P, which is the result of multiplying the sum of the rectangle’s length and width by 2. This calculates the rectangle’s perimeter. The correct response is D.

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b What value is printed at the end of the code?

Create a strategy Follow the pseudocode, applying the provided values for W and L.

Apply the idea 1

START

SET W = 4

SET L = 7 L = 7

SET P = 2*(W + L) P = 2 × (4 + 7)

W = 4

= 2 × 11 = 22 5

PRINT P metres

END

Print 22 metres

So the pseudocode prints 22 metres.

Example 3 Look at the pseudocode below: 1

START

SET a = 15

IF a / 2 IS whole number

4 THEN PRINT a is even 5 ELSE PRINT a is odd 6

END IF

END

What statement is printed at the end of the code? Note: the symbol / represents division.

Create a strategy Follow the pseudocode for the value of a.

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Apply the idea

Since

a 15

Step 2

a 7.5 2

Divide a by 2 for Step 3

a 7.5 is not a whole number, the code would skip Step 4 and PRINT: a is odd. 2

Idea summary Common notations for pseudocode: • • • •

INPUT - indicates a user will be inputting a command. OUTPUT or PRINT - indicates that an output will appear on the screen. SET - assign an initial value to an expression. IF - THEN - ELSE - a decision, where a choice is made.

4.01 Practice questions Understanding

Explorer 2, 3

What is the purpose of pseudocode?

Consider the pseudocode: 1

START

SET a = 6

SET b = 4

SET c = a * b

PRINT c

END

Adventurer 1–3

Trailblazer 1–3

Define what the pseudocode commands START, SET and PRINT mean. 3

166

What pseudocode notation would be used if there was a decision required in an algorithm?

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Fluency

Ex 1

4, 5, 7

5–7

6–8

Evaluate the ﬁnal output of these pseudocodes: a

START

SET a = 80

SET a = 29

SET b = 20

SET b = 9

SET c = a – b

SET c = a + b

PRINT c

END

START

SET a = 3

SET a = 6

SET b = a ∗ a

SET b = 4

PRINT b

SET m = (a ∗ b)/2

END

PRINT m END

START

SET a = 5

SET a = 82

SET b = 2

SET b = a + 18

SET c = 8

PRINT b

SET d = a ∗ b + c

END

PRINT d END Ex 2

START

For the pseudocode: What does the code compute? Volume

Units

SET L = 14

Area

Perimeter

SET A = W ∗ L

What is the output of this code?

PRINT A m2 END START

For the pseudocode: a

SET W = 11

What does the code compute?

SET W = 8

Angle

Diameter

SET L = 10

Area

Perimeter

SET P = 2 ∗ W + 2 ∗ L

What is the output of this code?

PRINT P m END

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Ex 3

Which statement is printed at the end of the pseudocode? 1

START

SET a = 21

IF a/2 gives a whole number result

4 THEN PRINT a is even 5 ELSE PRINT a is odd

END IF

END

Which statement is printed at the end of the pseudocode? 1

START

SET a = 42

IF a/3 gives a whole number result

4 THEN PRINT a is divisible by 3 5 ELSE PRINT a is not divisible by 3 6

END IF

END

Reasoning

9, 11

10, 11

The pseudocode to determine the product of three numbers is displayed: Step A B C D E F

Instruction SET a = 3 END SET b = 4 SET c = 2 PRINT a ∗ b ∗ c START

Jonas believes that the pseudocode sequence is in the right order, while Lyla argues that the sequence is not correct. Who is correct? Provide a reason for your answer. 10

In a pseudocode, there is a statement that says “PRINT a/b”. What potential problem can arise if the variable b is set as 0?

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The provided pseudocode seems to determine the perimeter of a square. Why is the S (the side of the square) being multiplied by 4? 1

START

SET S = 5

SET P = 4 * S

PRINT P m

END

Problem-solving 12–14 12

12–15

12, 14–16

Consider this algorithm: START Start with a number

Is it a whole number?

YES

Is it an even number?

YES

Add one

Subtract one

Divide by two

Round to nearest whole number END

b 13

Will you always get a whole, even number out when you start with: i

A decimal number?

An even number?

iii

An odd number?

Give examples of any numbers that do not give an even, whole number.

How could you change this algorithm so that it does always give a whole, even number?

Modify the pseudocode such that it calculates the average of a and b and prints the result. 1

START

SET a = 3

SET b = 7

SET c = a + b

PRINT c

END

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Draft a pseudocode that computes and gives the output of a number (m) cubed.

You’re given two pseudocodes. The ﬁrst determines the area of a rectangle and the second determines the volume of a rectangular prism. Draft the two pseudocodes.

Write a pseudocode to calculate the average of three numbers (x, y and z) and display the result. Then, swap the values of x and y and display the new values of x and y.

4.02

Loops

After this lesson, you will be able to… • recognise the structure and function of a WHILE loop. • analyse and interpret pseudocode with iterations. • apply the concept of loops to create and modify algorithms. • connect the understanding of loops to real-world scenarios and problems.

Loops Loops, also known as iterations, are where certain steps in an algorithm are executed multiple times. Let’s look at the steps involved in watering a garden. Instead of listing out each plant, we can use a loop to simplify. START

START

Go to the tap and turn on the hose.

Walk to the first plant with the hose.

Start with the first plant.

Water the plant for 30 seconds.

Move to the next plant. Water the plant for 30 seconds.

Are there plants left?

Move to the next plant.

Water the plant for 30 seconds.

END

Repeat this until all plants are watered. END

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Yes

Move to the next plant.

In pseudocode, we often use a WHILE loop. The WHILE loop has a condition at the beginning that decides how many times the loop should run. This pseudocode contains a WHILE loop: 1

START begin

SET b = 3

b starts at 3

SET total = 0

we start with total as 0

WHILE b < 6

do the next steps while b is under 6

5 total = total + b

add b to the current total

6 b = b + 1

add 1 to b

END WHILE

stop when b hits 6

PRINT total

show the final total

END done

The loop includes these steps: total = total + b b=b+1 The steps repeat until b reaches or passes 6. The table shows how b and total, change with each loop: at the start after 1st loop after 2nd loop after 3rd loop

total 0 0+3=3 3+4=7 7 + 5 = 12

b 3 3+1=4 4+1=5 5+1=6

After the third loop, b is 6, so the loop ends. The ﬁnal result for total is 12. This is the number shown at the end of the pseudocode.

Example 1 Examine the pseudocode provided: 1

START

SET b = 4

SET product = 1

WHILE b < 7

5 product = product * b 6 b = b + 1 7

END WHILE

PRINT product

END

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a What is the initial value of b?

Create a strategy Find the value SET for b.

Apply the idea In line number 2, it is stated that the starting value SET for b is 4.

b What will the OUTPUT of this code be?

Create a strategy Construct a values table for the loop at Steps 5 and 6.

Apply the idea Creating a values table for every loop until b = 7 is achieved: to start post first loop post second loop post third loop

product 1 1∗4=4 4 ∗ 5 = 20 20 ∗ 6 = 120

b 4 4+1=5 5+1=6 6+1=7

After the third loop with b = 7, the loop concludes. The value of product is now 120. Given that the ﬁnal command is to PRINT product, the code’s output will be 120.

Example 2 Consider the algorithm represented by the ﬂow chart: START sum = 0 a=2 a = 10

Yes PRINT sum END

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sum = sum + a

a=a+2

C:a=a+2 E : SET a = 2 A : WHILE a ≠ 10 F : PRINT sum B : sum = sum + a G : END WHILE D : SET sum = 0

Rearrange the letters A, B, C, D, E, F, and G so the corresponding pseudocode correctly represents the ﬂow chart’s algorithm.

Create a strategy Use the given ﬂow chart to ﬁnd the matching code and letter.

Apply the idea The first action is sum = 0, so the first action is D. The second action is a = 2, so the next action is E. Then, we have a decision where if a = 10 we are going to PRINT sum if not, we are going to do two operations. But since we started with a number less than 10 we are going to repeat the two operations until the value is a = 10. This corresponds to the WHILE loop in A. Since the prior step has a loop, the next lines must be the operations within the loop, which are sum = sum + a and a = a + 2 in the flow chart. These correspond to B and C. After reaching a = 10 we can END WHILE, which is G. Finally, we can PRINT sum, which is F. So the correct order is D, E, A, B, C, G, F. 1

SET sum = 0

SET a = 2

WHILE a not equal to 10

sum = sum + a

a = a + 2

END WHILE

PRINT sum

Idea summary Loops in algorithms require steps to be executed multiple times. In pseudocode, we often use the WHILE loop. It has a condition at the start that decides how many times the loop will run.

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4.02 Practice questions Understanding

Explorer

Adventurer

1, 2

Trailblazer

1–3

What helps decide when a loop ends?

Given the ﬂow chart:

1–3

START sum = 0 a=5 a < 10

Yes

sum = sum + a

a=a+1

No PRINT sum END

What is the initial value of a?

What condition must be met for this loop to run?

Look at the pseudocode: a

On which lines of code is the loop?

START

What condition must be met for this loop to run?

SET b = 9

How does the line total = total + b function in the pseudocode?

SET total = 0

WHILE b ≤ 15

5 total = total + b 6 b = b + 2

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END WHILE

PRINT total

END

Fluency

4, 5a, 7, 8, 9a

4, 5b, 8, 9bc, 10

4–6, 8, 9b-f, 10, 11

START

Consider this pseudocode:

SET b = 12

What is the starting value of b?

What will be the ﬁnal value of b by the time the loop ends? SET total = 0 What will be the value of PRINT total after the loop has WHILE b > 5 ﬁnished? total = total + b

b= b– 1 END WHILE PRINT total END Ex 1

For each pseudocode: i

What is the starting value of b?

1 START

What is the output after execution? 1

START

SET b = 5

SET b = 4

SET total = 0

WHILE b < 13

SET m = 0

5 total = total + b

WHILE b < 10

6 b = b + 1

6 total = total + b

END WHILE

7 b = b + 1

PRINT total

8 m = m + 1

END

END WHILE

10 PRINT total/m 11 END 6

Find the ﬁnal value of m.

START

SET b = 8

SET m = 0

WHILE b > 2

5 total = total + b 6 b = b/2 7 m = m+1 8

END WHILE

END

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For the ﬂow chart: START

a=7

a > 11

Yes

PRINT a

a=a−1

No END

Arrange these lines of pseudocode in the correct order for the algorithm. Step A B C D E Ex 2

Code WHILE a > 11 END WHILE PRINT a SET a = 7 a=a−1

For the ﬂow chart: START sum = 0 a=1 a = 12

sum = sum + a

a=a+1

Yes PRINT sum END

Arrange these lines of pseudocode in the correct order for the algorithm. Step A B C D E F G

176

Code a=a+1 SET a = 1 WHILE a ≠ 12 PRINT sum sum = sum + a END WHILE SET sum = 0

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Identify the ﬁnal printed output for each pseudocode segment: a

START

SET b = 18

SET b = 2

SET b = 4

SET b = 1

WHILE b > 6

WHILE b < 30

WHILE b < 8

WHILE b < 12

b= b− 2

PRINT b

END WHILE

b= b∗ 3

b= b+ 2

b= b+ 3

PRINT b

END WHILE

END

START

SET b = 4

WHILE b < 25

PRINT b

b= b∗ 3

PRINT b

END WHILE

END

The flow chart shows the algorithm for finding the sum of the first n counting numbers. START sum = 0 a=1 sum = a + sum a = n? Yes

a=a+1

PRINT sum END

Find the output of this algorithm if n = 6.

Arrange the lines of pseudocode for adding the ﬁrst 50 counting numbers in the correct order for the algorithm. Step A B C D E F G

Code sum = a + sum PRINT sum SET a = 1 a=a+1 GO TO LINE 3 SET sum = 0 IF a = 50 THEN GO TO LINE 7

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The factorial of a non-negative integer n is given by n! = n × (n − 1) × (n − 2) × … × 2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The ﬂow chart represents the algorithm for ﬁnding n!: START p=1 a=1 p=a∗p a = n? Yes

a=a+1

PRINT p END

Find the output of this algorithm if n = 5.

Arrange the lines of pseudocode in the correct order for ﬁnding 50!. Step A B C D E F G

Reasoning 12

Instruction p=p∗a a=a+1 PRINT p SET a = 1 IF a = 50 THEN GO TO LINE 7 SET p = 1 GO TO LINE 3 12

In the algorithm, explain what the “WHILE” loop does and what it is responsible for.

12, 13

12, 14

START

SET b = 15

SET sum = 0

WHILE b > 8

5 sum = sum + b 6 b = b – 2

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END WHILE

PRINT sum

END

Why does the value of b increase by 2 in each loop?

START

SET b = 4

SET sum = 0

WHILE b < 15

5 sum = sum + b 6 b = b + 2

If the condition was changed from b < 17 to b ≤ 17, how would this change the output? Explain your reasoning.

END WHILE

PRINT sum

END

START

SET b = 6

SET sum = 0

SET m = 0

WHILE b < 17

6 sum = sum + b 7

b= b+ 2

m= m+ 1

END WHILE

10 PRINT sum / m 11 END

Problem-solving 15 15

15, 16

Modify the provided pseudocode to compute the product of odd numbers instead of the sum. What will be the initial value of the product variable, and how will you determine if a number is odd?

15–17

START SET a = 10 SET sum = 0 WHILE a > 2 sum = sum + a a= a− 1 END WHILE PRINT sum END

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Modify the pseudocode to calculate the sum of cubes of the numbers.

START SET a = 8 SET sum = 0 WHILE a > 1 sum = sum + a a= a− 1 END WHILE PRINT sum END

Develop a pseudocode that calculates the sum of a series of prime numbers and also counts the number of primes in the sequence.

4.03

Debugging

After this lesson, you will be able to… • understand the term ‘bugs’ in the context of programming and their impact on the functionality of code. • apply practical debugging tips to identify and fix bugs in the code. • critically analyse algorithms and pseudocode for potential bugs. • debug faulty algorithms and pseudocode by identifying and correcting bugs.

Debugging It is quite common for programmers to make mistakes when they are writing computer programs in code. These mistakes are known as ‘bugs’. They stop the computer program (code) from doing what it is supposed to. An important part of programming is testing your program and debugging (which means removing or ﬁxing the bugs). Debugging tips: • Test different inputs and see what outputs they give. Are the outputs what you expected? • Identify what is going wrong with the output - how is it different to what you expect? • Go through the section of the code step by step thinking about what each command does and write down the result of each step to check. You can even add in the print code after each step to see if the calculations match yours.

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Example 1 The linear search algorithm will ﬁnd the position of the number 6 in the list of ﬁve numbers 9, 4, 6, 7, 3. a This algorithm contains a bug. Step Number 1 2 3 4 5 6

Algorithm Step Value = 6 If first number in list = Value, then PRINT 1 If second number in list = Value, then PRINT 2 If third number in list = Value, then PRINT 3 If fourth number in list = Value, then PRINT 4 If third number in list = Value, then PRINT 5

OUTPUT

Which steps of the algorithm produce an output?

Create a strategy Apply each of the steps of the algorithm to determine which steps produce an output (PRINT a value). Use the list of numbers given as the input.

Apply the idea The original list of numbers was 9, 4, 6, 7, 3. Step Number 1 2 3 4 5 6

OUTPUT

3 5

Steps 4 and 6 produced outputs.

b Which step contains the bug?

Create a strategy Use the answer in part (a) and compare it to the desired ouput.

Apply the idea Our desired output is 3 since the number 6 is in the third position. Looking at the answers in part (a), Step 4 will give us that desired output while Step 6 will give us an incorrect output of 5. So Step 6 contains the bug.

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Example 2 This pseudocode prints the multiplication table for 2. 1

START

SET n = 2

SET i = 1

WHILE i ≤ 10 DO

5 SET product = n * i 6 PRINT product 7 i = i - 1 8

END WHILE

END

In it’s current form the code operates like this with outputs: n i printed values

2 1, 0, −1, −2 2, 0, −2, −4

And, it doesn’t actually end. a What line contains the bug?

Create a strategy Consider what is happening in each output.

Apply the idea By looking at the outputs, it should be incrementing up instead of down. So line 7 should be i = i + 1 to produce the output we want: i i=1 i=1+1 i=2+1 i=3+1 i=4+1

i×n 1×2 2×2 3×2 4×2 5×2

Printed value 2 4 6 8 10

So there is a bug at line 7.

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b Correct the code.

Apply the idea 1

START

SET n = 2

SET i = 1

WHILE i ≤ 10 DO

SET product = n * i

PRINT product

i = i + 1

END WHILE

END

Idea summary Debugging tips: • Test different inputs and see what outputs they give. Are the outputs what you expected? • Identify what is going wrong with the output - how is it different to what you expect? • Go through the section of the code step by step thinking about what each command does and write down the result of each step to check. You can even add in the print code after each step to see if the calculations match yours.

4.03 Practice questions Understanding

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

What is referred to as ‘bugs’ in programming?

What is the process of ﬁxing or removing errors from a program called?

What are some debugging tips?

4.03 Debugging mathspace.co

183

Fluency

184

4, 7, 8

Consider this pseudocode:

4, 5, 7, 8

4–8

START

What is the output of this code?

What would the output of this code be if a was 17?

SET a = 18

IF a/2 IS integer

Which of the two outputs from parts a and b doesn’t make sense?

Which line contains an error? What should it be instead?

END IF

END

Consider this pseudocode which is meant to calculate the mean (average) of two numbers:

START

SET a = 7

SET b = 6

SET m = (a+b)/4

PRINT m

END

START

SET a = 3

SET sum = 0

SET n = 0

WHILE a < 10

What is the output of the code?

Which line contains an error? What should the line of code be to remove the error?

Consider this pseudocode which is meant to calculate the mean of a set of numbers: a

What is the ﬁnal value of sum?

What is the ﬁnal value of a?

What is the ﬁnal value of n?

Now, what is the output of this code?

Which line contains an error? What should the line of code be to remove the error?

Mathspace Victorian Curriculum – Year 8 mathspace.co

4 THEN PRINT a is odd 5 ELSE PRINT a is odd

6 sum = sum + a 7 a = a + 1 8 n = n + 1 9

END WHILE

PRINT sum / a

END

Ex 1

Consider this list of numbers: 11, 8, 5, 10, 7 a

Each step of the linear search algorithm is listed below. Step Number 1 2 3 4 5 6

Ex 2

Value = 10 If first number in list = Value, then PRINT 1 If second number in list = Value, then PRINT 2 If third number in list = Value, then PRINT 3 If fourth number in list = Value, then PRINT 4 If fifth number in list = Value, then PRINT 5

Which step of the algorithm produces an output?

What is the output of this algorithm?

This algorithm contains an error. Step Number 1 2 3 4

Algorithm Step

Algorithm Step If Value = first number in list, then PRINT 1 If Value = second number in list, then PRINT 2 If Value = third number in list, then PRINT 3 Value = 8

Which step of the algorithm produces an output?

What is the output of this step of the algorithm?

Now, which step contains the error?

This pseudocode prints the list of positive even numbers. 1

START

SET n = 2

SET i = 1

WHILE i ≤ 10 DO

5 SET product = n * i 6 PRINT product 7 i = i - 1 8

END WHILE

END

In it’s current form the code operates like this with outputs: n i printed values

2 1, 0, −1, −2 2, 0, −2, −4

And, it doesn’t actually end. Identify and correct the line that contains the bug.

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185

Reasoning 9

9, 10

Consider this pseudocode which determines the largest number in a list: 1

START

SET list = [3,18,12,8,2]

SET largest = first element in list

FOR each number IN list DO

5 IF number > largest THEN 6 SET largest = number 7 END IF

END FOR

PRINT largest

END

If we change “largest” to “smallest” in line 3, will the pseudocode determine the smallest number in the list? Explain your answer.

Explain what would happen if we remove lines 4 and 8.

Consider this pseudocode which meant to count the number of even numbers in a list: 1

START

SET list = [list of numbers]

FOR each number IN list DO

4 IF number/2 results in an integer THEN 5 SET count= count+1 6 END IF 7

END FOR

PRINT count

END

The pseudocode has an error. Identify the error and suggest a line of code. Explain your answer.

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Problem-solving 12 11

12, 13

The factorial of a number n is given by n! = n × (n − 1) × (n − 2) × … × 2 × 1. So 4! = 4 × 3 × 2 × 1 = 24. Consider the given pseudocode which is meant to calculate the factorial of a number n:

11, 13, 14

START

SET a = 1

SET f = 1

WHILE a < n

What is the output of the code if n = 8?

5 f = a * f

Which line contains the error? Explain your answer.

6 a = a + 1

Consider this pseudocode which is meant to calculate the sum of the ﬁrst n: a

What is the output of the code if n = 7?

Which line contains the error?

END WHILE

PRINT f

END

START

SET a = 1

SET sum = 0

WHILE a ≤ n

5 sum = a - sum 6 a = a + 1

END WHILE

PRINT sum

END

A store provides a discount to its customers based on the number of items they purchase. If a customer buys more than 10 items, they get a 10% discount. If they buy more than 20 items, they get a 20% discount. a

If a customer buys 15 items each priced at $10, how much would they need to pay after the discount?

Identify the error in the pseudocode and write the correct pseudocode to determine the discount based on the number of items and then to calculate the ﬁnal amount.

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START

SET items = user input ()

SET price per item = user input ()

SET total = items * price per item

IF items > 10

6 total = total * 0.8 7

ELSE IF items > 20

8 total = total * 0.9

END IF

PRINT total

END

Consider this pseudocode which sorts a list of ﬁve numbers from smallest to largest: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a

188

START SET list = [8, 5, 10, 2, 7] SET a = 0 FOR n FROM 1 TO 4 a = list element at position n FOR m FROM (n + 1) TO 5 IF list element at position m < a a = list element at position m END IF END FOR SWAP a WITH list element at position n PRINT list END FOR END

The pseudocode was changed to give these outputs: i

Which line most likely contained the error?

Write a replacement line that could have resulted in this output.

The pseudocode was changed to give these outputs: i

Which line most likely contains the error?

Write a replacement line that could have resulted in this output.

Mathspace Victorian Curriculum – Year 8 mathspace.co

OUTPUT: 2, 5, 10, 8, 7 2, 5, 10, 8, 7 2, 5, 7, 8, 10 2, 5, 7, 8, 10

OUTPUT: 2, 5, 10, 2, 7 2, 2, 10, 2, 7 2, 2, 2, 2, 7 2, 2, 2, 2, 2 OUTPUT: 2, 5, 10, 8, 7 2, 5, 10, 8, 7 2, 5, 8, 10, 7 2, 5, 8, 10, 7

4.04

Algorithms for divisibility

After this lesson, you will be able to… • understand and apply algorithms for testing divisibility by 2, 3, 4, 5, 6, 8, 9 and 10. • use divisibility algorithms to solve problems.

Divisibility algorithms A divisibility algorithm uses a type of search and sort algorithm, where you include a speciﬁed test and check to sort numbers. To test whether numbers are divisible by 10, 2, 4, 5, or 8, we use these algorithms: • If a number is divisible by 10, then the number ends with a zero. • If a number is divisible by 2, then the number ends with an even number. • If the number can be halved twice, then the number is divisible by 4. • If a number is divisible by 5, then the whole number ends in 0 or a 5. • If the number can be halved 3 times, then the number is divisible by 8. To test whether a number is divisible by 3, we use this algorithm: Step 1 2

Instruction Find the sum of the digits of the number. If the sum is a multiple of 3, then the number itself is a multiple of 3.

To test whether a number is divisible by 9, we use this algorithm: Step 1 2

Instruction Find the sum of the digits of the number. If the sum is divisible by 9, then the number is also divisible by 9.

To test whether a number is divisible by 6, we use this algorithm: Step 1 2

Instruction Test whether the number is divisible by 2. Test whether the number is divisible by 3.

Example 1 Consider these numbers: 940, 257, 8535, 486, 1923, 11 705, 21 735, 92 872, 98 941, 77 990, 2327, 6644, 985, 605, 8470 a List all the numbers that are divisible by 10.

Create a strategy

Apply the idea

Choose all the numbers that end in a 0.

940, 77 990, 8470

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189

b List all the numbers that are divisible by 5.

Create a strategy

Apply the idea

Choose all the numbers that end in a 0 or a 5.

940, 8535, 11 705, 21 735, 77 990, 985, 605, 8470

c List all the numbers that are divisible by 2.

Create a strategy

Apply the idea

Choose all the even numbers, or the numbers that end in a 0, 2, 4, 6, or 8.

940, 486, 92 872, 77 990, 6644, 8470

d List all the numbers that are divisible by 10, 5 and 2.

Create a strategy

Apply the idea

Choose the numbers that appear in all of the three lists from parts (a), (b) and (c).

940, 77 990, 8470

Example 2 To test whether a number is divisible by 6, we use this algorithm: Step 1 2 3

Instruction Test whether the number is divisible by 2. Test whether the number is divisible by 3. If the number is divisible by both 2 and 3, then the number is divisible by 6.

Consider the number 42 765. a Is 42 765 divisible by 2?

Create a strategy

Apply the idea

Check if the last digit is even.

The digit in the ones column, 5, is not even. So 42 765 is not divisible by 2.

b Is 42 765 divisible by 3?

Create a strategy Find the sum of the digits and check if the sum is divisible by 3.

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Apply the idea Sum of digits = 4 + 2 + 7 + 6 + 5

Add the digits together

= 24 24 is a multiple of 3 since 3 × 8 = 24. So 42 765 is divisible by 3.

c Is 42 765 divisible by 6?

Create a strategy

Apply the idea

Refer to answers in parts (a) and (b) to see if the number is divisible by 2 and 3.

The number is not divisible by 2 but is divisible by 3. Since it is not divisible by both numbers, it is not divisible by 6.

Idea summary • • • • •

If a number is divisible by 10, then the number ends with a zero. If a number is divisible by 2, then the number ends with an even number. If the number can be halved twice, then the number is divisible by 4. If a number is divisible by 5, then the whole number ends in 0 or a 5. If the number can be halved 3 times, then the number is divisible by 8.

4.04 Practice questions Understanding

Explorer 1, 2

Adventurer 1–3

Trailblazer 1–3

Which algorithm would you use to determine if a number is divisible by 5?

For a number to be divisible by 6, which two other numbers should it be divisible by according to the given algorithm?

The divisibility rule for dividing by 8 is to check if the number can be halved 3 times. For example: Check if 32 is divisible by 8: 32 16 (first time) 2 16 8 (second time) 2 8 4 (third time) 2 So 32 is divisible by 8. If a number was not divisible by 8 in the final step, how would you know? Show an example.

4.04 Algorithms for divisibility mathspace.co

191

Fluency Ex 1

4–7, 10–12, 15

4, 6–8, 11, 13, 15

4, 8, 9, 11, 13–15

For the given numbers: 780, 329, 6545, 692, 1328, 12 510, 22 735, 93 874, 89 942, 78 980, 2328, 6548, 985, 605, 8560

List all the numbers that are divisible by 10

List all the numbers that are divisible by 5.

List all the numbers that are divisible by 2.

List all the numbers that are divisible by 10, 5 and 2.

To test whether a number is divisible by 10, the following simple algorithm will be used: “If the number ends in a 0, then the number is divisible by 10.” Write down all the numbers from these lists that are divisible by 10.

313, 882, 710, 170, 479, 381, 860, 465, 256, 344

42 381, 92 176, 86 498, 59 433, 24 499, 21 880, 95 534, 61 110, 25 190, 39 137

6 459 620, 5 638 943, 2 878 605, 1 904 790, 5 037 628, 6 805 217

To test whether a number is divisible by 2, the following simple algorithm will be used: “If the number is even (that is, the digit in the ones column is even), then the number is divisible by 2.” Write down all the numbers from these lists that are divisible by 2.

63, 35, 16, 43, 71, 11, 97, 80, 74, 59

391, 138, 575, 385, 917, 787, 873, 992, 239, 564

6 750 025, 4 415 417, 8 603 042, 3 789 901, 7 103 764, 2 670 389

To test whether a number is divisible by 5, the following simple algorithm will be used: “If the number ends in a 0 or a 5, then the number is divisible by 5.” Write down all the numbers from these lists that are divisible by 5.

85, 38, 63, 35, 52, 90, 47, 76, 24, 99

865, 583, 324, 369, 766, 711, 920, 507, 555, 678

8 927 050, 2 770 753, 6 606 005, 9 200 311, 9 911 699, 7 548 014

To test whether a number is divisible by 4, the following simple algorithm will be used: “If the number can be halved twice, then the number is divisible by 4.” Write down all the numbers from these lists that are divisible by 4.

8934, 6236, 5943, 3130, 5122, 2025, 1528, 6632, 9217, 8339

1 805 159, 2 952 174, 9 951 756, 5 885 468, 7 486 650, 1 581 254, 4 993 860, 4 924 944

30 685 091, 45 590 085, 13 189 788, 29 003 798, 81 339 993, 92 123 492

To test whether a number is divisible by 8, the following algorithm will be used: “If the number can be halved 3 times, then the number is divisible by 8.” Determine whether these numbers are divisible by 8. a

192

42 344

62 106

Mathspace Victorian Curriculum – Year 8 mathspace.co

To test whether a number is divisible by 3, the following algorithm will be used: Step 1 2

Instruction Determine the sum of the digits of the number. If the sum is a multiple of 3, then the number itself is a multiple of 3.

Determine whether these numbers are divisible by 3. a Ex 2

34 257

7 352 941

946 357 246 188

To test whether a number is divisible by 6, the following algorithm will be used: Step 1 2 3

Instruction Test whether the number is divisible by 2. Test whether the number is divisible by 3. If the number is divisible by both 2 and 3, then the number is divisible by 6.

Determine whether these numbers are divisible by 6. a 12

52 779

51 329 784

32 156 894

943 162 547 886

To test whether a number is divisible by 9, the following algorithm will be used: Step 1 2

Instruction Determine the sum of the digits of the number. If the sum is divisible by 9, then the number is also divisible by 9.

Determine whether the following numbers are divisible by 9. a 13

49 536

27 648 391

354 427 126 389

For each number: i

Determine whether the number is divisible by 2.

Determine whether the number is divisible by 3.

iii

Determine whether the number is divisible by 9.

Then, determine whether the number is divisible by both 6 and 9.

835 794

6 712 854

To test whether a number is divisible by 7, the following algorithm will be used: Step 1 2 3 4

Instruction Remove the last digit and double it. Subtract this answer from the remaining digits. (Optional): Repeat Step 1 and Step 2 until the number is small enough for you to confidently divide by 7 If the final answer is divisible by 7, then the whole number is divisible by 7.

Determine whether the following numbers are divisible by 7. a

3185

1789

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193

To test whether a number is divisible by 11, the following algorithm will be used: Step 1

Instruction Remove the last digit and subtract it from the remaining digits. (Optional): Repeat Step 1 until the number is small enough for you to confidently divide by 11 If the final answer is divisible by 11, then the whole number is divisible by 11.

2 3

Determine whether the following numbers are divisible by 11. a

2948

Reasoning

2974

16, 17

16–18

The sum of the digits of a mystery number is 15. Could this mystery number be divisible by 3 and 9? Justify your answer.

If a number is divisible by both 4 and 5, what other number is it guaranteed to be divisible by and why?

A number is known to be divisible by 2, 3, and 9. What other number must it also be divisible by? Justify your answer by listing 3 numbers.

Problem-solving 19 19

19, 20

Space Station Signal: A space station sends data packets as large integers to Earth. For security, they want to send only numbers that satisfy the algorithm. Should the space station send the number 5231 ? 1

SET n = number

IF n is an integer that ends with 0 or 5

3 THEN PRINT “Yes” 4 ELSE PRINT “No”

END IF

END

For quick decision-making, an air traﬃc control tower only permits planes with tail numbers dictated by the algorithm to take off during peak hours. Can a plane with tail number 63 927 take off during peak hours? 1

SET n = tail number

SET SUM = sum of the digits of n

IF SUM/9 gives an integer result

4 THEN PRINT “Yes” 5 ELSE PRINT “No”

194

END IF

END

Mathspace Victorian Curriculum – Year 8 mathspace.co

4 Chapter review 1

What might this algorithm be used for? Step 1 2 3 4 5

Instruction Heat the oven to 190°C. Mix the dough ingredients in a large bowl. Roll out the dough on a ﬂat surface. Cut the dough into shapes and place on a baking tray. Bake for 10-15 minutes or until golden brown.

For each algorithm, work out the output at each step and complete the table: a

Step 1 2 3

Instruction Start with the number 150. Subtract 30 from 150. Divide the result by 2.

Output

Step 1 2 3

Instruction Start with the number 22. Add 8 to 22. Triple the result.

Output

Step 1 2 3

Instruction Start with the number 45. Reduce the number by 10%. Subtract 5 from the result.

Output

In an algorithm, if a speciﬁc step has been repeated 9 times, how many iterations of that step have occurred?

Given the instruction SET value = 6 followed by SET value = SET value × 2, what would be the ﬁnal value of the variable “value” after executing both instructions?

Regarding the given pseudocode:

START

For how many iterations does the loop run?

SET a = 5

What purpose does the sum = sum + a line serve in the pseudocode?

WHILE a < 20

SET sum = 0 sum = sum + a a = a + 3 END WHILE PRINT sum END

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195

This algorithm machine has been designed to add ﬁfteen. ⬚ ⟶ +15 ⟶ ⬚

If 35 is introduced into the algorithm machine, would it require a regrouping when adding units?

If 14 is put into the algorithm machine, would it require a regrouping when adding units?

Describe the types of numbers that would need regrouping.

Study this addition algorithm: 1.

Add the units.

2. Is the result at least 10? • If no - GO TO Step 3. • If yes - regroup into tens and units, then GO TO Step 3. 3. 4.

Add the tens. Is the result at least 100? • If no - GO TO Step 5. • If yes - regroup into hundreds and tens, then GO TO Step 5.

5. Add the hundreds. 6. Is the result at least 1000? • If no - GO TO Step 7. • If yes - regroup into thousands and hundreds, then GO TO Step 7. 7. a

Utilize the algorithm to determine the sum of 368 + 425. Record the outcome at each step: i

Step 2

Step 4

iii

Step 6

Step 2

Step 4

iii

Step 6

Apply the given algorithm to the number 54: • If the number is more than 50, subtract 7. • If the number is less than or equal to 50, triple it.

Apply the given algorithm to the number 180: • If the number is less than 150, multiply it by 3 then subtract 10. • If the number is more than or equal to 150, subtract 20.

196

Step 7

Apply the algorithm to check the result of 692 + 317. Record the outcome at each step: i

Write the ﬁnal answer by adding the thousands, hundreds, tens and units.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Step 7

Complete the tables below by determining the results of each algorithm: a

Step 1 2 3

Instruction Begin with the number 15. Subtract 4 from this number. Perform step 2 three additional times.

Step 1 2 3 4

Instruction Begin with the number 45. Add 6 to this number. Perform step 2 three additional times. Round this number down to the nearest 10.

Step 1 2 3

Instruction Begin with the number 40 000. Divide this number by 4. Perform step 2 three additional times.

Result

Consider this algorithm: 1.

Start with the number 11.

2. Add 11 to this number. 3.

Repeat step 2 three more times.

How many 11’s are added together?

Express this algorithm as a sum.

What is the resulting number?

Calculate 11 × 5.

What is the output of this algorithm if the input is 11?

Start Input a number Multiply the number by 10 Subtract 3 from the number Output the number End

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197

What is the output of this algorithm if the input is A = 6 and B = 12? Start Input A Input B

C=B−A

Is A > B

Yes

C=A−B

Output C

End

Determine the value that is printed at the end of the pseudocodes: a

START

SET a = 60

SET a = 4

SET b = 45

SET b = a2

SET c = a – b

PRINT b

PRINT c

END

END c

START

SET a = 7

SET a = 100

SET b = 3

a = a – 35

SET c = 5

PRINT a

SET d = a ∗ b − c

END

PRINT d END 15

What statement is printed at the end of the pseudocode?

START SET a = 15 IF a / 2 IS integer THEN PRINT “a is even” ELSE PRINT “a is odd” END IF END

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For each pseudocode: i

Find the initial value of a.

Find the OUTPUT of the code.

START

SET a = 5

SET sum = 0

WHILE a < 16

SET n = 0

sum = sum + a

WHILE a < 17

a = a + 1

sum = sum + a

END WHILE

a = a + 3

PRINT sum

n = n + 1 END WHILE

END

PRINT sum / n END 17

Determine the value that is printed at the end of each pseudocode: a

START

SET a = 16

SET a = 3

SET a = 8

SET a = 30

WHILE a > 9

WHILE a < 40

WHILE a < 12

WHILE a > 15

a = a − 1

PRINT a

a = a ∗ 4

a = a + 1

a = a − 4

END WHILE

PRINT a

END WHILE

END

Given this algorithm: • If a number is divisible by 5, add 5 then multiply it by 10. • If a number exceeds 50, add 4 and then divide it by 4. What is the issue with this algorithm? Justify your response using the number 55.

Mia developed an algorithm to multiply a number by 10, but she made a mistake. Instead, she added 10 to the number ten times. When she applied her algorithm to the number 6, what was her outcome?

When using the subtraction method, subtracting 57 from 93, how many instances would you need to regroup?

A tech company determines the price of a digital item based on its storage capacity. The tech company uses the algorithm: 1.

For data under 20MB, the price is $9.

2. For data between 20MB and 60MB, the price rises by $0.30 for every MB over 20MB. 3.

For data more than 60MB, there’s a base charge of $21 plus $0.20 for every MB exceeding 60.

Determine the price for a data size of 73MB.

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199

A video rental store follows a late fee system. The system is: 1.

If a movie is returned within 4 days, there’s no charge.

2. If a movie is returned between 4-8 days, the fee is $3. 3.

If a movie is returned after 8 days, the fee is $6 plus $0.75 for every day beyond 8.

Determine the penalty for a movie that is returned after 10 days. 23

Anna has a routine for charging her collection of smart gadgets: 1.

Plug in the charging station.

2. Charge a gadget. 3.

Move to the next gadget.

Follow steps 2 and 3 until all gadgets are charged.

5. If the charging station runs out of ports, unplug fully charged gadgets and go back to step 2. If Anna has 7 gadgets and the charging station has ports for only 4 gadgets at once, how many times will she need to swap gadgets on the charging station? 24

A flow chart for an online quiz game: Start

SET a = answer Provide another question Is a correct?

Deduct 1 point

Yes Add 2 points

End

Based on the flow chart if you’ve answered four questions wrong consecutively, what would be your score change based on the ﬂow chart before the quiz ends? 25

Construct a pseudocode that inputs two numbers A and B, swaps their values, and displays the swapped numbers A and B.

Design a pseudocode that calculates the sum of a sequence of numbers and also determines the smallest and largest number in the sequence.

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"The science of operations, as derived from mathematics, offers an infinity of possible outcomes—coding transforms this infinite potential into reality through algorithms." Ada Lovelace

Big ideas Understanding time units, time representations, and time zone conversions can help efficiently measure and calculate time differences.

Curriculum Outcomes VC2M8M04 Solve problems involving time and duration, including using 12- and 24-hour time across multiple time zones

Time Chapter outline 5.01 5.02 5.03

Elapsed time 12 and 24-hour time Time zones and the date line Investigation: Plan with time zones Chapter 5 review

204 209 213 227 231

Pilots and aircrew have to adjust to different time zones during flights. A flight from Sydney to Los Angeles crosses over 17 time zones!

5.01

Elapsed time After this lesson, you will be able to… • convert between seconds, minutes, and hours. • calculate time differences.

Time conversions We use these conversions to change between different units of time: • 1 minute = 60 seconds • 1 hour = 60 minutes • 1 day = 24 hours Remember there are 7 days a week and 52 weeks a year. A year, deﬁned by the time for Earth to orbit the Sun, is approximately

days. Every year has 365 days, except every fourth year,

called a leap year, which has 366 days to account for the extra quarter day. Leap years have an additional day in February. As we do for other types of conversions, we use multiplication and division to change between time units as follows: ÷60

Seconds

÷60

Minutes

×60

÷24

Hours

×60

Days

×24

For example, if we want to know how many seconds are in 5 minutes of time, we can multiply the number of minutes by 60 because there are 60 seconds in each minute.

So, there are 300 seconds in 5 minutes.

Example 1 It takes around 300 minutes to drive from Sydney to Thredbo. How many hours is this?

Create a strategy Divide the number of minutes by the number of minutes in an hour.

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Apply the idea 1 hour = 60 minutes, so we need to divide by 60. Write as a fraction Remove the zeros Divide 30 by 6 The drive takes 5 hours.

Example 2 Han boils an egg for 360 seconds. How many minutes is this?

Create a strategy

Apply the idea

Divide the number of seconds by the number of seconds in a minute.

1 minute = 60 seconds, so we need to divide by 60. Write as a fraction Divide 360 by 60 Han boils an egg in 6 minutes.

Idea summary Conversions between the different units of time: ÷60

Seconds

÷60

Minutes

×60

÷24

Hours

×60

Days

×24

Elapsed time Now let’s explore how to calculate changes in time using a clock.

Interactive exploration Discover this concept in action online

mathspace.co

5.01 Elapsed time mathspace.co

205

When adding or subtracting times, we need to be aware that there are 60 minutes in an hour. If the minutes add to 60 or more, increase the number of hours. The time will change between a.m. and p.m. whenever the hour changes from 11 to 12.

Example 3 Which of these represents 30 minutes after 11:40 p.m.? A 12:10 a.m.

11:10 p.m.

11:70 p.m.

12:10 p.m.

Create a strategy

Apply the idea

When adding times, if the minutes add to 60 or more, increase the number of hours. The time will change between p.m. to a.m. whenever the hour changes from 11 to 12.

Add the 30 minutes by adding 20 and then 10 minutes. 11 : 40 p.m. + 20 minutes = 12 : 00 a.m. Add the remaining 10 minutes: 12 : 00 a.m. + 10 minutes = 12 : 10 a.m. 30 minutes after 11:40 p.m. will be 12:10 a.m., option A.

Example 4 Kate leaves her home at 11:15 a.m. to visit her grandmother in a different state. She arrives at her grandmother’s house at 5:15 p.m. How long was Kate’s trip in hours?

Create a strategy

Apply the idea

Since both times have the same number The number of hours from 11:00 a.m. to of minutes, we can focus on the difference 12:00 midday is 1. The number of hours between between the hours. Find the difference between 12:00 midday and 5:00 p.m. is 5. 11:00 a.m. and 5:00 p.m. Length of trip = 1 + 5 Add the hours = 6 hours

Evaluate

Idea summary Remember that when adding times, if the minutes add to 60 or more, increase the number of hours. The time will change between a.m. to p.m. whenever the hour changes from 11 to 12.

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5.01 Practice questions Understanding 1

⬚ seconds = 1 minute ⬚ hours = 1 day

A number of hours to minutes

A number of seconds to minutes

A number of hours to days

A number of minutes to hours

A number of hours to seconds

A number of days to minutes

Describe how to convert these units of time.

⬚ days = 1 week

Is the time after these changes written with a.m. or p.m.? a

6:30 p.m. + 12 hours

1:00 a.m. − 2 hours

9:05 a.m. − 9 hours and 5 minutes

4:20 p.m. + 24 hours and 20 minutes

Fluency 4

Trailblazer 2, 3

60 minutes = ⬚ hour

Adventurer 2, 3

Complete the statements: a

Explorer 1, 2

4a–e, 5–8, 10aceg, 12, 14, 16

4–8, 10aceg, 13, 14, 16

4, 7–9, 10bdfh, 11, 14, 15

Convert the times: a

9 days to hours

11 minutes to seconds

36 hours to minutes

24 hours to seconds

240 seconds to minutes

360 minutes to hours

780 seconds to minutes

780 minutes to hours

Ex 1

It takes 150 minutes from Melbourne to Queensland by air. How many hours (and minutes) is this?

Ex 2

Jean can run a lap for 180 seconds. How many minutes is this?

Neil’s swimming lesson went from 11:15 a.m. until midday. How many minutes was Neil’s lesson?

A YouTube video started playing at 1:40 p.m. and ﬁnished at 2:18 p.m. How long is the video in minutes?

How many hours are there in a non-leap year?

Find the new time:

Ex 3

30 minutes after 9:30 a.m.

40 minutes after 11:40 p.m.

9:30 a.m. plus half an hour

7:00 p.m. plus 24 minutes

45 minutes after 11:20 a.m.

20 minutes after 3:50 p.m.

10:30 a.m. plus two and a half hours.

9:10 p.m. plus 63 minutes

A song clip starts playing at 7:55 p.m. and finishes at 8:11 p.m. How long is the song clip in minutes? 5.01 Elapsed time mathspace.co

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Ex 4

Sally’s cruise ticket says that it “begins at 10:00 a.m. and ends at 2:00 p.m. How long is the cruise in hours?

James leaves home at 4:00 p.m. and gets home at 8:00 p.m. How many hours was James out?

Jake leaves his home at 8:10 a.m. to visit his aunt in a different state. He arrives at his aunt’s house at 4:10 p.m. How long was Jake’s trip in hours?

Eileen starts watching a movie at 11:15 a.m., which goes for 3 hours and 45 minutes. What time will the movie ﬁnish?

A computer starts updating at 1:00 p.m. The update will go for 130 minutes. Find the time that the update will be ﬁnished.

Reasoning 17

17, 18

18, 19

Josie is trying to calculate the time 42 minutes after 9:45 a.m. Opie tells her to just add 42 and 45. Josie does this and gives her answer as 9:87 a.m. a

Explain why this answer cannot be correct.

What would you add to Opie’s instructions to help Josie get the correct answer.

We usually use 52 weeks to represent a year. Is this the closest number of weeks for both a normal year and a leap year? Explain.

Complete the calculation: a

Add 2 hours and 20 minutes to 1 hour and 15 minutes.

Add 2 hours and 20 minutes to 1 hour and 40 minutes.

Add 2 hours and 30 minutes to 1 hour and 40 minutes.

For each of the conditions for the sum of b and d minutes given below add a hours and b minutes to c hours and d minutes given a, b, c and d are integers. d

b + d < 60

b + d = 60

b + d > 60

Problem-solving 20–22 20

20–23

22–25

Today is Dave’s birthday and he just turned 9 years old. Assume there are 365 days in a year. Find his age in: a

Days

Hours

Minutes

Seconds

The average person sleeps for 8 hours a night. How many weeks does the average person spend asleep over 5 years? Assume there are 365.25 days in a year. Give your answer to the nearest week.

If an average person showers for 6 minutes per day, ﬁnd how many hours they spend showering in: a

208

One year (assume non-leap year)

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55 years (ignoring leap-years)

A year on Mars is 687 Earth days. Xavier lives in a Martian settlement and has just celebrated his 5th birthday, measured in Martian years. Assuming each Earth year is 365 days, how old would Xavier be in Earth years, in years and months, to the nearest month.

A day on Mars is 24 Earth hours and 37 Earth minutes. A year on Mars is 687 Earth days. How long is a Martian year in Martian days? Give your answer to the nearest Martian day.

Augustus is working on a scheme to decimalise the day. He wants each day to be divided into 10 decimal hours, each decimal hour to be divided into 100 decimal minutes and each decimal minute to be divided into 100 decimal seconds. a

Find the length of a decimal hour in minutes.

Find the length of a decimal minute in seconds.

Find the length of a decimal second in seconds.

5.02

12 and 24-hour time

After this lesson, you will be able to… • convert between 12 and 24-hour times.

12 and 24-hour time There are two ways we represent time over a day: 12-hour and 24-hour time. The following diagram shows how 12 and 24-hour time are related: 24 hour time 00:00

04:00

08:00

12:00

16:00

20:00

00:00

12:00 am

4:00 am

8:00 am

12:00 pm

4:00 pm

8:00 pm

12:00 am

12 hour time

12-hour time represents a day as two blocks of 12 hours using “a.m.” and “p.m.” to represent before and after midday respectively. 24-hour time represents time as one block of 24 hours. There are no “a.m.” or “p.m.” indicators in this format. Instead, the hours are numbered from 00 (midnight) to 23 (11 p.m.). For example: • 8:45 a.m. is 08:45 • 3:15 p.m. is 15:45

Interactive exploration Discover this concept in action online

mathspace.co

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To convert between 12 and 24-hour times: • In 24-hour time 12:00 a.m. and 12:00 p.m. are 00:00 and 12:00 respectively • Other a.m. times are the same in 24-hour format with the addition of a zero before the digit representing the hour • Other p.m. hours are converted by adding twelve to the hours digit to get 24-hour time • These processes are reversed to convert between 24 and 12-hour times

Example 1 Convert 3:41 p.m. to 24-hour time.

Create a strategy Add 12 to the given hour.

Apply the idea 3:41 p.m. to 24-hour time = 3 : 41 + 12 : 00 = 15 : 41

Add 3 and 12 Evaluate

So, the 24-hour time of 3:41 p.m. is 15:41.

Example 2 Convert 14:54 to 12-hour time.

Create a strategy Subtract 12 from the given hour and add “p.m.” at the end.

Apply the idea 14:54 to 12-hour time = 14 : 54 − 12 : 00

Subtract 12 from 14

= 2 : 54 p.m.

Evaluate and add p.m. at the end

So the 12-hour time of 14:54 is 2:54 p.m.

Idea summary Conversion between 12 and 24-time: 24 hour time 00:00

04:00

08:00

12:00

16:00

20:00

00:00

12:00 am

4:00 am

8:00 am

12:00 pm

4:00 pm

8:00 pm

12:00 am

12 hour time

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5.02 Practice questions Understanding

Describe how to convert these time formats.

A 12-hour time (p.m.) to 24-hour time

A 24-hour time (after midday) to 12-hour time

How does the 12-hour time format differ from the 24-hour time format?

4cdgh, 5b–d, 6ac, 7, 8, 10, 11, 15, 17, 18

10:30 a.m.

9:15 p.m.

4:12 a.m.

8:40 a.m.

7:25 p.m.

4:31 a.m.

12:47 a.m.

6:13 p.m.

00:45

13:59

00:56

Convert these times to 12-hour time format: 05:30

18:30

Convert these 24-hour clock times to a.m. or p.m. times: 09:48

12:28

22:47

Find the time and day that is: a

19 hours and 20 minutes after 20:15 on Thursday

17 hours and 15 minutes before 15:25 on Thursday

22 hours and 58 minutes after 19:25 on Monday

18 hours and 39 minutes before 14:12 on Monday

Saturday 13:15 − 15 hours 18 minutes

Thursday 13:20 + 21 hours 59 minutes

Express these times in 24-hour time: a

Quarter to midnight

Twenty past one in the afternoon

Half past ten at night

Quarter past eleven in the morning

Ten past eight in the morning

Ten to four in the afternoon

A clock shows the time 19:39. If it is known that the clock is running slow by 31 minutes, what is the actual time: a

4abef, 5ab, 6acd, 7a–d, 8, 10–12, 14, 16, 18

a 7

4abef, 5ab, 6, 7ab, 8, 9, 11, 13, 14, 16, 17

Convert these times to 24-hour time format :

a 6

Trailblazer 1–3

Why do we use “a.m.” and “p.m.” in the 12-hour clock system?

Fluency

Ex 2

Adventurer 1–3

Ex 1

Explorer 1–3

in 24-hour time

in 12-hour time

A clock shows the time 17:27. If it is known that the clock is running fast by 40 minutes, what is the actual time: a

in 24-hour time?

in 12-hour time?

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Write the time shown on each of the clocks in 24-hour time: a

A taxi departs at 13:32 from Sydney and arrives at 20:14 at Melbourne. How long did the taxi take for the trip?

David and his family visited his grandfather for his birthday lunch. They left at 2:50 p.m. What is this time in 24-hour time?

Amy baked cookies for an afternoon tea party for her friends. The cookies take 35 minutes to cook. Amy puts them in the oven at 3:05 p.m. What time did she take the cookies out of the oven? Express your answer in 24-hour time.

Mary’s mum said that she would meet her at the mall for coffee at 5:20 p.m. What is this time in 24-hour time?

Gabby had friends over for a dinner party that went on for 3 hours. If they started eating at 18:20, at what time did they ﬁnish? Express your answer in 24-hour time.

A train ride between two cities takes 6 hours and 49 minutes. If you leave one city at 21:40, at what time will you arrive at the other city? Express your answer in 24-hour time.

A ship arrives at 17:28 in Melbourne. If the trip took 2 hours and 16 minutes, at what time did the ship depart from Sydney? Express your answer in 24-hour time.

Reasoning 19

19, 20

Kirk says that it is easy to convert from 12-hour times to 24-hour time, all you need to do is add 12 to the hours. a

Explain why Kirk is incorrect.

Give one example where his method would work and one where it would not.

Make a correction to Kirk’s statement so that it will always work.

Put these times in order from earliest in the day to latest in the day: 12:25 p.m., 00:30, 19:30, 12:25 a.m., 8:25 p.m., 13:25

Problem-solving 21, 23

22, 23

22–24

A shop opens at 10:00 a.m. and closes at 6:30 p.m. If it takes the shop owner 30 minutes to set up and 20 minutes to close down, what is the total time the owner spends at the shop?

Indi is taking a ﬂight from Sydney to Tokyo which departs at 9:40 a.m. local time and takes 9 hours and 50 minutes. If she had to be at the airport in Sydney 3 hours before her ﬂight, and it takes her one and a half hours to collect luggage and clear immigration in Tokyo, how long is her journey from arriving at the airport in Sydney to leaving the airport in Tokyo?

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A digital clock shows 22:50 and is losing 2 minutes every hour.

What time will it show in 6 hours?

What time did it show 2 hours ago?

A clock is showing 8:50 p.m. and is gaining 3 minutes every hour. a

What time will it show in 7 hours?

What time did it show 4 hours ago?

5.03 Time zones and the date line After this lesson, you will be able to… • find the time difference between two cities using time zones.

Time zones 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. Coordinated Universal Time (UTC for short) helps us to compare the time in different places. Greenwich, a city in England, was chosen to be the time zone with UTC + 0 because it lies at 0° longitude. Every other location is given a UTC offset from Greenwich allowing it to be compared to other places in the world.

Exploration Study the following world map. Pay attention to the time indicated at various locations around the world. 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

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Answer the following questions: 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?

Countries east of the UTC + 0 time zone are ahead in time compared to UTC + 0 and will have a + offset. Countries to the west of the UTC + 0 will have time zones that are behind and will have a − offset. 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. When comparing time zones across the International Date Line, the change in time will be big enough that the day will normally change as well. We can use UTC offsets to calculate time differences between places. For example, if it is 2:00 p.m. on Sunday, in UTC − 11, then to ﬁnd the time and day in UTC + 12, would need to add 11 + 12 = 23 hours to get 1:00 p.m. on Monday.

Example 1 Consider the map shown:

Svalbard

When it is 17:00 in Cape Town, what is the time in San Francisco?

Nuuk Moscow

New York San Francisco Caracas

Bamako

Cape Town

+13 +14 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 UTC +1 +2 +3 +4

Create a strategy Find the time difference between the two cities.

Apply the idea Based on the map, Cape Town is in UTC + 2 so it is 2 hours ahead UTC + 0. While San Francisco is in UTC − 8 so it is 8 hours behind UTC + 0. So San Francisco is 10 hours behind Cape Town. So we need to subtract 10 hours from the given time. San Francisco time = 17 : 00 − 10 hours = 7 : 00 It will be 07:00 in San Francisco. 214

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Subtract the 10 hours Evaluate

Example 2 Use the table of time zone information. When it is 9:00 a.m. in Mexico City, what is the time in Sri Lanka?

City Tijuana Mexico City Havana Kiev Sri Lanka Singapore Seoul

UTC –8 –6 –5 +2 +5.5 +8 +9

Create a strategy Find the time difference between the two cities.

Apply the idea Mexico City is in UTC − 6 so it is 6 hours behind UTC + 0. While Sri Lanka is in UTC + 5.5 which means it is 5.5 hours ahead UTC + 0. So Sri Lanka is 6 + 5.5 = 11.5 hours or 11 hours and 30 minutes ahead of Mexico City. We need to add 11.5 hours to the time in Mexico City. Sri Lanka time = 9 : 00 a.m. + 11.5 hours = 8 : 30 p.m.

Add the 11.5 hours Evaluate

When it is 9:00 a.m. in Mexico City, it is 8:30 p.m. in Sri Lanka.

Idea summary The world is divided up into time zones which are labelled with a UTC offset. This speciﬁes the number of hours behind or ahead that time zone is from UTC + 0. We can use the numbers and sign of the UTC to ﬁnd the time difference between two places. The International Date Line is an imaginary line that runs from the North Pole to the South Pole and is roughly at 180° longitude. When comparing time zones across the International Date Line, the change in time will be big enough that the day will normally change as well.

Time zones in Australia Australia is divided into three time zones as shown in the image: • 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)

2h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

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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 QLD

QLD

WA SA

SA NSW

NSW

VIC

TAS

NO DST

DST

Example 3 When the time is 14:15 in Perth what is the time in: 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

a Sydney during non ADST?

Create a strategy Refer to the left side of the image above.

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TAS DST

Apply the idea Sydney (in NSW) is UTC + 10 while Perth (in WA) is UTC + 8. This means that Sydney is 2 hours ahead of Perth. Sydney time = 14 : 15 + 2 hours

Add the 2 hours

= 16 : 15

Evaluate

b Sydney during ADST?

Create a strategy Refer to the right side of the image above.

Apply the idea Sydney (in NSW) is now in AEDT which is UTC + 11 while Perth (in WA) is still in UTC + 8. This means that Sydney is 3 hours ahead of Perth. Sydney time = 14 : 15 + 3 hours

Add the 3 hours

= 17 : 15

Evaluate

Idea summary Australia is divided into three time zones: • Australian Eastern Standard Time (AEST) UTC + 10. • Australian Central Standard Time (ACST) UTC + 9.5. • Australian Western Standard Time (AWST) UTC + 8. Some states in Australia observe two Daylight Saving Time (ADST) zones from October to April: • Australian Central Daylight Time (ACDT) UTC + 10.5. • Australian Eastern Daylight Time (AEDT) UTC + 11.

5.03 Practice questions Understanding

Adventurer 1ab, 2, 3

Trailblazer 1cd, 2, 3

Describe these times as a number of hours ahead or behind UTC + 0: a

Explorer 1ab, 2

Cities at UTC + 1

Cities at UTC − 9 c

Cities at UTC − 3.5 d

Island at UTC + 12

Do these Australian states or territories observe Daylight Saving Time (ADST)? a

Victoria

Western Australia c

Queensland

South Australia

New South Wales d

Northern Territory

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Consider the map shown:

Vancouver (DST)

Moscow

Lisbon (DST) Timbuktu

Riyadh

Lima

Manila

Mumbai Perth

Auckland (DST)

Santiago +13 +14 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1UTC +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12

Are these statements true or false? a

Perth is 3 hours 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. 4–6, 8, 9ac, 11ab, 12ab, 13

Fluency

5, 7, 8, 9ab, 11ab, 12ab, 13

4 Moscow

Abakan

Paris

Tokyo

Accra Dobo

Hobart −1 UTC +1

218

Which two cities are in the same time zone?

How many hours is Hobart ahead of Paris?

Find the time in Cairo when it is 11:00 in Beijing.

Find the time in Hobart when it is 2:00 p.m in Moscow.

Beijing

Cairo

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+9 +10 +11

5, 7, 8, 9bc, 10, 11cd, 12cd, 13

Ex 1

Nuuk

How many hours is San Francisco behind Cape Town?

Find the time in San Francisco when it is 17:00 in Cape Town.

What time is it in Bamako when it is 10:00 a.m. in Nuuk?

Moscow

San Francisco New York

Bamako Caracas

Cape Town

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1 UTC +1

• Perth (UTC + 8)

Which city will have a time of 11 p.m. when it is 4 p.m. UTC + 0? • Cairo (UTC + 2) • Singapore (UTC + 7)

Ex 2

Which location will be the ﬁrst to celebrate New Year’s Eve? • Canberra (UTC + 10) • Darwin (UTC + 9)

• Rio de Janeiro (UTC − 3)

Consider the tables of time zones: City UTC Los Angeles –8 Tijuana –8 Vancouver –8 Chicago –6 Mexico City –6 San Salvador –6 Havana –5 New York –5 Toronto –5 Athens +2 Cairo +2 Kiev +2

City Baku Dubai Yerevan Afghanistan Sri Lanka Hong Kong Perth Shanghai Singapore Seoul Tokyo Auckland

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?

Find the time and day in Tokyo when it is 13:15 on a Thursday in Tijuana.

UTC +4 +4 +4 +4.5 +5.5 +8 +8 +8 +8 +9 +9 +12

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This table shows the time of several locations relative to UTC + 0? UTC −5 −4 −3 −2 −1 0 +1 +2 +3 +3.5 +4 +4.5 +5 +5.5 a

220

Location New York, Cuba, Peru Chile, Barbados, Brazil (west), Bolivia Argentina, Brazil (east), Uruguay, Greenland South Sandwich Islands 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 Iran Mauritius, United Arab Emirates, Armenia Afghanistan Pakistan, Maldives, Kazakhstan India, Sri Lanka

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 is it in Algeria?

When it is 21:25 on the 13th day of the month in Uruguay, what 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 is it in Mauritius?

Mathspace Victorian Curriculum – Year 8 mathspace.co

AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

2h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

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) 6:15

South Australia (SA)

Queensland (QLD) 11:45

10:15 5:25 c

Complete the table to show the equivalent time in each time zone: AWST

ACST 5:25

AEST

5:11 12:25 7:17 11

Convert these Daylight Savings Time in Sydney to Eastern Standard Time: a

16:14 AEDT

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

14:35 AEST

4:00 a.m. AEST

9:45 a.m. AEST

5:13 p.m. AEST

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Ex 3

When the time is 19:20 in Perth determine the time in: a

Sydney during No ADST.

Sydney during ADST.

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

Reasoning 14

TAS DST

14, 15

14–16

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?

During daylight saving, Jasper caught a flight from Melbourne to Perth at 9 p.m. Melbourne time on a Thursday. After a 5.5 hour flight he arrived in Perth to find that it was still Thursday. Explain why this could not happen with the same length of flight if he had flown from Brisbane. 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

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TAS DST

Consider the table of time zones: City Vancouver

UTC −8

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 13 hours. Explain how she can work on Friday in the Vancouver office.

Explain why this cannot happen when Kiki ﬂies from Vancouver to Auckland.

27a, 28, 29ab, 30a

18, 20, 22, 24, 26b, 27b, 28, 29bd, 30c

19, 21, 23, 25, 26c, 27c, 29ace, 30b

Han, who is in Melbourne (UTC + 10) at the moment, has an interview over Skype with the human resources manager of an American trading ﬁrm at 4 p.m. on Sunday, Memphis (UTC − 6) time. a

Calculate the time difference between the two cities.

Find the time in Melbourne when it is 4 p.m. on Sunday in Memphis.

What 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 is located 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 p.m. in San Francisco. Determine the time it would be in the other two cities: i

Rome

Cairo

Buzz suggests 8 a.m. in Rome. Determine the time it would be in the other two cities: i

UTC +12

Problem-solving 17, 18, 20, 22, 24, 26a, 17

City Auckland

Cairo

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:

San Francisco Person Sophia Quentin Harry

They want to organise an hour long online meeting within everyone’s working hours.

City Los Angeles Chicago Melbourne

Time zone UTC − 8 UTC − 6 UTC + 10

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

A b 20

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?

Valerie in Hobart has a phone conversation with Oprah in Manila. Manila is 2 hours behind Hobart time. a

If Valerie calls at 10:30 a.m., determine the time it is for Oprah.

If Oprah calls at 7:30 p.m., determine the time it is for Valerie. 5.03 Time zones and the date line mathspace.co

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The table shows the current time in major capital cities beginning with the letter M: City Madrid Managua Manila Melbourne Mexico City Miami Minneapolis Minsk Montevideo Montreal Moscow Mumbai

Time Thu 1:58 a.m. Wed 5:58 p.m. Thu 7:58 a.m. Thu 9:58 a.m. Wed 6:58 p.m. Wed 7:58 p.m. Wed 6:58 p.m. Thu 2:58 a.m. Wed 8:58 p.m. Wed 7:58 p.m. Thu 3:58 a.m. Thu 5:58 a.m.

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.

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 in Brazilia will the plane arrive?

What day will the plane arrive in Brazilia?

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 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.

This table displays the departure and ﬂight times for three Air China ﬂights: Flight Number CA994 CA927 CA950

224

Airport Code Departure Time San Francisco 1:13 p.m. UTC -8 (SFO) Shanghai Pudong 5:56 p.m. UTC + 8 (PVG) Beijing 7:21 p.m. UTC + 8 (PEK)

Airport Code Beijing UTC + 8 (PEK) Beijing UTC + 8 (PEK) San Francisco UTC -8 (SFO)

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 Victorian Curriculum – Year 8 mathspace.co

Flight Time 12 hours and 31 mins. 8 hours and 31 mins. 14 hours and 37 mins.

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 1 2 3

Station Moscow UTC + 3 Novosibirsk UTC + 6 Ulaanbaatar UTC + 8

Departure Time 8:56 p.m Monday 1:25 p.m. Thursday 6:46 a.m. Saturday

How long does the ﬁrst train take?

How long does the second train take?

How long does the third train take?

Station Novosibirsk UTC + 6 Ulaanbaatar UTC + 8 Beijing UTC + 8

Arrival Time 2:29 p.m. Wednesday 9:47 p.m. Friday 2:27 p.m. Sunday

Consider the cities Hong Kong (GMT + 8) and Paris (GMT + 1). a

Calculate the time difference between the two locations if Hong Kong starts operating on daylight saving time (clocks are moved forward an hour) and Paris remains on standard time.

What time is it in Hong Kong when it is 8:00 a.m. in Paris and if Hong Kong has started operating on daylight saving time and Paris remains on standard time.

What time is it in Hong Kong when it is 8:00 a.m. in Paris and if Paris has started operating on daylight saving time and Hong Kong remains on standard time.

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 −8

Big Diomede

At 3:00 p.m. on Wednesday, Hannah rows from Big Diomede to Little Diomede in 5 minutes.

What day will it be on Little Diomede when Hannah arrives? b

UTC + 12

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?

Little Diomede

International Date Line

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The map shows the time zones across the Australian States: 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

DST

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 airplane 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 airplane 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 airplane 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.

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 mins, 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 mins, determine the time he arrived in Darwin local time.

226

TAS

Maria boards a train from Adelaide, South Australia at 4:55 p.m. Sunday towards Perth, Western Australia. If the trip took her 19 hours and 20 mins, determine the time she arrived in Perth local time.

Mathspace Victorian Curriculum – Year 8 mathspace.co

AWST (UTC + 8)

ACST (UTC + 9.5)

AEST (UTC + 10)

2h behind

0.5 h behind

Compared to AEST

NT QLD WA SA NSW VIC TAS

INVESTIGATION

Plan with time zones Investigate how to plan a trip across multiple time zones.

Objectives To convert between multiple time zones. To read and interpret a time zone map. To apply understanding of time zones to real-world situations.

Travelling west from Australia

Imagine you are planning a trip from Australia to the Azores islands of Portugal. The Azores are located about halfway around the world, and you will need to take 3 flights to get there.

Here is your flight itinerary: Leaves Melbourne at 1:05 p.m. Arrives in Singapore at 6:40 p.m. Leaves Singapore at 1:35 a.m. the next day Arrives in Berlin, Germany at 8:35 a.m. Leaves Berlin, Germany at 1:05 p.m. Arrives in Ponta Delgada, Portugal at 4:50 p.m.

The time zone map is based on standard time, not on daylight savings time.

C D B

A +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 A - Melbourne

B - Singapore

C - Berlin

D - Ponta Delgada

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227

City

UTC

Melbourne

Use the time zone map to complete the table with the UTC offset for each city you land in.

Singapore Berlin UTC + 0

Ponta Delgada

Investigate To make sure you arrive safely, your family member in Melbourne wants you to call or text each time you land. 1.

What time will it be in Melbourne when you land in Singapore? Should you call or text your family member?

How long was your flight from Melbourne to Singapore?

What time will it be in Melbourne when you land in Berlin? Should you call or text your family member?

How long was your flight from Singapore to Berlin?

What time will it be in Melbourne when you land in the Azores? Should you call or text your family member?

How long was your flight from Berlin to the Azores?

Did you cross the international date line?

What was the total travel duration for your trip (including layovers)?

Travelling east from Australia

Pacific Ocean

USA

Hawaiian Islands Australia

Fiji

Suppose you have a friend who lives in California, and they invite you to visit them in Los Angeles. Since most of the trip is over the Pacific Ocean, you only have one layover in Fiji.

Here is your flight itinerary: Leaves Sydney at 1:15 p.m. Arrives in Nadi, Fiji at 6:55 p.m. Leaves Nadi, Fiji at 9:40 p.m. Arrives in Los Angeles, USA at 12:55 p.m.

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The time zone map is based on standard time, not on daylight savings time.

B A +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 A - Sydney

City

B - Nadi

UTC

Sydney UTC + 12

Nadi

C - Los Angeles

Use the time zone map to complete the table with the UTC offset for each city you land in.

Los Angeles

Investigate Your friend in LA wants you to text them when you leave Sydney and when you land in Fiji. Your family member in Sydney wants you to text them when you land in Fiji and when you land in America. 1.

When you leave Sydney, what time is it in California?

When you land in Fiji, what time is it in Sydney? What time is it in Los Angeles?

How long was your flight from Sydney to Fiji?

When you land in Los Angeles, what time is it in Sydney?

How long was your flight from Fiji to Los Angeles?

Did you cross the international date line? Does that affect any of your answers above? Explain your reasoning.

What was the total travel duration for your trip (including layovers)?

Explain why you left and arrived on the same day.

Planning a meeting Imagine you are working for a company that has employees that work virtually from all over the world. You are in charge of two different teams.

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229

Team A The members of this team live in the following locations: Adelaide, Australia Canberra, Australia Melbourne, Australia Perth, Australia Darwin, Australia

Team B The members of this team live in the following locations: Adelaide, Australia Chicago, IL, USA Cairo, Egypt Rome, Italy To be respectful of everyone’s time, your company has a policy that requires meetings only be scheduled between 9 a.m. and 5 p.m. in each person’s time zone. If a meeting is scheduled outside of these times, you are not required to attend. You need to schedule two meetings; one with Team A and one with Team B. These meetings are important, so you are hoping to find a time that is suitable for everyone to attend. Remember, you need to attend the meeting too, so include your location when considering time zones.

Investigate 1. 2. 3. 4.

What range of times would be best for scheduling a meeting with Team A? Include the time zone you are using as the standard. Is it possible to find a meeting time for Team B that will be between 9 a.m. and 5 p.m. in each person’s time zone? Explain your answer. If you had to pick a time that would allow most of you to meet, what time would that be? Which person in which city would be left out? For the person for which the meeting was scheduled outside of the 9 a.m. to 5 p.m. timeframe, what would be some ways that they could still receive the information from the meeting? How would your answers to the previous questions be different if the meetings were in June versus December?

Discussion 1. 2. 3.

230

Explain the importance of time zones. Why wouldn’t it make sense for the entire world to be in the same time zone? What are the difficulties of communicating with people across the world? What could be some benefits of working for a company with employees around the world?

Mathspace Victorian Curriculum – Year 8 mathspace.co

5 Chapter review 1

Describe the difference between 12 and 24-hour time.

If it is currently 2:00 p.m. in Sydney (AEST), what is the current time in London (GMT)?

Convert: a

7 days to hours

15 minutes to seconds

48 hours to minutes

12 hours to seconds

300 seconds to minutes

360 minutes to hours

900 seconds to minutes

600 minutes to hours

A train ride from Sydney to Canberra takes 180 minutes. How many hours and minutes is this?

Where is the International Date Line and what happens when you cross it?

Emily’s cooking class starts at 1:30 p.m. and ends at 4:00 p.m. How long is the cooking class in hours?

John starts reading a book at 8:30 a.m., which takes 2 hours and 20 minutes to ﬁnish. What time will John ﬁnish reading the book?

A software installation on a server starts at 9:15 p.m. The installation will last for 105 minutes. Find the time that the installation will be completed.

Express the times in 24-hour time:

Five to seven in the evening

Twenty-ﬁve past two in the afternoon

Half past ten at night

Quarter to eleven at night

Quarter to three in the afternoon

Five past six in the morning

Find the time and day that is represented by: a

21 hours and 30 minutes after 19:45 on Monday

16 hours and 45 minutes before 12:30 on Friday

18 hours and 40 minutes after 20:15 on Sunday

24 hours and 20 minutes before 13:55 on Tuesday

A bus leaves at 14:25 from Los Angeles and arrives at 22:05 in San Francisco. How long did the bus take for the journey?

Sarah and her family set out for a day trip to the beach. They departed at 3:15 p.m. State this time in 24-hour time.

Mike starts grilling steaks for a dinner party. The steaks take 45 minutes to cook. Mike puts them on the grill at 4:35 p.m. What time did Mike take the steaks off the grill? Express your answer in 24-hour time. Chapter 5 review mathspace.co

231

A ﬂight between two cities takes 7 hours and 15 minutes. If you leave one city at 21:10, at what time will you arrive at the other city? Express your answer in 24-hour time.

If an average person reads for 20 minutes per day, ﬁnd how many hours they spend reading in: a

One year (assume non-leap year)

40 years (ignoring leap-years)

A day on Jupiter is 9.9 Earth hours and a year on Jupiter is 4332.59 Earth days. How long is a year on Jupiter in Jupiter days? Give your answer to the nearest Jupiter day.

Today is Alice’s birthday and she just turned 12 years old. Assume there are 365 days in a year. Find her age in: a

Days

Hours

Minutes

Seconds

Put the times in order from earliest in the day to latest in the day: 02:15, 18:45, 12:45 p.m., 12:45 a.m., 10:30 p.m., 16:30

Do these Australian states or territories observe Daylight Saving Time (DST): a

Western Australia b

New South Wales c

Tasmania

South Australia

Which location will be the ﬁrst to celebrate New Year’s Eve? • New Delhi (UTC + 5) • Washington (UTC − 5) • Rio de Janeiro (UTC − 3)

Convert the Daylight Savings Time in Sydney to Eastern Standard Time: a

2:49 a.m. AEDT

11:50 p.m. AEDT

19:43 AEDT

22:56 AEDT

When the time is 18:15 in Perth determine the time in: a

Sydney during No DST

Sydney during DST

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

232

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TAS DST

The ﬂight time from Los Angeles to Tokyo is 11 hours and from London to New York is 8 hours. a

Why would you gain time if you travelled from Los Angeles to Tokyo?

Why would you lose time if you travelled from London to New York?

Jenny in Hobart has a phone conversation with Joanne in Jakarta. Jakarta is 3 hours behind Hobart time. a

If Jenny calls at 10:45 a.m., determine the time it is for Joanne.

If Joanne calls at 4:45 p.m., determine the time it is for Jenny.

Eiichiro 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 1 2 3

Station Moscow UTC + 3 Novosibirsk UTC + 6 Ulaanbaatar UTC + 8

Departure Time 6:44 p.m. Wednesday 5:13 p.m. Saturday 6:46 a.m. Monday

How long does the ﬁrst train take?

How long does the second train take?

How long does the third train take?

Station Novosibirsk UTC + 6 Ulaanbaatar UTC + 8 Beijing UTC + 8

Arrival Time 1:24 p.m. Friday 9:50 p.m. Sunday 2:16 p.m. Tuesday

Did you know?

Time plays a critical role in baking! For example, if you leave bread in the oven for too long, it becomes too dry, and if you remove it too early, it remains undercooked. Timing the process ensures the perfect loaf. The right baking time allows the dough to rise properly, creating a soft and airy inside while forming a crisp, golden crust. Precise timing also helps lock in moisture, ensuring the bread stays fresh for longer. Mastering baking times can elevate the texture and flavor of baked goods, turning a simple recipe into a bakery-worthy creation. Chapter 5 review mathspace.co

233

Big ideas Algebraic expressions can be simplified and manipulated using various strategies, allowing us to simplify and solve mathematical expressions and equations.

Curriculum Outcomes VC2M8A01 Create, expand, factorise, rearrange and simplify linear expressions, applying the associative, commutative, identity, distributive and inverse properties VC2M8A02 Graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution

Algebra Chapter outline 6.01 6.02 6.03 6.04 6.05 6.06

Substitution Add and subtract algebraic terms Multiply and divide algebraic terms Simplify algebraic expressions The distributive law Numeric factors Chapter 6 Review

236 241 246 252 256 261 266

Emoji are symbols used to represent different words, objects and feelings. In algebra we use letters as symbols to represent unknown numbers.

6.01 Substitution After this lesson, you will be able to… • substitute positive and negative values for variables.

Substitution Substitution is a method used to replace variables with known values. By substituting these values, we can simplify an algebraic expression or verify the solution to an equation. To substitute numerical values in an algebraic expression we: • Determine which values that will be substituted for each variable. • Replace each variable with the corresponding value. It’s important to remember to keep the sign of the number. • Simplify the expression by following the order of operations. • Calculate the ﬁnal value of the expression. The order of operations, when evaluating an expression with substitution are the same as we have explored in whole numbers, fractions, decimals and integers: • Evaluate operations inside grouping symbols such as brackets. • Evaluate any powers. • Evaluate any multiplication or division, reading from left to right. • Evaluate any addition or subtraction, reading from left to right. • Consider and check the sign of the final answer, especially if negative values were involved in the calculation.

Example 1 Find the value of m + n when m = 6 and n = −4

Create a strategy

Apply the idea

Replace m with 6 and n with −4

m + n = 6 + (−4) =2

Substitute the values Evaluate

Example 2 Find the value of

when p = −28 and q = −7

Create a strategy

Apply the idea

Replace p with −28 and q with −7

236

Mathspace Victorian Curriculum – Year 8 mathspace.co

Substitute the values

Evaluate the denominator

Evaluate the quotient

Example 3 Find the value of k2 when k = −7

Create a strategy

Apply the idea

Replace k with −7

k2 = (−7)2

Substitute the values

= 49

Evaluate the power

Reﬂect and check When substituting a negative, it is important to use brackets to make sure that we apply the operation to both the negative sign and the number. If we had not used brackets in this example, we could have forgotten to square the negative and gotten the wrong answer: k2 = −72

Forgot to use brackets

= −49

Incorrect answer

Idea summary It is important that when we substitute values, the value that we substitute fits into exactly the same position as the variable, and that we follow the order of operations.

6.01 Practice questions Understanding 1

Trailblazer 3

There are ⬚ years in a century. ⬚ − 175 = −75

2.5 × ⬚ = 250

−5 + 2 (△ − 3)

If △ = 5, calculate the value of: a

Adventurer 1–3

Substitute the number 100 into the expressions: a

Explorer 1–3

△+△+△

12 × △

If 5a = 500, a = ⬚. d

What number does −9k equal, if k is 6?

6.01 Substitution mathspace.co

237

4, 5a, 6a, 7a, 8a, 9–11, 12a–c, 13, 14, 15ab, 16–18, 21ab

Fluency

Ex 1

3c + 9 when c = 6

45 − 7x when x = 6

z + 2 when z = −7

4 − t when t = 3

m = 7 and n = 9

m = 4 and n = 3.5

m = −10 and n = −2.5

p = −9, q = 21 and r = 3

a = 4, b = 16 and c = 2

u = 37, a = 2 and t = 14

u = 14, v = 5.5 and w = 3.6

238

a = 2, b = 3 and c = 4

u = 18, a = 2 and t = 4

Evaluate the expression u + vw when: u = 59, v = 3 and w = 15

Evaluate: a

when:

Evaluate u + at when:

a Ex 2

when:

p = 5, q = −3 and r = −9

Evaluate the expression

a 9

and n = 72

Evaluate the expression

a 8

when r = 12

Evaluate the expression mn when:

a 7

4–10, 11h–s, 12c–h, 13, 14, 15cd, 16, 19, 20, 21cd, 22

Evaluate:

e 5

4–10, 11e–n, 12b–f, 13, 14, 15cd, 16, 18, 19, 21ab

when k = 8

when k = 3

when k = 15

Evaluate: a

3y + 5w when y = 6 and w = 5

4x − 2y − 6 when x = 3 and y = 2

7 − p − q when p = 4 and q = 5

4 ( p + q) when p = 7 and q = 8

when p = −28 and q = −1

st when s = 7 and t = −8

4xy when x = −6 and y = −5

when a = 56 and b = −8

when m = 12 and n = 20

when u = −2 and v = 3

−m + n when m = 9 and n = −2

5 + a + b when a = −7 and b = 4

m −6x + 4y + 6 when x = 3 and y = 5

m − n when m = 2 and n = −9

−5 − s + t when s = 8 and t = 15

7x + y when x = 6 and y = −36

−4 (s − t) when s = 6 and t = 15

m − 2n when m = −19 and n = −7

when m = 8 and v = −15

Mathspace Victorian Curriculum – Year 8 mathspace.co

Ex 3

Evaluate: a

p + q + r when p = 5, q = 9 and r = 6

3j + 5k + 6l when j = 3, k = 8, and l = 7

abc when a = 8, b = 9 and c = 6

2pqr when p = 3, q = 4 and r = 5

6a − 3b + 4c when a = 8, b = 6, and c = 7 f

x − y − z when x = −4, y = 5 and z = 6

abc when a = −3, b = −9, and c = 3

[ b − a ]2 when a = 9, b = 4  

Find 18.6 − 3x when x is equal to 4.1

Evaluate

Evaluate: a

when r = −1.6 and s = 2.8, rounding your answer to three decimal places.

k2 when k = −8

c2 when c = 9

s3 when s = 5

k3 when k = −9

x=5

x = −1

x = −2

Evaluate x2 + 6x + 9 when: a

x=2

Verify if x = 4 is a solution to the equation 3x + 2 = 5x − 6 using substitution.

Verify if x = −1 is a solution to the equation 8 (x − 2) = 4 (2x − 3) using substitution.

Verify if x = 4 is a solution to the equation x + 1.5 =

Verify if x = 3 is a solution to the equation

Given the ordered pairs, use substitution to verify whether each is a solution to the given equation: a

(−1, 5)

using substitution.

Equation: y = −3x + 8 c

x − 0.5 using substitution.

(3, −2) Equation: y = x − 3

(0, 4)

(4, 8)

Equation: y = − x + 4 Equation: y = 1.5x + 2 22

Substitute and simplify the expressions when x = y2: a

Reasoning

b 25, 26

(4x)2

c 25, 26

−2x2 + (3x)2 23, 24

What is the largest whole number value that you can substitute for p so that the expression 81 − p2 is positive?

What is the smallest whole number value that you can substitute for p so that the expression 64 − p2 is negative?

Given the equation x − 2y = 6, ﬁnd the value of x when y = 3. Verify if your answer is correct by substitution.

6.01 Substitution mathspace.co

239

Explain how substitution can be used to ﬁnd the value of the expression x2 − y2 when x = 5 and y = 3.

Problem-solving 27 27

27–29

28–30

The area, A, of a triangle is given by the formula:

where h is the height of the triangle and b is the length of its base.

Find the area of a triangle that has a base of 7 cm and a height of 5 cm.

Find the area of a triangle that has a base of 25 cm and a height of 16 cm.

When the heating system in a house is on a setting of s, the temperature, T, of the house within the ﬁrst 30 minutes can be estimated by using the formula

where a is the initial temperature and t is the number of minutes since turning the heating system on. Calculate the temperature of the room after 18 minutes if the initial temperature is −3 °C and the setting on the heater is 5 °C. 29

Energy can be measured in many forms. A quantity of energy is given in units of joules (J). The kinetic energy, E, of an object in motion is calculated using the formula:

where m is the mass of the object in kilograms and v is the speed of the object in metres per second. Find the kinetic energy, E, of an object with a mass of 6 kg, travelling at a speed of 19 metres per second. 30

Valerie stands at the top of a cliff and launches a tennis ball across the valley. To estimate the vertical position, y, of the ball compared to herself she uses the formula:

where v is the initial vertical velocity and t is the number of seconds since the ball is launched. a

Find the vertical position of the ball after: i

240

2 seconds

3 seconds

iii

At which of the times is the ball above Valerie?

Mathspace Victorian Curriculum – Year 8 mathspace.co

8 seconds

6.02 Add and subtract algebraic terms This subtopic After this lesson, you will be able to… • add and subtract algebraic terms.

Addition and subtraction of algebraic terms An algebraic term is made up of a coeﬃcient and a single or multiple variables with powers (also called exponents). The variable u with a coeﬃcient of 3 and a power of 2 is written as:

3u2 = 3 × u × u The variable is u with a coefficient of 3 and power of 2. For example, these are all algebraic terms: 2x, 4y, −7ab, −12xy2 z3 Terms are classified as like terms if they have the same variables and each variable has the same power. Coeﬃcients can be different. For example, the following are like terms: −3xy2 and 5xy2 The following two terms are not like terms, even though they have the same variables, as the powers for each variable are different: −6ab3 and 2a2 b Only like terms can be added to and subtracted from one another. To collect like terms we look at the coeﬃcients of the terms and add or subtract them normally, keeping the variables and their powers the same.

Example 1 Simplify the expression 10x + 3x.

Create a strategy Collect like terms through addition.

Apply the idea 10x + 3x = 13x

Add like terms

6.02 Add and subtract algebraic terms mathspace.co

241

Example 2 Simplify the expression 12n − 9c − 7n.

Create a strategy Collect like terms through subtraction.

Apply the idea Only 12n and −7n are like terms, so we should leave −9c as it is. 12n − 9c − 7n = (12n − 7n) − 9c

Group like terms

= 5n − 9c

Subtract like terms

Example 3 Simplify the expression −6vw − 4v2w + 2v2w − 8wv

Create a strategy Collect like terms.

Apply the idea −6vw − 4v2w + 2v2w − 8wv = (−6vw − 8wv) + (2v2w − 4v2w)

Group like terms

= −14vw − 2v w

Combine like terms

Idea summary Two algebraic terms are called like terms if they have exactly the same combination of variables and each variable has the same power (exponent). To collect like terms means to simplify an expression by combining all like terms together through addition and/or subtraction.

6.02 Practice questions Understanding 1

Explorer 1, 2a–h, 3

Adventurer 1–3

Trailblazer 1–3

Describe the options using the best word from this set: term, coefficient, variable, constant a

242

The number 3 in the term 3x

The letter x in the term 6x

4x in the expression 4x + 7

The number 7 in the expression 5x + 7

The number −5 in 4 − 5x

The letter u in the term −12u

The number 8 in 6z + 8

−3y in 2 − 3y + 4z

Mathspace Victorian Curriculum – Year 8 mathspace.co

Describe the pair of terms as like or unlike. a

10p and 5p

3 and y

5p and 5

10a and −9a

15p and 15q

8 and 8z

8z and −8z

12b and −21b

5n2 and 8n

2ab and 6ba

7m2 and 7m

13xy and 14yx

Donna earns $15 per hour of work and is paid double for every hour worked on the weekend. At the end of a week, Donna calculates her pay for that week to be 15x + 30y dollars. a

What variable represents the number of hours Donna worked during the weekdays?

What variable represents the number of hours Donna worked during the weekends? 4a–m, 5a–g, 7

Fluency 4 Ex 1, 2

4n–x, 5g–t, 6h–r, 9

Simplify: a

2a + 5a

10x + 6x

4b + 3b

12y − 3y

9x + 4x

12p − 9p

4m − 4m

3c + 4c + 7c

15x − 6x − 2x

19b − 12b − 6b

8x − 3x + 7x

x+9+7

m 17n − 9m − 8n

−3n + 6n + 3n

8p − 7p

6x − 9x

−5m − 10m

7x − (−10x) − 8x

−6x − 5x − 3 + 8

4xy + 10xy − 9xy

10ab + 3ab − 8ba v

2u2 + 9u2

−5x2 − 2x2

−5pq + (−4pq)

Simplify: a

8x + 6y − 2y − 4x

10m + 9n + 5m + 10n

11m + 8n + 14m

6p + 8q − 6p

8x + 9y − 5x + 10y

12j + 9k − 7k + 11j

13m + 2n − 8m + 10n

10x − 9y − 7x + 11y

7a + 11a − 9b + b

8x − 7y − 6z + 10z

19a − 8b + 4c − 8a + b

18x − 9y + 7x + 11y

m 11y − 6z + y + 6z

−3s + 4t − 6t + 9s

9xy + 12yx

9p2 − 11p + 11p2

7n2 + n2 − 4m + 2m2 − 5n2

12x2 + 10x − 9x − 5x2

2ab − 7c + ab + 9c

s Ex 3

4h–s, 5a–n, 6a–g, 8

15y − 8y + 10y − y

Simplify: a

5x + 2y − 8x + 6z − 3y + 4z

20 − 9x + 4y + 6y − 7x + 21

10x + 9y − 3y − 2x

10p − 7q + 10 − 3p + 7q − 3

−2x + 2y − 6y − 10x

−5m + n − (−7m) − 6n − 2m + 7

−2m + n − (−6m) − 3n − 4m + 5

2xy + 5yz + 4xz − 8xy + 2xz − 3yz

8x2 + 10x − 6 + 4x2 − x + 11

9mn + 8m − 7n + 7n − 10m + 2mn

−6 + 8x − 9 + 5x + 16x

11s + 6t + 3st − 2ts + t − 4s

m 13ab − 8bc + 6ba − 9b + 11cb

10pq + 4p2q − 8qp + 7p2q

20x2 − 12x2 + 10x2 − x2

6zxy + 4xzy + 10xyz + 12xy + 11yx

−4rk + 3r − 6k − 2rk

−16vw + 4v2w + 2v2w − 8wv

6.02 Add and subtract algebraic terms mathspace.co

243

Which expressions are equal to 11y? a

9 + y + 10y − 9

6y − 5y

5y + 6y

11 + y

9y + 3y − 1

10y + 1

Which expressions are equal to 8r + 9? a

9 + 8r

9r + 8

17r

72r

9r + 8 − r + 1

10r + 2 − 2r + 7

Which expressions are equal to 9rt + 8? a

5r + 4t + 8

8 + 4rt + 5rt

9r + 8t

17rt

9tr + 8

2tr + 7rt − 2 + 10

Reasoning

10, 11

10, 12

12, 13

Tom simpliﬁed the expression 7x + 6p − 4x + 2 and found it to be 3x + 6p + 2. Xanthe simpliﬁed the same expression and found it to be 11x + 6p + 2. Explain who is correct and why.

In the expression 3x2 + 5x + 7x2 − 2x, which terms are like terms? Explain your answer.

Are these two expressions equivalent? Explain why.

•

12ab + 7c − 8ab − 9c

•

−6c + 2ab + 4c + 6ab − 4ba

Is it ever possible that 8m + 5n = 13mn? Justify your answer.

Problem-solving 14a–e, 15, 17, 18 14

14d–h, 16–18, 20

14d–h, 16, 17, 19–21

Write an algebraic expression for the perimeter of the shape: a

c 5m

2 cm 8n cm

5b m

2q m

f 8n − 4

5n + 4

9v cm 9n − 1 6v cm

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2a cm

9x + 5

6d − 5

8x + 4 4x − 5 6x + 6

5d − 4 7d + 1

An isosceles triangle has 2 sides of length 5a cm. If the third side has a length of a cm, write a simpliﬁed expression for the perimeter of the triangle.

Write a simpliﬁed expression for the perimeter of an equilateral triangle that has sides of 25a − 10b cm.

In a multiple choice quiz, x points are awarded for a right answer, and y points are deducted for a wrong answer.

If Lucas got 10 questions right and 4 questions wrong, write an expression for how many points he got.

If James got 9 questions right and 5 questions wrong, write an expression for how many points he got.

Write a simplified expression for how many points they scored all together.

Criza attends a dessert festival. Each ice cream costs $c and each milkshake costs $m. a

If Criza buys 4 ice creams and 8 milkshakes, write an algebraic expression for his total cost.

If David buys 2 ice creams less than Criza and double the number of milkshakes, write an expression for his total cost.

Tracy saves all of her $1 and $2 coins. After some time, she has 40 more $2 coins than $1 coins. If p represents the number of $2 coins, write a simplified expression for the total value of the coins that Tracy has in terms of p.

Laura decides to put a bird feeder in her back garden. The ﬁrst day she sees b birds use the feeder. The next day she sees b + 2 birds, and on the third day she sees b + 4 birds. a

If the number of birds continues to follow the same pattern, write an expression for how many birds Laura will see on the fourth day.

If the pattern continues, write a simpliﬁed expression for the total number of birds Laura will have seen after 5 days.

The width of a rectangle is 25 cm less than double the length. Let l be the length of the rectangle. Write an algebraic expression for the perimeter of the rectangle in terms of l.

6.02 Add and subtract algebraic terms mathspace.co

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6.03 Multiply and divide algebraic terms After this lesson, you will be able to… • multiply and divide algebraic terms.

Multiplication of algebraic terms Multiplication of algebraic terms does not use like terms. Our goal is to combine multiple terms into one. We can do this by: • Multiplying coefficients • Multiplying instances of the same variable and representing them with a power For example, consider: 3ab × 7ab2 We start by grouping the coefficients and variables and will write the variables with powers in expanded form: 3×7×a×a×b×b×b Which can be simplified to: 21a2 b3 Multiply all the coefficients together, keeping in mind any negatives if there are any. Rewrite the variables using the appropriate power.

Example 1 Simplify the expression 7 × u × v × 9

Create a strategy Multiply the coeﬃcients and combine the variables into one term.

Apply the idea 7×u×v×9=7×9×u×v = 63uv

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Rearrange the terms Evaluate the multiplication

Example 2 Simplify the expression (−5r) × (−4s)

Create a strategy Multiply the coeﬃcients and combine the variables into one term. Remember that the product of two negative numbers is a positive number.

Apply the idea (−5r) × (−4s) = (−5) × (−4) × r × s = 20rs

Rearrange the terms Evaluate the multiplication

Idea summary When we multiply algebraic terms our goal is to write multiple terms as one. We do this by multiplying coeﬃcients and variables and writing them in the simplest form possible.

Division of algebraic terms Our goal when we divide algebraic terms is the same as for multiplication, that is, combining multiple terms into one in its simplest form. We use a similar technique to multiplication as follows: • Divide and simplify coeﬃcients as if they were a fraction. • Divide instances of the same variable making use of the fact that a variable is representing a number and a number divided by itself always equals 1. For example:

Writing variables without powers in expanded form gives: 6× a× a×b 6× a × a× b = 2× a× b 2× a × b Dividing coeﬃcients means

is simplified.

We see that we can remove a common factor of a and b from both the numerator and denominator, which leaves us with: 3a

6.03 Multiply and divide algebraic terms mathspace.co

247

Example 3 Simplify the expression

Create a strategy Remove any common factors in the numerator and denominator.

Apply the idea The coeﬃcients 15 and 5 have a highest common factor of 5. The variable y is also a common factor for both the numerator and denominator. Divide the numerator and denominator by 5 Evaluate Remove the common factor of y

Example 4 Simplify the expression

Create a strategy Remove any common factors in the numerator and denominator.

Apply the idea The coeﬃcients 6 and 18 have a highest common factor of 6. The variable a is also a common factor for both the numerator and denominator. Divide the numerator and denominator by 6 Evaluate Remove the common factor of a

Idea summary We divide algebraic terms by considering coeﬃcients and variables separately, combining what is left into a simpliﬁed expression or term.

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6.03 Practice questions Explorer

Understanding 1

3 × x × y

2×x×x×y

−2 × x × x × y × y

5×x×x×y×y×y

Are these statements true or false? a

5abc = 5bca

−2 × 3 × x × y = −5xy

a4 = a × a × a × a

2x × 3y can be simplified even though there are not like terms because they are being multiplied.

Simplify:

34 ÷ 7

15x ÷ 4

6, 7a–f, 8a–g, 9a–d, 10a–j, 13

12y ÷ 7c

7e–h, 8d–l, 9, 10k–s, 11a–g, 12a–f

19abc ÷ 12cd

8h–l, 9e–h, 10k–v, 11h–l, 12g–l

10 × 6u

5x × 2

11 × 3y

7 × (−3u)

(−8x) × 9

(−12) × (−2u)

(−3w) × 2

10 × (−3y)

2x × 3x

4y × y

3a × 5a

(4x)2

10 × (−r) × s × (−5)

6u2 × 7v8

2a4 × (−5b3)

Simplify: x×x 8x × 2xy

9b × 3b

(10a)

Simplify: a

9×m×n×8

w×4×z×6

9r × 6s

(−2a) × (−4b)

3w × (−7z)

(−5r) × (−8s)

i 9

Ex 1, 2

Simplify:

Rewrite each division as an unsimplified fraction:

Fluency

Simplify:

1–5

a 5

Trailblazer

1–5

Simplify:

a 4

Adventurer

1–5

16p × 14q

(−10r ) × 6s

4p × (−3q )

Simplify: a

6r × 2 × 8s

7w × 9x × 10y

5x × 2y × (−9)

4a × (−2a) × 10a

(−2w) × (−4x) × (−10y)

(−8h) × 5k × (−3r) × (−4s)

(−5x) × (−2x) × (−3y) × (−3)

(−2h) × 4k × (−j ) × (−5i) 6.03 Multiply and divide algebraic terms mathspace.co

249

Ex 3

Ex 4

Simplify: a

5m ÷ 40

20wz ÷ 40wz

(−4rs) ÷ 44r

(−36uv) ÷ (−6uv)

10mn ÷ 5m

18xy ÷ 36y

(−50abc) ÷ (−500ab)

10r6 ÷ 5r6

(−24r4) ÷ 6r4

27r2 ÷ 9r

(−22x2 y) ÷ − 2xy

k 12

(−20x ) ÷ 10x

Simplify: a

9x × 4x ÷ 2x

(−5x) × 8y ÷ 10y

11xy × 3y ÷ y

20y ÷ 4 × 3y

(10y ÷ 2y) × (4x ÷ 2)

(6x × 3y) ÷ (9x × 2)

If the distance of a marathon is 20x km, what would be the distance of a half marathon?

Reasoning

14–16

Use a = 1, b = 2, c = 3 and d = −1 to show that 2ab × 4cd = 8abcd

Consider the expressions 3x2y and − 4yz2. Explain why the product of these two expressions is −12x2 y2 z2.

Consider the expression:

Explain why the simpliﬁed expression is 4z. 250

Mathspace Victorian Curriculum – Year 8 mathspace.co

14, 16, 17

When we simplify

we get 3. However, if we simplify

we get .

Why do we have to write the 1 on the top of a fraction when simplifying, but not on the bottom of the fraction?

Problem-solving 18a–c, 20–22 18

18, 20–22

18c–f, 19, 21–23

Write an algebraic expression for the area of the shape: a

5q cm

6 mm 3q m 10q cm

7v mm 4m

f 4q m 4n m 9q m 8n m

5n cm

Ariel has two times as many books as Harry does, and Harry has triple the number of books that Carl does. If Carl has n books, write an expression for the number of books that Ariel has.

Write an expression for the area of a square with side length of 5ab cm.

A rectangle has a width 5 times its length. If the length is 4y cm, write an expression for the area of the rectangle.

At Larry’s dinner party, pizza slices are shared evenly among his 10 guests.

If Larry shares x slices, write an algebraic expression for the number of slices given to each guest.

If Larry shares 20 pizza slices, ﬁnd the number of slices given to each guest.

While Judy is packing rectangular boxes into crates, she notices that each crate is 12 times wider than the width of one box, and 11 times longer than the length of one box. Judy wants to know the greatest number of boxes she can pack into each crate. Let the length of one box be L cm, and the width of one box be W cm. The volume of a rectangular prism is: V = length × width × height a

Find an expression for the volume of one box with a height of 44 cm.

Find an expression for the volume of a crate of height H cm.

Find an expression for the number of identical boxes that Judy can ﬁt into each crate.

If the crate is 88 cm high, calculate how many boxes Judy can ﬁt into each crate. 6.03 Multiply and divide algebraic terms mathspace.co

251

6.04 Simplify algebraic expressions After this lesson, you will be able to… • simplify algebraic expressions.

Simplify algebraic expressions We use our skills in adding, subtracting, multiplying and dividing algebraic terms to simplify algebraic expressions following the same order of operations as for numeric expressions. The process of ﬁnding a common denominator and creating equivalent fractions is used when simplifying expressions that involve fractions and variables.

Example 1 Simplify the expression 4s + 9 × 5s

Create a strategy Use the order of operations.

Apply the idea 4s + 9 × 5s = 4s + 45s = 49s

Evaluate the multiplication Evaluate the addition

Example 2 Simplify the expression 12s − 8s ÷ 4

Create a strategy Use the order of operations.

Apply the idea 12s − 8s ÷ 4 = 12s − 2s = 10s

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Divide −8s by 4 Evaluate the subtraction

Example 3 Simplify the expression (−7n) × 4 − (2n + 5n)

Create a strategy Use the order of operations.

Apply the idea (−7n) × 4 − (2n + 5n) = (−7n) × 4 − 7n

Simplify expression in brackets

= −28n − 7n

Perform multiplication

= −35n

Perform subtraction

Example 4 Simplify the expression

Create a strategy Evaluate each of the operations on the numerator and denominator separately.

Apply the idea Collect the like terms in the numerator Evaluate the multiplication in the denominator Remove the common factor of 18m in both numerator and denominator

Idea summary We simplify algebraic expressions using the familiar numerical techniques of order of operations and those associated with working with fractions.

6.04 Simplify algebraic expressions mathspace.co

253

6.04 Practice questions Understanding 1

Explorer 1–4

Adventurer 1–4

Trailblazer 1–4

Evaluate 2 + 4 × (5 − 2)2

Why is the expression 4x + 2 − 3x − 1 not considered to be simpliﬁed?

Simplify: a

Fluency

Ex 1, 2

254

5a–g, 6a–d, 7, 8, 9a–d, 10ab

5a–k, 6a–k, 7, 8, 9aceg, 10a–f

5h–p, 6e–n, 9, 10c–l

Simplify: a

2t + 4 × 8t

2c + 7c × 3 + 4c

61a − 6 × 9a − 2a

8 × 3a + 6 × 2a

20w − 10w ÷ 2 − 3w

22s − 16s ÷ 4

12u ÷ 3 − 2u

6p × 8q ÷ 2

8jk × 15k ÷ 10j

20pq ÷ 5q × 2p

8jk × 15k ÷ 20j

3n2 + 8n2 − 3n2

m 9n2 + 3n × 6n

5n2 + 5n2 × 2 + 7n2

5mn × 3n − 2mn2

60n2 ÷ 5n ÷ 3

o Ex 3

Simplify: a

(5u − 2u) × 3

(13a + 15a) ÷ 7

(19v − 5v) ÷ 2 × 3

6v + (15v − (9v − 2v))

9p ÷ (6p − 3p)

(−8n) × 4 − (9n + 4n)

30r ÷ (3r + 2r) + 5r

5v − 5 (6v − 2v)

−8t − (8t − 6t ÷ 2)

(7m − 5m) × 4m

(−4n) × 8 − (7n + 4n)

80mn ÷ (4m) ÷ (5n)

m (6u × 12vw) ÷ (6v × 32u)

((−8r) × 16st) ÷ (24s × (−9r))

Mathspace Victorian Curriculum – Year 8 mathspace.co

Ex 4

Consider a

Find the lowest common denominator of

Write

as a single fraction.

Consider a

Find the lowest common denominator of

Write

and

as a single fraction in simplest form.

Simplify: a

Simplify:

Reasoning 11

and

11, 12

11–13

Simplify the expression 12x − 5x × 2 in two different ways. Perform the multiplication ﬁrst and simplify, then repeat the problem and do the subtraction ﬁrst. Explain why an order of operations is needed when simplifying algebraic expressions.

Substitute n = 5 to show that 6n × 2 − (3n + 2n) is equal to 7n.

Describe the steps you would take to simplify the expression: (10x − 2x) × 5 + (11x + 3x − x) × 2

Problem-solving 14, 15 14

14–16

14–18

Consider the rectangular prism: a

Write an expression for its volume.

Write an expression for its surface area. 7 mm 2a mm 4a mm

6.04 Simplify algebraic expressions mathspace.co

255

Dylan, Jimmy and Valentina are travelling from Adelaide to Sydney at x, y and z km/h respectively.

Let S1 represent the average speed of Dylan and Jimmy’s vehicles, and S2 represent the average speed of Dylan and Valentina’s vehicles. Write a simpliﬁed expression for S1 + S2 in terms of x, y and z. Frank wants to determine the proﬁt he makes on each eraser he sells. Frank can sell 3 erasers for x dollars, and it costs y dollars to make 9 erasers.

Determine a simpliﬁed expression for the proﬁt Frank makes on each eraser. Leave your answer as a single fraction.

Frank realised he could sell 3 erasers for $3 more than they previously were, and reduce the cost of production of 9 erasers by $2. Find the new proﬁt per eraser.

Scott receives $y for his birthday. He spends

on going to the movies and

on computer

games. Write a simpliﬁed fraction for the amount of money he has left. Glen lived in a rectangular bedroom. His mother asked him to ﬁnd out the area of his room, and since Glen did not have a measuring tape or ruler, he counted how many of his steps each wall was.

If Glen’s foot length is x cm, and two adjacent walls were 12 and 10 footsteps long, write an expression for the area of his room.

6.05 The distributive law After this lesson, you will be able to… • apply the distributive law to simplify algebraic expressions.

The distributive law To evaluate an expression like 3(7 + 2) we can add the 7 and 2 in the brackets ﬁrst and then multiply by the 3. The distributive law allows us to evaluate an expression like 3(7 + 2) by distributing the 3 to both the numbers in the brackets as follows: 3(7 + 2) = 3 × 7 + 3 × 2 A way to think about the distributive law is by considering area of rectangles.

256

Mathspace Victorian Curriculum – Year 8 mathspace.co

13 m (5 + 8)

Area = 9 × (5 + 8)

Area = 9 × 5 + 9 × 8

= 9 × 13

= 45 + 72

= 117 m2

= 117 m

9 × (5 + 8) = 9 × 5 + 9 × 8 The distributive law states that if A, B, and C are any numbers then: A (B + C) = AB + AC The distributive law is particularly useful for expanding algebraic expressions where we can’t evaluate the expression in the brackets.

Interactive exploration Discover this concept in action online

mathspace.co

Example 1 Expand 9 (5 + w)

Create a strategy Use the distributive law A (B + C) = AB + AC.

Apply the idea 9 (5 + w) = 9 × 5 + 9 × w = 45 + 9w

Expand the brackets Evaluate each multiplication

Example 2 Expand −9 (n − 2)

Create a strategy Use the distributive law A (B − C) = AB − AC.

Apply the idea −9 (n − 2) = −9 × n − (−9 × 2)

Expand the brackets

= −9n − (−18)

Evaluate each multiplication

= −9n + 18

Combine the adjacent signs

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257

Idea summary Expand means to write an algebraic expression without brackets. We can use the distributive law to expand an algebraic expression brackets like so: A (B + C) = AB + AC and if the second term in the brackets is negative: A (B − C) = AB − AC where A, B and C are any numbers.

6.05 Practice questions Understanding

Explorer 1–4

Adventurer 1–4

Complete:

The distributive law in mathematics states a (b + c) = a × ⬚ + a × ⬚

4 5

258

Evaluate 5 (12 − 6)

Complete 5 (12 − 6) = 5 × ⬚ − 5 × ⬚

Trailblazer 1, 4–5

Evaluate 5 × 12 − 5 × 6

The area of the whole rectangle is 16 × 20 = 320 units2. a

Find the area of the shaded rectangle.

Find the area of the unshaded rectangle.

Find the sum of the two areas.

Does 16 (18 + 2) = 16 × 18 + 16 × 2?

Complete: a (b + c + d) = a × ⬚ + a × ⬚ + a × ⬚

The area of the rectangle is k (m + n). a

Write an expression for Area A.

Write an expression for Area B.

Write an expression for the sum of the two areas.

Does k (m + n) = kn + km?

Mathspace Victorian Curriculum – Year 8 mathspace.co

B m+n

6, 7a–e, 8a–d, 9a–d, 10a–c

Fluency Ex 1

Ex 2

5 (s + 7)

10 (5 + w)

3 (r − 6)

( y + 8) × 7

4 (4s + 5)

p (q + 4)

4 (2r + 3s + 5t)

6m (2n + 3)

r (s + 8)

9 (v + 6)

Expand: a

−8 (t + 10)

−8 (c − 5)

−10 (8 + 3r)

−8 (−2t + −3)

9 (u − v)

−6 ( 4m + 3n + 5p) g

−r (11s + 13t)

− (y + 2)

−10 (3 + 8u3)

3uv (v+ u)

− (2m − 3n) 5

m 4s (4t − 9)

4p (4q − 3+ 6r) 3

−10 (y + 7) 3

6p (2q − 9)

− (9 − s )

Expand: a

r (r + 7)

6m (4m + 7)

4wy (y + w)

−6a (2a+ 9)

9f (f − 4g−6h)

2f (f − 7g−6h)

Expand and simplify: a

4 (x + 8)−2

7 + 5 (3x + 4)

5 + 3 (x + 4)

8 (3x + 4)−6x

34 + 9 (4x − 5)

5x− 8 (2x − 3)

9x− 5 (2x + 3)

−8 (5x − 7) − 9

4y + 5 + 6 ( y − 9)

4 (5 (x − 3) + 2)

Expand and simplify: a

7 ( y + 9) + 5 ( y + 6)

10 (x − 4) + 3 (x + 8)

5 ( y + 7) − ( y − 9)

3 (6x − 5y) − 2 (9y + 4x)

−8 (x + 4) − 3 (x − 2)

11 (a + 4) − 6 (4 − a)

x (x + 4) + 8 (x + 9)

8y2 − 4y + 7y ( y − 5)

4u (6u + 9v) − 2u (8v + 5u)

9w (8w + 6z) − 5w (4z + 2w)

Reasoning 11

6e–j, 7g–o, 8, 9d–j, 10d–j

Expand:

6d–j, 7d–l, 8, 9c–h, 10c–h

11, 13, 17

11, 12, 14, 17

12, 14–17

The diagram shows a square with side length x. a

What is the area of a square with side length x?

Write an expression for Area I.

Write an expression for Area II.

Show that x (x − 4) + 4x = x2

x 4

Area I

Area II

Use the diagram and areas of rectangles to demonstrate that 4 (x + 6) = 4x + 24

6.05 The distributive law mathspace.co

259

A student incorrectly used the distributive property and wrote 6 (7x + 2) = 42x + 2 Explain the error and how to correct it.

A student incorrectly used the distributive property and wrote 2x + (4x − 1) = 8x2 − 2x Explain the error and how to correct it.

This rectangle has width x + 2 and length x + 6. Use the diagram and areas of rectangles to ﬁnd a simpliﬁed expression for (x + 2) (x + 6).

Use the distributive law to prove that 5 (3x − 4) = 4x + 2 (5x − 5) + x − 10.

The distributive law can be used to calculate 5 × 19 as follows:

Area I

Area II

Area III

Area IV

5 × 19 = 5 × 10 + 5 × 9 = 50 + 45 = 95 Use a similar method to calculate 8 × 107

Problem-solving 18–20

18, 19, 21

20–22

A solar energy company offers a range of solar panels, each with a power rating of (m + 8) kW, where m is the panel model number. Write an expression in terms of m for the total amount of power produced by 11 solar panels.

The representatives of 8 countries are attending a global summit, and each country is able to send (2r − 3) representatives. Write an expression for the total number of representatives attending the summit.

A forklift operator has enough time to make 5 trips between the two ends of a warehouse. A forklift can carry (2 + 3n4) boxes at a time. Write an expression for the total number of boxes that can be carried across the warehouse.

Consider the rectangle shown:

8st

Find an expression for the area of the rectangle in expanded form.

2 (s + t)

A rectangle is 8 m longer than it is wide. Its width is represented by the variable u. Form an expression for the area of the rectangle in expanded form.

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6.06 Numeric factors After this lesson, you will be able to… • factorise algebraic expressions.

Factorisation Interactive exploration Discover this concept in action online

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The distributive law says that for any numbers A, B, and C, A (B + C) = AB + AC. We saw how to use this rule to expand algebraic terms, but we can also use the rule in reverse. The reverse of expanding algebraic expressions is called factorising. Factorising an algebraic expression means writing the expression with the highest common factors (HCF) between the terms taken outside of the brackets. Consider the expression: 2x2 + 6y The HCF is 2, which we take outside the brackets to give the factorised form of: 2 (x2 + 3y) Factorising is a critical tool in algebra, as it simpliﬁes expressions and makes them easier to work with. It’s particularly useful when solving equations, graphing functions, and simplifying fractions.

Example 1 Factorise 42t + 35

Create a strategy Determine the highest common factor of the numbers 42 and 35.

Apply the idea 7 is a factor of both 42 and 35, so factorise out the multiple of 7. 42t + 35 = 7 × 6t + 7 × 5 = 7 (6t + 5)

Write the terms as products of 5 Factorise

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Example 2 Factorise the expression −2s − 10

Create a strategy Determine the highest common factor between the terms −2s and −10.

Apply the idea −2 is a factor of both −2 and −10, so we can factorise −2 out. −2s − 10 = −2 × s + (−2) × 5 = −2 (s + 5)

Write the terms as products of −2 Factorise

Reﬂect and check It is important to be careful about factorising a negative because the sign of each term will change. If the leading term in an expression is negative, then we should factorise the negative out.

Example 3 Factorise 10g − 35h + 15

Create a strategy Determine the highest common factor between the terms 10g, −35h and 15. As there are no similar algebraic facts, we can focus on finding a numeric factor. What is the highest common numeric factor between the numbers 10, −35 and 15?

Apply the idea 5 is a common numeric factor between the numbers 10, −35 and 15. So we can factorise the 5 out: 10g − 35h + 15 = 5 × 2g − 5 × 7h + 5 × 3 = 5 (2g − 7h + 3)

Write the terms as products of 5 Factorise

Idea summary The reverse of expanding algebraic expressions is called factorising. Factorising an algebraic expression means writing the expression with any common factors between the terms taken outside of the brackets.

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6.06 Practice questions Understanding

Adventurer 1–3, 4c–f

Trailblazer 1–3, 4g–j

In the expression 4 (x + 6), what is the numeric factor?

Complete:

5 is the common factor of 5a and 15, therefore 5a + 15 = ⬚ (a + 3)

Is 2 (4 − x) the same as −2 (−4 + x)? Can numeric factors be negative or positive?

Determine the highest common factor of the these terms: a

2 and 6x

3 and 15x

6x and 21

10 and 16u

35x and 14

4x and −16y

−15 and 5x

24m, 12n and 18q

27m, 24n and 15q

−45x, −9x and −90

Fluency 5

c e g i 6 Ex 1, 2

5–7, 8a–d, 9ab, 10a–d

5f–j, 6e–l, 7, 8c–h, 9, 10c–h, 11

6e–l, 7, 8g–l, 9–11

Complete the factorisation: a

Ex 3

Explorer 1–3, 4a–d

2a + 16 = ⬚ (a + 8)

9a − 36 = 9 (⬚)

5v − 30 = ⬚ (v − 6)

16a − 6 = ⬚ (8a − 3)

2r + 14 = 2 (⬚)

5y − 30 = ⬚ ( y − 6) 6v + 18 = 6 (⬚)

4r + 12 = ⬚ (r + 3)

18c − 24 = ⬚ (3c − 4) 5t − 5 = 5 (⬚)

Factorise: a

6v + 30

6a − 6

25t + 25

2c − 18

40c − 35

2m + 10n

6w + 30y

14y + 4w

4a − 14b

32 − 36jk

8jkt − 18

−3g − 30fh

Complete the factorisation:

Factorise:

2w − 10z + 8y = 2 (⬚ − ⬚ + ⬚)

4w + 4z + 4y

8g + 28h − 20

4u − 12v + 20w

10m − 25n − 15p

9w − 9y − 9k

5p − 5a − 5b

3m − 6n + 12p

20g − 15h + 25

60 − 25rt − 35r2t

12m − 32n − 20m2n

16z − 32yz − 24y2

25r2 − 15t2 − 45r2t2

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Factor out the negative coeﬃcient of the expressions: a

−7r − 7s

−8x − 80y

−4r − 3s

−2 − 9r

−18a + 16

Factorise: a

−8b − 16

−36 − 6x

−12s + 10

−80v − 30

−28b − 21

−4m − 32n

−49y − 21w

−12b + 18a2

−8v + 56

−63r + 28

−10u + 9v

Factorise: a

−18p − 9q − 45r

Reasoning

−10r − 25s + 20t

12, 14, 15

c −20u − 25v + 45w d

12–15

−8wy − 18y − 10z

13–16

Expand the expression −(2x + 3 − 5y). Explain what happens to the sign of each term when a negative is taken as the common factor.

Explain why the expression x2 − 5x + 6 cannot be factored with a numeric factor.

Compare the factored forms of 6x + 9 and 9x + 6. What can you say about their common factors?

Explain why 2x + 2 = 2 (x + 0) has not been factorised correctly, then correct the error.

Use the diagram and your knowledge of areas of rectangles to write the factored form of x2 + 6x + 4x + 24 x

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Problem-solving 17–19

18–20

19–23

A rectangular garden has a length represented by the expression 12x and a width represented by the expression 6y. Determine the perimeter of the garden and express in factorised form.

A bakery sells two types of bread, represented by the algebraic expressions 8x and 12y. The bakery decides to create a new combo package that includes both types of bread. Write an expression representing the total number of bread units in the combo package in factorised form.

A farmer has two types of fruit trees, with the number of fruits produced by each tree represented by the expressions 5x − 15 and 10x − 30. The farmer harvests one tree of each type. What is the expression representing the total number of fruits harvested after factoring out the numeric factor?

A truck driver has two routes with distances represented by the expressions 21x2 − 14 and 9x2 − 6. If the truck driver takes both routes, write an expression representing the total distance travelled in factorised form.

A factory produces two types of products, with the number of units produced per day represented by the expressions 14x − 8y and 26x + 3y. If the factory decides to merge the production of both products, write an expression representing the total number of units produced per day in factorised form.

We know that

, therefore

and

Factorise and simplify: a 23

The expression 12 (x + 1) + y (x + 1) has a common factor of x + 1 and can be written in factorised form as (x + 1) (12 + y). Factorise: a

k (b − 4) + 6 (b − 4)

m (x + 3) + n (x + 3)

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6 Chapter review 1

Simplify: a c

−3x − 4x

5x − (−7x) − 3x

7x2 + 9x − 5 + 3x2 − x + 10

8mn + 7m − 6n + 6n − 9m + 3mn

−3m + n − (−7m) − 4n − 5m + 6

6×a×a×b×b×b

Simplify: −4 × a × a × b × b

To get to school in the morning, Sally walks for 7 minutes to the bus stop and then waits for 2 minutes for the bus to arrive. She then catches the bus the rest of the way to school. a

If the bus trip takes y minutes, write an expression in terms of y for the total length of time it takes Sally to get to school.

Explain what the constant in your expression represents.

Simplify: a × 5 × b × 7 3

15 × (−p) × q × (−4)

7v × 8w

(−12s6) × 5t4

6u4 × (−2v2)

3r3 × (−4s2)

Simplify: a

Simplify:

266

8ab + 5ab − 12ba

3xy + 4yz + 5xz − 7xy + 3xz − 2yz

Simplify:

a 4

14b − 8b − 5b

8a × 3a ÷ a

(−6a) × 9b ÷ 18b

10ab × 2b ÷ b

(7a × 4b) ÷ (14a × 2)

Simplify: a

70m − 7 × 8m − 4m

4k2 + 9k2 − 3k2

30j − 15j ÷ 3 − 4j

(−8k) × 5 − (3k + 6k)

(23x − 2x) ÷ 3 × 4

((−9y) × 18zw) ÷ (30z × (−10y))

Mathspace Victorian Curriculum – Year 8 mathspace.co

Simplify: a

Simplify:

Expand and simplify: a

6 + 4 (2p + 3)

7 (2p + 3) − 4p

p ( p + 3) + 7 ( p + 8)

−7 ( p + 3) − 2 ( p − 1)

9 (b + 3) − 5 (3 − b)

3v (5v + 8w) − v (7w + 4v)

10mnp − 20

Factorise: a

70 − 140rs

6a + 6b + 6c

8p − 24q + 32r

5s − 10t + 20u

12x − 12y − 12z

20w − 40xy − 30x2

−4h − 40qr

80 − 30mn − 50m n

Evaluate: a

−a + b when a = 12 and b = −4

pq when p = 9 and q = −7

when x = −33 and y = −2

Substitute and simplify the following expressions when a = b2: a

Factorise:

e 14

8w + 32

4a2

(5a)2

−3a2 + (2a)2

Given the following ordered pairs, use substitution to verify whether each is a solution to the given equation: (4, −3)

(−2, 6)

Equation: y = −4x + 10

Equation:

(0, 5)

(5, 9)

Equation:

b d

Equation: y = 1.7x + 3

What is the largest integer value that you can substitute for x so that the expression 144 − x2 remains positive?

An isosceles triangle has 2 sides of length 6b m. If the third side has a length of b m, write a simpliﬁed expression for the perimeter of the triangle.

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The length of a rectangle is 35 mm less than triple the width. Let b be the width of the rectangle. Write an algebraic expression for the perimeter of the rectangle in terms of b.

Write an algebraic expression for the area of the shape: a

b 5 mm 7s m 8q mm 10s m

Samantha has three times as many pens as Ben does, and Ben has four times the number of pens that Charlie does. If Charlie has p pens, write an expression for the number of pens that Samantha has.

Laura, Jack, and Naomi are travelling from Houston to Dallas at a, b and c km/h respectively. Let S3 represent the average speed of Laura and Jack’s vehicles, and S4 represent the average speed of Laura and Naomi’s vehicles. Write a simpliﬁed expression for S3 + S4 in terms of a, b and c.

A wind energy company offers a range of wind turbines, each with a power rating of (n + 10) kW, where n is the turbine model number. Write an expression in terms of n for the total amount of power produced by 12 wind turbines.

A truck driver has enough time to make 6 trips between the two ends of a warehouse. A truck can carry (3 + 2m3) crates at a time. Write an expression for the total number of crates that can be carried across the warehouse.

A software company develops two types of applications, with the number of units produced per week represented by the expressions 18x − 9y and 32x + 4y. If the company decides to combine the production of both applications, write an expression representing the total number of units produced per week in factorised form.

In physics, potential energy is often used as a measure. It is given in units of joules (J). The potential energy, E, of an object at height is calculated using the following formula: E = mgh where m is the mass of the object in kilograms, g is the acceleration due to gravity and h is the height in metres. Find the potential energy, E, of an object with a mass of 8 kg, at a height of 21 metres. Assume that the acceleration due to gravity is 9.8 m/s2.

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When a car is driving at a constant speed s, the distance D the car will cover within the ﬁrst 40 minutes can be estimated by using the formula D = st where s is the speed in km/hour and t is the time in hours. Calculate the distance the car will cover after 25 minutes if the car is driving at a constant speed of 60 km/hour.

Did you know?

Algebra helps with budgeting and finance! Whether you’re planning a monthly budget, calculating interest on a loan, or figuring out how much to save for a future goal, you’re applying algebra. It enables you to forecast your savings, investments, and spending, giving you control over your financial future. By using algebraic equations, you can adjust your expenses and income variables to see how different scenarios affect your financial goals. For example, algebra helps you calculate how long it will take to pay off debt by making extra payments or how much interest you’ll earn on savings over time. It’s the key to making informed financial decisions, whether you're saving for a vacation or planning for retirement.

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Big ideas • Linear relationships can be represented and analysed using graphs and equations. • Linear equations and inequalities can be solved using various strategies to determine the values that satisfy the given conditions and represent the solutions.

Curriculum Outcomes VC2M8A02 Graph linear relations on the Cartesian plane using digital tools where appropriate; solve linear equations and one-variable inequalities using graphical and algebraic techniques; verify solutions by substitution VC2M8A03 Use mathematical modelling to solve applied problems involving linear relations, including financial contexts involving profit and loss; formulate problems with linear functions, and choose a representation; interpret and communicate solutions in terms of the context, and review the appropriateness of the model VC2M8A05 Experiment with linear functions and relations using digital tools, making and testing conjectures and generalising emerging patterns

Linear relationships Chapter outline 7.01 7.02 7.03 7.04 7.05 7.06

Solve equations Tables of values Linear graphs Investigation: Linear solutions Multi-step equations Intersecting lines Inequalities Investigation: Experiment with linear inequalities Chapter 7 review

272 278 288 302 304 312 325 341 344

Think of a linear equation as a balanced scale. Just as both sides of the scale have the same weight, both sides of the equation have the same value.

7.01 Solve equations After this lesson, you will be able to… • solve equations using algebra.

Inverse operations To solve equations using algebra, the two most important rules to remember are: • Use inverse (opposite) mathematical operations • What we do to one side of the equation we must do to the other Using inverse operations allows us to get the variable we are solving for on its own. When the variable is on its own we call it the subject of the equation. Applying the same mathematical operation to both sides of the equation keeps it balanced and ensures both sides remain equal. The following table reviews and provides examples of applying inverse operations to isolate a variable. Isolate a variable means to get it on its own. Operation Addition Subtraction Multiplication Division

Inverse operation Subtraction Addition Division Multiplication

Example x+4−4=x x−3+3=x y×4÷4=y y÷2×2=y

Example 1 Solve 24 + p = 59

Create a strategy The inverse (opposite) of addition is subtraction, so we need to subtract from both sides to isolate p.

Apply the idea 24 + p = 59 24 + p − 24 = 59 − 24 p = 35

Write the equation Subtract 24 from both sides Evaluate

Reﬂect and check To check the solution p = 35, we can substitute into the original equation. Left-hand side = 24 + p = 24 + 35

Substitute p = 35

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Idea summary When solving equations we use inverse operations to isolate the variable. We always apply the same operation to both sides of the equation to keep it balanced.

Multiple-step equations To solve multi-step equations, our goal is to isolate the variable and ﬁnd what value of the variable makes the equation true. To do this, we consider how they are built and then apply inverse operations in reverse order to both sides of the equation. Consider the equation: 5x + 3 = 18 It has been built by starting with x, ‘multiplying by 5’ and then ‘adding 3’. To solve for x we ﬁrst ‘subtract 3’ and then ‘divide by 5’ on both sides of the equation. Given equation Subtract 3 from both sides Evaluate Divide both sides by 5 Evaluate

Interactive exploration Discover this concept in action online

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Example 2 Consider the equation 2( p + 6) = −8 a Which pair of operations will make p the subject of the equation? A

Divide by 2, then subtract 6.

Divide by 6, then subtract 2.

Multiply by 2, then add 6.

Divide by 2, then add 6.

Apply the idea Consider the order of operations that you would use to evaluate if substituting in a value for p. First you would do ( p + 6), then you would multiply by 2. To make p the subject of the equation we reverse that entire process. Step 1, reverse the multiply by 2, which is to divide by 2.

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Step 2, reverse the add 6, which is to subtract 6.

The correct option is A. b Use the inverse operations identified in part (a) to solve the equation.

Apply the idea 2( p + 6) = −8 2( p + 6) −8 = 2 2 p + 6 = −4

Given equation

Divide both sides by 2

Evaluate

p + 6 − 6 = −4 − 6 p = −10

Subtract 6 from both sides

Evaluate

Example 3 Solve

Create a strategy Apply the inverse operations in the reverse order of how the equation was built. It was built by doing:

Apply the idea

Reﬂect and check

We need to undo adding 7 and then dividing by 2.

To check the solution u = 3, we can substitute into the original equation.

So the order of inverse operations will be:

u+ 7 2 3+7 = Substitute Substituteuu= =3 3 2 10 = 2 =5

Left-handside =

1. Multiply by 2 2. Subtract 7 Write the equation

Multiply both sides by 2

Evaluate

This is the same as the right-hand side of the Subtract 7 from both sides equation, so the value of u = 3 is correct.

Evaluate

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Idea summary To solve a multiple-step equation ﬁrst identify the operations used to build the equation, then apply the inverse operations in the reverse order to both sides of the equation.

Rearrange equations With the skills we now have, we can reverse as many operations as we need to isolate our variable. But what if we want to solve equations like 4 − x = 17 or

= 20?

In both of these cases it is harder to identify the operations used to build the equation. However, as variables represent numbers, we can apply inverse operations directly to them as well. In the equation 4 − x = 17, x is being subtracted from 4. If we added x to both sides of the equation we would end up with: 4 − x + x = 17 + x Which would give: 4 = 17 + x Subtract 17 from both sides: −13 = x We can also write this as: x = −13

Example 4 Solve

= 11

Create a strategy Reverse the division by p ﬁrst and then solve the equation.

Apply the idea Write the equation Multiply both sides by p Simplify Divide both sides by 11 Evaluate

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Reﬂect and check To check the solution p = −4, we can substitute into the original equation. Left-hand side =

−44 p

−44 Substitute pSubstitute = −4 p = −4 −4 = 11 This is the same as the right-hand side of the equation, so the value of p = −4 is correct. =

Idea summary We can apply inverse operations directly to variables in equations to solve them.

7.01 Practice questions Understanding 1

7 − ⬚ = −2

x−4=6

8 × ⬚ = −48

4x = 92

−21 ÷ ⬚ = 3

For each pair of operations: i

Write the expression if you start with x and apply the two given operations.

Write the operations required to isolate the variable.

Add 6 then multiply by 4

Multiply by − 3 then subtract 7

Subtract 9 then divide by 3

Divide by 11 then add 8

For each of the equations, name the inverse operation that must be applied to the variable to rearrange the equation and make the variable the subject. a

276

Trailblazer 3, 4

For each of the equations, name the inverse operation that must be applied to both sides in order to solve the equation: a

Adventurer 2–4

Write the equation with the correct missing value: a

Explorer 1–4

2 − x = −4

7 = −3 − a

Mathspace Victorian Curriculum – Year 8 mathspace.co

5a–d, 6a–i, 7a–f, 8a–d, 9, 11a

Fluency 5

Ex 1

Ex 2

Ex 3

Write the inverse operation needed to make the variable the subject of the equation.

Solve the equation.

y+5=9

t − 2 = 35

9p = 36

y + 3 = −8

t − 3 = −45

5p = −50

Solve: a

8k = 56

n + 27 = 11

r + 22 = 69

8k = −48

6 = −16 + n

p + 36 = 62

24 + p = −2

t − 15 = 0

9t = 3

3x + 11 = 5

4x − 6x = 4

−10 + 3k = 5

−w − 7 = 7

12q + 8 = 26

4s − 9 = 39

−6s + 5 = 47

6t + 2 = −22

8m + 9 = 65

5w − 8 = 2

−63 − 9y = 63

−72 − 8y = 64

Solve: a

2 (q + 11) = 20

3 (q + 3) = −18

2 ( p − 5) = −12

7 (11 − n) = 210

2 (u + 5) = 18

2 (u + 2) = −8

−4 (r + 5) = −44

4 (u − 4) = −36

Solve: a

−

d =3 7

−

y = −6 2

Solve:

Solve: a

Reasoning 12

Solve:

a Ex 4

6c–l, 7e–l, 8e–h, 9, 10, 11b–d

For each of the equations:

i 7

5e–h, 6b–j, 7c–j, 8d–f, 9, 10, 11b–d

12–14

12–15

Identify the step where Angus made an error and explain what he should have done: 1

4x − 8 = 12

4x = 4

Subtract 8 from both sides

x=1

Divide both sides by 4

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Identify the step where Emily made an error and explain what she should have done:

1 2

x + 5 = 20 3 x 5 from both sides = 15 Subtract 5 fromSubtract bothsides 3

3 x = 5

Multiply left side by 3 and divide right side by 3

Solve 2(x + 3) = 2x + 6. What do you think this solution means?

Solve 2x + 3 = 2x + 7. What do you think this solution means?

Problem-solving 16, 17

16–18

17–21

A rectangle’s length is 4 units more than its width. If the perimeter of the rectangle is 36 units, write an equation to represent this situation using x for the width, and solve it to ﬁnd the length and width of the rectangle.

The average of 3 and x is 14. Write an equation and solve it for x.

A kite has its long sides equal to 3 less than double the length of the short sides. If the perimeter of the kite is 24 cm, ﬁnd the length of the short sides.

The formula A = 2π r × (R − r) is used to calculate the surface area of a particular shape. Rearrange the formula to make R the subject.

The sum of three consecutive integers is 42. If the smallest integer is represented by x, write an equation to represent this situation, and solve it to ﬁnd the three integers.

A father is four times as old as his son. In twenty years, the father will be twice as old as the son. Write an equation to represent this situation using x for the son’s age, and solve it to ﬁnd the current ages of both the father and the son.

7.02 Tables of values After this lesson, you will be able to… • graph a linear relationship from a table of values. • determine if an equation is linear from the graph. • determine if an equation is linear by analysing the first differences.

Identify linear equations To identify if an equation is linear we can create a table of values and then plot and draw a line through the points on a Cartesian plane. If the line formed by the points is straight, the equation represents a linear relationship otherwise the relationship is called non-linear. 278

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Let’s use this process to check if this equation is linear: y = 3x − 5 First we create the table of values for the equation by choosing a set of x-values. These values are often determined by the context or given in the problem. x y

Next, substitute each x-value into the equation to ﬁnd the corresponding y-value. y=3×1−5

Substitute x = 1

=3−5

Evaluate multiplication

= −2

Evaluate subtraction

Substituting the remaining values of x allows us to complete the table: x y

1 −2

2 1

3 4

4 7

Organise the x and y values into a table, forming ordered pairs (x, y). x

−2

(3, 4),

(4, 7)

(1, −2), (2, 1), 8 7 6 5 4 3 2 1

In this table, the x-values are the independent variables, and the y-values are the dependent variables. The value of y depends on the value of x according to the given equation. x 1

−1 −2 −3

−2 −1

−1 −2 −3

Use the ordered pairs (x, y) from the table of values to plot points on the Cartesian plane.

−2 −1

8 7 6 5 4 3 2 1

The x and y value in each column of the table can be grouped together to form the coordinates of a single point, (x, y).

After plotting all the points, observe the pattern formed by the points.

We can sketch a straight line through the points to get the graph of y = 3x − 5. The line is straight and continuous (no breaks in the line) conﬁrming that the equation is linear.

x 1

Notice that when sketching a straight line through a set of points, the line should not start and end at the points, but continue beyond them, across the entire Cartesian plane.

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Interactive exploration Discover this concept in action online

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Example 1 Consider the equation y = 2x. a Fill in the blanks to complete the table of values.

x y

Create a strategy Substitute each x-value to the given equation.

Apply the idea For x = −1, y = 2 × (−1) = −2

Substitute x = −1 Evaluate

For x = 0, y = 2 × (0) =0

Substitute x = 0 Evaluate

For x = 1, y = 2 × (1) =2

Substitute x = 1 Evaluate

For x = 2, y = 2 × (2) =4

Substitute x = 2 Evaluate x y

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−1 −2

0 0

1 2

2 4

−1

b Plot the points in the table of values.

Create a strategy Plot each point using the coordinates from the table.

Apply the idea 5

The points from the table have the coordinates (−1, −2), (0, 0), (1, 2), (2, 4).

4 3 2 1 −4 −3 −2 −1

x 1

−1

−2 −3

c Is the relationship linear?

Create a strategy Connect the points plotted from part (b) to check if it makes a straight line.

Apply the idea 5

Since the line formed is a straight line, the relationship is linear.

4 3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3

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Idea summary We can identify if an equation is linear by creating a table of values, plotting and drawing a line through the points and checking that the line produced is straight and continuous.

First differences Given only a table of values, we can also determine whether it represents a linear or non-linear relationship. First we check that the x-values are increasing or decreasing by the same amount. We then focus on analysing the differences in consecutive y-values to determine if the relationship is linear. If the differences are the same, the relationship represented by the table is linear. These differences are called ﬁrst differences. Consider this table of values:

x y

Notice we have the y-values −2, 1, 4, 7.

1 −2

2 1

3 4

4 7

Calculating the ﬁrst difference between consecutive y-values, we have 1 − (−2) = 3 4−1=3 7−4=3 The ﬁrst difference between consecutive y-values is constant, 3. This value represents the increase in y for the increase in consecutive values of x. A constant change in the dependent variable, y in this case, is what makes a relationship linear and the line representing the equation a straight line.

Example 2 Consider the relationship between x and y in the table. x y

1 5

2 1

3 −3

4 −7

5 −11

Is the relationship linear?

Create a strategy In a linear relationship, the y-value must change by equal amounts as the x-value changes by the same amount.

Apply the idea We can see that the x-values are increasing by 1 each time. For the y-values: the change from 5 to 1 is 1 − 4 = −4. From 1 to −3 is −3 − 1 = −4. We can see that the y-values continue to decrease by 4 each time. Yes, the relationship is linear.

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Idea summary If the ﬁrst difference of any two consecutive y-values from a table with consecutive x-values are constant or the same, then the relationship represented by the table of values is linear. If the ﬁrst difference varies, the relationship is non-linear.

7.02 Practice questions Explorer

Understanding

Adventurer 2–4

1–4

Trailblazer 2, 4

Describe how to plot the point (5, −2) on a Cartesian plane.

Place these ordered pairs that satisfy the linear equation y = 2x in the table of values: (2, 4), (−1, −2), (5, 10), (0, 0) x y

−1

Consider these coordinates on the Cartesian plane: x y

−2 −4

0 0

2 1

6 5

3 6

4 3

Do the points form a straight line?

2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Do the points plotted on each of the Cartesian planes form a linear or non-linear relationship? a

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

x 1

2 3 4 5

6 5 4 3 2 1 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6

x 1 2 3 4 5 6

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y 3 2 1

−11−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 −1

x 1

−2

Ex 1

5b–e, 6c–l, 9–11

Complete the table of values.

Plot the points from the table on a Cartesian plane.

iii

Is the relationship linear or non-linear?

y = 3x

y = −x + 2

x −1 0 1 2 y e

284

y = 2x − 4

−1

y = −4x + 5 x y y=

Does the table show a linear relationship between x and y? a

x 1 2 3 4 y 4 8 12 16

5 20

x y

1 4

2 5

3 6

4 7

5 8

x 1 2 3 4 y 5 7 13 19

5 24

x y

1 −3

2 −3

7 −3

9 −3

13 −3

x −1 0 1 2 y 3 7 11 15

x y

−1 3

0 3

1 3

2 3

Mathspace Victorian Curriculum – Year 8 mathspace.co

5c–f, 6e–n, 8, 9, 12

y = 3x + 4

x x −2 0 2 4 y y 6

For each equation:

x x −1 0 1 2 y y

Ex 2

−1 −2 −3

−4

5a–d, 6a–i, 7, 10, 11

−4 −3 −2 −1

−3

Fluency

8 7 6 5 4 3 2 1

x −1 0 1 2 y 3 7 11 13

x 5 10 15 20 y 15 30 45 60

x y

3 0

3 3

3 6

3 9

x y

1 5

2 8

4 11

7 14

x 0 1 3 7 y 14 7 0 −7

x y

−6 0

−5 9

−3 18

1 27

m x −6 −3 0 6 y 27 18 9 0

x y

−1 7

0 11

1 7

2 3

25 75

Consider the table of values:

2 4

3 7

4 10

Plot the points in the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 6, ﬁnd the corresponding value of y.

Consider the table of values: a

Plot the points in the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 10, ﬁnd y.

Consider the table of values: a

Plot the points in the table on a Cartesian plane.

Is the relationship between x and y linear?

If we extend the table of values to include x = 6, ﬁnd y.

5 13

x y

−6 −3

−2 −2

2 −1

6 0

x y

−2 −2

0 0

2 2

4 4

Let the height of a pot plant be y cm. As the pot plant grows, the height increases according to the equation y = 3t + 6, where t is the elapsed time in weeks. a

Complete the table of values: Time (t weeks) Height of pot plant ( y cm)

x y

8 17

Plot the points in the table on a Cartesian plane.

Is the relationship linear or non-linear?

For each table: i

Plot the points from the table on a Cartesian plane.

Write the linear equation for the rule between x and y.

x 1 2 3 4 y 6 12 18 24

5 30

x y

1 8

2 9

3 10

x 1 2 3 4 y −7 −8 −9 −10

5 −11

x y

1 −2

2 −6

3 4 5 −10 −14 −18

4 11

5 12

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For each table: i

Find the linear equation between x and y.

Verify if the linear equation is correct by substituting values from the table.

x −1 0 1 2 y 5 2 −1 −4

3 −7

x y

1 7

x −8 −7 −6 −5 −4 y −36 −31 −26 −21 −16

x y

3 4 5 6 7 −12 −17 −22 −27 −32

Reasoning

13, 15

2 10

13, 14

3 13

4 16

5 19

13, 14

Explain at least two different ways to determine if a set of points forms a linear relationship.

Consider the three points A (2, −4), B (2, −6), and C (4, −1). Without plotting the points, explain whether the points form a linear relationship.

Sketch the graph of a non-linear relationship. Explain how you know that the relationship is not linear.

Problem-solving 16ab, 18–20 16

16bc, 18–21

16bd, 17, 19–21

For each linear equation: i

Complete the table of values.

Sketch the graph of the line.

y = 2x − 4

y = −2x + 4

x x 0 1 2 3 y y c

x x 0 2 4 6 y y 17

Sherin is taking a taxi from her home in Sydney to a restaurant outside of the city. The ﬂag fall fee is $4, and the rate per kilometre is $1.50. Create a table of values and graph the relationship between distance travelled and cost for the ﬁrst 20 km on a Cartesian plane.

Let the height of a candle be y centimetres. If the candle is lit, the height decreases according to the equation y = −2t + 8, where t is the elapsed time in minutes. a

Complete the table of values: Time (t min) Height of candle ( y cm)

Sketch the graph of y = −2t + 8.

The height of candle and the time elapsed must be non-negative. Using part (b), for what values of t does the equation y = −2t + 8 accurately model the height of the candle?

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Huda opens a bank account and deposits $300. At the end of each week she adds $10 to her account. a

Complete the table which shows the balance of Huda’s account over the ﬁrst four weeks: Week (W ) Account total (A)

0 $300

1 $310

Is the relationship between the number of weeks, W, and the account, A, linear?

Write the linear relationship for Huda’s account total, A, in terms of the number of weeks W, for which she has been adding to her account.

Yvonne has constructed the ﬁrst three triangles of a pattern using matchsticks: Yvonne makes a table comparing the number of triangles to the number of matchsticks needed to construct them:

Number of triangles Matchsticks

1 3

2 6

Complete the table for the pattern.

Is the relationship between the number of triangles, T, and the number of matchsticks, M, linear?

Write the linear relationship for the number of matchsticks, M, in terms of the number of triangles, T, for this pattern.

Peter is making a sequence of shapes out of matchsticks: Figure 1

Figure 2

Peter makes a table comparing the ﬁgure number to the number of matchsticks needed to construct it:

Figure 3

Figure 4

Figure number Matchsticks

1 5

2 6

Complete the table for the pattern.

Is the relationship between the ﬁgure number, F, and the number of matchsticks, M, linear?

Write the algebraic rule for the number of matchsticks, M, in terms of the ﬁgure number, F, for this pattern.

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7.03 Linear graphs After this lesson, you will be able to… • determine the x-intercept and y-intercept of a line from a given equation or a graph. • graph a line directly from an equation in the form y = mx + c without needing to create a table of values. • apply their understanding of intercepts and line equations to solve related problems. • identify whether a line is horizontal, vertical or neither from a given equation or a graph.

Axis intercepts Lines drawn on the Cartesian plane extend forever in both directions. If we ignore the special case of horizontal and vertical lines which we will look at next, all other lines will either cross both the x-axis and the y-axis or they will pass through both axis by passing through the origin (0, 0). We use the word intercept to refer to the point where the line crosses or intercepts an axis. • The x-intercept is the point where the line crosses the x-axis. The coordinates of the x-intercept will always have a y-coordinate of zero. • The y-intercept is the point where the line crosses the y-axis. The coordinates of the y-intercept will always have an x-coordinate of zero. Note: Every straight line must have at least one intercept but cannot have any more than two intercepts. Here are some examples: 4

1 −4 −3 −2 −1

−1

x 1

−4 −3 −2 −1

−1

−2

−3

−4

x-intercept: (−3, 0) ; y-intercept: (0, 2)

x 1

A line passing through the origin, (0, 0)

We only need to identify two points to sketch a straight line, and the x and y-intercepts are probably the most useful points to identify and plot. They are also two of the easier points to ﬁnd as we are substituting in either the values x = 0 or y = 0, which simpliﬁes the work needed to solve.

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Example 1 Consider this graph.

a Find the x-value of the x-intercept.

2 1 −4 −3 −2 −1 −1

x 1

2 3 4

−2 −3 −4

Create a strategy

Apply the idea

The x-intercept is the point where the line intersects the x-axis.

The line intersects the x-axis at (3, 0). So the x-value of the x-intercept is x = 3.

b Find the y-value of the y-intercept.

Create a strategy

Apply the idea

The y-intercept is the point where the line intersects the y-axis.

The line intersects the y-axis at (0, 4). So the y-value of the y-intercept is y = 4.

Example 2 A line has the equation y = 3x − 3. a Find the y-value of the y-intercept.

Create a strategy

Apply the idea

Substitute x = 0 into the given equation and then solve for y.

y = 3 × (0) − 3 = −3

Substitute x = 0 Evaluate

b Find the x-value of the x-intercept.

Create a strategy Substitute y = 0 into the given equation and then solve for x.

Apply the idea Substitute y = 0

Add 3 to both sides

Evaluate

Divide both sides by 3

Evaluate

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c Sketch the line of the equation y = 3x − 3.

Create a strategy

Apply the idea

Plot the intercepts and draw a straight line through them.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

We can see that the intercepts (1, 0) and (0, −3) were plotted, and then a straight line was drawn through the points.

Idea summary • The x-intercept is the point where the line crosses the x-axis. The coordinates of the x-intercept will always have a y-coordinate of zero. • The y-intercept is the point where the line crosses the y-axis. The coordinates of the y-intercept will always have an x-coordinate of zero. • Every straight line must have at least one intercept but cannot have any more than two intercepts.

Graph from an equation We have seen how to identify if a table of values represents a linear relationship using first differences. The first difference represents the amount the y value increases for a given increase in x. We can use this idea to graph a straight line from an equation. We can represent the equation of any straight line, except vertical lines, in the form: y = mx + c • m is equal to the gradient of the line or first difference in a table of values representing the equation. • c is where the line cuts the y-axis, also called the y-intercept.

Interactive exploration Discover this concept in action online

mathspace.co

We only need two points to graph a line. We can get those two points from a linear equation in the form y = mx + c by: • c is the y-intercept, so one point on the line is (0, c) • Substituting a value of x other than 0 will give another point on the line. 290

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Example 3 Consider the linear equation y = 2x + 9. What are the values of the gradient, m and the y-intercept, c?

Create a strategy

Apply the idea

m is the coeﬃcient of x and c is the constant The coeﬃcient of x is 2. So the gradient, m = 2. term when the equation is in the form y = mx + c. The constant term is 9. So the y-intercept, c = 9.

Example 4 Graph the linear equation y = 4x + 8. a What are the coordinates of the y-intercept?

Create a strategy

Apply the idea

When the equation is in the form y = mx + c, the The y-intercept is at (0, 8). value of c is the y-intercept.

b Sketch the equation y = 4x + 8.

Create a strategy We have one point already, the y-intercept (0, 8). We need one other point to be able to sketch the line. Substitute any value of x into the equation to find a coordinate on the line.

Apply the idea Let’s use x = 1 to find a second coordinate. y = 4x + 8 = 4 x 1 + 8

Substitute x = 1

= 12 Evaluate Now we have two coordinates. Plot them both on the Cartesian plane and draw a line through both points.

12 10 8 y = 4x + 8 6 4 2 −4 −3 −2 −1 −2 −4 −6 −8

x 1

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Idea summary We can graph any straight line in the form: y = mx + c. Knowing: • m is equal to the gradient of the line. • c is where the line cuts the y-axis.

Horizontal lines Let us ﬁrst explore the concept of horizontal lines, their properties, and how to represent them using linear equations.

Interactive exploration Discover this concept in action online

mathspace.co

A horizontal line is a straight line that runs parallel to the x-axis on a Cartesian plane. • The gradient of a horizontal line is always zero. This is because there is no change in the y-value as the x-value increases or decreases. • The equation of any horizontal line is y = c because m is zero in the equation of a line given by y = mx + c. • The x-axis is a horizontal line, and every point on it has a y-value of 0 so the equation of the x-axis is y = 0.

Example 5 4

What is the equation of this line?

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Create a strategy

Apply the idea

Recall that horizontal lines have equations of the The y-intercept of the horizontal line is y = 4, form y = c, where c is the y-intercept. so c = 4. So, the equation of the line is y = 4.

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Idea summary A horizontal line has a gradient of zero (m = 0), and an equation of the form: y = c where c is the y-intercept of the line. The x-axis is a horizontal line, and every point on it has a y-value of 0 so the equation of the x-axis is y = 0.

Vertical lines This time, we’ll explore the vertical lines.

Interactive exploration Discover this concept in action online

mathspace.co

In the case of vertical lines, the gradient is considered to be undeﬁned. A vertical line has an equation of the form x = d where d is the x-intercept of the line. The y-axis is a vertical line, and every point on it has an x-value of 0 so the equation of the y-axis is x = 0. 4

Here are two examples of vertical lines.

3 2 1 x = −1 −2 −1

−1

x=4 1

x 6

−2 −3 −4

Example 6 Is the graph of y = 2 a horizontal or vertical line?

Create a strategy

Apply the idea

If the equation is a y = a number, then it is a horizontal line, if the equation is a x = a number, then it is a vertical line.

The equation y = 2 is a horizontal line. It is saying that regardless of the x-values, y is always 2.

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Example 7 What is the equation of the line that is parallel to the y-axis and passes through the point (−8, 3)?

Create a strategy

Apply the idea

Any line that is parallel to the y-axis is a vertical line and all vertical lines are in the form of x = a number. Which means that regardless of the values for y, x is always that number.

The x-coordinate of the point (−8, 3) is −8. The equation is x = −8.

Idea summary A vertical line has an undeﬁned gradient, and an equation of the form: x = ‘a number’. The y-axis is a vertical line, and every point on it has an x-value of 0 so the equation of the y-axis is x = 0.

7.03 Practice questions Understanding 1

Explorer 1–8

Adventurer 1–8

Which of the coordinates represent x or y-intercepts? a

(7, 0), (5, 3), (0, −5), (−2, 4)

What is the gradient of a horizontal line? Explain.

What is the gradient of a vertical line? Explain.

For the points (3, 2), (3, –1) and (3, 9):

Trailblazer

1–8

x y

0 5

1 6

What is the x-coordinate of each point?

Would these points lie on a horizontal or a vertical line?

What is the equation of the line on which these three points lie?

−9 0

For the points (3, 2), (5, 2) and (−7, 2): a

What is the y-coordinate of each point?

Would these points lie on a horizontal or a vertical line?

What is the equation of the line on which these three points lie?

Is a line with equation y = k parallel to the x-axis, the y-axis, or neither?

Is a line with equation x = k parallel to the x-axis, the y-axis, or neither?

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3 9

The table shows some points on the line with equation x = 0. Does the line x = 0 represent the y-axis or x-axis? The table shows some points on the line with equation y = 0. Does the line y = 0 represent the y-axis or x-axis?

0 −1

0 4

0 6

x y

−6 0

−4 0

1 0

5 0

For each graph: i

Find the x-value of the x-intercept.

Find the y-value of the y-intercept. y 6

x −3

−2

−1

−1 −2 −1

−2 −3

−2

−2 −1

1 −4 −3 −2 −1

−1

y = 2x − 4

−1

x 4

−2

−3

−4

0 −8

9, 10, 11a–d, 12ab, 13, 9, 10, 11a–d, 12bc, 13–16, 9, 10, 11e–h, 12cd, 14, 14a–d, 15, 16, 17ab, 19–21, 17ab, 18–20, 22–24, 25ab 17cd, 18–22, 25cd, 26 22ab

Fluency Ex 1

x y

Will the lines cross the y-axis at y = 2? a

y = 5x + 2

y=2−x

y = 2x

y=x+2

y=x−2

y=2

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Ex 2

Ex 3

Ex 4

Find the value of the x-intercept for each line: a

y = 2x − 2

3x + y = −3

y = 4x − 8

y = 9x

2y + 2x = 4

3x − 5y = 1

For each equation: i

Find the coordinates of the y-intercept.

Find the coordinates of the x-intercept.

iii

Use the intercepts to sketch the graph of the line.

y = 2x − 4

y = −2x + 2

y = 3x − 6

y = −4x + 8

Find the coordinates of the y-intercept.

Find the coordinates of the x-intercept.

Find the value of y when x = 2

Sketch the graph of the line.

For each equation: i

Find the value of the gradient, m.

Find the value of the y-intercept, c.

y = 4x + 9

y = 5x − 6

y = −5x + 8

y = −4x − 2

2y = 8x − 1

3y = −6x − 2

y − 5x = 4

2y − 3x = 6

For y = 4x − 5, a

Find the coordinates of the y-intercept.

Using the y-intercept from part (a) and the point (2, 3), sketch the line y = 4x − 5.

For y = 2x + 1,

Complete the table of values. −2

−1

Plot the points from the table of values.

Sketch the graph of each equation using the gradient and y-intercept: a

y=x+1

y = 3x − 1

Which of these lines have the same gradient? • y = 2 + 7x • • y = 5 − 7x • y = 7x • y = 5x + 7 • y = 7x − 2

296

x y

For y = 5x,

2y + x = −3

Mathspace Victorian Curriculum – Year 8 mathspace.co

y = −x + 4

y = −3x − 4

Ex 5

Find the equation of each line: a

1 −4 −2

−1

x 2

−2

−3

−5 −4 −3 −2 −1 −1

−4

−2

1 −6 −5 −4 −3 −2 −1

Ex 6

−1

x 1

2 3 4 5

x 1

−3

−2

−1

x 1

−1

−2

−3

−4

Is the line given by each of these equations horizontal or vertical? a

y=5

x = −4

y=0

x=0

y=3

x = 17

For each table of values: i

Sketch the graph of the line that passes through the four points.

Find the equation of the line.

x 2 2 2 2 y −2 −1 0 1

x y

y = −2

−4 5

−2 5

0 5

2 5

y=0

Plot the lines on a Cartesian plane: a

y=8

x=7

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Ex 8

For the graphed line:

Identify the x-intercept.

Does this line have a y-intercept?

State the equation of the line.

2 1

−4 −3 −2 −1

−1

−2 −3

For the graphed line:

Identify the y-intercept.

Does this line have an x-intercept?

State the equation of the line.

2 1 −4 −3 −2 −1 −1

x 1

−2 −3

Ex 7

298

Find the equation of the line given each condition: a

The line is parallel to the x-axis and passes through the point (3, −5).

The line is parallel to the y-axis and passes through the point (−9, 3).

The line is perpendicular to the x-axis and passes through the point (3, −8).

The line is perpendicular to the y-axis and passes through the point (3, −4).

For y = −

Find the coordinates of the y-intercept.

Find the coordinates of the point on the line where x = 4.

Sketch the graph of the line.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Reasoning 27

27, 28

27, 30–32

29–33

For y = −2x + 3, a

What is the y-value, when x is 0?

If the equation of a line is y = mx + c, what is the value of the y-intercept?

−4 −3 −2 −1

7 6 5 4 3 2 1

−1 −2 −3 −4

These three equations are graphed on the Cartesian plane:

y = x + 4, y = 2x + 4, y = 4x + 4

What do all of the equations have in common?

What do all of the graphs have in common?

What conclusion can be made about all lines that have the form y = mx + 4?

−8 −6 −4 −2 −2 −4

If a line has equation y = mx + c, explain how you can tell if the line will pass through the origin.

For this line graph: 7

What is the y-value when x = 0?

What is the y-value when x = 1?

When the x-value increases by 1, by how much does the y-value change? This is the gradient of the line.

5 4

The equation of this line is y = 2x + 4. Explain how to ﬁnd the gradient from the equation of the line.

3 1 −4

−3

−2

−1

x 1

−2

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For the three linear equations and their corresponding graphs:

5 4 3 2 1

y = 4x + 3, y = 4x + 6, y = 4x − 3

What do all of the equations have in common?

What do all of the graphs have in common?

What conclusion can be made about all lines that have the form y = 4x + c?

300

2 3 4 5

What do the points A (−4, 3), B (−1, 3), C (3, 3), and D (7, 3) have in common?

Explain why the equation of the line in part (a) is y = 3.

A line goes through the points A (−5, q), B (−2, q), C (2, q), and D (8, q). What is the equation of the line.

What is the equation of the x-axis?

Consider these questions: a

What do the points A (−3, −5), B (−3, −1), C (−3, 3), and D (−3, 10) have in common?

Explain why the equation of the line in part (a) is x = −3.

A line goes through the points A ( p, −7), B ( p, −2), C ( p, 0), and D ( p, 4). What is the equation of the line.

What is the equation of the y-axis? 34, 36, 38, 39

35, 37–41

The temperature (in degrees Celsius) inside a freezer over a ﬁve-hour period was recorded: Time (h) Temperature (T )

Consider these questions:

Problem-solving 34, 36 34

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

0 −9

1 −9

2 −9

3 −9

4 −9

5 −9

Sketch the line that passes through each of the pairs of values from the table.

Write an equation for the temperature T, at time h.

A car rental company charges a ﬂat fee of $25 per day plus $0.20 per kilometre driven. The linear equation that represents the total cost ( y) for renting a car and driving x kilometres is y = 0.20x + 25. a

Determine the intercepts of the graph of the equation.

Graph the equation on the Cartesian plane. Use an appropriate scale and label the axes.

Interpret the meaning of the x-intercept of the graph.

Mathspace Victorian Curriculum – Year 8 mathspace.co

A water tank is initially full and starts leaking water at a constant rate. The tank loses 10 litres of water every hour. After 6 hours, the tank still has 200 litres of water left. The equation that represents this situation is y = −10x + 260 where x is hours and y is litres. a

Determine the number of litres in the tank initially.

After how many hours will the tank be empty?

Use the results from part (a) and part (b) to graph the equation on the Cartesian plane. Use an appropriate scale and label the axes.

Edin and Marius are saving money for a trip. Edin starts with no savings and saves $100 per month. Marius saves $80 per month after starting with savings of $60. Both start saving in January. a

Write linear equations to represent the total amount ( y) saved by each friend after x months.

Interpret the meaning of the y-intercept of each equation.

Graph both equations on the same Cartesian plane. Use an appropriate scale and label the axes.

Determine which friend will have more savings by June.

What are the coordinates of the point of intersection of the lines y = 7 and x = −3?

What are the coordinates of the point of intersection of the lines y = q and x = p?

What do we call the point that is the intersection of the lines y = 0 and x = 0?

Find the perimeter of the rectangular region enclosed by the lines x = −2, y = 3, x = 5, and y = −1.

A region on a Cartesian plane is a square of side length 6 units. The centre of the square is at (2, 1). Identify the equations of the horizontal and vertical lines that form the sides of the square.

The charge for parking at a city centre car park is $5 per hour. Parking for 0 to 1 hours costs $5, for 1 to 2 hours costs $10, and so on. Bernie thinks he can draw this on graph by using a series of horizontal straight line segments, for example using y = 5 for values of x from 0 to 1 and y = 10 for values of x from 1 to 2. He can then work out the charge for a given length of stay by intersecting an appropriate vertical line with his graph. a

Draw Bernie’s graph for times from 0 to 6 hours.

What vertical line should Bernie draw on his graph to ﬁnd the parking charge for a stay of 4.5 hours?

Why would Bernie have a problem if he tried to use his graph for a stay of exactly 2 hours?

How could Bernie solve this issue?

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INVESTIGATION

Linear solutions Investigate integer solutions to equations in the form ax + by = c.

Objectives To find integer solutions to linear equations in standard form

Linear solutions

Solutions to equations

The solution to an equation is the value that makes the equation true. For example, the solution to 3x = 12 is x = 4. 3x = 3(4) = 12 So x = 4 is a solution to the equation.

Two-variable equations An equation can have more than one variable, such as x + 2y = 6. In this case, the solutions will be x and y-values that, together, make the equation true.

Many solutions

A two-variable equation can have many solutions. Consider the previous equation, x + 2y = 6. Two solutions are x = 0, y = 3 and x = −2, y = 4. x + 2y = 0 + 2(3)

x + 2y = −2 + 2(4)

=0+6

= −2 + 8

So two solutions are x = 0, y = 3 and x = −2, y = 4.

Interactive exploration Discover this concept in action online

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Equation

Solution 1

Solution 2

Solution 3

Solution 4

Find four solutions to 5 different equations. Draw a table like the one shown to record your answers.

Investigate After filling out the table with your solutions, answer the following questions: 1.

What strategies did you use to find the solutions to each of the equations?

How many solutions does each equation have?

When plotted, what do the solutions form in the Cartesian plane?

Discussion 1.

Do you think the previous equations have non-integer solutions? Explain your reasoning.

What is the minimum number of solutions we would need to know to plot the line of the equation?

If we knew the solutions to the equation, how could we use them to write the equation in the form y = mx + c?

Consider the pair of equations −5x + 10y = 30 and

. These equations both

have the solutions (0, 3), (10, 8), and (20, 13). What does this information tell us about the graphs of these equations?

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7.04 Multi-step equations After this lesson, you will be able to… • solve three-step equations using different methods. • solve equations with variables on both sides.

Three-step equations To solve three-step equations we can still think about how the equation was built. Then we can apply inverse operations in reverse order to both sides of the equation until the variable is isolated and we have a solution for the equation. Consider the equation:

We can see that the expression

was formed by:

1. First multiply by 3 2. then adding 8 3. then dividing by 4 In order to isolate the variable, we want to undo these operations working backwards. This means to solve for x we: 1. First multiply by 4 (inverse operation of divide by 4) 2. then subtract 8 (inverse operation of add 8) 3. then divide by 3 (inverse operation of multiply by 3) Applying these inverse operations gives us:

Undo the division (multiply both sides by 4)

Undo the addition (subtract both sides 8)

Undo the multiplication (divide both sides by 3)

Following these steps isolates the variable x and solves the equation. We can always check our answer by substituting it back into the original equation. If both sides are equal, then our answer is correct. The solution must satisfy the equation.

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Example 1 Solve the equation: −3 (−2h − 5) = 33.

Create a strategy Consider the order of operations as what is happening to the variable h. h is: 1. being multiplied by −2 2. then 5 is subtracted 3. then it is multiplied by −3.

Apply the idea The reverse of the order of operations is: 1. First divide by −3 (inverse of multiply by −3) 2. then add 5, (inverse of subtract 5) 3. then divide −2 (inverse of multiply by −2). Now let's write it out mathematically: Write the equation

Divide both sides by −3

Evaluate

Add 5 to both sides

Evaluate

Divide both sides by −2

Evaluate

Reﬂect and check Substitute h = 3 into the original equation to check. Start with the left-hand side: −3 (−2h − 5) = −3 (−2 × 3 − 5) Substitute h = 3 = −3(−6 −5) = −3(−11) = 33 This is the right-hand side of the equation, so h = 3 is the solution.

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Example 2 Solve the equation:

− 9 = −4

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression on the left-hand side was built by applying the operations ‘divide by 8’, then ‘multiply by 5’, and ‘subtract 9’ to the variable x, so to solve we will: 1. Add 9 on both sides of the equation 2. Multiply by 8 on both sides of the equation 3. Divide by 5 on both sides of the equation Write the equation

Add 9 to both sides

Evaluate

Multiply both sides by 8

Evaluate

Divide both sides by 5

Evaluate

Reﬂect and check Substitute x = 8 into the original equation. Start with the left-hand side: 5x 5×8 Substitute − 9 Substitute x = 8x = 8 −9= 8 8 =5−9 = −4 This is the right-hand side of the equation, so x = 8 is the solution.

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Example 3 Solve the equation:

= −4

Create a strategy Apply the inverse operations in the reverse order.

Apply the idea We can see from the equation that the expression adding 4, and then dividing by 5, so to solve we will:

was built by multiplying x by 8, then

1. Multiply by 5 on both sides of the equation 2. Subtract 4 on both sides of the equation 3. Divide by 8 on both sides of the equation Write the equation

Multiply both sides by 5

Evaluate

Subtract 4 from both sides

Evaluate

Divide both sides by 8

Evaluate

Reﬂect and check Substitute x = −3 into the original equation. Start with the left-hand side: 8x + 4 8x − 3 + 4 Substitute Substitute = x = −x3= −3 5 5 −24 + 4 = 5 −20 = 5 = −4 This is the right-hand side of the equation, so x = −3 is the solution.

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Idea summary We can solve three-step equations by using inverse operations to isolate the variable, just like we do with one and two-step equations. Remember that what we do to one side of an equation, we must also do to the other side.

Equations with variables on both sides We can use inverse operations to gather terms containing the variable. Once all the terms containing the variable are on one side we can simplify that side by collecting like terms just as we do for expressions. Then we end up with a one, two or multi-step equation which we can solve using inverse operations. It is best to move the variable term with the smaller coeﬃcient over to avoid unnecessary introduction of negative coeﬃcients. To keep the equation balanced when moving terms, remember to do the same operations to both sides of the equation.

Example 4 Solve the equation: 4x + 3 = 5x

Create a strategy Apply reverse operations to rearrange the equation so that all of the x-terms are on the same side of the equation.

Apply the idea 4x + 3 = 5x 4x + 3 − 4x = 5x − 4x 3=x

Write the equation Subtract 4x from both sides Simplify

Example 5 Solve the equation: 2x − 1 = x + 5

Create a strategy Apply reverse operations to rearrange the equation so that all of the x-terms are on the same side of the equation.

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Apply the idea 2x − 1 = x + 5

Write the equation

2x − 1 − x = x + 5 − x x−1=5

Subtract x from both sides Evaluate

x−1+1=5+1

Add 1 to both sides

x=6

Evaluate

Idea summary Whenever we want to solve equations that have variables on both sides, our main goal is to move all the variable terms to one side of the equation using inverse operation on both sides of the equation. We can then combine the like terms and finish solving the equation.

7.04 Practice questions Understanding 1

2x = 8

Trailblazer 1–3

3x − 4 = 12

For a

Identify the three operations applied to x in the equation.

Write the inverse operations needed to solve the equation.

James says the solution to the equation

+ 3 = 7 is x = 10. Fiona says it is x = 20.

How can you ﬁnd out who is correct without actually solving the equation?

Who is correct?

Fluency 4

Adventurer 1–3

How many steps are required to solve each equation? a

Explorer 1–3

4ab, 5a–i, 6a–d, 7a–d, 8a–e, 9a–c, 10a–c, 11, 12a–c, 13ab

4bc, 5c–k, 6c–g, 7c–f, 8c–g, 9c–g, 10c–e, 11, 12b–e, 13ac

4bd, 5e–l, 6e–j, 7e–j, 8e–j, 9d–h, 10e–h, 11, 12c–f, 13cd

For each equation: i

Solve for x.

Verify the answer by substituting the value of x back into the equation.

2(x + 1) = 8

3(x − 2) = 6

21 = 3(2x + 1)

14 = 2(3x − 5)

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Ex 1

Solve: 3 (4s + 1) = −21

5 (2x − 6) = 100

6 (3x + 15) = −72

9 (8 − x) = 45

10 (7 − 3x) = 20

Ex 2

Ex 3

Ex 4

20 = 2 (3x + 11)

4 (5x − 1) = −24

2 (7 + 2x) = 38

2 (−3g + 7) = −16

−3 (−2x + 7) = −45

−3 (4 − x) = −12

Solve: a

3 (w + 8) + 5 = 35

9 (x − 7) + 8 = 26

4 (x + 6) − 8 = 48

2 (x − 9) − 4 = 16

−5 (g + 4) + 9 = −41

6 (q + 1) − 5 = −23

−8 (x + 4) − 3 = −59

−7 (x + 5) + 4 = −17

−2 (x − 2) − 7 = −29

5 (x − 3) − 7 = −52

Solve: a

Solve:

Solve: a

Ex 5

310

2 (3t − 7) = 16

7x = 6x + 2

6x = 5x + 37

10x = 9x + 50

39x + 125 = 40x

8x = 6x + 10

7x = 3x − 12

Solve: a

7x = x + 30

3x = x − 18

3x = −8x − 44

4x − 9 = 5x − 6

Mathspace Victorian Curriculum – Year 8 mathspace.co

Solve: a

8k + 2 = 15k − 12

8k − 2 = 4k + 14

10r + 8 = 74 − r

12r + 8 = 116 − 6r

Reasoning 14

14, 15

14–16

15–17

Ryan said that to solve 9(4x−6) = 18 he would first add 6 to both sides of the equation. Linda said that she would divide both sides by 9. Explain who is correct and who is incorrect and why.

For the equation 13(2x − 8) = −26: a

Expand the bracket first and then solve.

Start by dividing both sides by 13 and then solve.

Was one method easier than the other? Explain your answer.

For the equation 4x − 2 = 4x + 5: a

Attempt to solve the equation for x.

What do you think your solution means?

Estelle doesn’t know how to solve the equation

= 8.2 so she estimates the solution.

She says the solution is approximately x = 5 because 40 ÷ 5 = 8. Is her estimate reasonable? Explain how it could be improved.

Problem-solving 18 18

18–20

19–21

Jamie had x pencils in her pencil-case. She ﬁnds one more, and then shares all her pencils equally with her friend. Her sister then gives her 5 more pencils. Jamie now has 7 pencils left in her pencil-case. a

Write the three-step equation that can represent this situation.

Solve the equation.

The average of 7, x + 3 and 4x is 25. Find the values of x + 3 and 4x.

The lifespan of a mouse is 3 years less than a hedgehog. The lifespan of a deer is 2 times that of the mouse. Let x represent the age of the hedgehog. If the total lifespan of these three animals is 11 years, calculate the lifespan for the hedgehog.

Sarah has three times as many books as Tom. If she gives away 15 books to Tom, they will have the same number of books. How many books does each person have initially? Let x represent the number of books Tom has initially.

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7.05 Intersecting lines After this lesson, you will be able to… • solve linear equations graphically by finding the point of intersection of two lines.

Solve equations graphically Solving linear equations using the graphical method is best shown through an example. Consider the linear equation: 5x − 13 = x − 1 The ﬁrst step to solve this graphically is to treat the left-hand and right-hand sides of the equation as separate linear equations. The left-hand side can be written as follows: y = 5x − 13 And the right-hand side as: y=x−1 Let us explore how to find the solution of a linear equation graphically.

Exploration Study the graph of the lines y = 5x − 13 and y = x − 1 y 4 3 (3, 2)

2 1 −2 −1

y = 5x − 13 1

−1

y=x−1

Answer the following questions: 1. Based on your observations, can you describe the relationship between the two lines? Do they appear to intersect or are they parallel? 2. Describe the point (3, 2).

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3. Find the value of y if x = 3 in y = 5x − 13. 4. Find the value of y if x = 3 in y = x − 1. 5. What can you say about the answer to Questions 3 and 4?

When we graph the two equations on the Cartesian plane, we look at where the lines intersect. This point of intersection represents a point where the equations are equal and the x and y values are the same for both equations. We can use substitution to check the solution obtained graphically. Substituting x = 3 back into our original equation gives: y = 5 x − 13 = 5 × 3 − 13 = 15 − 13

Substitute in the value of 3 for x Evaluate the multiplication

= 2 =2

Evaluate the subtraction

Quickly checking the second equation: y = x−1 = 3−1 =2

Substitute in the value of 3 for x Evaluate the subtraction

This confirms that x = 3 is the solution to our equation. It is important to note that when two lines have the same gradient but different y-intercepts, they are parallel and will never intersect. When two lines are parallel, there is no solution to the linear equations because the lines do not share a common point. For example, consider the following equations: 1

y = 2x + 1

y = 2x − 3

The gradients of both lines are the same (m = 2), but they have different y-intercepts. (c = 1 and c = −3).

3 2 1 −4 −3 −2 −1 y = 2x + 1

−1

x 1

Since the lines are parallel, they will never intersect, and there is no solution to these equations.

−2 −3 y = 2x − 3 −4

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Example 1 In this question we will ﬁnd the point of intersection between the line y = x − 9 and the line y = −x − 7. a Sketch the line y = x − 9, together with the line y = −x − 7.

Create a strategy Use the y-intercept and another point to sketch the lines.

Apply the idea y

Here is the table of values for y = x − 9: x y

−1 −10

0 −9

1 −8

-2

Here is the table of values for y = −x − 7: x y

−1 −6

0 −7

1 -1

-1

x 1

-2 -3

1 −8

-4 y = -x - 7 -5

Now we can plot the points from the table of values and draw the line to graph each equation.

-6 -7 -8

y=x-9

-9 -10

b What is the point of intersection of these two lines?

Create a strategy Look for the point where the lines meet.

Apply the idea The point of intersection is at (1, −8).

Reﬂect and check This means that the values x = 1 and y = −8 make both equations true, so they are the solution to the linear equations. We can check this by substituting these values into both equations to see if they are true.

314

y = -x - 7

Write the equation

-8 = -1 - 7

Substitute the value of x and y

-8 = -8

Evaluate

Mathspace Victorian Curriculum – Year 8 mathspace.co

y=x-9

Write the equation

-8 = 1 - 9

Substitute the value of x and y

-8 = -8

Evaluate

Substituting the values of x and y to both the equations makes the solution correct.

Example 2 Use the sketch of the line

to find the solution of

13 12 11 10 9 8 7 6 5 4 3 2 1 −4 −3 −2 −1−1

x 1 2 3 4 5 6 7

Create a strategy Find the other equation of the line that must be drawn on the graph and determine the point of intersection.

Apply the idea Rewrite y =

+ 5 = 13 as two separate equations, then graph in the same Cartesian plane.

1 2 y = 13

y 13 12 11 10 9 8 7 6 5 y = 4x + 5 3 4 3 2 1

−4 −3 −2 −1−1

1 2 3 4 5 6 7

The point of intersection is (6, 13). The value of x that will make the equation

+ 5 = 13 “true”, is 6.

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Reﬂect and check To check if the solution is correct, we can substitute the value of x to the equation

+ 5 = 13.

Substitute x = 6 in the equation

Evaluate the multiplication

Evaluate the division

Evaluate the addition

The equation is true which means x = 6 is the solution to the equation.

Example 3 For the linear equations y = 5x − 7 and y = −x + 5: a Plot the lines of the two equations on the same graph.

Create a strategy Identify two points for each line to plot the graphs.

Apply the idea The first point of y = 5x − 7 is its y-intercept at (0, −7).

5 y 4 3 2 1 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8

x 1

y = 5 × 1 − 7 Substitute x = 1 y = −2 Evaluate So the second point is at (1, −2). The first point of y = −x + 5 is its y-intercept at (0, 5).

5 y 4 3 2 1 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8

316

Another point can be plotted by substituting x = 1 into the equation.

x 1

Mathspace Victorian Curriculum – Year 8 mathspace.co

Another point can be plotted by substituting x = 1 into the equation. y = −1 + 5 Substitute x = 1 y = 4 Evaluate So the second point is at (1, 4).

b Find the values of x and y which satisfy both equations.

Create a strategy Identify the coordinates of the point of intersection of the lines.

Apply the idea The point of intersection is (2, 3) from the graph in part (a). The values which satisfy both equations are x = 2 and y = 3.

Reﬂect and check This means that the values x = 2 and y = 3 make both equations true, so they are the solution to the linear equations. We can check this by substituting these values into both equations to see if they are true. y = 5x − 7

Write the equation

3 = 5(2) − 7

Substitute the values of x and y

3 = 3

Evaluate

y = −x + 5

Write the equation

3 = −2 + 5

Substitute the values of x and y

3 = 3

Evaluate

Substituting the values of x and y to both the equations make both equations true. The solution is correct.

Example 4 For

a To find the solution to this equation graphically, which two straight lines would need to be graphed? A

Equation 1: y =

Equation 3: y = 9x − 30

Equation 2: y = 4

Equation 4: y =

−5

Create a strategy Treat the left-hand and right-hand sides of the equation as separate linear equations.

Apply the idea The left-hand side can be written as: And the right-hand side as: So the answers are options A and D.

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b Graph the equations in part (a) on the same plane.

Create a strategy Identify two points for each line to plot the graphs.

Apply the idea 6 5 4 3 2 1

−3 −2 −1 −1 −2 −3 −4 −5 −6

x 1

2 3 4 5 6 7

3×2 Substitute x=2 y= −5 2 y = 3−5 Simplify the fraction Evaluate y = −2 So the second point is at (2, −2).

The first point of y = (0, 0).

is its y-intercept at

Another point can be plotted by substituting x = 3 into the equation. x 1

2 3 4 5 6 7

2×3 Substitute x=3 y= 3 y=2 Evaluate So the second point is at (3, 2).

c Find the values of x and y which satisfy both equations.

Create a strategy Identify the coordinates of the point of intersection of the lines.

Apply the idea The point of intersection is (6, 4) from the graph in part (b). The values which satisfy both equations are x = 6 and y = 4.

318

− 5 is its y-intercept at

Another point can be plotted by substituting x = 2 into the equation.

−3 −2 −1 −1 −2 −3 −4 −5 −6

6 5 4 3 2 1

The first point of (0, −5).

Mathspace Victorian Curriculum – Year 8 mathspace.co

Reﬂect and check This means that the values x = 6 and y = 4 make both equations true, so they are the solution to the linear equations. We can check this by substituting these values into both equations to see if they are true. Write the equation Substitute the values of x and y Simplify the numerator Simplify the fraction Evaluate Write the equation Substitute the values of x and y Simplify the numerator Evaluate Substituting the values of x and y to both the equations make both equations true. The solution is correct.

Idea summary To solve linear equations graphically, we represent each side of the equation as a linear relationship, graph each one and ﬁnd the point where the two lines intersect on a graph. This point is called the point of intersection and gives us the solution to the linear equation.

7.05 Practice questions Understanding

Explorer 1, 2

Adventurer 1–3

Trailblazer 1–3

The point (2, 5) satisﬁes the equations y = 2x + 1 and y = −3x + 11. What does this point represent if both equations are graphed?

In what situation do two different lines on a graph never cross each other?

Are the lines represented by the equations y = 2x + 1 and y = −3x + 4 parallel, intersecting, or the same line?

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Fluency 4

4–6, 7a–d, 9, 10, 13

4–6, 7c–g, 9, 10, 12, 13, 14a 4, 7g–l, 8, 9, 11, 12, 14

What are the coordinates of the point of intersection of the two lines? a

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

7 6 5 4 3 2 1

4 3 2 1 x

320

−3

y x 1

x 4

9 8 7 6 5 4 3 2 1

−2 −1 −1 −2

y 1

−4

−7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

−1

−2

−2 −1

−4 −3 −2 −1

1 2 3 4 5 6 7

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9 8 7 6 5 4 3 2 1 −2 −1 −1 −2

2 3 4 5 6 7 8

x 1

2 3 4 5 6 7 8 9

−3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

2 1

x 1 2 3 4 5 6 7 8

−8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

6 5 4 3 2 1 −4 −3 −2 −1

−1

x 1

−2

2 1

6 5 4 3 2 1 −1 −1 −2 −3 −4

x 1 2 3 4 5 6 7 8 9 10 11

9 8 7 6 5 4 3 2 1

Consider the horizontal line with equation y = 8. Find the point of intersection of the graph of y = 3x − 4 with the line y = 8

The graph of y = 3x − 4 is shown on the Cartesian plane: a

Find the solution to 3x − 4 = 8

−4 −3 −2 −1

−1 −2

x 1

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The graph of y = −2x − 4 is shown on the Cartesian plane:

Find the point of intersection of the graph with the line y = −12

−5 −4 −3 −2 −1 −2

Find the solution to −2x − 4 = −12

x 1 2 3 4 5 6

−4 −6 −8 −10 −12 −14

Ex 1

For each pair of equations: i

Sketch the graph of the two lines on the same Cartesian plane.

Find the coordinates of the point of intersection.

y=3

y = 3x − 4

y = −x + 6

y = −x + 2

x = −3 y = 5 y = x + 2 y = 2x − 4 e

y = −4x + 2

y = 5x + 6

y = 3x − 12 y = 3x − 2 y = −2x + 3 y = 2x + 12 i

y = 3x − 3

y=x−5

y = 2x − 7

y = −4x + 11 y = −x − 7 Ex 2

The graph of plane: a

l y = 3x

is shown on the Cartesian

9 8

What is the equation of the other line that must be graphed in order to solve the equation

7 6 5

Find the solution to

3 2

−6−5−4−3−2 −1 −1

1 2 3 4 5 6 7 8 9

Emma wishes to ﬁnd the point of intersection of the lines: y = −2x + 3 y = −1 a

Complete the table of values for y = −2x + 3: x y

322

−1

Sketch the graphs of both lines on a Cartesian plane.

Determine the point of intersection.

Verify the solution by substituting the values of x and y in the ﬁrst equation.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Scott wishes to ﬁnd the point of intersection of the lines: y = 2x + 1 y = −3x + 11

Complete the table of values for y = 2x + 1:

x y

−1

Complete the table of values for y = −3x + 11:

x y

−1

Sketch the graphs of both lines on a Cartesian plane.

Determine the point of intersection.

Verify the solution by substituting the values of x and y to both equations.

For the linear equations: • Equation 1: y = 4x − 3 • Equation 2: y = 4 − 3x x y

Ex 3

−3

−2

−1

Fill in the table of values using the line y = 4x − 3

Fill in the table of values using the line y = 4 − 3x

Sketch the graph of both lines on the same coordinate axes.

Find the values of x and y that satisfy both equations.

For the linear equations: • • −8y = 8x + 8

Ex 4

Sketch the graph of both lines on the same coordinate axes.

Find the values of x and y that satisfy both equations.

For 2x + 2 = −2x + 2, a

To find the solution graphically, which two straight lines would need to be graphed? A

y = 2x − 2

y = 2x + 2

y = −2x − 2

Sketch the graph of both lines on the same coordinate axes.

Find the values of x and y that satisfy both equations.

y = −2x + 2

For each pair of equations: i

Sketch the graph of both lines on the same coordinate axes.

Determine whether there is a value for x and y that solves both equations. If yes, provide the x and y-values.

y = −4x − 1

4x − 2y = 2

y = 5x − 7

y=x

y = −4x + 2 −2x + 4y = 2 y = −x + 5 y = −1

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Reasoning 15

15, 16

The graph of plane:

16, 17

is shown on the Cartesian

What is the equation of the other line that must be graphed in order to solve the equation

Find the solution to

Explain why the solution to the equation

10 9 8 7 6 5 4 3 2 1

represents the point of intersection on

−6 −5 −4 −3 −2 −1

the graph between the two lines.

−1

x 1

Write an equation that represents the intersection of the lines of the equations y = 3x. Explain why your equation represents the intersection.

Imagine you are given the gradients and y-intercepts of two lines. Explain how you can determine if the lines will intersect without graphing them.

Problem-solving 19

18–20

and

18–20

When comparing two mobile phone plans with different monthly fees and usage charges, explain how the point of intersection between the two linear equations representing the plans would be signiﬁcant for a consumer.

A small bakery produces two types of bread: whole wheat and sourdough. The bakery produces bread at different rates throughout the day. The whole wheat bread production is represented by the equation y = 3x + 10, where x is the number of hours since the bakery opened and y is the number of whole wheat loaves produced. The sourdough bread production is represented by the equation y = 4x + 5, where x is the number of hours since the bakery opened and y is the number of sourdough loaves produced. Here are the graphs of the two equations: a

At what point do the two lines intersect, and what does this point represent in the context of the problem?

How many loaves of whole wheat and sourdough bread are produced after 6 hours?

Which type of bread is produced at a faster rate, and how can you tell from the graph?

Bakery bread y, number of loaves 45 40 35 30

Whole wheat bread

25 20 15 10 5

Sourdough bread x, hours 5

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A taxi charges a flag fall fee (upfront fee) of $5 and an additional $0.80 per kilometre travelled. A rideshare service charges a $2 ﬂag fall fee and $1.20 per kilometre travelled. At what distance will the cost of both services be the same?

7.06 Inequalities After this lesson, you will be able to… • interpret graphs of simple linear inequalities. • graph linear inequalities on a Cartesian plane. • solve simple linear inequalities algebraically. • solve linear inequalities graphically. • solve real-world problems using linear inequalities.

Linear inequalities An inequality is a relationship that is described using any of the inequality symbols • < less than • ≤ less than or equal to • > greater than • ≥ greater than or equal to The inequality x ≤ 4 is read as, x is less than or equal to 4 and means that x can be any value that is less than or equal to the number 4. We would show this on a number line like this: −8

−6

−4

−2

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When graphing inequalities on a number line, if the inequality is: • ≤ or ≥ we fill the endpoint as our inequality could equal that point. • < or > we leave the endpoint unfilled to show that the inequality cannot equal that point.

Example 1 Write the inequality for x that is represented on the number line. −10

−5

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Create a strategy

Apply the idea

To determine the direction of the inequality, we The line starts at the point 1 and stretches out to can think about which values are covered by the the right, which means it is either x > 1 or x ≥ 1. line. Since the circle on the point 1 is hollow, it means that it is not included in the inequality. We can then look at whether the starting point is included or not to determine which inequality symbol to use.

The inequality represented by this line is x > 1.

Idea summary To graph an inequality on a number line, start by determining which direction the ray will point, right or left. This can be determined by making sure the ray covers all the values that make the inequality true. The end point of the ray will be an unﬁlled circle if the inequality has a < or >. The end point of the ray will be a ﬁlled circle if the inequality has a ≤ or ≥.

Graph simple linear inequalities We know the graphs of vertical lines are of the form x = a, where a is a constant. For example, the line x = 2. This shows graphically that everywhere on the graph the value of x is 2.

4 3 2 1

-4 -3 -2 -1 -1 -2 -3 -4

If we want to graph what x ≥ 2 looks like we need to include any x value that is greater than or equal to 2.

7 6 5 4 3 2 1 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

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If the inequality doesn’t include the equals sign, the boundary line is dashed. For example, the graph of x > 2 would be:

5 4 3 2 1

−5 −4 −3 −2 −1 −1

2 3 4 5

−2 −3 −4 −5

We can approach inequalities involving horizontal lines in a similar way. 5 4 3 2 1

-3 3 -2 2 -11 -11 -2 -2 -3 -3

y<4

x 1

5 4 3 2 1

-3 -2 -1 -1 -2 -3

x 1

5 4 3 2 1

-3 -2 -1 -1 -2 -3

y=4

y≥4

To graph a linear inequality in one variable, we follow these steps: 1. If the inequality involves the x variable, draw a vertical line at the x-value. If it involves the y-variable, draw a horizontal line at the y-value. 2. If the inequality includes “equal to” ( ≥ or ≤ ), draw a solid line to indicate all points on the line are included in the solution. 3. If the inequality does not include “equal to” (> or <), draw a dashed line to indicate all points on the line are not included in the solution. 4. Shade the region to the right or above the line for > or ≥ and to the left or below the line for < or ≤.

Example 2 Write the inequality that describes the shaded region.

7 6 5 4 3 2 1 −4 − 4 −3 −3 −2 −2 −1 −1 −1 −1 −2 −2 −3 −3 −4 −4 − 5 −5 −6 −6 −7 −7

x 1

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Create a strategy

Apply the idea

Identify the variable and constant value to determine the equation of the line, then choose the necessary inequality sign that satisﬁes the shaded region and the type of line.

Since the line is horizontal, it is an equation involving only the y-variable. The shaded region is below the line, so it’s either < or ≤. The line passes through y = 5 and points on the line are included in the shaded region. The inequality is y≤5

Idea summary To graph a linear inequality in one variable: • Identify the variable (x or y) and the inequality symbol (>, <, ≥, or ≤ ). • Determine the constant value (a or b). • Draw a vertical line at x-value (a) for x-variable or a horizontal line at y-value (b) for y-variable. • Use a solid line for “equal to” ( ≥ or ≤ ) and a dashed line for others. • Shade the region based on the inequality symbol.

Solve linear inequalities algebraically Solving linear inequalities uses similar techniques as solving linear equations. Let’s take a look at solving the inequality: 3x + 2 ≥ 14 The difference in solving inequalities is that we are looking for a range of values for x instead of a single value as we do for equations. Subtract 2 from both sides Simplify Divide both sides by 3 Simplify We found that x ≥ 4. This means when x is any number greater than or equal to 4 the inequality is true. We can show this on a number line. 0

We can check our answer by substituting. Let’s try the numbers just above and below 4, say x = 5 and x = 3. When x = 5: 3x + 2 = 3 (5) + 2 = 15 + 2 = 17 ≥ 14 328

This satisﬁes the inequality

Mathspace Victorian Curriculum – Year 8 mathspace.co

When x = 3: 3x + 2 = 3 (3) + 2 =9+2 = 11 ≱ 14

This does not satisfy our inequality

Our substitution to check conﬁrms that x ≥ 4 is the solution set and that our number line is correct.

Example 3 Solve: 3x + 27 > 3

Create a strategy

Apply the idea

Solve the inequality by isolating x on one side of the inequality.

Write the inequality Subtract 27 from both sides Simplify Divide both sides by 3 Simplify

Example 4 Solve:

Create a strategy

Apply the idea

Solve the inequality by isolating a on one side of the inequality.

Write the inequality Subtract 3 from both sides Simplify Multiply both sides by 5 Simplify

Idea summary When solving inequalities we apply inverse operations in reverse order similar to solving single variable equations. For inequalities, however, we end up with a range of values rather than a single solution for our variable.

Solve linear inequalities graphically We can use the ideas of the graphs of lines to graphically solve linear inequalities. Remember though, as we saw with the algebraic method, we will end up with a range of values as a solution. 7.06 Inequalities mathspace.co

329

Our starting point is to consider the left-hand side and right-hand sides of a linear inequality as lines. If we have the linear inequality 3x + 5 < 11 We think of left-hand side as the line: y = 3x + 5 And the right-hand side as the line: y = 11 To solve 3x + 5 < 11 we just need to ﬁnd for what values of x the line y = 3x + 5 is less than y = 11, or we can think about it as the values of x where the y values of the line y = 3x + 5 are less than when y = 11. Graphing both lines gives:

14 13 12 11 10 9 8 7 6 5 4 3 2 1

y y = 11 y = 3x + 5

x 1

We notice that the lines intersect at x = 2. Remember we are looking for all values of x where the line y = 3x + 5 is below the line y = 11. Therefore we can see that our solution to the inequality is x < 2. We can conﬁrm our solution using the algebraic method:

Subtract 5 from both sides

Simplify

Divide both sides by 36

Simplify

14 13 12 11 10 9 8 7 6 5 4 3 2 1

y y = 11 y = 3x + 5

x 1

Example 5 9

Consider the graph:

8 7 6 5

y = 3x + 9

4 3 2 1

−5 330

Mathspace Victorian Curriculum – Year 8 mathspace.co

−4

−3

−2

−1

a Find the solution to the linear inequality 3x + 9 ≥ 0.

Create a strategy Consider the left-hand side and right-hand sides of a linear inequality as lines. If we have the linear inequality 3x + 9 ≥ 0 We think of left-hand side as the line: y = 3x + 9 And the right-hand side as the line: y=0 To solve 3x + 9 ≥ 0 we just need to ﬁnd for what values of x the line y = 3x + 9 is greater than or equal to y = 0, or we can think about it as the values of x where the y-values of the line y = 3x + 9 are greater than or equal to when y = 0.

Apply the idea Graphing both lines gives:

Notice that the lines intersect at x = −3. Remember we are looking for all values of x where the line y = 3x + 9 is equal or above the line y = 0.

8 7 6

From the graph, we can see that the solution is x ≥ − 3.

y = 3x + 9

4 3 2

y=0 −5 −4

1 −3

−2

−1

Reﬂect and check We can conﬁrm our solution using the algebraic method: Subtract 9 from both sides

Simplify

Divide both sides by 3

Simplify

7.06 Inequalities mathspace.co

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b Plot the solution on a number line.

Create a strategy Plot the inequality solution from part (a).

Apply the idea To graph the inequality x ≥ −3, plot a solid dot on point −3 with a ray towards positive inﬁnity. −5

−4

−3

−2

−1

Example 6 Consider the graph:

y 14 12 10

y=9

8 y = 3x − 21

6 4 2

x 2

a Find the solution to the linear inequality 3x − 21 < 9.

Create a strategy To solve 3x − 21 < 9 we just need to ﬁnd for what values of x the line y = 3x − 21 is less than y = 9, or we can think about it as the values of x where the y-values of the line y = 3x − 21 is less than when y = 9.

Apply the idea

Reﬂect and check

Notice that the lines intersect at x = 10. We can conﬁrm our solution using the algebraic Remember we are looking for all values of x method: where the line y = 3x − 21 is below the line y = 9. Add 21 to both sides From the graph, we can see that the solution is Evaluate the sum x < 10. Divide both sides by 3

332

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Evaluate

b Plot the solution on a number line.

Create a strategy Plot the inequality solution from part (a).

Apply the idea To graph the inequality x < 10, plot a hollow dot on point 10 with a ray towards negative inﬁnity. 0

Example 7 Consider the graph of y = x − 6.

y 8 6 4 2

−8 −6 −4 −2 −2

−4 −6 −8

Using the graph, find the solution of the inequality x − 6 < 0.

Create a strategy

Apply the idea

The solutions to x − 6 < 0 corresponds to points on the line where y < 0. Find the points where the line lies below the x-axis.

Looking at the graph, the part of the line that lies below the axis is x < 6. y 8 6 4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

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333

Idea summary The graphical method of solving linear inequalities uses the idea of representing each side of the inequality as a line and determining the solution by inspection.

7.06 Practice questions Understanding 1

Less than or equal to

≤

Greater than

≥

Greater than or equal to

If x = 7, is the inequality true or false? x+3>8

2x ≥ 15

8≤y

22 ≥ m

x − 10 < 0

4 − 2x > −15

−10 < y

0>k

When graphing the inequality x > −2, which side of the vertical line x = −2 should be shaded? 5, 6, 7ab, 8abef, 9abef, 10a–d, 11ab, 12, 13ace, 16, 17

5, 6, 7cd, 8c–f, 9c–f, 10d–f, 11b–g, 13beg, 14–17

5, 6, 7cd, 8cdgh, 9cdgh, 10ef, 11e–j, 13dfg, 14–17

Write the inequality shown on each number line: a

b −5 −4 −3 −2 −1 0

2 3 4 5

−5 −4 −3 −2 −1 0

2 3 4 5

334

The inequality 7 < x can also be written as x > 7. Rewrite each inequality with the variable on the left:

Fluency Ex 1

Trailblazer 2–4

Less than

a 4

iii

a 3

Adventurer 2–4

Match the inequality sign with the correct description. i

Explorer 1–4

−5 −4 −3 −2 −1 0

2 3 4 5

−5 −4 −3 −2 −1 0

2 3 4 5

Mathspace Victorian Curriculum – Year 8 mathspace.co

Ex 2

Write the inequality that describes the shaded region: a

5 4 3 2 1

2 3 4 5

−4 − 4 − −3 3 − −2 2 − −11

−11 − −2 − 2 −3 − 3

d 4

−1

x 2

−7 −6 −5 5 −4 −4 − −3 3 −2 −2 −1 −1 −1 −1

− 2 −2

−2 −2

− 3 −3

−3 − 3

−4 − 4

−4 −4

x 1 2 3 4

y≤4

x < −3

x≥1

y > −5

Solve the inequality: a

x + 5 ≥ 11

9>y+4

n+8≤7

m − 2.5 ≥ 9.5

7 ≤ x −3

11 < x −8

x − 8 ≤ −4

−7 + x ≥ −11

Solve the inequality: a e

Ex 3

2 1

Sketch the graph of the inequality: a

5 4

−4 −3 −2 −1

7 6

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

10x < 90

b f

c 56 ≤ 8p

3x ≤ −18

21 + 7x ≥ 21

15x ≥ −60

Solve the inequality: a

6x − 54 ≥ 30

3x + 21 > 6

−14 + 5x ≤ 6

4 (x + 3) ≤ 12

−12 > 4 (4x + 5)

7.06 Inequalities mathspace.co

335

Ex 4

Solve the inequality: a

Consider the inequality 6 (x + 2) ≤ 48: a b

Solve the inequality. Does the value of x satisfy the inequality 6 (x + 2) ≤ 48: i

x = −6

x=6

iii

x=0

x = 12

8 < 2 (x + 7)

For each of the inequalities: i

Solve the inequality.

Plot the solution to the inequality on a number line.

2x − 2 < 6

3 < 4x − 1

3 (x − 1) < 9

Construct and solve an inequality for the situation: “The sum of 2 groups of x and 1 is at least 7.”

Consider the situation: “2 less than 4 groups of p is no more than 18”.

16 Ex 5, 6

Construct and solve the inequality.

Find the largest value of p that satisﬁes this condition.

Using the graph, ﬁnd the solution of the inequality: a

2x + 6 ≥ 0

4x − 3 < 17

18 16 14 12 10 8 6 4 2

8 6

y = 2x + 6

4 2 −8 −6 −4 −2 −2

x 2

−4 −6 −8

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Mathspace Victorian Curriculum – Year 8 mathspace.co

−2

−2 −4

y = 17

y = 4x − 3

x 2

Ex 7

Consider the graph of y = 5x + 3. Using the graph, solve the inequality 5x + 3 ≤ 0. 4

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

Reasoning 18

18–22

19, 20, 22

Write down the inequality shown in each diagram. What is the difference between a solid boundary and a dashed boundary? 5 4 3 2 1

5 4 3 2 1

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

19, 22, 23

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

2 3 4 5

The shaded regions in the graphs are deﬁned by a pair of inequalities. i

Write down the inequalities that deﬁne each region.

Is the point (2, 3) contained in any of the shaded regions?

5 4 3 2 1 − −5 5− −4 4− −3 3− −2 2 − −11 − −11 − 2 −2 − 3 −3 − 4 −4 − 5 −5

x 1

2 3 4 5

5 4 3 2 1 −5 −5 − −4 4− −3 3− −2 2 − −11 −1 −1 −2 −2 − −3 3 −4 −4 −5 −5

7.06 Inequalities mathspace.co

337

5 4 3 2 1 − −5 5− −4 4− −3 3− −2 2 − −11 − −11 − 2 −2 −3 − 3 −4 − 4 − 5 −5

5 4 3 2 1

x 1

−5 −5 − −4 4− −3 3− −2 2 − −11 −1 −1 −2 −2 − 3 −3 −4 −4 −5 −5

2 3 4 5

Isabelle tried to solve an inequality but made a mistake in her work. 1 2 3 Determine which step is incorrect and explain the error.

Explain the difference between the inequalities: 0 ≤ x ≤ 4 and 0 < x < 4

Percy tried to plot the solution to the inequality 4x + 28 ≥ −8 on a numberline, however his answer is incorrect. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1

9 10

Identify the error(s) and explain how to fix them. 23

338

Ryan wants to save enough money to buy a cricket bat that costs $40. He has $22.10 saved from his Birthday. In order to make more money, he plans to wash windows for $2 per window. a

Let w be the number of windows that Ryan washes. Construct an inequality that results in Ryan having at least enough savings to buy his cricket bat.

Solve the inequality for w, rounded to two decimal places.

Is each statement correct? Explain your thinking. i

Ryan must wash more than 9 windows to be able to afford the equipment.

Ryan must wash at least 8 windows to be able to afford the equipment.

iii

If Ryan washed 8 windows, and 95% of another window, he could afford the equipment.

The number of windows Ryan must wash to be able to afford the equipment must be greater than or equal to 9.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Problem-solving 26 24

24–27

25, 27–30

At a sports clubhouse the coach wants to rope off a rectangular area that is adjacent to the building. He uses the length of the building as one side of the area, which measures 26 m. He has at most 42 m of rope available to use. If the width of the roped area is W, form an inequality and solve for the range of possible widths.

Consider the graphs of y = x + 2 and y = x − 3:

ow many solutions does the inequality x + 2 ≥ x − 3 H have?

6 y=x+2

4 2 −8 −6 −4 −2 −2 −4

x 2

y=x−3

−6 −8

Skye was given $72 for a birthday present. This present, along with earnings from a Saturday job, is being set aside for a mountain bike. The job pays $5.00 per hour, and the bike costs $379. a

Set up and solve an inequality to ﬁnd the minimum number of hours that Skye needs to work to be able to buy the mountain bike, where h represents the number of hours worked. Round your answer to one decimal place.

If Skye can only work her job for a whole number of hours, ﬁnd the minimum number of hours she must work to afford her bike.

A particular bus can carry a maximum weight of 4750 kg on board. If the average adult passenger weighs 75 kg and the driver weighs 80 kg, then the number of passengers on board the bus at any one time must satisfy 75a + 80 ≤ 4750, where a represents the number of adult passengers. a

There are 58 adults already on board and 8 more at the bus stop waiting to get on. Find the combined weight of everyone on board if the driver lets all of the new passengers on.

Can the driver safely let all of the new passengers on board?

When breeding certain types of fish it is recommended that the number of female fish is more than double the number of male fish. a

Write the inequality for the recommended relationship. Let f be the number of female ﬁsh and m be the number of male ﬁsh.

Do these combinations satisfy the inequality from part (a)? i

f = 10, m = 8

f = 23, m = 9

iii

f = 13, m = 10

f = 8, m = 4

7.06 Inequalities mathspace.co

339

Alex makes two types of table, A and B. Each week, he has to make more than 2 and less than 6 of table A, and more than 5 and less than 8 of table B. He calls the number of table A that he makes x and the number of table B that he makes y. He plots the graph to show the possible numbers of each table that he can make. a

Write the inequalities that bound the region for possible production of each type of table.

Assuming he makes whole numbers of tables, what are the possible combinations of the number of tables of each type he can make? Give your answers as (x, y) coordinates.

8 6 4 2 x 2

If Alex makes $40 proﬁt on each table A that he makes, and $50 proﬁt on each table B that he makes: i

How many of each table should he make to make the most money?

What is the maximum proﬁt he can make in a week?

Tamsin makes two different types of sunshade, y P and Q. Each week, she has to produce at least 3 of 8 shade P and 5 of shade Q. Constraints on materials mean that the possible weekly construction levels are within the triangular region shown in the diagram. 6 She calls the number of shade P that she makes x and the number of shade Q that she makes y. She plots a graph to show the possible numbers of each shade that she can make. a

340

4 2

Write the inequalities that bound the region for possible production of each type of shade.

x 2

Assuming she makes whole numbers of sunshades, what are the possible combinations of the number of sunshades of each type she can make? Give your answers as x, y coordinates.

If Tamsin makes $250 proﬁt on each shade P that she makes, and $500 proﬁt on each shade Q that she makes: i

How many of each shade should she make? Is there only one answer?

What is the maximum proﬁt she can make in a week?

Mathspace Victorian Curriculum – Year 8 mathspace.co

INVESTIGATION

Experiment with linear inequalities Investigate linear inequalities in the Cartesian plane.

Objectives To graph linear inequalities using digital tools To recognise how changes in a linear inequality are shown on the graph

Exploring inequalities on the Cartesian plane

Our goal in this investigation is to make connections between equations and inequalities represented symbolically and graphically.

Interactive exploration Discover this concept in action online

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Investigate Based on this exploration, we want to draw connections between the inequality and the graph. 1.

Without using the applet, would the inequality y > −3x have a solid or dashed line? Would we shade above or below the line?

If we want the line to go through the origin and the point (1, 3), have a solid line, and be shaded below the line, what would the inequality be?

Use your findings to complete the table provided in the downloadable asset. Below the table in the downloadable asset, sketch a graph of each inequality listed in the table.

Investigation: Experiment with linear inequalities mathspace.co

341

Negative signs in inequalities

So far, we have explored inequalities in two variables where the y-term is positive. Now, we will explore what happens when the y-term is negative.

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Equation

Solution 1

Solution 2

Solution 3

y = 3x

−y = 3x

y = −3x

−y = −3x

y > 3x

−y > 3x

y > − 3x

−y > −3x

y < 3x

−y < 3x

y < −3x

−y < −3x

Use the applet to compare the graphs of the equations and inequalities in the table. Then, answer the questions below.

Investigate

342

Is there a point that satisfies both y = 3x and y > 3x? Explain why or why not.

What is the gradient of −y = 3x? What is the gradient of the boundary line of −y > 3x? What is the gradient of the boundary line of −y < 3x?

Is the graph of −y > 3x shaded above or below the line? Is this what you expected?

Which of the other inequalities has a graph that is equivalent to −y > 3x?

Is the graph of −y < 3x shaded above or below the line? Is this what you expected?

Which of the other inequalities has a graph that is equivalent to −y < 3x?

Based on your answers, what patterns do you notice about the negative sign and the direction of the inequality?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Discussion 1.

How can we determine which region to shade when graphing an inequality in two variables? Can you think of another way?

What does it mean if the boundary line is dashed? solid?

If a point is in the shaded region, will it satisfy the inequality? Give an example to justify your answer.

If we multiply both sides of an inequality by a negative, how does that affect the direction of the inequality symbol?

If we divide both sides of an inequality by a negative, how does that affect the direction of the inequality symbol?

Did you know?

Linear relationships are key in home improvement! When painting a wall, you can use a linear relationship to figure out how much paint you’ll need based on the wall’s area. If 1 litre covers 37 square metres, it’s possible to work out how many litres you’ll need for any room size, making your DIY projects more predictable. This helps avoid wasting money on extra paint or running out before the job is finished. You can apply the same concept when determining the amount of tiles, flooring, or wallpaper needed for a renovation. By knowing how area scales linearly with materials, you can accurately estimate costs and time, ensuring your project stays on budget and schedule. Investigation: Experiment with linear inequalities mathspace.co

343

7 Chapter review 1

For each equation: i

Write the inverse operation needed to make the variable the subject of the equation.

Solve the equation.

u + 27 = 2

t − 2 = 35

−7j = 56

y + 3 = −8

t − 3 = −45

7n = −35

Solve: a

2 (q + 11) = 20

3 (q + 3) = −18

3 ( p − 5) = −12

−4 (r + 5) = −12

4 (u + 2) = −8

2 (u + 5) = 18

For each equation: i

Complete the table of values.

Plot the points from the table on a Cartesian plane.

iii

Is the relationship linear or non-linear?

y = 2x x y

344

y = −x + 2 x y

−1

y = 4x + 2 x y

−1

x y

Does the table show a linear relationship between x and y? a

x 1 2 4 7 y 5 8 11 14

8 17

x y

1 −1

2 3

3 7

4 11

5 15

x 1 2 3 4 y 5 1 −3 −7

5 −11

x y

5 15

10 30

15 45

20 60

25 75

x 0 1 3 7 y 14 7 0 −7

x y

−6 0

−5 9

−3 18

1 27

Mathspace Victorian Curriculum – Year 8 mathspace.co

Vanessa opens a bank account and deposits $300. At the end of each week she adds $10 to her account. a

Complete the table which shows the balance of Vanessa’s account over the ﬁrst four weeks: Week (W ) Account total (A)

0 $300

1 $310

Is the relationship between the number of weeks, W, and the account, A, linear?

Write the linear relationship for Vanessa’s account total, A, in terms of the number of weeks W, for which she has been adding to her account.

For each graph: i

Identify the x-value of the x-intercept.

−2 −1

Identify the y-value of the y-intercept.

1 −4 −3 −2 −1

1 −1

x 1

−1

−3 −4

Find the value of the x-intercept for each line: a

−2

y = 9x

2y + x = −3

2y + 2x = 4

For y = 4x, a

Find the coordinates of the y-intercept.

Find the coordinates of the x-intercept.

Find the value of y when x = 2.

Sketch the graph of the line.

Consider the graphed line:

Identify the x-intercept.

Does this line have a y-intercept? Write down the equation of the line.

2 1 −4 −3 −2 −1 −1

x 1

−2 −3

Chapter 7 review mathspace.co

345

A taxi company charges a ﬂat fee of $25 per trip plus $0.20 per kilometre driven. The linear equation that represents the total cost ( y) for renting the taxi and driving x kilometres is y = 0.20x + 25. a

Determine the intercepts of the graph of the equation.

Graph the equation on the Cartesian plane. Use an appropriate scale and label the axes.

Interpret the x-intercept of the graph.

Solve: a

3 (3x + 15) = −63

9 (7 − 3x) = 20

2 (−3g + 7) = −23

4 (5x − 1) = −16

−3 (4 − x) = −12

16 = 3 (2x + 9)

−5 (g + 4) + 9 = −41

−7 (x + 5) + 4 = −17

6 (q + 1) − 5 = −29

4 (x + 6) − 8 = 48

−2 (x + 4) − 3 = −59

9x + 11 = 21 3

Solve the equation 13(2x − 8) = −26 in two different ways. a

Method 1: start by expanding the bracket.

Method 2: start by dividing both sides by 13.

Was one method easier than the other? Explain your answer.

Show that the equation 5y + 3 = 5y − 7 has no solutions.

What are the coordinates of the point of intersection of the two lines? a

y 9 8 7 6 5 4 3 2 1

−2 −1 −1 −2

346

x 1

2 3 4 5 6 7 8

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9 8 7 6 5 4 3 2 1 −2 −1 −1 −2

x 1

2 3 4 5 6 7 8 9

2 1

−3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8 −9

2 1

x 1 2 3 4 5 6 7 8

−8 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

f 6 5 4 3 2 1

6 5 4 3 2 1 −4 −3 −2 −1

−1 −1 −2 −3 −4

x 1

−1

−2

x 1

x 1 2 3 4 5 6 7 8 9 10 11

For each pair of equations: i

Sketch the graph of the two lines on the same Cartesian plane.

Find the coordinates of the point of intersection.

y = 3x + 3 x = −1

The graph of plane:

y = −x + 6

y=x+2 is shown on the Cartesian

Find the equation of the other line that must be graphed in order to solve the equation

Find the solution to

y = 3x − 3

y = −4x + 11

y = 3x 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

−6−5−4−3−2 −1 −1

x 1 2 3 4 5 6 7 8 9

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Ejay wishes to ﬁnd the point of intersection of the lines: y = −2x + 3 y = −1 a

Complete the table of values for y = −2x + 3: x y

−1

Sketch the graphs of both lines on a Cartesian plane.

Determine the point of intersection.

Verify the solution by substituting the values of x and y in the ﬁrst equation.

When evaluating two different internet packages with variable monthly fees and data charges, explain how the intersection point of the two linear equations representing the packages would be relevant for a customer.

Write the inequality that describes the shaded region: a 7

y 4

5 4

2 1

−7 −6 −5 5− −4 4− −3 3− −2 2− −11 − −11

−4 − 4 −3 −3 − −2 2 − −11

−11 − −2 − 2 −3 −3

Solve: a x + 5 ≥ 13 e

−9 + x ≥ − 11

i 21

−2 −2

− 3 −3 − 4 −4

9>y+6

11 < x − 10

3x ≤ − 24

15x ≥ − 60

x−8≤−2

For 50 + 5x > 50, a

Solve the inequality.

Do these values of x satisfy the inequality 50 + 5x > 50? i

348

x 1 2 3 4

x=0

x=1

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iii

x = 10

x=5

Consider the graph of y = 5x + 2. Using the graph, solve the inequality 5x + 2 ≤ 0.

3 2 1 −4 −3 −2 −1

−1

x 1

−2 −3 −4

When breeding certain types of fish it is recommended that the number of female fish is more than double the number of male fish. a

Write the inequality for the recommended relationship. Let f be the number of female ﬁsh and m be the number of male ﬁsh.

Do these combinations align with the recommendation? i

f = 14, m = 10

f = 9, m = 4

iii

f = 8, m = 6

f = 8, m = 4

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Big ideas • Understanding Fractions are the properties and relationships of angles, sides, congruence, similarity, essential for and transformations in geometric figures enable us to solve problems and establish mathematical connections between them. • developing proportional thinking and reasoning. •• Transformation of triangles can result in the formation of various types of developing proportional thinking and reasoning. quadrilaterals. • 3D coordinate systems enable accurate object positioning and description in three-dimensional space, facilitating spatial analysis and visualisation.

Curriculum Outcomes VC2M8SP01 Identify the conditions for congruence and similarity of triangles and explain the conditions for other sets of common shapes to be congruent or similar, including those formed by transformations VC2M8SP02 Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related problems explaining reasoning VC2M8SP03 Describe in different ways the position and location of three-dimensional objects in 3 dimensions, including using a three-dimensional Cartesian coordinate system with the use of dynamic geometry software or other digital tools VC2M8SP04 Design and test algorithms involving a sequence of steps and decisions that identify congruency or similarity of shapes, and describe how the algorithm works

Chapter outline

Geometry Chapter outline 8.01 8.02 8.03 8.04 8.05 8.06

Angles and triangles Transformations and congruence Congruent triangles Transformations and similarity Similar triangles Investigation: Determine congruence or similarity Properties of quadrilaterals Investigation: 3D coordinate systems Chapter 8 review

352 362 374 385 395 407 411 424 429

Every snowflake reveals a world of intricate geometric patterns. The unique design of each snowflake comes from the unique path it takes through the clouds.

8.01

Angles and triangles After this lesson, you will be able to… • identify and classify different types of angles. • solve problems involving angle relationships. • identify and classify different types of triangles based on their angles. • use the relationship between the sides and angles of a triangle to solve problems involving triangle sides and angles.

Angle relationships Adjacent angles share a common arm and common vertex.

Vertically opposite angles are equal in size.

Complementary angles add to 90° and supplementary angles add to 180°. Complementary

Forming right angle 90°

Supplementary

Forming straight angle 180°

Example 1 Find the value of the missing angle x in the diagram: x° 40°

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75°

Create a strategy Supplementary angles are angles that, when combined, form a straight angle equal to 180°.

Apply the idea x + 40 + 75 = 180

Equate the sum of the angles to 180

x + 115 = 180

Add the left-hand side

x = 65°

Subtract 115 from both sides

The missing angle is 65°.

Reﬂect and check Remember to always include relevant units, in this case, we are ﬁnding an angle, so we should use the degree symbol.

Idea summary Adjacent angles are two angles sharing a deﬁning line, ray, arm or vertex and do not overlap. Vertically opposite angles are two pairs of equal angles formed whenever two segments, lines, or rays intersect at a point. If two angles add to 90°, we say they are complementary. If two angles add to 180°, we say they are supplementary. A full revolution is 360°.

Angles and triangles Recall that the angle sum of a triangle will always be 180°, as seen in this triangle. Angle sum of a triangle

We can also take any of the triangle’s sides and extend it to create an exterior angle. The size of an exterior angle is always equal to the sum of the opposite interior angles (the internal angles on the opposite side of the triangle as the exterior angle).

Angle on straight line

Angle sum of a triangle

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Interactive exploration

Discover this concept in action online

mathspace.co

Example 2 Find the value of the missing angle x in the diagram. x° 70° 35°

Create a strategy Use the angle sum of a triangle to form an equation and solve.

Apply the idea x + 70 + 35 = 180 x + 105 = 180 x = 75

Equate the interior angles to 180 Add the left-hand side Subtract 105 from both sides

The missing angle is 75°.

Reﬂect and check Remember to always include relevant units, in this case, we are ﬁnding an angle, so we should use the degree symbol.

Example 3 Solve for the value of h. 40° 63°

h°

Create a strategy The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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Apply the idea h = 40 + 63 Add the interior angles = 103 Evaluate

Idea summary The angle sum of a triangle is 180°. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Angles and sides Largest angle

Largest side

In any triangle, the longest side will always be opposite the largest angle and the smallest side will be opposite the smallest angle.

Smallest side

Smallest angle

For any triangle, the sum of the lengths of each pair of sides must be greater than the remaining side length.

Example 4 Will three sides of length 3, 5, and 9 make a triangle?

Create a strategy Create a table to compare the side lengths.

Apply the idea Side 3 5 9

Sum of remaining sides 5 + 9 = 14 3 + 9 = 12 3+5=8

Compare 5+9>3 3+9>5 3+5<9

In the ﬁnal row, the side of 9 is longer than the sum of the other two sides, so the triangle is impossible. The three sides of length 3, 5, and 9 cannot make a triangle.

Reﬂect and check For scalene triangles, we only have to check that the smallest two sides are bigger than the largest side, in this case if 3 + 5 < 9 so the triangle is impossible.

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Idea summary For any triangle, the sum of the lengths of each pair of sides must be longer than the length of the remaining side. m+n>p

m+p>n

n+p>m

8.01 Practice questions Understanding

Explorer 1–3

Adventurer 1–3

What is the measure of a full revolution in degrees?

Match each of the angle names to the correct deﬁnition:

Vertically opposite angles

Supplementary angles

2. Angles that sum to 90°

Complementary angles

Angles that sum to 180° Angles on opposite sides of an intersection point

Complete the statements with ‘larger than’ or ‘smaller than’: a b

If one side of a triangle is longer than another side, then the angle opposite the longer side is ⬚ the angle opposite the shorter side.

If one angle of a triangle is larger than another angle, then the side opposite the smaller angle is ⬚ the side opposite the larger angle.

Fluency

Trailblazer

1–3

4–6, 7a–d, 8, 9, 10a–d, 4–6, 7e–h, 8, 9, 10c–f, 11a–d, 12, 13, 16a–c, 17, 18 11c–f, 12–15, 16a–c, 18

4–6, 7e–h, 8–10, 11bd–f, 13–15, 16d–f, 19

Among the marked angles in the diagram, ﬁnd: a

A pair of adjacent angles

A pair of vertically opposite angles

A pair of complementary angles

D F

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Are the angles supplementary? a

160°

140°

19°

40°

d 165° 35°

50° 40°

Are the angles complementary? a

25°

35°

66°

55°

32°

42°

68°

48°

Describe the angles as complementary or supplementary: a

90° & 90°

88° & 2°

47° & 133°

36° & 54°

25° & 155°

62° & 28°

78° & 102°

177° & 3°

Given that the angles in the diagram are complementary, ﬁnd the value of x: a

x°

x° 41°

43° x°

48°

x° 79°

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Ex 1

Given that the angles in the diagrams are supplementary, ﬁnd the value of x: a

b 84°

x°

c 75°

98°

x°

Find the value of x, y or z in each of the diagrams: a

x°

10° x° 40°

x°

y°

x°

z° z° z° z° z° z°

y°

y° y°

z°

x°

z°

x° 160°

x° x°

x°

h x°

x°

25°

x°

Ex 2

x°

65°

35°

x°

35°

Find the value of x: a

x°

40°

82°

x° 110° 53°

x°

f 3x°

68°

x°

72° (x + 62)°

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67°

Ex 3

Find the value of x: a

b 135°

63° 135°

Find the value of x.

Find the value of y.

x°

113°

x°

51° y° x° 64°

Find the value of a.

Find the value of b.

Find the value of x.

Find the value of m.

58° 19°

m°

x°

Ex 4

a°

61°

b°

56°

Is it possible to form a triangle with these side lengths? a

6, 3 and 8

9, 6 and 6

2, 3 and 15

14, 18 and 13

Find the sum of the interior angles in the triangle.

Is the largest angle opposite the shortest, middle or longest side?

10, 20 and 9

Is the smallest angle opposite the shortest, middle or longest side?

13, 7 and 6

102° 20 cm

27°

16 cm 51°

23 cm

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Name the longest side of this triangle.

Name the largest angle of this triangle. C

A 7 cm

72°

9 cm

A 55°

10 cm

52° B

Name the shortest side of the triangle. A 54°

64°

Name the smallest angle of the triangle. C

19 cm B

22 cm

61°

23 cm

B A

Order the sides of the triangle from smallest to largest.

72° 55°

52° B

Reasoning

20, 21

20–22 4

Explain why these lengths cannot make a valid triangle.

Explain why it is not possible to make this triangle.

98° 65° 33°

The hypotenuse is the longest side in a right triangle. Explain why if one of the angles is a right angle, then the other two angles must be acute angles.

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Problem-solving 23, 24

Find the value of x.

Using the angles, classify the triangle.

24–26

x° 65°

45°

What type of triangle is the outer triangle?

Find the value of a.

Find the value of b.

47°

b°

a°

Consider the given triangle: a

Find the value of x.

Find the measure of ∠CAB.

Find the measure of ∠ABC.

Find the measure of ∠BCA.

Which side is the longest side of the triangle?

C (6x + 10)° (7x − 2)°

(8x + 4)°

Determine the possible range of values for x in this triangle.

x−5

4x − 9

2x + 7

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8.02 Transformations and congruence After this lesson, you will be able to… • describe the properties of rigid transformations. • identify congruent shapes.

Rigid transformations There are three special kinds of transformations: Translations slide objects, without changing their orientation.

5 down 5 right

This shape has been translated by 5 units to the right and 5 units down.

Each point is translated by 5 units to the right and 5 units down.

It remains the same shape when translated 5 units down and 5 units right because, for translations, the order does not matter. Reﬂections ﬂip objects across a line.

A shape reﬂected across a line.

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Every point is the same distance from the reﬂecting line on the opposite side.

Rotations move an object around a central point by some angle.

This shape has been rotated 90° clockwise around the point A.

Each point will stay the same distance from the central point A.

Exploration Apply the transformations listed to the rectangle shown: 7 6 5 4 3 2 1 −7 −6 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7

• A reﬂection across the x-axis.

• A translation right 1 unit and down 2 units. • A rotation 90° anticlockwise about the origin. x 1 2 3 4 5 6 7

After applying the transformations, answer the following questions: 1. Did the transformations change the length of the sides of the rectangle? 2. Did the transformations change the area of the rectangle? 3. Did the transformations change the interior angles of the rectangle?

What makes reflections, translation, and rotations special is that the original shape and the transformed shape have the same properties: • They have the same area. • Every side length stays the same. • Every internal angle stays the same. For this reason, these three transformations are called rigid transformations.

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Example 1 Describe the single transformation shown in the images: a

Create a strategy To identify the transformation we check for which of the following apply: • If one triangle is moved vertically or horizontally, but is the same orientation as the first, then it is a translation. • If the two triangles mirror each other then the transformation is a reflection. • If one triangle is turned around a point, then the transformation is a rotation.

Apply the idea In this case we can imagine a line of reflection shown and can see that one triangle is a reflection of the other.

Apply the idea In this case we can see that the triangle has been turned, this is a rotation.

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Apply the idea In this case we can see that the triangle has moved horizontally towards to the right, this is a translation.

Idea summary Translations slide shapes around. Reflections flip shapes across a line and rotations rotate shapes around a point. Reflections, translations and rotations are sometimes called rigid transformations. They preserve the area, side lengths, and internal angles of the original shape.

Congruence We say two shapes, X and Y, are congruent if we can use some combination of translations, reﬂections and rotations to transform one shape into the other. We use the symbol ≡ to express congruence, so we read A ≡ B as ‘A is congruent to B’. A

Here △ABC ≡ △FDE because a translation of 4 units right and a reﬂection across the x-axis transforms △ABC to △FDE.

4 3 2

1 B C 1 −5 −4 −3 −2 −1 0 −1 D

x 2

−2 −3 −4 −5

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Each side and angle in △ABC corresponds to a side and an angle in △FDE: • ∠A corresponds to ∠F. • ∠B corresponds to ∠D. • ∠C corresponds to ∠E. • Side AB corresponds to side FD. • Side BC corresponds to side DE. • Side CA corresponds to side EF. To determine which parts are corresponding, we can look at how the transformations affected the individual parts of the figures, or we can look at the order in which the shapes are named. Because the figures are congruent, the corresponding angles and sides of the figures are equal.

Example 2 The diagram shows two triangles that are translations of one another:

P A

Which angle has the same size as ∠B? A ∠P

∠Q

∠R

Create a strategy Look for the angle in △PQR that is also opposite the side with 3 markings.

Apply the idea ∠B is the angle at point B. This angle is opposite the side with 3 markings. In △PQR point Q is opposite the side with 3 markings. So ∠Q has the same size as ∠B. Option B is the correct answer.

Example 3 △ABC is reﬂected along the dotted line and its image △XYZ is produced.

9 C

7 B

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a Find the length of each side of △XYZ.

Create a strategy

Apply the idea

Identify the corresponding sides.

CB corresponds to XY. BA corresponds to YZ. CA corresponds to XZ. So we have the length of each side: XY = 8 cm YZ = 7 cm XZ = 9 cm

b △XYZ is: A

congruent to △ABC

an enlargement of △ABC

Create a strategy

Apply the idea

A reflection of a figure is also known as a flip, and does not affect size.

Since △ABC is reﬂected along the dotted line and produced the image △XYZ, then have the same area, every side length stays the same, and every internal angle stays the same. So △XYZ is congruent to △ABC. So option A is the correct answer.

Idea summary Two shapes, A and B, are congruent if we can use some combination of rigid transformations to transform one into the other. We use the symbol ≡ to express this relationship, so we read A ≡ B as ‘A is congruent to B’. Corresponding sides and angles of congruent ﬁgures are equal.

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8.02 Practice questions Understanding 1

Explorer

Adventurer

1 −4 −3 −2 −1

−2 −3

−4

−4 4 −3 −3 − −2 −1

−1

−4 −3 −2 −1

−11

−2

−2 − 2

−3

−3 −3

−4

−4 4

Describe the relationship between rigid transformations and congruence.

Fluency

3–6, 8–10, 12, 14–16, 19ab

3–9, 11, 13, 14, 16–19

Name each transformation as a translation, rotation or reﬂection. a

368

x 1

−1

−3

−4 −3 −2 −1

−2

−1

Ex 1

1, 2

Describe the transformation as rigid or non-rigid. a

Trailblazer

Mathspace Victorian Curriculum – Year 8 mathspace.co

4–7, 9, 11, 13, 14, 17–19

What are two possible transformations that could be used to move the rectangle from position A to B?

9 8 7 6 5 4 3 2 1

−8 −6 −4 −2

Describe the transformations to move the triangle from position A to B.

Describe the transformation that moves the quadrilateral from position A to B.

x 2

−1

7 y 6 5 B 4 3 2 1 A −7−6−5−4−3−2−1−1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5 6 7

y x

−8 −6 −4 −2 −1 −2 −3 −4 A −5 − −6 − −7 − −8 −9

Describe the transformation that moves the triangle from position A to B.

7 6 5 4 3 2 1

A −7 −6 −5 − −4 4 −3 −2 −1

−1 −2 −3 −4

y B

x 1 2 3 4

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Ex 2

The diagram shows 2 triangles that are translations of one another:

Which angle in △PQR is the same as ∠BAC?

P A

The diagram shows 2 quadrilaterals that are rotations of one another:

J M

Which angle in quadrilateral MNOP is the same as ∠JKL?

L P K O

The diagram shows 2 triangles that are reﬂections of one another:

The diagram shows 2 triangles that are translations and reﬂections of one another:

Which angle in △ABC is the same as ∠RPQ?

Which angle in △PQR is the same as ∠CBA?

hat transformations can be applied to W parallelogram ABDC to move it into the same position as parallelogram FGJH?

Which angle in parallelogram △FGJH is the same as ∠BDC?

P A

D G

hat transformations can be applied to △ABC W to move it into the same position as △DFE?

Which side is equal to AB?

C 370

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Ex 3

△KLM is reﬂected along the dotted line, and its image △TVU is produced. a

Find the length of each side: i

iii

K 10

14 L 12

Describe the relationship between △TVU and △KLM.

△ABC is translated to the right to form △DEF:

Which side of △DEF corresponds to BC?

Which angle of △DEF corresponds to ∠BCA?

If AC = 14 cm, ﬁnd the length of DF.

If the smallest angle in △DEF is ∠EFD, name the C smallest angle in △ABC.

Given these quadrilaterals are congruent, find the value of x. x 12 12

5 4

Given these quadrilaterals are congruent, ﬁnd the value of z.

12 5 9

Given these quadrilaterals are congruent, ﬁnd the value of y. 2y

9 y+5

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Find the value of x given the shapes are congruent. a

11.6 41°

G F

9.5

139° x

L 5

2x − 1 13

5x − 3

3x + 1

Reasoning

20, 22

20, 21

21, 23

Explain why performing a combination of translations, rotations, and reﬂections will always produce a congruent ﬁgure.

Draw an example of two triangles that have one pair of congruent angles, and two pairs of congruent sides, but are not themselves congruent triangles.

A shape can be reﬂected and visually look exactly the same, true or false? Justify your answer.

A shape and its reﬂection can overlap each other, true or false? Justify your answer.

Problem-solving 24, 25 24

372

24–27

25–28 Q

Consider △PQR with midpoint X at side PR: a

What type of triangle is △PQR?

Are these statements correct? i

∠QXR > ∠QXP

∠QXR = ∠QXP

iii

∠QXR < ∠QXP

Is the line through Q and X a line of symmetry for △PQR?

Are ∠RQX and ∠PQX equal or not equal?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Consider △PQR with angle bisector QX at ∠PQR: a b

Are these statements correct? i

c d

What type of triangle is △PQR? PX > RX

PX = RX

iii

PX < RX

Is the line through Q and X a line of symmetry for △PQR?

Are △QXR and △QXP congruent or not congruent?

The shapes in the diagram are congruent. Find the value of p.

p+2

2( p + 5)

2p − 3

Farah and Kate are standing outside looking up to the top of a building. The angles between their lines of sight and the ground are equal, as shown in the diagram. It takes Farah 20 strides to reach the base of the building, while it takes Kate 18 strides to reach the base of the building. If Kate’s stride length is 76 centimetres, ﬁnd Farah’s stride length.

Consider △JKL with the line JM bisecting (cutting in half) ∠KJL and KL = 6: a

What type of triangle is △JKL?

Are these statements correct? i

KM > ML

KM = ML

J 3

iii

KM < ML

Is the line through J and M a line of symmetry for △JKL?

Are △KJM and △MJL congruent or not congruent?

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8.03

Congruent triangles

After this lesson, you will be able to… • use the four tests to determine congruence and non-congruence between triangles.

Test for triangle congruence Recall that two triangles are congruent if one can be moved by translation, rotation, and/or reflection to lie exactly on top of the other. For example, if we reflected one of these triangles, rotated it, and translated it, we could place it directly on top of the other. We can see that three pairs of corresponding sides are equal, and three pairs of corresponding angles are equal. When deciding whether or not two triangles are congruent, we do not need to know the lengths of all the sides and the measures of all the angles. We need only three pairs of sides or angles in one of the following combinations. Side-side-side congruence (SSS) If two triangles have three equal side lengths, then the triangles must be congruent. Try this yourself with three straight objects — once you put them together, you can rotate, translate, and reflect the triangle to make every other possible combination:

Since each triangle is made from the same three sides, they pass the side-side-side or SSS congruence test.

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Side-angle-side congruence (SAS) The side-angle-side (SAS) rule is a reliable method to determine if two triangles are congruent. This rule requires two triangles to have one pair of corresponding sides that are equal in length, and the angles enclosed by these sides must also be identical. When these conditions are met, the second pair of corresponding sides is guaranteed to be equal, confirming triangle congruence.

In the diagrams, you can see two triangles with these congruence conditions. Notice how each triangle has: • one side marked with a single hash, indicating these are equal in length • another side marked with a double hash, indicating these are equal in length • an angle between them marked with an arc, indicating these are equal in length The congruence of these elements according to the SAS rule ensures that the entire triangles are congruent, meaning every corresponding side and angle in one triangle exactly matches the corresponding side and angle in the other triangle.

Interactive exploration

Discover this concept in action online

mathspace.co

There is no side-side-angle (SSA) congruence. Right angle-hypotenuse-side congruence (RHS) If two right-angled triangles have hypotenuses with equal lengths and another pair of equal sides, then the triangles must be congruent.

Notice that the right angle is not included between the sides — this is an exception to the side-side-angle rule, but we do not call it SSA. Instead, this congruence test is called right angle-hypotenuse-side, or RHS.

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Angle-angle-side (AAS) and Angle-side-angle (ASA) congruence If you know two angles, then by the fact that the sum of angles in a triangle is 180°, then we know all three angles. The AAS and ASA congruence use the fact that the angles are equal, and that we have one pair of matching sides. ASA - is two angles and the included side between them. AAS - is two angles and a non-included side.

In summary, if two triangles do not satisfy these tests, one of two things could be true: • The two triangles are deﬁnitely not congruent, or • We don’t have enough information to know whether or not they’re congruent Two triangles are deﬁnitely not congruent if there is a pair of sides or angles that are in the same relative position but are not equal to each other.

Example 1 Consider the triangles: a Are these triangles congruent? 34° 9

56°

Create a strategy We can use the fact that the sum of the measure of the interior angles of a triangle is 180° to find the other angle of Triangle B. Then, use the appropriate congruence test to check whether they are congruent or not.

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Apply the idea We can find the unlabelled angle of Triangle B by using 180° − (56° + 34°). This means that the unlabelled angled is 90°. This is the same as the angle from Triangle A. Because both triangles are right angle, we can try the RHS congruence for checking. We see that both the hypotenuses measures 9 and both legs measures 5, and so the triangles must be congruent.

b What congruence test does this pair satisfy? A

SSS

SAS

AAS

RHS

Create a strategy Choose the congruence test used from part (a).

Apply the idea We found from part (a) that two triangles are congruent by right angle-hypotenuse-side or RHS. The answer is option D.

Example 2 D

Consider the diagram:

a Are △ABD and △CDB deﬁnitely congruent?

Create a strategy Take note of the markings used for congruence and determine if any of the congruence tests are satisﬁed.

Apply the idea Looking at △ABD and △CDB in the diagram, we have one pair of common sides, which tells that those two sides are equal, BD = DB. Based on the given markings in the diagram, we also have sides CD = AB. The parallel line marks on sides CD and AB also give us a pair of equal angles which are angles ∠ABD and ∠CDB, since they are alternate interior angles on parallel lines. So, we have enough information to deﬁnitely say that △ABD and △CDB are equal. So, △ABD and △CDB are definitely congruent.

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b What congruence test does this pair satisfy? A

SSS

SAS

AAS

RHS

Create a strategy Take note of what the triangles share in common.

Apply the idea △ ABD and △CDB have a common side DB = BD, another pair of equal sides CD = AB, and a pair of equal angles ∠CDB = ∠ABD which are between the equal sides. This gives us side-angle-side congruence or SAS. Option B is the correct answer. c Select the three statements that, when put together, establish congruence for this test. Make sure each reason is correct as well. A

∠ADB = ∠CBD (Alternate angles on parallel lines)

AD = CB (Corresponding sides on parallel lines)

∠ABD = ∠CDB (Corresponding angles on parallel lines)

AB = CD Given

BD is common

∠ABD = ∠CDB (Alternate angles on parallel lines)

Apply the idea We have previously established that: • BD is a common side, • AB = CD from the same markings on the diagram, and • ∠ABD = ∠CDB since they are alternate interior angles on parallel lines. So, the three correct statements are options D, E, and F.

Idea summary These are the four congruence tests that can be used to prove two triangles are congruent: • Side-side-side, or SSS: The triangles have three pairs of equal sides. • Side-angle-side, or SAS: The triangles have two pairs of equal sides with an equal included angle. • Right angle-hypotenuse-side, or RHS: Both triangles have a right angle, equal hypotenuses, and equal sides. • Angle-angle-side, or AAS: The triangles have two pairs of equal angles and one pair of equal sides.

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8.03 Practice questions Explorer

Understanding

Adventurer

1–3

Trailblazer

1–3

If two triangles are congruent, what do we know about their corresponding sides and angles?

Decide whether these conditions can be used to show two triangles are congruent:

Three congruent sides

Three congruent angles

Two congruent sides and a non-included angle

Two congruent legs of a right triangle

Name the triangle congruence test described by each statement. a

Two consecutive angles and a side length are equal.

Corresponding side lengths are equal.

Two pairs of corresponding angles and an included side are equal.

Have right angles and equal length of hypotenuse and another side.

Two corresponding sides and included angles are equal.

Fluency 4

4–10, 13, 14, 18

4, 6, 8, 10, 14, 16, 18

4, 6, 10–12, 14–19

Are the two triangles congruent? If yes, name the congruence test they satisfy. a

R S

4 R

63°

28°

63°

28°

84°

4 R

D 84°

U B

U 4 T

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I G

63°

28°

63°

28°

5 4

B F

Triangle B Triangle A

Triangle C

7 31° 73°

31°

73°

Ex 1

73°

31°

Which two triangles are congruent?

Which congruence test do the triangles satisfy?

Triangle B Triangle A Triangle C 9

9 6 6

Which two triangles are congruent?

Which congruence test is satisﬁed?

Triangle A Triangle B 4

8 4

380

Which two triangles are congruent?

Which congruence test is satisﬁed?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Triangle C

8 9

Triangle A Triangle B 85°

Triangle C 7

31° 7

85° 31°

Which two triangles are congruent?

Which congruence test is satisﬁed?

85° 31°

Triangle A Triangle B

Triangle C

5 27°

27°

Which two triangles are congruent?

Which congruence test is satisﬁed?

Triangle A

Triangle B

Triangle C

5 5

Which two triangles are congruent?

Which congruence test is satisﬁed?

Triangle A

41°

T riangle B

Triangle C Triangle D

41°

4 41°

7 41°

Which two triangles are congruent?

Which congruence test is satisﬁed?

Triangle A Triangle B 81° 42° 7

42° 81° 7

Which two triangles are congruent?

Which congruence tests are satisﬁed?

Triangle C

81°

Triangle D

7 57°

57° 81° 7

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Which congruence test do △ABD and △CDB satisfy?

Which two triangles are congruent?

Which congruence test does this pair satisfy?

8 6 3

E 5

6 D

KM and LN are straight line segments: a

Which two triangles are congruent?

Which congruence test does this pair satisfy?

Which two triangles are congruent?

Which congruence test does this pair satisfy?

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Which congruence test do △EFG and △EHD satisfy?

Name the sides and angles of the triangles that satisfy this congruence test.

Ex 2

hich congruence test do △PQS and △RSQ W satisfy?

Name the sides and angles of the triangles that satisfy this congruence test.

Which congruence test do △ABX and △DCX satisfy?

Name the sides and angles of the triangles that satisfy this congruence test.

R C A

Reasoning

20, 22

21, 22

22, 23

If we can use angle-side-angle (ASA) to show two triangles are congruent, explain why the triangles will also pass the angle-angle-side (AAS) congruence test.

Explain why angle-angle-angle (AAA) is not a valid triangle congruence test. Draw a diagram to support your answer.

Draw two line segments that bisect each other. Draw two new lines that connect the endpoints of the original segments. Will a pair of congruent triangles always be formed? Explain.

If we have △ABC and △PQR, where: • ∠A = ∠P • AB = PQ • BC = QR Is there enough information to know for sure that the two triangles are congruent? Draw a diagram to support your answer.

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Problem-solving 25, 26

24, 25, 27, 28

24, 26, 28 P

△PQR and △PSR are shown in the diagram. a

What additional information is needed so that the triangles are congruent by RHS?

What additional information is needed so that the triangles are congruent by SSS?

What additional information is needed so that the triangles are congruent by SAS? Q

△JKN and △MLN are shown in the diagram.

What additional information is needed so that the triangles are congruent by SSS?

What additional information is needed so that the triangles are congruent by SAS?

What additional information is needed so that the triangles are congruent by AAS?

Consider △PQX and △RSX where X is the centre of the circle. a

Explain how we know sides QX, XP, SX, and XR all have the same length.

What additional information is needed so that the triangles are congruent by SSS?

What additional information is needed so that the triangles are congruent by SAS?

S P

X is the centre of both circles. a

Which congruence test do triangles △ABX and △DCX satisfy?

Name the sides and angles of the triangles that satisfy this congruence test. Justify your reasoning.

X C

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For each diagram: i

Name the congruence test that the two triangles satisfy.

Solve for x.

35 − 6x

6x°

4x + 5

(5x − 7)°

(3x + 15)°

(8x − 18)°

d (3x + 5)°

(7x − 23)°

8.04 Transformations and similarity After this lesson, you will be able to… • define and explain the concept of dilations and how it results in similar shapes. • calculate the scale factor given a pair of similar shapes. • determine whether a given pair of shapes are similar by checking the ratio of corresponding sides and equality of corresponding angles.

Dilations A dilation will result in a shape that is enlarged or reduced, so all the side lengths will increase or decrease in the same ratio. The scale factor is the value by which we increase or decrease the side lengths of the shape. A scale factor of 3 means the new shape will have side lengths 3 times as big, whereas a scale factor of

means the new shape will be

as big as the original.

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Here, the smaller triangle is dilated by a scale factor of 2 to produce the larger triangle.

If any of these ratios were not equal to 2, then this would not be a dilation.

In general, • If the scale factor is greater than 1, the dilated shape will be larger than the original. • If the scale factor is greater than 0 but less than 1, the dilated shape will be smaller than the original.

Interactive exploration

Discover this concept in action online

mathspace.co

Example 1 Determine whether the given diagrams show a dilation. If yes, ﬁnd the scale factor.

4 5

15 3

Create a strategy To check if the diagrams represent a dilation, we need to check if the ratios of the corresponding sides of the two triangles are equal. If the ratios are equal, then the diagram shows a dilation.

Apply the idea We can then calculate the scale factor by ﬁnding the ratio of a side of the larger triangle to the corresponding side of the smaller triangle. The ratio of the sides of the larger triangle to the corresponding sides of the smaller triangle is: 12 9 15 = = = 3 4 3 5

So, the scale factor is 3.

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Idea summary A dilation will result in a shape that is enlarged or reduced, so all the side lengths will increase or decrease in the same ratio. A scale factor is the value by which we increase or decrease the side lengths of the shape. • A scale factor greater than 1 is an enlargement. • A scale factor between 0 and 1 is a reduction.

Similarity While reﬂections, rotations and translations result in a shape congruent to the original shape, dilations will result in a shape that is similar to the original shape. Compare these triangles. What do you notice about the sides? What about the angles?

60°

60° 2

You might have noticed that both triangles have the same angle measures, but they have different side lengths. Speciﬁcally, the smaller triangle has been dilated by a scale factor of 3. All of the corresponding sides between the two shapes are in the same ratio, or proportional, and all corresponding angles between the two shapes are equal. This means the two triangles are similar. Can we say that these two quadrilaterals are similar? 7

7 4

4 7

All corresponding angles between the two shapes are equal, but the corresponding sides are not proportional. If we look at the proportions for two pairs of sides, we have 7 and 8 . This is not 4 7 equal.

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The same scale factor has not been applied to each side of the smaller rectangle to create the larger one. Therefore, the two quadrilaterals are not similar. To prove that two figures are similar, one must be able to be transformed to the other using only rotations, translations, reflections, and dilations.

Example 2 B

Are these triangles similar?

24 26

A E 10

12 F 5

Triangle DEF is the image of triangle ABC.

Create a strategy Determine if the same scale factor has been applied to corresponding sides.

Apply the idea The hypotenuse is easy to identify in both triangles. Find the scale factor here: 26 =2 13 Test the scale factor on the other sides. AB = 24, so we need a side that has length of 12. AC = 10, so we need a corresponding side that has length of 5. AB 24 = = 2 12 DE AC 10 = 2 = DF 5 This confirms △ ABC is similar to △ DEF.

Example 3 Are these triangles similar? 42

3 8 388

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Create a strategy Identify whether or not all the corresponding sides are in the same ratio.

Apply the idea The first triangle has side lengths 48, 42 and 18. We've written the lengths in descending order. Let's do the same for the other triangle: 8, 7, 3 If these triangles have a constant scale factor, then we can compare the scale factors for corresponding side lengths.

48 = 6 8 42 = 6 7 18 = 6 3 So, there is a common scale factor of 6, and the triangles are similar.

Idea summary Two shapes are similar when: • All corresponding sides are in the same ratio. • All corresponding angles are equal.

8.04 Practice questions Understanding

Explorer 1–4

Adventurer 1–4

Trailblazer 2–5

What characteristics are required for two shapes to be similar?

What is the difference between congruent shapes and similar shapes?

Determine whether each statement is true or false. a

Any dilation with a scale factor less than or equal to 1 is a reduction.

Any dilation with a scale factor greater than 1 is an enlargement.

After dilating a ﬁgure, all corresponding angles will be congruent.

All rectangles are similar.

All squares are similar.

All equilateral triangles are similar.

All isosceles triangles are similar.

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Identify each transformation as a congruence transformation or a similarity transformation. a

A translation of 3 units left and 8 units down.

A reﬂection across the y-axis.

A dilation by a factor of 0.6.

A rotation of 90° anticlockwise.

Provide an example of a transformation that satisﬁes each statement: a

After the transformation, the side lengths of the transformed image are equal to the side lengths of the original object.

After the transformation, the angle measures of the transformed image are equal to the angle measures of the original object.

After the transformation, the side lengths and angle measures of the transformed image are equal to the side lengths and angle measures of the original object. 6ac, 7, 8, 9bc, 10, 11, 12ab, 13, 14, 16, 17, 19

Fluency

6ac, 7, 8, 9bc, 11, 12ac, 15, 16, 18, 20

6bd, 7, 8, 9ad, 11, 12cd, 15, 16, 18, 20

Draw a labelled diagram of the figure after it has been dilated by the given factor: a

Scale factor of 3

Scale factor of 4

Scale factor of

8 14

8 6

Ex 1

Determine whether the diagram shows a dilation. If yes, ﬁnd the scale factor of the smaller shape to the larger one. a

5 15

4 5

390

6 18

Mathspace Victorian Curriculum – Year 8 mathspace.co

3 6

3 10

For each diagram: i

Are the shapes congruent or similar?

Name the transformation that takes ﬁgure A to ﬁgure B.

A B

B A

A B

Ex 2

For each pair of ﬁgures: i

Identify the scale factor.

List the transformations that would take the original ﬁgure to the image.

A 4 B B′ 2

A′

B′

3 6

C′

D′ A

2 4

B C′

3 C

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B 3 C

B 24

3 9

D′

A′

A B′ 10

12 C′

C′

B′

A′

In the diagram, △ECD is similar to △LMN. a

Match the corresponding angles by ﬁlling in the blanks: ∠D corresponds to ∠⬚ ∠E corresponds to ∠⬚

∠C corresponds to ∠⬚ b

What must be true about these pairs of angles?

Which side in △LMN corresponds to CD?

Which side in △LMN corresponds to CE?

C N

D Ex 3

Determine whether each pair of triangles is similar. If they are, find the scale factor of the larger shape to the smaller one. a

3 2

2 4 9

Find the scale factor for each pair of similar ﬁgures: a

A B

b B

C′

Mathspace Victorian Curriculum – Year 8 mathspace.co

C′ 12

B′ 4

392

10 A′

D′

4 A

B′

12 A′

D′

B′ 25

B′ B

A′

C′ C

A′

C′

C 5

Consider the two similar triangles:

DE corresponds to which side in △LMN ?

DC corresponds to which side in △LMN ?

M C

N L

Consider the two similar triangles: a

What is the corresponding angle in △RPQ for: i

∠D

∠B

CD corresponds to which side in △RPQ?

CB corresponds to which side in △RPQ?

iii

∠C Q

B C

The smaller quadrilateral has been reﬂected, then enlarged and ﬁnally rotated to form the larger quadrilateral: a

Are the two shapes congruent, similar, or neither?

Which side in the larger shape is corresponding to AB?

Which angle in the larger shape is corresponding to ∠ADC?

E F

The two given triangles are similar: Find the missing length y.

10 cm 4 cm

y cm 7 cm

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For each pair of similar triangles determine the value of x: a

b 26°

8 cm

30° 6 cm

x°

12 cm x° 14

9 cm

The two triangles are similar: a

Comparing the side lengths of the two triangles, ﬁnd the ratio of corresponding sides.

Find the value of each variable: i

iii

14° 5

m° n° 4

122°

8 44°

s° 14

The two given trapeziums are similar: a

What is the enlargement factor?

Calculate the size of x.

5 127°

x 127° 44

In each diagram the two quadrilaterals are similar. a

If a = 6 cm, d = 11 cm and g = 5 cm, ﬁnd the value of k.

If r = 9 m, s = 17 m and c = 5 m, ﬁnd the value of d.

Reasoning

21, 22

21, 23

Mathspace Victorian Curriculum – Year 8 mathspace.co

s q

81 20 79°

394

22–24

Determine whether the two given shapes are similar. Explain your answer.

a b

78°

Explain how you know these pairs of ﬁgures are similar: a

b 1.15

60°

2.3 30° 2

60° 2.30

4.6 30° 4

If two shapes are congruent, are they also similar? Explain your answer.

If two shapes are similar, are they also congruent? Explain your answer.

Problem-solving 25, 26

25, 26

25–27

If a square with a perimeter of 20 cm is dilated by a factor of 0.6, ﬁnd the side length of the dilated square.

If a square with an area of 25 m2 is dilated by a factor of 0.4, ﬁnd the side length of the dilated square.

A video phone call includes a full-screen image of the person being called and a smaller image of the caller in the corner of the screen (see image for reference). Consider making a video call on the smartphone which is 1782 pixels by 828 pixels in dimension. Suppose the smaller image is

th the size of

the entire screen. What would be the dimensions of the smaller image in pixels?

8.05

Similar triangles

After this lesson, you will be able to… • find the missing sides of a similar triangle given the scale factor and the length of one side. • compare the side lengths and angle measures of two triangles to determine if they are similar.

Tests for similarity Recall two shapes are similar if all their corresponding angles are congruent and their corresponding sides are proportional. We do not need to check all angles and sides in order to tell if two triangles are similar. 8.05 Similar triangles mathspace.co

395

Side-side-side similarity (SSS) If two triangles have three pairs of corresponding sides that are in the same ratio, the triangles must be similar.

8 9

We can show △ABC was dilated by a scale factor of 3, which proves the triangles are similar.

18 24

D 27

Since all three pairs of corresponding sides are proportional, the triangles pass the side-sideside or SSS similarity test.

Angle-angle-angle similarity (AAA) If two triangles have three pairs of equal angles, then the triangles are similar.

Since all three pairs of angles are equal, the triangles pass the angle-angle-angle or AAA similarity test.

C N

Dilating by a scale factor does not change the angles in a shape. This means we can show the triangles are similar using congruency and similarity transformations.

Side-angle-side similarity (SAS) If two triangles have a pair of matching sides in the same ratio and the angles between them are equal, then the triangles must be similar. A

47º 6 R B 4 47º P

Using our understanding of dilations, we can use logical reasoning to understand why this works. • If two sides were dilated by the same scale factor, the angles opposite those sides were unchanged. • Since the included angle is the third angle in both triangles and the measures are equal, we know that all pairs of matching angles are equal. • This means all the angles are equal, and all the sides are in the same ratio.

Since two pairs of matching sides are proportional and the included angles are congruent, the triangles pass the side-angle-side or SAS similarity test.

Interactive exploration

Discover this concept in action online

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mathspace.co

Right angle-hypotenuse-side similarity (RHS) If two right-angled triangles have hypotenuses in the same ratio and another pair of matching sides in the same ratio, then the triangles must be similar.

X A

We can use Pythagoras’ theorem to ﬁnd the third side and show that it is in the same ratio as the other sides.

Notice that the right angle is not included between the sides. If the lengths of both legs were known, we could have used SAS, but this is a special case. This similarity test is called right angle-hypotenuse-side, or RHS.

Example 1 Consider the triangles: a Which of these triangles are similar? A

B 54°

52°

54° 24

74°

68°

38°

74°

52° 74°

Create a strategy Recall that triangles are similar when: • All corresponding sides are in the same ratio. • All corresponding angles are equal. • Two pairs of sides are in the same ratio and the included angles are equal. • Triangles are right-angled and the hypotenuses and another pair of sides are in the same ratio.

Apply the idea The corresponding angles of the triangles in options A and C are equal. The answers are options A and C. b Which of the following conditions satisﬁed the answer from part (a)? A

All corresponding angles are equal.

Two angles are equal, and one side is a multiple of the corresponding side of the other.

All corresponding sides are in the same ratio.

Apply the idea The answer is option A: All corresponding angles are equal.

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Example 2 These two triangles are similar.

6 R

30 8

C 18

Q A

a Find the value of m.

Create a strategy Use the matching angles indicated on the triangles and ﬁnd the scale factor between the two triangles.

Apply the idea

The matching sides are:

• PR and AC • QR and BC • PQ and AB

6 R

Using the matching sides PR and AC, the scale factor is 18 = 3.

Since m is a length in the smaller triangle, we can ﬁnd its value by dividing the length of the matching length by the scale factor. Divide 30 by 3 = 10

Evaluate

b Find the value of n.

Create a strategy Since n is a length in the larger triangle, we can ﬁnd its value by multiplying the length of its corresponding side by the scale factor.

Apply the idea n = 8 × 3

Multiply 8 by 3

= 24

Evaluate

Example 3 A 6 D

E 10

B 398

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a Determine if the △ABC and △ADE are similar.

Create a strategy These are the four similarity tests for triangles: • AAA: Three pairs of equal angles • SSS: Three pairs of sides in the same ratio • SAS: Two pairs of sides in the same ratio and an equal included angle • RHS: Both have a right angle, and the hypotenuses and another pair of sides are in the same ratio.

Apply the idea ∠ADE = ∠ABC due to corresponding angles (F rule) ∠AED = ∠ACB due to corresponding angles (F rule) ∠A common So, △ABC and △ADE are similar due to the AAA, three pairs of equal angles.

b Given that DE = 7, what will the length of BC be?

Create a strategy

Apply the idea

Since the triangles are similar, all pairs of matching sides will be in the same ratio.

AC = BC AE DE

Write the equation

This means that:

16 = BC 6 7

Substitute the values

BC DE

AC AE

16 × 7 = BC 6 BC = 56 3

= 18.67

Multiply both sides by 7 Evaluate Rounded to two decimal places

Idea summary These are the four similarity tests that can be used to prove similarity in triangles: • Angle-angle-angle, or AAA: The triangles have three corresponding congruent angles. • Side-side-side, or SSS: The triangles have all corresponding sides in the same ratio. • Side-angle-side, or SAS: The triangles have two pairs of corresponding sides in the same ratio and the included angle is congruent. • Right angle-hypotenuse-side, or RHS: Both triangles are right-angled and the hypotenuses and another pair of sides are in the same ratio.

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8.05 Practice questions Explorer

Understanding 1

Trailblazer 2, 3

1–3

Determine whether each statement is true or false: a

If two triangles are similar, then their corresponding sides are equal.

If two triangles are similar, then their corresponding angles are equal.

Decide whether the two triangles are similar: a

Two sides of one triangle are proportional to two corresponding sides of another triangle.

Three pairs of corresponding sides are proportional.

Three corresponding angles are equal.

Two pairs of corresponding sides are proportional and a pair of corresponding, non-included angles are equal.

Two pairs of corresponding legs of two right-angled triangles are proportional.

Name the triangle similarity test described by each statement. a

Two pairs of consecutive angles are equal and a pair of corresponding side lengths are proportional.

All pairs of corresponding side lengths are equal.

Two pairs of corresponding angles are equal and the pair of included sides are proportional.

Have right angles and the lengths of the hypotenuse and another side are proportional.

Two pairs of corresponding sides are proportional and the pair of included angles is equal.

Fluency Ex 1

Adventurer

1, 3

4a–d, 5a–c, 6–8, 10, 11

4b–e, 5, 7–9, 11

4d–g, 5, 7–9, 11, 12

For each set of triangles: i

Identify which two triangles are similar.

Name the similarity test they satisfy.

B 13

C 22

20 11

B 12

C 9 6

400

Mathspace Victorian Curriculum – Year 8 mathspace.co

7 10

27º

41º

D 34º

15 56º

57º 81º

43º 8 80º

81º

81º 56º

41º

44º

C 12

A 10 56º

21 27º

41º

42º

For each pair of triangles: i

Identify the similarity test that demonstrates similarity between the pairs of triangles.

Show all reasoning.

49 M

9 76°

14 L

U G

21 76° F

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401

143

85°

85° B

121

S 56 D

Which similarity test do these triangles satisfy?

Find the enlargement factor.

2 2

Ex 2

For each of the similar triangles, ﬁnd the value of the missing side lengths. a

f E

D 28

16 C

L x

42 M

30 g

G 33

44 H N

18 L

12 M

402

30 24

Mathspace Victorian Curriculum – Year 8 mathspace.co

Identify two similar triangles in the diagram.

Which similarity test do the triangles satisfy?

If KL = 20, what is the length of MN?

Suppose we are given the following information:

• AB = 2, BC = 3 • DE = 4, EF = 6 A

What additional information is needed to show △ABC is similar to △DEF by SAS similarity?

Ex 3

In the diagram, segment AB is parallel to segment DC. a

Write down the angle that is equal to ∠EAB. Give a reason as to why they are equal.

Write down the angle that is equal to ∠ECD. Give a reason as to why they are equal.

Which similarity test do these triangles satisfy?

In the diagram, segment JK is parallel to segment MN. a

Write down the angle that is equal to ∠KLJ. Give a reason as to why they are equal.

Write down the angle that is equal to ∠LJK. Give a reason as to why they are equal.

Which similarity test do these triangles satisfy?

K J

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I f sides AC and BA are proportional to DF and ED respectively, which similarity test would these triangles satisfy?

If sides AC and CB are proportional to DF and FE respectively, which similarity test would these triangles satisfy?

If angle CBA and angle EDF are complementary, which similarity test would these triangles satisfy?

Reasoning

13, 14

A C

13, 14

14, 15

Are these pairs of triangles similar? If yes, name the similarity test they satisfy. If not, explain why not. a

78 cm

38°

54°

63 cm

48 cm

36.9° 8

25 53.1°

39 cm

33 cm

57° 96

Y 42

85° E 24 cm

D G R 21 B

48 P

Consider the three triangles: A

A′

1.5 E

2 C

404

C′

1.5

B′

Describe a similarity transformation from △ABC to △A′B′C′.

Is △A′B′C′ congruent to △DEF ? Justify your answer.

Justify △ABC is similar to △DEF using transformations.

Mathspace Victorian Curriculum – Year 8 mathspace.co

2 D

Determine whether each statement is enough to prove △ABD is similar to △CBE. Justify your answer. A

Side DA is parallel to side EC.

AB is proportional to BC and AD is proportional to CE.

∠BCE and ∠BEC are complementary.

AB is proportional to CB and BD is proportional to EB.

Problem-solving 17, 19–21

B D

16, 17, 19, 20, 22

16, 18, 19, 21, 23 V

In the diagram, ∠S is equal to ∠V.

5x + 11

Find the value of x that makes △V UW similar to △SUT. U

18 U

△DCE is similar to △BCA. Solve for x.

12 − x C

x+7

4 A

△ABE is similar to △ACD. a

Solve for x.

Solve for AC.

A 10

6.25 E x−1

B x+2

△ABE is similar to △DCE. a

Solve for AE.

Solve for ED.

C A

x−1

2 B

x+5 D

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A large tree casts a shadow of 30 m. At the same time, a 2 m high stick, standing vertically, casts a shadow of 3 m. a

Why are △ABE and △DCE similar?

If the height of the tree is h metres, ﬁnd h.

h Sun Ray C 2m A

30 m

Ms. Shin is 167 cm tall. She stands near a ﬂagpole that casts a 6.7 m long shadow. If Ms. Shin’s shadow is 213 cm long, determine how tall the ﬂagpole is in metres. Round your answer to two decimal places.

A horizontal brace lies between the slanted sides of an A-shaped ladder. In this position, the top of the ladder is 220 cm from the ground. The brace is 44 cm in length and sits 71 cm from the top of the ladder. How far apart are the feet of the ladder? Round your answer to the nearest whole centimetre.

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Peter constructs a ramp in the shape of a rightangled triangle. To give the ramp more strength, he wants to place a brace from the base point across to the hypotenuse. The brace should meet the hypotenuse at a right angle, as shown in the diagram: a

Identify a triangle similar to △ABC.

How long does the brace, BD, need to be?

Find the length x.

Mathspace Victorian Curriculum – Year 8 mathspace.co

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INVESTIGATION

Determine congruence or similarity Investigate algorithms that identify congruency or similarity between shapes.

Objectives • To design, create, and test ﬂow charts that classify sets of shapes as congruent or similar

Materials • •

Pen or pencil Graph paper

•

Flow chart (Download from digital textbook)

Shapes that are congruent or similar Algorithm An algorithm may be deﬁned as: A process or set of rules to be followed in calculations or other problem-solving operations. Algorithms can exist outside of mathematical contexts. For example, the instructions for assembling a piece of furniture are an algorithm as it provides a step-by-step procedure that, when followed accurately, results in a fully assembled, functional item. We will create an algorithm that classifies two shapes as similar, congruent, both or neither. To do this, we need to plan the types of decisions we need to make when considering whether two shapes or similar or congruent. Flow Chart Our algorithm is going to be in the form of a flow chart. When creating the flow chart, we need to ensure it works for all shapes. Considering different cases can help identify mistakes and improve the algorithm.

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Some questions we need to consider are: What conditions must be satisﬁed for two shapes to be congruent? What conditions must be satisﬁed for two shapes to be similar? If we do not have enough information to determine if the shapes are congruent or similar, are there other questions we can ask that will help us determine more information?

Flow chart symbols The rounded rectangle is used for start and end points. Diamonds are used when a decision needs to be made. Parallelograms are used for input. Arrows indicate the direction of the diagram.

Analyse the ﬂow chart, and use it to answer the questions that follow. Start

Two shapes

All corresponding angles are equal

Neither similar nor congruent

Yes Similar

408

Use the ﬂow chart above to classify the quadrilaterals shown as congruent, similar, both, or neither. M

Did the algorithm accurately classify the quadrilaterals? Explain your reasoning. How can the ﬂow chart be improved?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Use the ﬂow chart above to classify the triangles shown as congruent, similar, both, or neither.

28°

63° H

N 63°

28°

Did the algorithm accurately classify the triangles? Explain your reasoning. How can the ﬂow chart be improved?

Investigate Work with a partner to complete the steps and answer these questions. 1.

With a partner, use each other’s findings from the activity to draw a flow chart that accurately classifies two shapes as congruent, similar, or neither.

How did you and your partner ensure that your algorithm will check every possible condition for any input?

If a shape does not have all angles listed, how might you identify the missing angles, if possible?

If a shape doesn’t have all side lengths listed, how could you determine the missing side lengths, if possible?

Suppose we cannot ﬁnd all the missing information (side lengths or angles). Can we still determine if two shapes are similar or congruent in such cases? Under what circumstances could this be possible?

Triangles that are congruent or similar

Recall that we do not need to know all sides and angles when determining whether two triangles are congruent or similar. Instead, we can use congruence or similarity tests to classify them.

Work with a partner List the triangle congruency and similarity tests. Create an algorithm in the form of a ﬂow chart for classifying two triangles as congruent, similar, or neither. Use the downloadable asset as a guide if you need assistance at any point. Use the Interactive investigation to consider various cases. Then, test your completed flow charts validity.

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Interactive investigation Discover this concept in action online

mathspace.co

Investigate Answer these questions once you have completed the procedure. 1.

How did you and your partner decide to start your ﬂow chart? What factors affected this decision?

What was the most challenging part of creating an algorithm for determining congruence or similarity between two triangles?

How did the testing phase impact your understanding of the eﬃciency and effectiveness of your ﬂow chart?

Discussion

410

Why is it important to consider different cases when creating an algorithm?

How were algorithms created by your classmates similar to or different from yours?

Algorithms are used in many real-world applications. Can you think of any professions that might use similar algorithms to the one you built today?

Mathspace Victorian Curriculum – Year 8 mathspace.co

8.06 Properties of quadrilaterals After this lesson, you will be able to… • identify and construct special quadrilaterals using congruent triangles. • relate between the properties of special quadrilaterals and their diagonals.

Construct quadrilaterals from triangles Recall the following quadrilaterals and the properties used to classify them:

Trapezium One pair of parallel sides

Many of these features can be explained by constructing the quadrilateral from a pair of congruent triangles.

Parallelogram Two pairs of parallel sides

Rhombus All sides equal

Kite Two pairs of adjacent equal sides

(Opposite sides are equal) Rectangle All angles right

Square

Interactive exploration

Discover this concept in action online

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If we want to make a quadrilateral from two congruent triangles, we need to connect a pair of corresponding sides of the triangles together. This makes sure that the points match up exactly. Another way to think about this construction is to start with one triangle and make a copy of the triangle with a rotation or a reflection. If we use a reflection to make the second triangle, the quadrilateral we form always has two pairs of equal adjacent sides. This means a reflection always makes a kite.

If we use a rotation to make the second triangle, the quadrilateral we form always has two pairs of equal opposite

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sides. This means rotation always makes a parallelogram. If we start with an isosceles triangle, then reﬂecting it across its base produces the same quadrilateral as rotating it around the midpoint of its base. The result is always a rhombus, which explains why rhombuses are both parallelograms and kites at the same time.

Example 1 Suppose we have a right-angled scalene triangle, then create another triangle by rotating the original around the middle of its longest side. If we join the triangles together, which of the following must be true of the resulting quadrilateral? A It will always be a parallelogram but may not be a rectangle. B It will always be a rectangle but may not be square. C It will always be a kite but may not be a parallelogram. D It will always be a square.

Apply the idea The kind of quadrilateral that you can make when you rotate the triangle is a parallelogram but may not be a square. So, option B is the correct answer.

Idea summary By transforming triangles, we can make different types of quadrilaterals. A reﬂection of a triangle always makes a kite. A rotation of a triangle always makes a parallelogram.

Diagonals of parallelograms When we make special quadrilaterals by joining two congruent triangles together, the sides of the triangles that were joined together form one of the diagonals of the shape. If we draw in the other diagonal of a quadrilateral, we will make four triangles in total. At least two of these triangles will be congruent to each other. We can use the congruent triangles to investigate some interesting properties of the diagonals. Let’s investigate parallelograms.

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If we rotate the triangle △DAB around the midpoint X of the side DB, this side becomes a diagonal of the parallelogram we create.

Drawing in the other diagonal AC splits the parallelogram into four triangles. We can use the AAS congruence test to show △AXB ≡ △CXD.

Since AX and CX are corresponding sides in these congruent triangles, they must be equal. Since XB and XD are corresponding sides in these congruent triangles, they must be equal too. In other words, X is the midpoint of the diagonal BD and the midpoint of the diagonal AC.

This means the diagonals of a parallelogram bisect each other. Now, let’s look at a rectangle. D

One pair of congruent triangles is △DBA and △CAB.

When we draw in the diagonals, we create a few pairs of congruent triangles.

We can use the SAS congruence test to show that △DBA ≡ △CAB. B

The sides DB and CA are corresponding sides and must be equal. Notice these are also the diagonals of the rectangle. This means the diagonals of a rectangle are equal in length.

Example 2 The triangles △ABC and △CDA are congruent.

a Fill in the blanks, to state pairs of equal angles. ∠CAB = ∠⬚ ∠BCA = ∠⬚

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Create a strategy Remember that corresponding angles in congruent triangles are equal.

Apply the idea The corresponding angle of ∠CAB in △CDA is ∠ACD. The corresponding angle of ∠BCA in △CDA is ∠DAC. So we have the pairs of equal angles: ∠CAB = ∠ACD ∠BCA = ∠DAC

b Which two statements are true about the quadrilateral ABCD? A

AD ∥ BC because ∠ACD = ∠CAB

AD ∥ BC because ∠DAC = ∠BCA

AB ∥ DC because ∠ACD = ∠CAB

AB ∥ DC because ∠DAC = ∠BCA

Create a strategy Consider AC as a transversal and identify the alternate angles.

Apply the idea Let’s look at AC as a transversal between AD and BC:

The alternate angles on this transversal are ∠DAC and ∠BCA which we know are equal from part (a). So: AD ∥ BC because ∠DAC = ∠BCA.

Now looking at AC as a transversal between AB and DC, then the alternate angles on this transversal are ∠ACD and ∠CAB which we know are equal from part (a). So: AB ∥ DC because ∠ACD = ∠CAB. Options B and C are the correct answers.

c Which statement is true of the quadrilateral ABCD?

414

ABCD must be a rhombus but is not necessarily a square.

ABCD must be a parallelogram but is not necessarily a rhombus.

ABCD must be a square.

ABCD must be a rectangle but is not necessarily a parallelogram.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Create a strategy Use the diagram as a guide to help us ﬁnd the name of the quadrilateral with features matching ABCD. Trapezium One pair of parallel sides

Parallelogram Two pairs of parallel sides

Rhombus All sides equal

Kite Two pairs of adjacent equal sides

(Opposite sides are equal) Rectangle All angles right

Square

Apply the idea We know from parts (a) and (b) that the quadrilateral ABCD has two pairs of opposite parallel sides and the opposite sides are equal. The type of quadrilateral that is deﬁned by this feature is a parallelogram. So, ABCD must be a parallelogram but is not necessarily a rhombus. So, option B is the correct answer.

d Suppose that ∠ACD = 36° and ∠BCA = 44°. Find the size of ∠ABC.

Create a strategy Remember that co-interior angles on parallel lines add to 180°.

Apply the idea Since AB and DC are parallel, the angles ∠ACD, ∠BCA and ∠ABC are co-interior angles. So we have: ∠ACD + ∠BCA + ∠ABC = 180 36 + 44 + ∠ABC = 180 80 + ∠ABC = 180 80 + ∠ABC – 80 = 180 – 80 ∠ABC = 100°

Co-interior angles on parallel lines Substitute the values Add like terms Subtract 80 from both sides Evaluate

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Idea summary We can use congruent triangles to prove properties of quadrilaterals. The diagonals of a parallelogram bisect each other. The diagonals of a rectangle are equal in length.

Diagonals of kites Let’s have a look at the diagonals of a kite. We could make this kite by reﬂecting △DAB across its side DB, and this side of the triangle becomes one of the diagonals of the kite we make.

D A

X C

Drawing in the other diagonal, AC, splits the kite into four triangles. We can call the point where the diagonals cross X. Using the SAS congruence test, we can show △ABX ≡ △CBX.

Since these triangles are congruent, we also know that ∠BXA ≡ ∠BXC. These angles are supplementary, so we know that ∠BXA + ∠BXC = 180°. Since ∠BXA and ∠BXC are equal and sum to 180°, they must both be right angles. This means the diagonals of a kite are perpendicular to each other. There are other properties we could prove as well — at least one diagonal of a kite bisects the other, some of the quadrilaterals have diagonals that bisect the angles they pass through, and so on. All of these properties can also be proved using the congruent triangles that are created from drawing the diagonals.

Example 3 R

The diagonals of this kite intersect at T, splitting the kite into four triangles. The diagonal PR bisects the angles ∠QRS and ∠SPQ.

T Q

P 416

Mathspace Victorian Curriculum – Year 8 mathspace.co

a Which triangle is congruent to △RST ?

Create a strategy

Apply the idea

Look for the triangle that has the same markings Based on the markings given in the diagram we and common sides. have RS = RQ and ∠SRT = ∠QRT. The two triangles also share a common side, RT. So, △RST ≡ △RQT by SAS.

b Which statement is true? A

∠RTS = ∠TQR

∠RTS = ∠TRQ

Create a strategy

∠RTS = ∠RTQ

Apply the idea

Find the angle in △RQT that corresponds to the The angle in △RQT that corresponds to angle ∠RTS in △RST. the angle ∠RTS in △RST is ∠RTQ. So ∠RTS = ∠RTQ. Option C is the correct answer.

c Which statement is true? A

∠RTS + ∠RTQ = 180°

∠RTS + ∠RTQ = 90°

∠RTS + ∠RTQ = 360°

Create a strategy

Apply the idea

Identify the kind of angle they form together.

Looking at the two angles, ∠RTS and ∠RTQ, in the diagram we can see that together they form a straight angle. A straight angle measures 180° so we have: ∠RTS + ∠RTQ = 180°. So, option A is the correct answer.

d Which two statements are true about the diagonals of this kite? A

RP bisects SQ.

RP is perpendicular to SQ.

RS is perpendicular to RQ.

SQ bisects RP.

Create a strategy Use the ﬁndings from the previous parts.

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Apply the idea We know from parts (b) and (c) that ∠RTS and ∠RTQ are equal and add to 180°. So we have: Replace ∠RTQ with ∠RTS Simplify Divide both sides by 2 Evaluate So, we can say that RP is perpendicular to SQ. We know in part (a) that △RST ≡ △RQT so ST and QT are corresponding sides. So ST = QT. We can say that RP bisects SQ. So, options A and B are the correct answers.

Idea summary We can use congruent triangles to prove properties of quadrilaterals. The diagonals of a kite are perpendicular to each other. At least one diagonal of a kite bisects the other. This table summarises the properties of the diagonals of special quadrilaterals: Kite Diagonals are perpendicular One diagonal bisects the other Both diagonals bisect each other One diagonal bisects the angles it passes through Both diagonals bisect the angles they pass through Diagonals are equal in length

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Parallelogram

Rectangle

✓ ✓

Square

✓

Mathspace Victorian Curriculum – Year 8 mathspace.co

Rhombus

✓

8.06 Practice questions Understanding

Adventurer 2, 3a–c, 4

Trailblazer 1, 2, 3c–e, 4

Complete the statements by determining the rigid transformation of the triangles to form quadrilaterals. A

Reflection

⬚ through a point passing through a side of the triangle.

b 2

Explorer 2, 3a–c, 4

Translation

Rotation

⬚ across a side of the triangle.

Name the quadrilaterals that always have: a

Diagonals that are equal in length.

Diagonals that are perpendicular.

At least one diagonal that bisects the angles it passes through.

Both diagonals that bisect the angles they pass through.

Diagonals that bisect each other.

Identify the type of quadrilaterals that have diagonals with these properties: • Their diagonals are equal in length. a b • Their diagonals are equal in length. • Their diagonals bisect each other. • Their diagonals bisect each other. • Their diagonals are perpendicular. • Their diagonals are not perpendicular. c

• Their diagonals are not equal in length. d • Their diagonals bisect each other. • Their diagonals are not perpendicular.

• Their diagonals are not equal in length. • One diagonal bisects the other. • Their diagonals are perpendicular.

• Their diagonals are not equal in length. • Their diagonals bisect each other. • Their diagonals are perpendicular.

Use the appropriate geometrical symbols to represent each of the following statements: a

Parallelogram PQRS where, i

opposite sides are parallel,

opposite sides are equal,

iii

opposite angles are equal. Q

Trapezium WXY Z where, i

one pair of opposite sides is parallel,

other pair of opposite sides is not parallel

R W

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Fluency 5

Construct another triangle by reﬂecting the original across its base.

If we join the triangles together, what type of quadrilateral will be formed?

10 8.4

Construct another triangle by rotating the original by 180° around the middle of its base.

If we join the triangles together, what type of quadrilateral will be formed?

10 10

Consider the obtuse scalene triangle: Construct another triangle by reﬂecting the original across its shortest side.

18.6 11.3

If we join the triangles together, what type of quadrilateral will be formed?

8.8

Consider the acute scalene triangle:

7–10, 12–17

Consider the isosceles triangle:

Ex 1

6–8, 10–14, 17, 18

Consider the isosceles triangle:

5–7, 9–12, 14, 17, 18

Construct another triangle by rotating the original by 180° around the middle of its longest side.

If we join the triangles together, what type of quadrilateral will be formed?

Consider the right-angled scalene triangle:

Create another by rotating the original by 180° around the middle of its longest side.

If we join the triangles together, what type of quadrilateral will be formed?

If the length of PX is 6, ﬁnd the length of PR.

If the length of QS is 16, ﬁnd the length of QX.

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If the length of XT is 6, ﬁnd the length of YT.

If the length of BT is 12, ﬁnd the length of AC.

D T C

If the size of the highlighted angle ∠KLN is 53°, ﬁnd the size of ∠KLM.

If the size of ∠SPQ is 78°, ﬁnd the size of the highlighted angle ∠SPT.

R S

T Q

Given quadrilateral ABCD is a parallelogram, with ∠ADC = 134°, ∠BAC = 22°, and ∠CBD = 52°, ﬁnd the size of: a

∠BCD

∠BDC

∠ACD

∠ABD

D M

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Given quadrilateral ABCD is a rhombus, with ∠ADC = 116°, and BD = 10, ﬁnd the size of: a

∠DBA

∠BCD

∠BMC

Given quadrilateral ABCD is a square, with AC = 14, ﬁnd the size of: a

∠BCD

∠ABD

Given quadrilateral ABCD is a rectangle, with ∠ABD = 58°, and AC = 20, ﬁnd the size of: a

∠CBD

∠BCD

C B

A M

Reasoning

Ex 2

19, 20

20, 21

△RPQ and △PRS are congruent: a

Which angle is equal to: i

19, 20

∠RPQ

∠PRS

Explain why PQ ∥ SR.

Explain why PS ∥ QR.

What type of quadrilateral is PQRS?

Suppose that ∠RPQ = 47° and ∠SPR = 24°. Find the size of ∠RSP.

Given that ABCD is a rhombus, solve for x.

Justify your solution.

(8x − 3)°

E A

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(5x + 2)° B

Complete the proof for the statement: If a parallelogram is a rectangle, then its diagonals are congruent. Proof: Parallelogram ABCD is a rectangle. We want to show that ⬚ = BD.

If ABCD is a rectangle, then ∠ABC, ∠BCD, ∠CDA, and ∠DAB are all ⬚ angles. This also tells us that AB = ⬚ and ⬚ = AD.

This means that we have two congruent right triangles, △ABC and △DCB, using the ⬚ theorem. By ⬚ parts of congruent triangles, we get that AC = BD.

Problem-solving 22, 23

Ex 3

22, 23

22–24

The diagonals of this kite intersect at E, splitting the kite into four triangles: The diagonal AC bisects the angles ∠DAB and ∠BCD. a

What can be said about △AEB and △AED

Are these equalities correct?

∠DEA = ∠EAB

∠AEB = ∠AED

iii

∠BEA = ∠EAD

∠AEB + ∠AED = 180°

∠AEB + ∠CED = 90°

∠BEC + ∠CED = 360°

Are these statements true about the parts of this kite? i

AC bisects BD.

iii

BC is perpendicular to DC.

AC is perpendicular to BD.

BD bisects AC. A

Given: • ABCD is a rhombus. • ∠DAC = 4x + 9 • ∠DAB = 11x − 3 a

Solve for x.

Solve for ∠BAC.

Solve for ∠ABC.

C A

Let AT = 4x − 10 and BD = 5x + 20. a

Classify the quadrilateral.

Solve for x.

T B

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INVESTIGATION

3D coordinate systems Investigate the different ways of locating the position of objects in 3-dimensional space.

Objectives •

Describe the position and location of objects in 3D in different ways.

•

Use the 3D coordinate system and other digital tools to locate points on 3D objects.

3D coordinate systems

In this investigation, we will explore some different ways of specifying the position of objects in 3-dimensional space.

This diagram represents a room in the shape of a rectangular prism.

To specify the position of a point in the room, we could measure from the position of the red dot. This would make the red dot the origin of our coordinate system. Suppose we want to specify the position of the blue dot, which is on the floor of the room.

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Now suppose we want the position of the green dot, which is in the air above the blue dot.

1 43

We could do this using our existing knowledge of coordinates. We can define x and y-axes, with units in metres. By doing this, we can specify the position of the blue dot as (4, 2). The red dot, being the origin, has coordinates (0, 0).

3 2 1

1 43

We can introduce a new axis, the z-axis, to measure the vertical height. If the green dot is one metre above the floor, we can give its coordinates as (4, 2, 1): 4 metres along the x-axis, 2 metres along the y-axis, and 1 metre along the z-axis. The red dot, the origin is now (0, 0, 0).

We have extended from 2-dimensions, where the coordinates are (x, y), to 3-dimensions, where the coordinates are (x, y, z).

Interactive investigation

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Where would we need to place four additional points to form a cube, with all coordinates positive (known as the first octant)?

Investigation: 3D coordinate systems mathspace.co

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8 cm

How many vertices do you need to draw this shape? 4 cm

What are the coordinates of the vertices? 4 cm

6 cm

How many vertices do you need to draw this shape? 3 cm

What are the coordinates of the vertices?

4 cm

3 cm 7 cm

How many vertices do you need to draw this shape? What are the coordinates of the vertices?

5 cm 6 cm

Investigate After creating the shapes, try creating other shapes of your choice. Then, answer the following questions.

426

How did you determine the coordinates of the vertices?

How can you verify that the lengths of the sides of the shapes you drew in the applet match the lengths of the sides of the shapes in the images? Give two strategies to do so.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Latitude and longitude We saw in the previous section that we needed 3 coordinates to specify the position of an object in 3D space. A point on the Earth’s surface can be specified with two numbers. A common system used is latitude and longitude. Latitude

60 30

North (+)

Longitude

60 30

Equator

30 60

30 90 South 90

(–)

150

180

150

120

West 90 (–)

East (+)

60 30 0 30 Prime median

The equator divides the Earth into northern and southern hemispheres. The prime meridian divides the Earth into eastern and western hemispheres. It passes through the geographical north and south poles and Greenwich, which is in England. For example, the GPS coordinates of central Perth are 31.9523° S, 115.8613° E. As latitude and longitude coordinates, we would describe Perth’s location as (−31.9523, 115.8613). Notice the latitude angles are between 0 and 90°, and the longitude angles are between 0 and 180°.

Interactive investigation

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Altitude Not everything is on the surface of the Earth. Aeroplanes and satellites are examples of things that are above the surface of the Earth. Two aeroplanes will not collide just because they have the same latitude and longitude. For example, two aeroplanes could be at 31.9523° S 115.8613° E (directly over central Perth) and they would not collide if one was 5000 m above the ground and the other was 8000 m above the ground.

To describe objects that are not on the surface of the Earth, we would need a third coordinate. P = (Latitude, Longitude, Altitude) It is important to note that altitude is not the same as height. Investigation: 3D coordinate systems mathspace.co

427

height

altitude

elevation

Height versus altitude When we talk about height, we usually use the ground as a reference. For example, your height is a measurement from the ground to the top of your head. But altitude is a measurement from the mean sea level to the object in the air.

Mean Sea Level

https://www.astroviewer.net/iss/en/ Use this link to look up the current position of the International Space Station (ISS).

Interactive investigation

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Discussion

428

Compare the two-dimensional and three-dimensional views of shapes and maps. What is similar between the views?

Contrast the two-dimensional and three-dimensional views of shapes and maps. What is different about the views?

Why do we need to introduce a third axis for describing three-dimensional objects?

Mathspace Victorian Curriculum – Year 8 mathspace.co

8 Chapter review 1

Determine whether each pair of angles are: • Complementary angles • Supplementary angles • Neither a

160° 19°

25° 65°

140° 40°

34° 55°

Find the value of x: a

83°

b 94°

x°

130°

d 150° x°

x°

50° 50°

x°

f 50°

x° x°

x°

55°

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Find the value of x: a

b 30°

x°

110° x°

55° 70°

d x°

38° x°

34°

73°

f x°

135°

52°

112°

x° 108°

Determine whether each diagram shows a reﬂection across the given line: a

x 1

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x 1

The given diagram shows two triangles that are translations of one another:

What is the angle that is congruent to ∠CBA?

P A

△ABC is reﬂected along the dotted line and its image XYZ is produced. a

i b

Find the length of each side: XY

iii

Describe the relationship between △XYZ and △ABC.

7 B

Consider the given diagram:

What information is needed so that △PQR and △PSR are congruent?

Consider the given diagram:

What information is needed so that △JKN and △MLN are congruent?

Consider the given diagram: a

Which congruence test do △ABD and △CDB satisfy?

What are the three ways in which the triangles satisfy this congruence test?

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Consider the triangles: Triangle A

Triangle B

Triangle C

4 9

8 9

Triangle D

Identify the two congruent triangles.

State the congruence test that this pair satisﬁes.

The dimensions of the triangle shown are to be enlarged using a scale factor of 2.

Find the enlarged length of: a

20 m

26 m B 23 m C

Consider the quadrilaterals: a b

Is quadrilateral A′B′C′D′ an enlargement of quadrilateral ABCD?

Find the scale factor.

5 D

C’ 12

B’

12 A’

Triangle ABC has been reduced to form a smaller triangle A′B′C′. Find the scale factor.

B 6 cm A

D’

6 cm

B’ A’

7 cm

C’ 7 cm

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The two quadrilaterals in the diagram are similar. Find the value of b.

b 8

Determine whether each pair of triangles are similar, and if so, state the similarity test to prove this. a

8 A C

84°

P Q

R 16

84° 10

D 4

P 14

8 5

C L

56 35

M 24

Consider the diagram: a b

Identify two similar triangles and state the similarity test that they satisfy.

Given that DE = 7, what is the length of BC?

E 10

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Consider the diagram where TZ and UY are non-parallel straight lines: a

Explain why the triangles △TUX and △Y ZX are not similar.

What length does Y Z need to be for the two triangles to be similar?

T 6 4

If the length of PX is 4, ﬁnd the length of PR.

If the length of QS is 18, ﬁnd the length of QX.

X 6

Z Q

If the length of AT is 20, ﬁnd the length of BD.

If the size of ∠SPQ is 81°, ﬁnd the size of the highlighted angle ∠SPT.

R S

T Q

Given quadrilateral ABCD is a rhombus, with ∠ADC = 128°, and BD = 12, ﬁnd the size of: a

∠DBA

∠BCD

∠BMC

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The diagonals of this kite intersect at T, splitting the kite into four triangles: The diagonal PR bisects the angles ∠QRS and ∠SPQ. a

What can be said about △RST and △RQT?

Are these equalities correct?

∠RTS = ∠TQR

∠RTS = ∠RTQ

iii

∠RTS = ∠TRQ

∠RTS + ∠RTQ = 360°

∠RTS + ∠RTQ = 180°

∠RTS + ∠RTQ = 90°

T Q

Are these statements true about the parts of this kite? i

RP bisects SQ.

RP is perpendicular to SQ.

iii

RS is perpendicular to RQ.

SQ bisects RP.

Did you know?

Artists and architects often use geometry to create visually appealing structures! The famous pyramids of Egypt are prime examples of geometric precision, where the architects used triangles and squares to build a stable, symmetrical structure that has lasted for thousands of years.

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Big ideas Pythagoras’ theorem provides a fundamental relationship in right-angled triangles, allowing for the calculation of side lengths leading to real-world problem-solving.

9 Pythagoras’ theorem Chapter outline 9.01 9.02 9.03 9.04 9.05

Squares and square roots Investigation: Proof of Pythagoras’ theorem Pythagoras’ theorem Find the hypotenuse Find a short side Applications of Pythagoras’ theorem Chapter 9 review

438 444 449 456 461 468 477

If you spot a ship disappear on the horizon while surfing at the beach, you can use Pythagoras’ theorem to work out how far away it is!

9.01 Squares and square roots After this lesson, you will be able to… • define square numbers. • describe what it means to square a number. • understand and identify perfect squares. • find square roots using both numerical and symbolic methods.

Square numbers When a whole number is multiplied by itself we say we are ‘squaring the number’. The resulting number is known as a square number or perfect square. For example, 3 squared is written as 32 = 3 × 3 and the result of 9 is called a square number. Square numbers can be represented as shown in the diagram: 12

Each formation of dots forms a square. 12 = 1 × 1 = 1 22 = 2 × 2 = 4 32 = 3 × 3 = 9 42 = 4 × 4 = 16 and so on. The numbers 1, 4, 9, 16, 25 … are the square numbers.

Interactive exploration

Discover this concept in action online

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Example 1 Evaluate 92

Create a strategy Use the fact that squaring a number means multiplying by itself.

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Apply the idea 92 = 9 × 9 = 81

Multiply the base by itself Evaluate

Example 2 Evaluate 42 + 92

Create a strategy

Apply the idea

Use the fact that squaring a number means multiplying by itself.

42 + 92 = 4 × 4 + 9 × 9 Multiply the base by itself = 16 + 81

Evaluate the multiplication

= 97

Evaluate

Idea summary A square number or perfect square is the result of multiplying a whole number by itself. For example, 3 squared is written as 32 = 3 × 3 and the result of 9 is called a square number.

The square root If we are asked to ﬁnd the square root of a value, we are being asked, “What number multiplied by itself would give this value?” For example, the square root of 25 is written as

and is equal to 5 because 5 × 5 = 25.

The square root of a perfect square is a whole number.

To ﬁnd the square root of a number that is not a perfect square we need to use a calculator.

Example 3 Evaluate

Create a strategy Perform the operation inside the square root symbol then ﬁnd the square root of the result.

Apply the idea Multiply 5 and 12 by itself Evaluate the multiplication Evaluate the addition Find the square root

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Reﬂect and check since 132 = 169.

Idea summary To ﬁnd the square root of a number, we need to ﬁnd the number that when multiplied by itself would give the original number. e.g.

= 5 since 52 = 25.

To ﬁnd the square root of a number that isn’t a perfect square we use a calculator.

9.01 Practice questions Understanding

2 3

Ex 1

Complete: 15 × 15 = 225, therefore the square root of 225 is equal to ⬚. Given that

= 26, what is 26 × 26?

4a–d, 5a–f, 6a–d, 7a–d, 8, 10a–f, 11, 12a–d, 14, 16

4a–d, 5c–h, 6b–e, 7c–f, 9, 10c–h, 11, 12a–d, 14, 16

4e–h, 5eg–l, 6e–h, 7e–h, 9, 10h–l, 12e–h, 13, 15–17

Is the number a perfect square? a

3.8

0.44

121

(−12)2

Evaluate: a

(−8)2

132

152

202

0.72

1.22

502 + 102

112 – 72

(−12)2 − 82

1002 – (–10)2

Evaluate: a e

440

Trailblazer 2, 3

1, 4, 9, 16, ⬚, ⬚, ⬚

i Ex 2

Adventurer 2, 3

Continue the pattern:

Fluency

Explorer 1–3

42 + 102

52 + 72

Mathspace Victorian Curriculum – Year 8 mathspace.co

10 Ex 3

Evaluate: a

42 + 15

(0.1 + 0.7)2

(9 − 5)2 + 28

4 (22 + 1)2

Frida was working out 22 + 72 and thought that she could simplify the expression using the fact that 2 + 7 = 9. a

Evaluate 22 + 72, by ﬁrst expanding and evaluating each square.

Now, using the fact 2 + 7 = 9, evaluate (2 + 7)2 to find 92.

Is 22 + 72 = (2 + 7)2 a true statement?

Chael was working out 102 − 62 and thought that he could simplify the expression using the fact that 10 − 6 = 4. a

Evaluate 102 − 62, by ﬁrst expanding and evaluating each square.

Now, using the fact 10 − 6 = 4, evaluate (10 − 6)2 to find 42.

Is 102 − 62 = (10 − 6)2 a true statement?

Evaluate: a

Consider the number a

102 − 3 × 52

Complete the inequality with two consecutive perfect square numbers that 40 lies between: ⬚ < 40 < ⬚

Complete the inequality with two consecutive whole numbers that

lies between:

<⬚

⬚<

Determine the two consecutive whole numbers between which each of these roots lies. a

What integer lies between

Harriett was working out the fact that 36 × 4 = 144.

and

and thought that she could simplify the expression using

Evaluate

, by ﬁrst evaluating each square root.

Now find

, to evaluate

Calculate

a true statement?

using the same method.

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441

Nigel was working out fact that 36 ÷ 4 = 9.

and thought that he could simplify the expression using the

Evaluate

, by ﬁrst evaluating each square root.

Now find

, to evaluate

Calculate

a true statement?

using the same method.

Find the area of a square having a side of 18 cm.

At a graduation ceremony, Aaron notices the chairs are arranged into a square grid with an equal number of rows and columns. If there are 196 chairs in total, how many chairs are in each row?

Reasoning 18

18, 20

18, 19, 21, 22

What is 102? b What is (−10)2?

If a square has an area of 100 square metres, what is the length of each of its sides? Is there only one possible answer? Explain your reasoning.

What is the smallest perfect square that is divisible by both 12 and 16? Justify your answer.

Explain why it is not possible to ﬁnd the square root of a negative number.

A perfect square is an integer that is the square of an integer. For example, 9 is a perfect square as it is 32. Explain why these numbers are not perfect squares: 8, 0.09, −64,

For each student: i

Explain the error made.

Ralph tried to evaluate the expression 2 (3 + 2)2. He thought that he had the right answer and submitted the following to his teacher:

Write the correct work and answer.

2 (3 + 2)2 = 2 (5)2 = 102 = 100 b

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Ida tried to evaluate the expression to her teacher:

Mathspace Victorian Curriculum – Year 8 mathspace.co

(8 ÷ 4 − 3 × 4 + 42 + 3). She submitted this answer

Problem-solving 24–26

24–26

23, 25–27

Find two perfect squares which sum to a third perfect square.

Find the perimeter of a square whose area is 64 mm2.

Find the perimeter of a square room with an area of 132.25 m2.

Find the cost of fencing a square garden that has an area of 49 m2, if fencing costs $26.95 per metre.

Consider the pattern: 12 = 1 22 = (2 + 1) + 1 32 = (3 + 2 + 1) + (2 + 1) 42 = (4 + 3 + 2 + 1) + (3 + 2 + 1) a

Using the pattern, write the sums for 52 and 62.

How many new numbers are in each new row?

If you sum the new numbers in each row, what pattern is formed?

Using a visual diagram, explain why this pattern occurs.

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INVESTIGATION

Proof of Pythagoras’ theorem Investigate Pythagoras’ theorem visually to prove the relationship between the side lengths of right-angled triangles.

Objectives • To prove Pythagoras’ theorem • To use technology to explore proofs for Pythagoras’ theorem

Materials •

Plain paper

•

Pen or pencil

• Coloured pencils or markers

•

Straight edge

•

Scissors

•

Grid paper

Prove Pythagoras’ theorem - hands on activity Key terms Before you begin this activity, deﬁne each of these terms. Right-angled triangle Hypotenuse Leg Area Area of a Square Why prove it? The study of mathematics is like a huge pyramid, with things which are proven forming the base of more complicated work above it. We can help build our justiﬁcation and discussion skills through proofs. A proof is a logical argument that shows a statement is true, using established facts and reasoning.

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We will do our ﬁrst proof by hand using paper, scissors, and pencils. Using the diagrams shown, we want to show that the white space in the ﬁrst square is equal to the white spaces in the second square. a

b c

c c

a b

c2 = a2 + b2

In the ﬁrst square, notice that: Area of big square = Area of white space + Area of triangles

c b

c c

Use coloured pencils to make each of the four triangles a different colour. To be sure that the four triangles we are creating are equal, it may be helpful to trace the ﬁrst one and create a stencil of it. Then, you can use the stencil to form the other three triangles.

b a

c b

Start with a square with four equal right- angled triangles drawn into it, so that the hypotenuses form the square in the middle. Each of these triangles has sides a and b, and hypotenuse c.

Discuss this question with a partner.

c Area c

How could we express the area of the white square in the middle, with its side length equal to c?

b b

Cut out the four coloured triangles and rearrange the triangles as shown in the diagram. Then cut the white square that was in the middle of the ﬁrst diagram so that it ﬁts into the spaces available.

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Area b

Find the area of the two white squares with side lengths equal to a and b.

Area a

Discuss this question with a partner.

Why does this mean that a2 + b2 = c2?

Investigate Consider the following questions once you have completed the above procedure. 1.

How did you calculate the area of the white square in the centre with its side length equal to c? Did any of your classmates use a different strategy?

How did you calculate the area of the two white squares with side lengths equal to a and b? Did any of your classmates use a different strategy?

Based on your answers to the previous questions, as well as the procedure above, do you think that the equation a2 + b2 = c2 will always be true?

Prove Pythagoras’ theorem - applet activity

a2 + b2 = c2

Now, let’s explore another way to verify Pythagoras’ theorem is true for any right-angled triangle.

Interactive exploration

Discover this concept in action online

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mathspace.co

After exploring the applet, answer these questions: What is the length of side a? What is the area of the square next to side a? What is the length of side b? What is the area of the square below side b? After sliding the slider, what do we know about the area of the square next to side c? How does this relate to Pythagoras’ theorem? Try it! Using grid paper, draw a right-angled triangle with the given leg lengths and draw squares with the given side lengths. a = 6, b = 8 a = 5, b = 12 a = 8, b = 15

Similar to the applet above, cut and arrange the squares with side lengths of a and b to create a third square of side length c. Is the side length of this new square equal to the lengths of the hypotenuses of your right-angled triangles?

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Investigate Consider the following questions once you have answered the questions and explored both applets. 1.

Write in words what you think is happening in the second applet. Then, discuss your answer with a partner.

Do you think this is true for any right-angled triangle with side lengths a and b?

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Discussion 1.

Which activity showed that Pythagoras’ theorem is always true the most clearly for you? Discuss with a partner why you felt the representation in the activity you selected made the most sense to you.

Do you think the relationship a2 + b2 = c2 will work for triangles that are not rightangled? Explain your reasoning.

Before the internet or global travel, maths concepts were often discovered independently and without the knowledge that they had already been used by people around the world. Research and explain a way Pythagoras’ theorem was used or a proof of Pythagoras’ theorem that originated in Egypt, Greece, India, or China.

Did you know?

Pythagoras’ theorem is used in real life to determine distances! For example, when an architect designs a sloping roof, they use the theorem to calculate the exact length of the beams that will support it. Pilots rely on it when plotting the shortest flight path between two points, taking into account altitude changes.

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Pythagoras’ theorem

9.02

After this lesson, you will be able to… • define and explain the Pythagoras’ theorem. • understand the properties of right-angled triangles and their relationship to the Pythagoras’ theorem. • apply the Pythagoras’ theorem to find the length of the missing side of a right-angled triangle. • identify Pythagorean triples and know how to generate them. • solve problems involving the Pythagoras’ theorem and Pythagorean triples.

The Pythagoras’ theorem The Pythagoras’ theorem states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written algebraically: a2 + b2 = c2 a

c 2

c = a2 + b2 b

where c represents the length of the hypotenuse and a and b are the two shorter sides. The theorem is proved in the investigation: Proof of Pythagoras’ theorem. We can use the formula to ﬁnd the unknown length of a side of the triangle if we know the lengths of the other two sides.

Example 1 Which side of the triangle in the diagram is the hypotenuse? A X

Z Z

Create a strategy

Apply the idea

Use the fact that the hypotenuse is the longest side in a right-angled triangle.

Based on the given diagram, the hypotenuse is side X as it is the longest side of the rightangled triangle and opposite the right angle. The correct answer is Option A.

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Example 2 Consider the right-angled triangle. What equation can we construct from the given information? A 132 + x2 = 192

13 + x = 19

C 13 + 19 = x

132 + 192 = x2

19 13

Create a strategy Use the Pythagoras’ theorem: c2 = a2 + b2.

Apply the idea In the given triangle, the length of the two shorter sides are 13 and x, and the side opposite the right angle is 19. So a = 13, b = x and c = 19. c2 = a2 + b2

Use the Pythagoras’ theorem

Substitute the values

19 = 13 + x

The correct answer is option A.

Idea summary The Pythagoras’ theorem states that in a right-angled triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides:

a2 + b2 = c2 c

is the length of the hypotenuse

is the length of a shorter side

is the length of the other shorter side

Pythagorean triples A Pythagorean triple (sometimes called a Pythagorean triad) is an ordered set of three numbers (a, b, c) which are three positive integers that satisfy the equation a2 + b2 = c2. (3, 4, 5) is a Pythagorean triple because 32 + 42 = 52. (4, 3, 5) is also a Pythagorean triple because 42 + 32 = 52. If (a, b, c) is a triple then (b, a, c) is also a triple, since b2 + a2 is the same as a2 + b2. So the order of the ﬁrst two numbers in the triple doesn’t matter. (5, 12, 13) is another Pythagorean triple because 52 + 122 = 132. 450

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Exploration We have seen that a Pythagorean triple is a set of three positive integers that satisfy Pythagoras’ theorem. A triangle, drawn with side lengths equivalent to the Pythagorean triple, will be right-angled. 7

Here are two examples of Pythagorean triples. 1. Can you find any other Pythagorean triples? 2. How many do you think there are?

θ 24

3. Is there any easy way to find an infinite number of triples? θ

8 6

If we multiply each number in a Pythagorean triple by a constant, we get another Pythagorean triple. For example, if we multiply (3, 4, 5) by 2 we get the triple (6, 8, 10) which is also a Pythagorean triple. If the side lengths of a triangle form a Pythagorean triple then the triangle must be a right-angled triangle.

Example 3 The two smallest numbers in a Pythagorean triple are 20 and 21. What number, c, will complete the triple?

Create a strategy Use the Pythagoras’ theorem: c2 = a2 + b2.

Apply the idea We have the given values for a = 20 and b = 21. Use the Pythagoras’ theorem

Substitute the values

Evaluate the squares

Evaluate

Apply the square root to both sides

Evaluate the square root

So, the value for c is 29, forming the triple (20, 21, 29).

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Example 4 Luke knows the two greatest numbers in a Pythagorean triple, which are 37 and 35. What number, a, does Luke need to complete the triple?

Create a strategy Use the Pythagoras’ theorem: c2 = a2 + b2.

Apply the idea We are given b = 35 and c = 37. c2 = a2 + b2 2

Use the Pythagoras’ theorem 2

37 = a + 35 2

37 − 35 = a

1369 − 1225 = a2

Subtract 352 from both sides of the equation Evaluate the squares

Evaluate the subtraction

Apply the square root to both sides

144 = a =a

Substitute the values

12 = a

Evaluate the square root

So, the number that will complete the triple is 12.

Idea summary A Pythagorean triple (sometimes called a Pythagorean triad) is an ordered triple (a, b, c) of three positive integers so that a2 + b2 = c2. If we multiply a Pythagorean triple by a constant then we get another Pythagorean triple. If the side lengths of a triangle form a Pythagorean triple then the triangle must be a right-angled triangle.

9.02 Practice questions Understanding

Explorer 2, 3

Adventurer 1–3

Trailblazer 1, 2

Pythagoras found that in a right-angled triangle with sides a, b and c, the relationship between the sides is c2 = a2 + b2. Which side is the hypotenuse?

Consider the right-angled triangle with sides (7, 24, 25).

452

What is the length of the hypotenuse?

What lengths are the two sides that are next to the right angle?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Ex 1

Name the hypotenuse of each triangle: a

N Y

Y a

Fluency 4

4, 5ac, 6ac, 7ac, 8ab, 9ab, 10b–d

5bc, 6ac, 7ac, 8ac, 9ac, 10abd

5bd, 6bd, 7bd, 8bd, 9cd, 10abd

Use these steps to determine if (3, 4, 5) is a Pythagorean triple a

Calculate 32 + 42

Is (3, 4, 5) a Pythagorean triple?

Calculate 52

Use Pythagoras’ theorem to determine whether the triangles are right-angled: a

b 12

35 10

23 37

16 9

9.02 Pythagoras’ theorem mathspace.co

453

Ex 2

For each of the right-angled triangles, write an equation involving the side lengths. a

c cm

9 cm

3 cm

Determine whether the set represents the side lengths of a right-angled triangle: a

Ex 3, 4

(3, 4, 5)

(2, 4, 6)

(5, 12, 13)

⬚, 15, 17

7, ⬚, 25

28, ⬚, 53

14, 48, ⬚

(8, 13, 19)

(9, 16, 25)

(20, 21, 29)

Determine whether these are Pythagorean triples: a

Complete the Pythagorean triples: a

(6, 8, 13)

Find the number that completes a Pythagorean triple given: a

The two greatest numbers are 100 and 96.

The two smallest numbers are 22 and 120.

The two greatest numbers are 10 and 8.

The greatest and smallest numbers are 17 and 8.

Reasoning

11, 12

11–13

A right-angled triangle has side lengths x, y, and hypotenuse z. Will a triangle with side lengths of 2x, 2y, and 2z make a right-angled triangle as well? Explain your answer.

Gisela asked her friend Chenzi if he can construct a right-angled triangle with side lengths of 26 cm, 10 cm, and 24 cm. Chenzi said it is not a right-angled triangle. Describe Chenzi’s error and correct his mistake. 1

262 + 102 = 676 + 100 = 776

2 2

24 = 576

3 4 5 454

a + b2 = c2 776 = 576

Mathspace Victorian Curriculum – Year 8 mathspace.co

Pythagoras’ theorem

Are there inﬁnitely many Pythagorean triples? Explain your reasoning.

Problem-solving 14, 15 14

15, 16

15–17

Three towns Melba, Florey and Giralang are positioned as shown in the diagram:

Melba

Which two towns are furthest apart, assuming a direct route is taken? Florey

Giralang

A group of engineering students have made a triangle out of some wooden strips. The side lengths are 20, 48, and 52 m. a

Is the triangle they made a right-angled triangle?

How many metres of wooden strips did they use to make the triangle?

Determine whether right-angled triangles can be created using these lengths of wooden strips: i

(30, 40, 50)

(26, 41, 53)

iii

(14, 48, 50)

(24, 45, 51)

Given that the length of the hypotenuse of a right-angled triangle is 20, ﬁnd the two other side lengths that would complete a Pythagorean triple.

A right-angled triangle has a hypotenuse with a length of 65 units. If the lengths of the legs are integers, ﬁnd two possible pairs of lengths that can complete the triangle.

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9.03

Find the hypotenuse

After this lesson, you will be able to… • define the hypotenuse of a right-angled triangle. • state the formula for Pythagoras’ theorem. • apply Pythagoras’ theorem to find the length of the hypotenuse of a right-angled triangle given the lengths of the other two sides. • apply Pythagoras’ theorem to solve real-world problems involving right-angled triangles.

Find the hypotenuse Pythagoras’ theorem is: a2 + b2 = c2 a

where c represents the length of the hypotenuse and a and b are the two shorter sides.

We can use the theorem to ﬁnd the hypotenuse if we know the lengths of the other two sides.

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Example 1 Find the length of the unknown side c in the triangle. 7

Create a strategy Use Pythagoras’ theorem: a2 + b2 = c2.

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Apply the idea Write the formula Substitute a and b Evaluate the squares Evaluate the sum Square root both sides Evaluate the square root

Example 2 Find the length of the unknown side c in the triangle. c

Create a strategy

Apply the idea

Use Pythagoras’ theorem: a2 + b2 = c2.

Write the formula Substitute a and b Evaluate the squares Evaluate the sum Square root both sides Evaluate the square root

Idea summary To ﬁnd the length of the hypotenuse we can use Pythagoras’ theorem: a2 + b2 = c2 c

is the length of the hypotenuse

is the length of a shorter side

is the length of the other shorter side

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9.03 Practice questions Explorer

Understanding

Adventurer

1–3

Trailblazer

1–3

Consider the right-angled triangle: a

Name the hypotenuse.

Write Pythagoras’ theorem for this triangle.

Is a + b > c?

Evaluate to two decimal places: a

c2 = 9

c2 = 45

Fluency

Ex 1

The hypotenuse of a right-angled triangle is c. Find the value of c, rounded to two decimal places. a

4a–d, 5–7, 9, 10, 13

c2 = 25 + 64

4d–g, 5, 6, 8, 9, 11, 12

c2 = 92 + 122

4e–h, 5, 6, 8, 9, 11, 12

Find the length of the hypotenuse in each of the right-angled triangles: a

8 c

e c

9 40

12 9

h 48

14 458

Mathspace Victorian Curriculum – Year 8 mathspace.co

cm 21 m 20 m

5 Ex 2

Find the value of c for each right-angled triangle, rounded to two decimal places: a

21 9

c cm

17 cm

9 cm c cm

3 cm

For each right-angled triangle, ﬁnd the length of the hypotenuse, c, given the other two sides. Round your answers to two decimal places. a

9 cm

4 and 7

15 and 16

7 and 10

15 and 22

Calculate the length of the diagonal of the hand held device:

5 cm

x cm

8 cm

A farmer has a square ﬁeld of side length 12 metres. He wants to build a fence along the diagonal of the ﬁeld. How long will the fence be, to the nearest metre?

Consider the triangle: a

Find the length of the hypotenuse.

Find the perimeter of the triangle.

15 m

A right-angled triangle has two shorter sides measuring 15.4 mm and 17.7 mm. a

Find the length of the hypotenuse.

Find the perimeter of the triangle.

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A right isosceles triangle has two sides measuring 17 cm. a

Find the length of the hypotenuse.

Find the perimeter of the triangle.

Complete the table, rounding the values to one decimal place: Side length of square

Length of diagonal

1 3 7 11 249 13

Find the length of the diagonal of a rectangle with: a

Side lengths of 5 and 12

Reasoning

15, 16

Side lengths of 9 and 14

15–17

14, 17, 18

If you are given two sides of a right-angled triangle, can you always ﬁnd the length of the other side? If you are given only one side length, how many possible triangles could satisfy the Pythagoras’ theorem?

Prove that a triangle with sides of 3 cm, 4 cm and 6 cm is not a right-angled triangle.

Annette says a right-angled triangle with side lengths of 2 cm and 3 cm has a hypotenuse of 4 cm. Explain why Annette is incorrect, and determine the correct length of the hypotenuse.

A triangle has side lengths of 5 cm and 13 cm and the longest side is x cm, where x is an integer. Is it possible for the triangle to be a right-angled triangle? Explain your reasoning.

If two right-angled triangles have two pairs of corresponding sides in proportion, can we show they are similar by the side-side-side similarity test? Explain your reasoning.

Problem-solving 19, 21 19

19, 21

19–21

For each figure, find the length of the unknown side x, rounded to two decimal places: a

b x

12 6 8

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A right-angled triangle has sides of length a, b, and c, where a and b are positive integers and c is the length of the hypotenuse. If the sum a + b = 13 and the product ab = 30, what is the length of the hypotenuse? Give your answer rounded to two decimal places.

Find the perimeter of the trapezium, rounded to two decimal places. A

8 cm

x cm

a cm

6 cm

14 cm b cm

9.04

Find a short side

After this lesson, you will be able to… • use Pythagoras’ theorem to find the length of a short side given the length of the hypotenuse and the other short side. • solve real-world problems involving finding the length of a short side using Pythagoras’ theorem.

Find a short side

We can rearrange Pythagoras’ theorem to ﬁnd short sides a or b if we know the hypotenuse c, and one other side. a2 = c2 − b2 b2 = c2 − a2

We can take the square root of both sides to give us these formulas: or

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Example 1 Find the length of the unknown side in this right-angled triangle. Round your answer to two decimal places. b

Create a strategy Use the formula

Apply the idea Write the formula

Substitute c = 24 and a = 10

Evaluate the expression inside the square root sign

Take the square root and round answer to two decimal places

Example 2 Consider a cone with slant height of 13 m and perpendicular height of 12 m.

12 m

13 m

a Find the length of the radius, r, of the base of this cone.

Create a strategy The radius serves as the shortest side of the right-angled triangle. Use the formula

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Apply the idea Write the formula Substitute c = 13, b = 12 and a = r Evaluate the expression inside the square root sign Take the square root

b Find the length of the diameter of the cone’s base.

Create a strategy

Apply the idea Diameter = 2 × r

The diameter of the circle is equal to twice its radius.

Write the formula

=2×5

Substitute the value of r.

= 10 m

Evaluate

Idea summary Rearranging the Pythagoras’ theorem to ﬁnd a shorter side: a2 = c2 − b2 or b2 = c2 − a2 We can take the square root of both sides to give us these formulas: or

9.04 Practice questions Explorer

Understanding

1–3

Adventurer 1–3

Trailblazer 1–3

Consider the right-angled triangle. Which of these statements is incorrect? A

c2 = a2 + b2

b2 = c2 − a2 2

a =c +b

a2 = c2 − b2 a

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Evaluate and round your answer to two decimal places. a

a2 = 64.2

Fluency

Ex 1

Find the value of a, rounded to two decimal places. a

a2 = 121 − 64

4a–e, 5a–c, 6ac, 7, 8

a2 = 122 − 92

4b–f, 5a–c, 6ac, 7, 8

a2 + 52 = 122

4c–i, 5c–e, 6bd, 8, 9

Find the length of the unknown side in each of the right-angled triangles: a

c 15

20 12 y

b 24

20 m

29 m

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17 m

Find the length of the unknown side in the triangle, rounded to one decimal place: a

b 16

9 m

13 m s

Calculate the length of the unknown side s, rounded to two decimal places, of a right-angled triangle which has: a

A hypotenuse of length 6 and another side of length 5.

A hypotenuse of length 18 and another side of length 13.

A hypotenuse of length 7 and another side of length 4.

A hypotenuse of length 23 and another side of length 15.

Find, rounded to two decimal places: a

The value of x

The value of y

5 cm

15 cm

20 cm x cm

y cm

Consider the given triangle: Find, rounded to two decimal places: a

The value of x

The value of y

The length of the base of the triangle

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9 Ex 2

Consider the cone with slant height of 26 m and perpendicular height of 24 m: a

Find the length of the radius, r.

Find the length of the diameter of the cone’s base.

26 m

24 m

Reasoning 10

10, 11

11, 12

A triangle has side lengths of 2 cm, 4 cm, and 5 cm. A teacher wants to change the side of length 4 cm to make the triangle a right-angled triangle instead. What should the length 4 be changed to in order to form a right-angled triangle? Explain your reasoning.

A triangle has side lengths of 5 cm and x cm, where x is an integer. The hypotenuse is 8 cm. Is the triangle a right-angled triangle? Explain your reasoning.

Is it possible for a right-angled triangle to have one short side and the hypotenuse the same length? Explain your reasoning.

Problem-solving 13, 15–19, 21

13, 15, 17, 19–21

14, 15, 17, 19, 20, 22

Find the perpendicular height of an equilateral triangle given each side of the triangle is 22 cm. Round your answer to two decimal places.

Find the length of the sides of an isosceles right-angled triangle that has a hypotenuse of 18 cm, rounded to two decimal places.

A ladder is leaning against a wall. The base of the ladder is 6 metres away from the wall, and the ladder is 12 metres long. What is the height of the wall from the ground to the top of the ladder, rounded to two decimal places?

Find the value of k. Round your answer to two decimal places.

Consider the ﬁgure shown. Find the length of the unknown side x, rounded to two decimal places.

7 x 12

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Find the length of m and n in the triangle. Round each answer to the nearest tenth.

14 cm

Calculate the perimeter of this shape (diagram not to scale). Round your answer to two decimal places.

9 cm

18 cm

Solve for u. Round your answer to two decimal places. 14 mm

A pole is being held upright by two cables attached to the top of the pole and to the ground either side by pegs, as shown in the diagram. The length of the cables are 10 metres and 17 metres. The height of the pole perpendicular to the ground is 8 metres.

Skye notices that she can divide the diagram into two smaller right-angled triangles, each with a height of 8. What are the base lengths of these right-angled triangles?

How far apart (measured along the ground) are the pegs at the ends of the cables?

Two toy triangular tents are drawn in cross-sections from the side, showing the height of the tent and width of the tent base. Find the width of each tent, if the measurements are in cm: a

b 13

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9.05 Applications of Pythagoras’ theorem After this lesson, you will be able to… • apply Pythagoras’ theorem to solve a variety of real-world problems involving right-angled triangles.

Applications of Pythagoras’ theorem To apply the Pythagoras’ theorem to real-life situations, follow these steps: 1. Read the question and draw a carefully labelled diagram from the information. 2. Mark any right angles in the diagram with a small square in the corner. 3. Determine which side, hypotenuse or a short side, you are trying to find. 4. Write down the appropriate formula: To find the hypotenuse use c2 = a2 + b2 To find a short side use a2 = c2 − b2 or b2 = c2 − a2 5. Substitute the known values into the formula and solve for the unknown.

Example 1 The screen on a handheld device has dimensions 9 cm by 6 cm, and a diagonal of length x cm. x cm

9 cm

Find the value of x, rounded to two decimal places.

Create a strategy Use the Pythagoras’ theorem: a2 + b2 = c2

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6 cm

Apply the idea x2 = 92 + 62

Substitute a = 9, b = 6, and c = x

x = 81 + 36

Evaluate the squares

x = 117

Evaluate the sum

x = 117

Take the square root of both sides

x = 10.82 cm

Simplify and round to two decimal places

Example 2 A sports association wants to redesign the trophy they award to the player of the season. The front view of one particular design is shown in the diagram:

y cm 16 cm

x cm

3 cm 12 cm

a Find the value of x.

Create a strategy We can use the Pythagoras’ theorem: a2 + b2 = c2.

Apply the idea We can apply Pythagoras’ theorem to the bottom triangle that has side lengths of 16 cm and 12 cm, since the only unknown side is the hypotenuse. Substitute a = 12, b = 16, and c = x Evaluate the squares Evaluate the sum Take the square root of both sides Evaluate

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b Find the value of y, rounded to two decimal places.

Create a strategy We can use the rearranged Pythagoras’ theorem: a2 = c2 − b2.

Apply the idea Substitute a = y, b = 3, and c = 20 Evaluate the squares Evaluate the difference Take the square root of both sides Simplify and round to two decimal places

Idea summary To apply the Pythagoras’ theorem to real-life situations, follow these steps: 1. Read the question and draw a carefully labelled diagram from the information. 2. Mark any right angles in the diagram with a small square in the corner. 3. Determine which side, hypotenuse or a short side, you are trying to find. 4. Write down the appropriate formula: To find the hypotenuse use c2 = a2 + b2 To find a short side use a2 = c2 − b2 or b2 = c2 − a2 5. Substitute the known values into the formula and solve for the unknown.

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9.05 Practice questions Understanding

Explorer 1, 2ab

Adventurer 1, 2ab

Trailblazer 1, 2c–e

How do we know when we can use Pythagoras’ theorem to solve a problem?

Consider these statements. i

Can these problems be solved by Pythagoras’ theorem?

If Pythagoras’ theorem is applicable, what are the values of a, b, and c?

Finding the diagonal length (x cm) of a television with dimensions of 55.9 cm and 48.2 cm.

Finding the length of the third side of a triangular fence (x m) given the lengths of opposite sides that are equal (36 m).

Finding the perimeter of a rectangular garden given measurements of the diagonal (100 m) and width (65 m).

Finding the distance between the top of a building and a fire hydrant (x m) using building height (72 m) and hydrant distance from the building (26 m).

Finding the height of the wall reached by an inclining ladder (x m) given the ladder length (3 m) and the angle of inclination (45°).

Fluency

3 Ex 1

3, 6, 8, 9, 11–13

4–6, 8–10, 12–14

4, 5, 7, 10, 13, 14

The screen on a handheld device has dimensions 29 cm by 21 cm, and a diagonal of length x cm. Find the value of x. Round your answer to two decimal places. 29 cm

x cm

21 cm

4 Ex 2

A sports association wants to redesign the trophy they award to the player of the season. The front view of one particular design is shown in the diagram. a

Find the value of x.

Find the value of y. Round your answer to two decimal places.

y cm 15 cm

x cm

4 cm 8 cm 9.05 Applications of Pythagoras’ theorem mathspace.co

471

To help bear heavier loads, a support cable joins the end of one arm of a crane to the other, through a small tower that rises h m above the crane arm.

hm 19 m

Find, rounded to two decimal places:

The value of h.

The value of l.

The total length of the support cable.

21 m

Yuri and Dave are playing football together. At one point in the game, they are near the same corner of the ﬁeld. Yuri is on the goal line, 11 m away from the corner, while Dave is on the side line, 17 m away from the corner.

11 m

41 m

17 m

Find the shortest distance between Yuri and Dave. Round your answer to two decimal places. 7

Archaeologists have uncovered an ancient pillar which, after extensive digging, remains embedded in the ground. The lead researcher wants to record all of the dimensions of the pillar, including its height above the ground. However, the team can only take certain measurements accurately without risking damage to the artifact. These measurements are shown in the diagram. a

h cm

232.1 cm

Find the value of x. Round your answer to two decimal places.

x cm 178.3 cm 304.6 cm

Find h, the height of the pillar. Round your answer to two decimal places.

Two ﬂagpoles of height 14 m and 19 m are 22 m apart. A length of string is connected to the tops of the two ﬂagpoles and pulled taut. Find the length of the string, rounded to one decimal place. 19 m 14 m 22 m

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A soft drink can has a height of 13 cm and a radius of 4 cm. Find L, the length of the longest straw that can ﬁt into the can.

4 cm

Round your answer down to the nearest centimetre, to ensure it ﬁts inside the can. 13 cm

A movie director wants to shoot a scene where the hero of the ﬁlm ﬁres a grappling hook from the roof of one 54 m building to the roof of another. The shorter building is 37 m tall, the taller building is 54 m tall and the street between them is 10 m wide.

37 m

10 m

Find the minimum length of rope, l, needed for the grappling hook. Round your answer to two decimal places 11

The town Bunderidda is 3 km directly south of Appleby and 4 km directly west of Cottenham. Find the distance from Appleby to Cottenham.

The top of a flagpole is 4 m above the ground and the shadow cast by the flagpole is 9 m long. Find the distance from the top of the flagpole to the end of its shadow, rounded to two decimal places.

A ladder is leaning against a wall. The base of the ladder is 6 metres away from the wall, and the ladder is 7 metres long. How far up the wall does the ladder reach? Round your answer to two decimal places.

Find the value of d, rounded to one decimal place. 74.6 cm 182.4 cm

203.9 cm

d cm

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Reasoning 15

15, 16

16, 17

Sean’s car has run out of petrol. He walks 8 km directly west, then 6 km directly south looking for a petrol station. He is now h km directly from his starting point. a

Can Pythagoras’ theorem be used to ﬁnd the distance Sean is from his car? Explain your reasoning.

Find the value of h.

A ladder of height h cm is placed against a vertical wall. The bottom of the ladder is 80 cm from the base of the wall and the top of the ladder touches the wall at a height of 150 cm. a

Dana draws the diagram shown. Explain Dana’s mistake.

Find h, the height of the ladder against the wall.

150 cm

80 cm

A right-triangle has a hypotenuse that is 3 times longer than the shortest side, x. The second leg of the right-triangle is equal to y. Find a relationship between x and y.

Problem-solving 18–20, 22, 24

19, 20, 22–24

19–21, 23–26

Find the height of the house, h, rounded to two decimal places. 4.9 m

4.9 m

3.5 m

9.2 m 474

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Linda is setting up an electric fence around her rectangular yard to keep her dog from escaping. The yard measures 8 m by 10 m. There is also a diagonal fence across the yard. If the electric fencing costs $1.80 per metre, what will be the total cost of the fencing?

A city council plans to build a seawall xm and boardwalk along a local coastline. According to safety 2.43 m regulations, the seawall needs to be 5.25 m high and 7.66 m 7.66 m 13.59 m deep and will be built at the 5.25 m bottom of a 13.59 m long sloped section of shoreline. This means that the boardwalk will need to be built 2.43 m above the seawall, so that it is level with the public area near the beach. Find the width of the boardwalk, x m, rounded to two decimal places.

A ramp is being planned for a school entrance using these measurements. • The ramp is 22.3 metres long. • The ﬂat platform is 2.4 metres long. • The height of the ﬂat platform above ground level is 1.6 metres.

2.4 m Ramp 22.3 m 1.6 m Ground Level

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a b

Find the gradient of the ramp. Round your answer to the nearest hundredth.

Ramp length

The School Accessibility Guidelines require a

Vertical height

wheelchair ramp to have a maximum gradient of . Does the ramp meet the School Accessibility

Horizontal run

Guidelines’ requirements? gradient =

Vertical height Horizontal run

A 125 m long cable is connected to the top of a flagpole that is 18 m high. It is to be pulled tight and anchored into the ground so that the flagpole will be vertical. How far will the cable reach from the base of the flagpole if you are on level ground and the flagpole is to be vertical? Round your answer to the nearest metre.

V UTR is a rhombus with perimeter 112 cm. The length of diagonal is 46 cm.

Find the length of

, rounded to two decimal places.

Find the length of the other diagonal decimal places.

R W

, rounded to two

Find the area of the rhombus where one of the diagonals has a length of 32 cm and the side lengths are 20 cm.

32 cm

20 cm

Two similar triangles are created by cables supporting a yacht’s mast. Find the length of the longest cable. Round your answer to one decimal place.

Mast

1.3 m 5.4 m

476

1.8 m

In the diagram △ABD and △ECD are similar right-angled triangles, with AE = 10, AB = 6 and EC = 1.2. a

Let x be the length of interval ED. Find x.

Find the radius of the circle. Round your answer to four decimal places.

Mathspace Victorian Curriculum – Year 8 mathspace.co

B C 1.2 D

9 Chapter review 1

Write the equation that can be used to ﬁnd the value of x:

Use these steps to determine if (6, 8, 10) is a Pythagorean triple: a

Calculate 62 + 82

Calculate 102

Is (6, 8, 10) a Pythagorean triple?

Determine whether these measurements represent the sides of a right-angled triangle: a

(3, 7, 9)

(7, 24, 25)

(2, 4, 6)

(5, 12, 13)

If x, y, and z are three sides of a right-angled triangle, will side lengths of 6x, 6y and 6z make a right-angled triangle as well?

Complete these Pythagorean triples: a

⬚, 15, 17

21, 72, ⬚

= 353 2

15 = 225

3 4

a + b2 = c2

Pythagoras’ theorem

353 = 225

For each right-angled triangle, ﬁnd the length of the hypotenuse, c, given the other two sides. Round your answers to two decimal places. a

4 and 5

6 and 11

8 and 10

18 and 26

A right-angled triangle has two shorter sides measuring 13.6 mm and 3.5 mm. a

28, ⬚, 53

172 + 82 = 289 + 64

Carlos asks his friend Maria if she can construct a right-angled triangle with side lengths of 17 cm, 8 cm, and 15 cm. Maria says it is not a right-angled triangle. Describe Maria’s error and correct her mistake. 1

9, ⬚, 41

Find the length of the hypotenuse.

Find the perimeter of the triangle.

Calculate the perimeter of the triangle.

For each triangle: i

Calculate the length of the hypotenuse.

Round your answers to two decimal places. a

A right-angled triangle has two shorter sides measuring 15.4 mm and 17.7 mm.

A right isosceles triangle has two sides measuring 17 cm. Chapter 9 review mathspace.co

477

Consider the triangle: a

Find the length of the hypotenuse.

Find the perimeter of the triangle.

c cm

8 cm

12 cm

Calculate the length HF, rounded to two decimal places.

Calculate the length DF, rounded to two decimal places.

Joanne says a right-angled triangle with side lengths of 2 cm and 3 cm has a hypotenuse of 4 cm. Explain why Joanne is incorrect, and determine the correct length of the hypotenuse.

Consider this trapezium:

A square prism has dimensions of 12 cm by 12 cm by 15 cm:

C H G

Find the value of a.

Find the value of b.

Find x, rounded to two decimal places.

Find the perimeter of the trapezium, rounded to two decimal places.

15 cm

8 cm

6 cm

a cm

C 14 cm

x cm

D b cm

Find the value of b, to two decimal places, where b is the length of one of the shorter sides of a right-angled triangle which has: a

A hypotenuse of length 11 mm and another side of length 8 mm.

A hypotenuse of length 16 cm and another side of length 5 cm.

Calculate the value of a in each triangle, rounded to two decimal places: a

a mm

10 mm

82 m

8 mm 80 m

14 m

am 11 m 13 m

478

E 12 cm

Mathspace Victorian Curriculum – Year 8 mathspace.co

17 m

Barry wants to measure the length of his ladder, but he doesn’t have a long enough tape measure. Instead, he leans the ladder up against a wall of vertical height 2.4 m. He measures that the foot of the ladder rests at a horizontal distance of 1.8 m away from the base wall. Find the length of the ladder, to the nearest metre.

Neville is making a rectangular shaped pencil case. He decides to make one side of the pencil case 18 cm wide, in order to ﬁt his pens, and determines that the diagonal will need to be 20 cm long, in order to ﬁt his ruler. Find the length of the other side of the pencil case. Round your answer to one decimal place.

A soft drink can has a height of 11 cm and a radius of 4 cm. Find L, the length of the longest straw that can ﬁt into the can.

4 cm

Round your answer down to the nearest cm, to ensure it ﬁts inside the can.

11 cm

L cm

Two ﬂag posts of height 12 m and 17 m are erected 20 m apart. Find the length, l, of the string needed to join the tops of the two posts. Round your answer to one decimal place.

17 m 12 m 20 m

Edgar’s car has run out of petrol. He walks 12 km west and then 9 km south looking for a petrol station. If he is now h km directly from his starting point, ﬁnd the value of h.

A wooden plank of the greatest possible length is placed in a garden shed. Calculate the length of the plank of wood rounded to one decimal place.

2.4 m 4.2 m

The screen on a handheld device has dimensions 9 cm by 5 cm, and a diagonal of length x cm. Find the value of x. Round your answer to two decimal places.

0.94 m

5 cm

x cm

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Big ideas • Geometric figures have properties and relationships that allow us to calculate and analyse their sizes and dimensions. • Understanding volume and capacity, as well as the ability to estimate and convert between units, is essential in real-world situations that involve measuring and comparing quantities.

10 Measurement Chapter outline 10.01 10.02 10.03 10.04 10.05 10.06 10.07 10.08 10.09 10.10 10.11

Perimeter Circumference of a circle Perimeter of composite shapes Estimation and units of area Area of rectangles, triangles and parallelograms Area of composite shapes Area of trapeziums, kites, and rhombuses Investigation: Area of circles Area of a circle More composite areas Units of volume and capacity Volume of right prisms Chapter 10 review

482 491 499 507 515 523 530 544 548 554 561 569 579

The distance a bicycle travels in one spin of the wheel is equal to the circumference of the wheel. We can calculate that using π and the diameter of the wheel!

10.01

Perimeter

After this lesson, you will be able to… • define what perimeter is and explain how it is calculated for various 2D shapes, including rectangles, squares, and regular hexagons. • interpret and solve problems involving perimeters of 2D shapes with given measurements. • extend the concept of perimeter to include algebraic expressions. • apply understanding of perimeters to solve problems where side lengths are represented by variables.

Perimeter Perimeter is the total length of the sides around the outside of a 2D shape. To calculate the perimeter we add the sides around the border of the shape.

Interactive exploration

Discover this concept in action online

mathspace.co

Some polygons have a formula that can be also be used to ﬁnd perimeter: Consider a rectangle.

Length

If the width is w and the length is l, then: Width

Perimeter = l + w + l + w Perimeter = 2 × (l + w) Consider a square.

Side length

If each side has length l, then: Perimeter = l + l + l + l Perimeter = 4 × l

Side length

Consider a regular hexagon. If each side is l, then: Perimeter = l + l + l + l + l + l Perimeter = 6 × l

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Example 1 Find the perimeter of a rectangle with a length of 20 cm and a width of 5 cm.

Create a strategy Use the formula for the perimeter of the rectangle.

Apply the idea We are given length = 20 cm and width = 5 cm. Perimeter = 2 × (20 + 5)

Substitute the length and width

= 2 × 25

Find the sum in the brackets

= 50 cm

Evaluate

Example 2 Find the perimeter of the square shown.

10 cm

Create a strategy Use the formula for the perimeter of the square.

Apply the idea We are given length = 10 cm. Perimeter = 4 × Length

Use the formula

= 4 × 10

Substitute the length

= 40 cm

Evaluate

Example 3 This shape has a perimeter of 64 cm. What is the length of each side?

P = 64 cm

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Create a strategy

Apply the idea

Divide the perimeter by the number of sides. Use the fact that the shape shown is a regular octagon which has 8 sides of equal length.

Substitute the perimeter Divide both sides by 8 Evaluate

Idea summary To ﬁnd perimeter add the side lengths around the outside of a 2D shape. Some shapes have a formula that can also be used to ﬁnd perimeter: The perimeter of a rectangle with width w and length l is Perimeter = 2 × (l + w) The perimeter of a square with side length l is Perimeter = 4 × l The perimeter of a regular n-gon (a polygon with n-sides) is Perimeter = n × Length of side

Algebraic expressions and perimeter Perimeter can also be applied to solve problems where unknown side lengths are represented by variables.

Example 4 Write an algebraic expression for the perimeter of this shape.

2x − 5

x+3

Create a strategy Add the lengths of the sides and simplify by collecting like terms.

Apply the idea

484

P = x + (2x − 5) + (2x − 5) + x + (x + 3)

Add the ﬁve sides

P = x + 2x − 5 + 2x − 5 + x + x + 3

Brackets were unnecessary

P = 7x – 7

Collect like terms

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 5 The equilateral triangle and the rectangle have the same perimeter. Find the value of x.

2x x 18

Create a strategy Find the perimeter of each shape. Set them equal to each other and solve.

Apply the idea P = 18 × 3

Calculate the perimeter of the triangle

P = 54 cm

The perimeter of the triangle is 54

P = 2x + x + 2x + x

Calculate the perimeter of the rectangle

P = 6x

The perimeter of the rectangle is 6x

6x = 54

The two shapes have equal perimeters

x=9

Solve the equation

Idea summary The perimeter is the sum of the sides around the outside of a 2D shape. This idea can be used to solve algebraic problems.

10.01 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

Complete: a c

5 metres = ⬚ centimetres 9 kilometres = ⬚ metres

b d

7 centimetres = ⬚ millimetres 2.5 metres = ⬚ millimetres

A regular hexagon has six equal sides. One way to calculate the perimeter is to add the six side lengths. What is another way to ﬁnd the perimeter of this shape?

Identify the shape given the information: a

Perimeter of 20 cm and sides of length 5 cm, 5 cm, 5 cm, and 5 cm

Perimeter of 95 m and sides of length 43 m, 26 m, and 26 m

Perimeter of 30 mm and sides of length 5 mm each

Perimeter of 170 cm and sides of length 17 cm each

Perimeter of 22 m and sides of length 6 m, 5 m, 6 m, and 5 m 10.01 Perimeter mathspace.co

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Fluency

4b, 6b, 7, 8cd, 9b, 11, 12b, 4b, 6b, 7, 8cd, 9c, 10, 11, 13, 14, 16, 17, 19, 20a, 21, 12c, 13, 14, 16, 17, 19, 20c, 22, 24b 21–23, 24b

Find the perimeter, given each square is 1 unit in length: a

4a, 5, 6a, 7, 8ab, 9a, 11, 12a, 13a, 14, 15, 17, 18, 20b, 24a

Find the perimeter: a

10 m 10 m

10 m

3m 5m

Ex 1

Find the perimeter of a rectangle given the dimensions: a

Length: 20 m, Width: 7 m

Find the perimeter of the tile:

Length: 15.5 m, Width: 12.5 m 11 cm

9 cm

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Find the perimeter: a

11 cm C

55 cm 48 cm

9 cm

10 cm

73 cm

19 cm C 23 cm

x+q

x+p 22 cm B x

Ex 2

Find the perimeter of: a

An equilateral triangle with a side length of 15 m

A scalene triangle with side lengths 18 mm, 25 mm and 32 mm

An isosceles triangle where the 2 equal side lengths are 25 mm and the third side measures 5 cm

Find the difference between the perimeter of the square and the perimeter of the rectangle:

18 m 12 m 5m

Find the perimeter:

Find the perimeter of: a

A regular hexagon where the length of each side is 9 cm

A regular decagon where the length of each side is 16 mm

A regular pentagon where the length of each side is 12 m

5 mm

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Given the perimeter, P, of the shape, ﬁnd the value of the variable: a

4y + 8

4y + 3 26

P = 86 m

P = 53 cm

Ex 3

A rectangle has a perimeter of 110 cm. If it has a length of 26 cm, ﬁnd its width.

Find the side length of a square with perimeter of 32 m:

P = 32 m

A square has a perimeter of 44 mm. Find the side length of the square.

Use Pythagoras’ theorem to ﬁnd the missing side and then calculate the perimeter of the triangle, rounded to two decimal places. a

9 y

Ex 4

Find the side length of an equilateral triangle that has a perimeter of 57 m.

An isosceles triangle has equal sides of length 25 cm and a perimeter of 60 cm. Find the length of the third side.

Find the side length of:

A regular pentagon with a perimeter of 55 cm

An equilateral triangle with a perimeter of 36 m

A regular octagon with a perimeter of 720 mm

Write an algebraic expression for the perimeter of quadrilateral AGHF where F is the centre of the circle:

A 2x − 3

−2 + x F

G x+7 H

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Write an algebraic expression for the perimeter of this triangle: 5.2 + 1.24x

2.3x + 7 4.03x − 3.52

Write an algebraic expression for the perimeter of this rectangle:

x2 − 8x + 2 x2 + 3x − 1

x2 + 3x − 1 x2 − 8x + 2

Ex 5

Each pair of rectangles has the same perimeter. Find the value of x. a

3 cm

b 4 cm

8 cm x

6 cm 3 cm x

6 cm

Reasoning 25

25, 27, 28

26–28

A rectangle has a perimeter of 36 cm. Find all possible dimensions of the rectangle, given that the sides are whole numbers. Explain how the dimensions can be found.

12 cm 6 cm

3 cm

P = 36 cm

Give the whole number dimensions of a square and a rectangle with the same perimeter. Write an expression for the relationship between the side lengths of the square and rectangle.

In a polygon with n sides, what is the relationship between the perimeter and the length of each side?

Determine if each statement is true or false. Justify your answer. a

The perimeter of a square with side length 6 cm is equal to the perimeter of a rectangle with length 9 cm and width 3 cm.

If the perimeter of a regular hexagon is 48 cm then the length of each side is 8 cm.

The perimeter of a rectangle can never be equal to the perimeter of a square.

The perimeter of a triangle is always less than the perimeter of a rectangle with the same base and height.

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Problem-solving 29–32

29–32

30–34

Francesca has a picture that is 18 cm long and 8 cm wide. The picture is in a frame with sides that are 1 cm longer than each of the picture’s sides. What is the perimeter of the picture frame?

A square has the same perimeter as an equilateral triangle. If the triangle has sides of length 55 mm, ﬁnd the side length of the square rounded to two decimal places.

A rectangular piece of astroturf is placed inside a rectangular area of lawn as shown.

If both the astroturf and the grass area are to be fully fenced, calculate the total amount of fencing required.

If the fencing costs $25.60 per metre, calculate the total cost of the fencing.

16 m

20 m

26 m 30 m

Find the value of x given the perimeter of the billboard is 74 m. x

A path surrounds a swimming pool. Find the value of x given the perimeter of the outside of the path is 68 m.

x x

16 m x

A carpenter is calculating the amount of timber required to build a roof truss frame as shown by triangle ABC in the diagram. The carpenter knows that the base BC is 12 m and the perpendicular height is 4 m. Calculate the amount of timber required for the roof truss frame, rounded to two decimal places.

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A B

10.02 Circumference of a circle After this lesson, you will be able to… • identify and define the different features of a circle, including the radius, diameter, and circumference. • apply the formulas for calculating the circumference of a circle in terms of the radius and diameter. • use the formulas for calculating the length of an arc for different fractions of a circle. • solve problems involving the circumference of a circle, lengths of arcs, including situations where they need to add in any relevant radius lengths.

Circumference The circumference of a circle is the distance around the outside of the circle.

Interactive exploration

Discover this concept in action online

mathspace.co

The formula for π is:

The amazing thing about this is that the value for π is the same regardless of the circle chosen. For every circle, the ratio of the diameter to the circumference is the same. This value is known as π and has an approximation 3.141592… It can be rearranged to form the formula of the circumference of a circle: C = πd Where C is the circumference and d is the diameter of a circle. As the diameter is equal to two times the radius, the formula can also be written as: C = 2π r Where C is the circumference and r is the radius of a circle. In calculations involving π, there can be two ways to answer the question. One, as an approximation, where the calculation involving π is left to the last step and a calculator is used. Because π is irrational, any calculation with π will always need to be rounded to some number of decimal places. The other way is as an exact value, which means we leave π in the answer. For example, 3π is an exact answer, and 9.425 is the value of 3π rounded to 3 decimal places.

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Example 1 A circle has a radius of 22 cm. What is its exact circumference?

Create a strategy

Apply the idea

We can use the formula C = 2π r.

C = 2π r

Use the formula

= 2π × 22

Substitute r

= 44π cm

Evaluate

Example 2 Find the circumference of the circle rounded to two decimal places. 13 cm

Create a strategy

Apply the idea

Use the formula C = π d.

C = πd

Write the formula

C = π (13) cm

Substitute d into the formula

C = 40.84 cm

Evaluate

Example 3 A circle has a circumference of 20 cm. What is the radius rounded to two decimal places?

Create a strategy Use the formula C = 2π r.

Apply the idea Use the formula Substitute the circumference Rearrange Evaluate using a calculator Round the answer

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Idea summary The diameter of a circle is double the radius:

d = 2r d r

is the diameter is the radius

The circumference, C, of a circle can be found using one of the following formulas:

C = 2π r r

is the radius of the circle

C = πd d

is the diameter of the circle

Fractions of a circle An arc is a fraction of the circumference of a circle. To ﬁnd the arc length of one-quarter, half or three-quarters of a circle we multiply the formulas of the circumference by the appropriate fraction. This arc is half of a circle.

Arc length =

× 2π r

Arc length =

× πd

This arc is a quarter of a circle.

r r

Arc length =

× 2π r

Arc length =

× πd

This arc is three-quarters of a circle. Arc length =

× 2π r

Arc length =

× πd

Example 4 Find the length of the arc rounded to one decimal place. 15 km

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Create a strategy Use the formula Arc length =

× 2π r.

Apply the idea Write the formula Substitute r into the formula Evaluate

Example 5 Calculate the perimeter of the shape rounded to two decimal places.

11 mm

Create a strategy The perimeter of the shape is made of two straight line sections of length 11 mm and three quarters of the circumference of a circle with radius 11 mm. We can use the formula C = 2π r.

Apply the idea Write out the strategy Substitute the radius Evaluate using a calculator Round the answer

Idea summary An arc is a fraction of the circumference of a circle. To ﬁnd the arc length of a fraction of a circle, such as ,

or , ﬁnd the value of the

full circumference and then ﬁnd the corresponding fraction of that. When ﬁnding the perimeter, don’t forget to add in any relevant radius length.

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10.02 Practice questions Understanding 1

Explorer 1–3

Adventurer 2–4

Trailblazer 2–4

What is the diameter of this circle in terms of x?

Write the circumference of the circle in terms of π and x. x

Write the diameter of the semicircle in terms of r.

What is the length of the semicircular arc in terms of π and r?

Write the perimeter of the semicircle in terms of π and r.

Write the diameter of the quarter circle in terms of r.

What is the length of the quarter circle arc in terms of π and r?

Write the perimeter of the quarter circle in terms of π and r.

b c d e

π is equal to 3.142. π is equal to 3.14 when rounded to two decimal places. π can be expressed as a fraction using whole numbers. The formula of the circumference of a circle with diameter d is C = 2π d. To calculate the length of a semicircular arc we can work out the circumference of the whole circle and then divide by 2.

Fluency

Ex 1

Are these statements true about π ? a

5ab, 6ab, 7, 8ac, 9, 10ac, 11a, 12ab, 13ab

5ab, 6ab, 7, 8ac, 9, 10b, 11b, 12ab, 13ab

5cd, 6cd, 7, 8bd, 9, 10d, 11cd, 12cd, 13cd

Given x, calculate the circumference of the circle. Round your answer to two decimal places. a

x = 7 cm

x = 15 m

x = 12.5 mm

x = 2.25 km

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6 Ex 2

Given d, calculate the circumference of the circle. Round your answer to two decimal places. a

d = 10 m

d = 3.4 mm

d = 17.5 cm

d = 0.68 km

A circle has a circumference of 50 cm. Calculate the radius rounded to one decimal place.

Ex 3

Calculate the radius of the circle, rounded to two decimal places: a

A circle with circumference of 30 cm

A circle with circumference of 11 cm

A circle with circumference of 25 cm

A circle with circumference of 100 cm

A circle has a circumference of 30 cm. Calculate the diameter rounded to two decimal places.

Calculate the diameter of the circle, given the circumference. Round your answer to two decimal places.

Circumference = 44 cm

Circumference = 6 cm

Circumference = 55 cm

Circumference = 200 cm

Calculate the diameter of the circle, given the circumference. Round your answer to one decimal place. a

Circumference = 37 cm

Circumference = 44 cm

Circumference of = 70 cm

Circumference = 24.5 cm

For each value of r: i

Find the length of the semicircular arc.

Find the perimeter of the semicircle. r

Round all answers to two decimal places.

13 Ex 4, 5

r = 11 m

r = 4.8 cm

r = 9.2 cm

r = 22 mm

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.

496

r = 14 cm

r = 10.5 m

r = 6.8 cm

r = 18 km

Mathspace Victorian Curriculum – Year 8 mathspace.co

Reasoning 14

14, 15

14–16

16–18

Are these statements true or false? Justify your answer. a

The circumference of a circle is always greater than its diameter.

The circumference of a circle is always twice its radius.

The circumference of a circle with diameter 2 cm is always greater than the circumference of a circle with radius 1 cm.

Caitlin and David calculate the circumference of this circle using different formulas. a

Caitlin uses the formula C = π d to calculate the circumference.

10 cm

Find her result rounded to two decimal places. b

David uses the formula C = 2π r to calculate the circumference. Find his result rounded to two decimal places.

c 16

Compare the results of their calculations and explain what you notice.

Calculate the value of 2π rounded to two decimal places and then consider: a

The radius of a sphere is 32 m. Two cables are wrapped around the sphere, one at the surface of the sphere and one 1 m above the surface of the sphere. Find the difference in the length of the two cables, rounded to two decimal places. What do you notice?

Would the relationship you noticed in part (a) hold if we had a much larger sphere, such as wrapping the cable around the equator of the Earth, radius 6400 km?

The perimeter of the quarter circle is p cm. If we were ﬁnding the radius, explain how to derive the equation

The perimeter of the semicircle is p cm. If we were ﬁnding the radius, explain how to derive the equation

Problem-solving 19, 21, 23 19

19, 21, 23

20–24

The perimeter of the semicircle is 50 m. Find the radius of the semicircle rounded to two decimal places.

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The perimeter of the quarter circle is 25 cm. Find the radius of the quarter circle rounded to two decimal places. r

Find the perimeter of the shaded region. Round your answer to two decimal places. 10 cm

Find the perimeter of the shaded region. Round your answer to two decimal places.

15 cm 15 cm

A fairground roundabout has seats which are 5 m and 6 m away from the centre. For a ride, Jasper sits on the inner seat and Juliet sits on the outer seat. If the ride consists of 50 complete revolutions, how much further does Juliet travel than Jasper? Round your answer to the nearest metre.

Kiki trains on a circular track with 8 lanes. The inner lane has a radius of 20 m and each lane is 1 m further out than the previous one.

498

One week, Kiki’s training each day consists of running 80 laps in the inside lane. How far does she run over the course of the seven days? Round your answer to the nearest metre.

The next week, Kiki decides to run 10 laps in each of the 8 lanes each day. How much further does she run in the second week compared to the ﬁrst week? Round your answer to the nearest metre.

Mathspace Victorian Curriculum – Year 8 mathspace.co

10.03 Perimeter of composite shapes After this lesson, you will be able to… • apply understanding of perimeter calculations to composite shapes that include parts of circles. • apply Pythagoras’ theorem to calculate unknown side lengths within composite shapes as part of determining their perimeters. • solve complex problems involving the perimeter of composite shapes, including those that contain parts of circles.

Perimeter of composite shapes The perimeter of a composite shape is found by adding the lengths of the sides around the boundary of the shape. If the boundary includes parts of a circle then the appropriate formulas will need to be used. Sometimes parts of circles can be combined. For example, the arc lengths of two semi-circles may combine to make the circumference of one circle. Sometimes a length may need to be calculated using subtraction or addition or using Pythagoras’ theorem.

Example 1 Find the perimeter of the shape, giving 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.

10.03 Perimeter of composite shapes mathspace.co

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Apply the idea he top straight edge is 17 cm, the bottom T straight edge is 17 + 6 cm and we have the semi-circle and the quarter circle

= 40 + 9.4247… + 9.4247…

valuate each part then add them, or put E the whole expression into your calculator

= 58.8495…

Evaluate

= 58.85 cm

Round the answer

Example 2 An outline of a block of land is shown:

17 m xm

a Find the length of the side labelled x m.

8m 32 m

Create a strategy Visualise the block of land as a combination of triangle and rectangle as shown. 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

Square root 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 all the lengths of the sides.

Apply the idea Perimeter = 17 + 17 + 8 + 32 m

Add together the four straight side lengths

Perimeter = 74 m

Evaluate

Idea summary The perimeter of a composite shape is found by adding the lengths of the sides around the boundary of the shape. If the boundary includes fractions of a circle then the appropriate formulas will need to be used. With some shapes it may be necessary to work out unknown lengths before we can calculate the perimeter. This may involve using concepts such as Pythagoras’ theorem.

10.03 Practice questions Understanding

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

What is the difference between a simple shape and a composite shape?

Which is the correct approach for finding the perimeter of a composite shape?

There is no formula, it depends on the shape

Add the lengths of all the exterior sides together

Multiply the length of one side by the number of sides

Which of these statements is true about a composite shape? A

It can be made up of only one simple shape

It can be made up of two simple shapes that do not touch each other

It can be made up of two or more simple shapes that are touching each other

It cannot be made up of circles

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Fluency 4 Ex 1

4, 5, 7, 8a–c, 9a–f, 11, 12

4, 6, 7, 8d–f, 9c–g, 10, 12

4, 6, 7, 8ef, 9d–h, 10, 12

For each of the composite shapes: i

What are the basic shapes that make up the composite shape?

Find the perimeter rounded to two decimal places, where appropriate.

22 cm 11 cm

20 cm 10 cm 18 cm

16 cm 7 cm 18 cm

16 cm

Calculate the perimeter: 2 mm x 5 mm 13 mm

Calculate the perimeter: a

4m 9m

18 m 5m

x y 2m

12 m

y 3m 3m

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Calculate the perimeter: 15 m 4m 10 m 11 m

Find the perimeter: a

11 cm

18 cm

4 mm 6 mm

8 mm

4 cm 8 cm

4 mm

2 mm 10 mm

8 cm

10 cm 8 cm

7 cm

9 cm

6 cm

11 cm

7 cm

7 mm

16 m

12 m

3 mm

10 m 15 mm

31 m

17 mm

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Find the perimeter, rounded to one decimal place: a

6 cm

7 cm

3m 94.8 3m

5 cm

34 cm

46 cm 20 cm

26 cm

17 cm

Find the perimeter of the shaded arc, rounded to two decimal places:

A rectangle has two semicircles of radius 3 m cut out at each end as shown: Find the perimeter of the shape, rounded to two decimal places.

12 Ex 2

12 m

An outline of a block of land is pictured: a

Find the value of x.

Find the perimeter of the block of land.

11 m xm

6m 19 m

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Reasoning 13

13, 14

13–15

Vivek is calculating the perimeter of the ﬁgure. He writes down this calculation: P = 7 + 18 + 7 + 18 + π × 18 Vivek has made two mistakes. What are they?

7 cm 18 cm

Determine whether the statements are 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.

Use two different methods to calculate the perimeter of the shape to the nearest centimetre.

18 cm

12 cm 8 cm

Problem-solving 16–18

17–20

treet Cedar S 127

11.5

The diagram shows the image of 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: 4000.

Main Stre

17–19

Line of wall

Building

17.5 17.5

Property Line 127

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Ellie is fencing a paddock. The length across the paddock is 90 m. a

Find the value of x rounded to two decimal places.

What length of fencing does Ellie need in total?

If the cost of fencing is $20.21 per metre, how much will the fencing cost?

x 90

100

A farmer wants to build a fence around the entire perimeter of his land, as shown in the diagram. The fencing costs $37 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 him to build the fence?

A 1m F x

5m 7m

E 2m

y C 3m

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 20 m of wall. a

7m 1.5 m

3.5 m

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, to one decimal place.

The insecticide costs $18 per litre, the labour cost for the job is $70 and there is a call-out fee of $35. Find the total cost of the treatment.

According to regulations, a standard 400 m running track consists of a straight section with length 84.39 m and two semicircular sections. The radius of the inner edge of Lane 1 is 36.80 m and the radius of the inner edge of Lane 2 is 38.02 m. a

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.

506

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.

Mathspace Victorian Curriculum – Year 8 mathspace.co

38.02 m 36.80 m

Lane 1 Lane 2 84.39 m

Start

10.04 Estimation and units of area After this lesson, you will be able to… • describe and provide examples of different units of area measurement and correlate these units with real-life objects or spaces. • select the most appropriate unit of area for measurement. • convert a unit to a different unit of area as required.

Units of area When we have a 2D shape, the amount of space inside the shape is called the area of the shape. The area is a measure of how many squares ﬁt inside the shape.

The unit square

In the metric system the squares to measure area have side lengths of 1 mm, 1 cm, 1 m or 1 km.

1 unit

From smallest to largest, the common units used for measuring area are: • square millimetres (mm2) • square centimetres (cm2) • square metres (m2) • square kilometres (km2)

1 unit2 of area 1 unit

To be able to visualise and estimate areas, it can be useful to identify common objects that have an approximate area of these units. For example: • 1 mm2 is about the size of a piece of glitter. • 1 cm2 is about the area of a ﬁngernail. • 1 m2 is roughly the area of the floor below a shower, or that of a school desk. • 1 km2 is the surface area of 800 Olympic swimming pools, or the area of several city blocks. A hectare is a special unit of area often used to measure the area of a piece of land. It is equal to a piece of land that is 100 m long by 100 m wide. Therefore 1 ha = 10 000 m2. It’s also still common to see the word acre used to measure the size of land in Australia. 1 hectare is approximately equivalent to 2.47 acres.

Example 1 Choose the most appropriate unit of measurement. a Which of these areas is best measured in square metres (m2): A Your classroom

An exercise book

C A matchbox

Australia

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Create a strategy Think about what units would be appropriate to measure each object.

Apply the idea Square metres (m2) is suited for big sizes, so we could use it to measure the area of your classroom. Square centimetres (cm2) is suited for smaller objects, so we could use it to measure the area of a matchbox or an exercise book. Square millimetres (mm2) is suited for tiny objects, so we could use it to measure the area of a matchbox. Square kilometres (km2) is suited for bigger places, so we could use it to measure the area of Australia. The answer is option A.

b Which of these areas is best measured in square centimetres (cm2): A Your street

A school playground

C Your pencil case

Japan

Create a strategy Think about what units would be appropriate to measure each object.

Apply the idea Square metres (m2 ) is suited for big sizes, so we could use it to measure the area of your street, and a school playground. Square centimetres (cm2) is suited for smaller objects, so we could use it to measure the area of your pencil case. Square kilometres (km2 ) is suited for bigger places, so we could use it to measure the area of Japan. The answer is option C.

Idea summary The area of a 2D shape is the amount of space inside the boundaries of the shape. Area is measured in square units: mm2, cm2, m2 and 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.

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Estimate area Estimation of areas, and understanding area benchmarks are useful skills that will help us to be able to appropriately choose units for area and to be able to visualise the size of something if we are told the units of area. There are some a few different ways we can estimate area: • Use a mental image of common objects of size 1 mm2, 1 cm2, 1 m2 and/or 1 km2 and try to picture how many it would take to cover the object. • For shapes that are rectangular (or close to rectangular), estimate the length and width of the object and then multiply: Area = length × width. • If a picture of the item is available, you could draw a scale grid over it and estimate by counting the number of squares needed to cover the item.

Example 2 Estimate the total area of the artwork above the sofa if the sofa is 2.2 m in length. A 1.7 m2

0.02 m2

C 5 m2

0.4 m2

2.2 m

Create a strategy Use the dimensions of the sofa to estimate the area of the artwork above it.

Apply the idea Let’s start by estimating the length of the artwork. Since the length of the sofa is 2.2 m, 2 m is a reasonable assumption for the length of the artwork. We could estimate the width of the artwork to be half of the length. So the largest area the artwork could take up would be 2 m × 1 m = 2 m2. This rules out option C. The closest to 2 m2 is option A at 1.7 m2.

Example 3 Each square on the grid has an area of 3 mm2. Estimate the area of the curved shape shown: A 111 mm2

72 mm2

C 26 mm2

48 mm2

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Create a strategy

Apply the idea

Count the number of squares that are inside or almost completely inside the curved shape, multiply by the area of 3 mm2, and then choose the option that is closest to this result.

Area = 21 × 3 = 63 mm

Multiply 21 by 3 2

Evaluate

Since option B is the nearest and higher than 63 mm2, the estimated area of the curved shape is 72 mm2.

Idea summary Techniques we can use to estimate area: • Use a mental image of common objects of size 1 mm2, 1 cm2, 1 m2 and/or 1 km2 and try to picture how many it would take to cover the object. • For shapes that are rectangular (or close to rectangular), estimate the length and width of the object and then multiply: Area = length × width. • If a picture of the item is available, you could draw a scale grid over it and estimate by counting the number of squares needed to cover the item.

Convert units of area When we convert units of area we are changing the size of the squares that the area is measured in.

Interactive exploration

Discover this concept in action online

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mathspace.co

The conversion factors are summarised in the conversion chart: ÷ 10 000

÷ 100

mm²

cm²

× 100

÷ 10 000

hectares

m²

× 10 000

hectares

acres

× 2.47

Example 4 Convert 6 m2 to cm2.

Create a strategy Sketch a square with side length of 1 m. We want to convert to centimetres, how many centimetres will each side length be and work out how many square centimetres is in 1 square metre.

Apply the idea 6 m2 = 6 × (100 × 100) 2

= 60 000 cm

Convert the square metre into centimetres Evaluate

Idea summary When converting between units of area: • Multiply if converting to a smaller unit - more smaller squares will be needed to cover the same area. • Divide if converting to a larger unit - fewer larger squares will be needed to cover the same area.

10.04 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

Fill in the blanks. There are ⬚ mm in 1 cm, ⬚ cm in 1 m and ⬚ m in 1 km. 10.04 Estimation and units of area mathspace.co

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Order these units of area from smallest to largest: • km2 • cm2 • mm2 • m2

Name an object that has an area of approximately 1 m2.

Name an object that has an area of approximately 500 cm2.

Fluency Ex 1

4, 6–9, 11ab, 12

4, 6–9, 11bc, 12

4–6, 8, 10, 11cd, 12

For these areas, choose the most appropriate unit of measure from the list: • Hectares • Square centimetres • Square metres • Square kilometres a

Postage stamp

Bathroom wall

Paper

Cabinet

City

Television

Kitchen

Farm

Netball court

Table

National park

Door

Find the area of a square with sides 290 cm in length. Give your answer in square metres.

This garden bed has three rows of four plants. Each plant is spaced about 24 cm apart, and sits 12 cm away from the edge of the garden bed. Estimate the area of the garden bed in square centimetres.

Ex 2

Estimate the area of the Mathspace logo on the box. A

150 cm2

1 m2

750 cm2

1000 cm2

50 cm 1.5 m

Ex 3

Each square on the grid has an area of 4 mm2. Estimate the area of the curved shape.

Estimate the area of the shape shown if each axis is measured in metres.

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Mathspace Victorian Curriculum – Year 8 mathspace.co

16 14 12 10 8 6 4 2 0

50 cm

2 4 6 8 10 12 14 16

Estimate the area of the butterﬂy if each axis is measured in millimetres.

10.4 9.1 7.8 6.5 5.2 3.9 2.6 1.3 0

1 2 3 4 5 6 7 8

Estimate the area: a

10 cm2

200 cm2

50 cm2

55 000 mm2

6 m2

100 mm2

6000 cm2

7000 cm2

7000 mm2

6000 mm

2 m2

7 m2

70 cm2

400 mm2

2000 mm2

b 10 cm 4 cm

78 cm

70 cm

18 mm

66 mm Ex 4

Convert: a

3 m2 to square centimetres

7 m2 to square centimetres

0.56 km2 to square metres

6 ha to square metres

10.4 ha to square metres

20 000 m2 to hectares

Reasoning

13, 15, 16

13, 14, 16

There are 100 cm in 1 m but to convert from square metres to square centimetres we must multiply by 10 000. Explain why this is correct.

Determine whether these statements are true or false. Explain your reasoning. a

The unit of estimation must always be the same as the unit of measurement used to ﬁnd the exact area.

The area of a square is equal to the perimeter of the square divided by 4.

The area of a rectangle with length 5 cm and width 2 cm is equal to the area of a rectangle with length 2 cm and width 5 cm.

The area of a shape can never be greater than its perimeter.

When estimating area, it is always best to round the measurements to the nearest whole number.

Why is it a good idea to estimate the area of a shape before calculating it exactly?

A hectare is approximately the size of an athletics arena. Why are blocks of land, farms and large properties measured in hectares? 10.04 Estimation and units of area mathspace.co

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Problem-solving 17–19

17–22

17–24

These road signs are pictured with relative sizings. If the GIVE WAY sign is an equilateral triangle with a side length of 75 cm, estimate the area of the BICYCLES EXCEPTED sign.

A park has an area measuring 19 acres. Given that 1 acre is approximately 4000 m2, ﬁnd the area of the park in square metres.

A sample of rectangular farms around Australia were measured and their dimensions, in metres, are recorded in the table:

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?

Which farm has an area of exactly 1 ha?

Farm 1 2 3 4 5

Length, m Width, m Area, m2 200 100 350 21 100 24 400 40 100 100

A property covers an area of 7.22 ha. Given that 1 ha is 10 000 m2, calculate: a

The area of the property in square metres.

The area of the property in acres, given that 1 acre is approximately 4000 m2.

A large cattle station covers an area of 320 acres. Given that 1 acre is approximately 0.4 ha, and 1 ha is 10 000 m2, calculate: a

The area of the cattle station in hectares.

The area of the cattle station in square kilometres.

A farm covers an area of 51 000 m2. Given that 1 ha is 10 000 m2, and 1 acre is approximately 4000 m2, calculate: a

The area of the farm in hectares.

The area of the farm in acres. Round your answer to two decimal places.

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, ﬁnd 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:

514

The area of the farm in hectares.

The area of the farm in square metres.

Mathspace Victorian Curriculum – Year 8 mathspace.co

10.05 Area of rectangles, triangles and parallelograms After this lesson, you will be able to… • calculate the area of rectangles and parallelograms using the appropriate formula. • find the missing dimension of a rectangle or parallelogram given its area and one of its dimensions. • apply knowledge of area calculation to solve real-world problems that involve rectangles and parallelograms.

Area formulas The area of a rectangle is given by width

Area = length × width, or A=l×w

length

The area of a square is given by Area = side × side, or A = s × s = s2

side

The area of a triangle is given by height

base

The area of a parallelogram is given by height base

Area = base × height, or A=b×h

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Example 1 Find the area of a rectangle whose length is 11 cm and width is 5 cm.

Create a strategy

Apply the idea

Use the area of a rectangle formula.

Area = l × w Start with the formula for the area of a rectangle = 11 × 5

Substitute l = 11 and w = 5 2

= 55 cm

Evaluate

Example 2 Find the area of the given triangle with base length 10 m and perpendicular height 8 m.

8m 10 m

Create a strategy Use the formula for the area of a triangle: A =

×b×h

Apply the idea For this triangle, the base is b = 10 m and the height is h = 8 m. Start with the formula for the area of a triangle

Substitute b and h

Evaluate

Example 3 Find the area of a parallelogram whose base is 15 cm and height is 7 cm.

Create a strategy Use the area of a parallelogram formula.

Apply the idea A=

×b×h

= 15 × 7

Substitute b = 15 and h = 7 2

= 105 cm

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Start with the formula for the area of a parallelogram Evaluate

Mathspace Victorian Curriculum – Year 8 mathspace.co

Idea summary The area of a rectangle is given by:

A=l×w A l w

is the area of the rectangle is the length of the rectangle is the width of the rectangle

The area of a square is given by:

A=s×s A s

is the area of the square is the side length of the square

The area of a triangle is given by:

A b h

is the area of the triangle is the base of the triangle is the perpendicular height of the triangle

The area of a parallelogram is given by:

A=b×h A b h

is the area of the parallelogram is the base length of the parallelogram is the perpendicular height

Provide all answers with appropriate units.

Find dimensions from the area To ﬁnd a missing dimension, we substitute the area and the known dimension into the area formula and solve for the unknown. The parallelogram shown has an area of 792 m2, and a base length of 88 m. How can we determine the perpendicular height? From the formula, we know that h

Area = b × h Substitute known values for A and b: Solve for h: h

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Example 4 A rectangular park has an area of 36 m2. If the park is 4 m wide, how long is it?

Create a strategy Use the area of a rectangle formula.

Apply the idea Start with the formula for the area of a rectangle Substitute A = 36 and w = 4 Divide both sides by 4 Evaluate

Idea summary Find a missing dimension of a rectangle, square or parallelogram by substituting the area and given dimension into the area formula, and then solve for the unknown variable.

10.05 Practice questions Understanding 1

Explorer 1–3, 4ac, 5

Adventurer 2, 3, 4bd, 5

Trailblazer 2, 3, 4bd, 5

For the image shown: a

The length is a and the width is b.

b cm

Explain how we would work out the area of the rectangle.

Write an expression for the area of the rectangle.

What would the units be for the area?

If the lengths had been in metres instead of centimetres, what would the units be for the area?

Casper, Ollie and Stella are working out the area for the rectangle shown, given that x = 2 and y = 20. a Stella suggests multiplying 2 and 20. Explain why Stella is incorrect. b Ollie plans to change the length into centimetres, so he will multiply 200 and 20. i Is Ollie using a valid method? ii What would be the units for Ollie’s answer? c

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a cm

y cm xm

Casper intends to change the width into metres, so he will multiply 2 and 0.2 i Is Casper using a valid method? ii What would be the units for Casper’s answer?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Alex and Beth are calculating the area of the square shown, where a = 4. a Alex says that a square is a special case of a rectangle, so he can work out the area by calculating 4 × 4. Is Alex correct? b Beth says that the formula for the area of a square is A = a2, so she calculates 42. Is Beth correct? c After realising that they both got an answer of 16, they conclude that they could use either method. Are they correct? d What would be the units for their answer?

Identify the base and height of the triangle: a

3 cm

32 mm

5.3 cm 2.65 cm 20 mm

13 mm

22 mm

7 cm

8 cm

d 18.5 cm

10 cm

29 cm

6.35 cm 14.5 cm 11 cm

10.5 cm

A parallelogram with a base of 12 cm and a height of 9 cm is formed into a rectangle: a

Find the length of the rectangle.

Find the width of the rectangle.

Calculate the area of the rectangle.

Explain why the area of the parallelogram and the rectangle are the same.

9 cm

12 cm 9 cm

Find the base of the parallelogram.

Find the height of the parallelogram.

With reference to the diagram, explain why the formula for the area of a parallelogram is A = bh. 6ace, 7, 10, 11ace, 12a, 6bdf, 8, 10cd, 11bdf, 12bc, 6ef, 8, 9, 10cd, 11ef, 12d, 13ace, 14ac, 15, 17ac, 18ac 13bdf, 14bd, 16, 17bd, 18bd 13ef, 14d, 16, 17bd, 18bd

Fluency

Ex 1

12 cm

Calculate the area of the rectangle using each of the given dimensions, in each of the units provided.

width length

length = 7 cm, width = 4 cm

length = 12 mm, width = 5 mm

length = 15 m, width = 6 m

length = 6 km, width = 3 km

length = 12 cm, width = 6 mm

length = 5 m, width = 40 cm i

in cm2

in m2

in cm2

in mm2

10.05 Area of rectangles, triangles and parallelograms mathspace.co

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A square has a perimeter of 20 cm. Calculate the area of the square.

A rectangle has a perimeter of 20 cm and a length of 8 cm. Calculate the area of the rectangle.

A rectangle has an area of 20 cm2. The length and width are integer values, and the length is greater than the width. Write down the dimensions of three different rectangles that satisfy these conditions.

Find the area:

Ex 2

b 6 cm

5 cm

8 cm

6 cm

50 cm 45 cm

9 cm

73 cm

4 cm Ex 3

Calculate the area of the parallelogram using each of the given dimensions, in each of the units provided a

base = 9 m, height = 7 m

base = 10 mm, height = 5 mm

base = 20 cm, height = 3 cm

base = 14 m, height = 5 m

base = 8 km, height = 400 m i

14 Ex 4

520

base

in m2

base = 9 m, width = 4 cm i

in cm2

in m2

Find the width of the rectangles, given the area and the length. a

in km2

height

area = 14 m2, length = 7 m 2

area = 36 mm , length = 9 mm

area = 28 cm2, length = 14 cm

area = 50 cm2, length = 12.5 cm

Find the area of the triangle that has the dimensions: a

Height = 10 cm and base = 7 cm

Height = 5 m and base = 3 m

c e

Height = 6 mm and base = 4 mm

Height = 2 cm and base = 8 cm

Base = 15 cm and height = 80 mm

Base = 6.2 m and height = 180 cm

Find the height of the parallelograms, given the area and the base length. a

area = 40 cm2, base = 8 cm

area = 56 m2, base = 7 m

area = 39 km2, base = 3 km

area = 60 mm , base = 15 mm

Mathspace Victorian Curriculum – Year 8 mathspace.co

Complete the table for lengths, widths and areas of rectangles. Length (cm) Width (cm) Area (cm2)

16 13 52

25 5

Area = 24 cm2 and height = 6 cm

Area = 35 cm2 and height = 7 cm

Area = 14 mm2 and height = 7 mm

Area = 300 mm2 and height = 3 cm

Find the height measurement for the triangle given the area and base: a

Area = 50 cm2 and base = 10 cm

Area = 100 cm2 and base = 25 cm

Area = 27 mm2 and base = 3 mm

Area = 800 mm2 and base = 40 cm

19, 20

20, 21

raw a diagram showing how we can re-form this parallelogram D into a rectangle.

If the parallelogram is formed into a rectangle, ﬁnd:

Find the base measurement for the triangle given the area and height:

Reasoning 19

9 54

16 4

Complete the table for bases, heights and areas of parallelograms. Base (cm) Height (cm) Area (cm2)

The length of the rectangle

The width of the rectangle

20–22

6 cm

10 cm

Find the area of the parallelogram.

The dimensions of the large rectangle are double the size of the small rectangle. a

Find the value of x.

Is the area of the large rectangle double the area of the small rectangle?

If the side lengths of a rectangle are doubled, by what factor is the area increased?

If the side lengths of a rectangle are tripled, by what factor is the area increased?

10 cm

12 cm x cm2 6 cm

Two right triangles have the same area. If one triangle has a base of 7 cm and a height of 9 cm, what are the possible dimensions of the other triangle given the base and height are whole numbers?

10.05 Area of rectangles, triangles and parallelograms mathspace.co

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A rectangle and parallelogram each have sides of a cm and b cm. Which shape would have the greater area? Explain.

Problem-solving 23, 25, 27 23

24, 26, 27

24, 26–29

A rectangular swimming pool measures 12 metres by 8 metres. a

What is the area of the pool in square metres?

If the pool deck surrounding the pool is 2 metres wide, what is the total area of the pool and deck in square metres?

A rectangular garden measures 9 metres by 13 metres. A path runs around the outside of the garden, with a width of 1 metre. What is the area of the path in square metres?

The ﬂoor of a room measures 8 metres by 10 metres. What is the area of the ﬂoor in square metres if the room is in the shape of a parallelogram?

A farmer wants to install a fence around his rectangular field. One of the longer sides of the field is 60 metres, and the shorter side is 30 metres. He wants to add a rectangular-shaped area to the field that extends 20 metres out from the shorter side. What is the area of the rectangular-shaped area he wants to add?

A rectangular room has a floor that measures 4 m by 3 m. The floor is to be tiled with tiles that are a 25 cm square. How many tiles are required to completely cover the floor?

Amy wants to paint her bedroom walls. Two of the walls measure 4 m high and 6 m wide, while the other walls measure 4 m high and 5 m wide. There is a triangular window on one of the walls that has a base of 4 m and height of 2 m. If each litre of paint can cover an area of 8 m2, how many whole litres of paint will Amy need to buy?

Ryan is re-tiling his rectangular kitchen ﬂoor. The kitchen measures 3 m by 5 m, but there is an island in the centre in the shape of a right triangle with a base of 6 m and a height of 2 m. Each tile measures 30 cm by 30 cm.

522

Calculate the number of tiles that Ryan needs to buy.

If Ryan needs to account for 15% wastage, how many tiles does he need to buy?

Mathspace Victorian Curriculum – Year 8 mathspace.co

10.06 Area of composite shapes After this lesson, you will be able to… • identify the individual shapes that make up the composite shape. • calculate the total area of the composite shape by accurately finding and adding the areas of each individual shape. • calculate the total area of the composite shape by accurately finding the area of the larger shape and subtracting the areas of the cutaways or holes. • solve problems that require the calculation of the area of composite shapes.

Area of composite shapes using addition Irregular shapes have different length sides and internal angles. We can think of a composite shape as one that is made from a number of smaller shapes. Irregular shapes can be considered composite shapes if they can be decomposed into simpler shapes. Dashed lines can be used to visualise which simple shapes make up a composite or irregular shape.

We ﬁnd the total area by adding the areas of each of the simple shapes.

Example 1 Find the area of the ﬁgure shown. 4 mm

9 mm

15 mm

Create a strategy Calculate the area of the triangle and rectangle, then add them.

10.06 Area of composite shapes mathspace.co

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Apply the idea Area of the triangle: Use the area of triangle formula

Substitute b = 15 and h = 4

Evaluate

Area of the rectangle: A=l×w

Use the area of rectangle formula

= 15 × 9

Substitute b = 15 and h = 9 2

= 135 mm

Evaluate

Total area: A = 30 + 135 2

= 165 mm

Add the areas Evaluate

Example 2 Find the total area of the ﬁgure shown.

6 cm 15 cm

19 cm

6 cm

Create a strategy Divide the shape into three. Use the area formulas for a parallelogram and rectangle.

Apply the idea Area of parallelograms: A=b×h

Use the area of parallelogram formula

= 19 × 6

Substitute b = 19 and h = 6

= 114

Evaluate

Both parallelograms = 114 × 2

Multiply the area by 2 2

= 228 cm

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Mathspace Victorian Curriculum – Year 8 mathspace.co

Evaluate

Area of rectangle: A=l×w

Use the area of rectangle formula

= 19 × 15

Substitute l = 19 and w = 15 2

= 285 cm

Evaluate

Total Area: A = 228 + 285 2

= 513 cm

Add the areas Evaluate

Idea summary To find the area of an irregular or composite shape we can add the areas of the individual shapes it is made up of.

Area of composite shapes using subtraction We can also ﬁnd the area of a composite shape using subtraction. =

−

Area of = Area of composite shape rectangle

−

Area of triangle

Example 3 Find the shaded area in the ﬁgure shown. 6 cm 3 cm 14 cm

Create a strategy Subtract the area of the triangle from the area of the parallelogram. −

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Apply the idea Area of triangle: Use the area of triangle formula Substitute b = 14 and h = 3 Evaluate Area of parallelogram: A=b×h

Use the area of parallelogram formula

= 14 × 6

Substitute b = 14 and h = 6

= 84

Evaluate

Area of composite shape: A = 84 – 21

Subtract the area of triangle from the area of parallelogram

= 63 cm

Evaluate

Idea summary To find the area of an irregular or composite shape with holes and cutaways we can subtract the cutaway areas from the larger area.

10.06 Practice questions Understanding 1

Explorer 1–2

Adventurer 1–2

Name the two geometric shapes that together form this composite shape.

Explain how to ﬁnd the total area.

Trailblazer 1–2 6 cm 5 cm 9 cm

How would you calculate the total area of this shape? 14 cm 8 cm 9 cm

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Fluency 3 Ex 1, 2

3ace, 4a–c, 5a, 6a

3bdf, 4bc, 5b, 6b

3bdf, 4cd, 5b, 6b

Find the area of the composite shapes: a

6 cm

2 cm

10 cm 5 cm

7 cm 6 cm

7 cm

6 cm 2 cm

8 cm

2 cm

6 cm

7 cm

14 cm

0.35 m

5 cm 9 cm 10 cm

14 cm

0.25 m 1m

4 Ex 3

Find the shaded area in the diagrams. Round your answer to one decimal place. a

9 cm

3 3 cm

10 m

2m 3m

13 m

17 m

10 m 20 m

For each ﬁgure: i

Find the value of x.

3 cm

Find the area of the shaded region. 3 cm x cm

x cm

7 cm

10 cm

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For each ﬁgure: i

Find the value of x.

iii

Calculate the area of the shaded region.

y cm x cm

Find the value of y.

x cm

A = 25 cm2

A = 16 cm2

15 cm A = 100 cm2

Reasoning 7

7, 8

y cm

7, 8

Calculate the shaded area using the addition method and then the subtraction method. Comment on which method is the most eﬃcient. 15 3 7 11 4 5

Use the images in Figures 1 and 2 to write steps explaining how to calculate the shaded area in Figure 3. Note that you do not need to calculate the actual areas. • Figure 1: • Figure 2: • Figure 3: 13 cm

13 cm

4 cm 9 cm

9 cm

4 cm 9 cm

6 cm 6 cm

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Mathspace Victorian Curriculum – Year 8 mathspace.co

Problem-solving 9, 10

9, 10, 12

9–12

A piece of origami paper, originally in the shape of a parallelogram, is folded along its shortest diagonal as shown. What is the total area covered by the folded paper? 17 cm

6 cm

The back cover of a mobile phone is shown in the diagram. The rounded corners are quadrants: Find the area of the back cover, excluding the circular camera lens. Round your answer to two decimal places.

10 cm 0.5 cm

5 cm

0.2 cm

The diagram shows a parallelogram-shaped area of lawn. There are sprinklers placed at points A and B, which are conﬁgured to water the indicated sectors of the lawn.

Find the area of lawn that is covered by sprinkler A in square metres. Round your answer to one decimal place.

Find the area of lawn that is covered by sprinkler B in square metres. Round your answer to one decimal place.

If each sprinkler uses 2 L of water per square metre in one hour, ﬁnd the amount of water the sprinklers use in an hour, in litres. Round your answer to one decimal place.

5m 70° B 1.71 m

This tile pattern is made up of two different parallelograms in a tessellated pattern: a

Find the area covered by the smaller tiles.

Find the area covered by the larger tiles.

Find the total area covered by the tiles.

24 m

10.06 Area of composite shapes mathspace.co

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10.07 Area of trapeziums, kites, and rhombuses After this lesson, you will be able to… • define and identify a rhombus, kite, and trapezium. • understand and apply the formula for the area of a rhombus. • understand and apply the formula for the area of a kite. • understand and apply the formula for the area of a trapezium. • compute the area of a rhombus, kite, and trapezium given specific dimensions. • deduce a missing dimension of a rhombus, kite, or trapezium when given the area and one other dimension.

Area of a rhombus A rhombus is a four-sided shape with four equal sides and opposite sides parallel. It is similar to a square, but doesn’t necessarily have right angles. This means a square is a special type of rhombus, and a rhombus is a special type of parallelogram. If we are given the base and perpendicular height, then we can use the formula for the area of a parallelogram to find the area of a rhombus. If we are given the lengths of the diagonals, we need another method for finding the area of a rhombus. We can use the similarities between rhombuses and rectangles to find the formula for area.

Interactive exploration

Discover this concept in action online

We can turn a rhombus into a rectangle that has one dimension equal to the length of one of the diagonals while the other dimension will be half the other diagonal. Notice that it does not matter which diagonal we halve.

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Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 1 Find the area of the rhombus.

A 4 mm C

8 mm

Create a strategy We can ﬁnd the area of rhombus using the formula:

Apply the idea We can see from the diagram that the diagonals have lengths of 4 mm and 8 mm. Start with the formula for the area of a rhombus Substitute the values Evaluate the product

Idea summary The area of a rhombus can be found using the formula:

x y

is the length of one diagonal is the length of the other diagonal

Area of a kite A kite is similar to a rhombus, but it has two pairs of equal sides rather than four equal sides. This means a rhombus is a special type of kite.

Interactive exploration

Discover this concept in action online

mathspace.co

10.07 Area of trapeziums, kites, and rhombuses mathspace.co

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We can use the same formula for the area of a kite as the area of a rhombus.

x y

Example 2 Find the area of the kite.

18 m 10 m

Create a strategy We can use the formula:

Apply the idea Start with the formula for the area of a kite Substitute the diagonal lengths Evaluate

Idea summary The area of a kite can be found using the formula:

x y

532

is the length of one diagonal is the length of the other diagonal

Mathspace Victorian Curriculum – Year 8 mathspace.co

Area of a trapezium Trapeziums (or “trapezia”) have one pair of parallel sides.

Interactive exploration

Discover this concept in action online

mathspace.co

We can turn a trapezium into a rectangle, as demonstrated in the image.

6 cm 8 cm

The length, l, will be the average length of the two parallel sides.

13 cm

So 8 cm

The width is 8 cm. A = l × w = 9.5 × 8 = 76 cm2

l cm

The area of a trapezium is given by: Area =

× (side a + side b) × height h

Where a and b are the lengths of the parallel sides, and h is the perpendicular height.

Example 3 5 cm

Find the area of the trapezium.

3 cm 8 cm

Create a strategy We can ﬁnd the area of trapezium using the formula:

Apply the idea The side lengths are a = 5 cm and b = 8 cm, and the height is h = 3 cm. Start with the formula for the area of a trapezium Substitute the values

Evaluate the product

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Idea summary The area of a trapezium is given by:

a b h

is a parallel side length is the other parallel side length is the perpendicular height

Unknown dimensions If we know the area of the shape and one other dimension, then we can ﬁnd an unknown dimension using the appropriate area formula and solving for the unknown dimension.

Example 4 The kite has an area of 48 cm2. The length of one of its diagonals is 12 cm.

k cm 12 cm

If the other diagonal has a length of k cm, solve for the value of k.

Create a strategy Substitute the area, the known diagonal, and the variable k into the formula for the area of a kite.

Apply the idea Start with the formula for the area of a kite Substitute the values and k Evaluate the product Divide both sides by 6 Evaluate

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Mathspace Victorian Curriculum – Year 8 mathspace.co

Idea summary We can ﬁnd an unknown dimension of a quadrilateral if we know the area and the other dimension. We substitute the known values into the area formula and solve for the unknown dimension. A summary of the area formulas for quadrilaterals: Rectangle

Square

Parallelogram

A = lw

A = s2

A = bh

Rhombus A=

Kite

1 xy 2

Trapezium

1 xy 2

1 ( a + b)h 2

a s

w l

x y

10.07 Practice questions Understanding 1

Explorer 1–3

Adventurer

Trailblazer

1–3

What is a trapezium? A

A quadrilateral with all sides equal

A quadrilateral with one pair of parallel sides

A quadrilateral with all angles equal

A quadrilateral with no parallel sides

What is a kite? A

A quadrilateral with no parallel sides

A quadrilateral with one pair of parallel sides

A quadrilateral with all angles equal

A quadrilateral with all sides equal

Here is a rhombus and a kite. What is the difference between them?

0.9 mm 4 mm Total height

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4, 5a–d, 6ac, 7, 9ab, 10, 12, 13ace, 14a–d, 15a, 16, 18, 19ac, 20a, 21, 23–25

Fluency

5c–f, 6bd, 8, 9bc, 11, 13bdf, 14c–f, 15b, 17, 18, 19ab, 20bc, 22, 26, 27

5c–f, 6bd, 8, 9cd, 11, 13bdf, 14e–h, 15b, 17, 19bd, 20cd, 22, 26, 28, 29

This rhombus can be split into two triangles: a

Find the area of one triangle.

Find the area of the rhombus.

8m 14 m

4m 14 m 14 m 4m Ex 1

Find the area: a

b 6 cm

4m A

6 mm

8 cm

12 mm

6 cm 9 cm

8 mm

4 mm

10 cm

12 cm

536

Determine whether the pair of values could be the diagonal lengths, x and y of a rhombus with an area of 9 m2. a

x = 2 m and y = 9 m

x = 6 m and y = 3 m

x = 12 m and y = 3 m

x = 6 m and y = 6 m

Mathspace Victorian Curriculum – Year 8 mathspace.co

This kite can be split into two triangles: a

Find the area of one of the triangles.

Find the area of the kite.

2m 7m

This kite is formed into a rectangle: a

Find the length of the rectangle.

Find the width of the rectangle.

Find the area of the kite.

8 cm 28 cm

9 Ex 2

Find the area: a

7 cm

12 cm

4 cm 3 cm

21 cm

8 cm

17 m 9m

The trapezium is split into a rectangle and a right-angled triangle: a

Find the area of the rectangle.

Find the area of the triangle.

Find the area of the trapezium.

10 m

9m 19 m

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This trapezium is formed into a rectangle: a

Find the length, l, of the rectangle.

Find the area of the trapezium.

5 cm 8 cm 14 cm 8 cm l cm

Two identical trapeziums are put together to make a rectangle: 8 cm

10 cm

2 cm

10 cm

a Ex 3

8 cm

Find the area of the entire rectangle.

Find the area: a

8 cm

10 mm

5 cm

6 mm 4 mm

2 mm

11 cm

10 mm

6 mm 12 m

9 cm 6 cm

5m 13 cm 8m

538

Find the area of one trapezium.

Mathspace Victorian Curriculum – Year 8 mathspace.co

10 m 4m 14 m

Find the area: a

5 cm

8 cm 9 mm

9 cm

11 mm

4 cm

6 mm

e 4 mm 10 mm

11 cm 8 mm

17 cm

16 m

h 4m

14 m

8m 30 mm

15 mm

For each rhombus, ﬁnd the value of the variable: a

A = 64 cm2

x cm

8 cm

A = 128 cm2

8 cm

b cm

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Rhombus ABCD has an area of A = 55 cm2:

Given the diagonal BD = 11 cm and AC = x cm, ﬁnd the value of x. x cm

11 cm C

Rhombus ABCD has an area of 13 cm2:

If diagonal AC = 2 cm and diagonal BD = y cm, ﬁnd the value of y. A

2 cm y cm

B Ex 4

The kite has an area of 15 cm2. The length of one of its diagonals is 3 cm. Find the length of the other diagonal, k.

k cm 3 cm

Find the value of k: a

A = 42 cm2

A = 22.5 cm2

k cm

k cm 12 cm

A = 56 cm2

A = 137.5 cm2

k cm

11 cm k cm

8 cm

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5 cm

Find the value of the variable: a

A = 42 mm2

A = 36 cm2

a mm

a cm

7 mm 6 cm 8 mm

8 cm

A = 20 m2

A = 24 cm2

b mm

5 cm h cm

4 mm 7 cm 2 mm

Find the value of x if the area of the trapezium is 65 cm2:

12 cm

14 cm

The area of a kite is 640 cm2 and one of the diagonals is 59 cm. If the length of the other diagonal is y cm, ﬁnd the value of y, rounded to two decimal places.

Complete the table of the lengths of diagonal x and diagonal y for three kites that all have an area of 36 mm2: Area (mm2) 36 36 36

Diagonal, x (mm)

Diagonal, y (mm) 18

24 12

A ﬁeld has a rhombus shape with diagonals of 120 metres and 80 metres. What is the area of the ﬁeld in square metres?

A gardener wants to create a ﬂower bed in the shape of a rhombus. The length of one diagonal is 6 metres, and the other diagonal is 8 metres. What is the area of the ﬂower bed in square metres?

A farmer wants to plant a ﬁeld in the shape of a trapezium. The shorter base of the trapezium is 80 metres and the longer base is 120 metres. The height of the trapezium is 50 metres. If the farmer can plant 2500 seeds per square metre, how many seeds will he need to plant the entire ﬁeld? 10.07 Area of trapeziums, kites, and rhombuses mathspace.co

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A carpenter needs to cut a piece of wood in the shape of a trapezium. The longer base of the trapezium is 60 centimetres and the shorter base is 40 centimetres. The height of the trapezium is 24 centimetres. How many square centimetres of wood will the carpenter need to make the piece?

A kite-shaped water feature has diagonals of 30 metres and 20 metres. The owner wants to place tiles on the pool ﬂoor. If each tile is a square with an area of 0.25 square metres, how many tiles are needed to cover the entire pool ﬂoor?

A kite-shaped playground has an area of 24 square metres. The length of one diagonal is 10 metres. What is the length of the other diagonal?

Reasoning

30, 32, 34, 35

30, 31, 33, 36

30, 34, 36, 37

What is a trapezium? Can you explain its properties and how it differs from a parallelogram or rectangle?

What is a rhombus? Can you explain its properties and how it differs from a square?

Determine whether each statement is true or false. Justify your answer.

The area of a trapezium is always equal to the sum of the lengths of its parallel sides multiplied by its height divided by 2.

A rhombus is always a square.

A rhombus is a type of kite.

The area of a kite can be found by multiplying the lengths of its diagonals and dividing by two.

This rhombus is formed into a rectangle: a

Find the length of the rectangle in terms of y.

Find the width of the rectangle in terms of x.

Find the area of the rectangle in terms of x and y.

Find the area of the rhombus in terms of x and y.

Width Length

542

This kite is formed into a rectangle: a

Find the length of the rectangle in terms of y.

Find the width of the rectangle in terms of x.

Find the area of the rectangle in terms of x and y.

Find the area of the kite in terms of x and y.

Mathspace Victorian Curriculum – Year 8 mathspace.co

x y

Justin is trying to work out the area of a rhombus, but he has been given the base and height of the shape. Gaynor says that is ﬁne, he can just use A = bh. Is Gaynor correct? Explain.

Use the image to show that the area of the parallelogram is equivalent to twice the area of the trapezium. 7 mm

10 mm

2 mm

7 mm

10 mm

A kite is cut from a rectangular sheet as shown. Explain why it is not possible to cut a kite spanning the length of the sheet with a larger area.

Problem-solving 38ab

38, 39

38–41

A gardener charges $2 per square metre to mow lawns. a

How much would she charge to mow a trapezoidal lawn with parallel sides 8 m and 10 m, the distance between the parallel sides being 5 m?

How much would she charge to mow a kite-shaped lawn with diagonals 12 m and 8 m?

She charged $120 to mow a lawn in the shape of a rhombus with one diagonal of length 10 m. What was the length of the other diagonal?

For the rhombus in the diagram, one of the diagonals is of length 9 cm. How long is the other diagonal?

8 cm 9 cm

A trapezium has parallel sides that are 4 cm apart. The parallel sides are so that one is 4 times as long as the other. The area of the trapezium is 50 cm2. What are the lengths of the parallel sides?

A company that makes kites produces them with diagonals of length 50 cm and 30 cm. The kites need to be coated with a resin that costs $2.30 per m2. How much does it cost them to apply the coating to 1000 kites?

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INVESTIGATION

Area of circles Investigate how the formula of the area of a circle can be created.

Objectives • To derive the formula for the area of a circle • To recognise irrational numbers in an applied context • To solve problems involving circumference and area of a circle

Area of circles History Archimedes was the ﬁrst to prove the area formula. We’ll look at a different version that is sometimes called the “onion proof”. Imagine that a circle is an onion, with lots and lots of layers. We can unpeel the layers to ﬁnd the area.

Let’s start with a circle with a radius of 10 cm, made up of 10 layers each 1 cm thick.

The outermost layer will have a length equal to the circumference: 2 × π × 10 cm and a height of 1 cm.

We can start by unrolling the outermost layer which is a rectangle 2 × π × 10 cm wide and 1 cm high. 2 × π × 10

The next layer will be a bit shorter, but will still have the same height. We can keep unrolling one layer at a time. 2 × π × 10

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10 cm

2 × π × 10

This pattern will continue, until we get to the innermost layer, which will have a height of 1 cm but a very small length. If we stack these layers on top of each other, we get something that looks like a triangle. It’s not quite a triangle, because it has a few missing pieces.

20 layers

What if we were more accurate, and used a circle with 20 layers?

20 layers

10 cm

The edge of the “triangle” would appear much smoother, and the missing pieces would be much smaller. 2 × π × 10

Infinitely many layers

We could make a circle with inﬁnitely many layers. If we were to unroll an inﬁnite number of tiny layers, then what you would end up with is a perfectly smooth triangle.

Infinitely many layers 10 cm

If we have a perfectly smooth triangle, then we can use our triangle area formula to work out the area. The base of the triangle is equal to the circumference, and the height is equal to 10 cm.

2 × π × 10

Investigation: Area of circles mathspace.co

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Since the area of a triangle is: height base

The total area is:

This idea of cutting up a complicated shape into inﬁnitely many layers and adding all the layers up is something you’ll be seeing again in a few years time, as it forms the basis of integral calculus.

Investigate We don’t want to go through the process of unraveling a circle every time to ﬁnd its area. Let’s summarise our ideas.

546

We used a circle with a radius of 10 cm, what would be the area if the radius had been 5 cm? Is this half of the area of a circle with a radius of 10 cm? Explain why or why not.

What would be the area if the radius was 3 cm?

In general, for a circle with radius r cm, what would be the area?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Area of circles - applet activity Area of a circles We want to further explore our ﬁndings from the previous activity by using an applet. Rather than unravelling a circle with a known radius, let’s look at any circle in general with radius r.

Interactive investigation

Discover this concept in action online

mathspace.co

Investigate After completing the activity, discuss the questions below with a classmate. 1.

When a circle is cut into slices and rearranged, what shape(s) can be formed?

How does the difference between the area of the circle and area of the sectors change as the number of slices increases?

In general, for a circle with radius r cm, what would the area be?

Discussion 1.

Use the formula to ﬁnd the exact area of a circle with a radius of 6 cm. Then, use a calculator to approximate the area to 2 decimal places. Is the area rational or irrational?

Do you think the area of a circle will always be irrational? Explain your reasoning.

Investigation: Area of circles mathspace.co

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10.08

Area of a circle

After this lesson, you will be able to… • understand and explain the concept of a circle’s area and how it relates to its radius. • apply the formula for the area of a circle in various contexts. • calculate the area of a full circle, a semicircle, a quarter circle, and threequarters of a circle, using the given radius. • solve problems requiring the calculation of a circle’s area, given its radius, and vice versa. • deduce an unknown radius when given the area of a circle. • solve real-life problems involving areas of circles and segments of circles. • calculate the circumference of a circle given its area, applying mathematical skills to solve for radius first. • demonstrate precision in calculation and rounding to a specified number of decimal places.

Area of a circle Interactive exploration

Discover this concept in action online

mathspace.co

We can calculate the area of a circle using the formula: A = π r2 where A is the area and r is the radius of the circle. To find the area of a semicircle (half a circle) we multiply the area by . To find the area of a quarter of a circle we multiply the area by . To find the area of three-quarters of a circle we multiply the area by .

Example 1 Find the area of the circle, rounded to one decimal place.

6 cm

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Create a strategy Use the formula: Area of circle = π r2.

Apply the idea A = π r2

Start with the formula for the area of a circle 2

= π (6)

Substitute r = 6

= 113.1 cm

Evaluate, rounded to one decimal place

Example 2 Calculate the area of the ﬁgure, rounded to one decimal place.

2 cm

Create a strategy The area of the ﬁgure is equivalent to three-quarters the area of the circle.

Apply the idea Remember: Area of a circle = π r2 Start with the formula for the area of a circle Substitute r = 2 Evaluate, rounded to one decimal place

Idea summary We can calculate the area of a circle using the formula:

A = π r2 A r

is the area is the radius of the circle

We multiply the area formula by the relevant fraction to ﬁnd the area of a shape that is a quarter, half or three quarters of a circle.

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Find unknown radius The area formula can also be used to ﬁnd the radius if we know the area of the circle. To solve for the radius we will need to rearrange the formula for the area of a circle:

Swap the sides about the equals sign

Divide both sides by π

Take the square root of both sides

Example 3 A circle has an area of 49 mm2. What is its radius, rounded to two decimal places?

Create a strategy Substitute the area into the formula A = π r2.

Apply the idea Start with the formula for the area of a circle

Substitute A = 49

Divide both sides by π

Take the square root of both sides

Evaluate, rounded to two decimal places

Example 4 A circle has an area of 47 mm2. What is its circumference, rounded to two decimal places?

Create a strategy Use the formula A = π r2 to ﬁnd the radius so we can then ﬁnd the circumference.

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Apply the idea Start with the formula for the area of a circle Substitute the area Divide both sides by π Take the square root of both sides Now we can substitute the radius into the circumference formula. Start with the formula for the circumference of a circle Substitute the radius Evaluate, rounded to two decimal places

Idea summary We can use the area formula to ﬁnd the unknown radius of a circle if we know the area.

A r

is the area is the radius of the circle

10.08 Practice questions Understanding 1

12 m

6.8 cm

177 mm

19 m

5.7 cm

24.5 mm

25π

Calculate, rounding answers to two decimal places: a

Trailblazer 1cd, 2cd, 3cd

Find the diameter of a circle with radius: a

Adventurer 1bd, 2bd, 3bd, 4

Find the radius of a circle with diameter: a

Explorer 1ab, 2ab, 3ab, 4

7×π

π × 32

15 ÷ π

What fraction of a circle is shown in the diagram? a

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5a–c, 6ab, 7ab, 8, 10–12, 14a, 15

Fluency 5 Ex 1

5de, 6cd, 7cd, 8, 9a, 11, 14b

5ef, 6cd, 7cd, 8, 9b, 11, 13, 14cd

Calculate the area of each circle, rounded to two decimal places. a

14 cm

8 cm

10 cm

13 mm

8.6 mm 18.5 cm

8 Ex 2

Calculate the area of the circle with given dimension, rounded to one decimal place: a

Radius = 4 mm

Diameter = 10 mm

Radius = 12 mm

Diameter = 0.44 m

For each circle with given information, round answers to two decimal places: i

Calculate the radius.

Calculate the area.

Diameter = 12 mm

Circumference = 18 cm

Diameter = 22 mm

Circumference = 43.98 cm

Find the area, rounded to one decimal place: a

5 mm

11 m

d 7 cm

3 cm

For each circle with given area: i

Calculate the radius, rounded to three decimal places.

Calculate the circumference, rounded to one decimal place.

Area = 25 cm2

Area = 121 cm2

A circle has an area of 144 mm2. Calculate the radius, rounded to three decimal places.

Ex 3

11 Ex 4

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The area of a circle is 352 cm2. a

If its radius is r cm, ﬁnd r, rounded to two decimal places.

Find the circumference of the circle, rounded to one decimal place.

Mathspace Victorian Curriculum – Year 8 mathspace.co

A circle has an area of 144 cm2. a

Calculate the radius, rounded to two decimal places.

Calculate the diameter, rounded to one decimal place.

A circle has an area of 169 mm2. a

Calculate the diameter, rounded to three decimal places.

Calculate the circumference, rounded to one decimal place.

Find the radius of the circle with the given area, rounded to one decimal place. a

Area = 64 cm2

Area = 81 cm2

Area = 36 cm2

Area = 16 cm2

The radius of a circular baking tray is 10 cm. Find its area, rounded to two decimal places.

Reasoning 16

16, 17

alculate the area of two circles, one with a radius of 4 cm and the other with a radius of C 8 cm.

By what factor is the area increased by when the radius is doubled?

What happens to the area of a circle if we halve its diameter?

What happens to the area of a circle if we triple its circumference?

What happens to the area of a circle if we increase the radius by a factor of q?

What happens to the radius of a circle if we increase the area by a factor of q?

Problem-solving 18, 20

18–20

A pizza has a radius of 13 cm. Javier cuts himself a slice that is one quarter of the pizza as shown in the diagram. What area of pizza, to the nearest whole number, is remaining?

The engineering team at Rocket Surgery are building a rocket for an upcoming Mars mission.

18–21

A critical piece is the circular connective disc that connects the booster rocket to the rest of the spacecraft. This disc must completely cover the top of the booster rocket. The booster rocket has a diameter of precisely 713.5 centimetres. Round answers to two decimal places. a

Find the required area of the connective disc.

Instead of using a calculator, an engineer uses the approximation 3.14 for π . Find the area using the engineer’s approximation for π .

If the connecting disc is more than 100 cm2 too small, the disc will malfunction, resulting in catastrophic launch failure. Will the disc malfunction if it is built according to the engineer’s calculation? Explain your answer. 10.08 Area of a circle mathspace.co

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Danielle deciding between buying a large pizza with a diameter of 46 centimetres for $18, or two medium pizzas with a diameter of 31 centimetres for $9 each. She thinks that buying two medium pizzas will give her more pizza for the same price. Is her decision justiﬁed? Explain your reasoning.

A wind turbine has blades that are R m long which are attached to a tower 60 m high. When a blade is at its lowest point (pointing straight down), the distance between the tip of the blade and the ground is 20 m. a

Calculate the value of R.

Find the distance travelled by the tip of the blade during one full revolution, rounded to two decimal places.

A factor in the design of wind turbines is the amount of area covered by their blades. The larger the area covered, the more air can pass through the blades.

60 m

Find the area inside the circle deﬁned by the rotation of the blade tips, rounded to two decimal places.

10.09

More composite areas

After this lesson, you will be able to… • find the area of composite shapes made up of rectangles, triangles, circles and other polygons using addition and subtraction methods.

Area of composite shapes A composite shape is one that is made from a number of smaller shapes. Sometimes shapes can be combined. This shape consists of a square with side lengths of 6 cm and two semicircles, which is equal to one full circle. 6 cm

Area = Area of square + Area of circle

To calculate the area of a composite shape, we can use either of two methods: • Addition method - Divide the composite shape into basic shapes, work out the area of each basic shape, then add them together. • Subtraction method - Work out the area of the larger shape that encloses the composite shape, then subtract the areas of smaller basic shapes as necessary.

Interactive exploration

Discover this concept in action online

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mathspace.co

Example 1 Consider the composite shape. 7 cm 5 cm 5 cm 15 cm

a Which basic shapes make up this composite shape? A

A rectangle minus two triangles

One rectangle and two trapeziums

Two parallelograms

Two trapeziums

Create a strategy We can construct a line on the composite shape to break it up into its components.

Apply the idea As we can see there are two trapeziums in this construction. So, the correct option is D.

b Find the area of the composite shape.

Create a strategy Since the two trapeziums are identical, we can find the area of one trapezium and then double it.

Apply the idea Area

1 ( a b) h 2

Start with the formula for the area of a trapezium

1 ( 7 15) 15 Substitute 2 55 cm2 The area of both trapeziums 55 2

Evaluate Multiply the area by 2

110 cm2

Evaluate

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Example 2 Find the area of the composite shape rounded to two decimal places. 6 cm 5 cm 12 cm

Create a strategy To ﬁnd the area of this shape, subtract the area of the semicircle from the area of the trapezium.

Apply the idea Start with the formula for the area of a trapezium Substitute the values Evaluate Start with the formula for the area of a circle, and divide it by 2 Substitute the radius Simplify Note: We do not evaluate at this step. The calculation using π on the calculator is done at the end. This is so that we don’t introduce rounding errors. Subtract the areas Evaluate and round the answer

Idea summary To calculate the area of a composite shape, we can use either of two methods: • Addition method - Divide the composite shape into basic shapes, work out the area of each basic shape, then add them together. • Subtraction method - Work out the area of the basic shape that encloses the composite shape, then subtract the areas of smaller basic shapes as necessary.

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10.09 Practice questions Understanding

Explorer 1–3

Adventurer 2, 3

Trailblazer 2, 3

Complete the statement to describe how to ﬁnd the shaded area: a

7 cm 18 cm 6 cm

Total area = Area of ⬚ + Area of ⬚

Total area = Area of ⬚ − Area of ⬚

Which method, addition of areas or subtraction of areas, would be the most eﬃcient method to calculate the shaded area?

5 cm

Name the two shapes that make up the shaded area.

Which method, subtraction or addition, should be used to calculate the area? 6 cm

Fluency 4 Ex 1

4a, 5a, 6, 7ab, 8ac

4a, 5b, 6, 7acd, 8bd

4b, 5b, 6d, 7c–e, 8cd

For each of these composite shapes: i

What basic shapes make up the composite shape?

Find the area, rounded to two decimal places.

9 cm

8 cm

7 cm 5 cm 7 cm 17 cm

16 cm

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For each of these composite shapes: i

What basic shapes make up the composite shape?

Find the area, rounded to two decimal places.

16 cm 7 cm 18 cm

16 cm

Find the area of the composite shapes. Round your answer to two decimal places. a

6 cm

7 cm

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10 cm

5 cm

Find the shaded area in the diagrams. Round your answer to one decimal place. a

c 4 cm

16 cm 16 cm

20 cm

6 cm 5 cm

18 cm

8 Ex 2

Find the shaded area, rounded to two decimal places: a

14 cm

8 cm

12 cm

4 cm 16 cm

14 cm

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Reasoning 9

9, 10

Determine if each statement is true or false. Justify your answers. a

The subtraction method is always faster than the addition method for ﬁnding the area of a composite shape.

The area of a composite shape can be found by dividing it into multiple basic shapes and using the addition method.

The area of a composite shape is always equal to the sum of the areas of its component shapes.

If two composite shapes have the same perimeter, they will have the same area.

If you know the area of a composite shape and the area of one of the shapes that makes it up, you can ﬁnd the area of the remaining shapes.

The shape between two concentric circles as shown in the diagram is called an ‘annulus’. a

Calculate the area of the annulus rounded to one decimal place.

If the radius of the large circle is R and the radius of the small circle is r, write a formula to ﬁnd the area of an annulus.

6 cm 7 cm

Write your rule from part (b) in factorised form.

Problem-solving 11, 12

11–13

11, 13, 14

Tricia is drawing a Yin Yang symbol. She chooses to make the radius of the overall circle 16 cm and the radius of the small circles 2 cm: Find the area of the shaded part, rounded to two decimal places.

560

The front door shown is 1 m wide and 3 m high and has four identical glass panels, each 76 cm long and 12 cm wide. a

Calculate the total area of the glass panels.

The door is to be painted inside and outside. Calculate the total area to be painted.

Mathspace Victorian Curriculum – Year 8 mathspace.co

1m 12 cm 76 cm

The leadlight panel shown depicts a sunrise over the mountains. The mountain is represented by a green triangle 40 cm high. The yellow sun is represented by a sector of a circle with an 20 cm radius. There are 10 yellow sunrays in the shape of isosceles triangles with a base of 4 cm and a height of 12 cm. The sky is blue. Calculate the area of the leadlight panel. Round your answer to two decimal places. a

green glass

yellow glass

blue glass

120 cm

80 cm

On each of the four sides of a square an equilateral triangle is drawn, the ﬁnal ﬁgure resembling a four pointed star. If the sides of the square are the area of the equilateral triangle is

cm long and the formula for

, what is the area of the complete ﬁgure?

10.10 Units of volume and capacity After this lesson, you will be able to… • understand and explain the difference between volume and capacity. • identify and list common units of measurement for volume and capacity. • use the appropriate conversion factors to convert between different units of volume. • use the appropriate conversion factors to convert between different units of capacity. • convert units of volume to units of capacity, and vice versa. • solve problems requiring conversion between units of volume and capacity, applying appropriate conversion factors.

Volume and capacity Volume is the measure of space occupied by a three-dimensional object, while capacity refers to the amount of substance, which can be solid, liquid or gas, that a container can hold. Volume and capacity use different units of measurement. Common units for volume from smallest to largest are: • cubic millimetres (mm3) • cubic metres (m3) • cubic centimetres (cm3) • cubic kilometres (km3) Common units for capacity from smallest to largest are: • millilitres (mL) • kilolitres (kL) • litres (L) • megalitres (ML) 10.10 Units of volume and capacity mathspace.co

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Idea summary Volume is the measure of space occupied by a three-dimensional object, while capacity refers to the amount of substance, which can be solid, liquid or gas, that a container can hold.

Convert units of volume The following diagram shows the conversion factor between different units of volume: The conversion factors for volume involve very large numbers, so index notation is used in this conversion chart. 103 = 1000, or “1 followed by 3 zeros”.

÷106

÷103

mm3

cm3 ×103

÷109

m3 ×106

km3 ×109

However, rather than remembering these factors it is useful to understand that to convert units of volume we are converting the three length dimensions of a cube unit. We can do this 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

×100

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

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Multiply 8.12 by 1 000 000 Evaluate

÷1000

×1000

Idea summary The following diagram shows the conversion factor between common units for volume: ÷103

mm3

÷106

cm3

×103

÷109

×106

km3

×109

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).

Convert units of capacity The following diagram shows the conversion factor between common units for capacity: Similar to converting units of volume, multiply by the conversion factor when going from a larger unit to a smaller unit, and divide by the conversion factor when going from a smaller unit to a larger unit.

÷1000 mL

÷1000 L

×1000

÷1000 kL

×1000

ML ×1000

Example 2 Convert 4920 mL to litres.

Create a strategy Use the fact 1 L = 1000 mL.

Apply the idea mL ÷ 1000 = 4920 ÷ 1000 = 4.92 L

Divide 4920 by 1000 Evaluate

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Idea summary The following diagram shows the conversion factor between common units for capacity: ÷1000

÷1000

×1000

÷1000

×1000

Convert between units In some cases we may want to convert from units of volume to units of capacity, or vice-versa. For example, to ﬁnd the capacity of a swimming pool, we need to ﬁrst measure the dimensions of the pool in metres and then convert to litres. To convert between volume and capacity we can use the equality 1 cm3 = 1 mL. Another useful conversion to know is how many litres there are in a cubic metre. Starting with the equality: 1 cm3 = 1 mL Multiplying both sides of the equality by 1000 and using the conversion 1000 mL = 1 L gives: 1000 cm3 = 1 L Multiplying both sides of the equality by 1000: 1000 cm3 × 1000 = 1 L × 1000 1 000 000 cm3 = 1000 L Using the conversion 1000 L = 1 kL gives: 1 000 000 cm3 = 1 kL Converting a cubic metre into cubic centimetres: 1 m3 = 1 m × 1 m × 1 m

Equate 1 m3 to a cube of side length 1 m

= 100 cm × 100 cm × 100 cm 3

= 100 × 100 × 100 cm 3

Combine the units

1 m = 1 000 000 cm Therefore:

Convert each metre to centimetres Evaluate

1 m3 = 1000 L = 1 kL

To convert from capacity back into volume, we can simply reverse these steps.

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Mathspace Victorian Curriculum – Year 8 mathspace.co

Exploration It is helpful to visualise the space occupied by a certain capacity. We will use the equivalency statements to do this. Use a ruler if necessary and picture a 1 cm × 1 cm × 1 cm cube. This is the space occupied by 1 mL of water. 1 cm3 = 1 mL Now picture a 1 m × 1 m × 1 m cube. This is the space occupied by 1000 L or 1 kL of water. 1 m³ 1000 L = 1 kL Thinking about the size of 1 m3 or 1 kL, consider the capacity of one of the largest dams in the world, the Kariba dam in Zimbabwe, which has a storage capacity of 185 000 000 000 kL of water. Can you picture the volume of space this occupies? 1. How far into space would a tower with an equivalent volume and a 1 m2 base reach? 2. What is an object that is close in size to the volume occupied by 1 mL of water? 3. What is an object that is close in size to the volume occupied by 1 kL of water?

Example 3 Convert 6750 cubic centimetres (cm3) to litres (L).

Create a strategy Use the conversion 1 L = 1000 cm3. To convert cubic centimetres to litres, divide by 1000.

Apply the idea cm3 ÷ 1000 = 6750 ÷ 1000 = 6.75 L

Divide 6750 by 1000 Evaluate

Example 4 A small pond contains 3900 L of water. What is the volume of the pond in m3?

Create a strategy Use the fact 1000 L = 1 m3.

Apply the idea Pond’s volume = 3900 ÷ 1000 3

= 3.9 m

Divide 3900 by 1000 Evaluate

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Idea summary To convert between volume and capacity, we can use the equalities 1 cm3 = 1 mL and 1 m3 = 1000 L = 1 kL.

10.10 Practice questions Understanding 1

Explorer

Adventurer

1–6

Trailblazer

1–6

Are these measures of capacity or measures of volume? a

Litres

How many mL are in 1 L?

Complete the conversions: a b c

Cubic metres

cm3

1 km = ⬚ m, therefore to convert from km3 to m3 multiply by ⬚3 which is ⬚.

1 m = ⬚ cm, therefore to convert from m3 to cm3 multiply by ⬚3 which is ⬚.

1 cm = ⬚ mm, therefore to convert from cm3 to mm3 multiply by ⬚3 which is ⬚.

What operation is required to convert cubic centimetres to cubic metres?

What is the most appropriate unit for measuring the item’s volume? a

Sand pit

Wardrobe

Water glass

Air in a hall

Garden pond

What is the most appropriate unit for measuring the item’s capacity? a

Water jug

Fluency

Bucket

Swimming pool

7, 8ad, 9a–e, 10, 12, 13, 15 7, 8ab, 9b–f, 11–15

7, 8cd, 9c–h, 11–14

This cube has a volume of 1 cm3. Find its volume in cubic millimetres.

1 cm

1 mm Ex 1

Convert to cubic millimetres: a

9 Ex 2, 3

566

3.5 cm3

4.2 m3

4 m3

5000 mL

9 kL

19 cm3

1.67 kL

800 cm3

Convert to litres: a

5360 mL

18.62 m

Mathspace Victorian Curriculum – Year 8 mathspace.co

7850 cm

The volume of the suitcase is 88 500 cm3. Estimate the volume of the shirts.

The volume of the bookshelf is known to be 0.64 m3. Estimate the volume of the books on the bookshelf.

Convert: b

52 L to mL

1 cm to mL

1 m3 to kL

m 0.4 L to mL

Ex 4

Millilitres

88 000 L to kL

63 kL to L

1 mL to cm

36 cm to mL

720 mL to cm3

1 kL to m3

83 m3 to kL

270 kL to m3

3.8 mL to L

1.67 kL to L

84.6 L to kL

Litres

Kilolitres

A rectangular prism has dimensions 3 m, 8 m and 6 m. a

Find the volume of the prism in cubic metres.

Find the capacity of the prism in kilolitres.

A small pond contains 4300 L of water. What is the volume of the pond in m3?

Reasoning

A container has a volume of 1 cubic metre. Find its capacity in: a

68 000 mL to L

16, 18

17–19

A baby bottle holds 100 mL of milk. a

Find 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.

Find 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.

Find an expression for the volume of the cube in L.

Find an expression for the volume of the cube in kL.

What does this indicate about the conversion between m3 and kL?

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Calculate the volume of this rectangular prism in mm3 in two different ways. Method 1: First convert each side into mm and then ﬁnd the volume. Method 2: First ﬁnd 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.

Problem-solving 20–22 20

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:

21, 22

22–25

Find its volume in cm3.

Calculate the length of the juice box if it is 8 cm tall and 5 cm wide.

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:

568

7m 3m

Find 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.

If the pool is ﬁlled to a height 10 cm below the top, 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. Find the amount of water that is leaking out each week, to the nearest litre.

Assuming that water continues to leak at this rate, ﬁnd how many whole weeks it will take to empty the pool.

Mathspace Victorian Curriculum – Year 8 mathspace.co

0.8 m

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?

10.11

Volume of right prisms After this lesson, you will be able to… • calculate the volume of various types of right prisms, such as triangular, square, rectangular, pentagonal, hexagonal, and octagonal prisms. • identify composite solids and explain how they can be broken down into simpler solids. • calculate the volume of composite solids by adding the volumes of the individual solids that compose them. • use the formula for the volume of a right prism to solve problems requiring the calculation of an unknown dimension.

Volume of right prisms A right prism has two congruent and parallel polygon shaped ends connected by rectangles. A right prism gets its name because the ends are parallel and form a right angle with the rectangular faces connecting them. Prisms

Triangular

Square

Rectangular

Pentagonal

Hexagonal

Octagonal

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In general, the formula for volume V for any right prism is given by: V = Abase × h where Abase is the area of the prism’s base and h is the height of the prism. Take note that the height is the perpendicular distance between the ends.

Example 1 Find the volume of the triangular prism shown.

4 mm 11 mm 4 mm

Create a strategy Find the area of the triangle, then use the volume of a right prism formula.

Apply the idea Use the area of a triangle formula Substitute b = 4 and h = 4 Evaluate V = Ah

Use the volume of a right prism formula

= 8 × 11

Substitute A = 8 and h = 11 3

= 88 mm

Evaluate

Example 2 Find the volume of the ﬁgure shown. A = 30 cm2 8 cm

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Create a strategy Use the volume of a right prism formula.

Apply the idea V = Ah

Use the volume of a right prism formula

= 30 × 8

Substitute A = 30 and h = 8 3

= 240 cm

Evaluate

Idea summary The volume of any prism is given by:

V = Abase h Abase h

is the area of the base of the prism is the height of the prism

Take note that the height is the perpendicular distance between the ends.

Volume of composite solids Once we are familiar with ﬁnding the volume of right prisms, the same idea can be used to determine the volume of more complicated solids. We can think of a composite solid as being built out of a number of smaller, simpler solids. The volume of the composite solid is the sum of the volume of each solid it is built from. This complicated solid is built from simple rectangular prisms. Depending on the conﬁguration of the composite solid, sometimes it may be useful to think of building it from a large rectangular prism that then has smaller volumes taken away.

Example 3 Find the volume of the composite solid by breaking it up into rectangular prisms.

12 cm 3 cm

3 cm 7 cm 3 cm

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Create a strategy Break up the composite solid into two smaller rectangular prisms, as shown.

12 cm 3 cm

3 cm

The volume of the entire solid is found by adding together the volume of the two smaller rectangular prisms.

7 cm 3 cm

3 cm

Apply the idea Vcomposite = Vtop prism + Vbottom prism

Add the two smaller volumes

= (12 × 3 × 3) + (3 × 3 × 7)

Substitute the values

= 108 + 63

Evaluate the multiplications

= 171 cm

Evaluate

Example 4 18 mm

Find the volume of the composite solid shown.

14 mm

9 mm

12 mm

11 mm

Create a strategy

18 mm

Write an expression for the volume of the larger rectangular prism and subtract from this an expression for the volume of the smaller one.

14 mm

Use the fact that the volume of a rectangular prism is given by V = length × width × height.

12 mm 9 mm 11 mm

Apply the idea Vcomposite = Vlarger prism − Vsmaller prism = (18 × 12 × 14) − (11 × 12 × 9)

Substitute the values

= 3024 – 1188

Evaluate the multiplications

= 1836 mm

572

Subtract the two prisms

Mathspace Victorian Curriculum – Year 8 mathspace.co

Evaluate

12 mm

Idea summary The volume of a composite solid is the sum of the volume of each solid it is built from. Depending on the conﬁguration of the composite solid, sometimes it may be useful to think of building it from a large rectangular prism that has smaller volumes taken away.

Unknown dimension of a prism Sometimes we might know the volume of a prism but we are missing one measurement such as the area of the end or height. Using division, we can work backwards from the formula to ﬁnd out the missing value.

Example 5 A prism has a volume of 990 cm3. If it has a base area of 110 cm2, ﬁnd the height of the prism.

Create a strategy Use the volume of a right prism formula, V = Ah.

Apply the idea Use the volume of a right prism formula Substitute A = 110 and V = 990 Divide both sides by 110 Evaluate

Idea summary If we know the volume of a right prism, we can ﬁnd a missing dimension by substituting the volume and given dimensions into the appropriate volume formula.

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10.11 Practice questions Explorer

Understanding

1, 2

Adventurer 2, 3

How do you ﬁnd the volume of a rectangular prism?

What is the shape of the prism’s base? a

b 7 cm

5 cm

4 cm

5 cm

23 cm 6 cm

Trailblazer 2–4

4 cm

6 cm

13 cm

2 cm

8 cm

12 m 4m 6m

4 cm

What basic shapes are used to make each of these composite solids? a

10 cm 2 cm

2 cm 7 cm

10 m

2 cm

d 3m

14 cm

6m 11 m

7 cm 8 cm

574

How can the volume of a composite solid be found if it is made up of rectangular prisms?

Mathspace Victorian Curriculum – Year 8 mathspace.co

5, 6, 8, 10ab, 11, 13, 15–18

Fluency 5 Ex 1, 2

5, 6, 9, 10bdf, 12, 13, 15–18

Find the volume of the prisms: a b

5, 7, 9, 10c–f, 12, 14–18

3 cm 2 cm 8 cm

8 cm 2 cm

6 cm

4 cm

13 cm

f A = 21 cm2

5 cm

11 cm

3 cm 16 cm

h A = 30 cm2

4 cm

3 cm

10 cm

9 mm 10 mm

8 mm 50 cm

9 cm

9 cm 5 cm

9.9 m

8.7 m 7.5 m

Find the volume of these cubes: a

6 cm 12 cm

Find the volume of a cube that has a side length of 14 m.

Find the volume of a triangular prism with a base area of 20 cm2 and a height of 10 cm.

Find the volume of an octagonal prism with a base area of 80 mm2 and a height of 120 mm.

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10 Ex 3, 4

Find the volume of these composite solids. Round your answers to two decimal places where necessary. a

10 cm 2 cm

2 cm

3 cm 14 cm

7 cm

2 cm

24 cm

5 cm

7 cm

2 cm

12 cm

f 5 cm

7 cm

4 cm 6 cm

18 cm

12 cm 16 cm

5 cm 3 cm 6 cm

3 cm

6 cm 8 cm

3 cm 7 cm

This rectangular prism has a volume of 168 mm3: Find the length, a, of the rectangular prism.

6 mm

a mm 7 mm

The rectangular prism has a volume of 1680 mm3: Find the width of the rectangular prism in millimetres. 14 mm

20 mm

The volume of the triangular prism shown is 231 cm3. Find the value of k. 7 cm k cm 6 cm

The volume of this triangular prism is 349.965 cm3. Find the value of y, rounded to one decimal place. 7 cm y cm 9.9 cm

A prism has a volume of 770 cm3. Given it has a height of 11 cm, ﬁnd the base area of the prism.

A prism has a volume of 1089 cm3. Given it has a base area of 121 cm2, ﬁnd the height of the prism.

Ex 5

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Find the side length of a cube with a volume of 27 cm3.

Find the length of the rectangular prism with a volume of 162 cm3, a width of 6 cm and a height of 3 cm.

Reasoning

19, 20

20, 21

20–22

Two right prisms have the same height but different base areas. How would you compare their volumes?

A student claims that two rectangular prisms with the same volume will always have the same dimensions. Do you agree or disagree? Explain your reasoning.

How many different rectangular prisms can you ﬁnd that have a volume of 72 m3, if all edges have integer lengths? What are the dimensions of each prism that you ﬁnd?

A prism has a cross section that is a right-angled triangle. The triangular end has a base of a cm and a height of b cm. The length of the prism is c m. a

Write an expression in terms of a, b and c for the volume of the prism in cm3.

Write an expression in terms of a, b and c for the volume of a prism in which all lengths are twice as long as those of the original prism.

Problem-solving 23–25, 29, 31 23

23, 24, 26, 28, 29, 31

26–28, 30, 32, 33

A garden bed is 5 m in length, 2 m in width and 20 cm in height. Find the volume of soil in cubic metres that will be needed to ﬁll up the garden bed.

A regular box of cereal is in the shape of a rectangular prism and measures 20 cm by 12 cm by 18 cm. a

Find the volume of the box of cereal.

The company that makes these cereal boxes also makes a jumbo size box, which is twice as long, twice as wide, and twice as tall as the regular size boxes. i

Find the volume of the jumbo box of cereal.

Is the volume of the jumbo box twice the volume of the regular box?

A water tank in the shape of a rectangular prism has a length of 4 m, a width of 3 m and a height of 2 m. If a hose can ﬁll the tank at 80 L/min, how long will it take to ﬁll the tank from empty?

1 L of water has a mass of 1 kg. What mass of water would each of these containers hold?

Length 4 cm, width 3 cm and height 2 cm

Length 7 m, width 5 m and height 2 m

A swimming pool has the shape of a trapezoidal prism: a

Find the volume of the pool in cubic metres.

If the pool is three-quarters full what is the volume of the non ﬁlled space of the pool.

Find h, the distance of the water level to the top of the pool when it is is three- quarters full.

51 m 10 m

1.4 m

3.3 m 50.7 m

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A box of tissues is in the shape of a rectangular prism. It has a length of 39 cm, a width of 19 cm and a height of 11 cm. a

Find the volume of the box.

If the shelf at the supermarket is 95 cm long and has a depth of 40 cm, ﬁnd the maximum number of tissue boxes that can ﬁt on the shelf.

A special refrigerator is used to store medical samples and has dimensions 18 cm by 70 cm by 12 cm. The samples are stored in small containers that have dimensions 20 mm × 50 mm × 20 mm. Assuming both the refrigerator and the sample containers are rectangular prisms,

Find the dimensions of the sample containers in centimetres.

Find the volume of a sample container.

Find the volume of the fridge.

How many containers can be stored in the fridge?

This nesting box needs to have a volume of 129 978 cm3, a height of 83 cm and a width of 54 cm. Find the depth, d, of the box. 83 cm

d cm

54 cm

The volume of the tent is 4.64 m3. Find the height, h, of the tent. h cm

290 cm 200 cm

A solid piece of metal is in the shape of a triangular prism with base area 8 cm2 and length 10 cm. The triangular prism is melted down and used to make solid metal cubes of side 2 mm. How many such cubes can be made if we assume no loss of metal in the process?

578

Ethan plans to construct a mini tunnel in his diorama project, starting from a height of 5 cm. The interior roof of the tunnel is in the shape of a semi-circle. a

Determine the area of the shaded region rounded to two decimal places.

Since steel reinforcing covers 30% of the wall and roof space, only 70% of the volume needs to be ﬁlled with concrete. Find the volume of concrete required, rounded to the nearest litre.

Mathspace Victorian Curriculum – Year 8 mathspace.co

14 cm

Semi-circle arch

11 cm 5 cm 4 cm

26 cm 4 cm

10 Chapter review 1

Find the perimeter of each triangle: a

An equilateral triangle with a side length of 10 m

A scalene triangle with side lengths 12 mm, 16 mm and 20 mm

An isosceles triangle where the 2 equal side lengths are 15 mm and the third side measures 10 cm

Find the side length of each regular polygon: a

A regular hexagon with a perimeter of 60 cm

An equilateral triangle with a perimeter of 24 m

A regular octagon with a perimeter of 400 mm

Determine if each statement is true or false. Justify your response. a

The perimeter of a square with a side length of 10 cm is equal to the perimeter of a rectangle with a length of 20 cm and a width of 5 cm.

If the perimeter of a regular hexagon is 45 cm, then the length of each side is 10 cm.

The perimeter of a rectangle can never be equal to the perimeter of a circle.

The perimeter of a triangle is always less than the perimeter of a rectangle with the same base and height.

Calculate the circumference of the circle each value of x. Round your answer to two decimal places. a

x = 5 cm

x = 16 m

x = 18.2 mm

x = 20.1 km

Calculate the diameter of each circle, rounded to two decimal places: a

A circle with a circumference of 8 cm

A circle with a circumference 26 cm

A circle with a circumference of 59 cm

A circle with a circumference of 256 cm

Calculate the radius of each circle, rounded to two decimal places: a

A circle with a circumference of 12 cm

A circle with a circumference of 52 cm

A circle with a circumference of 77 cm

A circle with a circumference of 300 cm

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Find the perimeter of each shape: a

9 cm

13 m 4m

5 cm 6 cm 12 m 10 cm 5m

7 cm

2 mm 1 mm 6 mm

2 mm

1 cm

Find the perimeter of each figure, rounded to one decimal place: a

35°

12 cm

52° 32°

32° 26 cm

580

Joanne 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 9 cm. a

Is Joanne correct? Explain your answer.

Find the exact perimeter of the shape.

Consider the trapezium: a

Find the value of a.

Find the value of b.

Find the value of x. Round your answer to two decimal places.

Find the perimeter of the trapezium. Round your answer to two decimal places.

Mathspace Victorian Curriculum – Year 8 mathspace.co

60°

60° 60°

A 6 cm

8 cm

9 cm

E x cm

a cm

C 14 cm

b cm

Each square on the grid has an area of 3 mm2. Estimate the area of the curved shape:

An amusement park has an area measuring 22 acres. Given that 1 acre is approximately 3000 m2, ﬁnd the area of the park in square metres.

A farm covers an area measuring 280 acres. Given that 1 acre is approximately 0.3 ha, and 1 ha is 10 000 m2, calculate: a

The area of the farm in square metres

Find the width of each rectangle, given the area and the length. a

The area of the farm in hectares Area = 16 m2, length = 8 m 2

Area = 72 mm , length = 4 mm

Area = 36 cm2, length = 4 cm

Area = 60 cm2, length = 6.2 cm

Complete the table of base and height measurements for different parallelograms with an area of 48 m2:

Base (m)

Height (m)

Area (m2)

24 2

A rectangular garden measures 15 metres by 10 metres. a

What is the area of the garden in square metres?

If the walkway surrounding the garden is 3 metres wide, what is the total area of the garden and walkway in square metres?

A rectangular kitchen has a floor that measures 6 m by 5 m. The floor is to be covered with tiles that are a 30 cm square. How many tiles are required to completely cover the floor?

Find the area of each quadrilateral: a

6 mm

4 mm 8m

10 mm

16 m 10 m

6 cm 15 cm

f 6 cm 30 mm

8 cm

15 mm

Chapter 10 review mathspace.co

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The given rhombus, with diagonal lengths x and y, is formed into a rectangle: a

Find the length of the rectangle in terms of x and/or y.

Find the height of the rectangle in terms of x and/or y.

Find the formula for the area of the rhombus.

x y

Width Length

A gardener wants to plant a ﬂower bed in The shape of a trapezium. The shorter base of the trapezium is 60 metres and the longer base is 100 metres. The height of the trapezium is 40 metres. If the gardener can plant 2000 seeds per square metre, how many seeds will he need to plant the entire ﬂower bed?

A landscaper charges $3 per square metre to maintain gardens. Answer these questions:

How much would he charge to maintain a rectangular garden with dimensions 10 m by 7 m?

How much would he charge to maintain a circular garden with a radius of 4 m?

He charged $450 to maintain a garden in the shape of a square. What was the length of the side of the square garden?

Calculate the area of each circle, rounded to two decimal places. a

A circle with a radius of 3 m

A circle with a diameter of 5 cm

A circle with a radius of 7 mm

A circle with a diameter of 1.2 m

Find the radius of each circle, giving your answers to two decimal places: a

A circle with area of 54 cm2

A circle with area of 81 cm2

A circle with area of 42 cm2

A circle with area of 22 cm2

Richard is considering between buying a giant pizza with a diameter of 60 centimetres for $22, or two small pizzas with a diameter of 40 centimetres for $11 each. He assumes that buying two small pizzas will yield more pizza for an equivalent price. Is Richard’s assumption correct? Provide an explanation for your answer.

Find the area of each composite shape: a

10 cm

5 cm

12 cm 14 cm

7 cm

18 cm

14 cm 17 cm

8 cm 6 cm

5 cm

40 cm

7 cm 5 cm

19 cm 9 cm

15 cm 5 cm

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Mathspace Victorian Curriculum – Year 8 mathspace.co

14 cm

10 cm

For each of the following ﬁgure: i

Find the area of rectangle B.

Find the total area of the given shape.

12 mm

6 cm

2 mm 17 cm

4 mm 3 cm

4 cm

A B 3 mm

5 mm

8 cm

Consider each figure where all measurements are in centimetres: Figure 1

Figure 2

7 11 5

Figure 3

Figure 4

15 3 7

11 5

4 5

Find the area of the entire rectangle shaded as shown in Figure 1.

Find the area of the interior rectangle shaded as shown in Figure 2.

Find the area of the corner rectangle shaded as shown in Figure 3.

Calculate the area of the shaded region as shown in Figure 4.

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The volume of the suitcase has a total capacity of 72 580 cm3. Estimate the volume of the contained shirts.

Convert: a

36 cm3 to mL

720 mL to cm3

52 L to mL

83 m3 to kL

270 kL to m3

0.4 L to mL

3.8 mL to L

84.6 L to kL

Consider a cube with a side length of a m. a

Explain a process for converting a volume in m3 into a capacity in L.

Find an expression for the volume of the cube in L.

Find an expression for the volume of the cube in kL.

What does this indicate about the conversion between m3 and kL?

This juice box contains 400 mL of juice: a

Find its volume in cm3.

Calculate the length of the juice box if it is 4 cm tall and 5 cm wide.

For each of the following composite solid: i

Find the basic shapes used to make this composite shape.

Find the area of the solid’s base, rounded to three decimal places.

iii

Find the volume of the solid, rounded to two decimal places.

10 cm

2 cm

2 cm 7 cm

10 m

2 cm

d 3m

14 cm 7 cm

6m 5m

11 m

8 cm

A prism has a volume of 540 cm3. Given it has a height of 6 cm, ﬁnd the base area of the prism.

584

Mathspace Victorian Curriculum – Year 8 mathspace.co

A regular package of cookies is in the shape of a rectangular prism and measures 25 cm by 10 cm by 15 cm. a

Find the volume of the package of cookies.

The company that makes these packages of cookies also makes a party size package, which is twice as long, twice as wide, and twice as tall as the regular size packages. i

Find the volume of the party size package of cookies.

I s the volume of the party size package four times the volume of the regular package?

Two right prisms have the same height but different base areas. How would you compare their volumes?

Did you know?

When you’re planning a road trip, measurement plays a big role in determining how far you’ll travel. For instance, if you’re driving from Sydney to Melbourne, the distance is about 880 kilometres. Knowing this helps you estimate how long the trip will take and how much fuel you’ll need. Measurement doesn’t just help with distance – it also comes into play when checking your car’s tyre pressure or setting the air conditioning to the perfect temperature. Accurate measurements keep your journey safe and comfortable!

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Big ideas • Probability helps us analyse the chance that an event will occur. • Venn diagrams and two-way tables provide visual tools for organising and analysing data, enabling meaningful insights and interpretations.

Curriculum Outcomes VC2M8P01 Recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts VC2M8P02 Determine all possible outcome combinations for 2 events, using two-way tables, tree diagrams and Venn diagrams, and use these to determine probabilities of specific events in practical situations VC2M8P03 Conduct repeated chance experiments and simulations, using digital tools to determine probabilities for compound events, and describe results

11 Probability Chapter outline 11.01 11.02 11.03 11.04 11.05

Calculate probability Intersection and union Venn diagrams Two-way tables Mutually exclusive events Investigation: Probability simulations Chapter 11 review

588 601 615 627 637 647 652

Just because it’s cloudy doesn’t mean it will rain! We can use Venn diagrams to help visualise probabilities for weather patterns.

11.01

Calculate probability

After this lesson, you will be able to… • understand and explain the concept of probability as a number, expressed either as a fraction, decimal, or percentage between 0 and 1 (or 0% and 100%). • identify and explain probability terminology, including trials, outcomes, sample space, and events. • use the formula for calculating probability with equally likely outcomes:

• determine the probabilities of different outcomes in multi-step experiments using a tree diagram. • apply the concepts of probability and tree diagrams to real-world scenarios.

Probability as a number The probability of an event happening is a measurement of its likelihood. It can be expressed as a number from 0 to 1 using fractions or decimals, or from 0% to 100% using percentages: Impossible

Even chance

Certain

Unlikely

0 0%

2 3 9 10 8%

Likely

0.4

1 2 0.5 50%

0.63

6 7 72%

1 100%

A probability can never be less than 0 or more than 1. The larger the number, the more likely it is, and the smaller the number, the less likely it is. An experiment is something that is done to observe different possible results. A trial is performing an experiment one time. Rolling a die one time is a trial. An outcome is a possible result of a trial. When you roll a six-sided die, one outcome is rolling a 6. The set of all possible outcomes is called the sample space. When you roll a six-sided die the sample space is {1, 2, 3, 4, 5, 6}. If the outcomes in a sample space are equally likely, then we can use the formula:

When rolling a six-sided die, the probability of rolling an even number is:

as out of the six possible outcomes, three are even: 2, 4, and 6. 588

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Interactive exploration

Discover this concept in action online

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Example 1 A probability of

means the event is:

A Impossible

Unlikely

Likely

Certain

Create a strategy

Apply the idea

Compare the value to 0 and 1.

Probabilities range from 0 to 1. Events with probabilities close to 0 are unlikely, and events close to 1 are likely. A probability of

is close to

1 so the event is likely. The correct option is C.

Example 2 Select all events that have a probability of 25% on this spinner:

Select all correct options. A Apple

Ball

Pig

Star

Create a strategy Write the percentage into a fraction.

Apply the idea The 25% is equal to . It means the sector will take up one-quarter of the spinner. We can see that the apple and pig sectors both take up one-quarter of the spinner. The correct options are A and C.

Example 3 Consider this list of numbers: 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 7, 7, 7, 7, 9, 9

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a A number is chosen from the list at random. What is the probability it is an odd number?

Create a strategy We can use the rule:

Apply the idea There are 12 odd numbers in the list and 17 numbers in total.

b A number is chosen from the list at random. What number has the highest probability of being chosen?

Create a strategy

Apply the idea

Choose the number that appears more often than any other number.

The number that appears the most times in the list is 7 which appears 4 times. So, 7 has the highest probability of being chosen.

Example 4 Amy picks a whole number between 4 and 8 inclusive. a What are all the possible numbers she could pick?

Create a strategy

Apply the idea

Start at 4 and write all the numbers up to and including 8.

Possible outcomes = 4, 5, 6, 7, 8

Inclusive means that the numbers 4 and 8 are included in the sample space.

b How many different possible outcomes are there when Amy picks a number?

Create a strategy

Apply the idea

Count all the possible numbers that she can pick.

All the possible numbers that she can pick are the outcomes. In part (a), we said that all the different possible outcomes are 4, 5, 6, 7 and 8. So, there are 5 different possible outcomes.

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c Find the probability that Amy picked the number 5.

Create a strategy

Apply the idea

We can use the rule:

There is one number 5 in the list, and five numbers in total.

d Find the probability that Amy picked the number 1.

Create a strategy

Apply the idea

We can use the rule:

There is no number 1 in the list, which we can represent as 0, and five numbers in total.

Idea summary Probability is a number between 0 and 1 that measures the likelihood that an event will happen.

Probability notation An event is a grouping of outcomes from a sample space. When you roll a six-sided die, events might include “rolling an even number”, or “rolling more than 5”. We write the probability of a certain event occurring as: P (event) Where P represents probability and event represents the outcome or group of outcomes. For example, when rolling a die, the probability of “rolling an even number” is written: P (even) Or “rolling more than 5” is: P ( X > 5) Where X represents the event.

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Example 5 A 20-sided die is rolled. Write the following using probability notation. a The probability of an event A, getting a number greater than 15.

Create a strategy

Apply the idea

Write the probability of a certain event occurring We write event A, getting a number greater than as: 15 as: P ( A > 15)

P (event)

b The probability of an event A, getting a number less than 7.

Apply the idea We write event A, getting a number less than 7 as: P ( A < 7)

c The probability of getting an odd number.

Apply the idea We write the probability of getting an odd number as: P (odd)

d The probability of getting an even number.

Apply the idea We write the probability of getting an even number as: P (even)

Idea summary An event is a grouping of outcomes from a sample space. We write the probability of a certain event occurring as: P (event)

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Tree diagrams A tree diagram is useful to represent the sample space in experiments with more than one step. Each possible outcome is written as a branch of the tree diagram. H

2 3

T The outcomes for tossing a coin once

Outcomes for randomly selecting the top, middle, or bottom shelf

6 Outcomes from rolling a standard die

When more than one trial is carried out, we have two (or more) columns of branches. T

This tree diagram shows the outcomes and sample space for the experiment of tossing a coin twice. The ﬁrst column represents the outcomes for the ﬁrst trial (T, H). The second column represents the outcomes for the second time the coin is tossed (T, H). We read the outcomes along the branches from left to right. The tree diagram shows the sample space of four possible outcomes: {TT, TH, HT, HH} The probability of tossing two heads is:

The probability of tossing one head would be:

Which evaluates to:

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Example 6 G Y R

G Y R

On the island of Timbuktu, the probability that a set of traffic lights shows red, yellow or green is equally likely. Christa is travelling down a road where there are two sets of traffic lights. What is the probability that both sets of traffic lights will be red?

Create a strategy

Apply the idea

Find the number of favourable outcomes red and red or RR in the probability tree.

From the tree diagram, we can see that there is only 1 outcome for “red and red” or RR, which is shown in the bottom branch. We can also see that there are 9 possible outcomes.

Idea summary A tree diagram is useful in listing the outcomes and sample space for multi-step experiments. The outcomes are listed by reading the branches from left to right.

11.01 Practice questions Understanding 1

Explorer 1–3

Adventurer 1–4

Trailblazer 1–4

Are these statements true or false? a

The probability of an event occurring is always between 0 and 1, inclusive.

The sum of the probabilities of all possible outcomes in a sample space is always equal to 1.

An event includes all outcomes in a sample space.

A bag contains 5 red balls, 3 green balls, and 2 blue balls. The total number of outcomes is three, {(red, green, blue)}, representing the three distinct colours of balls in the bag.

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Put these probabilities in order from least likely to most likely:

Consider the tree diagram which represents the sample space when there are three marbles of different colours in a jar. One marble is drawn out of a jar, the colour noted, the marble replaced and then another marble drawn out of the jar:

G Y R

What colours could the three marbles be?

How many equally likely outcomes are there in total?

How many outcomes are there where both of the marbles are the same colour?

G Y R

Are these statements true or false? a

A tree diagram represents events that occur one after the other.

In a tree diagram, the branches represent the different outcomes at each stage of the process.

At each level of a tree diagram, the probabilities should always add up to 1.

Tree diagrams are only useful for representing 2 events.

Fluency

Ex 1

G Y R

5, 6, 7ab, 8ab, 9, 10ad, 11, 12, 14ad, 15, 16

5, 7bc, 8bc, 9, 10bd, 11, 12, 14ab, 15, 16, 18

5, 7cd, 8c–e, 9bc, 10cd, 12, 13, 14ce, 15, 17–19

Describe the likelihood that corresponds to each of these probabilities using the words impossible, unlikely, even chance, likely or certain. a

0.2

0.5

85%

0.7

25%

The jar contains 7 red, 3 blue and 10 green marbles: If a marble is drawn out of the jar without looking, list all the possible colours that the marble might be.

A standard six-sided die is rolled. a

List the sample space.

List the sample space for rolling an even number.

List the sample space for rolling a number strictly less than 3.

List the sample space for rolling a number divisible by 3.

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Ex 2

Find the probability of: a

Rolling a 10 on a standard die.

An event that will occur 3 times every 20 trials.

Rolling a 5 or more on a standard die.

An event that will occur 5 times every 100 trials.

An event that will occur 13 times every 500 trials.

For each spinner, ﬁnd the probability of spinning a star. Write your answer as a percentage. a

Ex 3

A number is randomly selected from the list: {2, 3, 3, 5, 5, 7, 7, 7, 7, 7, 12, 12, 12, 12, 15}

Ex 4

What is the probability of selecting a 5, P (5)?

What is the probability of selecting a 12, P (12)?

What is the probability of selecting a 15, P (15)?

Which number is most likely to be selected?

A single marble is drawn from the jar 5 red, 7 blue and 8 black marbles: a

Find the probability of drawing a red marble, P (red). Give your answer as a percentage.

Find the probability of drawing a blue marble or a black marble, P (blue marble or black marble). Give your answer as a percentage.

Consider the spinner shown:

List the sample space.

Find the probability of spinning a number less than 30, P ( N < 30).

Find the probability of spinning a number in the thirties, P ( N ≥ 30).

29 31

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A jar contains 15 marbles in total. Some of the marbles are blue and the rest are red. a

If the probability of picking a red marble is jar?

Find the probability of picking a blue marble.

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, how many red marbles are there in the

14 Ex 5

Ex 6

Write the probability notation for the event: a

Event A : getting heads when a coin is tossed.

Event A : getting a number greater than 3 when a die is rolled.

Event A : getting a sum greater than 11 when two dice are rolled.

Event A : a baby born being a girl.

Event A : a card randomly selected from a deck being a club.

While driving to work, Denis encounters two sets of traﬃc lights. Each set of traﬃc lights shows green, orange, or red. a

List the sample space

Find the probability that both sets of traﬃc lights are green, P (GG).

Find the probability that the two lights are orange or red, in any order, P (Orange or Red, in any order).

Find the probability that neither light is red, P ( Neither Red).

G O R G O R G O R

A fair die is rolled and then a coin is tossed. Die 1 2 3 4 5 6

Coin H T H T H T H T H T H T

Find the probability of getting an even number and a head.

Find the probability of getting an even number, a head, or both.

Find the probability of getting an even number or a head, but not both.

In a bag, there are 3 red marbles and 2 yellow marbles. You draw one marble, note its colour, and replace it. a

Find the probability of drawing two red balls, P (RR)

Find the probability of drawing one red ball and one yellow ball in any order, P (Red and Yellow, in any order).

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R1 R2 R3 Y1 Y2 R1 R2 R3 Y1 Y2 R1 R2 R3 Y1 Y2 R1 R2 R3 Y1 Y2 R1 R2 R3 Y1 Y2 597

The spinner shown is spun, then a fair coin is ﬂipped. a

Construct a tree diagram of the sample space.

Find the probability of spinning an odd number and getting heads.

Find the probability of spinning an odd number, getting heads, or both.

Every morning, Mae has toast for breakfast. Each day, she either chooses honey or jam to spread on her toast, with an equal chance of choosing either one. a

Construct a tree diagram for three consecutive days of Mae’s breakfast choices.

Find the probability that Mae chooses jam for her toast three days in a row.

Find the probability that Mae chooses honey for her toast on two days and jam another day, in any order.

Reasoning 20

20, 21

21, 22

A bag of marbles contains different coloured marbles. The probabilities of drawing a particular colour are listed: • Red: 25% • Blue: 35% • Green: 5% • Orange: 25% Determine whether the bag could contain marbles of a colour that is not listed. Explain your reasoning.

What are the two equally likely events on this spinner? Justify your answer with probability.

Write a scenario to match this tree diagram:

Car

Car Bus

Bus

Car Bus

Bike

Car Bike

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Problem-solving 24a, 25, 27, 28 23

23, 25–27, 29

A book has pages numbered from 1 to 100. If the book is opened to a random page, what is the probability that the page number: a

Is a multiple of 8?

Find the probability of spinning these shapes. Simplify your answers.

24, 25, 27, 28

A pig or a ball

Anything but an apple

iii

An elephant, a star or a teddy bear

Anything but a ball or teddy bear

Has the digit 8 in the page number?

Determine whether these events have a probability of less than 10%. i

Spinning a teddy bear

Spinning a star

iii

Spinning an apple

Spinning a ball or an elephant

A card is drawn from a standard deck of 52 cards: a

Find the probability of drawing the Jack of Spades or the Jack of Diamonds.

Leanna draws a 4 of Clubs and puts it back in the deck and asks you to draw a card. Find the probability that you will draw a card with a number larger than 4, and the same colour as Leanna’s card.

Sally is drawing 2 cards from a deck of 52 cards. She draws the ﬁrst card and checks whether it is red (R) or black (B). She replaces her ﬁrst card, then draws the second card and records its colour. a

What is the probability that Sally draws a black card, then a red card?

Determine if each of the following has an equal probability of drawing a black and then a red card (your answer to part a). i

Drawing a red card and then a black card

Drawing a red card and then another red card

iii

Drawing at least one black card

Drawing a black card and then another black card

Drawing one black card and one red card in any order

R B

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The 26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is to be picked out of the bag at random, ﬁnd the probability of picking a: a

C, T or A

A letter in the word MATHEMATICS

M, E, V, R, K or N

A letter in the word TEXTBOOK

Suppose there is a food truck that sells ice cream sundaes. You are given these choices, and you can choose only one from each category: • Ice cream types: vanilla, chocolate, or strawberry • Toppings: cherries, sprinkles, cookie crumbles, or peanuts • Sauces: hot fudge or caramel Ice cream types

Toppings Hot fudge

Vanilla Caramel Hot fudge Chocolate Caramel Hot fudge Strawberry Caramel

Sauces Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles Peanuts Sprinkles Cherries Cookie crumbles

How many different types of sundaes could be created?

What is the probability that someone randomly chooses a strawberry sundae?

What is the probability that someone chooses a sundae with hot fudge?

What is the probability that someone chooses a sundae with peanuts?

A circular spinner is divided into unequal parts. The green sector takes up an angle of 75° at the centre. The red sector takes up an angle of 80° at the centre and the blue sector takes up the remainder of the spinner. What is the probability that the spinner will land on blue?

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11.02

Intersection and union

After this lesson, you will be able to… • use the mathematical notation for complementary events. • apply the concept of complementary events in real-world scenarios and problems. • use mathematical notation for intersection and union. • apply knowledge of intersection and union in real-world problems related to sets. • interpret Venn diagrams and how they illustrate intersections, unions and complements. • apply the knowledge of Venn diagrams to solve problems and represent scenarios involving intersections, unions and complements of sets.

Complementary events The complement of an event is the opposite of the event. The complement of the event ‘rolling an even number’ is ‘NOT rolling an even number’, which we can also see is the same as ‘rolling an odd number’. Complementary events have probabilities that add up to 1. We can write this using probability notation as: P (even) + P (odd) = 1 Sometimes it is easier to calculate the complement of an event and use the following relationship to calculate the probability of an event: P (event) = 1 − P (complement) We write the complement of an event A as A′: P ( A) = 1 − P ( A′)

Example 1 The probability of the local football team winning their grand ﬁnal is 0.36. What is the probability that they won’t win the grand ﬁnal?

Create a strategy We can use the rule: Probability of complementary event = 1 − Probability of event

Apply the idea P (not winning) = 1 − P (winning)

Use the rule

= 1 − 0.36

Substitute the probability

= 0.64

Evaluate the difference

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Example 2 A number between 1 and 100 inclusive is randomly picked. a What is the complement of drawing a number greater than 61? A

Drawing a number greater than or equal to 61

Drawing a number less than 61

Drawing a number less than or equal to 61

Create a strategy Recall that the complement of the number being ‘greater than 61’ must include all other possible outcomes. If the number is not greater than 61, it must be 61 or lower.

Apply the idea So the correct answer is “Drawing a number less than or equal to 61”, which is option C.

b Find the probability that the number is greater than 61. Express your answer as a fraction.

Create a strategy We can use the rule: Probability of complementary event = 1 − Probability of an event

Apply the idea Consider the complement of the number to be greater than 61. The number must be less than or equal to 61. There are 61 numbers that are less than or equal to it, so the probability of picking a number is

. Use the rule Substitute the probability Evaluate the difference

Idea summary Complementary events can be thought of as opposite events. We use the notation A′ to represent the complement of event A.

Intersection and union The term set is used to describe a collection of things, numbers, objects or outcomes that have something in common. Consider the integers from 1 to 10. We call this the universal set. We write this as: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 602

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The null set is an empty set that has nothing in it. It is represented as {} or ∅. Note that {} is not the same as {0}. {0}, is actually a set of the value 0. The intersection of two sets is the elements both sets have in common. The term union describes the combination of outcomes for multiple events. Let the set of even numbers be A: A = {2, 4, 6, 8, 10}. And B be the set of integers less than 6: B = {1, 2, 3, 4, 5} The intersection of the two sets is the numbers that are in both A AND B. We write A ∩ B for intersection: A ∩ B = {2, 4}. The union of the two sets is the numbers that are in A OR B or both. We write A ∪ B for union: A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}.

Example 3 a What is meant by the intersection of sets A and B? A

The set of elements that are members of set A or set B or both sets.

The set of elements that are common to set A and B.

The set of elements that are only members of one of set A or set B.

The set of elements that are not members of either set A or set B.

Create a strategy Recall that the intersection of the two sets is the numbers that are in both A and B.

Apply the idea Consider the sets {7, 8, 9, 10, 11} and {6, 8, 10, 12}: The intersection of the two sets is {8, 10}. From the example, the intersection of the two sets includes elements that are in both sets so the answer is option B.

b Suppose set A = {3, 4, 5, 6, 7} and set B = {3, 7, 8, 9}. Which answer correctly shows the intersection of the two sets? A

{3, 4, 5, 6, 7} ∪ {3, 7, 8, 9} = {3, 7}

{3, 4, 5, 6, 7} ∩ {3, 7, 8, 9} = {3, 4, 5, 6, 7, 8, 9}

{3, 4, 5, 6, 7} ∪ {3, 7, 8, 9} = {3, 4, 5, 6, 7, 8, 9}

{3, 4, 5, 6, 7} ∩ {3, 7, 8, 9} = {3, 7}

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Create a strategy

Apply the idea

Recall that A ∪ B represents the union of set A and set B, while A ∩ B represents the intersection of set A and set B.

The intersection of the two sets includes elements that are in both sets. The elements that belong to both sets A and B are 3 and 7. So, the correct answer is option D.

Example 4 If A is the set of factors of 24, and B is the set of factors of 36: a List the elements in sets A and B

Apply the idea A = the set of factors of 24

B = the set of factors of 36

The factors of 24 are:

The factors of 36 are:

1, 24

1, 36

2, 12

2, 18

3, 8

3, 12

4, 6

4, 9

A = {1, 2, 3, 4, 6, 8, 12, 24}

B = {1, 2, 3, 4, 9, 12, 18, 36}

b List the elements of B ∪ A.

Create a strategy ∪ is the union symbol which means OR. So B ∪ A is asking what values are in either list, B or A.

Apply the idea B ∪ A = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}

c List the elements of A ∩ B.

Create a strategy

Apply the idea

Use the factors in part (a). Remember that intersection represent as ∩, and list the set of elements that are common to set A and set B.

Factors of 24: A = {1, 2, 3, 4, 6, 8, 12, 24}

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Factors of 36: B = {1, 2, 3, 4, 6, 9, 12, 18, 36} A ∩ B = {1, 2, 3, 4, 6, 12}

Idea summary Deﬁnitions and notation: • Universal set: The set of everything relevant to the question is called the universal set. • Null set: An empty set that has nothing in it. It is represented as {} or ∅. • Intersection: F ∩ G: only the elements that are in both F and G. • Union: F ∪ G: the elements that are in F or G or both.

Venn diagrams A Venn diagram provides a visual way to look at a sample space and understand the terms intersection and union. Each circle represents a set. Four Legs

Stripes

Turtle Elephant

Zebra

Lion

Tiger

Fish

Hippo Crow

Four Legs

Kangaroo

Stripes

In this diagram, we have two sets, the set of animals with four legs and the set of animals with stripes. The universal set comprises of 9 animals, as there are nine animals in total. The Zebra and Tiger belong to both sets and therefore are written in the section where both circles or sets overlap.

The intersection of the sets is: Four Legs ∩ Stripes = {Zebra, Tiger}

Turtle Elephant Zebra Lion Tiger Hippo

Fish

Crow Kangaroo

Four Legs Turtle Elephant Lion Hippo

Stripes

The union of the sets is: Four Legs ∪ Stripes = {Turtle, Elephant, Lion, Hippo, Zebra, Tiger, Fish}

Zebra Tiger

Fish

Crow

Kangaroo

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Stripes

Four Legs Turtle Elephant Lion Zebra Tiger Hippo

Crow

The complement of the set of animals with Four Legs is: (Four Legs)′ = {Fish, Crow, Kangaroo}

Fish

Kangaroo

Exploration Use the following Venn diagram to answer these questions: 1. How many traits belong to fish and dolphins? 2. How many traits belong to fish or dolphins? Fish

3. How many traits do not belong to fish and dolphins?

Scales

Dolphins Swim

Breath air

Have Have teeth Lay eggs fins Have live birth Gills

A Venn diagram allows us to visualise the overlaps and differences between these two categories.

Two legs

Example 5 The following diagram is used for sorting numbers according to two categories: • Numbers that are divisible by 7 • Numbers that are even

a In which area would you place the number 8? A

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Divisible by 7

Even number

Create a strategy

Apply the idea

Think about each circle one at a time, and Since 8 is not divisible by 7, we can’t place it in determine whether or not “the number 8” would any part of the left circle. be inside or outside that circle. Since 8 is an even number, we can place it in the right part of the right circle. So, the answer is option A.

b In which area would you place the number 14? A

Create a strategy

Apply the idea

Think about each circle one at a time, and Since 14 is divisible by 7, we can place it in any determine whether or not “the number 14” would part of the left circle. be inside or outside that circle. Since 14 is an even number, we can place it in any part of the right circle. In this case, both circles satisfy the condition, the answer is the intersection between the left and the right circle which is option A.

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c In which area would you place the number 15? A

Create a strategy

Apply the idea

Think about each circle one at a time, and Since 15 is not divisible by 7, we can’t place it in determine whether or not “the number 15” would any part of the left circle. be inside or outside that circle. Since 15 is not an even number, we can’t place it in any part of the right circle. In this case, we will place it outside the two circles which is option C.

Example 6 B

List the elements in each of these subsets:

16 9 1 14

24 18 21 15 11

a A∩B

Create a strategy

Apply the idea

Recall that the intersection of two sets are the numbers that are both in set A and set B.

13 is in the section where both circles overlap.

In a Venn diagram, the intersection of two sets is represented by the section where both circles overlap.

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So, A ∩ B = {13}

b A∪B

Create a strategy Recall that the union of two sets are the numbers that are both in set A and B. In a Venn diagram, the union of two sets is represented by the section covered by both sides. This includes: • the part of circle A that does not overlap with circle B • the part of circle B that does not overlap with circle A • the overlapping area where circle A and B intersect

Apply the idea In the Venn diagram, the following are found: • left section (only set A): {1, 9, 14, 16} • right section (only set B): {15, 18, 21, 24} • overlapping section: {13} So, A ∪ B = {1, 9, 13, 14, 15, 16, 18, 21, 24} c A ∩ B′

Create a strategy To find A ∩ B′, identify first set A and set B. Then determine the universal set U. Next is to find the complement of B which includes all elements in U that are not in B. Lastly, identify the elements which are common to both A and B′.

Apply the idea Firstly identify set A and B as: A = {1, 9, 13, 14, 16} B = {13, 15, 18, 21, 24} Then, we determine the universal set U as: U = {1, 4, 9, 11, 13, 14, 15, 16, 18, 21, 24} Next is to find the complement of B: B′ = {1, 4, 9, 11, 14, 16} 1, 9, 14, and 16 are the numbers common to both A and B′. So, A ∩ B′ = {1, 9, 14, 16} d (A ∪ B)′

Create a strategy We know A ∪ B from the answer in part (b). To find (A ∪ B)′, we find all elements in the universal set that are not in A ∪ B.

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Apply the idea We know that A ∪ B = {1, 9, 13, 14, 15, 16, 18, 21, 24} and U = {1, 4, 9, 11, 13, 14, 15, 16, 18, 21, 24}. 4 and 11 are the only numbers in U but not in A ∪ B. So, (A ∪ B)′ = {4, 11}

Idea summary A Venn diagram provides a picture of the sample space and can be used to calculate probabilities. The intersection of two sets is represented by the area that overlaps.

11.02 Practice questions Understanding

Explorer 1–3, 5

Adventurer 2, 3, 5, 6

Trailblazer 4, 6

What is the sum of the probability of an event and the probability of its complement?

Set A = {2, 4, 6} and Set B = {1, 3, 6} Does the number {6} represent the union or intersection of sets?

Set A = {1, 2, 3} and Set B = {1, 4, 6} Does {1, 2, 3, 4, 6} represent the union or intersection of sets?

Are the two events complementary? a • Event 1: Selecting a positive number • Event 2: Selecting a negative number b • Event 1: Drawing a red card from a standard deck of cards • Event 2: Drawing a black card from a standard deck of cards c

• Event 1: Drawing a club from a standard deck of cards • Event 2: Drawing a spade from a standard deck of cards

d • Event 1: Rolling a number greater than 3 on a die • Event 2: Rolling a number less than 3 on a die 5

Complete the sentence using the words ‘Dog’ or ‘Snake’: a b c d

610

The two sets in the diagram are ⬚ owners and ⬚ owners. 13 + 2 is the number of people who have a ⬚. 2 + 3 is the number of people who have a ⬚.

2 is the number of people who have a ⬚ AND a ⬚.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Dog Owners

Snake Owners

A constellation is randomly selected from these six constellations: Consider the events:

Caelum

• Event 1: The name of the constellation begins with a consonant • Event 2: The constellation has an even number of Indus stars • Event 3: The constellation has less than 5 stars • Event 4: The name of the constellation has more than 6 letters

Reticulum

Horologium

Pictor

Phoenix

Which two events are complementary?

Fluency

7–11, 12ab, 13ac, 14–16, 18, 19

7–11, 12bc, 13ab, 14, 15, 17, 18, 20

7–9, 11, 12cd, 13cd, 14, 17, 18, 20

Ex 1

The probability that James kicks a goal in his football game is 0.7. What is the probability that he doesn’t kick a goal?

Ex 2

A number between 1 and 50 inclusive is randomly picked. The probability that the number is less than 42 is

Describe the complement of drawing a number less than 42.

Describe the complement of drawing a number that is at least 43.

Find the probability that the number is at least 42.

The 26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is picked out of the bag at random, ﬁnd the probability of: a

Selecting a C.

Not selecting a C.

Selecting a K, L or M.

Not selecting a K, L or M.

Not selecting an A, E, I, O or U.

Not selecting a T, L, Q, A, K or Z.

Selecting a letter that is not in the word PROBABILITY.

Selecting a letter that is not in the word WORKBOOK.

A letter is chosen at random from the word ORDERED. a

Which letter has the highest probability of not being chosen?

Find the probability that the chosen letter is not D.

For each of these events, ﬁnd the probability of the complementary event: a

The probability that it will rain today is 0.43.

The probability of the local basketball team winning their grand ﬁnal is 0.49.

The probability of ﬂipping tails on a certain biased coin is P (tails) = 0.56.

The probability that Victoria will win the major prize in a raﬄe at the school fete is 0.053.

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Ex 3

Determine the intersection of each of these pairs of sets: a

A = {0, 1, 4, 9, 16, 25, 36, 49}

P = {1, 3, 5, 7, 9, 11, 13}

B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Q = {0, 2, 4, 6, 8, 10, 12} c

M = {−25, −5, −1, 0, 1, 5, 25}

W = the set of multiples of 3

N = {0, 1, −1, 5, −5, 25, −25} X = the set of multiples of 9 13

Determine the union of each of these pairs of sets: a c

A = {9, 99, 999, 9999}

F = {m, a, t, h}

B = {0, 1, 99, 100, 101}

G = the set letters in the word ‘math’

S = {1, 2, 3, 5, 8, 13, 21, 34}

J = the set of multiples of 5

T = {1, 2, 3, 12, 13, 21, 23, 31, 32} K = the set of multiples of 10 Ex 4

Ex 5

If A is the set of prime numbers less than 40 and B is the set of positive integers less than 12: a

List the elements in sets A and B.

List the elements of B ∪ A.

List the elements of A ∩ B.

A constellation is randomly selected from the eight shown:

Tucana

Reticulum

Indus

Crux

Volans

Phoenix

Pictor

Find the probability that the name of the constellation begins with a vowel.

The complementary event to the event in part (a) is selecting a constellation that begins with a consonant. Find the probability of this event.

Find the probability that the constellation has 6 or more stars.

The complementary event to the event in part (c) is selecting a constellation that has less than 6 stars. Find the probability of this event.

The Venn diagram is used for sorting numbers according to two categories: • Numbers that are divisible by 4 • Numbers that are odd

612

Caelum

Which section does the number 7 go in?

Which section does the number 24 go in?

Which section does the number 18 go in?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Divisible by 4

Odd number

Ex 6

The Venn diagram shows the overlap of teachers who own a pet and teachers who own a car: a

Construct a Venn diagram and shade the area that represents the teachers who own a pet and own a car.

Construct a Venn diagram and shade the area that represents the teachers who own a pet but don’t own a car.

Owns a pet

Owns a car

List the elements in each of these subsets: a

A∩B

A∪B

A ∩ B′

A 2

( A ∪ B)′

B 8

100 students were asked whether they studied either History or Geography. a

Determine how many students studied both History and Geography.

Find P (History ∩ Geography).

Determine how many students studied neither History nor Geography.

Find P (History ∪ Geography)′.

19 History 31

Geography 26

A bag contains 50 black marbles, 37 orange marbles, 29 green marbles and 23 pink marbles. If a marble is selected at random, ﬁnd these probabilities, in simplest form: a

P (orange)

Reasoning

b 21, 23

P (orange)′

c 21, 24

P (orange ∪ pink)

P (orange ∪ pink)′

22, 24

If an event is impossible, what is the probability of the complementary event? Explain your answer.

Given a set A and a sample space S, determine an expression for ( A′)′ — the complement of the complement of A. Explain your answer.

Given any two sets A and B, determine whether A ∪ B or A ∩ B has more elements. Explain your reasoning.

Set A contains every element of the sample space. Will the complementary set of A, which can be written as A′, have any elements? Explain your answer.

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Problem-solving 26, 27 25

25, 27

A jeweller has arranged his gems based on the cut and colour of the gem:

26–28

Amethyst Circle 2

• Cuts - triangular, rectangular, and octagonal • Colours - amethysts, emeralds, and rubies Write an appropriate category name for “Circle 2” in the diagram.

A marble is chosen at random from a box containing 4 different colour marbles, red, blue, green and purple. The probability of selecting different colours is given in the table. Find: a

P (Green)′

P (Blue)′

P (Red ∪ Purple)

P (Purple)′

P (Blue ∪ Green)′

Probability

Red Blue Green

Fifteen pool balls labelled 11 to 15 are placed in a bag. The first eight balls are solid colours and the next seven balls are striped. If one ball is chosen at random, find the probability: a

It is a solid coloured ball.

It is an odd number ball or a striped ball.

It is a solid coloured ball and an odd number.

It is a solid coloured ball with a prime number on it.

Consider the shaded region in the Venn diagram shown: Express this region in terms of P and Q, using unions, intersections, and/or complements.

614

Colour

Mathspace Victorian Curriculum – Year 8 mathspace.co

11.03

Venn diagrams

After this lesson, you will be able to… • interpret the different parts of a Venn diagram. • draw and label a Venn diagram for two or three categories. • use Venn diagrams to solve problems and answer questions related to categories and subsets.

Venn diagrams Venn diagrams provide a visual picture of two sets of data and can be used to represent the sample space in a probability question. Gymnastics

Hockey

In this Venn diagram, the total of the four numbers 5 + 6 + 3 + 15 = 29 represents the total number of people in the survey also called the universal set. The numbers in the shaded area represent the people who play hockey or do gymnastics or do both.

Gymnastics

Hockey

The circle on the left represents students who play hockey, there are 5 + 6 = 11 students.

The circle on the right represents students who do gymnastics, there are 6 + 3 = 9 students. 15

Hockey

Gymnastics

The number outside the circles is the number of students who do not do either sport. This means there are 15 students who do not participate in either hockey or gymnastics.

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Hockey

Gymnastics

The middle section highlighted represents the students that are in both categories. This section represents the 6 students who play both hockey and gymnastics.

Hockey

Gymnastics

The part of the circle highlighted is the students who play hockey but do not do gymnastics. 5 students only play hockey.

Example 1 A group of students were asked why they skipped breakfast. The two reasons given were that they were “not hungry” and they were “too busy”.

Not hungry

Too busy

a How many of the students skipped breakfast because they were not hungry?

Create a strategy Find the “Not hungry” circle in the diagram and add the number of students inside it.

Apply the idea Not hungry students = 8 + 11 = 19

Add numbers in the not hungry circle Evaluate

b How many of the students only skipped breakfast because they were too busy?

Create a strategy

Apply the idea

Find the number that is only in the too busy circle.

13 is in the “Too busy” circle and not in the “Not hungry” circle. Too busy students = 13

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c How many of the students skipped breakfast because of one reason?

Create a strategy Add the number of students that are in the “Not hungry” circle or in the “Too busy” circle but not in both.

Apply the idea Students with one reason = 8 + 13 = 21

Add the numbers Evaluate

Example 2 Carl makes a Venn diagram about the possible pets he could get from the local shelter.

Dogs

Fluffy

a Carl is told that a pet will be selected at random. What is the probability that his new pet will be a dog that isn’t ﬂuffy?

9 4

Create a strategy To ﬁnd the probability of a particular outcome we can use the rule:

Apply the idea The “number of favourable outcomes” is the number of dogs that aren’t ﬂuffy. The “total number of outcomes” is the total number of animals who could be selected. So the probability is

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b Carl is told that a pet will be selected at random. What is the probability that his new pet won’t be ﬂuffy?

Apply the idea The “number of favourable outcomes” is the number of animals that are not ﬂuffy. The “total number of outcomes” is the total number of animals that could be selected.

c Carl is told that for a donation they will select his pet from just the dogs at random. What is the probability that his new pet will be ﬂuffy?

Create a strategy Consider that the sample space for this selection is now the dogs and the favourable outcome is a ﬂuffy dog.

Apply the idea The “number of favourable outcomes” is the number of dogs that are ﬂuffy. The “total number of outcomes” is the total number of dogs that could be selected.

Idea summary Venn diagrams provide a visual picture of a problem and can be used to represent the sample space in a probability question.

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Three circle Venn diagrams Venn diagrams can also be drawn with three circles representing three different categories. Comedy

Action

This diagram represents a survey where people were asked if they liked three different movie genres. Horror

This highlighted circle represents people who said they like comedy movies.

This section represents people who said they like comedy, action and horror movies.

This region represents people who said they like comedy and horror, but not action.

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Interactive exploration

Discover this concept in action online

mathspace.co

Venn diagrams don’t always need to be drawn using circles. For scenarios that contain more than 3 categories, it may be beneficial to use rectangles to help visualise the different intersections.

Example 3 Joanne is struggling to decide what to watch online. She decides to pick one movie at random from the streaming website. A Venn diagram of her options sorts movies into three categories based on their genre: Comedy, Action and Horror.

Comedy 10

Action 4

20 6

13 Horror

a How many of the movies are horror ﬁlms?

Create a strategy Add all the numbers inside the “Horror” circle.

Apply the idea Horror films = 13 + 3 + 5 + 6 = 27

Add the numbers Evaluate

b How many of the movies ﬁt into only one genre?

Create a strategy Add the numbers in these parts where the movies ﬁt only one genre:

Apply the idea Number of movies = 10 + 20 + 13 = 43

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Add the numbers Evaluate

c How many of the movies are action films combined with at least one other genre?

Create a strategy Add the numbers in these parts where the movies are action combined with at least one other genre:

Apply the idea Number of movies = 4 + 5 + 6

Add the numbers

= 15

Evaluate

Idea summary Three circle Venn diagrams show how components of three sets are related using three overlapping circles.

11.03 Practice questions Understanding 1

Explorer 1, 2

Adventurer 1, 2

Name the two sets of numbers represented in the Venn diagram.

List the Multiples of 6 in the diagram.

List the Multiples of 8 in the diagram.

Which number is a multiple of 6 and 8?

Trailblazer 1, 2 Multiples of 6

Multiples of 8

6 12

8 24

18 36

16 32

The Venn diagram shows the brand of phone carried by students in a school. a

What does 68 represent?

What does 246 + 68 + 185 represent?

What does 36 represent?

What does 36 + 185 + 68 + 246 represent?

Sumsang

jPhone 68

246

185 36

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Fluency 3

3–8, 10, 12, 14, 15

4–8, 10, 12, 14, 16

A jeweller has arranged his gems based on the cut and colour of the gem:

5, 7, 9, 11, 13, 14, 16–18

Ruby

Triangular

Ruby

Triangular

• Cuts - triangular, rectangular, and octagonal • Colours - amethysts, emeralds, and rubies

How many of the gems are not rubies?

How many of the gems are both ruby and triangular?

How many of the gems are rubies that are not triangular?

A jeweller has arranged his gems based on the cut and colour of the gem: • Cuts - triangular, rectangular, and octagonal • Colours - amethysts, emeralds, and rubies

Ex 1

What fraction of the gems are rubies?

What fraction of the gems are triangular?

What fraction of the gems are both rubies and triangular?

A group of students were asked about their siblings. The two categories show if they have at least one brother and if they have at least one sister: a

How many of the students have at least one sibling?

How many of the students have at least one brother?

How many of the students don’t have a sister?

Have a sister

Have a brother

A student creates this diagram of his favourite animals: a

How many of the animals are aquatic?

How many of the animals are aquatic and have teeth?

How many of the animals are aquatic or have teeth, but not both?

Aquatic

Have teeth

Plankton Goldfish Seahorse Jellyfish Dolphin

Dog Tiger

Octopus Earthworm

A group of 60 people were surveyed on whether they enjoyed skateboard riding or bike riding. • 16 people said they enjoyed skateboard riding but not bike riding • 25 people said they enjoyed bike riding only • 12 people said they enjoyed both activities • 7 people said they did not enjoy either activity Complete the Venn diagram with the given information.

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Mathspace Victorian Curriculum – Year 8 mathspace.co

Skateboard

Bike

Sloth

A student is making a Venn diagram from the information gathered from her classmates: 18 people play cricket, 15 play softball, and 6 play both softball and cricket. a

How many people only play softball?

Construct a Venn diagram that shows how many students play each sport.

Avril makes a Venn diagram about the possible dance partners in her dance class: a

How many students are there in Avril’s class, not including Avril?

Avril thinks her partner is suitable if they are both good dancers and the right height for her. How many suitable partners are there in the class?

Avril is told that her partner will be randomly assigned. Find the probability that her partner will be suitable.

Play rugby league only.

Play rugby league.

Play rugby union.

Play rugby union only.

Do not play rugby league.

Do not play rugby union.

Rugby League

Find the number of students that chose: i

To party.

iii

Neither to study nor party.

Rugby Union 138 124

Party

Study 21

Not to study.

38 10

Find the probability that a student chosen at random chose: i

Not to party.

To study and party.

iii

To study only.

To study or party.

Dan makes a Venn diagram about the possible pets he could get from the local shelter: a

169

The Venn diagram shows the number of students choosing to study on the night before an exam, and the number of students choosing to party: a

Ex 2

Good dancer

The given Venn diagram shows the number of students in a school playing rugby league, rugby union, both or neither: Find the number of students who:

Right height

Dan is told that a pet will be selected at random. Find the probability that his new pet will be a cat that isn’t ﬂuffy.

Cats

Fluffy

Dan is told that a pet will be selected at random. Find the probability that his new pet won’t be ﬂuffy.

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Consider the Venn diagram shown: Find: a

P ( A)

P (B)

P ( A)′

P (B)′

P (B ∩ A′)

B 13

19 18

Consider the Venn diagram shown: Find: a

P ( A ∩ B)

P ( A ∩ B′)

P ( A ∪ B)

P ( A ∪ B)′

B 7

19 16

The Venn diagram shows the decisions of 535 consumers choosing to buy an iPhone and consumers choosing to buy a Blackberry: If a consumer is selected at random, ﬁnd the probability that he chose to buy:

Ex 3

624

A Blackberry.

A Blackberry but not an iPhone.

Both phones.

An iPhone or a Blackberry.

Neither phones.

An iPhone but not a Blackberry.

Charlie is struggling to decide what movie to watch. He decides to pick one at random from his collection. A Venn diagram of his collection sorts movies into three categories: Comedic ﬁlms; Romantic ﬁlms, and runtime is over 2 hours: a

How many movies are there in Charlie’s collection?

Charlie wants to watch a comedy. Find the probability he will select a comedy.

Blackberry

iPhone 68

246

185

Comedy

Romantic

Charlie is after a romantic comedy that goes for less than 2 hours. Find the probability that he will select a suitable ﬁlm.

Charlie is after a comedy ﬁlm that goes for longer than 2 hours. Find the probability that he will select a suitable ﬁlm.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Over 2 hours

A ﬂorist collected a sample of her ﬂowers and divided them into the appropriate categories as shown in the Venn diagram:

Red flowers

Find the probability that a ﬂower is: a

Not red but has thorns.

Not red and does not have thorns.

Flowers with thorns 14

4 9

10 Flowers with long stems

A group of 94 people were asked about whether they liked cake and/or ice cream. Ice cream

Cake

In total, 76 people said that they liked the cake. 12 people said that they liked ice cream but not cake.

Use this information to ﬁll in all missing values for the Venn diagram.

Reasoning 19

19, 20

19, 21

A student is making a Venn diagram about a recent catch-up test for those who missed or did poorly on the original test. In the class of 30 students, 27 sat for the original test and 10 sat for the catch-up test. a

Given that everyone in the class sat for at least one test, how many people sat for both tests? Explain your answer.

Construct a Venn diagram that shows how many students sat for each test.

A group of people were asked about whether they liked watching basketball and/or baseball. The results are shown in the Venn diagram:

31 Basketball

Baseball 49

Eiichiro makes this claim:

“In total 70 people like watching basketball, and 63 people like watching baseball. When added to the 31 people who do not like watching either sport, this means that 164 people in total were surveyed.” Explain why Eiichiro’s reasoning is incorrect, and ﬁx his mistake(s).

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At a particular school, the languages available for students to study are German, Italian, and Spanish. The languages studied by one year group are shown in the Venn diagram.

Five new students joined the school in this year’s group. One of these students studies German, one studies Italian and German, one studies Spanish and Italian, and two of them don’t study any languages.

4 16

13 2 12

22–24

A student is arranging a list of seven numbers into two categories. One category is his lucky numbers. Write an appropriate category name for “Circle 1” in the diagram.

9 36 Spanish

Draw a new Venn diagram representing the updated information and explain your reasoning.

Problem-solving 22, 23

Italian

German

22–25

Circle 1

Lucky numbers 31

9 30

A group of people were randomly selected and asked which modes of transport they used to get to work. All people catch either the bus or the train, so no one rides a bike or walks. The Venn diagram shows the results: If one of them is chosen at random, the probability that they catch a bus only is and a train?

Bus

Train

. How many people said they catch a bus

A class of 31 students were surveyed about playing 3 sports. • 16 students play tennis • 14 students play hockey • 2 students play none of these sports • 8 play both tennis and cricket • 7 play both cricket and hockey • 6 play both tennis and hockey • 3 play all three

626

Construct a Venn diagram for this situation.

Find the probability that a randomly selected student plays all three sports.

Consider the application of a Venn diagram to classify quadrilaterals based on their properties. a

Create a Venn diagram that classiﬁes quadrilaterals by their properties.

Create 3 questions that you could ask a classmate that would require using the Venn diagram to solve.

Use your Venn diagram to answer the questions.

Mathspace Victorian Curriculum – Year 8 mathspace.co

11.04

Two-way tables

After this lesson, you will be able to… • read and interpret two-way tables. • create two-way tables from given data and use them to analyse and interpret information. • convert between two-way tables and Venn diagrams by identifying the similarities between them. • solve problems involving two-way tables, including calculating percentages and conditional probabilities based on the information provided in the table.

Two-way tables Two-way tables represent data in columns or rows. The ﬁnal column and row display the total. This two-way table displays the handedness of students in a class and how they feel about English.

Enjoys English Doesn’t enjoy English Total

Righthanded 9

Lefthanded 4

Total 13

The columns show the writing hand and the rows show the enjoyment of English. The number in the bottom right cell is the total number of students in the class.

In this table, there are 30 students in total. Reading across the rows: 13 students enjoy English. Of these, 9 are right-handed and 4 are left-handed. 17 students don’t enjoy English. Of these, 15 are right-handed and 2 are left-handed. Reading down the columns: 24 students are right-handed. Of these, 9 enjoy English and 15 do not enjoy English. 6 students are left-handed. Of these, 4 enjoy English and 2 do not enjoy English.

Example 1 In a study, some people were asked whether they lie. A partially completed two-way table of the results is shown. a Complete the table of values.

Children Adults Total

Lie 15 10 ⬚

Don’t lie ⬚ ⬚ ⬚

Total 25 ⬚ 60

Create a strategy The values in these cells can be calculated by setting up equations with the values around them.

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Apply the idea To find the number of children who don’t lie, subtract the number of children who lie from the total number of children surveyed: 25 − 15 = 10 To find the total number of adults surveyed, subtract the total number of children from the total number of people surveyed: 60 − 25 = 35 To find the number of adults who don’t lie, subtract the number of adults who lie from the total number of adults surveyed: 35 − 10 = 25 Adding both the number of children and adults who lie will give a total of all liars. 15 + 10 = 25 Adding both the number of children and adults who don’t lie will give the total of those who do not lie. 10 + 25 = 35 So, the completed table is: Children Adults Total

Lie 15 10 25

Don’t lie 10 25 35

Total 25 35 60

b Of those in the study, one is chosen at random. What is the probability that they said they never lied?

Create a strategy To ﬁnd the probability of a particular outcome we can use the rule:

Apply the idea The “number of favourable outcomes” is the number of people who say they don’t lie. This includes children that don’t lie and adults that don’t lie. The “total number of outcomes” is the total number of people surveyed.

Idea summary Two-way tables represent data in columns and rows.

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Two-way tables and Venn diagrams We can convert between a two-way table and a Venn diagram by matching up the numbers in the table to the corresponding part of the Venn diagram.

Entered show Didn’t enter show Total

Lefthanded 4

Righthanded 9

To represent this information in a Venn diagram, we choose one row and one column to become circles. Here we chose the column marked “Left” and the row marked “Entered show”.

Entered

Left-handed

This two-way table represents the handedness of students, and whether or not they entered the talent show.

Total

• The number that is in both chosen categories (4) goes in the overlap of the two circles. • The total of the “Entered show” row is (13) which means the Entered circle must total 13. Therefore the second number in the Entered circle is 9. • The total of the “Left-handed” column is (6) which means the Left-handed circle must total 6. Therefore the second number in the Left-handed circle is 2. • The number outside the circles (15) is the number of students who do not belong to either circle, the Right-handed students who did not enter the show.

Example 2 100 random people in Australia were surveyed, examining their carbon footprint and the city they lived in. The people were then categorised as living in either an urban or regional location, and whether that person has a high carbon emission or low carbon emission.

High emission Low emission Total

Urban 37 24 61

Regional 13 26 39

Total 50 50 100

Which of the following Venn diagrams represents the information provided in the two-way table? A

High emission

Urban

High emission

Urban

100

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High emission

Urban

High emission

Urban

100

Create a strategy The categories represented in the table will form the basis for the Venn diagram.

Apply the idea 13 people have high emissions and live in regional locations. These people belong in the “High emissions” circle but not the “Urban” circle. This will be represented by this region: Determine which regions the remaining three cells correspond to. Remember that the numbers in the Venn diagram should add up to the totals given in the table. So, the answer is option C.

Idea summary We can convert between a two-way table and a Venn diagram by matching up the numbers in the table to the corresponding part of the Venn diagram. The total of a row or column in the table will equal the total of a circle in the Venn diagram.

11.04 Practice questions Understanding

630

Explorer 1–3

Adventurer 1–3

Consider the two-way table: a

How many students were surveyed in total?

What two questions were they asked?

Complete the table.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Brown hair Blonde hair Total

Trailblazer 1–3

Blue eyes 25 20 45

Brown eyes

Total 50 120

The two-way table displays data from a cafe, showing which types of drinks were ordered on specific days.

Tea Saturday Sunday Total

Which two drinks were ordered?

Which two days were surveyed?

Fill in the missing values in the two-way table.

Coffee 52

Total 64

30 120

The two-way table and Venn diagram show the same information:

Piano Not Piano Total

Guitar 20 90 110

Not Guitar 40 50 90

Total 60 140 200

Piano

Guitar B

Find the values of A, B, C, and D.

Fluency

4, 6, 8, 9, 10ac, 12, 14, 15, 17, 19, 20

5, 7, 8, 10, 11, 13, 16, 18, 20

50 students were asked whether or not they were allergic to nuts and dairy. The two-way table displays their responses:

Allergic to Dairy Not Allergic to Dairy Total

5, 7, 8, 10, 11, 13, 15, 17, 19, 20

Allergic to Nuts 6 6 12

Not Allergic to Nuts 11 27 38

Describe the category which has 11 students.

How many students are allergic to nuts?

How many students are allergic to nuts or dairy, or both?

Some students were asked if they are left or right-handed. The results are provided in the two-way table:

Female Male Total

How many of the students are righthanded males?

How many of the students are lefthanded?

How many of the students are not right-handed males?

Total 17 33 50

Lefthanded 10 5 15

Righthanded 39 50 89

Total 49 55 104

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

Complete the two-way table.

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

How many students picked music?

How many boys are in Tobit’s class?

A healthy living initiative asked people to describe how often they go to the gym. Their responses are shown in the table:

Girls 12 18 11 10

Total

Female 35 35

Male 37 13

Total

Maths Music Science English Total

Frequently Rarely Total

Complete the two-way table.

How many people were surveyed?

If one person is chosen at random, ﬁnd the probability that they are a male who frequently attends the gym.

If one person is chosen at random, ﬁnd the probability that they attend the gym rarely.

Members of a gym were asked what kind of training they do. The two-way table shows the results:

Male Female Total

Cardio 11 47

Weight 30 12

Total

Complete the two-way table.

How many gym members were asked altogether?

How many members do weight training?

If a member is chosen at random, what is the probability that they do weight training?

According to the table, are men or women more likely to do weight training?

A nonsmoking initiative asked smokers to describe how often they smoke:

Frequently Rarely Total

Female 57 35

Male 34 14

Complete the two-way table.

How many people were surveyed?

If one person is chosen at random, what is the probability that they are a frequent male smoker?

If one person is chosen at random, what is the probability that they smoke rarely?

Consider the two-way table: a

A∩B

A′ ∩ B′

B B′

A∪B

A′ ∪ B

Total

Find:

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Boys 11 8 13 15

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A′

5 12 17

10 13 23

Total

Total 15 25 40

A town has two campsites to choose between which both offer tent and cabin accommodation. Sunny This two-way table records the campers’ Campground choices in one summer: Platypus Creek a Complete the two-way table. Total b How many people stayed in each of the town’s campsites?

Cabin

178

101

281

What fraction of people stayed in a tent?

What fraction of the people staying in cabins were in Sunny Campground?

What fraction of people stayed at Platypus Creek?

The table shows the number of trains arriving either on time or late at a particular station: a

How many trains were late on Friday?

How many trains passed through the station on Wednesday?

How many trains were on time throughout the entire week?

What fraction of trains were on time over the whole week?

Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

This table describes the departures of trains out of a train station for the months of May and June: a b c

Tent

Arrived on time 23 20 27 28 15 22 26 161

Arrived late 9 5 8 14 12 6 13 67

Departed Delayed on time 123 32 124 47 247 79

How many trains departed during May and June? May What percentage of trains in June were delayed? June Round your answer to one decimal place. Total What percent of the total number of trains departed on time in May? Round your answer to one decimal place.

Total

Total 32 25 35 42 27 28 39 228 Total 155 171 326

Find the probability that a train selected at random in June would have departed on time.

Find the probability that a train selected at random from the 2 months was delayed.

A small cafe sells two types of sandwiches: ham and cheese, and Ham and Cheese turkey and avocado. The cafe Turkey and Avocado records the number of sandwiches Total sold during lunchtime for two days. The results are shown in the two-way table:

Day 1 12 6 18

Day 2 8 14 22

Total 20 20 40

Which day had more ham and cheese sandwiches sold?

Which type of sandwich was more popular on Day 2?

Calculate the probability that a randomly chosen sandwich sold on Day 1 is a turkey and avocado sandwich.

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15 Ex 1

In a study, a group of people were asked about their preferred transportation mode: Car or Bicycle. A partially completed two-way table of the results is shown.

Men Women Total

Car 30 25

Bicycle

Total 50 110

Complete the table.

Of those in the study, one is chosen at random. Find the probability that they are a woman who prefers using a bicycle.

Sophia asked some people in her community whether they were vegetarian or not. 29 responders said they were vegetarian, of which 8 were children. 14 children said they were not vegetarian, and 11 adults said they are not vegetarians. a

Construct a two-way table based on the results of Sophia’s survey.

What fraction of responders are vegetarian?

What fraction of adults are vegetarian?

Use the Venn diagram to complete the table of values. Rugby League

Rugby Union

179

185 146

Play Rugby League

Don’t play Rugby League

Total

Play Rugby Union Don’t play Rugby Union Total 18

A student makes a Venn diagram of students who are late to school, and students who catch the bus to school.

Late to school

Caught bus

Construct a two-way table based on the Venn diagram. 13

A vet has 28 pets visiting their practice in a day. The pets are categorised based on whether they have been vaccinated and whether they have been Vaccinated Microchipped microchipped. Construct a two-way table based on the Venn diagram. 3

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Ex 2

A total of 120 gym members were surveyed about their workout preferences. The members were categorised based on their preferred time of day to exercise (Morning or Evening) and the type of workout they preferred (Cardio or Strength Training). Morning 45 32 77

Cardio Strength Training Total

Evening 15 28 43

Total 60 60 120

Construct a Venn diagram that represents the information provided in the two-way table.

Reasoning

21, 22

22, 23

Describe how each section of a two-way table can be represented in a Venn diagram. Use an example to support your explanation.

176 primary school and middle school students were asked whether or not they had a smartphone.

Primary school Middle school Total

Phone 36 87 123

No phone 45 8 53

Total 81 95 176

Uri makes the claim: “Adding up the values in the ‘No phone’ column gives us that 45 + 8 + 53 = 106 students didn’t own a smartphone. This means that 176 − 106 = 70 students did own a smartphone.” Explain why Uri’s reasoning is incorrect, and ﬁx his mistake(s). 23

A poll was given where people were asked: “Do you like pineapple on your pizza?” This information was recorded: • 40 people responded to the poll • 14 adults responded “yes” • 16 of the people who responded were adults • The number of people who responded “yes” was four times the number of people who responded “no” Find and explain the error made in each statement. a

Since 14 adults responded yes, there were 40 − 14 = 26 adults who responded no.

There were 14 adults who said yes and 16 of the people were adults so the total of adults was 30

The number of people who responded “yes” was four times the number of people who responded “no” so we can take the total number of people, 40, and divide by 4 to ﬁnd the number of people who said no.

Construct a two-way table to represent the information from the poll.

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Problem-solving 25, 26 24

25, 26

24, 25, 27

Students were asked if they prefer chocolate (C) or vanilla (V ) ice cream. Each of the students was also asked if they prefer sprinkles (S) 30 or hot fudge (F ) as an ice cream topping. The results are displayed in the tree diagram: If Chocolate and Vanilla make up the columns, and Sprinkles and Hot fudge make up the rows, construct a two-way table based on the tree diagram.

60 residents of a city were asked “Do you support the construction of the new train station?” The residents questioned were also classiﬁed as living in the north, south or in the inner city.

North

Construct a two-way table based on the Venn diagram.

Voted “Yes”

South

A group of tourists was asked whether they spoke Mandarin or Spanish.

Mandarin Not Mandarin

Spanish 58 15

Not Spanish 10 17

Total

Complete the table.

How many people speak both languages?

If one person is chosen at random, ﬁnd the probability that they speak neither language.

If one person is chosen at random, ﬁnd the probability that they speak only one of the languages.

At a local university, students were asked what their favourite subject at high school was and what they have decided to major in after 3 years of university. The results are shown in the table: Maths favourite Maths major Science major Music major Art major Total

76 64 64 6 210

Science favourite 20 46 11 19 96

Music favourite 61 53 67 38 219

Art favourite 43 59 59 74 235

Total 200 222 201 137 760

One student is chosen at random. Find:

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The probability that a student’s favourite subject was mathematics at high school.

The probability that a student is majoring in music or arts at university.

The probability that a student’s favourite subject was music and they studied something different at university.

The probability that a student’s major is the same as their favourite subject.

Mathspace Victorian Curriculum – Year 8 mathspace.co

11.05 Mutually exclusive events After this lesson, you will be able to… • define mutually exclusive events and give examples of mutually exclusive scenarios in real-life contexts. • explain the mathematical relationship of mutually exclusive events in terms of probabilities • calculate probabilities for mutually exclusive events. • calculate probabilities for non-mutually exclusive events. • solve a variety of problems related to mutually exclusive and non-mutually exclusive events. • interpret Venn diagrams to visually represent mutually exclusive and nonmutually exclusive events.

Mutually exclusive events If events are mutually exclusive, it means they cannot happen at the same time. For example: • Tossing a coin - Consider the events ‘flipping a head’ and ‘flipping a tail’. We cannot flip a head and a tail at the same time. • Rolling a die - Consider the events ‘Rolling an even number’ and ‘rolling an odd number’. We cannot roll any number which is both even and odd. • Picking a card from a deck of cards - Consider the events ‘Drawing a 7 card’ and ‘Drawing a 10 card’. They have no outcomes in common. There is no card that is both a 7 and a 10. A

The Venn diagram of two mutually exclusive events looks like this, and so therefore

P (A and B) = 0 as there is no intersection. Some events can happen at the same time. They are called non-mutually exclusive. For example: • Picking a card from a deck of cards - Consider the events ‘drawing a Club card’ and ‘drawing a 7’. They have an outcome in common. We could pick a card that is a Club and a 7 because we could get the 7 of clubs. Clubs 7’s

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Example 1 Consider the following events: • Event 1: “A six-sided die is rolled, and the result is 3 or less” • Event 2: “A six-sided die is rolled, and the result is 4 or more” Are the events mutually exclusive?

Create a strategy To determine whether or not two events are mutually exclusive, ask these questions: • If Event 1 happens, do you know for sure that Event 2 did not happen? • If Event 2 happens, do you know for sure that Event 1 did not happen?

Apply the idea If Event 1 happens, do you know for sure that Event 2 did not happen? Yes, because a number that is 3 or less cannot also be 4 or more. If Event 2 happens, do you know for sure that Event 1 did not happen? Yes, because a number that is 4 or more cannot also be 3 or less. The events are mutually exclusive.

Idea summary Mutually exclusive events are events that cannot happen at the same time. A

P ( A and B) = 0

P ( A ∩ B) = 0

Probability with mutually exclusive events Consider an experiment using a standard deck of 52 cards with the events A: ‘Drawing a 7 card’ and B: ‘Drawing a 10 card’. What is P ( A or B)? We know that the events are mutually exclusive as it’s not possible to draw a 7 and 10 at the same time, therefore: 7’s

10’s

10 7

7 7

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10 10

Example 2 Two events A and B are mutually exclusive. If P ( A) = 0.37 and P ( A or B) = 0.73, what is P (B)?

Create a strategy Use the formula P ( A or B) = P ( A) + P (B).

Apply the idea P ( A or B) = P ( A) + P (B)

Write the formula

0.73 = 0.37 + P (B)

Substitute the values of P (A), P (A or B)

0.36 = P (B)

Subtract 0.37 from both sides

P (B) = 0.36

Make P (B) the subject

Idea summary If two events are mutually exclusive:

P ( A ∪ B) = P ( A) + P (B) P( A ∪ B)

means P ( A or B)

Probability with non-mutually exclusive events Exploration Consider the following two experiments using a standard deck of cards: • Experiment 1. With the following two events: A: Drawing a 2 card B: Drawing an 8 card • Experiment 2. With the following two events: A: Drawing a Heart card B: Drawing an 8 card Answer the following questions for Experiment 1 and then repeat for Experiment 2: 1. How could we find P ( A or B)? 2. Are there any cards that appear in both events A and B? 3. How will your answer to question 2 affect your answer to question 1?

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Now, let’s consider a card experiment with a standard deck of 52 cards and the events A: ‘Drawing a Club card’ and B: ‘Drawing a 7 card’. What is P ( A or B)? These events are non-mutually exclusive as the 7 of Clubs belong to both events. If we calculate P (Club or 7) = P (Club) + P (7), the 7 of Clubs will be counted twice. Therefore, we must subtract the probability of the intersection. Our formula becomes: Clubs

7’s

7 7

In actual fact, the relationship above holds for ALL probability events. It’s just that when two events are mutually exclusive, P ( A ∩ B) = 0 and so we get the shorter formula.

Example 3 A random card is picked from a standard deck. a Find the probability that the card is red or a diamond.

Create a strategy Use the formula P (R ∪ D) = P (R) + P (D) − P (R ∩ D).

Apply the idea There are 26 red cards out of 52 cards so P (R) = There are 13 diamond cards of 52 cards so P (D) =

. .

There are 13 diamonds and they are all red. Therefore P(R ∩ D) =

Write the formula

Substitute the values of P (R), P (D), P (R ∩ D)

Evaluate the addition

Evaluate

Simplify into the lowest term

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b Find the probability that the card is an ace or a diamond.

Create a strategy Use the formula P ( A ∪ D) = P ( A) + P (D) − P ( A ∩ D).

Apply the idea There are 4 ace cards out of 52 cards so P ( A) =

There are 13 diamond cards out of 52 cards so P (D) = Since there is 1 ace of diamond, P ( A ∩ D) =

. Write the formula

Substitute the values of P ( A), P (D), P ( A ∩ D)

Evaluate the addition

Evaluate

Simplify into the lowest term

c Find the probability that the card is an ace of spades or an ace of clubs.

Create a strategy Since each card is unique, use the formula P ( AS ∪ AC) = P ( AS) + P ( AC).

Apply the idea There is 1 ace of spades out of 52 cards so P ( AS) = There is 1 ace of clubs out of 52 cards so P ( AC) =

. .

Write the formula

Substitute the values of P ( AS), P ( AC)

Evaluate

Simplify into the lowest term

d Find the probability that the card is a black or a face card.

Create a strategy Use the formula P (B ∪ F ) = P (B) + P (F ) − P (B ∩ F ).

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Apply the idea There are 26 black cards out of 52 cards so P (B) =

There are 12 face cards out of 52 cards so P (F ) = Since there are 6 black face cards, P (B ∩ F ) =

. . Write the formula Substitute the values of P (B), P (F ), P (B ∩ F ) Evaluate the addition Evaluate Simplify into the lowest term

Idea summary The following is the formula for the probability of non-mutually exclusive events:

P ( A ∪ B) = P ( A) + P (B) − P ( A ∩ B) P( A ∪ B) means P ( A or B) P( A ∩ B) means P ( A and B)

11.05 Practice questions Understanding 1

Explorer

Adventurer

1–3

Explain why events A and B are mutually exclusive.

Complete the statement:

Trailblazer 1–3

P ( A or B) = P (⬚) + P (⬚)

Consider the diagram: Dogs

Fluffy

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Explain why the events in the Venn diagram are non-mutually exclusive.

Complete the statement: P (Dog or Fluffy) = P (Dog) + P (⬚) − P (⬚ and ⬚)

When picking a random card from a standard pack, which two events share no common outcomes? • Event A: picking a black card • Event B: picking a King

Fluency

4–7, 9, 11, 13

• Event C: picking a spade • Event D: picking a club 4–8, 10, 11, 13

5–8, 10, 12, 14, 15

A constellation is randomly selected from these six constellations. Consider these events: • Event 1: The constellation has 7 or more stars • Event 2: The constellation has more than 4 stars • Event 3: The name of the constellation begins with a C • Event 4: The name of the constellation begins with a consonant

Phoenix

Reticulum

Crux

Triangulum Australe

Volans

Tucana

Which two events are mutually exclusive? Ex 1

A standard six-sided die is rolled. Are these pairs of events mutually exclusive? a c e

Event A: Rolling a 4

Event A: Rolling an even number

Event B: Rolling an odd number

Event B: Rolling a prime number

Event A: Rolling an even number

Event A: Rolling a factor of 9

Event B: Rolling a factor of 6

Event B: Rolling an even number

Event A: Rolling a number less than 3

Event A: Rolling an even number

Event B: Rolling a number greater than 5 Ex 2

Ex 3

Event B: Rolling a multiple of 3

Two events A and B are mutually exclusive. a

If P ( A) = 0.45 and P ( A ∪ B) = 0.73, ﬁnd P (B).

If P ( A) =

and P (B) = , ﬁnd P ( A ∪ B).

A random card is picked from a standard deck. For each pair of events: i

Determine whether the events are mutually exclusive.

Find the probability of the union of the events.

A black card or a heart

A spade or a diamond

An ace of diamonds or an ace of clubs

A red card or a face card

For the Venn diagrams: i

Calculate the value of x.

Are events A and B mutually exclusive?

0.7

0.1

B 0.4

0.2

0.5

0.1

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The Venn diagram shows the party preferences of voters in a sample of the population. a

Find the probability that a random voter has a preference for both Labour and Liberal.

Are the preferences of voters in this population mutually exclusive?

If P ( A) = , P (B) =

and P ( A ∩ B) =

, ﬁnd P ( A ∪ B).

If P ( A) = , P (B) =

and P ( A ∪ B) =

, ﬁnd P ( A ∩ B).

If P ( A ∪ B) = , P ( A ∩ B) =

If P ( A) = 0.68, P (B) = 0.72 and P ( A and B) = 0.4896, ﬁnd P ( A or B).

and P ( A) = , ﬁnd P (B).

A group of 20 people were asked which colours they liked. Of this group, there were 9 people who only liked green, 6 who only liked pink, and 2 who liked neither. Create a visual model to represent this scenario. Find the probability of a person liking: i

Green and pink

Green or pink but not both

In a particular high school where there are 91 year 12 students, 50 students study Extension 2 English and 46 students study Extension 2 Mathematics. Five students study neither. a b

Liberal

Two events A and B are non-mutually exclusive.

Labour

Create a visual model to represent this scenario. If a student is randomly chosen, ﬁnd the probability that this student is studying: i

Extension 2 English and Extension 2 Mathematics.

Extension 2 English only.

iii

Extension 2 Mathematics only.

Extension 2 Mathematics or Extension 2 English.

Neither subject.

Among a group of 61 students, 14 are studying Philosophy, 41 are studying Science, and 7 are studying neither subject. a

How many students are studying Philosophy and Science?

If a student is picked randomly, ﬁnd the probability that they study at least one of these subjects.

Find the probability that a randomly selected student studies only one of the subjects.

Among a group of students studying Economics and/or Law, a single student is randomly chosen, and the probabilities are as follows: • P (studies Economics) = 0.5 • P (studies neither) = 0 • P (studies Law) = 0.7 Find the probability that a student chosen at random studies both Economics and Law.

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On a particular day of industrial action, 448 workers decided to work or strike. According to the data, 146 workers decided to work, 290 workers decided to strike, and 12 did neither. a b

Are the events mutually exclusive? Find the probability that a worker selected randomly chose to: i

P (Strike)

P (Strike ∩ Work)

P (Strike ∪ Work)

P (Strike′ ∪ Work′)

Reasoning 16

16, 18

iii

16, 18

P (Strike′ ∩ Work)

17, 18

A box of cupcakes contains six chocolate cakes and four cheesecakes. Two of the chocolate cakes and three of the cheesecakes have vanilla ﬁlling inside. a

Are the events “choosing a chocolate cake” and “choosing a cheesecake” mutually exclusive? Explain your answer.

Are the events “choosing a chocolate cake” and “choosing a cake with vanilla ﬁlling” mutually exclusive? Explain your answer.

Consider the formula for finding P ( A ∪ B) of mutually exclusive events and the formula for finding P ( A ∪ B) of non-mutually exclusive events. Which formula can be used to find P ( A ∪ B) of both mutually exclusive and non-mutually exclusive events? Explain your reasoning.

Enrica is given this information about a sample space containing events A and B: • The sample space has 34 elements. • Event A has 18 elements. • Event B has 14 elements. • There are 6 elements that exist in neither A nor B. Enrica states P ( A ∪ B) =

. Explain Enrica’s error and write the correct answer.

Problem-solving 20, 21 19

20–22

19, 21–23

For each of the groups of probabilities: i

Find P ( A ∩ B′).

Are events A and B mutually exclusive?

P ( A ∪ B) = 0.6, P ( A′) = 0.6 and P (B′) = 0.7

P ( A ∪ B) = 0.3, P ( A′) = 0.8 and P (B′) = 0.9

P ( A′ ∩ B′) = 0.1, P ( A) = 0.8 and P (B) = 0.3

P ( A′ ∩ B′) = 0.2, P ( A) = 0.1 and P (B) = 0.7

Find P (B ∩ A′).

iii

Find P ( A ∩ B).

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In a survey, 40 people were asked whether they walk (W ) or run (R). The results are: • W ∩ R′ = 12 • (W ∪ R)′ = 22 • W∩R=6

Construct a Venn diagram with this information.

Are the events mutually exclusive?

If one person is chosen randomly, ﬁnd the probability that they walk for exercise.

In an experiment, a number is chosen randomly from the numbers listed: {2, 3, 5, 6, 7, 10, 12, 14, 15, 16, 19, 20} • Event A: odd number is chosen. • Event B: multiple of 4 is chosen. a

Are the events mutually exclusive?

Which of these has the highest probability?

A∪B

B′

A∩B

P (( A ∩ B)′)

iii

P ( A′ ∩ B′)

P ( A′ ∪ B)

Find: i

P ( A ∩ B)

In a survey, 59 students were asked to select all the subjects they enjoyed out of Maths, English and Science. • 35 enjoyed Maths • 18 enjoyed Maths and Science • 32 enjoyed English • 17 enjoyed Maths and English • 36 enjoyed Science • 9 students only enjoyed Science • 9 enjoyed all three a

Construct a visual model to represent this situation.

Find the probability that a student likes Maths and only one other subject.

Find the probability that a student likes only one of the subjects.

The employees of Squiggle were discussing where they should hold their end-of-year party. Of all the employees: • 51 would not go to a restaurant. • 58 would not go to a bowling alley. • 57 would not go to a theme park. • 22 would go to neither a restaurant nor a bowling alley. • 25 would go to neither a bowling alley nor a theme park. • 23 would go to neither a restaurant nor the theme park. • 4 would not go to a restaurant, bowling alley or a theme park. • 6 would go to all three places. How many employees work at Squiggle in total?

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INVESTIGATION

Probability simulations Investigate and describe the results of probability simulations.

Objectives • To use digital tools to explore repeated chance experiments and simulations • To determine probabilities for compound events and describe results

Materials • Two dice

•

Tally chart

Probability Simulations If we roll two dice and add the numbers that show on the uppermost faces: The lowest and highest scores possible are 2 (getting a double 1) and 12 (double 6). Every score between 2 and 12 is possible, as shown in the table. Assuming the dice are fair, each of the 36 outcomes are equally likely to happen.

Probabilities with the sum of two dice To ﬁnd the probability of each of the sums, P (Sum), we can divide the number of times that outcome appears in the table n (Sum) by the total, 36.

Sum

P (Sum)

Investigation: Probability simulations mathspace.co

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A graph of these theoretical probabilities would look like this: Relative Frequency

7 36 6 36 5 36 4 36 3 36 2 36 1 36 1 2 3 4 5 6 7 8 9 10 11 12

Total

Theory versus reality This is a theoretical distribution, so what would happen in reality? We will investigate this by rolling some dice using a simulation.

Interactive investigation

Discover this concept in action online

mathspace.co

If you do not have internet access, take a pair of regular six-sided dice and roll them 36 times, then note the sum of the two dice. Plot your results on a relative frequency diagram or bar chart.

Interactive investigation

Discover this concept in action online

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Application to Game Play

The popular board game Settlers of Catan has players collect resources by rolling a pair of dice.

3 10 8

Players build settlements at the junction of three hexagonal tiles. If a dice roll, by any player, results in a total of 3, 8 or 10, the player with a settlement at the position shown by the red dot will receive a resource.

The average number of rolls to complete a game of Catan is 80. Theoretically, the probability of receiving resources from this settlement is:

So, a player could expect to get:

approximately 22 resources from this settlement over the course of an average game.

These 3 tiles have the same theoretical probability: 4 11 6

So, over the course of a game, a player with this settlement should expect the same number of resources as a player at the junction of 3, 8 and 10.

Interactive investigation

Discover this concept in action online

mathspace.co

Investigation: Probability simulations mathspace.co

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Record your findings in a table like this: Player

Sum of winning frequencies

Number of resources

A B Player A gets a resource if a 3, 8, or 10 is rolled. Player B gets a resource if a 4, 6, or 11 is rolled.

Investigate After doing the simulation with the applet or manually, consider the following questions: 1.

How closely does it match the theoretical model of each player receiving 22 resources over the course of a game with 80 rolls?

Compare your results with other students in your class. Did anyone get results that matched the theoretical model?

Combine your results with your classmates. Do the combined results give a better representation of the theoretical model than the individual results?

Choosing your numbers Player A

4 5 Player B 10

Consider other combinations of numbers that players could have such as:

11 12

With a partner, discuss the following questions: If you could be Player A or Player B, which would you choose? Explain your thinking. After 80 rolls, how far ahead would your chosen player be? Explain your thinking. Use a simulation, or other strategy, to see what might happen experimentally versus theoretically.

Investigate

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If you could pick any three numbers for Catan, which would you choose? Explain.

Give a three number combination for Catan you would want to avoid. Explain.

Would your answers to the ﬁrst two questions change depending on the length of the game? Explain.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Discussion 1.

Is the theoretical probability always reﬂected in reality? Explain.

When does a simulation begin to look like the theoretical probability? Explain.

For more complex probability problems, can we use simulations to determine probabilities? If so, explain how. If not, explain why not.

Did you know?

Did you know that the chances of flipping a coin and getting heads is always 50%, no matter how many times you flip it? It’s a common misconception that previous flips influence future outcomes, but each flip is independent! This idea is known as the “Gambler’s Fallacy,” where people believe that if heads has come up multiple times in a row, tails is “due” next. In reality, the probability stays the same, and each flip is an isolated event with no memory of previous outcomes. Whether you’re on your first flip or your hundredth, the chances are still 50-50! This principle applies not only to coins but to many other independent events in probability theory. Mini-Game: Coin Flip Streak Challenge! Try this interactive coin flip game to see how the Gambler’s Fallacy plays out: 1.

Flip a coin 10 times (you can use a real coin or a coin flip app).

Track your results: Write down each result (heads or tails) as you flip.

Look for patterns: Did you flip heads or tails multiple times in a row? How many times did you flip the same result consecutively?

Now, guess the next flip. If you’ve just flipped heads three times in a row, do you think tails is “due”? Flip again and find out!

Reflect: After completing the 10 flips, compare the number of heads and tails. Were your guesses influenced by previous flips? Did the outcome really change? Investigation: Probability simulations mathspace.co

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11 Chapter review 1

Are the two events complementary? • Event 1: Selecting an even number. a • Event 2: Selecting an odd number. b

• Event 1: Drawing a vowel from the English alphabet. • Event 2: Drawing a consonant from the English alphabet.

• Event 1: Selecting a prime number less than 10. • Event 2: Selecting a composite number less than 10.

• Event 1: Rolling a number greater than 4 on a die. • Event 2: Rolling a number less than 4 on a die.

Imagine there is a beverage stall that offers customised smoothie options. You are given the following choices, and you can choose only one from each category: • Fruit options: banana, apple, or orange • Supplement additions: protein powder, ﬂax seeds, chia seeds, or oats • Sweeteners: honey or agave syrup Honey Banana Agave syrup Honey Apple Agave syrup Honey Orange Agave syrup

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Oats Flax seeds Protein powder Chia seeds Oats Flax seeds Protein powder Chia seeds Oats Flax seeds Protein powder Chia seeds Oats Flax seeds Protein powder Chia seeds Oats Flax seeds Protein powder Chia seeds Oats Flax seeds Protein powder Chia seeds

How many different types of smoothies could be created?

What is the probability that someone randomly chooses a smoothie with banana?

What is the probability that someone chooses a smoothie with honey?

What is the probability that someone chooses a smoothie with oats?

Consider a student who can either choose to study or relax on a given day, with an equal chance of choosing either one. Let’s analyse this situation for three consecutive days. a

Construct a tree diagram for the student’s choices for three consecutive days.

Find the probability that the student chooses to study three days in a row.

Find the probability that the student chooses to study for two days and relax on another day in any order.

A letter is chosen at random from the word COCOON. a

Which letter has the highest probability of not being chosen?

Find the probability that the chosen letter is not C.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Determine the intersection of each of the following pairs of sets: a

A = {−10, −5, 0, 5, 10, 15, 20, 25} b P = {2, 4, 6, 8, 10, 12, 14} B = {−10, −8, −6, −4, −2, 0, 2, 4, 6, 8} Q = {1, 3, 5, 7, 9, 11, 13}

M = {−30, −6, −2, 0, 2, 6, 30} N = {0, 2, −2, 6, −6, 30, −30}

Determine the union of each of the following pairs of sets: a c

A = {8, 88, 888, 8888}

F = {s, c, i, e, n, c, e}

B = {0, 2, 88, 102, 104}

G = the set letters in the word ‘science’

S = {2, 4, 6, 10, 14, 22, 26, 38}

J = the set of multiples of 7

T = {2, 4, 6, 16, 22, 26, 30, 36, 40} 7

W = the set of multiples of 5 X = the set of multiples of 10

K = the set of multiples of 14

A marble is chosen at random from a box containing 5 different colour marbles, red, blue, green, yellow, and purple. The probability of selecting different colours is given in the table. Find: a

P (Green)′

P (Blue)′

P (Red ∪ Purple)

P (Purple)′

P (Blue ∪ Green)′

A survey was conducted among pet owners. The two categories show if they own at least one dog, and if they own at least one cat: a

How many of the participants own at least one pet?

How many of the participants own at least one dog?

How many of the participants don’t own a cat?

Colour

Probability

Red Blue Green Yellow

Own a cat

Own a dog

A student is making a Venn diagram from the information gathered from her classmates: 24 people read fantasy novels, 20 read science ﬁction, and 8 read both fantasy and science ﬁction. a

How many people only read science ﬁction?

Construct a Venn diagram that shows how many students read each genre.

The given Venn diagram shows the number of students in a school playing the piano, the violin, both or neither: Find the number of students who: a

Play the piano only

Play the piano

Play the violin

Play the violin only

Do not play the piano

Do not play the violin

Piano

Violin 50

195

170 170

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Consider the Venn diagram shown. Find: a

P ( A ∩ B)

P ( A ∩ B′)

P ( A ∪ B)

P ( A ∪ B)′

B 10

20 16

A librarian collected a sample of books from the library and divided them into the appropriate categories as shown in the Venn diagram: Find the probability that a book is: a

Not a fiction, but a hardcover

Not a ﬁction and does not have a hardcover

Fiction 18

Hardcover 16

7 12

14 Long decade

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Consider the two-way table shown. Find: a

A∩B

A′ ∩ B′

A∪B

A′ ∪ B

A′

Total

B′

Total

An art shop sells two types of paintbrushes: round and ﬂat. The shop records the number of paintbrushes sold over two days. The results are shown in the two-way table: a

Which day had more round paintbrushes sold?

Which type of paintbrush was more popular on Day 2?

Calculate the probability that a randomly chosen paintbrush sold on Day 1 is a ﬂat paintbrush.

Day 1

Day 2

Total

Round

Flat

Total

At a local book club meeting, club members were asked if they preferred fiction or nonfiction books. 36 members said they preferred fiction, of which 10 were teenagers. 16 teenagers said they preferred non-fiction, and 18 adults said they preferred non-fiction. a

Construct a two-way table based on the results of the book club survey.

What is the fraction of responders that prefer fiction?

What is the fraction of adults that prefer fiction?

Mathspace Victorian Curriculum – Year 8 mathspace.co

A total of 150 employees were surveyed about their lunch preferences. The employees were categorised based on the type of meal they preferred (Vegetarian or Non-Vegetarian) and where they preferred to eat (Cafeteria or Outdoors). Cafeteria 35 60 95

Vegetarian Non-Vegetarian Total

Outdoors 20 35 55

Total 55 95 150

Construct a Venn diagram that represents the information provided in the two-way table. 17

A standard 10-sided die (numbered from 1 to 10) is rolled. Determine whether the two events are mutually exclusive. a c e

Event A: Rolling a 6

Event A: Rolling an even number

Event B: Rolling an odd number

Event B: Rolling a prime number

Event A: Rolling an even number

Event A: Rolling a factor of 4

Event B: Rolling a factor of 8

Event B: Rolling an even number

Event A: Rolling a number less than 5

Event A: Rolling an even number

Event B: Rolling a number greater than 7 18

Event B: Rolling a multiple of 5

Two events C and D are mutually exclusive. a

If P (C) = 0.35 and P (C ∪ D) = 0.68, ﬁnd P (D).

If P (C) =

and P (D) = , ﬁnd P (C ∪ D).

A random card is picked from a standard deck. For each of the following: i

Determine whether the events are mutually exclusive.

Find the probability of the union of the events.

A black card or a club

A heart or a diamond

An ace of spades or an ace of hearts

A black card or a face card

Two events A and B are non-mutually exclusive. a

If P ( A) = , P (B) =

and P ( A ∩ B) =

, ﬁnd P ( A ∪ B).

If P ( A) = , P (B) =

and P ( A ∪ B) = , ﬁnd P ( A ∩ B).

If P ( A ∪ B) = , P ( A ∩ B) =

If P ( A) = 0.78, P (B) = 0.82 and P ( A and B) = 0.62, ﬁnd P ( A or B).

and P ( A) = , ﬁnd P (B).

Among a group of individuals engaging in painting and/or dancing, a single individual is randomly chosen, and the probabilities are as follows: • P (painting) = 0.6

• P (neither) = 0.1

• P (dancing) = 0.8

Find the probability that an individual chosen at random engages in both painting and dancing.

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Lily owns six red dresses and four black dresses. She randomly selects one of the dresses for herself and her friend.

R R

A probability tree diagram of this scenario is shown:

Find the probability that Lily selects a black dress for herself.

If we know that Lily selected a black dress for herself, ﬁnd the probability that she selects a red dress for her friend.

B B

Construct a tree diagram of all possible options.

If, at every stage, the possible outcomes of each choice are equally likely, ﬁnd the probability Alice reads Fiction and two Mysteries.

A standard deck of playing cards contains 52 cards: 26 are red (hearts and diamonds) and 26 are black (spades and clubs). Among each colour, there are 13 cards: 10 numbered cards (1 to 10), and 3 face cards ( jack, queen, and king). If one card is chosen at random, ﬁnd the probability that: a

It is a red card.

It is an odd numbered card or a face card.

It is a red card and an odd number.

It is a face card or a black card.

It is a red card with a prime number on it.

A group of people were randomly selected and asked which beverages they drank in the morning. The Venn diagram shows the results: drink coffee is

. There were no respondents that did not

answer coffee or tea. How many people said they drink both coffee and tea?

Coffee

Tea

In a survey of 36 students, it was found that: • 18 students play guitar • 16 students play piano • 3 students play none of these instruments

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Alice usually reads three different genres of books each week: She starts with either Fiction or Non-ﬁction, and then the next two books are either Mystery, Romance, or Fantasy. She never reads more than one Fantasy book, and if she reads Fiction, she will follow this immediately with a Mystery.

If one of them is chosen at random, the probability that they

• 9 play both guitar and drums • 8 play both drums and piano • 7 play both guitar and piano • 4 play all three instruments

Construct a Venn diagram for this situation.

Find the probability that a randomly selected student plays all three instruments.

Mathspace Victorian Curriculum – Year 8 mathspace.co

A group of international students were asked whether they spoke French or German.

French Not French Total

German 48 25

Not German 12 15

Total

Complete the following table.

How many students speak both languages?

If one student is chosen randomly, ﬁnd the probability that they speak neither language.

If one student is chosen randomly, ﬁnd the probability that they speak only one of the languages.

At a local software company, employees were asked what their favourite coding language was when they started coding, and what language they primarily use now after 5 years of working. The results are shown in the following table: Python now JavaScript now Java now PHP now Total

Python start 120 84 82 10 296

JavaScript start 35 76 16 28 155

Java start 91 79 99 56 325

PHP start 67 88 88 110 353

Total 313 327 285 204 1129

One employee is chosen at random. Find:

The probability that an employee’s favourite coding language was Python when they started.

The probability that an employee is now primarily using Java or PHP.

The probability that an employee’s favourite starting language was Java and they are using a different language now.

The probability that an employee’s current language is the same as their starting language.

In a survey, 60 people were asked whether they enjoy gardening (G) or reading (R). The results are shown below: • G ∩ R′ = 18 • (G ∪ R)′ = 32 • G ∩ R = 10

Are the events mutually exclusive?

If one person is chosen randomly, ﬁnd the probability that they enjoy gardening.

The members of “The Literary Squad” book club were discussing where they should hold their annual book review gala. Of all the members: • 63 would not go to a library. • 74 would not go to a coffee shop. • 67 would not go to a community centre. • 28 would go to neither a library nor a coffee shop. • 33 would go to neither a coffee shop nor the community centre. • 30 would go to neither a library nor the community centre. • 6 would not go to a library, coffee shop or community centre. • 8 would go to all three places. How many members are in the “The Literary Squad” book club in total? Chapter 11 review mathspace.co

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Big ideas • Statistical data can be analysed and described using different measures such as mean, mode, median and range. • Sampling methods can be a powerful tool for making inferences about a population, considering potential biases are essential to ensure the reliability of results.

Curriculum Outcomes VC2M8ST01 Distinguish between a population and a sample, and investigate techniques for data collection including census, sampling, experiment and observation, and explain the practicalities and implications of obtaining data through these techniques VC2M8ST02 Analyse and report on the distribution of data from primary and secondary sources using random and non-random sampling techniques

VC2M8ST03 Compare variations in distributions and proportions obtained from random samples of the same size drawn from a population and recognise the effect of sample size on this variation VC2M8ST04 Plan and conduct statistical investigations involving samples of a population; use ethical and fair methods to make inferences about the population and report findings, acknowledging uncertainty

12 Statistics Chapter outline 12.01 Measures of centre and spread Investigation: Random sampling 12.02 Sampling methods 12.03 Bias 12.04 Compare samples to populations Investigation: Plan and conduct statistical investigations Chapter 12 review

660 678 680 689 698 710 712

City planners use the statistics gathered in the Australian census every four years. It helps them decide where to build new roads, schools, or parks.

12.01 Measures of centre and spread After this lesson, you will be able to… • identify the mean, median and mode of a data set and understand the conditions when each is most appropriately used. • understand the range of a data set and how it is used to describe spread. • recognise the effects of outliers on measures of centre and spread. • identify outliers in a data set. • recognise and interpret the shape of different data distributions, including symmetrical, asymmetrical, positively skewed, and negatively skewed. • analyse histograms and other visual representations of data to identify their shape and skew.

Summary statistics and outliers Recall that we can use the summary statistics of mean, median or mode to describe the centre of a data set and the range to describe the spread. • The mean is often called the average and is calculated by adding all the scores in a data set and dividing by the number of scores. • The median is the number in the middle of an ordered numerical data set. • The mode is the most frequently occurring value in a numerical data set. Note that there can be more than 1 mode for a given data set. • The range of a numerical data set is the difference between the highest and the lowest score. An outlier is an extremely low or high score that is very different from the other scores in the data set.

Interactive exploration Discover this concept in action online

mathspace.co

Outliers can affect the summary statistics and skew or change the shape of our data. This can be a problem (especially for small data sets) because the mean, median and range might not properly represent the situation. We can counteract this by removing outliers. Removing outliers will have the following effects: Removing a low outlier The range will decrease. The median might increase. The mean will increase. The mode will not change.

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Removing a high outlier The range will decrease. The median might decrease. The mean will decrease. The mode will not change.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Example 1 Consider the following set of data: 53, 46, 25, 50, 30, 30, 40, 30, 47, 109 a Find the mean, median, mode and range.

Create a strategy To ﬁnd the mean, use the formula:

To find the median, find the middle score. To find the mode, find the most frequent score. To find the range, use the formula: Range = Highest score − Lowest score

Apply the idea Sum of scores Number of scores 53 + 46 + 25 + 50 + 30 + 30 + 40 + 30 + 47 + 109 = 10 460 = 10 = 46

Mean =

Use the formula Substitute all the scores Evaluate the addition Evaluate the division

To ﬁnd the median, order the scores: 25, 30, 30, 30, 40, 46, 47, 50, 53, 109 The middle two scores are: 40 and 46

Find the average of the middle values

Evaluate To ﬁnd the mode, choose the score which occurs most often. Mode = 30 To ﬁnd the range: Range = Highest score – Lowest score

Use the formula

= 109 – 25

Substitute the values

= 84

Evaluate the subtraction

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b Which data value is an outlier?

Create a strategy Choose the value that is much greater or much smaller than the rest of the data set.

Apply the idea Outlier = 109

c Find the mean, median, mode and range after removing the outlier.

Apply the idea Sum ofscores scores Sum Sum ofofscores Mean Mean Mean= = = Number ofscores scores Number Number ofofscores +30 +30 +30 +40 +46 +4747 +50 +53 25 30 30 30 40 46 50 53 25 25 + +30 + +30 + +30 + +40 + +46 + +47 + +50 + +53 = == 9 99 351 351 351 === 9 99 39 39 = = =39

Use the formula Substitute all the scores Evaluate the addition Evaluate the division

To find the median, order the scores: 25, 30, 30, 30, 40, 46, 47, 50, 53 The middle score is 40. Median = 40 To find the mode, choose the score which occurs most often. Mode = 30 To find the range: Range = Highest score – Lowest score

Use the formula

= 53 – 25

Substitute the values

= 28

Evaluate the subtraction

d Let A be the original data set and B be the data set without the outlier. Complete the table using the symbols >, < and = to compare the statistics before and after removing the outlier. Mean Median Mode

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With outlier A A A

⬚ ⬚ ⬚

Mathspace Victorian Curriculum – Year 8 mathspace.co

Without outlier B B B

Create a strategy Compare the summary statistics from part (a) and part (c).

Apply the idea Comparison table:

Statistics from parts (a) and (c): Mean Median Mode Range

With outlier 46 43 30 84

Without outlier 39 40 30 28

Mean Median Mode Range

With outlier A A A A

> > = >

Without outlier B B B B

When the outlier was removed, the values for the mean, median, and range all decreased.

Idea summary Mean • Sometimes called the average of a data set, this is the sum of the data values divided by the number of data values. Median • The middle value of a data set ranked in order. Mode • The data value that occurs most frequently. Range • The difference between the largest and smallest scores in the set. Removing outliers will have the following effects on the summary statistics: Remove a low outlier The range will decrease The median might increase The mean will increase The mode will not change

Remove a high outlier The range will decrease The median might decrease The mean will decrease The mode will not change

Best choice of centre Sometimes one measure may better represent the data than another. When deciding which to use we need to ask ourselves “Which measure would best represent the type of data we have?” The following summarises when each choice of centre is most appropriate. It’s a good idea to verify this using different data sets.

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Measure Mean Median Mode

Best choice when There are no extreme values or outliers in the data set. There are extreme values or outliers in the data set. There are no big gaps in the middle of the data. The data set has repeated numbers.

Example 2 The salaries of part-time employees at a company are given in the dot plot. Which measure of centre best reﬂects the typical wage of a part-time employee? Salaries (in $1000’s)

18 19 20 21 22 23 24 25 30 35 36 37 38

Create a strategy

Apply the idea

Choose the measure that is appropriate for extreme values.

The median is the best measure of centre that reﬂects the typical wage of a part-time employee due to the presence of the three extreme values in the data set.

Example 3 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 a Determine the mean number of times a speed camera issued a ﬁne in that month. Give your answer rounded to one decimal place.

Create a strategy

Apply the idea

To ﬁnd the mean, use the formula:

Add all the number of times a speed camera issued a ﬁne and divide by the total number of cameras: Find the sum of the scores

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Evaluate the division

b Determine the median number of times a speed camera issued a ﬁne in that month. Give your answer rounded to one decimal place.

Create a strategy The median in a data set with an even number of values is the average of the two middle data values once the data is ordered.

Apply the idea First half of the set: 101, 102, 115, 115, 121, 124, 127, 128, 130 Second half of the set: 143, 146, 162, 162, 163, 178, 183, 194, 977 The middle scores of the set: 130, 143 Find the average of the middle scores

Evaluate the addition

Evaluate the division

c Which measure is most representative of the number of ﬁnes issued by each speed camera in one month?

Create a strategy

Apply the idea

Use the measure that is less affected by outliers. The correct measure is median, because it is less affected when the outlier is removed.

d Which score causes the mean to be much greater than the median?

Create a strategy

Apply the idea

Choose the score which is really far away from the other score.

Outlier = 977

e The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which statement should she make to support the impression she wants to give? A

A sample of 20 speed cameras found that the median number of ﬁnes in one month was 136.5

A sample of 20 speed cameras found that, on average, 182.2 ﬁnes were issued in one month.

Create a strategy

Apply the idea

Choose the option which uses the larger number The correct option is B. out of the mean and median.

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Idea summary Mean • Appropriate for sets of data where there are no values much higher or lower than those in the rest of the data set. Median • A good choice when data sets have a couple of values much higher or lower than most of the others. Mode • A good descriptor to use when the set of data has some identical values, when data is nonnumeric (categorical) or when data reﬂects the most popular item.

Shapes of distributions Data may be described as symmetrical or asymmetrical. There are many cases where the data tends to be around a central value with no bias left or right. In such a case, roughly 50% of scores will be above the mean and 50% of scores will be below the mean. In other words, the mean and median roughly coincide. The normal distribution is a common example of a symmetrical distribution of data.

Percentage

The normal distribution looks like this bellshaped curve.

This picture shows how a data set that has an approximate normal distribution may appear in a histogram.

30 25 20 15 10 5

The dark line shows the symmetrical curve that can be drawn over the histogram that the data roughly follows. 75 80 85 90 95 100 105 110 Weight

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In this distribution, the peak of the data represents the mean, the median and the mode which are all equal.

Frequency

A uniform distribution is a symmetrical distribution where each outcome is equally likely, so the frequency should be the same for each outcome. For example, when rolling dice the outcomes are equally likely. While we might get an irregular column graph if only a small number of rolls were performed, if we continued to roll the dice the distribution would approach a uniform distribution as shown.

Score

If a data set is asymmetrical instead (i.e. it isn’t symmetrical), it may be described as skewed. A data set that has a positive skew (sometimes called a ‘right skew’) has a longer tail of values to the right of the data set. The mass of the distribution is concentrated on the left of the ﬁgure. A positively skewed graph has this general shape with the right side stretched out.

Frequency

General shape shown over a histogram of positively skewed data.

Score

A data set that has a negative skew (sometimes called a ‘left skew’) has a longer tail of values to the left of the data set. The mass of the distribution is concentrated on the right of the ﬁgure. A negatively skewed graph has this general shape with the left side stretched out.

Frequency

General shape shown over a histogram of negatively skewed data.

Score

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Example 4 The stem-and-leaf plot shows the age of people to enter through the gates of a concert in the ﬁrst 5 seconds. Stem

Leaf

4 5

4 Key: 1|2 = 12 years old

a What was the median age?

Create a strategy

Apply the idea

Find the middle score.

We know that there are 19 values recorded, so the 10th value will be the median. Median = 20 years old

b What was the difference between the lowest age and the median?

Create a strategy

Apply the idea

Subtract the lowest value recorded from the median.

The lowest age recorded was 10 years old. Difference = 20 – 10

Subtract the values

= 10 years

Evaluate

c What is the difference between the highest age and the median?

Create a strategy

Apply the idea

Subtract the median from the highest value recorded.

The highest age recorded was 54 years old. Difference = 54 – 20 = 34 years

Subtract the values Evaluate

d What was the mean age? Round your answer to two decimal places where necessary.

Create a strategy

Apply the idea

Use the formula:

Add the scores using a calculator. 432. Use the formula

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Evaluate the division

e Is the data positively or negatively skewed?

Create a strategy

Apply the idea

If most of the scores are relatively low, the Most of the scores appear in Stem 1 and Stem 2, distribution is positively skewed whereas if most so they are mostly low. So the distribution is of the scores are relatively high, the distribution positively skewed. is negatively skewed.

Example 5 State whether the scores in each histogram are positively skewed, negatively skewed or symmetrical (approximately).

Frequency

a 10 9 8 7 6 5 4 3 2 1 0

14 15 Score

Create a strategy

Apply the idea

Check where the bulk of the data sits and look at the general shape of the distribution.

The scores are roughly even in both the high and low end, so the distribution is symmetrical.

Frequency

10 9 8 7 6 5 4 3 2 1 0

14 15 Score

Apply the idea The scores have higher-frequency columns on the left (the lower scores), so the distribution is positively skewed.

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Frequency

10 9 8 7 6 5 4 3 2 1 0

14 15 Score

Apply the idea The scores have higher-frequency columns on the right (the higher scores), so the distribution is negatively skewed.

Idea summary A distribution is said to be symmetrical if its left and right sides are mirror images of one another. A normal distribution is symmetrical and shaped like a bell. A uniform distribution is a symmetrical distribution where each column is the same height. A data set that has a positive skew (sometimes called a ‘right skew’) has a longer tail of values to the right of the data set. A data set that has a negative skew (sometimes called a ‘left skew’) has a longer tail of values to the left of the data set.

12.01 Practice questions Understanding 1

670

Explorer 1–3

Adventurer 1–3

Trailblazer 1–3

Match each term with the correct deﬁnition: a

Mean

The average of all the values in the data set

The most common value in the data set

The difference between the highest and lowest values in the data set

The middle value of the data set when ranked in order

Median

Mathspace Victorian Curriculum – Year 8 mathspace.co

Mode

Range

Give an example of a ﬁve number data set where the median would be a better measure of centre than the mean. Justify your answer.

What is an outlier, and why is it important to identify them in a data set before analysing the data?

Fluency

Ex 1

4cd, 5–7, 8bd, 9, 11–15, 17, 19, 20

For each set of data: i

Find the mean, median, mode and range. Round your answers to two decimal places where necessary.

Identify the 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.

53, 46, 25, 50, 30, 30, 40, 30, 47, 109

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

Bar Chart

Consider the bar chart: a

Find the range.

Find the mode.

Determine the mean, rounded to two decimal places.

20 18 16 14 12 10 8 6 4 2 0

Consider the dot plot:

4ac, 5–7, 8bc, 9, 11–15, 17, 19

Frequency

4ab, 5, 6, 8ac, 10, 12–16, 18

38 39 40 41 Score

9 10 11 12 13 14 15 16 17 18 19

Find the total number of scores.

Find the median score.

Find the mode.

Find the range.

Consider the data provided in the table: a

Calculate the range.

What is the mode?

Determine the median.

Score 68 69 70 71 72 73

Frequency 16 41 30 31 49 29

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For each data set, calculate: i

Mean

Median

Leaf

Mode

iii b

Range

Leaf

Key: 6|2 = 62 c

Numbers of sit-ups per student

Key: 2|4 = 24

0 5 10 15 20 25 30 35 40 4550 55

Frequency

10 9 8 7 6 5 4 3 2 1 0

Price

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The mean for the data in the table is 13.7. a

If the outlier is removed, will the new mean be higher or lower than the current mean?

If the outlier is removed, calculate the new mean, rounded to one decimal place.

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Weight in kilograms 12 13 14 15 16 17 21

Frequency 2 5 1 2 0 0 1

The mean for the data in the stem-and-leaf plot is 48.50. a

If the outlier is removed, will the new mean be higher or lower than the current mean?

If the outlier is removed, calculate the new mean, rounded to two decimal places.

Leaf 3 4 4 9 4 6 6 8 9 5 1 4 6 6 7 8 Key: 2|3 = 23

12 Ex 2

Mean, mode and median are all measures of centre. Which best represents the centre for each data set? a

12, 15, 16, 21, 22, 25

15, 13, 16, 17, 15, 15, 15

8, 10, 14, 18, 19, 91

1, 1, 2, 1, 1, 3, 1, 1

What is the most appropriate measure of centre for the scenario? a

For a particular street in Sydney, house prices for houses sold in the last 36 months (in 1000’s) were: 920, 920, 709, 694, 667, 657, 630, 610, 584, 567, 555

The number of animal races that were won by a trainer over the years shown are listed in the table: Year Races won

2003 116

2004 105

2005 102

2006 108

2007 113

The number of minutes spent studying by each member of a group of students is: 85, 85, 95, 70, 60, 70, 105

A netball team has played 8 games in the season. Their ﬁnal score in each game is: 36, 46, 49, 40, 52, 44, 39, 37

The number of songs Oprah has downloaded in each of the last seven months is: 6, 12, 18, 9, 0, 3, 15

13 Ex 3

The weight of ﬁsh caught in a “weigh and release” ﬁshing competition, in kilograms, are given: 12.5, 15.1, 13, 14.2, 14.5, 14.9, 12.5, 14.3, 1.5 a

Find the mean weight.

Find the median weight.

Recalculate the mean weight excluding the outlier.

Recalculate the median weight excluding the outlier.

Which measure of centre best identiﬁes the typical ﬁsh weight? Explain your answer.

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Ex 4

What is the median age?

ind the difference between the lowest age and the F median.

ind the difference between the highest age and the F median.

Find the mean age, rounded to two decimal places.

Is the data positively or negatively skewed?

Frequency

1 0 1 2 3 4 5 6 6 2 0 1

1 2 2 5

3 4 7 9 4 4 5

1 Key: 1|2 = 12 years old

Score

25 20 15 10 5

2 3 4 5 6 Score

30 25 20 15 10 5 0

8 9 10 11 1213 14 15 16 17

Frequency

12 10 8 6 4 2 0

5 6 7 8 9 10 11 12 Score

14 Frequency

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Leaf

Determine whether the graph is positively skewed, negatively skewed or symmetrical:

Frequency

Ex 5

The stem-and-leaf plot shows the age of people to enter through the gates of a concert in the ﬁrst 5 seconds:

Frequency

4 5 Score

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1 2 3 4 5 6 7 Score

The shoe sizes of all the students in a class were measured and the data was presented in a bar graph: a

Are there any outliers?

Find the modal shoe size.

Describe the shape of the distribution of the data.

Frequency

10 5

Consider the data shown in the histogram: a

Are there any outliers? If so, what is the value?

What is the mode?

Describe the shape of the distribution of the data.

Frequency

7 8 9 Shoe size

45 40 35 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 10 11 Score

Consider the dot plot shown: a

Describe the shape of the distribution.

Are there any outliers? If so, what is the outlier?

1 2 3 4 5 6 7 8 9 Scores

Temperatures were recorded over a period of time and presented in the given dot plot: a

Are there any outliers? If so, what is the value?

Find the modal temperature.

Describe the shape of the data. 40 41 42 43 44 45 46 47 48 49 Temperatures

Consider the given dot plot: a

Are there any outliers? If so, what is the value?

Find the modal score(s).

Describe the shape of the data.

Reasoning 21

21, 22

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Scores

21, 22

23, 24

Explain why this data set does not have a mode: 4, 8, 1, 2, 5, 9, 3, 6

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These six numbers have a mean of 107: 102, 103, 111, 107, 108, 111 If a new number is added that is larger than 111, will the mean will be higher or lower? Explain your reasoning.

The median house price in Humbleton is $950 000 with a mean price of $1 000 000, and the median house price in Brockway is $950 000 with a mean price of $880 000. Which town is most likely to have some very expensive houses? Explain your reasoning.

When an outlier is removed from a data set, describe the effect on the statistic, and explain your reasoning: a

Mode

Range

Problem-solving 25, 28–30

Mean

25, 26, 28–30

Median

25–27, 29, 31

Write a set of 6 numbers whose mean is 10, range is 15, and mode is 8.

The mean of a set of 41 scores is 18.6. If a score of 71.8 is added to the set, ﬁnd the new mean. Round your answer to two decimal places.

The beaks of two groups of birds are measured, in millimetres, to determine whether they might be of the same species: Group 1 Group 2

41 54

32 44

49 44

34 54

43 43

47 47

37 41

38 39

Calculate the range for Group 1.

Calculate the range for Group 2.

Calculate the mean for Group 1. Round your answer to one decimal place.

Calculate the mean for Group 2. Round your answer to one decimal place.

How can we tell that the two groups of birds are most likely different species?

Two English classes, each with 15 students, sit a ten question multiple choice test. Their class results, out of 10, are shown: Class 1 Class 2

676

40 55

3 8

3 7

3 10

1 8

5 8

1 6

2 9

4 8

2 7

4 9

2 9

3 8

3 10

2 8

1 9

Calculate the mean, median, mode and range for Class 1. Round your answers to one decimal place if necessary.

Calculate the mean, median, mode and range for Class 2. Round your answers to one decimal place if necessary.

Which class was more likely to have studied effectively for their test?

Which statistical calculations support your answer?

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The selling price of recently sold houses are given: $467 000, $413 000, $410 000, $456 000, $487 000, $929 000

What is the mean selling price, rounded to the nearest thousand dollars?

Which of the prices raised the mean so that it is not reﬂective of most of the prices?

An outlier is a data value that signiﬁcantly differs from other data values in a given data set. Recalculate the mean selling price by excluding the outlier.

Is the mean calculated in part (c) more accurate as a measure of the average of the house prices?

Consider the stem-and-leaf plot showing the number of customers browsing an online store, recorded every ten minutes, over a 100 minute period:

Leaf 3 4 4 9

What are the largest and smallest number of customers recorded browsing the store over this time period? What is the range?

4 6 6 8 9

Find the mean of the data set, rounded to the nearest customer. Find the new mean of the data set with the outlier removed. Is this value more representative of the expected number of people browsing the store at any time? Why?

5 1 4

6 7 8 4 Key: 1|2 = 12

The glass windows for an aeroplane are rolled to a certain thickness, but machine production means there is some variation. The thickness of each pane of glass produced is measured (in millimetres), and the dot plot shows the results:

Glass thickness (mm)

The current median is 11.15. If the outlier is removed what is the new median?

The current mean is 11.1. If the outlier is removed what is the new mean? Round your answer to two decimal places.

Based on the data, what would you expect the average glass thickness for aeroplanes to be? Why?

10.5

10.7

10.9

11.1

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11.3

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INVESTIGATION

Random sampling Investigate methods of sampling using candy.

Objectives To use and compare various sampling methods To draw conclusions about the reliability of various sampling methods

Materials • • • •

Bags of multicoloured candies (with at least ﬁve distinct colours) Dice Pen or pencil Paper

Random sampling When taking a sample from a population, there are many different sampling methods that can be used. The ones we will explore today are the following: simple random, systematic, stratiﬁed, quota, clustered, convenience and judgement.

Procedure We will be working in groups for this investigation. Open one bag of candy; this represents the population of candies. Write down the colours in the bag and the number of candies there are of each colour. Each group will choose a sampling method. Use your group’s sampling method to select 10 different candies from the population. Write down the colours in the sample and the number of candies there are of each colour.

Simple random sample Assign a number on the dice to each colour of candy. Then, roll the dice 10 times to randomly select 10 candies.

Systematic sample Roll a dice and record the number. Line up the candies in rows. Then, select every nth candy where n is the number from the dice roll.

Stratified Group the candies from the bag by colour. Then, have a group member close their eyes, and randomly select a candy from each group. Continue until you have 10 candies.

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Quota sample Determine the population proportions of each colour. Then, use this proportion to select a certain number of candies of each colour so that the sample proportion is approximately equal to the population proportion.

Clustered sample Group the candies by colour. Assign each colour a number on the dice. Then, roll the dice to randomly select entire groups until you have 10 candies in your sample.

Convenience sample Have one group member pour the candies from the bag onto the table. The group member should choose the 10 candies that are closest to them.

Judgement sample Group members should take turns choosing candies based on their personal preference. On a piece of paper, record your group’s sampling method, the colour distribution of the population, and the colour distribution of the sample. Each group will share these results with the rest of the class.

Investigate Discuss these questions with your group after completing the procedure. 1.

How did your sampling method affect the distribution of candy colours?

Was your sampling method random or non-random? Explain your reasoning.

Was your sampling method representative of the entire population (the whole bag of candies)? Why or why not?

Discussion 1.

With a classmate, discuss the advantages and disadvantages of each sampling method and when each one might be most appropriate to use.

Think about real-world situations where your assigned sampling method might be used. What are the advantages and potential disadvantages in those situations?

Consider a scenario where you’re asked to make a decision about a large group (a school, a town, a country) based only on a sample. How would the choice of sampling method impact your decision and its reliability?

How does randomness affect the reliability of a sample? What are other factors that can improve a sample’s reliability?

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12.02 Sampling methods After this lesson, you will be able to… • define and distinguish between a census and a sample. • identify the advantages and disadvantages of using a sample. • define and apply the concept of random sampling. • define and execute systematic sampling. • understand and illustrate stratified sampling. • evaluate different sampling methods.

Census vs sample In a census, data is collected from every member of a population. In maths, a population means every member of a group. It may be a school’s population, a sports club’s population and so on. Collecting data from every member of a population is the most accurate way of gathering information, but it is not always the most practical and can be very expensive or time consuming. Typically, a sample survey is instead done on a subset of the population to make it quicker and less expensive.

Example 1 Hannah has chosen to collect information using a sample instead of a census. a What are the advantages of doing a sample? Select all that apply. A

It is cheaper to conduct.

There will be no sampling bias.

It is more accurate.

It takes less time.

Create a strategy

Apply the idea

Use the fact that samples require asking less people than a census.

Since a sample requires surveying less people, it would be cheaper to conduct and it would take less time. The correct options are A and D.

b What are the disadvantages of doing a sample? Select all that apply.

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It is less accurate.

It takes more time.

It is more expensive to conduct.

There can be sampling bias.

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Create a strategy

Apply the idea

Use the fact that samples do not survey the whole population being considered.

Because the whole population is not being surveyed, the results of a sample would be less accurate and there could be sampling bias depending on how the sample is collected. The correct options are A and D.

Example 2 For each of the following, determine whether they are a census or a sample: a Lucy has asked everyone in her oﬃce what snacks should be provided in the oﬃce.

Create a strategy

Apply the idea

Determine the population of interest for each scenario. Then decide if everyone in the population has been surveyed or only a smaller subset of the population has been surveyed.

The population in this example are the people in Lucy’s oﬃce. Lucy is receiving information about every person in this population, so she has conducted a census.

b James asks a few of his friends how they did in the test to see if he is above average in his class.

Apply the idea The population in this example are all the students in James’s class. James only asks a few friends, so he has conducted a sample survey.

c Joanne ﬁnds the height of the entire class to try to ﬁnd the average height of 15 year old students in Australia.

Apply the idea The population in this example are all 15 year old students in Australia. Joanne only collects information from her class, so she has conducted a sample survey.

Idea summary In a census, every member of a population is surveyed. In an unbiased sample, a representative proportion of the population is surveyed.

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Random sampling In this method, a sample is formed by selecting members from the population at random, where each member of the population has an equally likely chance of being selected. In simple cases, a sample could be created by drawing names from a hat. For most samples though, it is more common to use a random number generator. Calculators and spreadsheet applications usually have a random number generator that can generate random decimals between 0 and 1. These can be used to create random numbers within any range of values. While simple random sampling can be one of the cheapest and least time-consuming methods, it may not adequately represent every sub-group within larger populations.

Example 3 Sam wants to randomly select ﬁve people to use in a sample from a population of 20. He begins by assigning each person in the population, a number from 1 to 20. Using the random number generator on his calculator, he generates these random numbers: 0.532, 0.805, 0.686, 0.774, 0.272, 0.112, 0.145, 0.009, 0.196, 0.024, 0.018 Help Sam by examining the first two digits of each generated number to identify the five numbers within the range of 1 to 20. These numbers will be used to select numbers for this sample.

Apply the idea To identify the sample, first, look at the first two digits of each generated random number. If the number falls within the range of 1 to 20, keep it as part of the sample. Continue this process until you have selected five valid numbers, ensuring they all fall within the desired range. Applying this, the five suitable numbers would be: 0.112, 0.145, 0.196, 0.024 and 0.018 Truncating to two decimal places, we can create the following numbers that range between 1 and 20: 11, 14, 19, 2 and 1 This means that members of the population, randomly selected for the sample will be 1, 2, 11, 14 and 19.

Idea summary Random sampling selects people randomly from a population. Each person is equally likely to be selected.

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Systematic sampling To use systematic sampling, a starting point is chosen at random, and then items are chosen at regular intervals, such as, selecting every nth item from a list. For example, we may call every tenth business in the phone book or select every ﬁfth bottle from a production line. In other words, it uses a ‘system’ for selection. This method is often favoured by manufacturers for sampling products on a production line. While systematic sampling is better than simple random sampling, it can still under or over represent a subgroup within the population.

Systematic Sampling

Example 4 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 Divide the population size by the sample size Evaluate Every 16th item should be selected.

Idea summary Systematic sampling selects people at regular intervals or every nth on an ordered list.

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Stratified sampling Stratiﬁcation is the process of dividing a group into subgroups with the same characteristics before we draw our random sample. Then we look at the size of each subgroup (or strata) as a fraction of the total population. The number of items from each subgroup that are included in the sample should be in the same ratio as the amount they represent of the total population. For example, if a survey was being conducted on the student population of a school, and it was important to represent all year levels, a sample might be formed by randomly selecting 10% of students from each year group. While stratiﬁed sampling can be more expensive and time-consuming than other methods, it can also provide the best representation of the wider population.

Whole population

Groups sampled according to some criteria

Example 5 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. How many primary students should be selected?

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

Apply the idea The proportion of total students selected is selected. Find

. So

of primary students

Evaluate

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of the primary students should be

Reﬂect and check In general, we can calculate the number required for a given subgroup in a stratiﬁed sample using:

For a stratiﬁed sample no individual should ﬁt into more than one subgroup, and no group of the total population should be excluded.

Idea summary Stratiﬁed sampling selects a proportional amount of people from the different strata in a population.

12.02 Practice questions Understanding

Adventurer 2–5, 8

Trailblazer 4–8

True or false: In a sample survey, information is obtained from the entire population.

What is the main difference between random and systematic sampling?

Ex 1

Explorer 1–5

Random sampling involves selecting individuals at regular intervals, while systematic sampling involves selecting individuals randomly.

Random sampling involves selecting individuals randomly, while systematic sampling involves selecting individuals at regular intervals.

There is no difference between random and systematic sampling.

When might stratiﬁed sampling be a better choice than random sampling? A

When the population is too large to survey everyone.

When the population of individuals are all fairly similar.

When the population is divided into subgroups that are expected to have different characteristics.

In which type of sampling does every individual in the population have an equal chance of being selected? A

Random sampling

Systematic sampling

Stratiﬁed sampling

Peter has chosen to collect information using a census instead of a sample. a

What are the advantages in doing a census? Select all that apply. A

It is cheaper to conduct.

It’s more accurate.

There will be no sampling bias.

What are the disadvantages in doing a census? Select all that apply. A

It’s less accurate.

It’s more expensive to conduct.

It takes more time.

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David is conducting a research study to investigate the eating habits of people in a particular city. He opts for stratiﬁed sampling. What is an advantage of using this method?

Sarah is conducting a survey to gather data about the shopping preferences of customers in a large retail store. She chooses to use systematic sampling. What is a disadvantage of using this method?

John is conducting a research study on the satisfaction levels of employees in a company. He decides to use random sampling. Which option best describes the process of random sampling and provides one advantage of using this method? A

Random sampling involves dividing the population into distinct subgroups based on speciﬁc characteristics and then randomly selecting participants from each subgroup. One advantage of using this method is that it ensures representation from each subgroup, allowing for more accurate estimates for each group.

Random sampling involves selecting participants from a population in a completely random manner, ensuring each individual has an equal chance of being chosen. One advantage of using this method is that it reduces the chances of bias and provides a representative sample of the population.

Random sampling involves selecting participants from a population at regular intervals, using a predetermined starting point. One advantage of using this method is that it simpliﬁes the sampling process, making it more eﬃcient.

Fluency Ex 2

11 Ex 3

9, 10, 12–17

9, 10, 12–15, 17, 18

9–11, 13–15, 17, 18

Determine whether the scenario is a census or sample: a

Justin has determined the age of 10% of houses from each suburb in Sydney.

Valentina tests every engine the factory produces.

Oprah checks every dog brought to her vet to assess the treatment of dogs in the city.

A random selection of some people at a mall.

A stocktake of all the goods in a store.

A crash test of all the new cars manufactured by a factory.

Asking all the teachers at your school whether they approve of a new class timetable.

An election to decide the premier of New South Wales.

A taste test of a large batch of cookies you have just baked.

A body scan of randomly selected passengers at Melbourne International Airport.

For each scenario, determine the type of sampling method used: a

Drawing out the winning ticket in a lottery

Choosing every 50th person on the class roll to take part in a survey

Choosing 5% of the students in each year group for Years 7–12

Four lucky people from a group of 215 each stand to win a tablet. Every contestant is given a different number between 1 and 215. Using a random number generator, these numbers were generated: 0.1521, 0.5342, 0.3524, 0.7951, 0.0032, 0.2118, 0.4002, 0.0215 Examine the first three digits of each generated number and determine the four that fall within the range of 1 to 215. These numbers will determine the four winning contestants.

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12 Ex 4

A factory produces 3432 phones every day. One in every how many phones needs to be tested, if the factory is to test a systematic sample of: a

13 Ex 5

How many males should be selected?

88 phones per day

How many females should be selected?

Users of a particular streaming service can be in one of four categories - Standard, Family, Premium or Business. The table shows the number of people in each category:

Team Standard Family Premium Business

Number of People 3500 1500 2000 3000

How many customers are there across all the categories?

If a stratiﬁed sample of 400 is to be taken from the group, what proportion of people will be chosen?

For the sample to be stratiﬁed, ﬁnd the number of customers that should be chosen in each category: i

In a group of 2160 students, 960 are male and 1200 are female. A stratiﬁed sample of 18 is to be selected from the group based on gender. a

13 phones per day

Standard

Family

iii

Premium

Business

At a music concert there is expected to be a crowd of 3240 people. The band is giving away shirts for some people that attend based on what order they enter the venue. a

If the band is to give away 18 shirts, ﬁnd the proportion of people that will get shirts.

If the band is to give away 60 shirts, ﬁnd the proportion of people that will get shirts.

The owner of a cinema wants to use stratiﬁed sampling in their survey of people who come to their cinema. Are these methods considered to be stratiﬁed sampling?

Interview 10% of the people who used the candy bar and 10% of people who didn’t.

Interview every person that sees a romantic movie.

Interview 10% of the people from each movie.

Interview every 10th person that purchases a ticket.

A television station wants to estimate the number of viewers it had for a new show. They know that the country’s population is 13 620 000. When they randomly selected 5000 people and asked them if they had watched the show, they found that 400 of them said ‘Yes’. Estimate the number of people who watched the show in the entire population.

Out of 2160 students in a school, 216 were chosen at random and asked their favourite colour out of red, blue and yellow with 99 choosing red, 63 blue and 54 yellow. a b

One in every how many students at the school was sampled? Estimate the total number of students in the whole school who prefer the colour: i

Red

Reasoning 19

ii 19, 22

Blue

iii

Yellow

19, 20, 22

20, 21, 23

Determine whether a census or sample would be better for these scenarios, and explain your choice. a

A school teacher wants to ﬁnd out what grade level his students are reading at.

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A hospital wants to ﬁnd out which heart rate machines are faulty.

A store manager wants to know the exact number of stolen items from the store.

A farmer wants to ﬁnd the average height of the wheat stalks in his ﬁeld.

For each sample, give an example of a population that the sample could have been chosen from: a

A sample containing 50 people who drive white cars.

A sample of 50 people drawn from a population. In this sample, the youngest is 18 years old, and the oldest is 64.

A sample containing the ﬁrst 50 people to enter a train station on a given day.

You want to survey a group of n students about their favourite sports, but you only have time to survey 20 students. How would you use systematic sampling to select the 20 students for the survey?

You want to conduct a survey of the reading habits of students at your school. How would you use stratiﬁed sampling to ensure a representative sample?

You are conducting a survey of customer satisfaction at a popular restaurant. The restaurant has a customer database of over 10 000 people. How would you use systematic sampling to select a sample of customers for the survey?

Problem-solving 24, 26, 27

24, 26–28

25, 26, 28

A company wants to conduct a survey on the salaries of its employees. There are 500 employees in the company, and they want to survey a sample of 50 employees. They decide to use systematic sampling, and they want to start with a random employee number. If the random number generator selects the number 10, which employee numbers should be surveyed?

A city council wants to conduct a survey on the opinions of its residents regarding a new policy. There are 50 000 residents in the city, and they want to survey a sample of 1000 residents. They decide to use simple random sampling. If they use a random number generator to select the ﬁrst resident, how should the remaining residents to be surveyed be selected?

A university wants to conduct a survey on the satisfaction of its students. There are 10 000 students in the university, and they want to survey a sample of 500 students. They decide to use stratiﬁed sampling, with three strata based on the student’s major: Arts, Science, and Business. The proportions of students in each stratum are 40%, 30% and 30% respectively. How many students should be sampled from each stratum?

Prior to an election, a news editor wanted to view the opinions of randomly selected people. Some two-digit numbers were randomly generated, as shown in the table, and starting with 37, the editor moved 3 to the right, 1 down (so that the next number chosen would be 57). The numbers chosen were the ages of the people she would survey. 37 85 74 37 23

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49 79 71 33 82

19 84 40 84 64

72 57 49 34 22

38 46 42 35 21

33 49 17 51 76

80 53 50 50 38

83 55 68 87 67

90 51 27 65 43

23 36 15 47 75

List all the random numbers the editor chose.

Find the age range of the people who will be surveyed.

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78 66 47 38 39

21 61 70 78 76

29 86 47 80 72

72 66 63 39 48

40 41 32 60 33

In the table is the gender of 50 birds at a protected breeding site: Birds 1 – 10 Birds 11 – 20 Birds 21 – 30 Birds 31 – 40 Birds 41 – 50

M, F, M, F, F, M, M, M, F, M M, F, M, M, F, M, F, M, F, F F, F, M, M, F, M, M, F, F, F M, F, M, F, F, F, M, M, M, F F, M, F, M, M, F, M, F, M, F

Calculate the proportion of females in the sample.

What proportion of the ﬁrst 5 birds were female?

What proportion of the ﬁrst 10 birds were female?

What proportion of the ﬁrst 20 birds were female?

In a systematic sample, every second bird is chosen, in the order of appearance, from the first 20 birds. How many males will be chosen in the sample?

In a systematic sample, every third bird is chosen, in the order of appearance, from the first 40 birds. How many females will be chosen in the sample?

The breeding site has a population of 440 birds. If the proportion of males and females in the sample is indicative of the whole breeding site, how many female birds are there in the breeding site?

12.03 Bias After this lesson, you will be able to… • define and understand sampling bias. • identify potential sources of bias in a sampling procedure. • evaluate and determine whether these situations are likely to lead to bias and explain why. • define observational and experimental bias. • identify and provide examples of potential sources of observational and experimental bias. • propose strategies to minimise the potential for bias in sampling and observational or experimental methods.

Sampling bias When we take a survey it is important that the results are representative of the population. This means that the results that we get for any question we ask of the sample would be the same as if we asked it on a census.

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If a sample is not representative, we call it biased. There are a number of potential sources of bias that we should avoid: • Poor sampling techniques: If the people being surveyed do not resemble the population, the survey is likely to be biased. • Too small of a sample: In general, the bigger the number of people being surveyed, the closer the results will be to a census. Speciﬁc sampling methods are more or less prone to causing bias as follows: • Random sampling may not adequately represent every sub-group within larger populations. • Systematic sampling is better than simple random sampling but it can still under or overrepresent a subgroup within the population. • Stratiﬁed sampling can also provide the best representation of the wider population.

Exploration A local council mayor needs to decide which intersections in the city need to be improved. The mayor decided to conduct a survey to determine public sentiment regarding current road conditions. The intersections causing the most frustration for drivers will be prioritised for improvements. 1.

Decide whether or not the results from the following surveys accurately represent the population.

If they do not accurately represent the population, explain why. • 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.

Example 1 Georgia wants to know how the people in Australia are going to vote in an upcoming referendum. She selects 50 random people from her city to interview. Determine whether the following are the reasons why this sampling might give poor results: A The sample doesn’t include anyone from New Zealand. B Random sampling always leads to a very biased sample. C The people of this city are not a good representation of Australia. D The sample size is too small for such a big population.

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Apply the idea For option A, the sample not including anyone from New Zealand is not relevant to the study on how people in Australia are going to vote in the referendum. In option B, random sampling does not always lead to a biased sample; it can be a fair and representative sampling method. Option C means that by selecting random people from her city Georgia’s sample may not be representative of the entire population of Australia. People in her city may have different opinions and voting behaviors compared to people in other cities or regions of the country. Therefore, the results obtained from this sampling method may not accurately reﬂect how the entire country is going to vote in the referendum. In option D, the sample size being small does not necessarily lead to poor results if it is properly representative and selected. The correct answer is option C.

Idea summary There are a number of potential sources of bias that we should avoid: • Poor sampling techniques: If the people being surveyed do not resemble the population, the survey is likely to be biased. • Too small of a sample: In general, the bigger the number of people being surveyed, the closer the results will be to a census.

Observational and experimental bias When we collect data through either observation or experimental methods, bias can be introduced in the following ways: • The personal beliefs, expectations or characteristics of the person conducting the observations of experiments impacts the results. • Environmental conditions (the conditions under which the experiment is carried out) are not consistent or inﬂuence results when conducting observations or experiments.

Example 2 Which of the following is an example of bias caused by inconsistent experimental conditions? A A researcher measures the growth of plants in different rooms with varying temperatures. B A researcher only selects participants for a study who share their personal beliefs. C A researcher records the number of cars passing by a street corner every day at the same time. D A researcher asks participants to rate their satisfaction with a product without providing any information about the product.

Create a strategy Identify the example that shows how the experiment’s results may not be fair or accurate because something was done in a way that could affect the outcome.

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Apply the idea For option A, the researcher is testing how plants grow in rooms with different temperatures. However, because the temperatures are different, it is not a fair test. The plants might grow better or worse simply because of the temperature differences, not because of any other reason. This makes the results biased. In option B, the researcher is only choosing people who have the same opinions as them for a study. While this is not fair, it is not an example of inconsistent experimental conditions. The researcher in option C is counting how many cars pass by a street corner every day at the same time. This is a consistent way of collecting data and does not involve inconsistent experimental conditions. The researcher in option D is asking people to say how satisﬁed they are with a product, but they don’t give any information about the product itself. While this might not give accurate results, it is not an example of inconsistent experimental conditions. So the correct answer is option A.

Example 3 An experienced researcher only listens to classical music and dislikes other genres. She is doing an experimental study on the effects of music on plant health. How could this introduce bias into the results? A The researcher might get confused and play classical music to all the plants in the study instead of having a control and a treatment group. B The researcher’s belief may inﬂuence their data collection, leading to subjective observations or biased interpretations of the plant health. C The personal beliefs of the researcher have no impact on the results of the study.

Create a strategy Choose the option that shows how the experiment’s results may not be fair or accurate due to the researcher’s love of classical music.

Apply the idea For option A, the researcher doesn’t have a control group. This is a poorly designed experimental study. This would not be due to their preference of classical music, but not knowing how to run an experimental study. This likely wouldn’t be the case for an experienced researcher. In option B, the researcher’s belief that classical music is superior can inﬂuence their data collection. They may subjectively observe or interpret the plant health outcomes based on their bias, potentially overlooking or downplaying any positive effects of other music genres on plant health. Option C states that the personal beliefs of the researcher have no impact on the results of the study. This statement is incorrect as personal beliefs can introduce bias into the study. So option B is the correct answer.

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Idea summary When conducting observations and experiments as part of a statistical investigation, the personal beliefs of researchers and inconsistent environmental conditions can cause the results to be biased.

12.03 Practice questions Understanding

Adventurer 1–7

Trailblazer 1–7

True or False: Stratiﬁed sampling ensures that each subgroup of a population is represented in the sample, therefore eliminating the possibility of sampling bias.

True or False: Systematic sampling can be biased if the members of the population are arranged in a speciﬁc pattern.

True or False: Bias caused by the personal beliefs and expectations of researchers can be completely eliminated from any study.

True or False: Bias caused by inconsistent experimental conditions can impact the validity and reliability of research ﬁndings.

True or False: In an experiment testing the effects of different temperatures on plant growth, maintaining consistent temperature conditions is important to minimise bias.

True or False: The personal beliefs and expectations of researchers can introduce bias in their observations or experiments.

True or False: Researchers’ personal beliefs and expectations can inﬂuence the interpretation of study results.

Fluency

Ex 1

Explorer 1–7

8–13, 16, 18

9–12, 14, 16–18, 20

9–12, 15–17, 19

Sean wants to know what 8th graders think of their English class, so he polls 70 random 8th graders. Is the sample chosen biased or fair?

Alex wants to understand the favourite types of music among teenagers in Australia. He decides to interview 60 students from his high school. Are these valid reasons why this sampling might give poor results?

Ex 2

The sample is only taken from one high school, which might not represent all teenagers in Australia.

The sample size is too large for a study of this nature.

The students interviewed may have similar tastes in music, leading to a biased result.

The sample doesn’t include anyone over the age of 20.

Which of the options is an example of bias caused by inconsistent experimental conditions? A

A researcher tests the memory retention of students but gives one group the test in a quiet room and the other group in a noisy environment. 12.03 Bias mathspace.co

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A researcher selects participants for a survey based only on their willingness to participate.

A researcher measures the weight of different fruits using the same scale every day.

A researcher asks participants to rank their preference for different foods without specifying portion sizes.

Which of the options is an example of bias that can be introduced by inconsistent sampling methods? A

A researcher randomly selects participants from a list of registered voters in one state to conduct a political survey.

A researcher selects participants from different age groups for a study on memory, but the number of participants in each age group is not proportional to their representation in the population.

A researcher conducts interviews with customers in a shopping mall during weekdays, excluding weekends.

A researcher distributes surveys to students in a particular college, but only collects responses from those who voluntarily choose to participate.

Justin wants to know how the students at his school might vote for the next school captain. He leaves three ﬂyers in each class to be completed by whichever students want to complete them. Are these valid reasons why this sampling might give poor results? a

Sampling within each class is self selected.

Stratiﬁed sampling always leads to a very biased sample.

The sample includes people from outside the population.

The students are under 18 so can’t legally vote.

Determine if the statement describes sampling bias for random sampling: A survey of school teachers was conducted, and the participants were selected by drawing names out of a hat.

Determine if the statement describes sampling bias for systematic sampling: A researcher selected every 5th patient on a hospital ward for a study on pain management.

Which of the options is NOT a way to minimise bias caused by personal beliefs and expectations of researchers?

694

Randomly assigning participants to groups

Having multiple researchers independently analyse the data

Only selecting participants who agree with the researcher’s hypothesis

In an experiment, a researcher wants to test the effect of a new drug on reducing pain. To do this they give some patients the drug and others a placebo (a placebo has no medical value). Which of the options describes the best way to minimise bias caused by the researcher's expectations? A

The researcher should only include participants who have a high pain tolerance.

The researcher should know which participants are receiving the drug and which are receiving a placebo.

The researcher should not know which participants are receiving the drug and which are receiving a placebo.

The researcher should only include participants who have a low pain tolerance.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Determine whether the scenario represents bias caused by inconsistent experimental conditions: a

A researcher measures the effects of different study techniques on test scores but allows participants to choose their own study techniques.

A researcher compares the effectiveness of two pain-relief medications but uses different dosage levels for each medication.

A researcher observes animal behaviour but only records behaviours they personally ﬁnd interesting.

Determine whether the scenario represents bias caused by the personal beliefs and expectations of researchers: a

A researcher investigates the impact of different teaching methods on student achievement but only selects participants who perform well academically.

A researcher studies the effects of a new drug on reducing anxiety levels but does not disclose their own beliefs about the drug’s effectiveness.

A researcher observes animal behaviour but only records behaviours they personally ﬁnd interesting.

In a study on the effects of a new teaching method on student performance, one group of students receives additional resources and support while the other group does not. However, the students in the ﬁrst group also have access to private tutoring outside of school hours. How might this inconsistent experimental condition introduce bias into the results?

Ex 3

The students in the ﬁrst group may experience greater academic improvement due to the additional resources and private tutoring, which can skew the results in favour of the new teaching method.

The students in the second group may feel disadvantaged and exert less effort in their studies, affecting their performance.

The inﬂuence of private tutoring on student performance cannot be accurately determined, making it challenging to isolate the true impact of the new teaching method.

The inconsistent experimental condition does not introduce bias in the study.

In a study examining the effects of different study techniques on memory retention, a researcher believes that one particular technique is superior and unintentionally gives more attention and encouragement to participants using that technique. How might this personal bias affect the outcomes of the study? A

Participants using the favoured study technique may perceive higher levels of support and motivation, which can positively inﬂuence their performance.

The researcher’s behaviour, such as unintentionally providing more attention and encouragement to participants using the favoured technique, can inadvertently affect their performance and outcomes.

The study may produce biased results that support the superiority of the favoured study technique due to the unintentional bias introduced by the researcher.

The personal bias of the researcher does not affect the outcomes of the study.

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Reasoning 21

22, 23, 26

22, 23, 25, 27

21, 23, 24, 27

A research team conducted a survey to understand the preferences of moviegoers for different film genres. The survey was administered online, and participants were asked to rank their top 3 favourite genres. After analysing the results, the team noticed a trend that the most popular genres were the ones listed first on the survey. Identify the issue that might have influenced the survey results and suggest how it could have been avoided.

Casper is conducting a survey at his school regarding opinions about the food offered by the school canteen. He allocates a random number to every student and selects 20% of each year group at random. a

What type of sample has Casper taken?

When collecting the surveys, Casper found that only 15% of those to whom surveys were sent had responded. How can this introduce bias into the results?

Explain why the sample is biased: a

Hannah is surveying customers at a shopping precinct. She wants to know which stores customers shop at the most. She walks around an entertainment store and chooses 30 customers from the store for the survey.

A TV station wants to know what the most popular type of music is, so they ask listeners to contact them and vote for their favourite type of music.

Consider these two conclusions: Pablo has concluded that about 36% of people would drive an electric car based on his survey results: Would drive an electric car Would not drive an electric car

58 104

Jenny has concluded that about 44% of people would drive an electric car based on her survey results: Would drive an electric car Would not drive an electric car

224 285

Which claim do you think is more valid? Explain your reasoning. 25

A researcher is studying the effect of a new teaching method on student performance. They ﬁnd that students who received the new teaching method performed better on a test than students who did not. However, the researcher also ﬁnds that the students who received the new teaching method had more experienced teachers. How might this affect the results of the study?

A researcher is conducting a study on the relationship between exercise and mental health. They ﬁnd that people who exercise more frequently report better mental health. However, the researcher also ﬁnds that people who exercise more frequently tend to have higher incomes. How might this affect the results of the study and what could the researcher do to minimise this potential bias?

A researcher is studying the effect of a new drug on reducing the symptoms of a disease. They ﬁnd the drug is effective in reducing symptoms. However, the researcher also ﬁnds that the participants who received the drug were more likely to report their symptoms to the researcher. How might this affect the results of the study and what could the researcher do to address this potential bias?

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Problem-solving 28, 29 28

28, 29, 31

28, 30, 32, 33

Lachlan asks 120 grade 12 students at his school how much time they spend on homework per night. 78 grade 12 students say they do more than 3 hours. At a student council meeting, Lachlan reports “65% of students at this school do too much homework.” Which statement explains why Lachlan’s report is misleading?

The survey does not represent the population of the school.

The question should have been multiple choice.

The question was biased.

The sample size was too small.

The conclusion does not align with the survey question.

A school principal wants to estimate the number of students who ride a bicycle to school. Would these samples avoid bias?

All students who are in the school band

Eight students in the hallway

Ten students from each grade, chosen at random

130 students during the lunch periods

Does each scenario use biased sampling methods? a

A community nurse wants to know the average height of all 7th graders that attend the school where she visits, so she measures the height of all the basketball players.

A city councilman asks members of the ice hockey team if they would prefer a new skateboard park or a new ice-skating rink to be built as the new building project.

The lifeguard of a water park wants to determine which water rides are enjoyed the most so he asks every tenth person who leaves the park to list their three favourite rides.

A researcher is conducting a study on the effect of a new diet on weight loss. They ﬁnd that participants who followed the new diet lost more weight than those who did not. However, the researcher also ﬁnds that the participants who followed the new diet were more likely to exercise regularly. Which of these methods can the researcher use to account for this potential bias in their study? A

Ignore the exercise habits of the participants.

Match participants in the two groups based on their exercise habits.

Exclude participants who exercise regularly from the study.

Only include participants who do not exercise regularly in the study.

A researcher is studying the effect of a new intervention on reducing stress levels. They ﬁnd that participants who received the intervention reported lower stress levels than those who did not. However, the researcher also ﬁnds that the participants who received the intervention were more likely to have social support. Which of these methods can the researcher use to account for this potential bias in their study? A

Ignore the social support levels of the participants.

Exclude participants with high social support from the study.

Include social support as a common variable in the statistical analysis.

Only include participants with low social support in the study.

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A researcher is conducting a study on the relationship between sleep and academic performance. They ﬁnd that students who sleep more hours per night have better academic performance. However, the researcher also ﬁnds that students who sleep more hours per night tend to have less demanding extracurricular activities. Which of these statements is true based on the study?

The improved academic performance is solely due to increased sleep.

The improved academic performance is solely due to less demanding extracurricular activities.

The researcher should control for extracurricular activity demands in their analysis to determine the true effect of sleep on academic performance.

The relationship between sleep and academic performance is not affected by extracurricular activity demands.

12.04 Compare samples to populations After this lesson, you will be able to… • perform statistical comparisons between sample and population data using measures of central tendency (mean, median, and mode) and variability (range). • analyse sample data, evaluating the size of the sample in relation to the population, and the potential implications of this for the accuracy of the investigation. • interpret statistical comparisons, determining which data set has higher or lower results on average and which data set has more consistent scores. • compute and interpret the mean proportion from multiple samples as a prediction of population proportions.

Statistical comparisons We can compare samples to populations using statistical measures like mean, median, mode and range. These measures can help us determine which data set had lower or higher results on average or which set had more consistent scores. We can also compare the size of the sample to the population to gain insight into the accuracy of the investigation.

Example 1 The runs scored by each player in a cricket match are displayed in the table. Team A Team B

698

70 28

0 33

73 12

25 60

31 51

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52 24

1 4

20 19

2 1

7 5

8 7

a Find the mean runs for Team A to one decimal place.

Create a strategy Use the formula:

Apply the idea Substitute the values Evaluate b Find the mean runs for Team B to one decimal place.

Apply the idea Substitute the values Evaluate c Which team scored more runs?

Create a strategy

Apply the idea

Since each team has the same number of players, choose the team with the higher mean.

Team A had a higher mean than Team B, so Team A scored more runs.

Idea summary We can use the mean, median, and mode to compare data sets to determine which set had higher results on average. We can use the range to compare data sets to determine which set had more consistent results. We can use all these measures to compare the effectiveness of the size and makeup of the sample.

Estimating population proportions When we have multiple sets of sample data for a population we can estimate the proportion of a population associated with the investigation by considering the mean proportion of all the samples.

Example 2 A school wants to predict how many students in the whole school wear glasses. The year coordinates for Years 7, 8 and 9 calculated that the proportions of students in each year who where glasses are: • Year 6 : 0.25 • Year 7 : 0.245 • Year 8 : 0.375 12.04 Compare samples to populations mathspace.co

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a What is the estimated proportion of students in the school who wear glasses?

Apply the idea 0.25 + 0.245 + 0.375 3 0.87 = 3 = 0.29

Estimated Estimatedproportion proportion =

Find the average of the 3 proportions Evaluate the sum Evaluate the division

b If there are 368 students in the school, what is the estimated number of students who wear glasses?

Create a strategy Apply the estimated proportion of 0.29 to the total number of students.

Apply the idea Estimated number of glass wearers = 0.29 × 368

Multiply number of students by the estimated proportion

= 106.72

Evaluate the multiplication

≈ 107

Round to the nearest whole number

Idea summary We can use the mean proportion of multiple samples to predict population proportion for a statistical investigation.

12.04 Practice questions Understanding

Explorer 1–4, 6

Adventurer 1–3, 5, 6

Trailblazer 1–3, 5, 7

What is the difference between a sample and a population?

Why do we take samples instead of surveying or collecting data from the entire population?

What is the difference between a sample mean and a sample proportion?

When comparing the ranges of two different samples, what does a larger range indicate about the data sets?

If the range of a sample is much smaller than the range of the population, what might this suggest?

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You have a sample of 100 students from a school. The proportion of students who own a smartphone in the sample is 0.75. How can you use this sample information to estimate the proportion of students who own a smartphone in the entire school population?

Suppose you have collected two samples, Sample A and Sample B, from two different populations. The mean of Sample A is 25, and the mean of Sample B is 30. What can you infer about the population means based on these sample means?

Fluency 8

8, 10, 12, 14, 16–19, 21

9, 12, 13, 15, 17, 20, 21

The runs scored by each player in a cricket match are displayed in the table.

Ex 1

8, 10–12, 14, 16–19

Team A Team B

80 38

10 23

63 22

35 50

41 61

42 34

11 14

30 29

Find the mean runs for Team A rounded to one decimal place.

Find the mean runs for Team B rounded to one decimal place.

Which team scored more runs? Explain your answer.

12 11

17 15

18 17

The hours of sleep per night for two people over a two week period are shown. • Person A: 8, 5, 10, 7, 9, 7, 6, 10, 6, 9, 7, 7, 10, 5 • Person B: 8, 8, 8, 7, 7.5, 8, 7.5, 7, 7, 7, 7.5, 7, 7, 7.5 a

Complete the table. Round values to one decimal place where necessary. Mean

Median

Mode

Range

Person A Person B

Which person is the least consistent in their sleep habits? Explain your answer.

Which person has the most sleep over the 14 nights? Explain your answer.

In a survey of students’ favourite colours, this data was collected: red, blue, green, red, yellow, blue, red, blue, purple, green, red, yellow, blue, red, blue, orange, red, green, yellow, purple, blue, green, red, yellow, orange, purple, pink, orange, pink, pink What is the mode favourite colour for students?

The number of pizzas sold by two pizza shops during a week is displayed in this table. Pizza Shop A Pizza Shop B

12 Ex 2

15 25

20 18

18 25

15 22

22 18

15 22

Find the mode of pizzas sold for Pizza Shop A.

Find the mode of pizzas sold for Pizza Shop B.

Which pizza shop sold more pizzas? Explain your answer.

20 25

A school wants to know how many students prefer chocolate ice cream. Three years were surveyed about their favourite ice cream flavor. • Year 6: 0.254 of the students prefer chocolate ice cream • Year 7: 0.351 of the students prefer chocolate ice cream • Year 8: 0.205 of the students prefer chocolate ice cream

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What is the estimated proportion of students who prefer chocolate ice cream overall?

If the school has a population of 950 students, how many are expected to prefer chocolate ice cream?

A teacher gave all students in each of his classes a speed drill, which required the students to complete the assignment as fast as they could. The teacher found that the average time and the median time (in seconds) it took students to complete the task were both 120 seconds. Two samples, A and B, each of size 10, are taken from the population and tabulated.

Calculate the mean for sample A.

Calculate the mean for sample B.

Which sample mean is more like the population?

Calculate the median for sample A to three decimal places.

Calculate the median for sample B to three decimal places.

Which sample median is more like the population?

Sample A 141.88 131.53 126.36 108.49 116.79 123.34 110.09 90.37 115.13 123.46

Sample B 121.04 116.66 108.68 130.62 106.74 134.58 108.83 111.65 131.4 133.1

A shop owner wanted to analyse the time customers spend in their shop. They collected data on the duration of customer visits and found that, on average, customers spend 15 minutes in the shop. Using computer software, the owner simulated 10 samples, each of size 10, and the results are tabulated. Sample

Sample mean a

2 19.8 10.2 16.5 10.4 13.8 18.2 14.8 18.8 19 14.8 15.63

3 11.4 14.1 12.1 11.4 15.4 13.7 11 10.5 13.6 12.4 ⬚

4 10.9 11.6 18.9 14.2 18.5 19.6 19.3 15.2 12.8 17.6 15.86

Calculate the mean of each of the samples: i

702

1 19.1 13.8 19.3 13.2 12.7 15.5 17.6 18.9 19.8 17.4 16.73

Sample 3

Sample 5

iii

5 18 18.1 14.9 12.5 12 11 19.8 15.9 10.1 15.3 ⬚

6 10.3 11.2 15.1 19 11.4 11.9 18.5 14.7 16.6 11 13.97

Sample 8

7 16.7 15.1 18.3 13 18.9 19.6 13.5 19.5 19.4 19.7 17.37

8 12.4 13.3 18.4 19 12.7 18.2 17 16.2 16.6 14.1 ⬚ iv

9 11.9 14.5 11.1 12.9 17.1 13.2 11.5 11.9 13.9 18.2 13.62

10 17.5 15 12.8 13 19.3 10.3 17.3 19.3 18.9 11.7 ⬚

Sample 10

Calculate the mean of the sample means.

Calculate the difference between the mean of the sample means from part (b) and the population mean.

Mathspace Victorian Curriculum – Year 8 mathspace.co

Every day at practice for two months, a basketball player shoots 15 free throws. The player consistently writes down how many shots they make, and ﬁnd the mean is 10.5. Ten random samples, each of size 10, were taken from the distribution and the results are tabulated. Sample

Sample Mean a

3 11 6 9 12 11 8 13 11 11 10 10.2

4 12 9 10 10 11 ⬚ 14 10 9 11 10.9

5 10 12 11 11 12 11 11 12 9 11 11

6 11 13 12 6 11 6 11 10 8 11 9.9

Sample 1

Sample 4

iii

Sample 7

7 9 9 13 11 11 12 9 9 8 11 ⬚

8 11 9 11 13 9 11 14 9 10 12 10.9

9 14 10 11 13 8 11 10 10 9 11 10.7

10 ⬚ 11 7 7 7 10 10 9 13 14 9.9

Sample 10

Calculate the mean of the sample means. Give your answer to two decimal places. Find the median of the sample: i

2 11 9 12 10 13 11 11 11 13 11 11.2

Find the missing values from the listed samples: i

1 13 10 10 10 10 11 10 11 10 11 ⬚

Sample 1

Sample 3

iii

Sample 6

Sample 9

Comparing the population and sample means, which of these statements is true? A

he samples exhibit no bias and seem to be indicative of the whole (population) data T set.

he samples exhibit bias and seem to not be indicative of the whole (population) T data set.

A census for a particular country showed that 28% of people used public transport at some point during a regular week. At about the same time as the census, a sample of 5500 people in a region of the country showed that 1573 of those people used public transport at some point during a regular week. a

Determine the proportion of residents who use public transport at least once a week. Express your answer as a decimal.

Determine the sample proportion of the residents who use public transport at least once a week. Express your answer as a decimal, rounded to two decimal places.

Comparing the population and sample proportions, which of these statements is true? A

The sample exhibits no bias and seems to be indicative of the larger population.

The sample exhibits bias and does not seem to adequately represent the larger population.

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All mathematics students in a particular year are given a timed test. The goal is to answer as many questions as possible before the time runs out. Each question is worth 4 points. The teachers collected the data and found the range of scores was between 36 points and 84 points, and the mean and median score was 60. a

A sample of size 25 is taken, and a graph and table of results created. Calculate the mean and median of this sample, rounded to two decimal places if necessary. Value Frequency

6 4 2

36 44 52 60 68 76 84

Sample B 35 30 25 20 15 10 5

36 44 52 60 68 76 84 1 3 23 48 17 7 1

By considering the three samples and the original distribution, select all statements that apply. A

36 44 52 60 68 76 84 1 6 34 35 20 4 0

A sample of size 100 is taken from this distribution and the graph and table of results created. Calculate the mean and median of this sample, rounded to two decimal places if necessary. Value Frequency

36 44 52 60 68 76 84 1 1 7 9 4 3 0

Sample A

The larger the sample, the more likely the mean and median are close to the population distribution. If another sample of size 25 was taken, the mean and median would be the same as the above sample of size 25. The larger the sample, the more likely the graph is close to the graph of the population distribution. A larger sample must have a closer mean and median to the population than a smaller sample.

36 44 52 60 68 76 84

Sample C 45 40 35 30 25 20 15 10 5 36 44 52 60 68 76 84

A census for a particular country showed that 94% of people used public transport at some point during a regular week. At about the same time as the census, a sample of 2420 people in a region of the country showed that 1001 of those people used public transport at some point during a regular week.

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Determine the population proportion of the residents who use public transport at least once a week. Express your answer as a decimal.

Determine the sample proportion of the residents who use public transport at least once a week. Express your answer a decimal, rounded to two decimal places.

Comparing the population and sample proportions, which of these statements is true? A

The sample exhibits no bias and seems to be indicative of the larger population.

The sample exhibits bias and does not seem to adequately represent the larger population.

The spinner shown was spun 160 times and B was the result 56 times. a

Determine the population proportion of landing on B. Express your answer as a decimal.

Determine the sample proportion of landing on B. Express your answer as a decimal, rounded to two decimal places.

How could we use samples in a different way to more accurately calculate the population proportion? Select all correct answers.

We can take more samples of size 160 and ﬁnd the mean of the sample proportions.

There is nothing that can be done, since this is a game of chance.

We can decrease the size of the sample.

We can increase the size of the sample.

Sample A

Employees from a large company were asked how many days of the week they exercised. The answers ranged from 1 to 7, and 4 was both the average and the median number of days of exercise. a

A sample of size 25 is taken from this distribution and the graph and table of results created. Calculate the mean and median of this sample rounded to two decimal places if necessary. Value Frequency

1 2

2 4

3 7

4 3

5 4

6 3

1 16

2 7

3 13

4 21

5 18

6 13

6 4 2

7 2

A sample of size 100 is taken from this distribution and the graph and table of results created. Calculate the mean and median of this sample rounded to two decimal places if necessary. Value Frequency

7 12

Sample B

20 16 12 8 4 1

12.04 Compare samples to populations mathspace.co

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Another sample of size 100 is taken from this distribution and the graph and table of results are created. Calculate the mean and median of this sample rounded to two decimal places if necessary. Value Frequency

1 12

2 16

3 16

4 13

5 17

6 12

7 14

Sample C

20 16 12

By considering the three samples and the original distribution, select all statements that are true:

If another sample of size 25 was taken, the mean and median would be the same as the above sample of size 25.

The larger the sample, the more likely the graph is close to the graph of the population distribution.

The larger the sample, the more likely the mean and median are close to the population distribution.

A larger sample must have a closer mean and median to the population than a smaller sample.

A company produces two types of cereal: Cereal A and Cereal B. The company wants to determine the proportion of customers who prefer Cereal A over Cereal B. To do this, they survey customers at 160 different stores that stock the cereal. At each store, they randomly select 20 customers and ask them which cereal they prefer. The proportion of customers who prefer Cereal A is then recorded for each store. This sampling distribution represents these proportions for the 160 retail stores. The mean of the distribution is 0.65, which suggests that on average, 65% of customers at each store prefer Cereal A over Cereal B. Distribution of sample proportions 35

Frequency

30 25 20 15 10 5 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Proportion

706

What is the range of sample proportions?

How many samples showed that 90% of customers prefer Cereal A over Cereal B?

What fraction of the samples showed that 65% of customers prefer Cereal A over Cereal B?

Mathspace Victorian Curriculum – Year 8 mathspace.co

Reasoning

22, 25, 26

23, 25, 28

24, 25, 27, 28

In a study of the number of children per family, the population mode is found to be 2. A random sample of 100 families is taken, and the mode of the sample is found to be 2. What can you conclude from this information?

You want to compare the ages of customers at two different stores. Store A had a mode of 30 years old and Store B had a mode of 40 years old. What conclusion can you make about the age distribution of customers at each store?

In a survey conducted among a group of people, the mode of the respondents’ ages is 30. However, upon further investigation, it is revealed that the majority of the people in the survey are in their 40s and 50s, with only a few individuals in their 30s. How can you explain the mode being 30 despite the majority of respondents being in their 40s and 50s?

Classes 7M and 9C were surveyed about whether they should receive more maths homework. The results are displayed in the given table:

7M 9C

Agreed 6 8

Disagreed 24 24

What conclusion can we make from the data?

Are these results likely to be representative of the student population? Explain your answer.

A survey of 115 randomly selected people in Busan found that 6 of them were aged over 55. A second survey of 2183 randomly selected people in Busan found that 475 of them were aged over 55. a

Considering the ﬁrst survey, what is the sample proportion of people in Busan over the age of 55?

Considering the second survey, what is the sample proportion of people in Busan over the age of 55?

Which sample proportion is likely to be the better estimate of the population proportion? Explain your answer.

Phinosis Pty. Ltd. monitors the population of chameleons in the local area twice every month. Last year, a number of the chameleons were tagged with brown tags. a

The ﬁrst survey in March involves 551 chameleons and 110 of these have brown tags. In this survey, what is the sample proportion of chameleons with brown tags?

The second survey in March involves 203 chameleons and 15 of these have brown tags. In this survey, what is the sample proportion of chameleons with brown tags?

Assume that there is no change in the population of chameleons between the two surveys. Which sample proportion is likely to be the better estimate of the population proportion? Explain your answer.

It is a common belief that the ability to roll your tongue is a genetically inherited trait. Isaiah would like to determine what proportion of students at his university have the ability to roll their tongues. In a random sample of 100 students, Isaiah determines that 70 can roll their tongue.

12.04 Compare samples to populations mathspace.co

707

The dot plot shows the results of a simulation used to predict the proportion of people who can roll their tongue. 500 random samples of 100 students each from a population where 70% can roll their tongue were simulated. a

0.55

0.6

0.7

0.75

0.8

0.85

Simulated Sample Proportion Who Can Roll Tongue Based on the dot plot, what is the range of proportions in the simulated samples?

How would you expect the spread of the proportions to change if the sample size were increased? Explain your reasoning.

Based on the simulation, what is the approximate mean proportion of people who can roll their tongue in the population?

Isaiah’s Biology professor claims that 73% of students at the university can roll their tongue. Is her claim plausible? Explain your reasoning.

Problem-solving 29 29

0.65

29, 31

30, 32, 33

Over the course of one day, ten samples of customers at a café were surveyed. Each sample involved 25 customers who had ordered at least one item from the menu. a

The number of customers in each sample who ordered a tea is shown in the table. Complete the table by calculating the sample proportion for each sample. Sample Ordered Tea Proportion

1 8

2 18

3 14

4 5

5 12

6 4

7 4

8 1

9 12

10 24

Calculate the mean of the sample proportions, writing your answer as a decimal.

Estimate the population proportion of customers who ordered a tea, writing your answer as a decimal.

At an awards ceremony for Sun Valley Grammar School, Neville wants to ﬁnd out how many students studied Geography. To do this, he surveys several samples of 40 students. a

The number of students in each sample who studied Geography is shown in the table. Complete the table by calculating the sample proportion for each sample. Sample Studies History Proportion

708

1 34

2 5

3 20

4 18

5 24

6 9

7 6

8 7

9 21

10 37

Calculate the mean of the sample proportions, leaving your answer as a decimal.

Estimate the population proportion of students who studied Geography, leaving your answer as a decimal.

Mathspace Victorian Curriculum – Year 8 mathspace.co

The number of sales made by two sales teams during a week is shown in the table. Sales Team A Sales Team B

7 9

9 11

5 7

7 9

5 7

9 11

Find the mean, median, mode and range of test scores for Team A.

Find the mean, median, mode and range of test scores for Team B.

Compare the performance of Sales Team A and Sales Team B based on their mean, median, mode and range.

A population is sampled several times to investigate the proportion that purchase a new pot plant each year. Each sample is equal in size and the values of the sample proportions are recorded in the graph. a

How many samples were taken?

Estimate the population proportion of customers who purchased a new pot plant. Give your answer to three decimal places.

10 9 8 7 6 5 4 3 2 1 0 0.6 0.625 0.65 0.675 0.7 0.725 0.75 0.775 0.8 Sample Proportion

A grocery store owner wants to sell apples in bags of 10 and wants to ﬁnd the mean weight of an apple to put on the label of the bags. a

A random sample of 10 apples is taken and their weights, in grams, were as follows: 79.3, 81.5, 81.1, 86.6, 82.7, 81.2, 82.7, 79.2, 84.1, 86.4 Calculate the sample mean.

A sampling distribution is created by taking 100 samples of size 10. A graph of this distribution is shown. Use the graph to estimate the mean of the distribution. Distribution of sample means 16 14 Frequency

5 11

Number of Samples

12 10 8 6 4 2 70

Mean weight (g)

12.04 Compare samples to populations mathspace.co

709

INVESTIGATION

Plan and conduct statistical investigations Investigate the impact of using primary or secondary data and ethical and fair methods.

Objectives To understand the concept of food miles and their environmental implications To analyse primary and secondary data To conduct statistical investigations using random and non-random sampling techniques To infer about the population based on samples, considering ethical fairness and uncertainty

Materials •

At least ten labels from pantry items

•

Spreadsheet program

Plan and conduct statistical investigations What is it?

Food miles

The distance food travels from the time of its production until it reaches the consumer.

Why does it matter? The more miles it travels, the greater its environmental impact due to carbon emissions from transportation. This makes it essential for us to study and understand this in order to make more environmentally friendly choices in our daily lives.

Primary and secondary data sources

710

Primary data source: the original source of data that has been collected firsthand. Data that you gather yourself, directly from observation, surveys, experiments, or interviews Secondary data source: data that has already been collected and analysed by someone else. This could be in the form of books, articles, reports, or statistics that you find online or in the library

Mathspace Victorian Curriculum – Year 8 mathspace.co

Collect primary data: survey the food items in your home, selecting 10 items and recording the product name, origin, and estimated distance it travelled to reach you on a spreadsheet. Collect secondary data: use data sources approved by your teacher to find information about average food miles (or kilometres) for various types of food. Add this to your spreadsheet. Use the address http://www.foodmiles.com to collect some secondary data. Use your spreadsheet to calculate the mean, median, mode and range of your food distances from your primary source and secondary source. Compare these calculations and draw conclusions from your analysis, making sure to identify any patterns or anomalies. Prepare a short report summarising your findings.

Investigate 1.

How did you ensure fairness when selecting food items for your primary data? Did you use a random selection method or did you choose them yourself? Explain why you chose this method.

Which food item in your sample travelled the farthest and which travelled the least? Explain any factors that might have influenced this.

Considering your sample data, what inferences can you make about the overall food miles associated with the food items in your home? How confident are you in these inferences?

Discussion 1.

How might the concept of food miles influence our choices as consumers? What could be some of the challenges in reducing food miles?

Give an example of a different type of investigation where primary and secondary data sources are relevant.

Investigation: Plan and conduct statistical investigations mathspace.co

711

12 Chapter review 1

Describe the difference between a census and a sample.

Determine whether the following is a census or sample: a

Rochelle has asked everyone in her oﬃce what snacks should be provided in the oﬃce.

Luigi asks a few of his friends how they did in the test to see if he is above average in his class.

Mario ﬁnds the height of the entire class to try to ﬁnd the average height of 13 year old students in Australia.

A crash test of new cars just manufactured by a factory.

Asking all the teachers at your school whether they approve of a new class timetable.

An election to decide the premier of Queensland.

The mean of a set of 28 scores is 81.6. If a score of 73.5 is added to the set, ﬁnd the new mean. Round your answer to two decimal places.

For each set of scores:

Sort the scores in ascending order.

Find the median.

23, 19, 23, 25, 30, 45

85.2, 90.5, 70.3, 95.8, 80.6, 88.9

−10.6, 20.9, −30.7, 45.2, −15.8, 25.3

In a school, the number of books read over summer break by nine students are: 5, 8, 6, 7, 7, 8, 5, 7, 20

Find the mean number of books read.

Find the median number of books read.

Recalculate the mean number of books read excluding the outlier.

Recalculate the median number of books read excluding the outlier.

Which measure of centre best identiﬁes the typical number of books read? Explain your answer.

When an outlier is removed from a data set, describe the effect on the following, and explain your reasoning: a

712

Mode

Range

Mean

The stem-and-leaf plot shows the batting scores of two cricket teams, England and India: a

What is the highest score from England?

What is the highest score from India?

Find the mean score of England.

Find the mean score of India.

Calculate the combined mean of the two teams.

England

India 1

6 6 5 5 5 5

0 2 9

7 3

2 5

Explain why the following data set does not have a mode: 6, 11, 3, 8, 2, 15, 9, 7 Mathspace Victorian Curriculum – Year 8 mathspace.co

Median

2 4 7

Key: 0|2|4 = 20 and 24

Consider the dot plot shown: a

Describe the shape of the distribution.

Are there any outliers? If so, state the value. 1

Consider the data shown in the histogram: a

Are there any outliers? If so, what is the value?

What is the mode?

5 6 Score

45 40 35

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

In a study, a group of people were shown 30 names, and after 1 minute they were asked to recite as many names by memory as possible. The results are presented in the dot plot: a

What does each dot represent?

How many people took part in the study?

What is the largest number of names someone remembered?

What was the smallest number of names someone remembered?

What is the range?

11 12 13 14 15 Names Remembered

Four lucky people from a group of 259 each stand to win an iPad. Every contestant is given a different number between 1 and 259, and the winners are selected by generating a random number uniformly between 0 and 1. To ensure there is an equal chance of each contestant winning, the number is multiplied by 259 and then rounded up. In this case, the numbers generated were: 0.168, 0.567, 0.327, 0.862 Convert the generated numbers into the numbers of the four winning contestants.

The salaries of teachers and administrators in a school district are: • Administrators: $80 000, $80 000, $75 000, $80 000, $75 000, $70 000, $80 000 • Teachers: $70 000, $70 000, $75 000, $70 000, $70 000, $80 000, $75 000 a

Complete the following table: Mean

Median

Mode

Range

Administrators Teachers b

Who seems to be getting the higher salary, the teachers or the administrators? Explain your answer.

Chapter 12 review mathspace.co

713

Users of a particular streaming service can be in one of four categories - Standard, Family, Premium or Business. The table shows the number of people in each category: How many customers are there across all the categories?

If a stratiﬁed sample of 600 is to be taken from the group, state the proportion of people who will be chosen.

For the sample to be stratiﬁed, give the number of customers that should be chosen in each category: Standard

Family

iii

Premium

Business

A bakery produces 2250 bread loaves every day. How many bread loaves are tested daily if the bakery tests a systematic sample of: a

Number of People 3000 1000 2500 3500

i 15

Team Standard Family Premium Business

Every 15th loaf

Every 90th loaf

The owner of a theme park wants to use stratiﬁed sampling in their survey of visitors who come to their park. Determine whether these methods would be considered as stratiﬁed sampling: a

Interview 15% of the people who rode the roller coaster and 15% of people who didn’t.

Interview every person that visits the haunted house.

Interview 15% of the people from each ride.

Interview every 20th person that enters the park.

A school wants to conduct a survey on the satisfaction of its students with the school’s facilities. There are 800 students in the school, and they want to survey a sample of 8 students. They decide to use systematic sampling, and they want to start with a random student. If the random number generator selects the number 15, which students should be surveyed?

Lucy wants to know how the people in Australia are going to vote in an upcoming referendum. She selects 50 random people from her city to interview. Determine whether the following are the reasons why this sampling might give poor results:

714

The people of this suburb are not a good representation of Australia.

The sample doesn’t include anyone from New Zealand.

Random sampling always leads to a very bias sample.

The sample size is too small for such a big population.

A city mayor wants to estimate the number of residents who use public transportation daily. State whether the following samples would avoid bias: a

All residents who work in the city council.

Fifteen residents at the public park.

Fifty residents from each neighborhood, chosen at random.

200 residents during the morning rush hours.

Mathspace Victorian Curriculum – Year 8 mathspace.co

A recent survey for a particular country showed that 35% of people exercised at least three times a week. Concurrently with the survey, a sample of 4000 people in a region of the country showed that 1460 of those people exercised at least three times a week.

Determine the population proportion of the residents who exercise at least three times a week. Express your answer as a decimal.

Determine the sample proportion of the residents who exercise at least three times a week. Express your answer rounded to two decimal places.

Comparing the population and sample proportions, which of the following statements is true? A

The sample exhibits no bias and seems to be indicative of the larger population.

he sample exhibits bias and does not seem to adequately represent the larger T population.

A coach timed all players in her basketball team on a 100 metre sprint, and found that the average time and the median time (in seconds) it took for players to complete the run were both 15 seconds. Two samples, C and D, each of size 8, are taken from the team and tabulated.

Calculate the mean for sample C.

Calculate the mean for sample D.

Which sample mean is more like the population?

Calculate the median for sample C to three decimal places.

Calculate the median for sample D.

Which sample median is more like the population?

Sample C 14.88 15.53 15.36 14.49 15.79 15.34 14.09 15.13

Sample D 14.54 14.66 15.68 15.62 14.74 14.58 15.65 14.4

An environmental study sampled 50 trees from a forest to measure their height. If the researchers were to sample another 50 trees, do you expect the measures of centre and spread to be exactly the same, similar, or very different? Explain your answer.

Chapter 12 review mathspace.co

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Answers

Problem-solving 14 −2, 8

1.01 Add and subtract integers

15 Floor 8

Understanding 1 a Right

16 24 °C b Left

c Left

d Right

2 Subtracting a negative number is equivalent to adding a positive number. 3 a −6

b −3 + 6 − 2 + 7 − 14 = −6

Fluency

17 −22 °C 18 102 points 19 285 m 20 $1850 proﬁt 21 a i 141 L

ii 167 L

4 a 10

b 4

c −3

d 0

e 12

f 7

g 3

h 3

5 a 22

b 7

c −11

d 0

c One possible answer is STEP

e −5

f 62

g −10

h 0

d One possible answer is PART

i −4

j 20

k 0

l 14

6 a −3

b −9

c 0

d −12

e 2

f −1

g −2

h −24

Understanding

7 a 3

b 0

c −85

d −23

1 a True

b False

e 8

f −40

g −3

h −22

e True

f True

i −23

j 3

k 16

l 24

8 a 7−5=2

b 4+1=5

c −10 + 10 = 0

d 15 + 12 − 18 = 9

e 25 + 26 = 51

f 21 − 29 − 9 = −17

g −12 − 5 + 10 = −7

h −13 + 43 − 60 = −30

i −7 − 8 = −15

j −3 + 9 = 6

k 13 − 11 = 2

b 60 L below average 22 a −1

b −9

1.02 Multiply and divide integers

c False

2 a Negative

b Negative

c Positive

d Positive

3 a Negative

b Positive

c Negative

d Positive

d False

4 a −24

b −24

c 24

l −8

5 a −30

b 30

c 30

m −20 + 35 − 19 = −4

n −3 − 9 − 4 = −16

o −7 + 4 + 8 = 5

p 35 − 72 + 45 = 8

6 a −6

b −6

c 6

7 a −33

b −33

c 33

c 0

d −64

Reasoning

d −30

Fluency

9 He has removed the negative sign from the ﬁrst 3. It should be −3 − (−3) = −3 + 3 = 0.

8 a −36

b 120

10 No, if the sum of two integers is negative, at least one of the integers must be negative.

e 44

f 16

g 144

h −45

i −9

j −44

k 16

l −140

m −64

n −144

o 512

p −125

q 36

r −144

s −80

t −720

11 The smallest value of a is 5. 12 The largest value of c is −3. 13 a Positive

b Positive

c Negative

d Positive

716

Mathspace Victorian Curriculum – Year 8 mathspace.co

9 a −11 × (−5) = 55

b −6 × 11 = −66

c −7 × 8 = −56

d 7 × (−12) = −84

10 a r = 6

b n = 12

11 a −7

b 4

c q = −4

d t = −6

b −3

c 1

d −2

Problem-solving

e −4

f 0

g −1

h −10

i −6

j −5

k −3

22 −7, 5

l 5

n 4

o −2

p −6

12 a 1

13 a 54 ÷ (−6) = −9

b −35 ÷ 5 = −7

c −55 ÷ 5 = −11

d (−72) ÷ (−8) = 9

14 a n = 9

b p = −88 c t = −8

15 a −27

d k = −48

23 a Quadrants 1 and 3

b Quadrants 2 and 4

c Quadrants 2 and 4

d Quadrants 3 and 4

24 a 12

b −5

c 11

25 a $13 200

b $2400

c $10 800

26 x = 9.5

b 20

16 a 6

b 105

c −16

d −72

e 8

f −30

g 3

h −7

1.03 Four operations with integers Understanding

Reasoning

1 a False

b True

d False

e True

17 When we square a number the result is always positive. As −1 is negative there is no number x which we can square and get −1.

2 A

18 Yes, if one or both of the integers are zero, their product will be zero. This is because any number multiplied by zero is zero.

4 A

19 a 64

b 64

c In both cases, the product of the three numbers is the same because it doesn’t matter what order you do multiplication in. 20 The answer to both expressions is the same because of the associative property of multiplication, which states that the grouping of factors does not affect the product. In both cases, the product of the three numbers is the same: 9, -6, and 3. The only difference is the grouping in which they are multiplied, but due to the associative property, the result will be the same. 21 a Product −1 × −1 −1 × −1 × −1 −1 × −1 × −1 × −1 −1 × −1 × −1 × −1 × −1 −1 × −1 × −1 × −1 × −1 × −1

Number of negative Result terms 2 3 4 5 6

1 –1 1 −1 1

b If there is an even number of negatives then the answer is positive. If there is an odd number of negatives then the answer is negative.

c True

3 C

5 Multiplication 2 × 3, Subtraction 5 − 6, Addition −1 + 1 Fluency 6 a 2

b −5

c 16

d −7

e 5

f −8

g −11

h 0

7 a −49

b −12

c 21

d 8

e 43

f −66

g −34

h 80

i −48

j 31

8 2 + (−4) × 5 = −18 9 a −3

b 6

c −2

d −5

e −12

f −7

g 1

h 9

i −2

j −3 c −6

d 40

c 82

d 160

10 a 28

b −2

e 9

f −6

11 a 49

b 32

Reasoning 12 The order of operations have not been applied correctly. The division should have been performed before the subtraction. The correct answer is −14. 13 Yes. The distributive property of multiplication over addition is applied in both expressions.

Answers mathspace.co

717

14 The mistake occurred in step 1, where addition was performed before multiplication. The correct working step should be:

7 a −1

8 a

15 a The correct answer is 29.

b Two possible ways to change the expression are: • (7 − 4) × (3 − 9) ÷ 2 + 5 × 2 for the answer 1 • 7 − 4 × (3 − 9 ÷ 2 + 5) × 2 for the answer −21

9 a

i −35

10 a

20 $10

11 a

21 $107

1 (−3) + 10 × 2 − (−5) = (−3) + 20 − (−5) 2

= 17 − (−5)

= 17 + 5

= 22

Problem-solving 16 28 ripe apricots 17 $15 18 a 3 °C

b 9 °C

19 $116

22 5400 m south

Reasoning

1.04 Negative fractions

12 a Avery’s method is more eﬃcient because and

Understanding

have related denominators.

b Yes, both methods lead to the same result.

1 True

Wade’s method:

2 B 3 D 4 A, B 5 a Multiply

and

b Divide

c Divide

d Multiply

and

e Add

and

f Multiply

and

g Subtract

from

h Divide

b 0

c −1

Avery’s method:

Fluency 6 a

718

Mathspace Victorian Curriculum – Year 8 mathspace.co

13 In his second step, Weston switched the order

9 a −13.69

b 40.96

of the fractions without keeping their signs.

c 50.653

d −262.144

was positive and was negative, so when he switched them, the expression should have

10 a −9.5

been

. This would give him the correct

Reasoning

answer of

11 There are four decimal points in total in the question, so there will be four decimal points in the answer. Therefore, the answer will be −0.0900 or −0.09.

14 We subtract the sum of

from 1. It means

of his pay left.

Paul has 15 We divide

and

by 20. This means that each staff of the cake.

will receive Problem-solving

17 °C

1.05 Negative decimals Understanding 1 a True

b True

d False

e True

c False

d −12.1

12 Answer may look like: −3.066 42 because there is a total of 5 decimal places in the question, so there will be 5 decimal places in the answer. When you multiply a negative number by a positive number, the result is negative.

14 First, we evaluate the left-hand side of the expression: −3.83 − (−11.63) = 7.8

2 C

This simpliﬁes the equation to 7.8 = d ÷ 4.95

3 A 4 a Yes

c −5.0

13 Mia started by adding the ﬁrst two terms: 13.8 + (−14.52) = −0.72 Then, she subtracted the third term: −0.72 − 22.03 = −22.75 This is the same answer Mia got, so Mia is correct. James started by subtracting the second and third terms: (−14.52) − 22.03 = −36.55. This is not the answer James got, so James is incorrect.

b −0.5

b No

c No

d Yes

To solve for d, we need to multiply both sides by 4.95. So, we have

Fluency

d = 7.8 × 4.95 = 38.61

5 a −9.3

b 17.3

c −11.5

d 7.7

e 7.4

f 6.86

g −7.1

h −4.64

Problem-solving

i 0.8

j 11.43

k −7.56

l −9.25

15 a 6.6

b A b Dudinka to Alaska

6 a 3.8

b −6.0

c −0.4

d 0.9

16 a −32.4 °C

e −9.8

f −1.2

g −16.4

h −2.1

c 20.6 °C

i −0.7

j −2.3

k −17.1

l 2.3

17 a $24.74

7 a −19.2

b −59.28

c −0.54

d −49.92

e 38.22

f 57.6

g −41.71

h −15.92

i −65.34

j −30.34

k 5.54

l 35.77

8 a −4.6

b −9.8

c −0.5

d 0.3

e −0.9

f −52

g 92

h 4

b −$74.22

18 $1987.00 19 2.36 °C 20 a −5.15 m

b 1.2 m/s

21 $5.31

Answers mathspace.co

719

1.06 Rational and irrational numbers Understanding

f i Irrational × Irrational ii 3

1 It can be written as a fraction in the form where a and b are integers and b is not equal to zero. 2 Either.

iii Rational

g i Irrational × Irrational ii 3.16

iii Irrational

h i Irrational × Irrational ii 9

iii Rational

i i Rational − Rational

3 π

ii 1.75

iii Rational

j i Rational − Irrational ii −0.84

Fluency 5 a Rational

b Rational

c Irrational

d Rational

e Rational

f Irrational

g Rational

h Irrational

i Irrational

j Rational

k Rational

l Irrational

m Rational

n Irrational

o Irrational

p Rational

q Rational

r Rational

s Rational

t Rational

u Irrational

v Rational

w Rational

x Rational

iii Irrational

k i Irrational − Irrational ii 2.83

iii Irrational

l i Rational ÷ Rational ii 3

iii Rational

m i Rational ÷ Irrational ii 0.31

iii Irrational

n i Irrational ÷ Irrational ii 5

iii Rational

ii Approximation

Reasoning

b i 4

ii Exact value

10 0.345 can be written as a fraction of two

c i 4.25

ii Approximation

d i 3

ii Exact value

e i 7.75

ii Approximation

6 a i 1.73

f i 7.94

ii Approximation

g i 3.50

ii Approximation

h i 4.64

ii Approximation

integers,

, therefore it is rational.

11 No. It doesn’t have a repeating pattern. It cannot be expressed as either a repeating decimal or a terminating decimal, so the number cannot be rational.

b 12

c 18

d 11

f 3.46

g 1.73

h 2.24

12 Yes, all integers are rational numbers because they can be expressed as a fraction with a denominator of 1.

i 3.32

j 4.12

k 3.97

l 0.16

13 Because it cannot be expressed as a simple

n 24

o 108

p 2.6

7 a 4 e 2.65

8 a

b 9 a i Rational + Rational ii 0.92

iii Rational

b i Rational + Irrational ii 2.57

iii Irrational

c i Irrational + Irrational ii 3.15

iii Irrational

d i Irrational + Irrational ii 10.39

720

e i Irrational + Irrational ii 0 iii Rational

iii Irrational

Mathspace Victorian Curriculum – Year 8 mathspace.co

fraction (

is an approximate value) and its

decimal representation is non-terminating and non-repeating. 14 No, a number cannot be both rational and irrational. A rational number can be expressed as a fraction, while an irrational number cannot. 15 Every rational number can be expressed in the form where p is an integer and q is a positive integer. A whole number can also be expressed in this form where q = 1 and p ≥ 0.

Problem-solving

8 a

b A Rational numbers

c D Positive real numbers

9 a

10 a

11 a

16 Side length = number.

and it is an irrational

17 a C Whole numbers

d E All real numbers e B Integers f A Rational numbers 18 a Yes

b No

c No

d Yes

1.07 Recurring and terminating decimals

12 Fraction

Understanding

Recurring decimal

1 3.45 is terminating.

is recurring.

0.11 0.22

6.9 is terminating.

Rounded to two decimal places

0.33

is recurring.

2 a

3 a 0.67

b 1.23

c 5.88

d 7.89

4 a 0.25

b 0.9

c 0.4

d 0.72

e 0.6

f 8.611

g 0.65

h 1.5

0.44 0.56 0.67 0.78

Fluency

0.89

5 a

d 13

Fraction

1.00

Recurring decimal

Rounded to four decimal places

0.1429

6 a 0.50, Terminating

0.2857

b 2.33, Non-terminating

0.4286

c 0.4, Terminating d 4.75, Terminating

0.5714

e 0.11, Non-terminating f 0.17, Non-terminating

0.7143

g 0.45, Non-terminating 0.8571

h 0.29, Non-terminating 7 a

1.0000

Answers mathspace.co

721

Reasoning

1.08 Index laws

14 The value of π is a non-terminating decimal because it goes on forever without repeating.

Understanding

15 a No, because square roots of whole numbers can be either whole numbers or irrational numbers. b Square roots of whole numbers terminate if the number is a perfect square. c Yes, is irrational because its decimal representation is non-terminating and nonrecurring. 16 One possible answer: When the denominators of fractions are multiples of 3, the decimal representation often has a repeating pattern. For example,

is 0.333… and

is 0.666…,

where the digits 3 and 6 keep repeating. 17 a Recurring are: Terminating are:

1 a Subtract

b Add

c Multiply

d 1

2 a 32 × 33 = (3 × 3) × (3 × 3 × 3) = 35 b c d 53 ÷ 53 = 53 − 3 = 50 = 1 3 a i Yes

ii Yes

iii Yes

iv No

v No

vi No

b i False

ii True

c 6 4 a Positive

b Negative

c Negative

d Negative

e Negative

f Positive

g Negative

h Positive

Fluency

b 7=7 50 = 2 × 5 × 5

5 a 57

b 613

c 1792

9 = 3 × 3

d 4 × 513

e 226

f 29

16 = 2 × 2 × 2 × 2

g −9 × 28

h 9 × 1012

12 = 2 × 2 × 3 20 = 2 × 2 × 5 c The only prime factors that must be present in the denominator of a fraction for it to have a terminating decimal representation are 2 and/or 5. d

will be terminating as 2 and 5 are the

only prime factors of its denominator 40. 40 = 2 × 2 × 2 × 5 Problem-solving 2

18 0.3125 m

19 8, terminating metres, recurring.

j 6 a 2187

litres or

litres

22 4 pieces 23 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19 pieces because these numbers cannot be written as products of just 2s and 5s.

722

Mathspace Victorian Curriculum – Year 8 mathspace.co

d 18 750

7 a 32

b 67

c 102

d 156

8 a 189

b (−76)

c 108

d 76

f 5 × 46

9 a 243

b 64

c 16

d 80

e 16

f 64

g 27

h 150

10 a 512

b 618

c 720

d 940

e (−168)

e 3

f 10

g 5

h 428

i 815

j 1210

k 310

l 83760

11 a 1 21

b 4 782 969

c 84 375

b 1

c 1

d 1

e −1

f 27

g 2

h 0

i 9

j 2

k −9

l 32

n 1

o 1

p 4

12 a 0

b 1

c 0

d 6

e 4

f 4

g 256

h 3

13 a 2

b 4

c 3

d 19

3 4.37 is terminating.

e 3

f 8

g 3

h 8

is recurring.

i 9

j 50

is recurring.

14 a 12

b 2

c 6

d 34

e 0

f 10

g 0

h 4

Reasoning 15 3

16 5 × 5 = (5 × 5 × 5) × (5 × 5) = 5

17 The product of powers rule requires the bases to be the same. Since the bases in the expression are different, we cannot simplify it using the product of powers rule.

9.8 is terminating.

is recurring.

4 a −40

b −126

c −60

d −225

e 0

f −3

g −6

h 4

i 10

j 12

k 2

l −10

5 a 21

b −10

c −10

d −14

e −54

f 31

g −47

h −105

i 26

j −5

k 42

l −6

m 25

n 36

o 101

p 204

6 a −5.0

b −6.5

c −10.75

d −14.95

18 Sam’s solution is correct. When multiplying terms with the same base, we add the exponents: (43) × (42) × (45) = 4(3 + 2 + 5) = 410.

e −57.23

f 37.455

g −77.592

h −138.6

i −10.220

j −2007.495

19 Jane’s solution is incorrect. According to the quotient of powers law, when dividing terms with the same base, we subtract the exponents:

k 0.039

l −45.717

m 39.482

n 0.970

o 9.964

p 31.85

. 20 The student’s solution is incorrect because they subtracted the numerator from the denominator. According to the quotient of powers law, the exponent in the denominator must be subtracted from the exponent in the numerator. The expression should be 115 ÷ 1112 = 115 − 12 = 11 − 7 Problem-solving 21 n = 5

8 a Rational

b Rational

c Irrational

d Irrational

e Rational

f Rational

g Irrational

h Irrational

i Rational

j Irrational

k Rational

l Rational

9 a 10

22 n = 5

e 28

23 m = 4

10 a

24 a i 6 pages

ii 278 pages

b Yes

b 9

c 4.47

d 2.83

f 4.12

g 1.87

h 2.5

25 7 × 7 = 7 square units 9

26 2 × 2 × 2 = 2 cubic units 27 a 120 students

Chapter 1 review 1

7 a

b 153 students

11 a 0.4

f 2.25

12 a

Answers mathspace.co

723

13 a

b 15

b 2×3

c 6×8

d −2 × 6

e 48

f 93

g 198

h (−11)9

i −156

14 a 7

15 a 1

b 1

c 256

d 0

e 2

f 729

g 1

h 1

i 1

j 5

k 6561

l 1 677 721 6

m 1 176 49

n 1

ii 50%

b i

ii 3%

c i

ii 85%

d i

ii 22.5%

6 a 83%

b 42%

c 60%

d 35%

e 65%

7 a 180%

16 −8, 7 17 a $18 000

5 a i 8

b $3600

c $14 400

and it is an irrational

b 0.6

c 2.7

d 3.03

e 0.76

f 5

g 1.82

h 68.97

b 171%

c 1304%

d 110%

f 4.1%

g 800%

h 2400%

10 a 24% e 85.7%

= 0.694 square metres, recurring.

litres or

Fraction

litres.

24 q = 9 b ≈ 280 trees

25 a 200 trees

Decimal

Percentage

0.37

37%

1.21

121%

0.125

c ≈ 393 trees 26 32 × 33 × 34 = 39 cubic units

12 a

2.01 Percentage, decimal and fraction conversion

13 a i 50%, 100%, 30%

Understanding

and 0.8

2 a 100

b 53

3 a 71

b 71%

14 12% 15 25% c

16 70% 17 64%

Fluency 4 a

Reasoning 18 Multiplying by 100 and adding the percentage sign is the same as multiplying by 1. Therefore the size of the number does not change but the format changes from decimal to percentage. 19 a No

724

b 0.475 and

b i 37.5%, 20%, 80% ii c i 20%, 45%, 27%

1 “out of 100”

d 130%

9 a 0.34

20 $3851 21 Side length number.

c 218%

8 a 13%

18 $672 19 $30

b 270%

Mathspace Victorian Curriculum – Year 8 mathspace.co

b 67%

20 Converting a percentage to a decimal is the same as dividing by 100 and removing the percentage sign because a percentage is a number out of 100. A denominator of 100 is the same as dividing by 100. 21 a pq.r%

d 0.0pqr

Problem-solving 22 As a percentage:

3 a 20

b 10

4 a i 7

ii 10

b i 3

ii 4

c i 6

ii 5

d i 9

ii 50

As a fraction: 23 Filled portion as a percentage: 40%. As a decimal: 0.4.

5 a i 120

ii 1200

b i 5

ii 25

iii 500

c i 6

ii 60

iii 600 ii 600

b i 7

ii 700

c i 8

ii 800

d i 4

ii 400

e i 59

ii 5900

f i 500

ii 50 000

g i 48

ii 4800

25 36

h i 28.75

ii 2875

26 57%

7 a i $64

ii $6400

24 a 90%

b 65%

27 Investment C. 15% is the highest interest rate.

2.02 Find the whole Understanding 1 C 2 a

Fraction

Percentage 50% 25% 20% 10% 5%

Fraction

Percentage 100%

Fluency

6 a i 6

As a decimal:

c 4

b i $1.50

ii $150

c i 90 mL

ii 9000 mL

d i 115 mL

ii 11 500 mL

e i 520 g

ii 52 000 g

f i 4000 g

ii 400 000 g

g i 27 L

ii 2700 L

h i 140.5 cm

ii 14 050 cm

8 a 90

b 22

c 85

d 240

e 9

f 48

g 65

h 12

9 a 12

b 80

c 24

d 70

e 66

f 820

g 180

h 250

10 a 8400 g b $1.50

c $10

d 60 mL

11 a i 13

ii 130

b i 61

ii 610

c i $20

ii $200

d i $415

ii $4150

12 a i 6

ii 24

b i 42.5

ii 170

c i 12 500

ii 50 000

d i 150

ii 600

2% 1%

Answers mathspace.co

725

13 a 4550

b 40 000

c $32 000

d $72.75

e 1050 mL

f 8160 g

g $16 250

h 5150 m

b Subtract, because he wants to lower the value. 5 a 42

14 A and D 15 a

16 a (100 − p)%

b Add, because she wants to increase/raise the value. c 112

Problem-solving 17 a 400 ML

4 a 48

c 72

Reasoning

Fluency

6 a 84

b 132

c 100.8

d 400

e $152.60

f $36 000

g 92 km

h 540 mL

7 a 117% b 40 000 ML

b 105%

c 109.4%

d 312%

18 14.8 kg

8 a 128

b 580

c 972

19 $1200

d 112

e 1740

f 1296

20 The shop originally had 600 units in stock. 21 The whole book has 50 pages.

9 a 64.4 d 38.4 g 156 km

b 234

c 200

e $393.60

f $30 250

h 455 mL

22 500 000

10 a 83%

b 95%

c 60%

d 48%

23 1200

11 a 32

b 220

c 828

d 48

e 660

f 1104

24 2400 25 125 26 4700 27 8 km 28 80 000 29 240 30 1070

2.03 Percentage increase and decrease Understanding 1 A, C 2 A, D 3 a B b 3 more squares were added.

b 80%

13 a 27

b 15%

14 a 63

b 70%

15 a 60

b 30%

16 a i Decrease

ii 53.66%

b i Increase

ii 42.42%

c i Increase

ii 55.56%

d i Decrease

ii 10.66%

e i Increase

ii 98.73%

f i Increase

ii 0.71%

g i Decrease

ii 53.77%

h i Decrease

ii 37.04%

17 174 kg 18 42 m

c 100% + 3% = 103% = 1.03

19 40%

d A

20 30%

e 6 squares were taken away. f 100% − 6% = 94% = 0.94

726

12 a 48

Mathspace Victorian Curriculum – Year 8 mathspace.co

Reasoning

8 a $14

21 a 1950

d −$7500

b 1950

c Yes it is equivalent since 100% of 150 is 150.

9 a $68

b By adding 30% to 100%.

d $115

f $14

g $460

h $139

10 a $57

b $7800

c $730

d $1140

e $9

f $493

g $178

h $216

23 No, the discount is 0.8 × 1.10 = 0.88 which is a 12% discount.

11 Loss of 5% or $7.

24 No, the ﬁnal amount is:

12 Proﬁt of 25% or $9.

0.7 (original amount) × 1.3 = 0.91 (original amount) which is 91% of the original amount. So, the ﬁnal amount is not equal to the original amount. 25 Multiplier is Problem-solving

c $87

b $152.50 c $53

e $158

22 a 130%

b −$11 e $140 000

13 a $13

b 10%

14 a $15

b 30%

15 a $163

b 41.69%

16 a $4491

b 49.90%

17 a $21

b 105.00%

c 51.22%

26 200 minutes Reasoning

27 6000 m

18 They are similar because both ﬁnd the ratio of the change in a value to its original value, expressed as a percentage, when the value has increased or decreased.

28 25.5 kg 29 117 m 30 $40.91

The only difference is the proﬁt or loss percentage applies to business transactions only, and the units are in dollars. The increase or decrease percentage formula can be applied to any amount that has increased or decreased in any context.

2.04 Profit and loss Understanding 1 Proﬁt 2 Loss

19 Example answer:

3 a A proﬁt has occured. b A loss has occured. Fluency 4 a Loss of $56

b Proﬁt of $5

c Loss of $8

d Breaking even

e Proﬁt of $65 5 a $17

b $66

c $650

d $7

e $13

f $117

6 a $25

b $87

c $1620

e $7

f $75

d $474.50

7 a Proﬁt is $256.

b Loss is $1055.

c Proﬁt is $49.70.

d Loss is $1700.

e Proﬁt is $43.65.

f Loss is $710.

One reason could be that there are other expenses that Juliette and Alice did not originally account for. For example, if they are shipping the clothes they sell, they will need to purchase materials for shipping. They will also have to pay shipping costs. Another cost is the cost of transportation to the resale shops. 20 a Buzz’s friend found the amount of the loss instead of the selling price. He calculated 170 × 0.3 = 51 correctly, but he needs to subtract this from the purchase price to ﬁnd the selling price. b $119 21 a Harper’s friend found 20% of the selling price not the purchase price. b $1500

Answers mathspace.co

727

20 a $70

Problem-solving 22 a $5000

b $1999

c $3001

23 a $388.80

b $28.80

22 $72.40

24 a $72

b $472

23 $1617

25 a $32

b $168

26 a 112%

b $625

27 a 96%

b $625

b $4020.38

c No, because we apply discounts using multiplication, and the order of multiplication does not matter. d $3 820.25

30 25%

25 a $4250.88

31 a Yes, by $150.

b 50%

Understanding 1 a Mark-up

b Proﬁt

c Discount

d GST

b $4250.88

c Since we apply mark-ups using multiplication, the order of applying them does not matter. So both paintings received the same overall mark-up of the same original price.

2.05 Consumer percentages

26 a $8.40

2 a Increased by 10%

b Increased by 21%

3 a Decreased by 5%

b Decreased by 23%

Fluency 4 a $60

b $48

c $17.50

d $82.50

5 a $126

b $156

c $203

d $246.50

6 a $90

b $18

c $90

d $18

7 a $45.80

b $3.51

8 a $72

b 25.99%

9 a $74

b 23.64%

10 a $244.80

b $15 689.70

11 a $38.40

b $8685.60

12 a $113.40

b $98.60

b After you apply a mark-up of 40% the price is increased. So 40% of a larger price will be more than 40% of the original price. This means the discount is larger in value than the mark-up, and the new price will be lower than the original. 27 a No, because we must add each percentage to 100% and then multiply them. Elizabeth added 10% to 100%, but she did not add 40% to 100%. b Yes, because she ﬁrst added both percentages to 100% and then multiplied. c $924 28 a Yes, because he ﬁrst subtracted both percentages from 100% and then multiplied. b No, because we must subtract each percentage from 100% and then multiply them. Carl subtracted 20% from 100%, but he did not subtract 15% from 100%. c $74.80 29 a No, because we must subtract the percentage of the discount from 100% and add the percentage of the mark-up to 100% and then multiply them. Tom subtracted 15% from 100%, but he did not add 20% to 100%.

13 68 688 people 14 $229.41 15 $2473.48 16 $42.94 17 $42 665.36

728

21 $1694.00

24 a $4020.38

29 $1600

19 a 110%

c $43.90

Reasoning

28 $14 256

18 a $33

b $670

b $7.70 b 11

c $1177 c $62

d $60

Mathspace Victorian Curriculum – Year 8 mathspace.co

b Yes, because he ﬁrst subtracted the percentage of the discount from 100% and added the percentage of the mark-up to 100%, and then multiplied them. c $81.60

Problem-solving

13 219 pages

14 45.7 kg 15 77 litres

31 a $700

b $945

32 a $1422.90

b 20.95%

33 Next year’s deal 34 a $16.00

b $64.00

c $7.27

Chapter 2 review 1

Fraction

Decimal

Percentage

0.03

1.47

147%

0.625 2 a i 40%, 70%, 20%

c i 20%, 45%, 27%

f $100

17 a $20

b $60

e $10

f $75

ii 1100

d $1170

e $11

f $510

g $190

h $230

19 a $150

b 75.00%

c 42.86%

20 a $3000

b $4750

c −$1750

21 25% 22 $425

24 a $600

b $60

c $300

d $250

25 a $90

b $100

c $144

d $80

26 a $1104

b $412.50

ii $400

27 $862.50 28 $1617.00

d i 115 mL

ii 11 500 mL

e i 27 L

ii 2700 L

f i 140.5 cm

ii 14 050 cm

30 Next year’s deal

c 160 mL

3.01 Ratios

f 8400 g g $48

d $489.50

c $770

ii 5900 mL

b 45 kg

c $1300

b $9600

c i 59 mL

e $220

d $10

18 a $60

b i $4

5 a $44

c $500

b $240

3 Savings Account Z. 20% is the highest interest rate. 4 a i 11

b $50

e $15

23 a Alice’s sister found the amount of the loss instead of the selling price. She calculated 320 × 0.25 = 80 correctly, but she needs to subtract this from the purchase price to ﬁnd the selling price.

b i

16 a $20

d 180 m

29 a $250.00

b $749.99

c $166.67

h 12 mL Understanding

6 300

1 4:5

7 10 km

2 Divide by 3

8 a 36

b 310

c $648

d 112 mL

9 a 81

b 120

c 42

d 270

3 a

b 2

h 300 mL

4 a 3:8

c 8:3

d 11:5

e $136 10 a 1920

f $28 000 g 80 km b 1920

c Yes it is equivalent since 100% of 2400 is 2400. 11 40% 12 35%

5 2:10 and 1:5 Fluency 6 a 6:7

b 7:4

c 5:12

d 16:45

e 2:3

f 3:7

g 3:7

h 1:8

i 9:7

Answers mathspace.co

729

j 4:15:6

k 5:9

l 10:7

m 3:8

n 3:8

o 2:1

p 2:3

q 9:40

r 5:3

s 65:54

t 20:16:3

7 a 6

Problem-solving 20 Daniel will get 120 while Emma will get 60 jelly beans. 21 a $40 000

b 11

c 40

d 8

e 36

23 $180

k 1.12

l 45

24 $108 000

b 190:9

c 20:11

d 4:11

8 a 5:21 9 600 g 10 a $420

b $294

11 a $24 000

b $6000

12 a 3:4

b 20 m

13 a 18

b 2

14 a 8:13

b 224.77 Earth days

15 a 150 kg

b 360 kg

c 3:5

c 1:10:14

Reasoning 16 The ratio of the number of red marbles to the total number of marbles in the bag can be written as “red marbles:total marbles”. Since the total number of marbles is 5 + 3 + 6 = 14, the ratio can be written as 5:14. 17 We need to ﬁnd the total number of parts in the ratio, which is 2 + 3 + 4 = 9. Then, we divide the total number of pets in the store by the total number of parts in the ratio to ﬁnd the value of each part. 45 ÷ 9 = 5, so each part of the ratio represents 5 pets. This means that the number of rabbits is 4 × 5 = 20. 18 1 + 5 = 6. 20 is not divisible by 6. So if we make two piles, put 1 marble into one and 5 marbles into the other and repeat this twice more, we will have one pile with 3 marbles and one pile with 15 marbles, leaving us with two marbles. 19 a The number of games played must be an integer.

of the games must also be an

integer. So, the number of games must be a multiple of 5. b 3:2

b $125 000

22 19 hours and 12 minutes

25 a 15

b 110

26 a 105 mL

b 135 mL

27 a 15:8:5

b 39.80 Earth days

28 32 ml 29 a 12.61 hours

b 22.36:10.28:12.61

3.02 Maps and scale Understanding 1 2 cm represents 1000 km 2 a Larger

b Smaller

3 a 30 000 cm

b 0.3 km

b Real life item is 20 times larger. Fluency 5 a 1:300

b 1:5000

c 1:90 000

d 1:600 000

e 1:80 000

f 1:100 000

6 a 38 000 cm

b 380 m

7 a 6 250 mm

b 625 cm

8 8 cm 9 2 cm 10 8.874 km 11 1:7000 12 3500 km 13 a Approximately 514 m b Approximately 229 m c 400 m d Approximately 571 m f Approximately 2571 m 14 a 620 m

Mathspace Victorian Curriculum – Year 8 mathspace.co

c 20 cm

4 a Scale factor is 20.

e Approximately 1714 m

730

c 4:7:9

b Eli

Reasoning

9 45 words/min

15 To ﬁnd Carl’s house, start at E5 and and follow Top Lane for 300 m. A distance of 100 m is represented by the side length of 1 square, which means that 300 m is the side length of 3 squares. This means that the map reference of Carl’s house is D2.

10 37 litres per hour 11 $41/hr 12 a $2.60

b $1.80

13 a

Car

Litres used

Magnum Falcador

16 Tibsen park. Because Lesse park has an area of approximately 30 000 m2, Millenial park has an area of approximately 500 m2, and Tibsen has an area of approximately 59 000 m2 which is the largest.

Distance travelled (km)

L/km

13.405

38.3

0.35

20.71

54.5

0.38

Canyonero

6.336

19.2

0.33

Civil

2.754

8.1

0.34

b Canyonero

17 a Yes, both would be 8 m squares. b 1:200

14 a

c 1:9

Time taken (hrs)

Distance travelled (km)

d Yes. The ratio of areas would be a2:b2

320 160

b 80 km/hr

Problem-solving

15 80 toys/hour

18 7 cm apart

16 540 km

19 0.875 cm 20 a 1:45

b 5m

21 a Locations C and D

b 2080 km

17 400 km 18 80 calories per kilometre

22 The ﬁrst number should be 1 and the second number should be a multiple of 10 000. Example answer: 1:20 000 23 a 1:25

b 6m

3.03 Rates Understanding 1 i D

c Shop B

ii B

2 a dollars per litre c beats per minute

iii A

iv C

b words per minute d dollars per hour

19 23 min/episode 20 24 fps 21 a

b 45 m/min

22 a 6000 mL/h

b 41.67 cm/sec

c 4800 words/h

d 680 000 m/h

e 4500 beats/h

f 86.67 mL/min

g 500 L/kg

h 480 m/h

23 420 m/min 24 78 bpm 25 a 7200 mL

b 360 s

c 20 mL/s

3 A and C Reasoning Fluency 4 a 80 books/library c 52 km/hr 5 29 L/km 6 10 pages/hour 7 33 pavers per hour

b 17 people/bus d

cakes/guest

26 First, convert km to m by multiplying the value by 1000 (since there are 1000 metres in 1 kilometre). So, 120 km/hr is equal to 120 000 m/hr. Next, convert hr to min by dividing the value by 60 (since there are 60 minutes in 1 hour). So, 120 000 m/hr is equal to 2000 m/min. This means that 120 km/hr is equivalent to 2000 m/min.

8 $70/hr

Answers mathspace.co

731

27 To determine the earnings of Luke on Saturday, we add the total hours he rendered on his 3 jobs, 1.5 + 0.25 + 1.25 = 3 hours. Since he gets paid $64.50 per hour and he has 3 jobs, his total earnings on Saturday is (3 × $64.5) + (3 × $75) = $193.5 + $225 = $418.5. 28 a If Lisa will deposit $3.20 into her bank account for every 8 coffees she purchases, we can ﬁrst divide $3.20 by 8 coffees she purchases. Second, we simplify to $0.40 for every 1 coffee she purchases. b $0.40/cup of coffee

2 The Medicare levy is calculated as 2% of an individual’s taxable income for the ﬁnancial year. 3 a Australian dollar c Euro

b United States dollar d Indonesia Rupiah

4 a Multiply

b Divide

Fluency 5 a $1060.80

b $3840

c $457.20

d $4536

6 a 9.9%

b 9.2%

c 7.0%

d 9.6%

e 6.9%

29 The bucket will be full in 5 hours, but will not overﬂow until after that time. By calculating the amount of water in the bucket in 5 hours, we get 5000 mL. Converting millilitres to litres, we = 5 L in

know that there are 5000 mL ×

the bucket. This is how much water the bucket can hold. 30 John is faster, so he will be able to catch Avril. We can determine this by comparing each runner’s speed in metres per second. Avril runs , or 6 m/s. John runs

, or

7 m/s. John runs at a faster rate than Avril.

7 a $276

b 10

8 $133 200 9 $7530 10 a $0

b $2977.60

c $18 806.50

d $35 603.68

e $53 612.60

f $67 752.50

11 a $55 217.75

b $3959.10

c $59 176.85 12 189 HKD 13 10.86 AUD 14 58.96 USD

Problem-solving 31 540 runs

Reasoning

32 a 75 seconds b 75 s/m

15 3000 × 0.06 × 8 + 3000 × 0.02 × 8 = $1920

33 a 28 apples

b 11 days

c No

3000 × 0.04 × 16 = $1920

34 a 90 litres

b $11.70

c 24 minutes

Yes, Luke is correct.

c Increase

16 Yes, she is correct as she pays no income tax on $17 000 therefore her only payment is the Medicare levy of 2%.

35 662 36 a 108 bpm

b 120 bpm

37 a 15.2 km/h

b 10 km/h

c On his daily run 38 a 24 sheep per day c 23 sheep per day 39 a 18 kL/min

b 600 sheeps d Jenny b 9 kL/min

c 13 minutes 20 seconds

17 No, James is incorrect. He needs to multiply by 0.75 instead of divide because the exchange rate is in terms of 1 AUD not 1 USD. 18 a No. Low paid workers will pay less tax, but 18c for each $1 does not raise enough tax for those earning over $45 001 to pay the same tax.

3.04 Financial rates

b $25 000 × $0.18 = $4500, so they would pay $4500 − $4288 = $212 more.

Understanding

c $31 500 plus 37c for each $1 over $135 000 instead of $31 288 plus 37c for each $1 over $135 000.

1 B

732

Mathspace Victorian Curriculum – Year 8 mathspace.co

Cost per kilogram is $16.50 per kg

Problem-solving 19 a $2160

For the Tasmanian salmon, the cost per kilogram is:

b 6%

20 3.20% 21 a $20 064.70

b $1951.78

Cost per kilogram = Total cost ÷ Weight in kilograms Cost per kilogram = $51.00 ÷ 2 kg

c $7104.22 22 a $148 122

b $36 143.14

c $1394.50 per fortnight 23 a $52 354

b $6494.20

c $1047.08

d $10 036

e $2494.72 24 500 AUD

Cost per kilogram is $25.50 per kg From these calculations, we can see that the Atlantic salmon is the better buy, as it has a lower cost per kilogram than the Tasmanian salmon. Therefore, if Fiona is looking for the better deal in terms of price per kilogram, she should choose the Atlantic salmon. 12 The second apartment is the best value because it costs the same as the ﬁrst apartment but has 20m2 more space.

3.05 Best buys Understanding

13 a 0.71 L/$1

1 65 cents/egg 2 When the unit price of an item is cheaper than the unit price of another item, we consider it a better buy or the best buy.

b The unit price is the cost per unit of the item. So, for every litre of milk, the unit price is $1.4/L. The answer to part (a) is a unit rate, but not the unit price.

3 $1 buys 4 apples

Problem-solving

Fluency

14 a $1.86/L

4 a $2.65/L

b $3/L

c Apple juice

5 a $4.20/kg

b $3.60/kg

c Special Q

6 a 50 cents/egg

b 30 cents/egg

c Sunny Side Up 7 a $2.25

b $1.45

c Magnum

b $1.44/L

c Dunkilderry 15 Krispy Krunchers 16 a i 3

ii 3

b Barry’s Bargain Bins

Chapter 3 review

8 20 nails

1 a Larger

9 Greenery

2 $1.5 gets you 3 pears

10 a $7.50

3 a 12

b $8.00 c The Adventures Series

d $43.20

e 4 a 5:16

b Smaller

b 14

c 21

d 40

b 23:1

c 2:1

d 7:17

Reasoning

5 800 g

11 To determine which salmon is the better buy, we need to calculate the cost per kilogram of each type of salmon.

6 a 1:400

b 1:80 000

c 1:5000

d 1:90 000

e 1:600 000

f 1:100 000

For the Atlantic salmon, the cost per kilogram is: Cost per kilogram = Total cost ÷ Weight in kilograms

7 2 cm 8 a 380 m

b Jason

Cost per kilogram = $31.35 ÷ 1.9 kg

Answers mathspace.co

733

9 a 80 pages/printer

4.01 Algorithms in code

b 20 students/classroom Understanding

c 54 litres/day d

1 Pseudocode enables developers to write the steps of an algorithm in a human-readable manner, without including speciﬁc programming language syntax.

loaves/person

10 37.5 litres per hour 11 a

b 48 m/min

12 a 8000 mL/h

b 5400 words/h

c 720 000 m/h

d 5100 beats/h

e 100 mL/min

f 50 cm/sec

g 600 L/kg

h 540 m/h

13 a $1425

b $1820

14 a $310

b 12.5

c $720

d $2600

3 IF-THEN-ELSE Fluency

15 $10 770 16 a $4.80/kg

2 START indicates the start of the algorithm. SET initiates a variable and assigns a value to it. In this case, the value 6 is assigned to the variable a and 4 is assigned to the variable b. PRINT presents a value or message to the user.

b $5.00/kg

c Twisty Pasta

17 VeggieMart 18 137.5 ml 19 a 13.50 hours

b 24.50:12.25:13.50

20 5 cm apart 21 a 1:25

b 7m

4 a 60

b 38

c 9

d 12

e 18

f 100

5 a C

b 154 m2

6 a D

b 36 m

7 a is odd

22 A total of approximately 523 goals.

8 a is divisible by 3

23 a 200 seconds

b 200 s/m

Reasoning

24 a 20 kL/min

b 16.67 kL/min

c 13.64 minutes 25 a $2100

b 6%

26 a $58 680

b $8392.00

c $707.47 per month 27 a $58 776 c $1175.52

b $8420.80 d $10 560

e $963.68 28 384.62 GBP 29 26.4 GBP

33 a i 3

ii No b $1.63/L d $57.05 ii 3

b Danny’s Discount Depot 34 Crunchy Cookies

734

11 The multiplication by 4 is used to calculate the perimeter of a square. A square has four equal sides, so to determine the total perimeter, the length of one side (S) is multiplied by 4.

12 a i No

31 101.65 NZD c Stonebridge

10 Division by zero is undeﬁned in mathematics and programming. If b is initialized to 0, the pseudocode will attempt to divide by zero, leading to an error or unexpected outcome.

Problem-solving

30 112.5 AUD

32 a $1.93/L

9 Lyla is correct; the proper sequence should be F, A, C, D, E, B

Mathspace Victorian Curriculum – Year 8 mathspace.co

iii Yes iv Answers may vary. Sample answer: 2.4, 4.3, 6 and 6.2

b Remove “Add one” before the “Round to nearest whole number” then set the next step to be “Is it an even number?”. Then, instead of “Divide by two” we END the algorithm after we determined that the number is even. 13

1 START 2 SET a=3 3 SET b=7 4 SET c=(a+b)/2 5 PRINT c 6 END 1 START 2 SET m=4 3 SET cube = m ∗ m ∗ m 4 PRINT cube 5 END

15 Area of rectangle: 1 START 2 SET base, height 3 SET area = (base ∗ height) 4 PRINT area 5 END Volume of rectangular prism: 1 START 2 SET base, height, depth 3 SET volume = b ase ∗ height ∗ depth 4 PRINT volume 5 END 16

Step Instruction A START B SET x, y, z C

SET average = (x + y + z)/3

PRINT average

SET x = x + y

SET y = x − y

SET x = x − y

PRINT x, y

END

4.02 Loops Understanding 1 The condition set by the WHILE loop. 2 a 5 b The loop will continue to execute as long as the condition a < 10 holds true. 3 a Lines 4 to 7. b The loop will continue to execute as long as the condition b ≤ 15 holds true. c This line increases the “total” by the current value of b during each iteration of the loop. Fluency 4 a b = 12

b b=6

c PRINT total = 63 5 a i b=5

ii 68

b i b=4

ii 6.5

6 m=2 7 D, A, C, E, B 8 G, B, C, E, A, F, D 9 a 6 d 1, 4, 7, 10

b 2, 6, 18

c 4, 6

e 4, 12

f 12, 36

10 a 21

b F, C, A, G, D, E, B

11 a 120

b F, D, A, E, B, G, C

Reasoning 12 The condition b > 8 ensures that the loop will keep iterating as long as the value of b remains greater than 8. When b becomes 8 or less, the loop will stop. This condition is responsible for governing how many iterations the loop undergoes, contingent upon the starting value of b. 13 b is incremented by 2 in each loop iteration to progress through even numbers, starting from the initial value of b up until 14. This mechanism allows the loop to accumulate the sum of these sequential even integers.

Answers mathspace.co

735

14 If the condition changed to b ≤ 17, the integer 17 would be incorporated into the sum. This would enhance the cumulative sum by seventeen units and also upturn the count (m) by one. Consequently, the ultimate average value displayed would be different. Problem-solving 15 The initial value of the product variable should be 1. To determine if a number is odd, check if the remainder when divided by 2 is not zero. The modiﬁed pseudocode: START SET a = 10 SET product = 1 WHILE a > 2 IF remainder of a divided by 2 is not 0 product = product × a END IF a = a – 1 END WHILE PRINT product END 16 Introduce a new variable for the sum of cubes. In each iteration of the loop, cube the current number and add it to this new variable. The pseudocode: START SET a = 8 SET sum = 0 SET sumOfCubes = 0 WHILE a > 1 sum = sum + a sumOfCubes = sumOfCubes + a3 a = a − 1 END WHILE PRINT sum, sumOfCubes END

17 Use variables to track the sum of prime numbers and the count of primes. Update these variables inside the loop based on conditions. The pseudocode: START SET a = 2 SET primeSum = 0 SET primeCount = 0 WHILE a ≤ n IF isPrime(a) primeSum = primeSum + a primeCount = primeCount + 1 END IF a = a + 1 END WHILE PRINT primeSum, primeCount END

4.03 Debugging Understanding 1 In programming, ‘bugs’ refer to mistakes or errors that prevent a computer program or code from functioning correctly. 2 The process of fixing or removing errors from a program is called ‘debugging’. 3 Try to identify where the program or code is not working. Go through this section of your code step by step thinking about what each command does, writing down the result of each step if possible. If you are still having difficulty finding the error, take a break and look at it again, or ask a friend to look at your code. Fluency 4 a a is odd b a is odd c The output from part a, which should have been “a is even”. d Line 4, because it will say a is odd even though it is divisible by 2. It should be THEN PRINT a is even.

736

Mathspace Victorian Curriculum – Year 8 mathspace.co

5 a b Line 4, because to determine the mean of two numbers we should divide by 2 not 4. The line of code should be SET m = (a + b)/2. 6 a 42

We should add the line of code “SET count = 0” after line 2. Problem-solving 11 a 5040 b Line 4, it should be a ≤ n to include the value of n in the multiplication.

b 10 c 7

12 a 4

13 a $135

e Line 10, we should divide by n which is the number of the elements in the set, not a. The line of code should be PRINT sum/n. 7 a i Step 5

ii 4

b i None, because there is no value set first. ii None c Step 4 8 Line 7 contains the bug and it should be i = i + 1 to produce the list positive even numbers. Reasoning 9 a No, using smallest will not change the output of the pseudocode because “largest” or “smallest” works as a variable or value holder. The pseudocode would still determine the largest number in the list. b If we remove lines 4 and 8, the pseudocode would no longer iterate through the list of numbers. This means the variable “largest” would only be set to the first element of the list, and it would not be updated as the pseudocode doesn’t process the rest of the elements. The final value printed would always be the first element of the list, regardless of the actual values of the other elements. 10 T he pseudocode is missing the line of code setting the initial value for the variable “count”. Without this, there is a possibility that the pseudocode would assign a random initial value for “count” and print an incorrect output.

b Line 5

b The error is on lines 5 and 7. The conditions for discounts have been swapped. Correct pseudocode would check if items > 20 before checking if items > 10. The correct pseudocode is: 1 START 2 SET items = user input () 3 SET price per item = user input () 4 SET total = items ∗ price per item 5 IF items > 20 6 total = total ∗ 0.8 7 ELSE IF items > 10 8 total = total ∗ 0.9 9 END IF 10 PRINT total 11 END 14 a i Line 11 ii list element at position n = a b i Line 6 ii Possible answers: • FOR m FROM (n + 1) TO 4 • FOR m FROM n TO 4

4.04 Algorithms for divisibility Understanding 1 If a number ends in a 0 or a 5, then it’s divisible by 5. 2 The number should be divisible by both 2 and 3.

Answers mathspace.co

737

3 A number is not divisible by 8 in the final step if the quotient is a decimal. An example would be 20: 20 2 10 2 5

10 ( first time) 5 (second time)

2.5 (third time) 2 Since e the quotient in the third step is a decimal, 20 is not divisble by 8. Fluency 4 a 780, 8560, 12 510, 78 980 b 780, 8560, 78 980, 985, 605, 6545, 12 510, 22 735 c 780, 8560, 78 980, 692, 6548, 93 874, 1328, 2328, 12 510, 89 942 d 780, 8560, 12 510, 78 980 5 a 860, 710, 170

Reasoning 16 Yes, if the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3. However, for divisibility by 9, the sum of the digits should be divisible by 9. In this case, 15 is not divisible by 9, so the mystery number is not divisible by 9. 17 If a number is divisible by both 4 and 5, it’s also divisible by their least common multiple (LCM). The LCM of 4 and 5 is 20. Therefore, the number is guaranteed to be divisible by 20. 18 If a number is divisible by 9, it’s also divisible by 3 (since 9 is a multiple of 3). Given that it’s also divisible by 2, the number must be even. Thus, it’s guaranteed to be divisible by the least common multiple of 2, 3, and 9. Since 9 is a multiple of 3, the LCM is just the LCM of 2 and 9, which is 18. Therefore, the number is also divisible by 18. The three number could be: 36, 54 and 72. Problem-solving

b 61 110, 21 880, 25 190

19 No

c 1 904 790, 6 459 620

20 Yes

6 a 74, 80, 16 b 138, 564, 992

Chapter 4 review

c 8 603 042, 7 103 764

1 Example answer: Baking cookies.

7 a 85, 35, 90

2 a

b 920, 865, 555 c 8 927 050, 6 606 005 8 a 1528, 6236, 6632

Step

Instruction

Output

Start with the number 150.

150

Subtract 30 from 150.

120

Divide the result by 2.

b 5 885 468, 4 993 860, 4 924 944, 9 951 756 c 92 123 492, 13 189 788 9 a Yes

b No

10 a Yes

b No

c Yes

11 a No

b Yes

c No

12 a Yes

b No

c Yes

ii Yes

iii Yes

iv Yes

b i Yes

ii Yes

iii No

iv No

b No

15 a Yes

b No

738

Instruction

Output

Start with the number 22.

Add 8 to 22.

Triple the result.

d Yes

13 a i Yes 14 a Yes

Step

Mathspace Victorian Curriculum – Year 8 mathspace.co

Step

Instruction

Output

Start with the number 45.

Reduce the number by 10%.

40.5

Subtract 5 from the result.

35.5

3 The step has gone through 9 iterations.

Step

Instruction

4 After executing both instructions, the variable “value” would be 12.

Begin with the number 40 000

5 a 5

Divide this number by 4

Perform step 2 three additional times

b It aggregates the present value of a to the ongoing “sum” value.

Result 40 000 10 000 2500 625 156.25

6 a Yes b 11 + 11 + 11 + 11 + 11

11 a 5

b No c Numbers whose unit value ranges from 5 to 9 would require regrouping. 7 a i W e must regroup into tens and units, then proceed to Step 3. ii We will transition directly to Step 5.

c 55

d 55

12 107 13 6 14 a 15

b 16

c 16

d 65

iii We will progress to Step 7.

15 a is odd

iv The sum is 793

16 a i a = 5

ii OUTPUT = 110

b i a=5

ii OUTPUT = 9.5

b i Move on to Step 3 without regrouping. ii We must regroup into hundreds and tens, then advance to Step 5. iii We will transition directly to Step 7.

17 a 9

b 3, 12

c 8, 9, 10, 11

d 30, 26, 22, 18

8 47

18 The ﬂaw in the algorithm is the overlapping conditions. When considering the number 55, since it surpasses 50, it’s adjusted to

9 160

= 14.75. However, since it doesn’t satisfy

iv The result is 1009

10 a

Step

Instruction

Result

Begin with the number 15

Subtract 4 from this number

Perform step 2 three additional times

19 106

20 1 instance

−1

Step

Instruction

Result

Begin with the number 45

Add 6 to this number

Perform step 2 three additional times

Round this number down to the nearest 10

the ﬁrst condition, it remains unaffected by it. Clearly deﬁned, non-intersecting conditions are vital for an algorithm’s consistent outcomes.

51 57 63 69 60

21 Price = $21 + 13 MB × $0.20 per MB = $21 + $2.60 = $23.60 22 The movie is returned 2 days after the 8-day window. Hence, the ﬁne is $6 plus 2 days at $0.75 each, totalling to $7.50. 23 Anna will need to swap gadgets on the charging station 1 time. She charges 4 gadgets with the ﬁrst set of ports and then swaps to charge the remaining 3 gadgets.

Answers mathspace.co

739

24 Your score would decrease by 4 points. 25

1 2 3 4 5 6 7 8

5 2 hours and 30 minutes 6 3 minutes

START INPUT A INPUT B SET buffer=A SET A=B SET B=buffer PRINT A, B END

7 45 minutes 8 38 minutes 9 8760 hours 10 a 10:00 a.m.

26 Use variables to track the smallest and largest numbers. Update these variables inside the loop based on conditions. The pseudocode: START SET i = 1 SET sum = 0 SET smallest = ∞ SET largest = −∞ WHILE i ≤ n sum = sum + i IF i < smallest smallest = i END IF IF i > largest largest = i END IF i= i+ 1 END WHILE PRINT sum, smallest, largest END

b 12:20 a.m.

c 10:00 a.m.

d 7:24 p.m.

e 12:05 p.m.

f 4:10 p.m.

g 1:00 p.m.

h 10:13 p.m.

11 16 minutes 12 4 hours 13 4 hours 14 8 hours 15 3:00 p.m. 16 3:10 p.m. Reasoning 17 a The answer cannot be correct because time is represented in hours and minutes, not in decimal form. Adding the minutes together is correct, but the result should be converted back to hours and minutes. b Add the numbers of minutes together. If the answer comes to 60 or more, add one to the hours and subtract 60 from the minutes.

5.01 Elapsed time

18 Yes. 365 ÷ 7 = 52.14 which rounds to 52, 365 ÷ 7 = 52.29 which also rounds to 52.

Understanding

19 a 3 hours and 35 minutes

1 a 60

b 1

c 24

d 7

2 a Multiply by 60

b Dvide by 60

c Divide by 24

d Divide by 60

e Multiply by 3600

f M ultiply by 24 then multiply by 60

3 a a.m.

b p.m.

c a.m.

d p.m.

740

c 4 hours and 10 minutes d a + c hours and b + d minutes e a + c + 1 hours and 0 minutes f a + c + 1 hours and b + d − 60 minutes Problem-solving 20 a 3285 days

Fluency 4 a 216 hours

b 4 hours and 0 minutes

b 660 seconds

c 2160 minutes

d 86 400 seconds

e 4 minutes

f 6 hours

g 13 minutes

h 13 hours

Mathspace Victorian Curriculum – Year 8 mathspace.co

c 4 730 400 minutes

b 78 840 hours d 283 824 000 seconds

21 87 weeks

14 15:40

22 a 36.5 hours

b 2007.5 hours

16 21:20

23 9 years and 5 months

17 04:29

24 670 Martian days 25 a 144 minutes

15 17:20

b 86.4 seconds

c 0.864 seconds

18 15:12 Reasoning

5.02 12 and 24 hour time

19 a He is correct for times in the afternoon, but not for times in the morning when the hours are the same.

Understanding 1 a.m. is used for times from 12 midnight until 11:59 in the morning, p.m. is used for times from 12 midday until 11:59 in the evening. 2 a Add 12 to the given time b Subtract 12 from the given time 3 As there are 24 hours in a day, the hours can be numbered from 0 to 23 when using 24-hour time. With 12-hour time, each number must be used twice, which is why a.m. and p.m. are used to distinguish between morning and afternoon.

b For example, 3:15 p.m. is 15:15 in 24-hour time, but 9:15 a.m. is 09:15 not 21:15. c Add 12 to the number of hours for times between 1 and 11 p.m. subtract 12 for times that are 12 a.m., don’t change the hours for 12 p.m. 20 12:25 a.m., 00:30, 12:25 p.m., 13:25, 19:30, 8:25 p.m. Problem-solving 21 9 hours and 20 minutes 22 14 hours and 20 minutes

Fluency 4 a 10:30

b 21:15

c 04:12

d 08:40

e 19:25

f 04:31

g 00:47

h 18:13

5 a 5:30 a.m.

b 6:30 p.m.

c 12:45 a.m.

d 1:59 p.m.

6 a 09:48 a.m.

b 12:28 p.m.

c 10:47 p.m.

d 12:56 a.m.

7 a 15:35 on Friday

b 22:10 on Wednesday

c 18:23 on Tuesday

d 19:33 on Sunday

e 21:57 on Friday

f 11:19 on Friday

8 a 23:45

b 13:20

c 22:30

d 11:15

e 08:10

f 15:50

23 a 04:38

b 20:54

24 a 4:11 a.m.

b 4:38 p.m.

5.03 Time zones and the date line Understanding 1 a Cities are 1 hour ahead of UTC + 0. b Cities are 9 hours behind of UTC + 0. c Cities are 3 hours and 30 minutes behind of UTC + 0. b The island is 12 hours and 45 minutes ahead of UTC + 0. 2 a Yes

b No

c Yes

e No

f Yes

9 a 20:10

b 8:10 p.m.

d No

10 a 16:47

b 4:47 p.m.

3 a False

b False

c True

11 a 07:25 or 19:25

b 02:50 or 14:50

d False

e True

f False

c 08:11 12 6 hours and 42 minutes 13 14:50

d 13:18

Fluency 4 a Dobo and Tokyo c 5:00 a.m.

b 9 hours d 9:00 p.m.

Answers mathspace.co

741

5 a 10 hours

b 7:00

c 1:00 p.m.

b 3.5

c 9:00 p.m.

6 Canberra 7 Singapore 8 a 13 d 8:30 p.m.

e 6:15 on Friday

9 a i Algeria

ii 4 hours

iii 01:25

iv 14th

16 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 i South Sandwich Islands ii 2 hours c i Mauritius

iii 4:00 p.m.

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.

ii 1 hour

iii 18:25 10 a 4:42 p.m. b

Western Australia (WA) 6:15

South Queensland (QLD) Australia (SA) 7:45

15 Melbourne uses daylight saving, so the time difference is 3 hours. Adding 5.5 hours to 9 p.m. gives us 2:30 a.m. on Friday, but when we remove the 3 hour time difference the local time in Perth is 11:30 p.m. Queensland does not use daylight saving, so the time difference is only 2 hours.

8:15

Problem-solving 17 a 16 hours

b 8:00 a.m.

18 a Beth

c Monday

b Han

9:45

11:15

11:45

8:45

10:15

10:45

c i 5:00 a.m.

ii 6:00 a.m.

d i 9 a.m.

ii 11 p.m.

5:25

6:55

7:25

AWST

ACST

AEST

3:55

5:25

5:55

19 a C b Tuesday, Wednesday, Thursday, or Friday

5:11

6:41

7:11

20 a 8:30 a.m.

b 9:30 p.m.

10:25

11:55

12:25

5:47

7:17

7:47

21 a 14:10

b 5:20 p.m.

22 a 4 hours

b 2:00 p.m.

11 a 15:14 AEST c 1:49 a.m. AEST 12 a 15:35 AEDT c 10:45 a.m. AEDT 13 a 21:20

b 7:00 a.m. AEST d 10:50 p.m. AEST b 5:00 a.m. AEDT d 6:13 p.m. AEDT b 22:20

c 3:00 a.m.

d Saturday

23 a 4:50 p.m.

b Monday

24 22 hours 25 a 5:44 p.m.

b 2:27 a.m.

c 5:58 p.m.

26 a 38 hours and 33 mins. Reasoning

b 30 hours and 22 mins.

14 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.

c 31 hours and 41 mins.

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.

742

Mathspace Victorian Curriculum – Year 8 mathspace.co

27 a 8 hours

b 4:00 p.m.

28 a Tuesday

b Wednesday

29 a 12:25 p.m.

b 9:55 a.m.

c 2:00 p.m.

c 5 hours and 22 mins. d 1 hour and 26 mins. e 7 hours and 18 mins. 30 a 3:50 p.m.

b 10:21 p.m.

c 10:45 a.m.

20 New Delhi

Chapter 5 review 1 A 12-hour time divides the day into two periods - a.m. and p.m., each lasting 12 hours. 24-hour time, on the other hand, divides the day into 24 hours running from midnight to midnight. For example, 3 p.m. in 12-hour time would be 15:00 in 24-hour time. 2 London is typically 9 hours behind Sydney. Therefore, the current time in London would be 5:00 a.m. However, this can vary depending on daylight saving time. 3 a 168 hours

b 900 seconds

c 2880 minutes

d 43 200 seconds

e 5 minutes

f 6 hours

g 15 minutes

h 10 hours

21 a 1:49 a.m. AEST c 18:43 AEST 22 a 20:15

b 10:50 p.m. AEST d 21:56 AEST b 21:15

23 a Tokyo is 16 hours ahead of Los Angeles, so if you have an 11-hour flight, the local time when you arrive is only 5 hours later than the local time when you left. b There is a 5-hour difference between London and New York, so if you have an 8-hour flight, your local time when you arrive in New York will be 13 hours later than the local time when you left London. 24 a 7:45 a.m.

b 7:45 p.m.

25 a 39 hours and 40 mins.

4 3 hours and 0 minutes

b 26 hours and 37 mins.

5 The International Date Line is an imaginary line on the Earth’s surface, mostly following the 180° longitude, where the date changes as soon as one crosses it. 6 2.5 hours

c 31 hours and 30 mins.

6.01 Substitution Understanding 1 a There are 100 years in a decade.

7 10:50 a.m.

b 2.5 × 100 = 250 c 100 − 175 = −75

8 11:00 p.m. 9 a 18:55

b 14:25

c 22:30

d 22:45

e 14:45

f 06:05

10 a 17:15 on Tuesday

b 19:45 on Thursday

c 14:55 on Monday

d 13:35 on Monday

11 7 hours and 40 minutes

d If 5a = 500, a = 100. 2 a 15

b 60

c −1

b 3

c −5

d 1

b 14

c 9

d 25

3 −54 Fluency 4 a 27

12 15:15

e 4

13 17:20

5 a 63

14 04:25

6 a

7 a

8 a 26

b 65

9 a 104

b 33.8

15 a 121.67 hours

b 4866.67 hours

16 10 503 Jovian days 17 a 4380 days c 6 307 200 minutes

b 105 120 hours d 378 432 000 seconds

18 12:45 a.m., 02:15, 12:45 p.m., 16:30, 18:45, 10:30 p.m. 19 a No

b Yes

c Yes

d Yes

10 a

c 12

11 a 43

b 2

c −2

d 60

e 7

f −56

g 120

h −7

Answers mathspace.co

743

i 16

j −6

k −11

l 2

n 11

o 2

p 6

q 36

r −5

s 900

1 a Coefficient

b Variable

c Term

12 a 20

b 91

c 432

d 120

d Constant

e Coefficient

f Variable

e 58

f −15

g 9

h 1

g Constant

h Term

13 6.3 14 1.708 15 a 64

b 81

c 125

d −729

16 a 25

b 64

c 4

d 1

17 Yes

6.02 Add and subtract algebraic terms Understanding

2 a Like

b Unlike

c Unlike

d Like

e Unlike

f Unlike

g Like

h Like

i Unlike

j Like

k Unlike

l Like

3 a x

b y

Fluency

18 No

4 a 7a

b 16x

19 No

d 9y

e 13x

f 3p

20 No

g 0

h 14c

i 7x

21 a No

b Yes

c Yes

22 a 3y2

b 16y4

c 7y4

d Yes

Reasoning 23 8

c 7b

j b

k 12x

l x + 16

m 9n − 9m

n 6n

o p

p −3x

q −15m

r 9x

s −11x + 5

t 5xy

u 5ab

v 11u2

w −7x2

x −9pq

5 a 4x + 4y

b 15m + 19n

24 9

c 25m + 8n

d 8q

25 x = 12

e 3x + 19y

f 23j + 2k

Veriﬁcation:

g 5m + 12n

h 3x + 2y

i 18a − 8b

j 8x − 7y + 4z

k 11a − 7b + 4c

l 25x + 2y

m 12y

n 6s − 2t

12 − 2(3) = 6 12 − 6 = 6 6=6 26 Substitution is the process of replacing variables in an expression or equation with their corresponding values. To ﬁnd the value of the expression x2 − y2 when x = 5 and y = 3, we can substitute these values into the expression. x2 − y2 = (5)2 − (3)2 = 25 − 9 = 16 Problem-solving b 200 cm2

27 a

p 20p2 − 11p

o 21xy 2

q 3n − 4m + 2m

r 7x2 + x

s 25y − 9y2

t 3ab + 2c

6 a −3x − y + 10z

b −16x + 10y + 41

c 8x + 6y

d 7p + 7

e −12x − 4y

f −5n + 7

g −2n + 5

h −6xy + 2yz + 6xz

i 12x2 + 9x + 5

j 11mn − 2m

k 24x2 + 5x − 15

l st + 7s + 7t

m 19ab + 3bc − 9b

n 2pq + 11p2 q

28 6 °C

o 17x

p 20xyz + 23xy

29 1083 J

q −6rk + 3r − 6k

r −24vw + 6v2 w

30 a i 9.8 m

ii 0 m

iii −196 m

b 2s

744

Mathspace Victorian Curriculum – Year 8 mathspace.co

7 a Yes

b No

c Yes

d No

e No

f No

8 a Yes

b No

c No

d No

e Yes

f Yes

9 a No

b Yes

c No

d No

e Yes

f Yes

Fluency 6 a 60u

b 10x

c 33y

d −21u

e −72x

f 24u

g −6w

h −30y

b 6x2

c 4y2

7 a x2 2

e 16x y

Reasoning 10 Tom is correct as the sign on the left of the coeﬃcient tells us whether to add or subtract. −4x means we subtract 4x from 7x, not add. 11 In the expression 3x2 + 5x + 7x2 − 2x, the terms 3x2 and 7x2 are like terms because they both have the variable x raised to the power of 2. The terms 5x and − 2x are also like terms because they both have the variable x raised to the power of 1. 12 12ab + 7c − 8ab − 9c = 4ab − 2c −6c + 2ab + 4c + 6ab − 4ba = 4ab − 2c Therefore 12ab + 7c − 8ab − 9c = −6c + 2ab + 4c + 6ab − 4ba 13 Yes, if m = 1 and n = 1 the expression is true. Problem-solving 14 a (4 + 16n) cm

f 27b

h 16x2

8 a 72mn

b 24wz

c 54rs

d 50rs

e 8ab

f −21wz

g 40rs

h 42u2 v8

i 224p3 q3

j −60r8 s7

k −12p5 q5

l −10a4 b3

9 a 96rs

b 630wxy

c −90xy

d −80a3

e −80wxy

f −480hkrs

g 90x2y

h −40hijk

10 a x

b 3v

d xy

e 7q

g –6a

h –11

i −ac

j 12

p −4

t 10v4

v −3w5 b

d 6

e 2n

h 2

i −4

j 3r

k –2

l 11x

12 a 6

c 3x2

d 1

g 18x

h −4x

k 10x

l y

b (14 + 2q) m 11 a

c 15b m

d 24v cm

e 8a cm

f (22n − 1) units

g (27x + 10) units

h (23d − 12) units

15 11a cm 16 75a − 30b cm 17 a 10x − 4y

g 100a

d 15a2 2

b 9x − 5y

18 a 4c + 8m dollars

c 19x − 9y

b 2c + 16m dollars

j 15y

13 10x km

19 3p − 40 dollars 20 a b + 6 birds

i 33xy

b 5b + 20 birds

Reasoning

21 6l − 50 cm

14 2ab × 4cd = 2 × 1 × 2 × 4 × 3 × −1

6.03 Multiply and divide algebraic terms

Understanding

1 a 3xy

b 2x2 y

c −2x2 y2 d 5x2 y3

2 a True

b False

c True

d True

3 a 1

b 1

c 1

d 1

4 a 2

5 a

= −48

8abcd = 8 × 1 × 2 × 3 × −1 = −48

therefore 2ab × 4cd = 8abcd 15 When multiplying two algebraic terms, we need to multiply the coeﬃcients and combine like factors into a power. In this case: (3x2 y) × (−4yz2) = (3 × − 4) × x2 × ( y × y) × z2 = −12x2 y2 z2

Answers mathspace.co

745

16 When dividing algebraic terms, we need to divide the coeﬃcients and cancel any common factors. In this case:

17 A fraction represents a division. When dividing a number by 1 the answer is just the number so the 1 is unnecessary. However the 1 on the top of a fraction represents 1 part being divided by the bottom number, so it is important.

8 a 12

9 a

10 a 2 d 6

b 30t

c 3

e 2

f 3

g 12m

h 13v

i 15m + 10

j 9n2

k −m

l 18n2

Reasoning

Problem-solving

20 25a2 b2 cm2

11 In 12x − 5x × 2 if we do the multiplication ﬁrst we get the correct answer of 2x. If we do the subtraction ﬁrst the answer will be 14x which is incorrect. By following the order of operations, we ensure that we do not get two different answers to the same problem.

21 80y2 cm2

12 If n = 5,

18 a 42v mm2 d 25n2 cm2

b 6q m2

c 50q2 cm2

e 36q2 m2

f 16n2 m2

19 6n books

22 a

b 2

23 a 44WL cm3

b 132WLH cm3

c 3H

6n × 2 − (3n + 2n) = 30 × 2 − (15 + 10) = 60 − 25 = 35 Also, 7n = 7 × 5 = 35 Therefore, n × 2 − (3n + 2n) = 7n = 35

d 264

6.04 Simplify algebraic expressions Understanding 1 38 2 It is not written in simplest form as the two pairs of like terms can still be combined. 3 a 1

b 2

4 a

(10x − 2x) × 5 + (11x + 3x − x) × 2 = 8x × 5 + 13x × 2 Next, we simplify the multiplication in each term:

d 1

13 First, we can simplify the terms inside the parentheses:

8x × 5 + 13x × 2 = 40x + 26x Finally, we can combine the like terms:

d 4

40x + 26x = 66x So the simpliﬁed expression is 66x.

Fluency b 27c

c 5a

d 36a

Problem-solving

e 12w

f 18s

g 2u

h 24pq

14 a 56a2 mm3

5 a 34t i 12k

m 27n2

j 8p

k 6k

l 8n

n 22n2

o 13mn2

p 4n

6 a 9u

b 4a

c 21v

d 14v

16 a

e 3

f −45n

g 6 + 5r

h −15v

i −13t

j 8m2

k −43n

l 4

18 120x2 cm2

7 a 9

746

b (16a2 + 84a) mm2

Mathspace Victorian Curriculum – Year 8 mathspace.co

dollars

12 Total area = 4 (6 + x)

6.05 The distributive law

Area small rectangle = 4x Understanding

Area large rectangle = 24

1 a (b + c) = a × b + a × c 2 a 30

Total area equals Area small rectangle plus Area large rectangle

b 30

Therefore 4 (x + 6) = 4x + 24.

c 5 (12 − 6) = 5 × 12 − 5 × 6 3 a 288 units2

13 The student should multiply the second part of the sum, 2, by the number outside the brackets, 6 to get 12 as the second term on the right hand side.

b 32 units2

c 320 units2

d Yes

4 a(b + c + d) = ab + ac + ad 5 a km units2

b kn units2

c km + kn units

14 The 2x is added to the bracket, not multiplied. The correct answer should be 6x − 1.

d Yes

15 (x + 2) (x + 6) = x2 + 8x + 12

Fluency 6 a 5s + 35

b 50 + 10w

c 3r − 18

d 7y + 56

e 16s + 20

f pq + 4p

g 8r + 12s + 20t

h 12mn + 18m

i 9v2 + 54

j rs2 + 8r

16 LHS = 5 (3x − 4) = 15x − 20 RHS = 4x + 2 (5x − 5) + x − 10 = 4x + 10x − 10 + x − 10 = 15x − 20 LHS equals RHS therefore 5 (3x − 4) = 4x + 2 (5x − 5) + x − 10.

7 a −8t − 80

b −8c + 40

c −80 − 30r

d 16t + 24

e 9u − 9v

f −24m + 18n − 30p

= 800 + 56

g −11rs − 13rt

h −y − 2

= 856

i −2m + 3n

j 16pq − 12p + 24pr

k −10y2 − 70

l −30 − 80u3

m 16s5t4 − 36s5

n 12p3q5 − 54p3

o −9 + s3 2

8 a r + 7r

b 24m + 42m 2

c 4wy + 4w y

d 3uv + 3u v

e −12a2 − 54a

f 9f 2 − 36fg − 54fh

g 2f 2 − 14fg − 12fh 9 a 4x + 30

b 15x + 27

c 3x + 17

d 18x + 32

e 36x − 11

f −11x + 24

g −x − 15

h −40x + 47

i 10y − 49

j 20x − 52 b 13x − 16

c 4y + 44

d 10x − 33y

e −11x − 26

f 17a + 20

g x2 + 12x + 72

h 15y2 − 39y

i 14u + 20uv

j 62w2 + 34wz

Reasoning 11 a A = x2

18 (11m + 88) kW

20 (10 + 15n4) boxes 21 (16s2 t + 16st2) units2 22 (u2 + 8u) m2

6.06 Numeric factors Understanding 1 4

10 a 12y + 93

Problem-solving

19 (16r − 24) representatives

8 × 107 = 8 × 100 + 8 × 7

b x (x − 4)

2 5a + 15 = 5 (a + 3) 3 Yes, 2 (4 − x) = −2 (−4 + x). Numeric factors can be both positive and negative. 4 a 2

b 3

c 3

d 2

e 7

f 4

g 5

h 6

i 3

j −9

c 4x

d Area I plus Area II equals Area of Square. Therefore x (x − 4) + 4x = x2

Answers mathspace.co

747

Fluency 5 a 2

Problem-solving b 5

c 2

17 12 (2x + y)

d v+3

e a−4

f 4

g 5

h 6

i r+7

18 4 (2x + 3y)

j t−1 6 a 6 (v + 5)

b 6 (a − 1)

19 15 (x − 3) 20 10 (3x2 − 2)

c 25 (t + 1)

d 2 (c − 9)

21 5 (8x − y)

e 5 (8c − 7)

f 2 (m + 5n)

22 a 2

g 6 (w + 5y)

h 2 (7y + 2w)

i 2 (2a − 7b)

j 4 (8 − 9jk)

k 2 (4jkt − 9)

l −3 (g + 10fh)

7 2w − 10z + 8y = 2 (w − 5z + 4y) 8 a 4 (w + z + y)

d −2

23 a (b − 4) (k + 6)

b (x + 3) (m + n)

Chapter 6 review 1 a b

c −7x2

b ab

d 9x

b 4 (2g + 7h − 5)

2 a −4xy + 2yz + 8xz

b 10x + 8x + 5

c 4 (u − 3v + 5w)

d 5 (2m − 5n − 3p)

c 11mn − 2m

d −m −3n + 6

e 9 (w − y − k)

f 5 ( p − a − b)

g 3 (m − 2n + 4p)

h 5 (4g − 3h + 5)

i 5 (12 − 5rt − 7r2 t)

j 4 (3m − 8n − 5m2n)

k 8 (2z − 4yz − 3y2)

l 5 (5r2 − 3t2 − 9r2t2)

9 a −7 (r + s)

b −8 (x + 10y)

2 2

b 6a2b3

3 a −4a b

4 a 9 + y minutes b It represents that it takes Sally 9 minutes longer than the bus trip to get to school. 5 a 35ab

b 60pq

c −(4r + 3s)

d −(2 + 9r)

10 a −8 (b + 2)

b −6 (6 + x)

c −2 (6s − 5)

d −2 (9a − 8)

e −10 (8v + 3)

f −7 (4b + 3)

6 a 5u

d 15w3

g −4 (m + 8n)

h −7 (7y + 3w)

i −8 (v2 − 7)

j −7 (9r2 − 4)

e −7b

g −4t4

h 15

k − (10u2 − 9v2)

l −6 (2b − 3a2)

b 1

c a2

d 4

11 a −9 (2p + q + 5r)

b −5 (2r + 5s − 4t)

g 20ab

h b

c 21j

d −49k

c −5 (4u + 5v − 9w)

d −2 (4wy + 9y + 5z)

d −60s6t4

4 2

f −12r3s2

c 56v w

e −12u v

7 a 20 e 24a

f −3a

8 a 10m

e 28x Reasoning 12 −2x − 3 + 5y The sign of each term is reversed. Negative terms become positive terms and vice versa.

9 a 10 a 35q

13 The expression x2 − 5x + 6 cannot be factored with a numeric factor because the terms x2, −5x, and 6 do not have a common factor other than 1.

11 a 8p + 18

e 0

b 10k f 27w 50 3b 9 b b − 4 12 21r 2 34 f 15

3 2

d 10s2

b 10p + 21

c p + 10p + 56

d −9p − 19

14 Both 6x + 9 and 9x + 6 have the same common factor of 3.

e 14b + 12

f 11v2 + 17vw

12 a 8 (w + 4)

b 70 (1 − 2rs)

15 2 (x + 0) = 2x + 0. To factorise correctly the number inside the bracket should be 1. 2x + 2 = 2 (x + 1)

13 a 6 (a + b + c)

16 (x + 6)(x + 4)

748

Mathspace Victorian Curriculum – Year 8 mathspace.co

c 10 (mnp − 2) c 5 (s − 2t + 4u)

d −4 (h + 10qr) b 8 ( p − 3q + 4r) d 12 (x − y − z)

e 10 (8 − 3mn − 5m2n) f 10 (2w − 4xy − 3x2)

14 a −16

b 25b

16 a No

b No

15 a 4b

c −63

b 5.5

c b

Fluency

c Yes

d No

5 a i Subtract 5 from both sides ii y = 4

17 11

b i Add 2 to both sides ii t = 37

18 13b m

c i Divide both sides by 9 ii p = 4

19 8b − 70 mm 20 a 40q mm2

b 35s2 m2

d i Multiply both sides by 12 ii n = 132 e i Subtract 3 from both sides

21 12p pens

ii y = −11

f i Add 3 to both sides ii t = −42

23 (12n + 120) kW

g i Divide both sides by 5

24 (18 + 12m3) crates

ii p = −10

25 5(10x − y)

h i Multiply both sides by −9

26 1646.4 J

ii n = −36

27 25 km

6 a k=7

b n = −16

c r = 47

d k = −6

e n = 22

f p = 26

g p = −26

h t = 15

i q = 42

k t=8

l q = −20

b x = −2

c k=5

d w = −14

f s = 12

g s = −7

h t = −4

i m=7

b Divide both sides by 4

j w=2

k y = −14

l y = −17

c Multiply both sides by 5

8 a q = −1

b q = −9

c p = −1

d n = −19

e u=4

f u = −6

g r=6

h u = −5

7.01 Solve equations Understanding 1 a 7 − 9 = −2

b 8 × − 6 = −48

c −21 ÷ − 7 = 3

2 a Add 4 to both sides

d Subtract

to both sides

3 a i 4 (x + 6) ii Divide by 4 then subtract 6 b i −3x − 7 ii Add 7 then divide by −3 c i ii Multiply by 3 then add 9 d i ii Subtract 8 then multiply by 11 4 a Add x to both sides b Add a to both sides c Multiply both sides by p d Multiply both sides by b

7 a x = −2

b x = −5

d x = 56

e n = −14

f x = −21

g d = −21

h y = 12

10 a m = 23

b r = −18

c v = −28

c q = −6

9 a

d w = −16 11 a p = −5 d w=5 Reasoning 12 Step 2 should have been add 8 to both sides, not subtract from both sides.

Answers mathspace.co

749

13 Step 3 should have been multiply both sides by 3, not divide the right side by 3.

Fluency 5 a i

14 When we solve the equation we get x = x. This means x can be any number and therefore there are an inﬁnite number of solutions.

−1

−3

15 The equation 2x + 3 = 2x + 7 has no solution, because if you subtract 3 from both sides, you get 2x = 2x + 4, which is not true for any number.

6 5 4 3 2 1

−4 −3 −2 −1−1

−2 −3 −4 −5 −6

Problem-solving 16 The equation representing the situation is: 2(x + 4) + 2x = 36, where x is the width of the rectangle. Solving the equation, we get x = 7. Therefore, the width is 7 units, and the length is 11 units.

iii Linear b i

−1

17 ii

x = 25

10 9 8 7 6 5 4 3 2 1

18 The short sides are 5 cm long. 19 20 The equation representing the situation is: x + (x + 1) + (x + 2) = 42. Solving the equation, we get x = 13. Therefore, the three consecutive integers are 13, 14 and 15. 21 The equation representing the situation is: 4x + 20 = 2(x + 20), where x is the son’s age. Solving the equation, we get x = 10. Therefore, the son’s current age is 10 years, and the father’s current age is 40 years.

−3 −2

−1

x 1

−1 −2

iii Linear c i

−1

y 3

7.02 Tables of values

Understanding

1 Answers may look like:

x −2

Start at the origin (0, 0) then move 5 units to the right and then move 2 units down. 2

−1

−2

3 No c Non-linear

750

−1 −2

4 a Linear

−1

iii Linear d i

b Linear d Non-linear

Mathspace Victorian Curriculum – Year 8 mathspace.co

−1

−7

−15

−1 −2 −4 −6 −8 −10 −12 −14 −16

16 14 12

x 1

8 6 4 2 −2 −1

−2

−8

−4

y 4 3 2 1

x 1

−8 −6 −4 −2 −1

2 4 6 8

−2

−3 −4

b Yes

c 1

9 a

9 8 7 6 5 4 3 2 1

iii Linear

c 16

8 a

5 y 4 3 2 1 −3 −2 −1−1 −2 −3 −4 −5 −6 −7 −8 −9

f i

x 1

b Yes

iii Linear e i

7 a

10 y 8 6 4 2

−4 −3 −2 −1 −1

x 1

2 3 4

−2 −3

b Yes

10 a

x 1 2 3 4 5 6 7 8 9 −1

c 6

Time (t weeks)

Height of pot plant ( y cm)

y 16 14

iii Non-linear 6 a Yes

b Yes

c No

d Yes

e Yes

f Yes

g No

h Yes

i Yes

j No

k No

l No

m No

n No

12 10 8 6 4 2

t 1

c Linear

Answers mathspace.co

751

11 a i

ii The y-value is equal to negative four times the x-value plus two.

y 30

iii y = −4x + 2

12 a i y = −3x + 2

ii Verify by substitution:

y(−1) = −3(−1) + 2 = 5 y(0) = −3(0) + 2 = 2

6 x 1

y(2) = −3(2) + 2 = −4

ii y = 6x b i

12 11 10 9 8 7 6 5 4 3 2 1

y(1) = −3(1) + 2 = −1 y(3) = −3(3) + 2 = −7

b i y = 3x + 4 ii Verify by substitution: y(1) = 3(1) + 4 = 7 y(2) = 3(2) + 4 = 10 y(3) = 3(3) + 4 = 13 y(4) = 3(4) + 4 = 16 y(5) = 3(5) + 4 = 19 x 1

c i y = 5x + 4 ii Verify by substitution:

ii The y-value is equal to the x-value plus seven.

y(−8) = 5(−8) + 4 = −36

iii y = x + 7

y(−6) = 5(−6) + 4 = −26

c i

x −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11

752

y(−4) = 5(−4) + 4 = −16 d i y = −5x + 3 y(3) = −5(3) + 3 = −12 y(4) = −5(4) + 3 = −17 y(5) = −5(5) + 3 = −22 y(6) = −5(6) + 3 = −27 y(7) = −5(7) + 3 = −32

iii y = −x − 6 −2 −4 −6 −8 −10 −12 −14 −16 −18

y(−5) = 5(−5) + 4 = −21

ii Verify by substitution:

ii The y-value is equal to the x-value multiplied by −1 and then minus six. d i

y(−7) = 5(−7) + 4 = −31

Reasoning 13 Answers may look like:

5 x

Mathspace Victorian Curriculum – Year 8 mathspace.co

Choose any two points from the set and draw a straight line through them. If all the other points in the set also lie on this straight line, it indicates that the set of points forms a linear relationship. If the difference between the consecutive y-values is constant, then the relationship is linear. 14 No, because the two points with x-coordinate of 2 are in a vertical line, and the other point with x = 4 is not on the same vertical line. Therefore, they do not lie in a straight line.

15 If the points form a linear relationship then any two points will result in a straight line passing through all the points. That means any graph where this does not happen would be considered non-linear. One possible answer:

y 2 1 x 2

10 9 8 7 6 5 4 3 2 1 −3 −2

−1 −2

d i x

−1 −1

−5

−2

y 2 1

Problem-solving 16 a i

−2

−4

−2

−1

−2 −3 −4

−5

2 1

−3 −2 −1

−1

5.50

8.50

11.50

−2 30

−3

−4

b i

−2

10 5

y 5

2 4 6 8 10 12 14 16 18

18 a

3 2 1 −2

−1

x 1

−1

Time (t min)

Height of candle ( y cm)

−2

4 2

c i

−2

−8 −6 −4 −2 −2

t 2

−4

−6 −8

c 0≤t≤4

Answers mathspace.co

753

19 a

Week (W)

Account $300 $310 $320 $330 $340 total (A)

20 a

Number of triangles

Matchsticks

c x=2

d x = −3

e x=0

f x=2

h x=2

12 a i (0, −4)

ii (2, 0) 2

−2

−1

x 1

−1 −2

Figure number

−4

No. matchsticks

−5 −6

b i (0, 2)

7.03 Linear graphs

ii (1, 0)

iii

y 4

Understanding 1 a (7, 0), (0, −5)

b (0, 5), (−9, 0)

2 The gradient is 0.

Gradient is calculated as

. In the case of a

−2

1 −1

3 The gradient is undeﬁned.

−2

Gradient is calculated as

. In the case of a

vertical line the run is 0. We cannot divide by 0.

−1

horizontal line the rise is 0.

c i (0, −6)

ii (2, 0)

iii

4 a All of the points have x-coordinates of 3.

b A vertical line

c x=3

5 a All of the points have y-coordinates equal to 2.

−8 −6 −4 −2 −2

x 2

−4

b A horizontal line

−6

c y=2

−8

6 The line is parallel to the x-axis.

d i (0, 8)

7 The line is parallel to the y-axis. 8 a y-axis

ii (2, 0)

iii

9 8 7 6 5 4 3 2 1

b x-axis

Fluency 9 a i x=0

ii y = 0

b i x=2

ii y = 3

c i x=0

ii y = 0

d i x = 4.5

ii y = −3.5

−2

10 a Yes

b Yes

c No

d No

e Yes

f No

g Yes

h Yes

754

−3

c M=F+4

b Yes

c M = 3T

b Yes 21 a

b x = −1

iii

c A = 300 + 10W

b Yes

11 a x = 1

Mathspace Victorian Curriculum – Year 8 mathspace.co

−1

−1 −2

x 1

13 a (0, 0) d

11 10 9 8 7 6 5 4 3 2 1 −1

c y = 10

b (0, 0)

17 a

3 2 1 −4 −3 −2 −1

x 1

−1

−4

b ii c = 9

b i m=5

ii c = −6

c i m = −5

ii c = 8

d i m = −4

ii c = −2

e i m=4

f i m = −2

g i m=5

ii c = 4

h i

ii c = 3

5 4 3 2 1

x 1 2 3 4 5

−5 −4 −3 −2 −1 −1 −2 −3 −4 −5

15 a (0, −5)

1 2 3 4 5

y = 4x + 8

2 1

−4 −3 −2 −1

−3

14 a i m = 4

16 a

−2

−1 −2

−2

−1

−3

−1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

−1 −2 −3 −4 −5 −6 −7

x 1 2 3 4 5

x 1

18 y = 2 + 7x, y = 7x, y = 7x − 2 19 a x = 8

b y=3

c x = −4

d x=0

20 a Horizontal

b Vertical

c Horizontal

d Vertical

e Horizontal

f Vertical

g Horizontal

h Vertical

Answers mathspace.co

755

21 a i

4 3

1 −4 −3 −2 −1

y 4

x 1

−1

−2 −1

x 1

−1

2 3 4 5 6

−2

−2 −3

−3

−4

ii x = 2

b i

8 7 6 5 4 3 2 1

3 2 1 −3 −2 −1

x 1

−1

−2 −3

−6 −5 −4 −3 −2 −1 −1 −2

−4

1 2 3 4

23 a x = −2

b No

c x = −2

24 a y = 3

b No

c y=3

25 a y = −5

b x = −9

26 a (0, 0)

ii y = 5 22 a

−2 −1

x 1

d y = −4

b (4, −5)

−4 −3 −2 −1

c x=3

−1

x 1

−2

−3 −4

−5

−6

4 3 2

Reasoning

1 −2 −1 −1

x 1 2 3 4 5 6 7 8 9

−2

27 a y = 3

b y=c

28 a The constant term of 4 is the same. b All the graphs intersect the y-axis at the point (0, 4). c The line will pass through (0, 4). 29 There is no constant term as the value of c is zero.

756

Mathspace Victorian Curriculum – Year 8 mathspace.co

30 a y = 4

b y=6

c 2

d It is the coeﬃcient (number in front) of x.

280 260 240 220 200 180 160 140 120 100 80 60 40 20

31 a The coeﬃcient of x is the same, 4. b All the graphs have the same gradient of 4. c The line will have gradient of 4. 32 a All of the points have a y-coordinate of 3. b Since all of the points have a y-coordinate of 3, we call the line y = 3. c y=q

d y=0

Leaking water tank y, litres

x, hours 5

33 a All of the points have an x-coordinate of −3. b Since all of the points have an x-coordinate of −3, we call the line x = −3. c x=p

d x=0

Problem-solving 34 a

−1

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

h 1

37 a Edin’s savings: y = 100x Marius’s savings: y = 80x + 60 b The y-intercept of Edin’s savings equation is 0. This means she didn’t start with any money before saving. The y-intercept of Marius’s savings equation is 60. This means he started with $60 before saving each month. c

Edin and Marius save money y, dollars 500 400 300 200 100 x, months since January

b T = −9 −2 −1

35 a x-intercept: (−125, 0) y-intercept: (0, 25) b

d Edin

Car rental y, dollars 100 90 80 70 60 50 40 30 20 10 x, kilometres −100 −50

100

c If you drove −125 kilometres, it would cost $0 kilometres 36 a 260 litres b After 26 hours

1 2 3 4 5 6 7 8 9 10

b ( p, q)

38 a (−3, 7)

c The origin, coordinates(0, 0) 39 22 units 40 x = −1, x = 5, y = −2, and y = 4. 41 a

y 30 25 20 15 10 5 x 1

b x = 4.5

Answers mathspace.co

757

c He would intersect the line going from 1 to 2 and the line going from 2 to 3. d He needs some way to indicate which horizontal line exact integer values belong to.

7.04 Multi-step equations Understanding 1 a 1 step b 2 steps c 3 steps

9 a x = 16

b x=3

c x=3

d x=9

g x = 19

h x = 25

10 a x = 6

b x = 10

c x=0

d x = 60

e x = 80

f x = 30

g x=4

h x = −12

11 a x = 2

b x = 37

c x = 50

d x = 125

12 a x = 5

b x = −9

c x=5

d x = −3

e x = −4

f x = −3

13 a k = 2

b k=4

c r=6

d r=6

Reasoning

2 a Multiply by 3 then subtract 4 then divide by 2 b Multiply by 2 then add 4 then divide by 3 3 a Substitute the value for x into the equation

14 T he expression 9(4x − 6) was built by multiplying x, subtracting 6 then multiplying by 9. So the equation can be solved by: 1. Dividing by 9

b James is correct

2. Adding 6 Fluency

3. Dividing 4

4 a i x=3

b i

x=4

The first step is said by Linda so she is correct while Ryan is incorrect.

c i x=3

d i

15 a

13(2x − 8) = −26

26x − 104 = −26

26x = 78 x=3

x=4

13(2x − 8) = −26

2x − 8 = −2

2x = 6

x=3

c Method 2 is easier because it makes the numbers smaller. 5 a s = −2

b t=5

c x = 13

d x = −9

e x=3

h x = −1

i x=6

j g=5

k x = −4

l x=0

6 a w=2

b x=9

c x=8

d x = 19

e g=6

f q = −4

g x=3

h x = −2

i x = 13

j x = −6

7 a y = −21

b t = 36

c x=6

d c = −6

e x=5

f x = −8

g x = −14

h c = −12

i x = 16

j x = −24 c x=8

d x = 24

g x = −7

h x=8

8 a x = −22 b x = 6

758

e x=7

f x = −4

i t=5

j r=8

Mathspace Victorian Curriculum – Year 8 mathspace.co

16 a

b This is a contradiction because −2 is not equal to 5. Therefore, the equation has no solutions. 17 Yes, her solution is reasonable but it can be improved by thinking 40 ÷ 5 = 8 therefore, x + 1 = 5 so x is approximately 4. Problem-solving x+1 18 a +5 = 7 2

b x=3

19 x + 3 = 16 and 4x = 52

7 a i

20 x = 5

The lifespan of the mouse is x −3, the lifespan of the hedgehog is x, and the lifespan of the deer is 2 (x − 3).

2 1 −4 −3 −2 −1

The total of the three animals’ lifespans is 11 years, so x − 3 + x + 2 (x − 3) = 11. Expanding and simplifying, x − 3 + x + 2x − 6 = 11. So, 4x − 9 = 11. Adding 9 to both sides gives 4x = 20 and dividing both sides by 4 we have x = 5.

b i

6 5 4 3 2 1 −2

−1 −1

x = 15

8 7 6 5 4 3 2

Understanding 1 Answers may look like: (2, 5) is the point of intersection is where two lines cross each other. Two unique lines do not intersect each other on a linear graph when they have the same gradient and are parallel to each other.

5 4 3 2 1

c (−5, −6) d (7, 8)

b x=4

6 a (4, −12)

b x=4

x 1

ii (2, 4)

Fluency h (−3, −5)

−4 −3 −2 −1 −1

d i

3 The lines intersect.

5 a (4, 8)

2 Answers may look like:

g (5, −6)

ii (3, 5) c i

7.05 Intersecting lines

j (8, 0)

−2 −3 −4 −5 −6

So, Tom initially has 15 books, and Sarah initially has 3 × 15 = 45 books.

f (6, 8)

ii (−3, 3)

2x = 30

i (0, 4)

−3

3x − 15 = x + 15

e (2, −8)

−1

−4

Solving for x, we get:

b (−2, 3)

x 1

−2

21 Let x represent the number of books Tom has initially. Then, Sarah has 3x books. If Sarah gives away 15 books to Tom, they will have the same number of books, so:

4 a (2, 4)

−4 −3 −2 −1

−1 −2 −3 −4 −5

ii (2, 0)

Answers mathspace.co

759

e i

i i

y 2

y 12

x 1

−2

−4

−6

−8

−10

−1 −2

−12

x 1

2 3 4 5 6 7 8

ii (2, 3)

ii (2, −6) f i

j i

5 4 2

−5

x 1

−1

−7

−9

−2

ii (−1, −6)

ii (2, 4) g i

k i

3 1 −4 −3 −2 −1

x 1

−1

−2 −3 −4

l i

7 6 5 4 3 2 1

16 14 12 10 8 6

−4 −3 −2 −1

−1 −2

4 2 −4 −3 −2 −1

y x 1 2 3 4 5

ii (2, −3)

ii (2, −1) h i

2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5 −6 −7 −8

x 1

ii (2, 16)

760

−3

−4 −3 −2 −1

−7 −5 −3 −1 1 −1

Mathspace Victorian Curriculum – Year 8 mathspace.co

ii (2, 6) 8 a y = 10

b x=9

9 a

−1

y = 4x − 3

−6 −4 −2 −2

2 1 −4 −3 −2 −1

y 6

8 10

y = 4 − 3x

−4

−1

−6

−2

d x = 1, y = 1

−3 −4

12 a

c (2, −1)

3 2

−4 −3 −2 −1 −1

1 2 3 4 5 6 7

−2

10 a

−8y = 8x + 8

−3

−1

−4

b x = −1, y = 0

13 a B and D b

13 y 12 11 10 9 8 7 6 5 4 3 2 1 −4 −3 −2 −1−1

7 6 5 4 3 2 1

y = 2x + 2

−6 −4 −2

−1 −2 −3 −4

x 1

4 6 y = −2x + 2

c x = 0, y = 2

d (2, 5)

14 a i

3 2

11 a

y = −4x − 1 −4 −3 −2 −1

−3

−2

−1

−15

−11

−7

−3

−2 −4

−1

y = −4x + 2 1

x 4

−3

−2

−1

−2

−5

ii No

Answers mathspace.co

761

b i

of x is the x-coordinate of the point on the graph where the lines will intersect.

4x −2y = 2

−7 −5 −3

17 If the gradients are different, the lines will intersect, because they have different steepness and will eventually cross each other. If the gradients are the same, the lines are parallel and won’t intersect, unless they have the same y-intercepts.If they have the same y-intercepts, it means the lines are on top of each other, and they actually intersect at every point.

−2x + 4y = 2 x

1 −1 −1

−3

ii x = 1, y = 1 c i

9 8 7 6 5 4 3 2 1 −1

Problem-solving

y = 5x − 7

18 The point where the two mobile phone plan graphs intersect is important because it shows when both plans cost the same. This helps a consumer decide which plan is cheaper for their speciﬁc usage needs.

y = −x + 5 x 1

−1

19 a The two lines intersect when the bakery has produced the same number of loaves. On the graph, it’s the point where the lines cross each other, at (5, 25). So after 5 hours, there are 25 whole wheat loaves and 25 sourdough loaves.

ii x = 2, y = 3 d i

y 3 2

−3 −2

−1

x 1

−1

b After 6 hours, the bakery has made 28 whole wheat loaves and 29 sourdough loaves.

y=x

c Sourdough bread is produced at a faster rate because its line is steeper on the graph even though it started out with less loaves than the whole wheat, meaning more loaves are made per hour.

y = −1

−2 −3

20 C onstruct a table of values for each service and then plot the points on a graph, drawing a straight line through each graph.

ii x = −1, y = −1 Reasoning

Taxi:

15 a y = 8 b x = −4 c The solution to the equation

gives

us the x-coordinate at which the line crosses the line y = 8. This point is the intersection point because it is where both lines have the same x- and y-coordinates, meaning they share the same point on the graph. 16

. This equation represents the intersection of the two lines because the value

Rideshare:

Cost

Taxi vs. rideshare 14 y, cost 13 12 11 10 9 8 7 6 5 4 Rideshare 3 2 1 1

762

Mathspace Victorian Curriculum – Year 8 mathspace.co

Taxi

x, km 2 3 4 5 6 7 8 9

The cost of both services will be the same at 7.5 kilometres.

5 4 3 2 1

7.06 Inequalities Understanding 1 a <

b ≤

c >

d ≥

2 a True

b False

c True

d True

3 a y≥8

b m ≤ 22

c y > −10

d k<0

8 a x≥6

4 Right side Fluency 5 a x≥2 6 a x≥2 7 a

b x < −1 b y≤3 5 4 3 2 1

5 4 3 2 1

5 4 3 2 1 −5 −4 −3 −2 −1 −1 −2 −3 −4 −5

c y < −2

x 1 2 3 4 5

b y<5

c n ≤ −1

d m ≥ 12

e x ≥ 10

f x > 19

g x≤4

h x ≥ −4

d x<4

9 a n > 56

b x ≤ 24

c a < 66

d x > −5

d x > −3.5

e x<9

f p≥7

g x ≤ −6

h x ≥ −4

10 a x ≥ 14

b x≥0

c x > −5

d x≤4

e x≤0

f x < −2

11 a x ≤ 16

b x ≥ −24

c x > 11

d x≥4

e x < 19

f x ≥ 10

g x > 40

h x > 27

i x ≥ −12

j 12 a x ≤ 6 b i Yes

ii Yes

iii Yes

iv No

13 a i x < 4 ii −10 x

−5 − 5− −4 4− −3 3 −2 −1 −1 −2 −3 −4 −5

c x > −4

−5 − 5− −4 4− −3 3− −2 2− −11 − −11 − 2 −2 − −3 3 − −4 4 − 5 −5

−5 − 5− −4 4− −3 3− −2 2 −1 −1 −11 − −2 −2 −3 − 3 −4 −4 −5 5

1 2 3 4 5

−5

b i x>1 ii −10

c i x<4 ii −10

−5

d i x > −3 ii −10 x 1 2 3 4 5

e i x>1 ii −10

f i x≥8 ii 0

Answers mathspace.co

763

g i x ≥ −8 ii

−20

Problem-solving −15

−10

−5

24 W ≤ 8 25 Inﬁnite number of solutions

14 x ≥ 3 15 a p ≤ 5

b p=5

16 a x ≥ −3

b x<5

26 a h ≥ 61.4

b 62

27 a 5 030 kg

b No

28 a f > 2m b i No

ii Yes

iv No

29 a x > 2, x < 6, y > 5, y < 8

Reasoning 18 T he graph on the left shows x ≥ 2 and the graph on the right shows x > 2. The solid line shows that points on the line are included in the required region, the dashed line indicates that they are not. 19 a i x < 2, y < 3

ii No

b i x ≤ 2, y < 3

ii No

c i x < 2, y ≤ 3

ii No

d i x ≤ 2, y ≤ 3

ii Yes

20 Step 2 is incorrect. At this step she incorrectly combined −3 and −2. 21 0 ≤ x ≤ 4 includes both end points of 0 and 4 whereas 0 < x < 4 does not include the numbers 0 and 4. 22 The minimum should be at −9 instead of −8 and the dot at that point should be ﬁlled as −9 is included in the solution. 23 a 40 ≥ 22.10 + 2w

b (3, 6), (4, 6), (5, 6), (3, 7), (4, 7), (5, 7) c i 5 table A and 7 table B ii $550 30 a x ≥ 3, y ≥ 5, y ≤ 9 − 0.5x b (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7), (5, 5), (5, 6), (6, 5), (6, 6), (7, 5), (8, 5) c i T he answer is not unique, can choose 4 shade P and 7 shade Q, or 6 shade P and 6 shade Q, or 8 shade P and 5 shade Q. ii $4500

Chapter 7 review 1 a i Subtract 27 from both sides ii u = −25 b i Add 2 to both sides ii t = 37 c i Divide both sides by −7 ii j = −8 d i Multiply both sides by 3

b w ≥ 8.95

ii r = −15

c i N o, washing exactly 9 windows would generate $18, which with the $22.10 he already has is enough money to afford the equipment.

ii y = −11

ii No, washing 8 windows would only generate $16 which would give him a total of $38.10, which is not enough money to afford the equipment.

ii n = −5

iii No, he gets paid for washing the whole window, not parts. iv Yes, if Ryan washed 9 windows he would earn enough to buy the equipment. If he washed more than 9 windows, he would have more than enough money.

764

iii No

Mathspace Victorian Curriculum – Year 8 mathspace.co

e i Subtract 3 from both sides f i Add 3 to both sides ii t = −42 g i Divide both sides by 7 h i Multiply both sides by −9 ii n = −54 2 a q = −1

b q = −9

c p=1

d r = −2

e u = −4

f u=4

3 a i

−1

−2

6 5 4 3 2 1

y 2 1

−4 −3 −2 −1−1

−2 −3 −4 −5 −6

1 −1

iii Linear b i

iii Non-linear

−1

−2

10 9 8 7 6 5 4 3 2 1 −3 −2

4 a No

b Yes

c Yes

d Yes

e No

f No

5 a

Week (W)

x 1

iii Linear −1

ii y = 3 ii y = 0

7 a x=0

b x = −3

c x=2

d x=2

1 −1

−1

x 1

−1 −2

9 a None

−2

c y=8

b (0, 0) 11 10 9 8 7 6 5 4 3 2 1

$300 $310 $320 $330 $340

b i x=0

6 a i x=2

−1

c A = 300 + 10W

8 a (0, 0) x

−2

b Yes

−1 −1

Account total (A)

−2

c i

2 3 4 5 6 7 8 9

c y=3

b Yes

10 a x-intercept: (−125, 0) y-intercept: (0, 25) iii Linear d i

Car rental y, dollars 100 90 80 70 60 50 40 30 20 10 x, kilometres −100 −50

100

Answers mathspace.co

765

c If you drove −125 kilometres, it would cost $0 11 a x = −12 d

e x=0

g g=6

h x = −2

i q = −5

j x=8

k x = 24

l c = −6

10 3 p x=6

n c = −20

o x = 16

q x=0

r x = 64

m x=

12 a

13(2x − 8) = −26

26x − 104 = −26

26x = 78

x=3

13(2x − 8) = −26

2x − 8 = −2

2x = 6

x=3

b i

−4 −3 −2 −1

−1

8 6 4 2

−1 −2

2 3 4 5 6 7 8

ii (2, 3)

c (5, −6)

d (−3, −5)

e (0, 4)

f (8, 0)

7 6 5 4 3 2 1

−4 −3 −2 −1

−1 −2

b (6, 8)

17 a

3 2 1 −1

x 1

ii (2, 6) 16 a y = 13

−4 −3 −2 −1

This results in a contradiction because 3 is not equal to −7. Consequently, the equation does not have any solutions.

d i

15 a i

b x=6

−1

−3

ii (−1, 0)

−4 −3 −2 −1

−1

−2 −3 −4

Mathspace Victorian Curriculum – Year 8 mathspace.co

−2 −4

766

x 1

ii (2, 4) c i

c Method 2 is easier because it makes the numbers smaller.

14 a (7, 8)

8 7 6 5 4 3 2 1

x 1

c (2, −1)

18 The point where the two internet packages graphs intersect is important because it shows when both packages cost the same. This information can guide a customer in choosing the most cost-effective package based on their speciﬁc internet usage requirements. 19 a y ≤ 3

b x > −5

20 a x ≥ 8

b y<3

c x > 21

d x≤6

e x ≥ −2

f x ≤ −8

g x ≥ −4

h n > 84

i x ≤ 48

j a < 156 k x > −5.6

21 a x > 0 b i No

ii Yes

iii Yes

iv Yes

22 23 a f > 2m b i No

ii Yes

iii No

iv No

7 a Supplementary

b Complementary

c Supplementary

d Complementary

e Supplementary

f Complementary

g Supplementary

h Supplementary

8 a x = 49

b x = 42

c x = 47

9 a x = 96

b x = 82

c x = 40

10 a x = 25

b x = 20

c x = 60

d y = 72

e z = 10

f x = 40

g x = 130

h x = 55

11 a x = 30

b x = 37

c x = 49

d x = 44

e x=5

f x = 24

12 a x = 72

b x = 68

13 a x = 73

b y = 43

14 a a = 61

b b = 29

15 a x = 61

b m = 138

16 a Yes

b Yes

e No

f Yes

c No

d No

c BC

d ∠CAB

17 a 180° b The longest side c The shortest side

8.01 Angles and triangles

18 a BC Understanding

b ∠BCA

19 AC, AB, BC

1 360° 2 a 3

d x = 11

b 1

Reasoning

c 2

3 a If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. b If one angle of a triangle is larger than another angle, then the side opposite the smaller angle is smaller than the side opposite the larger angle. Fluency

20 The lengths cannot make a valid triangle because 4 + 7 is not greater than the third side, 12. 21 The sum of the angle measures exceeds 180°. 22 Since we know the hypotenuse is the longest side, this means that the right angle is larger than the other two angles. Since these two angles are smaller than the right angle, they are less than 90°, so they are acute. Problem-solving

4 a ∠BXC and ∠CXD or ∠AXF and ∠EXF b ∠BXC and ∠EXF c ∠AXF and ∠EXF or ∠AXF and ∠BXC 5 a Yes

b No

c No

d No

6 a Yes

b No

c No

d Yes

23 a x = 70

b Acute triangle

24 a Equilateral

b a = 60

c b = 13

25 a x = 8

b 68°

c 54°

d 58°

e BC

26 7 < x < 11

Answers mathspace.co

767

8.02 Transformations and congruence Understanding 1 a Rigid transformation

ﬁgure, the new ﬁgure will have identical angle measures and side lengths to the original shape. 21 Triangles ABC and DEF have one pair of congruent angles, and two pairs of congruent sides but they are not congruent.

b Non-rigid transformation c Non-rigid transformation

d Rigid transformation 2 When a shape is transformed through a rigid transformation, the angle and side lengths of the original shape and new image are still congruent. Fluency 3 a Translation

b Rotation

c Reﬂection

4 Translation or reﬂection 5 A translation of 3 units right and 2 units up 6 A reﬂection across the y-axis 7 A rotation of 270° anticlockwise around the origin, or equivalently, a rotation of 90° clockwise. 8 ∠QPR

Problem-solving 24 a Isosceles b i No

ii Yes

iii No

ii Yes

iii No

c Yes

10 ∠ACB

25 a Isosceles

11 ∠QRP

b i No

12 a Translation right and down

b ∠GJH

13 a Rotation and translation

b DF

14 a i 14

iii 10

ii 12

b △TUV congruent to △KLM b ∠EFD

c 14 cm

d ∠BCA

c Yes d Congruent 26 p = 8 27 Farah’s stride length is 68.4 centimetres. 28 a Scalene b i Yes

ii No

16 x = 8

c No

17 z = 4

d Not congruent

18 y = 5

iii No

8.03 Congruent triangles b x = 41°

c x=7

d x=3

Reasoning 20 These three types of transformations are all rigid transformations, meaning that they preserve segment lengths and angle measures. So if we use one or more of these transformations on a ﬁgure to produce a new

768

23 True, if the line of reﬂection overlaps the shape then the original and translated shapes will overlap each other.

d Equal

19 a x = 11.6

22 True, when the line of reﬂection is a line of symmetry the translated and original shape will look the same.

9 ∠NOP

15 a EF

Mathspace Victorian Curriculum – Year 8 mathspace.co

Understanding 1 All three pairs are corresponding sides are equal in length, and all three pairs of corresponding angles are equal in measure. 2 a Yes

b No

c No

d Yes

3 a AAS

b SSS

c ASA

d RHS

e SAS

Fluency

Reasoning

4 a Yes, they satisfy RHS.

20 If two pairs of angles of two triangles are equal, then the third pair of angles must be equal as well because the angle sum of any triangle is 180°. So if we know two angles and the included side are equal to the corresponding parts of another triangle, then the third angle is also equal and we have the information required to show Angle-Angle-Side.

b Unknown, there is not enough information. c Yes, they satisfy AAS. d No, they are not congruent. e Yes, they satisfy SAS. f Yes, they satisfy RHS. g No, they are not congruent. h Yes, they satisfy SSS.

21 Two triangles can have equal angles but have different side lengths. Consider these two equilateral triangles (where all angles are 60°):

i Yes, they satisfy AAS. j Yes, they satisfy SSS. 5 a Triangle A and B

b AAS

6 a Triangle B and C

b RHS

7 a Triangle A and C

b SSS

8 a Triangle A and B

b AAS

9 a Triangle B and C

b SAS

10 a Triangle A and C

b RHS

11 a Triangle A and C

b SAS

12 a Triangle A and D, Triangle B and C b ASA and AAS

60° 60°

1 cm 60°

8 cm 60°

60°

1 cm

b SSS

15 a △JKL and △JMN

b AAS

60° 8 cm

22 Yes. The intersecting lines form a pair of vertically opposite angles which are equal. Because the segments bisect each other, the triangles will be congruent by side-angle-side congruence (SAS) no matter how long the segments are.

13 SAS 14 a △ ACD and △BCE

8 cm

1 cm

16 a △PRN and △QRM b SSS and SAS 17 a SAS b EF = EH Given ∠FEG = ∠HED Vertically opposite angles

EG = ED Given 18 a SAS b PS = RQ Given QS is common ∠PQS = ∠RSQ Alternate angles on parallel lines

23 Consider the triangles shown in the diagram. With the given information there are two possible triangles, one of which is not congruent. So we do not have enough information to conclude that the triangles are congruent. B

19 a AAS b Example Answer: BX = CX Given

∠ABX = ∠DCX Alternate angles on parallel lines

C Q

∠BXA = ∠CXD Vertically opposite angles P

Answers mathspace.co

769

Problem-solving 24 a ∠QRP = ∠SRP

c b QR = RS

7 2

c ∠RPQ = ∠RPS

25 a JN = MN b Possible Answers:

c Possible Answers:

∠JKN = ∠MLN ∠KJN = ∠LMN 26 a They are all radii of the circle. b PQ = SR

7 a Yes

b Yes

Scale factor = 2 Scale factor = c No

c ∠QXP = ∠SXR

27 a SAS

d Yes

Scale factor = 8 a i Similar shapes

b ∠AXB = ∠DXC Vertically opposite angles

ii Dilation

b i Congruent shapes

ii Rotation

BX = CX Radii of the same circle

c i Congruent shapes

ii Reﬂection

AX = DX Radii of the same circle

d i Similar shapes

ii Dilation

28 a i AAS or RHS

ii x = 3

b i SSS or RHS

ii x = 9

c i SAS or RHS

ii x = 11

d i SSS or RHS

ii x = 7

9 a i ii Reﬂection and dilation by a scale factor of b i 3

8.04 Transformations and similarity

c i 3

Understanding 1 • Corresponding sides must be proportional to each other. • Corresponding angles must be equal. 2 Congruent shapes have equal side lengths, but the side lengths of similar shapes are proportional. 3 a False

b True

c True

e True

f True

g False

d False

4 a Congruence

b Congruence

c Similarity

d Congruence

5 a Any rigid transformation including a reflection, a translation, or a rotation b A dilation or any rigid transformation, including a reflection, a translation, or a rotation c Any rigid transformation including a reflection, a translation, or a rotation

ii Rotation, dilation by a factor of 3, and a translation OR a dilation by a factor of 3 and a horizontal and vertical reﬂection. d i ii Rotation and dilation by a scale factor of 10 a N

b Each pair of angles must be equal.

L M c MN

d ML

11 a Yes, the scale factor is 3. b No 12 a Scale factor =

b Scale factor = 3

c Scale factor =

d Scale factor = 6

13 a LN 14 a i ∠Q

b LM ii ∠R

b PQ

Fluency 6 a

ii Dilation by a scale factor of 3 and a translation

15 a Similar

iii ∠P c PR

b EF

c ∠EHG

16 y = 17.5 17 a x = 30

b x = 26

18 a 2 b i s = 14

770

Mathspace Victorian Curriculum – Year 8 mathspace.co

ii n = 122

iii m = 44

b x = 20

19 a 4 20 a

Reasoning 21 No, because even though all sides are in the same ratio, the matching angles are not equal. 22 a The two hexagons are each equilateral giving them proportional sides, and equiangular giving them equal angles. b All pairs of angles in the two triangles are equal and all pairs of sides are in the same proportion. 23 Yes, the conditions for similarity are equal angles and proportional sides. All corresponding sides and angles will be equal. All corresponding sides will have a scale factor of 1 which makes them proportional. 24 Not always, similar shapes are a scaled version of each other. Only if the scale factor is 1 are they also congruent.

ii SSS

d i A and C

ii SAS

e i B and C

ii SAS

f i C and D

ii RHS

g i B and D

ii AAA

5 a i SAS similarity test ii ∠LV M = ∠UV T and

and

b i SAS similarity test ii ∠U = ∠F and

and

c i AAA similarity test ii ∠A = ∠F and ∠B = ∠C (corresponding angles) d i SSS similarity test ii

and

6 a SAS

b 3

7 a g = 30, f = 5

b x = 42, y = 20

c a = 9, b = 66

Problem-solving

d x = 4, y = 5

8 a △JMN and △JKL

25 3 cm

b Angle-angle-angle similarity

26 2 m

c MN = 5

27 The dimensions of the smaller image would be 222 pixels by 103 pixels as we can’t have half a pixel.

9 ∠ABC is equal to ∠DEF 10 a ∠EDC, corresponding angles b ∠EBA, corresponding angles

8.05 Similar triangles

c AAA similarity 11 a ∠MLN, vertically opposite angles

Understanding 1 a False 2 a No

c i B and C

b ∠LMN, alternate angles on parallel lines

b True b Yes

c Yes

d No

c AAA similarity 12 a RHS similarity test

e Yes 3 a Angle-angle-angle similarity b Side-side-side similarity c Angle-angle-angle similarity d Right angle-hypotenuse-side similarity e Side-angle-side similarity

b SAS similarity test c AAA similarity test Reasoning 13 a △ AY P is similar to △BOX by the SAS similarity theorem. b △FDE is not similar to △GHK since they do not have equal angles.

Fluency 4 a i A and C

ii RHS

b i A and B

ii RHS

c △ZY G is similar to △BRP by the SSS similarity theorem.

Answers mathspace.co

771

c Kite, rhombus and square

14 a Dilation by a factor of b Yes, since A′C′ = DF, C′B′ = FE and ∠C′ = ∠F, △ A′B′C′ ≡ △DEF by side-angleside congruence (SAS). c △ ABC can be dilated by a factor of , then translated and rotated to get △DEF. Since a dilation is a similarity transformation and a rotation and translation are both congruence transformations, △ ABC and △DEF are similar. 15 a Yes. If side DA is parallel to side EC, then ∠DAB = ∠ECB and you can prove similarity using the AAA similarity test. b No, we would need more information about the ratio of the remaining pair of sides or another pair of equal angles in order to show similarity.

d Kite, rhombus and square e Rhombus and square 3 a Square

d Rhombus

e Kite 4 a i PQ ∥ RS and SP ∥ QR ii PQ = RS and SP = QR iii ∠P = ∠R and ∠Q = ∠S b i ZW ∥ XY ii Fluency 5 a

b Rhombus

10 10

c No, we would need more information about the angles in △ ABD or we would need to know side AD is parallel to side CE in order to show similarity. d Yes. AB is proportional to CB and BD is proportional to EB, you can prove similarity using the SAS similarity theorem.

b Rectangle

c Parallelogram

10 10

6 a

b Rhombus

10 10

Problem-solving

16 x = 11

17 18 a x = 6 19 a AE = 4

b AC = 18

7 a

b DE = 10

11.3

20 a All three pairs of corresponding angles are equal or AAA b h = 20 m

18.6

b Concave Kite

21 The ﬂagpole is about 5.25 m tall. 22 136 cm 23 a △ ADB or △BDC

18.6 11.3

8 a

14 12

b 12 units

c x = 9 units

8.06 Properties of quadrilaterals

b Parallelogram Understanding 1 a C

b A

2 a Rectangle and square b Parallelogram, rectangle, rhombus and square

772

Mathspace Victorian Curriculum – Year 8 mathspace.co

9 a

15 15

If ABCD is a rectangle, then ∠ABC, ∠BCD, ∠CDA, and ∠DAB are all right angles.

b Rectangle 10 a PR = 12

This also tells us that AB = DC and BC = AD.

b QX = 8

This means that we have two congruent right triangles △ ABC and △DCB, using the hypotenuse-leg theorem.

11 Y T = 6 12 AC = 24

By corresponding parts of congruent triangles, we get that AC = BD.

13 ∠KLM = 106° 14 ∠SPT = 39°

Problem-solving

15 a 46°

b 82°

c 22°

d 82°

16 a 58°

b 64°

c 5

d 90°

b i No

ii Yes

17 a 7

b 90°

c 14

d 45°

v No

vi No

18 a 32°

b 90°

c 20

d 10

c i Yes

ii Yes

22 a △ AEB and △ AED are congruent triangles.

23 a x = 7 Reasoning

iii No

iv Yes

iii No

iv No

b ∠BAC = 37°

c ∠ABC = 106°

19 a i ∠PRS

ii ∠SPR

b Because ∠RPQ = ∠PRS and these angles are alternate.

24 a Rectangle

Chapter 8 review

c Because ∠QRP = ∠SPR and these angles are alternate.

1 a Complementary angles

d PQRS must be a parallelogram, but is not necessarily a rhombus.

d Supplementary angles

b Neither

c Neither

2 a x = 97

b x = 86

20 Since ABCD is a rhombus, diagonals bisect each other at right angles, therefore,

e x = 40

f x = 75

3 a x = 55

b x = 40

∠CEB = 90°

e x = 67

f x = 56

e ∠RSP = 109°

Consider the △CEB. Recall that the sum of internal angles of a triangle is 180°. ∠CEB + ∠ECB + ∠CBE = 180° Triangle sum theorem 90° + (8x - 3)° + (5x + 2)° = 180° Substitute known values 13x + 89° = 180° Simplify 13x = 91° Subtract both sides by 89° x=7

ivide both sides D by 13

21 Parallelogram ABCD is a rectangle. We want to show that AC = BD.

4 a Yes

c x = 130

d x = 75

c x = 69

d x = 56

b No

5 ∠RQP 6 a i 8

ii 7

iii 9

b △XY Z congruent to △ ABC 7 Possible Answers: ∠RPQ = ∠RPS, QR = QS, ∠QRP = ∠SRP 8 Possible Answers: JN = MN, ∠JKN = ∠MLN, ∠KJN = ∠LMN 9 a SAS b AB = CD Given BD is common

Answers mathspace.co

773

∠ABD = ∠CDB Alternate angles on parallel lines

7 a 31

10 a Triangle C and D 11 a 40 m

b 46 m

12 a Yes

b 3

b SSS c 52 m

b 25

c 44

d 100

f 20

g 30

h 20

8 a 4 + 49 = 53

b 81

c No

9 a 100 − 36 = 64

b 16

c No

e 0.64

10 a 3

b 4

c 8

d 12

e 15

f 8

g 9

h 5

i 6

j 2

k 20

l 7

13 14

11 a 36 < 40 < 49

15 a Yes, SAS.

b Yes, RHS.

c No, they are not similar. 16 a △ ABC and △ ADE, AAA

d Yes, RHS.

b YZ=9 18 a PR = 8

12 a 3 and 4

b 1 and 2

c 9 and 10

d 7 and 8

e 11 and 12

f 6 and 7

g 15 and 16

h 14 and 15

17 a The triangles don’t have matching sides in the same ratio. b QX = 9

13 7 14 a 12

b 12

c Yes

d 12

15 a 6 ÷ 2 = 3 b 3

c Yes

d 2

16 324 cm

19 BD = 40

17 14 chairs

20 ∠SPT = 40.5° 21 a 64°

b 52°

c 6

d 90°

22 a △RST and △RQT are congruent triangles. b i No

ii Yes

v Yes

vi No

c i Yes

ii Yes

iii No

iv No

iii No

iv No

Reasoning 18 a 102 = 100 b (−10)2 = 100 c The side length is = 10 metres. There is only one possible answer since the length of a side of a square must be a positive number. 19 144 is divisible by both 12 and 16 and is a perfect square as 12 × 12 = 144.

9.01 Squares and square roots Understanding 1 1, 4, 9, 16, 25, 36, 49

20 A positive number multiplied by a positive number is a positive number.

2 15 × 15 = 225, therefore the square root of 225 is equal to 15.

A negative number multiplied by a negative number is a positive number.

3 676 Fluency 4 a No

b Yes

c No

d Yes

e No

f No

g Yes

h No

5 a 25

b 36

c 64

d 144

e 169

f 225

g 400

h 100

k 0.49

l 1.44

b 74

c 2600

d 72

g 80

h 9900

6 a 116 e

774

Mathspace Victorian Curriculum – Year 8 mathspace.co

Therefore it is not possible to ﬁnd a number a, such that a2 is negative. Therefore you cannot ﬁnd the square root of a negative number. 21

= 2.83 and 2.83 is not an integer.

= 0.3 and 0.3 is not an integer.

is not possible.

is not an integer.

22 a i R alph distributed the parentheses before taking the square. ii 2 (3 + 2)2 = 2 (5)2

= 2 × 25

= 50

b i I da moved the (9) into the square root without squaring it ﬁrst. She also did not extract the square root of 9 ﬁrst before multiplying to 3.

This layer can be viewed as adding two lots of the previous side length plus one corner square. +n

Problem-solving 23 There are inﬁnitely many answers (known as Pythagorean triples). The most famous Pythagorean triple is 32 + 42 = 52. Other examples are 52 + 122 = 132 and 72 + 242 = 25 and any multiples of these. 24 32 mm

This pattern increases by two at each stage starting with one. Hence, will create the odd numbers which are of the form 2n + 1 for n = 0, 1, 2, 3, ….

9.02 Pythagoras’ theorem Understanding

25 46 m

1 c is the hypotenuse.

26 $754.60

2 a 25

27 a 5 = (5 + 4 + 3 + 2 + 1) + (4 + 3 + 2 + 1) 62 = (6 + 5 + 4 + 3 + 2 + 1) + (5 + 4 + 3 + 2 + 1) b 2

b 7 and 24

3 a Z

b M

c Y

d c

Fluency

c The sequence of odd numbers: 1, 3, 5, 7, 9, … d Example answer:

4 a 25

b 25

c Yes, (3, 4, 5) is a Pythagorean triple.

The pattern of square numbers is shown.

5 a No 2

b Yes 2

c 142 + x2 = 172

To grow the square at each stage, we need to add an extra row at the top and an extra column on the side. 4 3 2 3

9 7 5 1

9 +5

d 32 + 92 = c2

7 a No

b Yes

c No

d Yes

8 a 8

b 24

c 45

d 50

9 a No

b No

c Yes

d No

10 a 28

b 122

c 6

d 15

11 Yes, because if z2 = x2 + y2 then 4z2 = 4x2 + 4y2. 12 Chenzi incorrectly assumed that the side with length 24 was the hypotenuse when he substituted the values into Pythagoras’ theorem. We can see that 242 + 102 = 262

3 4

Reasoning

This new, L-shaped layer is always the next odd number in the sequence of odd numbers.

d Yes

b 26 = b + 102

6 a c = 12 + 9 1

c No

25 +9

576 + 100 = 676 676 = 676 as required, so the hypotenuse is the side with length 26 centimetres.

Answers mathspace.co

775

13 Yes, as there are inﬁnitely many possible right-angled triangles. In addition, any multiple of a known Pythagorean triple gives another Pythagorean triple, and there are inﬁnitely many multiples of Pythagorean triples. Problem-Solving 14 Giralang and Florey 15 a Yes

b 120 m

c i Yes

ii No

iii Yes

iv Yes

Reasoning 14 Yes, if you are given two sides of a right-angled triangle, you can always find the length of the other side using the Pythagoras’ theorem. If you are given only one side length, there are infinitely many right-angled triangles that could satisfy the Pythagoras’ theorem. This is because there are infinitely many sets of three values (a, b, c) that satisfy the equation a 2 + b 2 = c2. 15 62 = 36

16 12 and 16

32 + 42 = 25

17 39 and 52 or 25 and 60

Therefore 32 + 42 ≠ 62 so the triangle cannot be right-angled as Pythagoras’ theorem does not hold.

9.03 Find the hypotenuse Understanding b c2 = a2 + b2

1 a c c Yes 2 a 3.87

b 7.81

c 7.62

d 4.92

3 a c=3

b c = 6.71

c c = 9.43 d c = 15

4 a c = 30

b c=5

c c = 17

d c = 15

e c = 41

f c = 20

g c = 50

h c = 29

16 By Pythagoras’ theorem, we have 22 + 32 = 42. The left side of the equation simpliﬁes to 4 + 9 = 13 while the right side simpliﬁes to 42 = 16. Since they are not equal, this combination is not possible. The hypotenuse length is actually

Fluency

5 a c = 10.30

b c = 24.19

c c = 9.49

d c = 19.24

6 a c = 8.06

b c = 21.93

c c = 12.21

d c = 26.63

17 We can use the Pythagoras’ theorem to check whether the triangle is right-angled. So, we have:

7 9.43 cm 8 17 m 9 a 17 m

b 40 m

10 a 23.46 mm

b 56.56 mm

11 a 24.04 cm

b 58.04 cm

Side length of square

13 a 13

776

Length of diagonal

1.4

4.2

9.9

15.6

249

352.1 b 16.64

Mathspace Victorian Curriculum – Year 8 mathspace.co

If we round this to two decimal places, we ﬁnd the hypotenuse to be = 13.93 cm. Since the length of the third side of the given triangle is an integer, the triangle is not a right-angled triangle. 18 Yes. If two pairs of corresponding sides are proportional, then the third side will also be proportional by Pythagoras’ theorem. So, Pythagoras’ theorem can be used to show SSS similarity in two right-angled triangles where any two sides are proportional.

11 We can use the Pythagoras’ theorem to check whether the triangle is right-angled. So, we have:

Problem-solving 19 a x = 11.70

b x = 10.00

20 21 36.49 cm

9.04 Find a short side Understanding 1 C 2 a 9.54

b 4.90

c 8.49

d 9.74

3 a a = 8.01

b a = 7.55

c a = 7.94

d a = 10.91

Fluency 4 a k = 12 units

b y = 12 units

c x = 9 units

d a = 10 units

e b = 21 m

f b = 7 units

g x = 4 units

h x = 16 units

i b = 15 m 5 a s = 14.4 units

b m = 14.1 units

c b = 9.4 m

d s = 12.8 units

e k = 24.2 units 6 a s = 3.32

b s = 12.45

c s = 5.74

d s = 17.44

7 a x = 14.14 cm

b y = 14.14 cm

8 a x = 4.90

b y = 8.66

c 13.56 9 a r = 10 m

If we round this to two decimal places, we ﬁnd the hypotenuse to be 6.24 cm. Since the length of the third side of the given triangle is an integer, the triangle is not a right-angled triangle. 12 No, it is not possible for a right-angled triangle to have one short side and the hypotenuse the same length. If one of the shorter sides (let’s say a) is equal to the hypotenuse (c), then we have: c2 = a2 + b2

Pythagoras’ theorem

Substitute a = c

c =c +b

This implies b2 = 0 which means the other short side has length zero, which is not possible. Therefore, a right-angled triangle cannot have one short side and the hypotenuse the same length. Problem-solving 13 19.05 cm 14 12.73 cm 15 10.39 m 16 k = 24.17

b 20 m

17 x = 7.21 units Reasoning

18 m = 15, n = 38.6

10 If the teacher changed the 4 to would be right-angled.

, the triangle

Pythagoras’ theorem

19 49.06 cm 20 u = 9.90 mm

Substitute known side lengths

21 a 6 m and 15 m

b 21 m

22 a 21 cm

b 52 cm

Evaluate the squares

Subtract 4 from both sides

Square root both sides

Answers mathspace.co

777

9.05 Applications of Pythagoras’ theorem Understanding

16 a Dana labelled the ladder with 150 cm, but the height of the ladder is unknown. The ladder should be labelled as h and the wall should be labelled as 150 cm.

1 When there is a right-angled triangle in the problem 2 a i Yes

ii c = x, a = 48.2, b = 55.9

150 cm

b i No c i Yes

ii c = 100, a = 65 and b = length

d i Yes

ii c = x, a = 26, b = 72

80 cm

b h = 170

e i No 17

Fluency 3 x = 35.81

Problem-solving

4 a x = 17

b y = 16.52

5 a h = 8.94

b l = 41.96

18 h = 5.19 c 62.96 m

6 20.25 m

19 $87.85 20 3.55

7 a x = 197.26

b h = 265.90

8 22.6 m

21 a 0.07 b Since 0.07 is less than

9 L = 15 cm

meets the requirements.

10 l = 19.72 m

22 124 m

11 5 km

23 a 28 cm

12 d = 9.85 m

b 23 cm

c 15.97 cm

d 31.94 cm

13 3.61 m

24 384 cm2

14 d = 128.7

25 h = 6.7 m

Reasoning

26 a x = 2.5

15 a Yes. If we consider viewing Sean’s walking distance from above, we can say Sean is walking to the left when he walks directly West. This is similar to drawing a horizontal line. When Sean walks directly South, he is walking downward, which is similar to drawing a vertical line. These two lines form a right angle. Connecting his start and end points forms a right-angled triangle.

Chapter 9 review

8 km 6 km

b 6.9327

1 182 + x2 = 252 2 a 100

b 100

c Yes, (6, 8, 10) is a Pythagorean triple because the square of the hypotenuse (102) is equal to the sum of the squares of the other two sides (62 + 82) 3 a No

b Yes

c No

d Yes

b 40

c 45

d 75

4 Yes h km

So Pythagoras’ theorem can be used. b h = 10

778

, the ramp

Mathspace Victorian Curriculum – Year 8 mathspace.co

5 a 8

6 Maria incorrectly assumed that the side with length 15 cm was the hypotenuse when she substituted the values into Pythagoras’ theorem. In fact, if we calculate correctly using Pythagoras’ theorem, we ﬁnd that 82 + 152 = 172(64 + 225 = 289), which means that the hypotenuse is the side with length 17 cm. Therefore, it is indeed possible to construct a right triangle with these side lengths. 7 a c = 6.40

b c = 12.53

c c = 12.81

d c = 31.62

8 a 14.04 mm

b 31.14 mm

9 a i c = 23.46 mm

ii 56.56 mm

b i c = 24.04 cm

ii 58.04 cm

10 a c = 14.42 cm

b 34.42 cm

11 a HF = 19.21 cm

b DF = 22.65 cm

12 By Pythagoras’ theorem, we have 22 + 32 = 42. The left side of the equation simpliﬁes to 4 + 9 = 13 while the right side simpliﬁes to 42 = 16. Since they are not equal, this combination is not possible. The hypotenuse length is actually

10.01 Perimeter Understanding 1 a 5 metres = 500 centimetres b 7 centimetres = 70 millimetres c 9 kilometres = 9000 metres d 2.5 metres = 2500 millimetres 2 Multiply the length of one side by six. 3 a Square or rhombus c Hexagon

b Triangle d Decagon

e Rectangle or parallelogram Fluency 4 a 16 units

b 22 units

5 a 30 m

b 26 m

6 a 54 m

b 56 m

7 40 cm 8 a 30 cm

b 176 cm d 3x + p + q units

c 64 cm 9 a 45 m

b 75 mm

c 10 cm or 100 mm 10 38 m 11 50 mm

13 a a = 6

b b=6

12 a 54 cm

b 160 mm

13 a a = 10 cm

b y=4m

14 29 cm

c x = 8.49

d 36.49 cm

15 8 m

14 a b = 7.55

b b = 15.20

16 11 mm

15 a a = 6

b a = 18

c a = 6.93 16 x = 3 17 Length = 8.7 cm 18 L = 13 cm 19 l = 20.6 m

d a = 9.64

c 60 m

17 a y = 6 units, P = 24 units b c = 10.30 units, P = 24.30 units 18 19 m 19 10 cm 20 a 11 cm

b 12 m

c 90 mm

21 5x 22 8.68 + 7.57x

20 h = 15

23 4x2 − 10x + 2 units

21 4.9 m

24 a x = 5 cm

b x = 7 cm

22 x = 10.30

Answers mathspace.co

779

Reasoning

33 x = 3 m

25 The possible dimensions of the rectangle are: • 1 cm by 17 cm

• 6 cm by 12 cm

• 2 cm by 16 cm

• 7 cm by 11 cm

• 3 cm by 15 cm

• 8 cm by 10 cm

• 4 cm by 14 cm

• 9 cm by 9 cm

34 26.42 m

10.02 Circumference of a circle Understanding

• 5 cm by 13 cm

1 a d = 2x

b C = 2π x

To ﬁnd the dimensions, we need to divide the perimeter by 2, which is 18, and then ﬁnd pairs of whole numbers that add up to that number.

2 a d = 2r

b Arc length = π r

26 Answer may look like: The square could have dimensions of 4 by 4 and the rectangle 3 by 5. Another possible answer is the square has dimensions of 5 by 5 and the rectangle 6 by 4. If the dimensions of the rectangle are l and w, and the side length of the square is s, we are looking for numbers so that 2s = l + w 27 The perimeter will be n × side length. 28 a True. Both shapes have a perimeter of 24 cm, since the square has four sides of length 6 cm and the rectangle has two sides of length 9 cm and two sides of length 3 cm. b True. Since a regular hexagon has six congruent sides, the length of each side is equal to the perimeter of the hexagon divided by 6, which is 8 cm. c False. It is possible for a rectangle and a square to have the same perimeter. For example, a square with side length 5 cm and a rectangle with length 6 cm and width 4 cm both have a perimeter of 20 cm. d True. It is not possible for a triangle to have a greater perimeter than a rectangle with the same base and height. For example, a right triangle with legs of length 6 cm and 8 cm has a perimeter of 24 cm, while a rectangle with base 6 cm and height 8 cm has a perimeter of 28 cm. Problem-solving

c P = π r + 2r 3 a d = 2r

b Arc length

c 4 a False

b True

c False

d False

e True Fluency 5 a 43.98 cm

b 94.25 m

c 78.54 mm

d 14.14 km

6 a 31.42 m c 54.98 cm

b 10.68 mm d 2.14 km

7 8.0 cm 8 a 4.77 cm c 3.98 cm

b 1.75 cm d 15.92 cm

9 9.55 cm 10 a 14.01 cm

b 1.91 cm

c 17.51 m

d 63.66 cm

11 a 11.8 cm

b 14.0 cm

c 22.3 cm

d 7.8 cm

12 a i 34.56 m

ii 56.56 m

b i 15.08 cm

ii 24.68 cm

c i 28.90 cm

ii 47.30 cm

d i 69.12 mm

ii 113.12 mm

13 a i 21.99 cm

ii 49.99 cm

b i 16.49 m

ii 37.49 m

c i 10.68 cm

ii 24.28 cm

d i 28.27 km

ii 64.27 km

29 60 cm Reasoning

30 41.25 mm 31 a 184 m

b $4710.40

32 x = 7 m

780

Mathspace Victorian Curriculum – Year 8 mathspace.co

14 a True. Since the diameter of a circle is twice its radius, and the circumference is π times the diameter, the circumference is always greater than the diameter.

b False. The circumference of a circle is equal to π times its diameter, not its radius. Therefore, the circumference of a circle is always twice its radius times π . c False. The circumference of a circle with diameter 2 cm is 2π cm, while the circumference of a circle with radius 1 cm is also 2π cm. The circumference of a circle is directly proportional to its diameter or radius, so if the radius is doubled, the circumference is also doubled. 15 a 31.42 cm

b 31.42 cm

c They got the same result because the diameter is always equal to twice the radius, so π d and 2π r are always equal. 16 a The answer is 6.28 m. 6.28 is 2π to two decimal places. b Yes.

2 B 3 C Fluency 4 a i Five rectangles

ii 140 cm

b i Two rhombuses

ii 66 cm

c i One semicircle and one rectangle ii 9π + 32 cm d i One square and four semicircles ii 32π cm 5 40 mm 6 a 42 m 7 60 m 8 a 49 cm

b 34 mm

c 54 cm

d 52 cm

e 64 mm

f 154 m

9 a 40.0 cm

17 The perimeter is given by:

This can be rearranged to:

b 60 m

b 33.4 cm

c 44.0 cm

d 31.4 cm

e 28.3 m

f 487.4 units

g 97.1 cm

h 239.9 cm

10 148.52 units 11 42.85 m

18 The perimeter is given by: p = π r + 2r This can be rearranged to:

Problem-solving 19 r = 9.72 m

b 46 m

Reasoning 13 He has included all four sides of the rectangle, but one of the sides of 18 cm is inside the shape. He has worked out the circumference of an entire circle of diameter 18 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.

20 r = 7.00 cm 21 29.85 cm 22 91.90 cm 23 314 m 24 a 70 372 m

12 a 10 m

b 12 315 m

10.03 Perimeter of composite shapes Understanding 1 A simple shape is a geometric shape that is made up of only one shape, while a composite shape is a geometric shape that is made up of two or more simple shapes combined.

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 to form a composite shape with the same perimeter as a hexagon made up of six equilateral triangles. c True. A composite shape with a hole in the middle has more sides than a solid shape

Answers mathspace.co

781

with the same outer dimensions because the perimeter of the hole is also added. Therefore, its perimeter will be greater. 15 P = l1 + 1 π d1 + l2 + 1 × π × d2 2 2 =18 + 1 × π × 10 + 12 + 1 × π × 8 2 2 = 58 cm or P = l1 + 1 × 2π r1 + l2 + 1 × 2 × π × r2 2 2 =18 + 1 × 2 × π × 10 + 12 + 1 × 2 × π × 8 2 2 2 2 = 58 cm

9 Approximately 130 m2 10 Approximately 26 mm2 11 a B

b D

c A

12 a 30 000 cm

b 70 000 cm2

c 560 000 m2

d 60 000 m2

e 104 000 m

d D

f 2 ha

Reasoning 13 Area has two dimensions, length and width and 100 × 100 = 10 000

Problem-solving 16 516 m 17 a x = 70.71

b 321.42 m

c $6495.90

18 a x = 5.39

b y = 5.39

c 21 m

19 a x = 9.8

b 2.8 L

c $155.40

20 a 400.00 m

b 407.67 m

c 95.88 m

d $777

10.04 Estimation and units of area Understanding 1 There are 10 mm in 1 cm, 100 cm in 1 m and 1000 m in 1 km. 2 Units of area from smallest to largest: • mm2

8 Approximately 200 mm2

• cm2

• m2

• km2

3 a Example answers: • Coffee table

• Car windscreen

• TV screen b Example answers: • Lid of a shoe box • iPad

14 a False. The unit of estimation can be different from the unit of measurement used to ﬁnd the exact area, as long as the conversion between units is taken into account. b False. The area of a square is equal to the length of one side squared. c True. The order of the length and width does not matter when calculating the area of a rectangle. d False. This statement is not always true. The area and perimeter are measured in different units (e.g., square units for area and linear units for perimeter), so they aren't directly comparable in this way. Moreover, for large shapes or shapes with small perimeters but large areas (like a large square), the area can be much larger than the perimeter. e False. When estimating area, it’s important to keep as much accuracy as possible. 15 It can help you check the reasonableness of your exact answer and catch any errors you may have made in your calculations.

Fluency

16 Hectares make it easy to visualise the size of the land.

4 a Square centimetres b Square metres

Problem-solving

c Square centimetres d Square metres f Square centimetres

g Square metres

h Hectares

18 76 000 m2

i Square metres

j Square metres

19 a

k Square kilometres

l Square metres

5 8.41 m2 2

6 Approximately 6912 cm 7 C

782

17 Approximately 3000 cm2

e Square kilometres

Mathspace Victorian Curriculum – Year 8 mathspace.co

Farm

Length, m

Width, m

Area, m2 20 000

200

100

350

7350

100

2400

400

16 000

100

10 000

b Farm 1 and Farm 4

c Farm 3 and Farm 2

d Farm 5

10 a 15 cm2

b 24 cm2

c 18 cm2

b 50 m2

c 60 cm2 d 70 m2

i 3.2 km

ii 3 200 000 m2

i 3600 cm2

ii 0.36 m2

12 a 2 m

c 4 mm

11 a 63 m2 2

20 a 72 200 m

b 18.05 acres

21 a 128 ha

b 1.28 km2

22 a 5.1 ha

b 12.75 acres

13 a 35 cm2

23 46 400 chickens

b 1 360 000 m2

24 a 136 ha

10.05 Area of rectangles, triangles and parallelograms

e 60 cm 14 a 5 cm 15

Understanding 1 a Multiply a and b c Multiply cm2

b Multiply A = ab d Multiply m2

2 a The length and width are in different units ii cm2 c i Yes, the units would now be the same. ii m2 b Yes

d m2

c Yes

4 a b = 7 cm, h = 2.65 cm

b 7.5 m2

d 4 cm

c 12 mm2 d 8 cm2 2

f 5.58 m b 8 m2

c 4 mm

d 13 km

Base (cm)

Height (cm)

Area (cm2)

Base (cm)

Height (cm)

125

Area (cm )

b i Yes, the units would now be the same.

3 a Yes

b 2 cm

d 1125 cm2

17 a Base = 8 cm

b Base = 10 cm

c Base = 4 mm

d Base = 20 mm

18 a Height = 10 cm

b Height = 8 cm

c Height = 18 mm

d Height = 4 mm

Reasoning

b b = 32 mm, h = 13 mm

19 a

c b = 6.35 cm, h = 10.5 cm d b = 18.5 cm, h = 11 cm 5 a 12 cm

c 108 cm2

b 9 cm

d The diagram shows that the parallelogram and the rectangle are made from the same two shapes. e 12 cm

f 9 cm

g The base of the parallelogram becomes the length of the rectangle and the height of the parallelogram becomes the width of the rectangle. Fluency

b i 10 cm

ii 6 cm

c 60 cm2 20 a x = 30

b No

c If dimensions are doubled, area is increased by a factor of 4. d If dimensions are tripled, area is increased by a factor of 9. 21 3 cm × 21 cm, 9 cm × 7 cm, 6 cm × 10.5 cm

6 a 28 cm2 b 60 mm2 c 90 m2 e i 20 000 cm2

ii 2 m2

f i 7.2 cm2

ii 720 mm2

7 25 cm2

d 18 km2

22 The rectangle. If we assume the length of the shapes was a cm, the width of the rectangle would be b cm, but the height of the parallelogram must be less than b cm.

8 16 cm2 9 length = 20 cm, width = 1 cm length = 10 cm, width = 2 cm length = 5 cm, width = 4 cm

Answers mathspace.co

783

Problem-solving

23 a 96 m

b 192 m

9 120 cm2

24 48 m2

10 65.75 cm2

25 80 m2

11 a 21.2 m2

b 15.3 m2

c 73.0 L c 1269 m2

26 600 m

12 a 504 m

27 4800

10.07 Area of trapeziums, kites, and rhombuses

28 10.5 litres 29 a 100 tiles

b 115 tiles

10.06 Area of composite shapes

1 a Rectangle and parallelogram b Example answer: Add the area of the rectangle and parallelogram together. 2 To calculate the total area of a composite shape composed of a rectangle and a triangle, you would ﬁnd the area of the rectangle and the area of the triangle separately, using their respective formulas, and then add the two areas together. Fluency b 26 cm2

c 143 cm2

d 231 cm2

e 0.625 m2

f 0.575 m2

4 a 220 cm2

b 76.5 cm2

c 62 m2

d 109 m2

5 a i x=4

ii 89 cm2

b i x=3

ii 99 cm2

6 a i x=4

ii y = 11

b i x=5

ii y = 10

1 B

3 A rhombus is a special case of a kite in which all four sides are equal in length. A kite has two pairs of equal sides. Both pairs of opposite angles of a rhombus are the same, whereas a kite only has one pair of opposite angles equal. Fluency 4 a 28 m2

b 56 m2

5 a 24 cm2

b 10 m2

c 36 mm2

d 27 cm2

e 32 mm2

f 60 cm2

6 a Yes

b Yes 2

c No

d No

7 a 7m

b 14 m

8 a 28 cm

b 4 cm

c 112 cm2

9 a 14 cm2

b 18 cm2

c 76.5 m2

10 a 90 m2

b 40.5 m2

c 130.5 m2

iii 44 cm2

11 a l = 9.5 cm

b 76 cm2

iii 50 cm2

12 a 36 cm2

b 18 cm2

13 a 47.5 cm2

b 16 mm2

c 32 mm2

d 50 m2

e 66 cm2

f 48 m2

d 84 cm2

Reasoning 7 A = 107 units2. The subtraction method is more efficient. 8 • Find the area of the rectangle in figure 1. • Find the area of the shaded triangle as shown in figure 2. • Then find the area of figure 3 by subtracting the area of the triangle from the area of the rectangle.

784

Understanding

2 A

Understanding

3 a 24 cm2

b 765 m

Mathspace Victorian Curriculum – Year 8 mathspace.co

14 a 22.5 cm2 2

b 60 mm2

c 16 cm2 2

d 36 mm

e 93.5 cm

g 84 m2

h 450 mm2

15 a x = 8 16 x = 10 17 y = 13

b b = 16

f 48 m2

18 k = 10 cm 19 a k = 7

b k=9

c k = 14

d k = 25

20 a a = 4

b a=4

c b=8

d h=4

21 x = 5 cm

36 Area Parallelogram = (7 + 10) × 2 = 34 mm2

22 y = 21.69 23

Area (mm2)

Diagonal, x (mm)

Diagonal, y (mm)

24 4800 m2

Therefore the area of parallelogram is equal to two times the area of trapezium. 37 The diagonals already span the length and width of the rectangle. So the product of the diagonals is already a maximum.

38 a $90

26 12 500 000 seeds

b $96

c 12 m

39 16 cm

27 720 cm2

40 5 cm and 20 cm

28 1200 tiles

41 $172.5

29 4.8 m

10.08 Area of a circle

Reasoning 30 A trapezium is a quadrilateral with one pair of parallel sides. It differs from a parallelogram or rectangle because those shapes have two pairs of parallel sides. 31 A rhombus is a quadrilateral with all sides of equal length. It differs from a square because a square has all sides of equal length and all angles of 90 degrees, while a rhombus has all sides of equal length but angles that are not necessarily 90 degrees. 32 a True. The correct formula for the area of a trapezium is the average of the lengths of its parallel sides multiplied by its height. b False. A rhombus is a parallelogram with all sides of equal length, but its angles are not necessarily all 90 degrees like a square. c True. A rhombus is a kite with all four sides equal in length. d True. The formula for the area of a kite is , where x and y are the lengths of the two diagonals.

34 a y

× 2 = 34 mm2

2 × Area Trapezium = 2 ×

Problem-solving

25 24 m2

33 a y

35 A rhombus is a special case of both a kite and a parallelogram. So, depending on the information given, it is possible to use the area formula for a kite or the area formula for a parallelogram.

b b

c c

d d

Understanding 1 a 6m

b 3.4 cm

d 88.5 mm

2 a 38 m

b 11.4 m

d 49 mm

3 a 21.99

b 28.27

c 4.77

d 78.54

4 a

Fluency 5 a 201.06 cm2

b 153.94 cm2

d 132.73 mm2

c 78.54 cm

e 1075.21 cm

f 58.09 mm2

6 a 50.3 mm2

b 78.5 mm2 2

c 452.4 mm 7 a i 6 mm

d 0.2 m2 ii 113.10 mm2

b i 2.86 cm

ii 25.70 cm2

c i 11 mm

ii 380.13 cm2

d i 7 cm

ii 153.94 cm2

8 a 14.1 cm2

b 19.6 mm2

c 95.0 m2

d 115.5 cm2

9 a i 2.821 cm

ii 17.7 cm

b i 6.206 cm

ii 39.0 cm

Answers mathspace.co

785

10 6.770 mm

Fluency

11 a r = 10.59

b 66.5 cm

4 a i Two trapeziums

12 a 6.77 cm

b 13.5 cm

ii 182 cm2

13 a 14.669 mm

b 46.1 mm

14 a r = 4.5 cm

b r = 5.1 cm

c 3.4 cm

d 2.3 cm

b i One trapezium and two semicircles ii 185.66 cm2 5 a i One semicircle and one rectangle ii 253.23 cm2

15 314.16 cm2

b i One square and four semicircles

Reasoning

ii 658.12 cm2

16 a Area of circle 1 is 50.27 cm2

6 a 64.27 cm2

Area of circle 2 is 201.06 cm2

c 176.55 units

b The area will increase by a factor of 4 c The area will be circle.

b 62.98 cm2 2

of the area of the original

7 a 113.1 cm2

d 83.90 cm2 b 146.1 cm2

d 19.6 cm2

c 261.8 cm

e 226.2 cm

17 a The area will increase by a factor of 9 2

b The area will increase by a factor of q

8 a 89.94 cm2

b 226.19 cm2

d 103.67 cm2

c 150.80 cm

c The radius will increase by a factor of Reasoning Problem-solving

9 a False. Depending on the complexity of the shape and the number of basic shapes involved, the addition method may be faster to use.

18 398 cm2 19 a 399 832.26 cm2 b 399 629.57 cm2 c Yes, because the approximate area is 202.69 cm2 smaller than the actual area. 20 No. The area of the large pizza is: 1661.9 cm2 The area of two medium pizza is: 1508.3 cm2 Therefore, the large pizza has a greater area than the two medium pizzas combined. So, if Danielle is looking to get more pizza for the same price, she should choose the large pizza. 21 a 40 m

b 251.33 m

c 5026.55 m2

10.09 More composite areas Understanding 1 a Total Area = Area of rectangle + Area of semi-circle b Total Area = Area of square − Area of circle 2 The subtraction method 3 a A square and a circle b The addition method

786

Mathspace Victorian Curriculum – Year 8 mathspace.co

b False. The area of some composite shapes can be calculated using the subtraction method. c False. While the area of a composite shape can be found by adding the areas of its individual shapes, there are some cases where this method may not work. For example, if the shapes overlap, the overlap area will be counted twice, and must be subtracted once. d False. The perimeter and area of a shape are not always directly related. For example, two shapes with the same perimeter may have different areas depending on how their sides are arranged. e True. Simply subtract the area of the known shape from the total area to ﬁnd the area of the remaining shapes. 10 a 40.8 cm2 b A = π R2 − π r2 c A = π (R2 − r2)

14 a 144 m3

Problem-solving 2

b 144 kL

11 402.12 cm

15 4.3 m 2

12 a 3648 cm or 0.3648 m

Reasoning

b 52 704 cm2 or 5.2704 m2 13 a 1600 cm2

16 a 100 cm3

b 1182.48 cm2

b v = lwh = 10 × 5 × 2 = 100 cm3

c 6817.52 cm2

c Answers may vary: length = 5 cm, width = 5 cm and height = 4 cm

10.10 Units of volume and capacity

17 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.

Understanding 1 a Capacity c Volume

b Volume d Capacity

b 1000 a3 L

2 1 L = 1000 mL 3 a 1 km = 1000 m, therefore to convert from km3 to m3 multiply by 10003 which is 1 000 000 000. b 1 m = 100 cm, therefore to convert from m3 to cm3 multiply by 1003 which is 1 000 000. c 1 cm = 10 mm, therefore to convert from cm3 to mm3 multiply by 103 which is 1000. 4 Divide by 1003 = 1 000 000. 5 a m3

b m3

c cm3

d m3

6 a mL

b L

c kL

d L

c a3 kL

d 1 m3 = 1 kL

18 Method 1: 40 × 60 × 150 = 360 000 mm3 Method 2: 4 × 6 × 15 = 360 cm3 360 cm3 = 360 × 103 mm3 = 360 000 mm3 19 1 L is equal to 1000 cm3, 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. Problem-solving 20 4.8 L 21 a 1600 cm3

b 40 cm

22 150 cups

Fluency

23 400

7 1000 mm3

24 a 36.469 m3 3

8 a 3500 mm b 4 200 000 000 mm 3 c 4 000 000 000 mm d 19 000 mm3 9 a 5.36 L d 1670 L g 7.85 L

b 5L e 18 620 L h 0.8 L

c 9000 L f 2000 L

11 Approximately 0.32 m3

13 a 1 000 000 mL c 1 kL

b 52 000 mL e 1 mL h 720 cm3 k 83 kL n 0.0038 L

b 36 469 L

c 34 015 L

e 20

25 7

10.11 Volume of right prisms Understanding

10 Approximately 44 250 cm3

12 a 68 L d 63 000 L g 36 mL j 1 m3 m 400 mL p 0.0846 kL

d 1717 L

c 88 kL f 1 cm3 i 1 kL l 270 m3 o 1670 L

1 The volume of a rectangular prism can be found by multiplying the length, width, and height of the prism: Volume = length × width × height. 2 a Triangle c Rectangle

b Trapezium d Kite

3 a A semicircle and a square b Two rectangles c Two rectangles

b 1000 L

d A triangle and a rectangle

Answers mathspace.co

787

4 To ﬁnd the volume of a composite shape made up of rectangular prisms, calculate the volume of each individual rectangular prism by multiplying their length, width, and height. Then, add the volumes of each prism together.

21 12 1 × 1 × 72, 1 × 2 × 36, 1 × 3 × 24, 1 × 4 × 18, 1 × 6 × 12, 1 × 8 × 9, 2 × 2 × 18, 2 × 3 × 12, 2 × 4 × 9, 2 × 6 × 6, 3 × 3 × 8, 3 × 4 × 6 22 a 50abc cm3

Fluency 5 a 216 cm3

b 48 cm3

c 32 cm3

d 217.5 cm3

e 330 cm3

f 84 cm3

g 270 cm3

b 400abc cm3 To compare the volumes of two right prisms with the same height but different base areas, we can compare their base areas. The prism with the larger base area will have a larger volume, and the prism with the smaller base area will have a smaller volume.

h 720 mm3 3

i 2250 mm 6 a 1728 cm3

j 645.975 m3 b 216 cm3

7 2744 m3

Problem-solving

8 200 cm

23 2 m3

9 9600 mm3

24 a 4320 cm3

10 a 68 cm3 3

d 1080 cm

b 384 cm3

c 399 cm3

e 828 cm

f 300 cm

11 a = 4 mm 12 p = 6 mm 13 k = 11 14 y = 10.1 15 70 cm2 16 9 cm 17 3 cm 18 9 cm

b i 34 560 cm3 25 300 min = 5 hrs 26 a 0.024 kg = 24 g

b 70 000 kg = 70 tonnes 27 a 1198.5 m3 28 a 8151 cm3 c 15 120 cm3

d 756

30 d = 29 cm 31 h = 160 cm

19 To compare the volumes of two right prisms with the same height but different base areas, we can compare their base areas. The prism with the larger base area will have a larger volume, and the prism with the smaller base area will have a smaller volume.

33 a 109.86 cm2

Mathspace Victorian Curriculum – Year 8 mathspace.co

b 5

29 a 2 cm × 5 cm × 2 cm b 20 cm3

32 10 000

788

b 299.625 m3

c h = 0.5875 m

Reasoning

20 I disagree. Two rectangular prisms can have the same volume but different dimensions. For example, a prism with dimensions 2 × 3 × 4 has a volume of 24 cubic units, and a prism with dimensions 1 × 4 × 6 also has a volume of 24 cubic units. The two prisms have the same volume but different dimensions.

ii No

b 2L

Chapter 10 review 1 a 30 m

b 48 mm

c 40 cm or 400 mm 2 a 10 cm

b 8m

c 50 mm

3 a False. The perimeter of the square is 4 × 10 cm = 40 cm. The perimeter of the rectangle is 2 × 20 cm + 2 × 5 cm = 40 cm + 10 cm = 50 cm. Since 40 cm is not equal to 50 cm, the perimeters are not equal. b False. Since a regular hexagon has six congruent sides, the length of each side is

equal to the perimeter of the hexagon divided by 6, which is 9 cm. c False. It is possible for a rectangle and a circle to have the same perimeter (or, more precisely, circumference in the case of the circle). The perimeter of a rectangle is 2 × (length + width) and the circumference of a circle is 2 × π × radius. Thus, these two can be made equal by selecting appropriate dimensions. d True. It is not possible for a triangle to have a greater perimeter than a rectangle with the same base and height. For example, a right triangle with legs of length 6 cm and 8 cm has a perimeter of 24 cm, while a rectangle with base 6 cm and height 8 cm has a perimeter of 28 cm. 4 a 31.42 cm

b 100.53 m

c 114.35 mm

d 126.29 km

5 a 2.55 cm

b 8.28 cm

c 18.78 m

d 81.49 cm

6 a 1.91 cm

b 8.28 cm

c 12.25 cm 7 a 30 cm

b $150.8

22 a 28.27 m

c 12.2 m

b 19.63 cm

c 153.94 mm 23 a r = 4.15 cm c 3.66 cm

d 1.13 m b r = 5.08 cm d 2.65 cm

24 No. The area of the giant pizza is: 2827.4 cm2 The area of two small pizzas is: 2513.3 cm2 Therefore, the giant pizza provides more area than the combined area of the two small pizzas. Thus, if Richard wants more pizza for the same price, he should choose the giant pizza.

26 a i 64 cm2

ii 132 cm2

ii 40 mm2

b i 16 mm

c 18 cm

8 a 94.0 cm

21 a $210

25 a 143 cm2 b 890 cm2 c 475 cm2 d 231 cm2

27 a 180 cm2 b 33 cm2

c 20 cm2 d 107 cm2

28 Approximately 36 290 cm3

d 47.75 cm b 50 m

20 6 400 000 seeds

d 22 mm

29 a 36 mL c 52 000 mL

b 720 cm3 d 83 kL

b 208.6 cm

e 270 m3

f 400 mL

9 a No

b 9π + 54 cm

g 0.0038 L

h 0.0846 kL

10 a a = 6

b b=6

c x = 8.49

30 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.

d 36.49 cm 2

11 Approximately 156 mm

b 1000 a3 L

12 66 000 m2 b 840 000 m2

13 a 84 ha 14 a 2 m 15

b 9 cm

c 18 mm

Base (m)

Height (m)

Area (m2)

16 a 150 m2

d 9.68 m

d 1 m3 = 1 kL 31 a 400 cm3

b 20 cm

32 a i A semicircle and a square ii 22.283 m2

iii 222.83 m3

b i Two rectangles ii 34 cm2

iii 68 cm3

c i Two rectangles

b 336 m2

ii 40 cm2

17 334

iii 280 cm3

d i A triangle and a rectangle

18 a 32 cm2

b 13 m2

ii 37.5 m2 2

c 80 m

d 45 cm 2

e 24 cm

f 450 mm

19 a y c

c a3 kL

iii 412.5 m3

33 90 cm2 34 a 3750 cm3 b i 30 000 cm3

ii No

Answers mathspace.co

789

35 To compare the volumes of two right prisms with the same height but different base areas, we can compare their base areas. The prism with the larger base area will have a larger volume, and the prism with the smaller base area will have a smaller volume.

d P (having a baby girl) e P (club) 15 a {GG, GO, GR, OG, OO, OR, RG, RO, RR,} b P (GG) = c P (Orange or Red, in any order) =

11.01 Calculate probability

d P (Neither Red) =

Understanding 1 a True

b True

c False

d False

16 a

17 a

b P (Red and Yellow, in any order) =

3 a Green, Yellow, Red b 9 outcomes

18 a

c 3 outcomes 4 a True

H 4

b True

c True

d False

T H

Fluency

T H

5 a Unlikely

T H

b Certain

c Impossible

d Likely

e Even chance

f Likely

g Unlikely

h Unlikely

i Likely

j Likely

k Unlikely

l Even chance

c {1, 2} 8 a 0

19 a

Honey

b {2, 4, 6}

Jam

d {3, 6}

Honey

Honey Jam

c Jam

9 a 100%

b 75%

10 a

Honey Jam

Honey

6 Red, Green, Blue 7 a {1, 2, 3, 4, 5, 6}

Honey Jam

c 25%

d 50%

Jam

b d 7

c 11 a P (red) = 25%

b P (blue marble or black marble) = 75% 12 a {27, 28, 29, 30, 31, 32, 33, 34}

Reasoning 20 The bag must contain another colour because the probabilities of the colours listed do not sum to 100%. 21 Spinning a pig or a star There are 8 total sections on the spinner. 2 are a pig and 2 are a star, so each have a

13 a 7

probability of

b P (greater than 3)

probability of occurring, they are equally likely.

14 a P (Heads)

c P (sum greater than 11)

790

Mathspace Victorian Curriculum – Year 8 mathspace.co

. Since they have the same

22 Answers may vary:

Fluency

Fred has three equally likely options to get to school. He can go by car, bus or bike. If he goes by car or bus then he comes home by car or bus. If he goes by bike then he comes home by car or bike.

23 a

b i Yes

8 a Drawing a number greater than or equal to 42 b Drawing a number that is less than 43 c

Problem-solving

24 a i

7 0.3

b ii

iii

ii Yes

iii No

iv No

25 a

9 a

10 a O

11 a P (rain)′ = 0.57

b P (lose) = 0.51

c P (heads) = 0.44 26 a b i Yes

12 a A ∩ B = {0, 1, 4, 9} ii Yes

iii No

iv Yes

v No 27 a

d P (lose) = 0.947 b P∩Q=∅

c M ∩ N = {−25, −5, −1, 0, 1, 5, 25} — that is, M∩N=M=N d W ∩ X = the set of multiples of 9 — that is, W∩X=X 13 a A ∪ B = {0, 1, 9, 99, 100, 101, 999, 9999}

b F ∪ G = {m, a, t, h} — that is, F ∪ G = F = G

28 a 24

c S ∪ T = {1, 2, 3, 5, 8, 12, 13, 21, 23, 31, 32, 34}

d J ∪ K = the set of multiples of 5 — that is, J∪K=J 14 a A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37} B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

11.02 Intersection and union

b B ∪ A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 19, 23, 29, 31, 37}

Understanding

c A ∩ B = {2, 3, 5, 7, 11}

1 1

15 a

2 The intersection of the sets

b Divisible by 4

4 a No because zero is not negative or positive c No

16 a Odd number

3 The union of the sets b Yes

d No

5 a The two sets in the diagram are Dog owners and Snake owners.

c Neither of the two sections 17 a

Owns a pet

Owns a car

b 13 + 2 is the number of people who have a Dog. c 2 + 3 is the number of people who have a Snake. d 2 is the number of people who have a Dog AND a Snake. 6 Event 3 and event 4

Answers mathspace.co

791

If either set contains at least one element that is not a member of the other set, however, then this extra element will be included in the union but will not be included in the intersection. So if the two sets are not exactly the same, then A ∪ B will contain more elements than A ∩ B.

Owns a pet Owns a car

24 No. The complement of a set contains all of the elements of the sample space which are not in the original set. In this case, A already contains every element of the sample space, so the complement will have no elements.

18 a A ∩ B = {8} b A ∪ B = {1, 2, 4, 6, 7, 8, 9} c A ∩ B′ = {1, 2, 4} d (A ∪ B)′ = {3, 5} 19 a 26

c 19

20 a

Problem-solving 25 Octagonal 26 a

Reasoning

21 The complementary event must be certain, therefore its probability is 1. 22 The complement of a set A includes every element of the sample space that is not an element of A. Graphically, we can show this as follows: A

27 a

8 15

11 15

4 15

28 Any set that matches the Venn diagram. For example, one possible way to represent this region is (P ∩ Q′) ∪ (P′ ∩ Q), which can be said as (P and not Q) or (not P and Q). Another way is (P ∩ Q)′ ∩ (P ∪ Q), which can be said as not (P and Q) and (P or Q).

11.03 Venn diagrams So if we then take the complement of this new set, we include every element not in the new set. That is, we include every element not in “the elements not in A”, which leaves us with all of the elements that were in A to begin with. Graphically, if we take the complement of this complement, we shade everything that was unshaded before: A

Understanding 1 a Multiples of 6 and multiples of 8 b 6, 12, 18, 24, 30, 18, 36 c 8, 16, 24, 32, 40 d 24 2 a Students who have both Sumsang and jPhones. b Students who have a Sumsang or jPhone or both. c Students who have neither Sumsang or jPhones.

So we can see that (A′)′ = A — that is, the complement of a complement is the original set. 23 If the two sets have the same elements, i.e. if A = B, then both the union and intersection will also have the same elements, i.e. A ∪ B = A ∩ B = A = B.

792

Mathspace Victorian Curriculum – Year 8 mathspace.co

d The total number of students or universal set. Fluency 3 a 3

b 1

c 4

4 a

5 a 20

b 14

c 17

6 a 6

b 2

c 6

make-up test, there were a total of 27 + 10 = 37 students who sat for both tests. Since there are only 30 students in the class, 37 − 30 = 7 students must have sat for both.

7 Skateboard Bike

b 16

Original test Catch-up test

25 20

8 a 9 b

Cricket

Adding 70 + 63 + 31 to get 164, however, is incorrect as this includes the 49 people who enjoy watching both sports twice.

9 a 30

b 6

10 a 169

b 223

c 192

e 262

f 293

11 a i 51

ii 40

iii 10

iii

b i

20 Eiichiro is correct in saying that 70 people like watching basketball in total, and that 63 people like watching baseball in total.

Softball

12 a

d 138

13 a

14 a

15 a

16 a 55

17 a 18

b 6 Cake 19

Ice cream 12

To correctly calculate the total number of people, we can just add the four numbers shown in the regions of the Venn diagram: 31 + 21 + 49 + 14 = 115 people. (This is also the result that we get by subtracting the extra 49 from Eiichiro’s calculation of 164.) 21 One student studies German, so we should add 1 into the top-left circle. Since the student studies no other languages, we should add it to the part of the circle which doesn’t overlap the other circles. So we should increase the 16 to a 17. One student studies both Italian and German, so we should increase the overlap section between German and Italian by 1, from 4 to 5. Similarly, one student studies Spanish and Italian, so we should increase this overlap section from 9 to 10. The last two students don’t study any languages, so they don’t belong in any of the circles and we should increase the 0 in the outer rectangle to be 2 instead. 2 German

Italian

13 2 12

Reasoning 19 a 7 students sat for both tests. Since 27 students sat for the original and 10 sat for the

10 36 Spanish

Answers mathspace.co

793

3. In the Venn diagram, the sections representing quadrilaterals that are not part of the parallelogram family would include trapezium and kite.

Problem-solving 22 Prime numbers 23 26 24 a

11.04 Two-way tables Tennis

Hockey

Understanding

1 a 120

3 4

b What hair colour and what eye colour they have.

5 Cricket

b 25 a Answers may vary. Example diagram:

Blue eyes

Brown eyes

Total

Brown hair

Blonde hair

Total

120

2 a Coffee and Tea

Trapezium One pair of parallel sides

b Saturday and Sunday c

Parallelogram Two pairs of parallel sides

Rhombus All sides equal

Coffee

Sunday

Total

120

3 a A = 50

b B = 20

Kite Two pairs of adjacent equal sides

(Opposite sides are equal) Rectangle All angles right

Tea Saturday

Square

b Answers may look like: 1. Which quadrilaterals have only one pair of parallel sides? 2. What properties do squares and rectangles share?

4 a Students who are allergic to dairy but not allergic to nuts. b 12

c 23

5 a 50

b 15

6 a

c Answers may look like:

Boys

Girls

Total

Maths

Music

English

Total

b 39

c 26

7 a

d 47

Female

Male

Total

Frequently

Rarely

Total

120

b 120

Mathspace Victorian Curriculum – Year 8 mathspace.co

c 54

Science

2. In the Venn diagram, both squares and rectangles have all angles equal to 90 degrees.

794

d D = 90

Fluency

3. Which quadrilaterals are not part of the parallelogram family? 1. In the Venn diagram, the section representing quadrilaterals with only one pair of parallel sides would be trapezium. Therefore, trapezium has only one pair of parallel sides.

c C = 40

Total

8 a

Cardio Weight Male

Female

Total

100

b 100

c 42

9 a

Female

Male

Total

Rarely

Total

140

Not vegetarian

Vegetarian

Total

Children

e Men

Frequently

b 140

16 a

Total

Adults

Total

Play Rugby League

Don’t play Rugby League

Total

Play Rugby Union

185

228

Don’t play Rugby Union

179

146

325

Total

222

331

553

b 13

c 27

d 28

Tent

Cabin

Total

Sunny Campground

178

240

Late

Not late

Total

Platypus Creek

101

281

382

Caught bus

Total

279

343

622

Didn’t catch bus

Total

Microchipped

Not Microchipped

Total

10 a 5 11 a

b Sunny Campground = 240 Platypus Creek = 382 c

Vaccinated

Not vaccinated

Total

12 a 12

b 35

c 161

13 a 326

b 27.5%

c 37.7%

d 20

Cardio

14 a More ham and cheese sandwiches were sold on Day 1.

b Turkey and avocado sandwiches were more popular on Day 2.

Morning

c The probability that a randomly chosen sandwich sold on Day 1 is a turkey and avocado sandwich is 15 a

Reasoning

Car

Bicycle

Total

Men

Women

Total

110

21 As an example, here is a two-way table showing whether a group of people enjoy music and/or skateboarding:

Skateboarding

Music

No music

Total

No skateboarding

Total

Answers mathspace.co

795

To represent this same information in a Venn diagram, we would want to set up a diagram with one circle for each category - in this case, music and skateboarding. Music

Skateboarding

The ﬁrst column of the two-way table shows the people who belong in the music circle. Similarly, the ﬁrst row shows the people who belong in the skateboarding circle. So the top- left number, in this case 15, is the number of people who belong to both circles, which goes in the middle overlap of the Venn diagram. Music

Skateboarding

Notice that when we represent information in a two-way table, the totals of each column and row are shown, but these totals are not visible in a Venn diagram representation. We can still ﬁnd the totals, however, by adding up the values inside/outside each circle. 22 Uri has incorrectly added all of the values in the ‘No phone’ column. The last value in this column, 53, is already the sum of the other two values and represents the total number of students who don’t have a smartphone. If we subtract 53 from 176 instead, we ﬁnd that 176 − 53 = 123 students own a smartphone, and this matches the value at the bottom of the ‘Phone’ column. 23 a The 40 people includes teenagers and children, not just adults, so it cannot be concluded that all 26 are adults who said no. b The 16 adults represents the total of adults. Using this information with the 14 adults saying “yes” tells us that 2 adults responded “no”.

c The ratio of yes to no is 4:1 so we need to divide the total by 5. Since the number of “no” responses is less than the number of “yes” responses we can conclude that the number of no responses was 8 and the number of yes responses was four times as much or 32.

The next number in the ﬁrst column, 21, still corresponds to people who like music, but not to people who like skateboarding. So this number belongs in the music circle but not the skateboarding circle. Similarly, the next number in the ﬁrst row, 13, corresponds to people who like skateboarding but not music. So this number belongs in the skateboarding circle but not the music circle. Music

Skateboarding

Adults

Children

Total

Yes

Total

Sprinkles

Hot Fudge

Total

Problem-solving 21

Finally the value in the second row and second column, 5, corresponds to people who like neither music nor skateboarding — that is, people who don’t ﬁt into either circle. So this number goes outside of the circles. 5

Music

Skateboarding

24 Chocolate

Vanilla

Total

North

South

Inner

Total

Yes

Total

Spanish

Not Spanish

Total

Mandarin

Not Mandarin

26 a 21

796

Mathspace Victorian Curriculum – Year 8 mathspace.co

b 58 27 a

12 a A Venn diagram or two-way table can be used to represent this scenario. The Venn diagram is shown below.

11.05 Mutually exclusive events

Extension 2 English

Understanding

Extension 2 Maths 10

1 a There is no overlap between A and B.

b P (A or B) = P (A) + P (B)

2 a It’s possible to be a ﬂuffy dog. b P (Dog or Fluffy) = P (Dog) + P (Fluffy) − P (Dog and Fluffy)

iii

3 C, D

13 a 1

Fluency

14 0.2

4 Event 1 and event 3 5 a Yes

b No

e Yes

f No

6 a 0.28

b i

c No

d Yes

15 a Yes b i

ii 0

v 1

iii

7 a i Mutually exclusive

Reasoning

b i Mutually exclusive

c i Mutually exclusive

16 a Yes, because choosing a cake that is a chocolate cake and (at the same time) a cheese cake is impossible in this scenario.

d i Non-mutually exclusive

8 a i x=0

ii Mutually exclusive

b i x=0

ii Mutually exclusive

9 a 0

b Yes

10 a

17 The formula for mutually exclusive events is P (A ∪ B) = P (A) + P (B) The formula for non-mutually exclusive events is P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

d P (A or B) = 0.9104

b No, because some chocolate cakes have vanilla ﬁlling.

11 a A Venn diagram or two-way table can be used to represent this scenario. The Venn diagram is shown below.

The formula for non-mutually exclusive events can be used for both types of events. If the events are mutually exclusive, then P (A ∩ B) will be 0, so: P (A ∪ B) = P (A) + P (B) − P (A ∩ B) = P (A) + P (B) − 0 = P (A) + P (B)

Green

Pink 3

6 2

b i

18 Enrica is not correct because the elements both events have in common (the elements in the intersection) have been counted twice. The sum of all the elements in event A, event B and (A ∪ B)′ is greater than the number of elements in the sample space.

Answers mathspace.co

797

To ﬁnd P (A ∪ B), we need to use the formula for the union of non-mutually exclusive events. We can calculate the intersection of the two sets using a Venn diagram and later solving for x.

Maths 9

English 6

9 9

9 9 Science

6 A 18 − x

B 14 − x

23 106 employees

Chapter 11 review Solving for x, we get x = 4 which means .

1 a Yes

b Yes

c No

d No

So, P (A ∪ B) = P (A) + P (B) − P (A ∩ B) = . Problem-solving 19 a i 0.3

ii 0.2

iii 0.1

iv Non-mutually exclusive

b i 0.2

2 a 24

3 a Study

ii 0.1

iii 0

Study

Relax Relax

iv Mutually exclusive

c i 0.6

iv Non-mutually exclusive

d i 0.1

Study

ii 0.7

iii 0

Relax

20 a

Study Relax

Relax

b No

4 a N

b P∩Q=∅ c M ∩ N = {−30, −6, −2, 0, 2, 6, 30} - that is, M∩N=M=N

b B ii 1

5 a A ∩ B = {−10, 0}

21 a Yes

b 0 22

c i 0

Study Relax

iv Mutually exclusive

Study Relax

ii 0.1

iii 0.2

Study

iii

22 a A Venn diagram or two-way table can be used to represent this scenario. The Venn diagram is shown.

d W ∩ X = the set of multiples of 10 - that is, W∩X=X 6 a A ∪ B = {0, 2, 8, 88, 102, 104, 888, 8888} b F ∪ G = {s, c, i, e, n} — that is, F ∪ G = F = G c S ∪ T = {2, 4, 6, 10, 14, 16, 22, 26, 30, 36, 38, 40} d J ∪ K = the set of multiples of 7 — that is, J∪K=J

798

Mathspace Victorian Curriculum – Year 8 mathspace.co

7 a

13 16

e 8 a 30

b 20

c 25

9 a 12 b

Science Fiction

Fantasy

19 a i Non-mutually exclusive

b i Mutually exclusive

c i Mutually exclusive

d i Non-mutually exclusive

20 a

b d P (A or B) = 0.98

c 16

21 0.5 22 a

23 a 10 a 195

b 245

e 340

f 365

c 220

d 170

Mystery Romance Fantasy Mystery Romance Fantasy Mystery Romance Mystery Romance Fantasy

Mystery

Fiction

Mystery

11 a

12 a 13 a 8

b 18

c 46

Romance

d 54

14 a More round paintbrushes were sold on Day 1. b Flat paintbrushes were more popular on Day 2.

Non-ﬁction

Total

26 a Guitar

Adults

Total

Vegetarian

Piano

5 4 5

4 6 Drums

Cafeteria

27 a 35

German Not German

17 a Yes

b No

c No

d No

e Yes

f No

Not French

Total

100

b 48 28 a

Total

French 35

18 a 0.33

Fiction

Teenagers

25 30

15 a

b 24 a

c The probability that a randomly chosen paintbrush sold on Day 1 is a ﬂat paintbrush is

Fantasy

Non-fiction

c b

d c

Answers mathspace.co

799

29 a No

iii

30 127 members

12.01 Measures of centre and spread d i

Understanding 1 a A

b D

c B

Median

1.5

Mode

0.9

Range

1.9

Mean

4540

Mode

iii

4 a i

iii

b i

iii

c i

800

Mean

Median

Mode

Range

Mean

Median

Mode

Range

Mean

Median

Mode

Range

102

Mean

Median

Mode

Range

Mean

1.92

Median

1.7

Mode

0.9

Range

3.8

ii 109

ii 1500

4700

Mean 4877.78 iv The mean and median increased, Median 4800 the range decreased, Mode 4700 but the mode stayed Range 800 the same.

5 a 4

b 42

c 40.33

6 a 15

b 15

c 15

7 a 5

b 72

c 71

ii 79

iii 72

ii 59

iii 35

iv 71

c i 27.73 ii 30

iii 30, 35

iv 45

d i 25.08 ii 26

iii 22

iv 9

8 a i 78 b i 58.2

Fluency

iv The mean, median and range decreased, but the mode stayed the same.

Range 3800

The median is a a better choice for the measure of centre as it is less affected by extreme values or outliers. 3 Outliers are data points that lie far from the majority of the data points in a data set. It is important to identify outliers because they can affect the interpretation of the data. Outliers can skew measures of centre (mean, median, mode) and measures of spread (range).

1.61

Median 4750

d C

2 An example of a set where the median is a better measure of centre than the mean is: 2, 3, 7, 8, 27

Mean

9 a Lower

b 13.3 kg

10 a Lower

b 44.56

11 a Mean

iv 29

b Mode

iv The mean, median and range decreased, but the mode stayed the same.

12 a Median

b Mean

c Mean

d Mean

ii 126

13 a 12.5 kg

c Median

d 15

d Mode or Median

e Median c 13.875 kg

b 14.2 kg d 14.25 kg

e Median, because it was the least affected by the outlier. iv The mean, median and range decreased, but the mode stayed the same. ii 4.7

Mathspace Victorian Curriculum – Year 8 mathspace.co

14 a 21 years c 30 years

b 11 years d 23.32 years

e Positively skewed 15 a Positively skewed

b Positively skewed

c Negatively skewed

d Positively skewed

e Symmetrical 16 a No c Negatively skewed

b 9

17 a No

b 3

29 a $527 000

c Positively skewed 18 a Positively skewed

b Yes, 9

19 a No

b 47 b 15 and 21

b Mean is 48.5 or 49 customers.

c Symmetrical Reasoning 21 Because every score occurs the same amount of times. 22 Higher, as a higher number will make the total larger when all the values are added, so the mean will be larger. 23 Humbleton, because the mean is greater than the median. This would mean that some very expensive houses have “dragged” the mean up, while the median indicates where the middle of most house prices are.

c 44.6 or 45 customers. This would be more representative for this store as the outlier is very different to the rest of the values (might be a spike in customers). 31 a 11.2 mm

12.02 Sampling methods Understanding 1 False 2 B 3 C

b The range always decreases.

4 A

c If the outlier removed was the largest value then the mean will decrease. If the outlier removed was the smallest value then the mean will increase.

5 a B and C

d If the outlier removed was the largest value then the median may decrease or remain the same. If the outlier removed was the smallest value then the median may increase or remain the same. Problem-solving 25 4, 6, 8, 8, 15, 19

b 11.15 mm

c Probably around where the centre is when the outlier is removed. So, around 11.1 mm

24 a The mode does not change.

b B and C

6 Stratiﬁed sampling ensures representation from different segments or groups within the population, allowing for more accurate analysis and conclusions about speciﬁc subgroups. 7 One disadvantage of using systematic sampling is that it may introduce a potential bias if there is a systematic pattern or periodicity in the population’s characteristics, leading to an underrepresentation or overrepresentation of certain individuals or groups in the sample. 8 B

26 19.87 27 a 17

c $446 600

30 a Smallest is 34 customers, largest is 84 customers. The range is 84 − 34 = 50 customers.

c Negatively skewed 20 a No

b $929 000

d This is more accurate as it is similar to all the values not including the outlier of $929 600.

b 16

c 39.5

d 47.5

e Although the ranges are similar, the mean values are signiﬁcantly different indicating that these two groups of birds are not of the same species. 28 a Mean = 2.6, Median = 3, Mode = 3, Range = 4 b Mean = 8.3, Median = 8, Mode = 8, Range = 4 c Class 2 d The mean, mode and median.

Fluency 9 a Sample b Census c Sample d Sample e Census f Sample g Census h Census l Sample j Sample 10 a Random sampling b Systematic sampling c Stratiﬁed sampling 11 3, 21, 152, 211

Answers mathspace.co

801

12 a 264

b 39

13 a 8

b 10

14 a 10 000

c i 140

ii 60

15 a

iii 80

iv 120

Problem-solving

16 a Yes

b No

c Yes

d No

17 1 089 600 people 18 a 10 students b i 990 students

23 Example answer: You would ﬁrst create a list of all the customers in the database, and then choose every nth customer on the list, where n is the total number of customers divided by the desired sample size.

ii 630 students

iii 540 students Reasoning 19 a Census, as the teacher needs to know what level every student is at.

24 The employees to be surveyed are 10, 20, 30… going up by 10 all the way to 500 25 The residents to be surveyed will depend on the random numbers generated. For example, if the ﬁrst random number is 174, the residents to be surveyed are 174, 1143, 2112, and so on. 26 For the Arts stratum, 200 students should be sampled. For the Science and Business strata, 150 students should be sampled from each. 27 a 37, 57, 50, 47, 72

b 35

28 a

e 6 males

f 6 females

b Sample, as it is an average. c Census, as they need to fix every faulty machine. d Census, as the manager wants to know the exact number. e Sample, as the farmer only needs the average. 20 a Example answers: • Drivers at a drive-in theatre. • People stuck in traffic due to roadworks. b Example answers: • People stuck in traffic due to roadworks. • Employees at a bank. c Example answers: • All people who catch the train in the morning on a given day. • All people who catch public transport on a given day. 21 Example answer: You would first create a list of all the students, and then choose every nth student on the list, where n is the total number of students divided by 20. 22 Example answer: You would first divide the students into different groups based on reading level, grade level, or other relevant factors. Then, you would randomly select students from each group to ensure representation from each subgroup.

802

Mathspace Victorian Curriculum – Year 8 mathspace.co

d g 220 females

12.03 Bias Understanding 1 False 2 True 3 False 4 True 5 True 6 True 7 True Fluency 8 Fair 9 a Yes

b No

c Yes

d No

b No

c No

d No

10 A 11 B 12 a Yes

13 No bias. This is an example of random sampling.

14 Example answer: Possible bias. If there is a pattern in which patients are assigned to the ward and the researcher only selects patients who are assigned to a speciﬁc ward, then there may be a bias. 15 C 16 C 17 a Yes

b Yes

c No

18 a Yes

b No

c Yes

19 A 20 C Reasoning 21 Answer may look like: The issue that might have influenced the survey results is the order in which the genres were listed, leading to a response bias. To avoid this issue, the survey could randomise the order of genres for each participant, ensuring that the order in which the genres are listed does not influence their choices. 22 a Stratified b Respondents tend to be people who have strong opinions or feelings about an issue. Casper is probably not getting much feedback from people who have no issues with the canteen. Casper has used only the statistics that are favourable to him. 23 a The sample is not representative of the target population. b The sample is from self-selecting participants, i.e. only those that made an effort to respond. 24 Jenny’s claim is the more valid claim of the two since she has surveyed more people, and by the law of large numbers, her claim most be more valid. 25 Sample response: The results of the study might be affected by the fact that the students who received the new teaching method also had more experienced teachers. This could mean that the improved performance is due to the experience of the teachers rather than the

new teaching method itself. To determine the true effect of the new teaching method, the researcher should control for teacher experience in their analysis. 26 Sample response: The results of the study might be affected by the fact that people who exercise more frequently also tend to have higher incomes. This could mean that the improved mental health is due to the higher income rather than the exercise itself. To minimise this potential bias, the researcher could control for income in their analysis or stratify their sample by income levels to ensure that the relationship between exercise and mental health is not confounded by income. 27 Sample response: The results of the study might be affected by the fact that participants who received the drug were more likely to report their symptoms to the researcher. This could mean that the apparent effectiveness of the drug is due to the increased reporting of symptoms rather than the drug itself. To address this potential bias, the researcher could implement a double-blind study design, where neither the participants nor the researchers know who is receiving the drug and who is receiving a placebo. This would help ensure that symptom reporting is not inﬂuenced by the knowledge of which treatment the participant is receiving. Problem-solving 28 a Yes

b No

c No

d No

29 a No

b No

c Yes

d No

30 a Yes

b Yes

c No

e Yes

31 B 32 C 33 C

12.04 Compare samples to populations Understanding 1 The population represents every single person or object being studied or surveyed, but the sample is only part of a population.

Answers mathspace.co

803

2 Taking a sample is often more feasible and cost-effective than surveying or collecting data from the entire population. 3 The sample mean is the average value of a numerical data set. The sample proportion is the percentage of samples that have a certain trait relative to all the samples. 4 A larger range indicates a greater spread or variability in the data sets. It suggests that there is more variation among the values in the sample, with some values being relatively far apart from each other. Conversely, a smaller range indicates less variability, with the values being closer together on average. 5 If the range of a sample is much smaller than the range of the population, it might suggest that the sample is not representative of the population and may be biased. 6 We can use the sample proportion to estimate the population proportion. Since 0.75 represents the proportion of students who own a smartphone in the sample, we can estimate that approximately 75% of students in the entire school population own a smartphone. This assumes that the sample is representative of the entire population. 7 Based on the given sample means, we can infer that the population mean of the ﬁrst population is around 25, and the population mean of the second population is around 30. However, we cannot determine the exact population means without additional information or conducting further analysis. Fluency 8 a 32.6

b 28.5

c Team A, because they have a higher mean. 9 a

Mean

Median

Mode Range

Person A

7.6

Person B

7.4

7.5

b Person A, because they have a signiﬁcantly higher range. c Person A, because their mean is higher. 10 Red 11 a 15

804

b 25

Mathspace Victorian Curriculum – Year 8 mathspace.co

c Pizza Shop B. Pizza Shop B sold more pizzas because they sold a total of 155 pizzas, while Pizza Shop A sold a total of 125 pizzas. 12 a 0.27

b 257 students

13 a 118.744

b 120.33

c Sample B

d 120.065

e 118.85

f Sample A

14 a i 12.56

ii 14.76

iii 15.79

iv 15.51

b 15.18 c 0.18 15 a i 10.6

ii 13

iii 10.2

iv 11

ii 11

iii 11

iv 10.5

b 0.29

c A

b 10.55 c i 10 d A 16 a 0.28

17 a Mean is 59.36 and median is 60 b Mean is 58.32 and median is 60 c Mean is 60.16 and median is 60 d A, C 18 a 0.94

b 0.41

c B

19 a 0.25

b 0.35

c A, D

20 a Mean is 3.8 and median is 3 b Mean is 4.05 and median is 4 c Mean is 3.99 and median is 4 d B, C 21 a 0.6

b 3

Reasoning 22 The sample mode is the same as the population mode, which suggests that the sample is likely representative of the population. 23 Store A has younger customers than Store B. 24 The mode being 30, despite the majority of respondents being in their 40s and 50s, suggests that there is a small subgroup of individuals who are in their 30s but have a higher representation within the sample. Although the frequency of individuals in their 40s and 50s is higher, the mode is determined by the value with the highest frequency, which happens to be 30 due to the concentrated presence of respondents in their 30s.

This highlights the signiﬁcance of understanding the distribution and composition of the data when interpreting the mode.

30 a

Sample

Studies History

Proportion

0.85

25 a Most students disagreed that they should do more homework.

0.125

0.5

b No, because only two classes out of the school were surveyed.

0.45

26 a

c The second sample. The larger the sample size, the closer the parameters of the sample resemble the parameters of the population. 27 a

28 a 0.26 b Example answer: The range or spread of the proportions would decrease because the proportion of the samples would be closer to the proportion of the population. c 0.7 d Yes, her claim is plausible. A proportion of 0.73 occurred 9 times in the simulation and 0.7 occurred 10 times. Since this is close to the proportion of Isaiah’s original sample proportion and occurs almost as often as 0.7, it is plausible that the actual proportion of students is 73%. Problem-solving Sample

Ordered Tea

Proportion

0.32

0.72

0.56

0.2

0.48

0.16

0.04

0.48

0.96

b 0.408

0.6

0.225

0.15

0.175

0.525

0.925

b 0.4525

c The ﬁrst sample. The larger the sample size, the closer the parameters of the sample resemble the parameters of the population.

29 a

5 6

c 0.408

31 a • • b • •

Mean: 6.71 Mode: 5 Mean: 9.29 Mode: 11

c 0.4525 • • • •

Median: 7 Range: 4 Median: 9 Range: 4

c Sales Team A has a lower mean, with a value of 6.71, compared to Sales Team B’s mean of 9.29. This indicates that, on average, Sales Team B performed better. Additionally, the median for Sales Team A is 5, which is lower than the median for Sales Team B at 9, further supporting Sales Team B’s superior performance. Sales Team A has a mode of 5, while Sales Team B has a mode of 11, indicating that Sales Team B had more frequent higher sales. Both teams have the same range which is 4, which does not provide additional information for comparison. In summary, based on the mean, median and mode, Sales Team B outperformed Sales Team A. 32 a 47

b 0.727

33 a 82.48 g

b 80.14 g

Chapter 12 review 1 A census involves collecting data from every individual in the population. In contrast, a sample involves collecting data from a subset of the population. The census provides a complete and accurate representation of the population but can be time-consuming and costly. A sample, if chosen well, can provide a good estimation of the population’s characteristics and is usually less time-consuming and costly.

Answers mathspace.co

805

2 a Census

b Sample

c Sample

d Sample

e Census

f Census

14 a 10 000 c i 180

3 Mean = 81.32

16 a Yes

ii 24 b i 70.3, 80.6, 85.2, 88.9, 90.5, 95.8 c i −30.7 − 15.8, −10.6, 20.9, 25.3, 45.2 ii 5.15 b 7 books d 7 books

e Median, because it was the least affected by the outlier. 6 a The mode does not change. b The range always decreases. c If the outlier removed was the largest value then the mean will decrease. If the outlier removed was the smallest value then the mean will increase. d If the outlier removed was the largest value then the median may decrease or remain the same. If the outlier removed was the smallest value then the median may increase or remain the same. 7 a 57

b 64

c 44.3

d 43.6

e 43.95 8 Because each number in the data set occurs only once. 9 a Positively skewed

b Yes, 9

10 a No

c Positively skewed

b 3

11 a One person in the group b 15 people

c 16 names

d 10 names

e 6

12 44, 147, 85, 224 13 a Mean

Median

Mode

Range

Administrators $77 142.86 $80 000 $80 000 $10 000 Teachers

$72 857.14 $70 000 $70 000 $10 000

b The administrators, because they have the higher mean, median and mode.

806

Mathspace Victorian Curriculum – Year 8 mathspace.co

iii 150

iv 210

b 25 b No

c Yes

d No

17 The students to be surveyed are 15, 115, 215, 315, 415, 515, 615 and 715

ii 87.05

c 6.625 books

ii 60

15 a 150

4 a i 19, 23, 23, 25, 30, 45

5 a 8.11 books

18 a Yes

b No

c No

d Yes

19 a No

b No

c Yes

d Yes

20 a 0.35

b 0.37

c A

21 a 15.076

b 14.984

c Sample D

d 15.235

e 14.700

f Sample C

22 The measures of centre (mean, median) and spread (range, interquartile range) would likely be similar but not exactly the same. This is because while the samples are taken from the same population, and we would expect similar characteristics, the speciﬁc trees (or individuals) chosen in each sample are different, leading to slight differences in the results.