CambridgeMATHS 12 Standard 2 – uncorrected sample pages by cambridge_aused

U N SA C O M R PL R E EC PA T E G D ES

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Contents viii

Introduction and overview

Acknowledgements

Ratios and rates

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About the authors

Calculations with ratios . . . . . . . . . . . . . . . . . . . . . . . . . . .

The capture-recapture technique . . . . . . . . . . . . . . . . . . . . . .

Dividing a quantity in a given ratio . . . . . . . . . . . . . . . . . . . . . .

Scale drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Building plans

Perimeter, area and volume from plans . . . . . . . . . . . . . . . . . . .

Rates and best buys . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Distance-time graphs

Energy consumption rates . . . . . . . . . . . . . . . . . . . . . . . . . .

Fuel consumption rates . . . . . . . . . . . . . . . . . . . . . . . . . . .

Review of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . .

Chapter checklist with success criteria . . . . . . . . . . . . . . . . . .

Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . .

Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . .

Extended-response questions . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Relative frequency and probability

Language of probability . . . . . . . . . . . . . . . . . . . . . . . . . . .

Definition of probability

Tables and tree diagrams . . . . . . . . . . . . . . . . . . . . . . . . . .

Range of probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Complementary events . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Relative frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Expected frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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Contents

Venn diagrams and two-way tables . . . . . . . . . . . . . . . . . . . . .

Review of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 110 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 111

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Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 113 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Extended-response question

. . . . . . . . . . . . . . . . . . . . . . . 115

Investments and loans

116

. . . . . . . . . . . . . . . . . . . . . . . 118

Simple interest on investments

Compound interest on investments . . . . . . . . . . . . . . . . . . . . . 123

Graphs of investments: Simple interest and compound interest

Appreciation and inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Shares and dividends

Straight-line depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Declining-balance depreciation . . . . . . . . . . . . . . . . . . . . . . . 152

Costs of borrowing money . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Reducing-balance loans . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Credit cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Credit card statements

Fees and charges for credit card usage . . . . . . . . . . . . . . . . . . . 181

. . . . . . 130

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

. . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Review of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 185 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 186 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 190 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Extended-response question

. . . . . . . . . . . . . . . . . . . . . . . 193

Practice paper 1

194

Trigonometry

198

. . . . . . . . . . . . . . . . . . . 200

Trigonometry in right-angled triangles

Angles of elevation and depression . . . . . . . . . . . . . . . . . . . . . 209

True bearings and compass bearings . . . . . . . . . . . . . . . . . . . . 216

The sine rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

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Contents

The cosine rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

The area of a triangle

Problem solving using trigonometry . . . . . . . . . . . . . . . . . . . . . 244

Compass radial surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

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Review of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 257 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 258 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 261 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Extended-response questions . . . . . . . . . . . . . . . . . . . . . . . 263

Simultaneous linear equations

264

Straight lines (CONSOLIDATING) . . . . . . . . . . . . . . . . . . . . . . . 266

Linear relationships (CONSOLIDATING) . . . . . . . . . . . . . . . . . . . 275

Solving simultaneous equations graphically . . . . . . . . . . . . . . . . . 281

Solving simultaneous equations algebraically . . . . . . . . . . . . . . . . 286

Solving practical problems using simultaneous equations . . . . . . . . . 291

Break-even analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Review of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 301 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 302 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 304 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Extended-response questions . . . . . . . . . . . . . . . . . . . . . . . 307

Bivariate data analysis

308

. . . . . . . . . . . . . . . . . . . . 310

Constructing a bivariate scatter plot

Using a bivariate scatter plot

Pearson’s correlation coefficient . . . . . . . . . . . . . . . . . . . . . . 322

Line of best fit

Least-squares regression line . . . . . . . . . . . . . . . . . . . . . . . . 336

Interpolation and extrapolation . . . . . . . . . . . . . . . . . . . . . . . 341

Statistical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

. . . . . . . . . . . . . . . . . . . . . . . . 315

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

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Contents

Review of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 354 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 355 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 359

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Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Extended-response questions . . . . . . . . . . . . . . . . . . . . . . . 361

Practice paper 2

362

Annuities

368

What are annuities? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

Tracking annuities every period . . . . . . . . . . . . . . . . . . . . . . . 376

Using tables for the future value of an annuity

Using tables for the present value of an annuity . . . . . . . . . . . . . . . 391

Solving problems involving superannuation and loans

. . . . . . . . . . . . . . . 384

. . . . . . . . . . . 398

Review of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 405 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 406 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 409 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Extended-response question

. . . . . . . . . . . . . . . . . . . . . . . 413

Exponential, quadratic and reciprocal relationships

414

Graphs of exponential relationships . . . . . . . . . . . . . . . . . . . . . 416

Exponential models in practical contexts . . . . . . . . . . . . . . . . . . 423

Graphs of quadratic relationships . . . . . . . . . . . . . . . . . . . . . . 427

Quadratic models in practical contexts . . . . . . . . . . . . . . . . . . . 435

Graphs of reciprocal relationships

Reciprocal models in practical contexts . . . . . . . . . . . . . . . . . . . 448

Inverse variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Problems involving exponential, quadratic and reciprocal relationships . . 458

. . . . . . . . . . . . . . . . . . . . . 442

Review of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 462 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 463 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 465 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 466 Extended-response questions . . . . . . . . . . . . . . . . . . . . . . . 467

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Contents

The normal distribution

468

Normally distributed datasets . . . . . . . . . . . . . . . . . . . . . . . . 470

The empirical rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

z-scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Converting z-scores into actual scores . . . . . . . . . . . . . . . . . . . 489

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Using z-scores to compare data . . . . . . . . . . . . . . . . . . . . . . . 494

Further applications of normal distributions

. . . . . . . . . . . . . . . . 500

Review of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 507 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 508 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 511 Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 512 Extended-response questions . . . . . . . . . . . . . . . . . . . . . . . 513

Network flow and critical path analysis

514

. . . . . . . . . . . . . . . . . . . . . . . . . . . 516

10A

Network flow problems

10B

Introduction to critical path analysis

10C

Constructing a network diagram given a precedence table . . . . . . . . . 536

10D

Dummy activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

10E

Forward and backward scanning . . . . . . . . . . . . . . . . . . . . . . 552

10F

Float times and the critical path . . . . . . . . . . . . . . . . . . . . . . . 561

10G

Minimum completion time . . . . . . . . . . . . . . . . . . . . . . . . . . 569

10H

Gantt charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576

Review of Chapter 10

. . . . . . . . . . . . . . . . . . . . 528

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

Key ideas and chapter summary . . . . . . . . . . . . . . . . . . . . . . 586 Chapter checklist with success criteria . . . . . . . . . . . . . . . . . . 587 Multiple-choice questions . . . . . . . . . . . . . . . . . . . . . . . . . 594

Short-answer questions . . . . . . . . . . . . . . . . . . . . . . . . . . 596 Extended-response questions . . . . . . . . . . . . . . . . . . . . . . . 598

Practice paper 3

600

Answers

606

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vii

About the authors

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Greg Powers is currently the Head of Mathematics at Cabramatta High School and the coordinator of the Mathematics Head Teacher Western Network. He is an experienced classroom teacher, having taught for over 30 years in a range of different schools. Greg has been a senior marker for the HSC, educational consultant for the Metropolitan South West Region and presented at numerous MANSW inservices. He has also enjoyed several curriculum roles with the Department of Education and Training. Greg is an experienced author who has written numerous texts on mathematics and technology.

Stuart Palmer was born and educated in NSW. He is a fully qualified high school mathematics teacher with more than 25 years’ experience teaching students from all walks of life in a variety of schools. He has been Head of Mathematics in two schools. He is very well known by teachers throughout the state for the professional learning workshops he delivers. Stuart also assists thousands of Year 12 students every year as they prepare for their HSC Examinations. At the University of Sydney, Stuart spent more than a decade running tutorials for pre-service mathematics teachers.

Christina Barbara has been a dedicated high school Mathematics teacher for over 20 years and is currently teaching at a Catholic high school in the Diocese of Parramatta. She has taught a wide range of students, from those requiring numeracy support in Year 7 to accelerated classes and Extension 2 Mathematics at the HSC level. She also prepared students for the HSC, serving as an HSC Study Session Facilitator at Western Sydney University. Before transitioning to teaching, Christina worked as a Building Estimator and Quantity Surveyor in Sydney’s construction industry.

Darren Thomas has worked in numerous high schools in the ACT and NSW for over 20 years. He previously owned a business specialising in supporting students with learning difficulties for nearly 15 years. He is currently the Head of Mathematics at a high school in north-west Sydney. He is passionate about mentoring staff and seeing them grow and develop. Since returning to the classroom in 2014, he has guided students to become independent learners who drive their own education.

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Acknowledgements

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The author and publisher wish to thank the following sources for permission to reproduce material:

To come

Every effort has been made to trace and acknowledge copyright. The publisher apologises for any accidental infringement and welcomes information that would redress this situation.

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

U N SA C O M R PL R E EC PA T E G D ES

Ratios and rates

How fast are we moving here on Earth? When we are standing still on Earth, we are actually moving very fast, in three different ways.

Our planet rotates 360 degrees every 24 hours, so a person standing on the equator is travelling at about 1700 km per hour. We are also orbiting around the Sun at a speed of approximately 30 km per second.

At this speed you could travel from Sydney to London in about 9.4 seconds.

In addition, the Earth and the Sun orbit the centre of our galaxy, the Milky Way, at a speed of approximately 200 km per second.

Our planet also moves because the universe is expanding, but the speed of this movement is very insignificant compared to those above.

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Chapter contents Calculations with ratios The capture-recapture technique Dividing a quantity in a given ratio Scale drawings Building plans Perimeter, area and volume from plans Rates and best buys Distance-time graphs Energy consumption rates Fuel consumption rates

U N SA C O M R PL R E EC PA T E G D ES

1A 1B 1C 1D 1E 1F 1G 1H 1I 1J

NSW Syllabus

In this chapter, a student:

Online resources

A host of additional online resources are included as part of your Interactive Textbook, including video demonstrations of all worked examples, auto-marked quizzes, downloadable spreadsheets, worksheets and calculator support, and much more.

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Chapter 1 Ratios and rates

1A Calculations with ratios Learning intentions

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• To consolidate prior learning about the meaning, use and applications of ratios • To reduce ratios to simplest form • To perform calculations and solve problems involving ratios Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Years 7 to 10 • Questions involving ratios regularly appear in HSC Examinations

What is ‘aspect ratio’?

These days, we all spend a lot of time looking at screens on phones, televisions and computers, which come in a huge variety of sizes.

Aspect ratio is used to describe the length of the horizontal side compared to the vertical side. Common aspect ratios include 16 : 9, 16 : 10, 3 : 2 and 4 : 3.

If the aspect ratio is 16 : 9, then:

• 16 and 9 have no common factors other than 1, so this ratio is in simplest form • the screen might be 16 cm wide and 9 cm high (16 cm by 9 cm) • we could multiply both numbers by 2 and get 32 cm by 18 cm • we could multiply both numbers by any number • we could divide both sides by any number

• we could divide both numbers by 16 and call this ratio 1 :

9 16

16 :1 9 • the sides are in the proportion as shown here, but the squares could be larger or smaller. • we could divide both numbers by 9 and call this ratio

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1A Calculations with ratios

Ratio A ratio is used to compare quantities with the same units, so the units are not written. For example, the ratio of 16 cm to 9 cm is written as 16 : 9 and pronounced “16 to 9”.

U N SA C O M R PL R E EC PA T E G D ES

Consider the diagram below.

The ratio of black squares to white squares is 4 : 3.

The ratio of white squares to black squares is 3 : 4. The ratio of white squares to all squares is 3 : 7.

Ratios are related to fractions in the following way: • the ratio of black squares to all squares is 4 : 7

! 4 of the diagram. • as a fraction, the black squares make up four-sevenths 7

The ratios above are in simplest form, because the two numbers have no common factors other than 1. Ratios which are not in simplest form can be expressed in simplest form by using division or multiplication. For example: 12 : 18

÷6

×3

2:3

Example 1

1 :2 3

Simplifying a ratio

Write these ratios in simplest form. 1 a 3: b 1.5 : 3.5 2

Solution

×2

3: 1 2

×2

1.5 : 3.5

×2

3:7

2 5 : 7 21

Explanation

6:1

×3

1:6

2 5 : 7 21 2×3 5 : 7 × 3 21 6 5 : 21 21 6:5

To convert one half to a whole number, multiply by 2. Check that ratio is in simplest form. To convert these decimals to whole numbers, multiply by 2 or 10. Check that ratio is in simplest form.

If the denominators were equal they could be ignored, so look for LCM of 7 and 21, which is 3.

Multiply the numerator and denominator of the first fraction by 3. Ignore the denominators. Check that ratio is in simplest form.

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Chapter 1 Ratios and rates

Now you try Write these ratios in simplest form. 1 a 1: b 1.4 : 1.2 4

5 3 : 6 4

Exercise 1A

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BUILDING

Note: Reduce all answers to simplest form. 1

Express each ratio in simplest form. a 15 : 3 e 8 : 12 i 17 : 51

d 14 : 30 h 48 : 32 l 27 : 9 : 3

b 3 kg to 12 kg

c 56 t : 16 t

The following ratios have different units on each side. Express each ratio in simplest form after first changing to the same units on each side. (For example, for a, first change the ratio to ‘40c to 100c’ or ‘0.4 : 1’).

a 40 c to $1 d 1 km : 250 m g 7 cm : 21 mm

c 24 : 16 g 81 : 27 k 5 : 10 : 20

Express each ratio in simplest form. a $24 : $18

b 10 : 40 f 49 : 14 j 9 : 18 : 9

b 3 h to 1 day e 3 m : 50 cm h 8 months: 4 years

c 2 mm to 1 cm f 6 km : 300 m i 2 L : 450 mL

There are 14 red blocks and 10 white blocks in a bag. What is the ratio of: a red to black

b black to red

c red to all blocks?

A clothing store has a discount sale. A dress marked at $250 is sold for $200. What is the ratio of the discount to the marked price?

Madeleine and Nathan invest $4500 and $2500 into a managed fund. What is the ratio of Madeleine’s share to Nathan’s share?

There were 80 blocks of land available for sale in a new land release. After one month 32 blocks have been sold. What is the ratio of sold blocks to the total number of blocks?

Write down 3 ratios which are equivalent to 2 : 3.

Write down 3 ratios which are equivalent to 16 : 10.

10 A square has a side length of 6 cm. If the side length is doubled, what is the ratio of the area of the original square to that of the larger square?

DEVELOPING

11 Express each ratio in simplest form. 1 1 2 3 3 a : b : c :1 2 5 3 7 4 1 1 1 1 2 3 d : e : f : 2 10 10 5 3 5 3 2 9 g 1: h :1 i :1 4 3 & Assessment © Powers, et al 2026 •10 Uncorrected 3rd sample pages • Cambridge University Press 978-1-009-76285-4 • (03) 8671 1400

Example 1

1A Calculations with ratios

4:3 : 60

4:3 60 :

c 4:3 1:

2:5 : 50

2:5 50 :

f 2:5 1:

Express each ratio in simplest form.

U N SA C O M R PL R E EC PA T E G D ES

Find the missing number in these equivalent ratios.

a 2x : x d 4a : 8a g 5d2 : 25d2

b 3y : 9y e 3xy : 12x h 22x : 33xy

c 4a : 2 f 7m2 : m i 16a2 b : 2ab2

In an election, Aiden scored 250 votes, Billie scored 175 votes and Chelsea scored 125 votes. Find the ratio in simplest form of Aiden’s votes : Billie’s votes : Chelsea’s votes.

3 15 Samantha and Mathilde own a restaurant. Samantha gets of the profits and Mathilde receives the 5 remainder. a What is the ratio of profits? b Last week the profit was $2250. How much does Mathilde receive? c This week the profit is $2900. How much does Samantha receive?

CHALLENGING

Express each of the following ratios in the form x : 1.

a 12 : 8

c 1.6 : 1.2

d 0.2 : 0.8

Express each of the following ratios in the form 1 : y. a 16 : 8

b 16 : 5

b 8 : 15

c 1.5 : 6

Dakari makes a blend of mixed lollies using 5 kg of jelly babies, 4 kg of liquorice and 1 kg of skittles. Their prices are given in the table. What is the cost of the blend per kilogram?

d 2.4 : 8.4

Lolly prices

Jelly babies

$5.95 per kg

Liquorice

$6.95 per kg

Skittles

$11.90 per kg

Communicate your reasoning, understanding and fluency 19

A rectangular screen is 160 cm long by 120 cm high. An image with an aspect ratio of 16 : 9 is projected onto the screen. The image is projected so that it fills up the entire length of the top of the screen. There will be some empty space at the bottom of the screen. What are the dimensions of that rectangle of empty space?

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Chapter 1 Ratios and rates

1B The capture-recapture technique Learning intentions

• To understand how the capture-recapture technique works • To use the capture-recapture technique to estimate an unknown population

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • Questions involving this technique regularly appear in HSC Examinations

The ‘capture-recapture’ technique is an application of ratio. It is sometimes used to estimate animal populations. This method can also be used to estimate the number of people needing particular services. The technique works by first capturing a random sample of the population. This initial sample is tagged and then released. At a later time a second sample is captured, and the number of recaptured or tagged members are recorded.

The capture-recapture technique

This diagram explains the technique. It is also a useful template for solving problems. In the capture:

Total population to be estimated

Tagged animals

Total captured

Tagged animals

In the recapture:

The two ratios are assumed to be equal.

Therefore, the three known numbers can be used to calculate the unknown number (i.e. the total population to be estimated).

Example 2

Using the capture-recapture technique

Adam is a biologist who is estimating the population of carp in a lake. He randomly captures and tags 120 carp. Two months later he samples 80 carp and finds he has recaptured 5 carp he had tagged. Estimate the number of carp in the lake.

Solution

Explanation

P : 120

Let P be the population to be estimated. 120 of these carp were tagged. In the recapture, 80 carp were caught, of which 5 were tagged.

80 : 5

P 120 = 80 5

Form an equation from the two ratios using fractions.

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1B The capture-recapture technique

P 120 × 80 = × 80 80 5 P = 1920 There are approximately 1920 carp in the lake.

Multiply both sides by 80 to give the estimate. Use the diagram to check that the answer seems reasonable: 1920 : 120

U N SA C O M R PL R E EC PA T E G D ES

= 80 : 5

1920 ÷ 80 = 24 ✓ 120 ÷ 5 = 24 ✓

Now you try

Emily is a biologist who is estimating the population of crabs in a lake. She randomly captures and tags 64 crabs. Three months later she samples 50 crabs and finds she has recaptured 8 of the crabs that she originally tagged. What is Emily’s estimate for the number of crabs in the lake?

Try this experiment

Simulate the capture–recapture technique using discs.

a Mix a large number of discs or blocks in a container. b First sample: Select 10 discs at random. Tag or mark each disc. c Replace these 10 discs into the container. Mix the discs. d Second sample: Select 10 discs at random. Determine the discs that have been selected again or ‘recaptured’. e Use the capture–recapture technique to estimate the number of discs. f Repeat the simulation by selecting 20 discs for each sample. g Repeat the simulation by selecting 30 discs for each sample.

Exercise 1B

Example 2

BUILDING

Patrick is a wildlife officer concerned about the number of rabbits in the park. He catches and tags 80 rabbits and then releases them. A week later he catches 70 rabbits in the park and finds that 10 of them are tagged. Estimate the number of rabbits in the park for Patrick.

Lucas has a large pond with goldfish. He would like to estimate the number of goldfish in the pond. He randomly captures and tags 24 goldfish. Five days later he samples 20 goldfish and finds he has recaptured 4 goldfish, as they are tagged. Estimate the number of goldfish in the pond for Lucas.

The capture-recapture technique was used to estimate the number of dollar coins in a large collection. The collector pulled out 12 dollar coins at random and tagged them. Two weeks later after the collection had been tipped out and put back a number of times, the collector pulled out 42 dollar coins. Three of these coins were tagged. Estimate the number of dollar coins in this collection.

Marjana would like to estimate the number of parrots in the national park. She randomly captures and marks 36 parrots. After three weeks, a second sample of 35 parrots is caught. Marjana finds she has recaptured 6 parrots. Estimate the number of parrots in the national park.

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Chapter 1 Ratios and rates

The number of grasshoppers appears to be increasing in a particular town. James needed an estimate of the grasshopper population. He spent the first day collecting 28 grasshoppers and tagging each insect. On the next day James collected another 28 grasshoppers and found that he had recaptured 4 grasshoppers. What is James’s estimate for the number of grasshoppers in the town?

U N SA C O M R PL R E EC PA T E G D ES

DEVELOPING 6

To determine the number of frogs in a pond, 20 frogs are caught and tagged and then released into the pond. Two days later 10 frogs are caught of which 4 have tags. Estimate the number of frogs in the pond.

The capture–recapture technique was used to estimate the number pencils in a large tub. A teacher pulled out 54 pencils and tagged them before putting them back. One month later, after many uses and shuffling around, another teacher pulled out 51 pencils and saw that 27 of them were tagged. Estimate the pencils in the large tub.

Jesse is a fisherman who is estimating the number of lobsters in a lake. He captures and tags 28 lobsters. Two months later he captures 15 lobsters and finds he has recaptured 5 of the tagged lobsters. What is Jesse’s estimate for the number of lobsters in the lake?

CHALLENGING

A local community is concerned about the number of cane toads. Sumana needs an estimate of the cane toad population. She spends the first day collecting 48 cane toads and tagging each animal. On the next day Sumana collects 44 cane toads and finds that she has recaptured 6 cane toads. What is Sumana’s estimate for the number of cane toads in the community?

Katerina would like to estimate the number of dingoes in the national park. She randomly captures and marks 22 dingoes. After four weeks, a second sample of 20 dingoes is caught. Katerina finds she has recaptured 4 dingoes. Estimate the number of dingoes in the national park.

Tim is a scientist who is concerned about the number of bats in the park. He catches and tags 60 bats and then releases them. Two weeks later he catches 85 bats in the park and finds that 25 of them are tagged. Estimate the number of bats in the park for Tim.

Communicate your reasoning, understanding and fluency 12

A ranger is concerned by the number of foxes in a wildlife park. She needs to estimate the population of the foxes, but she knows that the foxes that have been trapped and tagged previously learn to avoid the trap in the future. If the capture–recapture technique was used, would this lead to population estimate that is higher or lower than the actual population? Explain your answer.

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1C Dividing a quantity in a given ratio

1C Dividing a quantity in a given ratio Learning intentions

• To consolidate prior learning about ratios • To solve problems involving the division of an amount in a given ratio

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

When two people share an amount of money, they usually split it evenly. This is a 1 : 1 ratio or a 50−50 split. Each person gets half of the money. Sometimes, one person deserves a larger share than the other. For example, Olivia and William were paid $100 for doing a job, but Olivia worked for 3 hours and William only worked for 2 hours. In this instance we would be dividing $100 in the ratio 3 : 2.

Dividing a quantity in a given ratio To divide $100 in the ratio 3 : 2

• 3 parts and 2 parts makes 5 parts, so the ratio is

3 2 : 5 5

• Olivia gets

3 2 of $100 (which is $60) and William gets of $100 (which is $40) 5 5

Example 3

Dividing a quantity in a given ratio

Mikhail and Ilya were given $450 to share in the ratio 4 : 5. How much did each get?

Solution

Explanation

4+5=9 4 × 450 = 200 9 5 × 450 = 250 9

Add the two parts of the ratio. Change the ratio into fractions. Multiply each fraction by $450 to determine each share.

Mikhail got $200, Ilya got $250

Check that the answer is reasonable: $200 + $250 = $450

Now you try

Divide $1000 in the ratio 5 : 3.

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Chapter 1 Ratios and rates

Example 4

Dividing a quantity in a given ratio

A man left $6000 to be divided among his three children, Xia, Yui and Zi, in the ratio 5 : 8 : 7, in that order. How much did each child get? Explanation

5 + 8 + 7 = 20 5 × 6000 = 1500 20 8 × 6000 = 2400 20 7 × 6000 = 2100 20

Add the three parts of the ratio. Change the ratio into fractions. Multiply each fraction by $6000 to determine each share.

Xia got $1500, Yui got $2400, Zi got $2100

Check that the answer is reasonable. Do they add to $6000?

U N SA C O M R PL R E EC PA T E G D ES

Solution

Now you try

Divide $1000 in the ratio 1 : 3 : 4.

Exercise 1C

Example 3

Calculate how much each person receives if $100 is shared in the following ratios. a 7:3

b 2:3

c 11 : 9

d 7 : 8 : 10

c 1:5

d 7:5

Divide 240 into the following ratios. a 2:1

BUILDING

b 3:2

Share each amount in the ratio given. a $20 in the ratio 4 : 1 b $20 in the ratio 7 : 3 c $15 in the ratio 1 : 2 d 77 drinks in the ratio 3 : 4 e 100 lollies in the ratio 7 : 13 f 45 kg in the ratio 4 : 5 g 160 books in the ratio 5 : 3 h 360 pencils in the ratio 2 : 7 i 50 g in the ratio 1 : 3 j 60 km in the ratio 8 : 7

Example 4

A bag of 500 grams of chocolates is divided into the ratio 7 : 3. What is the mass of the smaller amount?

At a concert there were 7 girls for every 5 boys. How many girls were in the audience of 8616?

Hazel, Preetish and Andrew invest in a business in the ratio 6 : 5 : 1. The total amount invested is $240 000. How much was invested by the following people? a Hazel

b Preetish

c Andrew

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1C Dividing a quantity in a given ratio

The ratio of residential area to parks in a local community is 17 : 3. The total area of the local community is 40 km2 . What is the area of parks?

U N SA C O M R PL R E EC PA T E G D ES

FPO

Hayley is 15 years old and her brother is 5 years younger. If $200 is shared between them in the ratio of their ages, how much will Hayley receive?

In country town a census showed that there were 5 adults to every 7 children. If the population of the town was 7200, how many children lived there?

DEVELOPING

A punch is made from pineapple juice, lemonade and orange juice in the ratio 5 : 3 : 2.

a How much lemonade is needed if one litre of pineapple juice is used? b How much pineapple juice is required to make 15 litres of punch?

Angus, Ruby and Yuxi share an inheritance of $500 000 in the ratio of 7 : 5 : 4. How much will be received by the following people? a Angus

b Ruby

c Yuxi

In a fruit cake recipe the ratio of mixed fruit to flour to sugar is 5 : 3 : 2. A 250 g packet of mixed fruit is used to make the cake. How much sugar and flour are required?

A delivery load of 8.5 tonnes is to be divided between two stores in the ratio 11 : 6. How much will each store receive?

The load on a bridge is applied in three positions, A, B and C, in the ratio 5 : 7 : 5. If the total load on the bridge is 782 tonnes, what is the load taken at each point?

A jam is made by adding 5 parts fruit to 4 parts of sugar. How much fruit should be added to 1 2 kilograms of sugar in making the jam? 2

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Chapter 1 Ratios and rates

CHALLENGING 16

The perimeter of a rectangle is 42 cm. The ratio of length to breadth is 5 : 2. a What is the length? b What is the breadth? The ratio of $5 to $10 notes in Stephanie’s purse is 3 : 5. There are 24 notes altogether. What is the total value of Stephanie’s $5 notes?

The three sides of a triangle are in the ratio of 1 : 3 : 5. The longest side of the triangle is 16.2 mm. What is the perimeter of the triangle?

A chemical solution is made by mixing acid with water in the ratio 1 : 24. The solution is then taken and mixed with water in the ratio 1 : 3. How much acid would there be in 400 mL of this final solution?

The ratio of the populations of town A to town B is 3 : 8, and the ratio of town B to town C is 5 : 2. If the total population of the three towns is 92 655, find the population of each town.

U N SA C O M R PL R E EC PA T E G D ES

Communicate your reasoning, understanding and fluency 21

Solve the following problem and write a solution for Jarrah to convince him you are correct.

Jarrah has some fuel for his lawnmower in a container. The ratio of petrol to oil is 50 : 1. He has 2.5 litres of fuel already mixed at this ratio, but he needs a different ratio of 30 : 1 for the outboard motor on his boat. How many millilitres of oil does he need to add to this fuel if he wants to use all of it in his boat? Answer correct to the nearest millilitre.

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1D Scale drawings

1D Scale drawings Learning intentions

• To consolidate prior learning about ratios • To find unknown lengths using scale drawings and maps

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Years 7 to 10 • Questions involving scale drawing occasionally appear in HSC Examinations

A scale drawing is a diagram that represents an actual object. The scale factor of a scale drawing is the ratio of the size of the drawing to the actual size of the object. For example, a map is a scale drawing. It is not the same size as the area it represents. The measurements have been reduced to make the map a convenient size. The scale of a drawing may be expressed with or without units. For example, a scale of 1 cm to 1 m means 1 cm on the scale drawing represents 1 m on the actual object. Alternatively, a scale of 1 : 100 means the actual distance is 100 times the length of 1 unit on the scale drawing.

Scale drawing

Scale of a drawing = Drawing length : Actual length Scale is expressed in two ways:

• using units, such as 1 cm to 1 m (or 1 cm = 1 m) • no units, using a ratio such as 1 : 100.

Example 5

Using a scale

A scale drawing has a scale of 1 : 50.

a Find the actual length if the drawing length is 30 mm. b Find the drawing length if the actual length is 4.5 m. Answer to the nearest millimetere.

Solution

Explanation

a Drawing length : Actual length

Write the ratio.

1 : 50

× 30

Write the enlargement factor.

30 mm : 50 × 30 = 1500

Use the enlargement factor.

The actual length is 1500 mm.

State the length. Continued on next page

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Chapter 1 Ratios and rates

b Drawing length : Actual length

Write the ratio.

1 : 50 × 90

× 90 : 4500 mm

Find and use the enlargement factor. State the length.

U N SA C O M R PL R E EC PA T E G D ES

1 × 90 = 90 The drawing length is 90 mm.

Change 4.5 m to 4500 mm, as mm is an appropriate unit for measuring a map.

Now you try

The plans for a new house have been drawn using the scale 1 : 100.

a A door is 5 mm wide on the plan. How wide is the actual door? b A kitchen counter is going to be 2.5 metres long. How long will it be on the plan, in millimetres?

Exercise 1D

Example 5

BUILDING

A scale drawing has a scale of 1 : 100. What is the actual length of these drawing lengths? Express your answer in metres.

a 2 cm c 3.4 cm e 8.5 cm

b 10 mm d 28 mm f 49 mm

2 A scale drawing has a scale of 1 : 25 000. What is the drawing length of these actual lengths? Express your answer in millimeters. a 2 km c 4000 cm e 50 000 mm

b 750 m d 3.5 km f 1375 m

3 Express each of the following scales as a ratio in the form 1 : x. a 1 cm to 2 cm

b 1 mm to 5 cm

c 1 cm to 3 km

The scale on a map is 1 : 1000. Calculate the actual distances if these are the distances on the map. Express your answer in metres.

a Road 20 cm c Pathway 5 cm e Bridge 34 mm

b Shops 10 cm d Parking area 10 mm f Park 80 mm

5 The scale on a map is given as 1 cm = 15 km. What is the actual distance if the distance on the map is: a 2.5 cm? b 45 cm?

Woy Woy

Palm Beach

Penrith

Parramatta

Mona Vale

SYDNEY

Coogee

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1D Scale drawings

DEVELOPING 6

The scale on a map is 1 : 5000. Calculate the map distances if these are actual distances. Express your answer in millimetres. b 80 m e 150 m

c 100 m f 240 m

U N SA C O M R PL R E EC PA T E G D ES

a 50 m d 120 m 7

The scale on a map is given as 1 mm = 50 m. If the distance between two points is 350 m, what is the map distance between these points?

A scale drawing has a scale of 1 : 75. Find the:

a actual length if the drawing length is 15 mm b drawing length if the actual length is 3 m.

CHALLENGING

The Parkes radio telescope dish has a diameter of 64 metres. The image shown uses a photograph of the dish.

a Determine a scale for the image. b Estimate the height of the top of the dish above the ground.

The scale of a model is 2 : 150. For the actual lengths listed, calculate the model lengths. Express your answer in millimeters.

a 75 cm d 45 m

b 180 cm e 6m

c 300 cm f 36 m

A scale drawing of the space shuttle is shown here. The actual length of the space shuttle is 43 metres.

a What is the scale factor? b Calculate the length of the wing span to the nearest metre. c Calculate the breadth (b) of the shuttle to the nearest metre. d What is the length of the nose of the shuttle to the nearest metre?

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Chapter 1 Ratios and rates

Communicate your reasoning, understanding and fluency Sarah is going to make a scale model of the main arch of Sydney Harbour Bridge, without the pylons. The model will be displayed on a table which is 1.2 metres long. The highest point on the arch of the Sydney Harbour Bridge is approximately 134 metres above sea level. The arch spans 503 metres. Use this information above and the photograph of the bridge to answer these questions, showing all calculations.

U N SA C O M R PL R E EC PA T E G D ES

a Which scale would give Sarah the largest model that fits on the table: 1 : 100, 1 : 500 or 1 : 1000? Explain your answer, showing all calculations. b Using that scale, with the table as sea level, approximately how tall will her model be in millimetres? c The deck of the bridge is 49 metres wide. How many millimetres will this be on Sarah’s model? d According to the photograph below, what is the clearance between sea level and the middle of the road deck on the real bridge and Sarah’s model?

FPO

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1E Building plans

1E Building plans Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the relationship between building plans and elevations • To gather information from a building plan • To find unknown lengths and areas using building plans Past, present and future learning

• Some aspects of this section may be new to students • Questions involving building plans occasionally appear in HSC Examinations

A plan is a view of an object from the top. It is looking down on the object. A house plan is a horizontal section cut through the building showing the walls, windows, door openings, fittings and appliances. An elevation is a view of an object from one side, such as a front elevation or side elevation. House elevations are rarely a simple rectangular shape but show all the parts of the building that are seen from a particular direction. House elevations are a vertical section parallel to one side of the building.

Side elevation

Floor plan

Front elevation

Plans and elevations

A plan is a view of an object from the top.

An elevation is a view of an object from one side, such as a front elevation or side elevation.

Example 6

Drawing a plan and elevation

Draw the plan view, front elevation and side elevation of this object.

Plan view

Side elevation Front elevation

Continued on next page

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Chapter 1 Ratios and rates

Solution Plan

Explanation Front elevation

Imagine what you can see from above, the front and the side.

Side elevation

U N SA C O M R PL R E EC PA T E G D ES

Now you try

Draw the plan view of this object.

Building plans

A floor plan for a home is shown below. Building plans are a very common example of scale drawings. They are drawn using a scale factor such as 1 : 150. This allows the dimensions of a house to be determined by measurement and calculation. BEDROOM 4

RUMPUS ROOM

BEDROOM 3 robe

robe

LAUNDRY

VERANDAH

sliding door

FAMILY

sliding door

BATHR OOM

DINING

LEISURE

KITCHEN

BEDROOM 2

br robe

robe

penny

robe

W.I.R.

DOUBLE GARAGE

robe

ENSUITE

ENTRY

MASTER SUITE

PORTICO

Scale 1:150

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1E Building plans

Common floor plan symbols Shower: shower without a bathtub

Window: glass window in a solid wall

Toilet: toilet located on wall

Kitchen sink: two-compartment kitchen sink

Bathtub: bathtub showing location of drain

U N SA C O M R PL R E EC PA T E G D ES

Door swing: indicates direction the door opens

Example 7

Finding measurements from a house plan

A building plan is shown for the ground floor of a home. The scale is 1 : 150.

FAMILY

sliding door

a How many internal doors are there? b What is the meaning of PWDR? c What is the length of the house? d What are the internal dimensions of the double garage?

DINING

KITCHEN

pantry

SITTING

LAUNDRY

PWDR

DOUBLE GARAGE

STUDY

ENTRY

Scale 1 : 150

PORTICO

Solution

Explanation

a 4 internal doors

Pantry, laundry, powder room, garage.

b Powder room Continued on next page

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Chapter 1 Ratios and rates

c Drawing length is 12.6 cm Actual length = 12.6 × 150 cm = 1890 cm or 18.9 m d Drawing length is 4.1 × 3.5 cm Actual length = 4.1 × 150 cm

1 : 150 × 13.7

× 13.7 12.6 :

Use your ruler to measure the dimensions of the garage.

U N SA C O M R PL R E EC PA T E G D ES

= 615 cm or 6.15 m

Use your ruler to measure the length of the house.

1 : 150

× 4.3

4.1 :

Actual breath = 3.5 × 150 cm

= 525 cm or 5.25 m

1 : 150

× 3.6

Dimensions are 6.15 × 5.25 m.

3.5 :

Now you try

Using the plan in Example 7:

a how wide is the house at its widest point, in metres? b what are the internal dimensions (in metres) of the sitting room?

Exercise 1E

Example 6

BUILDING

Draw the plan, front elevation and side elevation for these objects. a

Plan view

Front elevation

Side elevation

Front elevation

Plan view

Side elevation

2 What are the objects represented by these views? Plan

Front elevation

Side elevation

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1E Building plans

Draw the plan view, front elevation and side elevation for these objects. a

U N SA C O M R PL R E EC PA T E G D ES

4 A section of a floor plan is shown opposite.

a What room is shown in the diagram? b What symbol is used for a shower? c What symbol is used for a door? d What does ‘WIR’ represent on the plan?

B.I.W.

W.I.R.

A section of a floor plan is shown opposite.

a What are the dimensions of the kitchen? b What symbol is used for the sink? c What symbol is used for the cooktop? d What does ‘Ref’ represent on the plan? e What does ‘P’ represent on the plan?

Space Kitchen 3150 × 2700 UO

Ref Space

A section of a floor plan is shown opposite.

a What are the dimensions of the bedroom? b What symbol is used for a window? c What does ‘BIW’ represent on the plan? d What is the length of the bedroom on the plan? e Calculate a scale for the floor plan.

Bedroom 3 3300 × 2800

Linen

B.I.W.

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Chapter 1 Ratios and rates

A section of a floor plan is shown opposite. BED 1

WALK-IN ROBE

4.15 × 3.80 13′8″ × 12′6″

A second-storey building plan is shown for a Masterton home.

a What are the dimensions of the third bedroom? Answer in metres. b What are the dimensions of the master bedroom? Answer in metres. c Use your ruler to help you estimate a scale for this plan. d Use your ruler and your scale from part c to find the breadth of the house. Answer in metres. e Calculate the area of the void. Answer correct to the nearest square metre. f Calculate the area of the ensuite. Answer correct to the nearest square metre.

Bedroom 3 3520 × 3520

B.I.W.

Example 7

B.I.W.

U N SA C O M R PL R E EC PA T E G D ES

a What are the dimensions of the bedroom? b What symbol is used for the toilet? c What is the length of the bedroom on the plan? d Calculate a scale for the floor plan. e What are the dimensions of the walk-in robe? f What is the area of the bedroom? Answer in square metres, correct to two decimal places.

Bedroom 2 2560 × 3520

Linen

Rumpus 5280 × 4320

Walk in Robe

B.I.W.

Bedroom 4 2280 × 3360

Void

Master Bedroom 4800 × 4160

A rumpus room is built measuring 5.5 m by 4.7 m. The floor plan uses a scale of 1 : 100. A concrete slab with a depth of 100 mm is used to build the rumpus room.

a What is the area of the rumpus room on the plan? Answer in square millimeters. b What is the volume of concrete for the rumpus room? Answer in cubic millimeters. c What is the volume of concrete for the rumpus room if the slab depth is 200 mm? Answer in cubic millimetres.

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1E Building plans

The front elevation of a house is shown opposite (scale 1 : 200).

U N SA C O M R PL R E EC PA T E G D ES

a What is the breadth of the house? Answer in metres. b What is the height of the chimney? Answer in metres. c What are the dimensions of the front door? Answer in metres. d What are the dimensions of the window on the right-hand side? Answer correct to the nearest centimetre. e What is the area of the window? Answer correct to the nearest square centimetre. f What is the area of the small triangular gable above the front door? Answer correct to the nearest square centimetre.

DEVELOPING

The front elevation of a house is shown.

TILED ROOF @ 22.5º PITCH

WP12 MOULDING

FACE BRICKWORK

12D

872

A 40 STEP TO BALCONLY

F.C SHEET NFLL OVER

NGL

SELECTED GLAZING TO SIDELIGHTS AND ENTRY DOORS

2698 GARAGE

C.L

2150 (NM) CAR DOOR OPENING

F.L

TILED ROOF @ 20º TO GARAGE ONLY

2162 HEIGHT

2590

C.L

3740 WALL

F.L

2248 OPENING

2440

4164 HEIGHT

C.L

F.L

SELECTED PANELIFT GARAGE DOOR WITH TIMBER LINTEL OVER

COLOURED RENDER FINISH TO PORTICO/BALCONY

a What is the height of the first storey? Answer in metres. b What is the height of the garage? Answer in metres. c What is the angle of the pitch of the roof? d How many windows are at the front of the house?

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Chapter 1 Ratios and rates

A building plan is shown below. The house length is 20 m and the breadth is 18 m.

U N SA C O M R PL R E EC PA T E G D ES

a What are the dimensions of the living room? Answer in metres. b What is the area of the living room? Answer correct to the nearest square metre. c The house and veranda were built on a concrete slab 200 mm thick. How much concrete was used? Answer correct to the nearest cubic metre.

CHALLENGING

A building plan is shown for a home. The house length is 24 m (includes portico) and the breadth is 15 m (includes garage).

BEDROOM 4

RUMPUS ROOM

BATHROOM

BEDROOM 3

DINING

FAMILY

LAUNDRY

Pantry

SITTING

BEDROOM 2

EN SUITE

DOUBLE GARAGE

ENTRY

MASTER SUITE

W.I.R

a What is a suitable scale for this plan? b What are the dimensions of the verandah? Answer correct to the nearest tenth of a metre. c What are the dimensions of the double garage? Answer correct to the nearest tenth of a metre. d What are the dimensions of bedroom 3? Answer correct to the nearest tenth of a metre. e Calculate the area of the sitting room. Answer correct to the nearest square metre. f What is the cost of carpeting the sitting room if the cost of the carpet is $140 per square metre? Answer correct to the nearest dollar.

VERANDAH

KITCHEN

PORTICO

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1E Building plans

A section of a building plan is shown opposite. The dimensions of the family room are 5.5 metres by 6.0 metres.

FAMILY

U N SA C O M R PL R E EC PA T E G D ES

a Estimate a suitable scale for this building plan. b What is the combined length of the family, kitchen and sitting rooms? Answer correct to the nearest tenth of a metre. c The family, dining, kitchen and sitting rooms are to be tiled. Calculate the combined area of these rooms. Answer to nearest square metre. d Ceramic tiles measuring 300 × 300 mm are to be laid in these rooms. How many tiles are required? e What assumption has been made to the answer in part d? f The family, dining, kitchen and sitting rooms are built on a concrete slab with a thickness of 0.15 m. What is the volume of concrete used for the slab?

Sliding door

DINING

KITCHEN

SITTING

Communicate your reasoning, understanding and fluency Planning a house on an A4 sheet of paper.

Take a blank A4 sheet of paper (210 mm by 297 mm), which will be the size of the plan of your new rectangular block of land and the one-storey house you will build upon it. The scale is 1 : 50. Answer the following questions, justifying every answer with calculations.

a What are the dimensions of your block of land, in metres? b What is the area of your block of land, in square metres? c Your house must not be closer than 900 mm to any of the boundaries of the block. Draw the footprint of the largest house you can build onto your A4 sheet. d What is the area of the largest house you can build, in square metres? e Search the internet to find some plans for a house in Australia with an area of 115 square metres. Also search for the typical building costs per square metre for a house in Australia. How much might it cost to buy this block of land and build this house in the suburb or town you like?

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Chapter 1 Ratios and rates

1F Perimeter, area and volume from plans Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To determine estimates of measurements from photos, diagrams and plans • To use estimates of measurements to calculate perimeter, area and volume • To estimate areas using the trapezoidal rule Past, present and future learning

• Some aspects of this section may be new to students • The trapezoidal rule was introduced in Year 11 • Questions involving these concepts regularly appear in HSC Examinations

Perimeter

The perimeter of a piece of land can be found using aerial photographs such as those obtained from Google Earth. It is much quicker than actually measuring the land.

Steps: 1 Determine the borders of the piece of land.

2 Access a land measurement website and view the aerial photograph of the land. 3 Using the online tools, click along the perimeter of the land.

Perimeter

When calculating the perimeter of land, multiply side lengths on the aerial photograph by the scale.

Example 8

Calculating distance using a scale

This aerial view on Google Earth shows a 50 m Olympic swimming pool outlined with the ‘path’ tool.

a What is the scale used on the image? b Use the scale to determine the length of the boat in the picture correct to the nearest metre. Solution

Explanation

a Length of swimming pool is 3.1 cm. Scale 3.1 cm to 50 m (or 5000 cm). Scale is approximately 1 : 1613.

3.1 cm : 5000 cm

÷ 3.1

1 cm : 1613 cm

b Boat length is 2.0 cm.

Boat length = 2.0 × 1613 = 3226 cm

1 cm : 1613 cm

×2

2 cm : 3226 cm

= 32 m Length of the boat is 32 metres, correct to the nearest metre. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

1F Perimeter, area and volume from plans

Now you try The next boat to arrive at the jetty is 3.5 cm long in the photograph in Example 8. How long is the actual boat, correct to the nearest metre?

Area

U N SA C O M R PL R E EC PA T E G D ES

The area of a block of land can be estimated and calculated directly from scale diagrams and aerial photographs. To estimate the area of the land, divide the diagram into square grids and count the number of squares. To calculate the exact area of land, use the appropriate formula. Composite shapes are divided into plane shapes. The dimensions of the shapes are determined using the scale.

Area

To estimate the area of land, divide it into square grids and count the number of squares. To calculate the exact area of land, use the appropriate formula.

Example 9

Calculating the area of land

A warehouse is shown on Google Earth. The red scale bar represents 42 m at the ground.

a Calculate the lengths of the sides of the pentagon occupied by the warehouse building using the scale. Answer correct to the nearest tenth of a metre. b Calculate the area of land occupied by the warehouse building. Answer correct to the nearest square metre. Solution

Explanation

a Scale bar is 1.2 cm represents 42 cm Scale is 1 cm to 35 m

÷ 1.2

AB = 4.2 × 35 = 147 m

1.2 cm : 42 m

÷ 1.2

1 cm : 35 cm

BC = 1.2 × 35 = 42 m

CE = 1.6 × 35 = 56 m

EF = 3.2 × 35 = 112 m

AF = 2.5 × 35 = 87.5 m

Total area = area of rectangle ABDF − area of triangle CDE

A = lb = AB × AF = 147 × 87.5 = 12 863 m2 1 1 A = bh = CD × DE = 45.5 × 35 = 796.3 m2 2 2 A = 12 863 − 796.3 = 12 066.2 m2

Area of warehouse is 12 066 m2

Area of ABDF Area of CDE

Area of warehouse = area ABDF − area CDE Write your answer, rounding to the nearest square metre.

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Chapter 1 Ratios and rates

Now you try The Pentagon is the headquarters building of the United States Department of Defence.

U N SA C O M R PL R E EC PA T E G D ES

a Do the five sides of the building appear to be equal? b Use the scale in the bottom right-hand corner to estimate the perimeter of the building, in metres. Answer correct to the nearest 100 metres. c By dividing the building into a trapezium and a triangle, estimate the area covered by the Pentagon, including the central courtyard. Answer correct to the nearest thousand square metres.

Volume of water

If water covers an area (such as the bottom of a pool, tank or reservoir), to a constant depth, its volume can be calculated using V = Ah

A = area of surface h = depth of water

b The trapezoidal rule for area A ≈ d f + dl is used to estimate the area of a piece of land or a body of water 2 with an irregular boundary such as a reservoir or dam. A is the area, b is the breadth between the parallel sides, d f is the distance along the first parallel side, dl is the distance along the last parallel side. Note that in Year 11, h was used for the breadth between the parallel sides, but here we will use b since h will be used for height or thickness of the shape. So if a body of water such as a pool has a surface area of 100 square metres and the area of its bottom is also 100 square metres (because it has a constant depth), and that depth is 2 m, then: V = Ah

V = 100 × 2 V = 200 m3

If the area of a reservoir is A square metres, and h metres of rain falls on it (with no water running in or out), then its volume will increase by Ah cubic metres.

Volume using V = Ah and the trapezoidal rule for area The trapezoidal rule: b A≈ d f + dl 2 A : Area of shape

b : Breadth between the parallel sides

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1F Perimeter, area and volume from plans

V = Ah A : Area of the body of water or the cross−section of the prism object

h : Height or thickness

U N SA C O M R PL R E EC PA T E G D ES

This rule only applies to bodies or prisms of uniform cross–section.

Example 10

Calculate volume with the trapezoidal rule for area

The shape of the water in a pool of constant depth with an irregularly-shaped bottom and top surface is shown.

a Apply the trapezoidal rule for area to approximate the volume of water in this pool. b If the pool is empty and 100 mm of rain falls into it, how many litres of water are collected in the pool?

Solution b a A≈ d f + dl 2 8 A ≈ (6 + 4) 2 A ≈ 40 m2 V = Ah

1.5 m

Explanation

Write the trapezoidal rule for area.

Substitute b = 8, d f = 6, and dl = 4 into the formula.

Evaluate to find the area of the pool in square metres. Write the formula for volume of a right prism.

V ≈ 40 × 1.5 3

V ≈ 60 m

The volume of water in the pool is approximately 60 cubic metres. 100 b 100 mm = m = 0.1 m. 1000

Substitute A ≈ 40 and h = 1.5 into the formula.

Evaluate to find the volume of water in the pool in cubic metres. Write the answer in words.

Since the area of the water added is the same as the area of the pool, we can repeat the steps in part a with the formula for the volume of a right prism, and just change h. It has been given in millimetres and needs to be changed to metres. There are 1000 mm in a metre. Divide 100 mm by 1000.

V ≈ 40 × 0.1 = 4 m3

Substitute A ≈ 40 and h = 0.1 into the formula.

The volume of water in the pool will be approximately 4 cubic metres. Every cubic metre holds 1000 litres 4 × 1000 = 4000 litres.

Write the answer in words.

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Chapter 1 Ratios and rates

Now you try The diagram below shows the top surface of a swimming pool. The water is 1.5 metres deep. 12 m

U N SA C O M R PL R E EC PA T E G D ES

a Use the trapezoidal rule to find the approximate area of the surface. b Hence, find the volume of the pool. c How many litres of water will fit in the pool?

10 m

Exercise 1F

Example 8

BUILDING

The length of the Olympic swimming pool shown opposite is 50 metres.

a Calculate a scale for the aerial photograph. b What are the dimensions of the aerial photograph? (Answer to the nearest metre.) c What is the length from A to B? d What is the length from B to C? e What is the length from C to A? f What is the perimeter of ABC? g Calculate the area of the land marked ABC. Assume a right angle at C.

Example 9

A sports field is shown opposite. The scale is marked by the red bar, which represents 50 m on the ground. Answer the following questions, correct to the nearest metre or square metre.

a What is the length of the grandstand highlighted by the yellow bar? b Calculate the length of the field. c Calculate the breadth of the field. d What is the perimeter of the field? e What is the area of the field? f What is the length of the diagonal of the field?

Two circular buildings, A and B, at Sydney airport are shown. The red scale represents 25 m on the ground. Answer the following questions, correct to the nearest metre or square metre.

a What is the radius of building A? b What is the diameter of building B? c What is the circumference of building A? d What is the area of building A? e What is the radius of the building B? f What is the circumference of building B?

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1F Perimeter, area and volume from plans

Calculate the distance between these places, to the nearest metre, using online tools.

U N SA C O M R PL R E EC PA T E G D ES

a Government House to the Sydney Conservatorium of Music b Wynyard and Town Hall stations c Tumbalong Park to the Archibald fountain d University of Sydney to Moore Park e Centennial Park to Bondi beach f Kings Cross to Woolloomooloo

The map below has a scale of 1 : 10 000. Estimate the area of the following sections of land. a A b B c Miles Island

Example 10

Find the volume of the shapes with the following dimensions using the trapezoidal rule for area and the formula for volume of a right prism.

a b = 14 m, d f = 28 m, dl = 36 m and h = 8 m b b = 9 m, d f = 45 m, dl = 25 m and h = 6

The diagram shows the base of a pool. Estimate the volume of the pool if its depth is everywhere 1.8 m. Answer correct to the nearest cubic metre.

10 m

A 3m

15 m

DEVELOPING

A farm is shown opposite. Answer the following questions correct to the nearest metre or square metre.

a What is the perimeter of paddock A? b What is the perimeter of paddock B? c What is the perimeter of paddock C? d What is the perimeter of the L-shaped paddock? e What is the area of paddock A? f What is the area of paddock B? g What is the area of paddock C? h What is the area of the L-shaped paddock?

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Chapter 1 Ratios and rates

A diagram has a scale of 1 : 100. What is the actual area of these shapes, given the following drawing lengths? Express your answer correct to the nearest square metre.

U N SA C O M R PL R E EC PA T E G D ES

a Square with a side length of 10 mm b Rectangle with a length of 2 cm and a breadth of 3 cm c Right-angled triangle with a base length of 25 mm and perpendicular height of 40 mm d Circle with a radius of 20 mm e Semicircle with a diameter of 50 mm

CHALLENGING

Asha took two measurements at 120 m intervals across the irregular-shaped reservoir in the diagram. The measurements were 100 m and 80 m.

a Estimate the surface area of the reservoir. Answer correct to the nearest square metre. b Asha measured the depth of water and found it to be 5 m everywhere. Estimate the volume of the reservoir. Answer correct to the nearest cubic metre.

100 m

120 m

80 m

120 m

150 mm of rain fell on the reservoir in Question 10, which has vertical banks so no water flows in or out and the depth does not vary.

a What is the volume of water in the reservoir after the rain? b After the rain, the farm that depends on the reservoir needs an extra 15 000 cubic metres of water for the next month. How much rain needs to fall in that time so that when the extra water has been pumped out, the water level in the reservoir remains the same? Answer correct to the nearest millimetere.

Communicate your reasoning, understanding and fluency 12

Maurice used three applications of the trapezoidal rule to estimate the area of this section of land. Explain why his estimate is larger than the actual area of the land.

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1G Rates and best buys

1G Rates and best buys Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the difference between ratios and rates • To convert rates between different units • To solve problems involving rates in a variety of practical contexts • To solve problems involving distance, speed and time Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Years 7 to 10 • Questions involving rates regularly appear in HSC Examinations

Rates

A ratio is a comparison of amounts of the same unit. The ratio 1 cm to 5 cm is written as 1 : 5, without the units.

A rate is a comparison of amounts with different units. For example, we may compare the distance travelled with the time taken. In a rate the units are different and must be specified. The order of a rate is important. A rate is written as the first amount per one of the second amount. For example, $2.99/kg represents $2.99 per one kilogram and 80 km/h represents 80 kilometres per one hour.

We are constantly interested in rates of change and how things change over a period of time. There are many examples of rates such as:

• Growth rate: The average growth rate of a child from 0 to 15 years of age. • Running rate: Your running pace in metres per second. • Typing rate: Your typing speed in words per minute.

• Wage rate: The amount of money you are paid per hour.

• Fuel consumption: Litres per 100 km or km per litres are both used.

The unitary method

Many problems can be solved using the unitary method. The word ‘unitary’ comes from ‘unit’ meaning ‘one’.

The unitary method involves finding one unit of a quantity by division. This result is then multiplied to solve the problem.

Using the unitary method Use division to get down to 1, then multiply to find the required amount.

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Chapter 1 Ratios and rates

Example 11

Using the unitary method for fuel consumption

A car can travel 360 km with 30 litres of petrol. At this rate, how far can it travel with 7 litres of petrol? Solution

Explanation 30 litres for 360 km

÷ 30

U N SA C O M R PL R E EC PA T E G D ES

1 litre for 12 km

Write the given information. Use division to get down to 1 litre. Multiply by 7 to get the required amount.

×7

7 litres for 84 km

The car can travel 84 km.

Answer the question and check it is reasonable (it should be considerably less than 360 km).

Now you try

a A car travelled 600 km with 50 litres of petrol. At this rate, how far could it drive with 37 litres of petrol? b A car travelled 600 km with 50 litres of petrol. At this rate, how many litres of petrol would it need for a 1000 km journey?

Example 12

Best buys

A shop sells the same cans of drink in the following ways:

One can for $2.40, six-pack of cans for $11.95 or 24-pack of cans for $39.95.

a Which is the best buy? b How much is saved when buying the 24-pack rather than 24 single cans? Solution

Explanation

a One single drink will cost $2.40

Start with the cost of a single can of drink.

$11.95 ÷ 6 ≈ $1.99

The drinks in the six-pack are cheaper.

$39.95 ÷ 24 ≈ $1.66

The drinks in the 24-pack are cheaper.

The best buy is the 24-pack. b 24 × $2.40 = $57.60

Compare 24 individual drinks to the 24-pack.

$57.60 − $39.95 = $17.65

This is a saving of $17.65.

Now you try

A breakfast cereal comes in three sizes: • 290 grams for $7

• 450 grams for $9.55

• 765 grams for $10.80.

What is the price per 100 grams for each size?

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1G Rates and best buys

Speed

U N SA C O M R PL R E EC PA T E G D ES

Speed is a rate that compares the distance travelled to the time taken. The speed of a car is measured in kilometres per hour (km/h). The speedometer in a car measures the instantaneous speed of a car. They are not totally accurate but have a tolerance of 5%. GPS devices are capable of showing speed readings based on the distance travelled per time interval. Most cars also have an odometer to indicate the distance travelled by a vehicle.

Speed S =

D D or T = or D = S × T T S

D = Distance

S = Speed

‘Road sign’ on the right is used to remember the formulas. Hide the required quantity to determine the formula.

T = Time

Example 13

Solving problems involving speed

a Find the average speed of a car that travels 341 km in 5 hours. b How long will it take a vehicle to travel 294 km at a speed of 56 km/h ? Solution D a S = T 341 = 5

Explanation

Write the formula.

Substitute 341 for D and 5 for T into the formula.

= 68.2 km/h

Evaluate.

D S 294 = 56

Write the formula.

= 5.25 h or 5 h 15 min

Evaluate and express the answer correct to the nearest hour.

b T=

Substitute 294 for D and 56 for S into the formula.

Now you try Usain Bolt ran 100 metres in 9.58 seconds. What was his average speed in metres per second, correct to one decimal place?

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Chapter 1 Ratios and rates

Converting between rates The speed of a moving car could be expressed in ‘kilometres per hour’ (km/h) or ‘metres per second’ (m/s). There are many different ways to convert a speed from one pair of units to a different pair. For example, you could:

U N SA C O M R PL R E EC PA T E G D ES

1 convert the number of kilometres to metres, and; 2 convert the number of hours to seconds, then; 3 reduce the rate to simplest form, in the same way you reduce a ratio to simplest form.

However, the aim is for the number of seconds to be 1, because m/s means ‘metres in 1 second’.

Note: Some people use rules like ‘divide the speed by 3.6’. These rules have limited applicability.

Example 14

Converting between rates

Convert 100 km/h into metres per second.

Solution

Explanation

100 km in 1 hour 100 000 m in 60 minutes

Write the rate given. Convert 100 km to metres and 1 hour to minutes.

100 000 m in 3600 seconds

Convert from minutes to seconds.

÷ 3600

_______ m in 1 second

Reduce, just like a ratio, to the form ___ : 1 In this case, divide both numbers by 3600.

100 km/h = 27.8 m/s (to one decimal place) Now you try

A hippopotamus can run at a speed of 30 km/h over short distances.

a At this speed, how many metres can it run in one second? Give your answer correct to one decimal place. b How many seconds would the hippopotamus take to run 100 metres? Give your answer correct to the nearest second.

Exercise 1G

Example 11

Convert to the rate shown.

a $100 in 4 h is a rate of $ /h c 700 L in 10 h is a rate of L/h e $1.20 for 2 kg is a rate of $ /kg g 1200 rev in 4 min is a rate of rev/min

BUILDING

b 240 m in 20 s is a rate of m/s d $39 in 12 h is a rate of $ /h f 630 km in 60 L is a rate of km/L h A rise of 20◦ in 4 h is a rate of ◦ /h

Express each rate in simplest form using the rates shown in the brackets. a 300 km on 60 L [km per L] c $640 for 5 m [$ per m] e 78 mg for 13 g [mg per g] g 20 g for 8 m2 [g per m2 ]

b 15 m in 10 s [m per s] d 56 L in 0.5 min [L per min] f 196 g for 14 L [g per L] h 75 mL for 5 min [mL per min]

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1G Rates and best buys

Use the rates provided to answer the following questions.

U N SA C O M R PL R E EC PA T E G D ES

a Cost of apples is $2.50/kg. What is the cost of 5 kg? b Carpet costs $28/m2 . What is the cost for 7 m2 ? c Cost savings are $35/day. How much is saved in 5 days? d Cost of a chemical is $65/100 mL. What is the cost of 300 mL? 1 e Cost of mushrooms is $5.80/kg. What is the cost of kg? 2 f Distance travelled is 1.2 km/min. What is the distance travelled in 30 minutes? g Concentration of a chemical is 3 mL/L. How many mL of the chemical is needed for 4 L? h Concentration of a drug is 2 mL/g. How many mL is needed for 10 g?

Example 12

4 Which is the best buy: 10 litres for $5 or 12 litres for $7.40?

Example 13

Use the information provided on speed to answer the following questions.

a I walk at 5 km/h. How far can I walk in 4 hours? b A car travels at 80 km/h. How far will it travel in 2.5 hours? c A plane is travelling at 600 km/h. How far will it travel in 30 minutes? d A train took 7 hours to travel 665 km. What was its average speed? e Ryder runs a 42.4 km marathon in 2 hours 30 minutes. Calculate his average speed. f A spacecraft travels at 1700 km/h for a distance of 238 000 km. How many hours did it take?

Example 14

Convert each rate to the units shown in the brackets.

a 39 240 m/min [m/s] c 88 cm/h [mm/h] e 0.4 km/s [m/s] g 6.09 g/mL [mg/mL]

b 2 m/s [cm/s] d 55 200 m/h [m/min] f 57.5 m/s [km/s] h 4800 L/kL [mL/kL]

Mia earns $37.50 per hour working in a cafe.

a How much does Mia earn for working a 9-hour day? b How many hours does Mia work to earn $1200? c What is Mia’s annual income if she works 40 hours a week? Assume she works 52 weeks in the year.

A delivery driver delivers 1 parcel on average every 20 minutes. How many hours does it take to deliver 18 parcels?

Water is dripping from a tap at a rate of 5 L/h. How much water will leak in one day?

10 A cricket team scores runs at a rate of 5 runs/over in a match. How many overs did it take to score 90 runs?

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Chapter 1 Ratios and rates

A bulldozer is moving soil at a rate of 22 t/h. How long will it take to move 55 t?

If Leo can march at 7 km/h, how far can he march in 2.5 hours?

Edward saves $40/week, how long should it take to save $1000?

Alexandra jogs 100 metres in 20 seconds. How many seconds would it take her to jog one kilometre?

U N SA C O M R PL R E EC PA T E G D ES

A car travels at a rate of 50 metres each second. How many kilometres does it travel in: a one minute?

b one hour?

Convert the following speeds to metres per second. Answer correct to the nearest whole number. a 60 km/h

b 260 km/h

17 An athlete runs 100 metres in 10 seconds. What is his average speed in kilometres per hour?

DEVELOPING

Natural gas is charged at a rate of 1.4570 cents per MJ.

a Find the charge for 12 560 MJ of natural gas. Answer to the nearest dollar. b The charge for natural gas was $160.27. How many megajoules were used?

Kirra’s council rate is $2915 p.a. for land valued at $265 000. Lucy has a council rate of $3186 on land worth $295 000 from another council.

a What is Kirra’s council charge as a rate of $/$1000 valuation? b What is Lucy’s council charge as a rate of $/$1000 valuation?

Fatima’s car uses 9 litres of petrol to travel 100 kilometres. Petrol costs $1.50 per litre.

a What is the cost of travelling 100 kilometres? b How far can she drive using $50 worth of petrol? Answer to the nearest kilometre.

CHALLENGING

A popular soft drink is sold in these ways: • a pack of 12 300 mL bottles for $12

• a pack of four 330 mL bottles for $8.50 • a 600 mL bottle for $4

• a 1.25 L bottle for $3.25 • a 2 L bottle for $4.20.

Which is the best buy?

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1G Rates and best buys

A motor bike is moving at a steady speed. When the speed is 90 km/h the bike consumes 5 litres of petrol for every 100 kilometres travelled.

U N SA C O M R PL R E EC PA T E G D ES

a The petrol tank holds 30 litres. How many kilometres can the bike travel on a full tank of petrol when its speed is 90 km/h? b When the speed is 110 km/h the bike consumes 30% more petrol per kilometre travelled. Calculate the number of litres per 100 kilometres consumed when the bike travels at 110 km/h.

A plane travelled non-stop from Los Angeles to Sydney, which is a distance of 12 057 kilometres, in 13 hours and 30 minutes. The plane started with 180 kilolitres of fuel, and on landing had enough fuel to fly another 45 minutes.

a What was the plane’s average speed in kilometres per hour? Answer correct to the nearest whole number. b How much fuel was used? Answer correct to the nearest kilolitre.

Communicate your reasoning, understanding and fluency 24

Sam bought a new car during 2026 for his business. Eight months later, he calculated the following: • kilometres driven: 15 900 km • servicing: $501.95 • petrol: $2094.76 • tolls: $961.92

• depreciation: $3501.50 • registration: $617.33

• compulsory third party insurance: $312.67 • comprehensive insurance: $1019.87.

Create a spreadsheet which Sam can use to track his spending on his car. Include a row for each of the expenses. Use the spreadsheet to answer the following and write down the formula used to do your calculations.

a Calculate the total amount spent on the car during those 8 months. b Calculate the cost of operating the car, in dollars per month. c Calculate the cost of operating the car, in dollars per 100 km. d Calculate the average distance travelled, in kilometres per month. e Based on these 8 months, how much will the car cost him for the whole year?

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Chapter 1 Ratios and rates

1H Distance-time graphs Learning intentions

• To interpret and analyse distance-time graphs to solve problems • To solve problems involving distance, speed and time

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Years 9 and 10 • Questions involving distance-time graphs occasionally appear in HSC Examinations

• a horizontal axis which shows the time taken, in hours or

minutes • a vertical axis which shows the distance from some place, in centimetres, metres or kilometres • a series of straight lines or curves which show the position of the object during the journey.

Distance from home

A distance-time graph usually contains:

Time

The distance-time graph above shows a journey away from ‘home’ then returning to the starting point.

• The horizontal sections indicate ‘breaks’ or ‘rests’ in the journey, in which the people were stationary. • The straight-line sections indicate that they moved at a constant rate. • The steeper the line, the faster they were moving.

Calculating speed from a distance-time graph

This distance-time graph shows a person travelling from Point A to Point B at a constant speed. a How long did the journey take? b How far was the journey? c What was the constant speed?

Distance (km)

Example 15

1 2 3 4 5 6 Time (hours)

Solution

Explanation

a 5 hours b 25 km c 5 km in 1 hour = 5 km/h

See the vertical red dashed line on the graph. See the horizontal red dashed line on the graph. In the first hour, the person moved from 0 km to 5 km. That is 5 km in 1 hour.

Alternative solution

Treat the information as you would treat a ratio.

25 km in 5 hours ÷5

÷5 5 km in 1 hour

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1H Distance-time graphs

Now you try Distance (km)

This distance-time graph shows a train travelling from Town A to Town B at a constant speed.

200

100

U N SA C O M R PL R E EC PA T E G D ES

a How long did the journey take? b How far was the journey? c What was the constant speed?

1 2 Time (hours)

Example 16

Interpreting and analysing a distance-time graph

This distance-time graph shows Mira’s cycling journey.

Distance from home (km)

a Describe what happened during the first hour, including the distance travelled and the speed. b How far did Mira ride altogether? c What was her average speed for the second (middle) part of the journey, after the rest? d Including the breaks, what was her average speed for the whole journey?

40 30 20

6 a.m. 7 a.m. 8 a.m. 9 a.m. 10 a.m.11 a.m. Time (hours)

Solution

Explanation

a She started at 6 a.m. and rode 10 km in one hour, so her speed was 10 km/h.

The first hour is from 6 a.m. to 7 a.m. She started from 0 km and made it to 10 km.

b 30 km + 30 km = 60 km

30 km away from home then 30 km back to home.

c She rode 10 km. From 8:30 to 9:00 is 30 min or 0.5 h 10 km in 0.5 hours

×2

__ km in 1 hour

Set it up as a rate. Simplify it like a ratio.

Her speed was 10 × 2 = 20 km/h.

Set it up as a rate. Simplify it like a ratio.

d She rode 60 km in 5 hours. 60 km in 5 hours

÷5

__ km in 1 hour

Her average speed was 60 ÷ 5 = 12 km/h.

Now you try

Using the distance-time graph in Example 16: a what was the average speed for the last part of the journey, correct to one decimal place? b excluding the breaks, what was Mira’s average speed for the whole journey?

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Chapter 1 Ratios and rates

Exercise 1H

Write a short description about what is happening at each point (A, B, C to F) along Aman’s journey.

B C D

U N SA C O M R PL R E EC PA T E G D ES

Distance from home

BUILDING

F G

Time

Draw a reasonably accurate distance-time graph which includes both of the following journeys.

a Enzo left home and walked 2 km in 30 minutes, then rested for 30 minutes, then walked home again in 1 hour. b Maria left the same home and walked 2 km in 45 minutes then turned and walked home in 1 hour.

DEVELOPING

Based on this graph, what was the average speed during the journey?

Distance (km)

Example 15

50 40 30 20 10

2 3 Time (hours)

CHALLENGING

This distance-time graph shows one of Jane’s training rides.

a Describe what happened during the first 90 minutes, including the distance travelled and the time taken. b How far did Jane ride altogether? c What was her average speed for the second part of the journey, after the first break? d What was the average speed for the last part of the journey? e Including the breaks, what was her average speed for the whole journey, correct to one decimal place? f Excluding the breaks, what was her average speed for the whole journey?

Distance from home (km)

Example 16

20 15 10

Noon 1 p.m. 2 p.m. 3 p.m. 4 p.m. Time (hours)

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1H Distance-time graphs

Communicate your reasoning, understanding and fluency Study the travel graph below. It describes the movements of a surfer over an 8-minute timeframe, starting 50 metres from the beach.

Distance from beach

U N SA C O M R PL R E EC PA T E G D ES

Time

Write a paragraph about the movements of the surfer, including estimates of the time spent moving or being stationary.

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Chapter 1 Ratios and rates

1I Energy consumption rates Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To recognise, understand and convert between units of energy • To apply units of energy to calculate kilowatt hours • To solve problems involving the consumption of energy Past, present and future learning

• Some aspects of this section may be new to students • This application of rates may not have been treated in Years 7 to 11 • Questions involving energy consumption occasionally appear in HSC Examinations

Energy is the capacity to do work. Energy exists in numerous forms, such as heat. The joule (symbol J) is the unit of energy used by the International System of Units (SI). Heat energy such as that produced by burning natural gas in the home is usually measured in megajoules (one million joules, symbol MJ).

Power

Power is the rate at which energy is generated or consumed. The watt is the International System of Units (SI) unit of power and is equal to one joule per second. The symbol for the watt is W. As for the joule, the standard SI prefixes such as milli, kilo and mega are then added and commonly used to measure power.

Electrical energy

The joule is not a practical unit for measuring electrical energy in most settings. It is more helpful to think of electrical energy in terms of the power drawn by an electrical device and the length of time the device is in use. For example, when a device with a power rating of 100 W is turned on for one hour, the amount of energy used is 100 watt-hours (Wh or W-h). This is the same amount of energy a 50 W device would use in 2 hours.

The kilowatt-hour (kWh or kW-h) is commonly used to measure electrical energy in household electricity meters.

It represents the amount of electrical energy when a 1000 W power load is drawn for one hour. It is the result of multiplying power in kilowatts and time in hours. Note the conversion 1 kWh = 3.6 MJ. Consumption is a rate expressed as an amount over time. Electrical energy consumption can be expressed in kilowatt-hours or megawatt-hours per unit of time, or megajoules per unit of time. The average annual energy use per household in Australia is 11 MWh/y or 3 MJ/y, which also equates to about eight tonnes of CO2 emissions.

Energy consumption

Energy consumption is the amount of energy consumed per unit of time.

Energy costs 1 Find the energy (kWh) by multiplying the power rating (kW) by the hours used. 2 Find the cost by multiplying the energy by the electricity charge. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

1I Energy consumption rates

Energy rating of appliances

U N SA C O M R PL R E EC PA T E G D ES

In Australia an energy rating label is provided for appliances such as fridges, TVs, washers, dryers and air conditioners. It allows consumers to compare the energy efficiency of similar products. Energy rating labels all have a simple star rating. The more stars on the label, the more energy efficient the appliance. The energy consumption figure is the number in the red box. It indicates the amount of electricity (kWh) the appliance typically uses to run in a year. The lower the number the less the appliance will cost to run. The running cost of the appliance is calculated by multiplying the energy consumption figure by the electricity price rate. Electricity suppliers usually give prices per kilowatt-hour. The screenshot below is from the Australian Government energy rating website (www.energyrating.gov.au/calculator).

Example 17

Calculating the cost of running appliances

Find the cost of running these appliances if the average peak rate for electricity is $0.21/kWh.

a A 3.5 kW air conditioner for eight hours. b A 200 W television for 20 hours. Solution

Explanation

a Electricity = 3.5 × 8

Determine the energy (kWh) by multiplying the power rating (kW) by the hours used.

= 28 kWh Cost = 28 × 0.21 = $5.88

b Electricity = 0.2 × 20 = 4 kWh Cost = 4 × 0.21 = $0.84

Determine the cost by multiplying the energy by the electricity charge.

Determine the energy (kWh) by multiplying the power rating (kW) by the hours used.

Determine the cost by multiplying the energy by the electricity charge.

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Chapter 1 Ratios and rates

Now you try Determine the cost of running the following appliances, given the average rate for electricity is $0.17 per kWh.

U N SA C O M R PL R E EC PA T E G D ES

a A dishwasher with an energy consumption of 670 kWh per year. b A 2.4 kW fan heater for five hours. c A 20 W LED light bulb for a year.

Energy-efficient housing

BASIX or Building Sustainability Index is a scheme to regulate the energy efficiency of residential buildings. It is an online program to assess and compare a house or unit design with energy and water targets. BASIX uses information such as site location, house size, building materials, water, thermal comfort and energy. Sustainable houses feature rainwater tanks, efficient showerheads, solar hot water, performance glazing and energy-efficient lighting.

Example 18

Calculating electricity costs

Christian is installing a solar PV (photovoltaic) system. His average daily consumption is 21.5 kWh. It is predicted the solar PV system will meet this need and export 6.3 kWh of energy daily to the grid. Christian’s energy retailer charges $0.2167 per kWh and pays $0.08 per kWh for energy. What is the expected saving from the solar PV system each day?

Solution

Explanation

Daily consumption = 21.5 × 0.2167

Multiply the energy by electricity charge for daily consumption.

= $4.65905

Exported = 6.3 × 0.08 = $0.504

Multiply the energy by electricity charge that is exported to the grid.

Saving = $4.65905 + $0.504

Add the two amounts to determine the saving.

= $5.16

Now you try

How much money would Christian have saved that day if the daily consumption was reduced by 20% and all the energy saved was exported to the grid?

Exercise 1I

Example 17

BUILDING

The Nguyen family has a clothes dryer with a 3.3 kW rating.

a How many kilowatts of electricity does it use in a week if it is switched on for an average of 2 hours per day? b What is the cost of running the dryer for a week if electricity is $0.2819 per kWh?

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1I Energy consumption rates

A dishwasher has an energy consumption figure of 250 kWh per year. Calculate the cost of running the dishwasher for a year using the following electricity tariffs.

U N SA C O M R PL R E EC PA T E G D ES

a $0.1985/kWh b $0.2019/kWh c $0.2754/kWh

Marcus uses a 25 W ceiling fan every hour of the day.

a How much energy does the ceiling fan use during the week? b What is the cost of using the ceiling fan for a week if electricity is $0.2999 per kWh?

Alice uses an oven with a rating of 1200 W for an average of 7 hours per week. Find the cost of the electricity for the oven at a rate of 24.11 cents per kilowatt-hour.

Luke received his natural gas account.

Amount used

Charges

First 6200 MJ

2.581 cents per MJ

Next 12 400 MJ

1.379 cents per MJ

Cost

a Find the cost of the first 6200 MJ. Answer to the nearest cent. b Find the cost of the next 12 400 MJ. Answer to the nearest cent. c What is the total charge? Answer to the nearest cent.

Hamish uses a 12 W laptop for a total of 10 hours each day. He is charged at a rate of 25.07 cents per kilowatt-hour.

a How many kilowatt-hours are used by the laptop during the year? b What is the cost of using the laptop for a year? Answer to the nearest cent.

Example 18

Eve’s daily consumption of electricity is 19.5 kWh. She has a solar PV system to meet this need and export 7.5 kWh of energy to the grid. Eve’s energy retailer charges $0.2425 per kWh and pays $0.10 per kWh for energy. How much is the solar PV system saving Eve each day?

The pie chart shows the sources of benefit to NSW of BASIX. The total benefit of BASIX for NSW to 2050 is estimated to be $294 000 000. 5% 4%

22%

69%

Household energy Household water Environmental benefits Avoided electricity expansion

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Chapter 1 Ratios and rates

a What percentage of the benefits of BASIX was environmental? b What percentage of the benefits of BASIX was household energy? c What is the household energy saving of BASIX for NSW to 2050? d What is the household water saving of BASIX for NSW to 2050? e What is the environmental saving of BASIX for NSW to 2050? The table shows the types of heating systems installed in new houses in one year.

Type

No active heating system

6332

a What is the total number of new houses? b Copy and complete the table by calculating the percentage of each heating system, correct to the nearest percent. c Which is the most popular heating system?

Gas fixed flued heater

2333

Gas hydronic system

327

1-phase air-conditioning

3357

3-phase air-conditioning

6867

Air-conditioning ducted

872

Wood heating

1635

Number

Percentage

U N SA C O M R PL R E EC PA T E G D ES

d How many houses installed air-conditioning? e Construct a pie chart showing the proportions of each heating system.

DEVELOPING

The power rating of some electrical appliances is shown below.

Electrical appliance

Power rating

Air conditioner 3500 W a How much energy is used by a clothes dryer for 6 Clothes dryer 1800 W hours? Answer in kilowatt-hours. b How much energy is used by an air conditioner Dishwasher 1600 W for half an hour? Answer in watt-hours. Clothes washer 1300 W c How many hours was the dishwasher used if the TV/Computer 400 W cost of operating the dishwasher was $13.44? The cost of electricity is $0.2625 per kilowatt-hour. Refrigerator 150 W d How many hours was the refrigerator used if the cost of operating the refrigerator was $1.38? The cost of electricity is 23 cents per kilowatt-hour.

CHALLENGING

An energy company charges for gas over a 3-month period are shown below. Usage charge

First 5000 MJ

$0.02790 per MJ

Additional MJ over 5000

$0.01675 per MJ

a Alexis used 4400 MJ of gas in this period. What is the cost of this gas usage? b Matthew used 5450 MJ of gas in this period. What is the cost of this gas usage? Answer correct to the nearest cent. c What percentage of Matthew’s gas usage was charged at the lower rate? Answer correct to one decimal place. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

1I Energy consumption rates

U N SA C O M R PL R E EC PA T E G D ES

d Stella received a gas bill for $117.18. How much gas did she use in this period? e Claire has decided to reduce her energy bills. She has a target of $70 for gas. What is the maximum number of MJ she is allowed in this period? Answer correct to the nearest whole number. f Jake used 6220 MJ of gas in this period. The gas charges are increasing by 5% in the next quarter. However, Jake has purchased a new gas heater with a 5-star rating that will reduce his consumption by 1000 MJ. What is Jake’s expected bill next quarter?

Communicate your reasoning, understanding and fluency 12

Answer the following questions using this electricity account, from the year 2017.

a When was this electricity account issued? b What is the total amount due? c What is the due date? d How much GST was charged? e What is the charge for the peak use of electricity? f How much has the bill increased from the last bill? g Express the cost increase as a percentage of the last bill. Answer correct to two decimal places. h The figures in the account are from the year 2017. How much would this cost in the current year? You could ask friends and family to show you their electricity invoices.

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Chapter 1 Ratios and rates

1J Fuel consumption rates Learning intentions

• To use fuel consumption rates to compare the efficiency of vehicles • To solve problems involving fuel consumption rates

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • Questions involving fuel consumption occasionally appear in HSC Examinations

Over the years, as petrol has become more expensive, car manufacturers are trying to lower the amount of fuel that cars use.

Fuel consumption rates

There are two different rates used to express fuel consumption of a car. They are:

• litres per 100 km (L/100 km) • kilometres per litre (km/L).

Both of these are rates, like all the other rates in the previous sections. The first one is unusual because the second number is 100 km, not 1 km. However, it is the rate most commonly used in car advertisements.

Example 19

Calculating fuel consumption rates

A car travelled 850 kilometres using 78.2 litres of petrol. What was the fuel consumption rate, in:

a litres per 100 kilometres? b kilometres per litre, correct to one decimal place? Solution

Explanation

Write down litres first, then km second. Treat the rate as if it is a ratio by dividing both numbers by 8.5, so that second number is 100. Alternatively, use the unitary method.

78.2 L for 850 km

÷ 8.5

____ L for 100 km

9.2 L/100 km

850 km for 78.2 L

÷ 78.2

Write down km first, then litres second. Treat the rate as if it is a ratio, by dividing both numbers by 78.2, so that the second number is 1.

÷ 78.2

____ km for 1 L

10.9 km/L

Now you try

A car travelled 500 kilometres using 40 litres of petrol. What was the fuel consumption rate, in: a litres per 100 kilometres?

b kilometres per litre?

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1J Fuel consumption rates

Exercise 1J Example 19

BUILDING

Find the fuel consumption for each of the following, in L/100 km and km/L.

U N SA C O M R PL R E EC PA T E G D ES

a Lincoln’s car uses 45.5 litres of petrol to travel 520 km. b A small car uses 36.9 litres of petrol to travel 600 km. c Gemma’s sedan uses 55.1 litres of LPG to travel 950 km. d A sports car travelled 250 km using 28.5 litres of petrol. e Anthony’s motor bike uses 167.5 litres of LPG to travel 2500 km. f Tahlia’s car uses 121.6 litres of petrol to travel 3200 km.

Maryam has bought a used car whose fuel consumption is 7.8 litres petrol per 100 kilometres. She is planning to travel around Australia. Calculate the number of litres of petrol Maryam’s car will use on the following distances. Answer correct to the nearest whole number.

a A trip of 4049 km from Darwin to Perth. b A trip of 982 km from Sydney to Brisbane. c A trip of 2716 km from Perth to Adelaide.

d A trip of 658 km from Melbourne to Canberra. e A trip of 732 km from Adelaide to Melbourne. f A trip of 309 km from Canberra to Sydney. g A trip of 3429 km from Brisbane to Darwin.

Charlie travels 45 km to work and 45 km from work each day.

a How many kilometres do they travel to and from work in a 5-day working week? b Charlie drives a four-wheel drive with a fuel consumption of 8 L/100 km to and from work. How many litres of petrol does Charlie use travelling to and from work? Answer correct to one decimal place. c What is Charlie’s petrol bill for work if petrol costs $1.20 per litre?

A family car uses LPG at a rate of 15 L/100 km and the gas tank holds 72 litres. How far can it travel on a tank of LPG?

Kai drives a truck with a petrol consumption of 16 L/100 km and a petrol tank that holds 90 litres. He is planning a trip from Moorebank to Melbourne. The distance from Moorebank to Melbourne is 840 km. Kai filled up the petrol tank at Moorebank. How many more times will he need to fill his tank before arriving at Melbourne? Give reasons for your answer.

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Chapter 1 Ratios and rates

DEVELOPING 6

Natalie is planning a trip from Parramatta to Canberra using a car with a fuel consumption of 9.6 L/100 km. The distance from Parramatta to Canberra via the highway is 278 km and avoiding the highway is 363 km. The cost of LPG is 68.5 cents per litre.

U N SA C O M R PL R E EC PA T E G D ES

a How much will the trip cost via the highway? b How much will the trip cost avoiding the highway? c How much money is saved by travelling via the highway?

Austin owns an SUV with a fuel consumption of 10.9 L/100 km in the city and 8.4 L/100 km in the country. Austin travels 12 000 km per year in the city and 20 000 km per year in the country. The average cost of petrol is $1.48 per litre in the city and 12 cents higher in the country.

a Find the cost of petrol to drive in the city for the year. b Find the cost of petrol to drive in the country for the year. c What is the total cost of petrol for Austin in one year? d What is the total cost of petrol for Austin in one year if the average cost of petrol increased to $2.00 in the city?

CHALLENGING

Eden buys a family car that uses ULP with a fuel consumption of 9.9 litres/100 km. Toby buys the LPG version of the family car with a fuel consumption 14.8 litres/100 km. Both Eden and Toby average 450 km in a week in the same conditions. The average price of ULP is $1.50 per litre and LPG is $0.85 per litre.

a How many litres of fuel are used by Eden in a week? b How many litres of fuel are used by Toby in a week? c Calculate each car’s yearly consumption of fuel. d What is Eden’s yearly fuel bill? e What is Toby’s yearly fuel bill? f Toby paid an additional $2400 for the LPG version of the family car. How many years will it take for the fuel savings to reach $2400 or the break-even point? Answer correct to the nearest whole number. g Research the current fuel prices using ULP and LPG. How long will it take for the fuel saving to exceed the initial costs?

Communicate your reasoning, understanding and fluency 9

Conduct some research about the fuel consumption of small, energy efficient cars, and larger cars, such as SUVs.

a What is the lowest fuel consumption you can find? What is the highest? b Fuel for a small car that can use E10 might be $1.50 per litre. For a much larger car that uses diesel or premium unleaded 98, fuel might be $1.80 per litre. Investigate the current prices for the two cars you chose. c How many kilometres does the average Australian drive per year? d Calculate the cost of petrol per year for the cars you chose. Is there much difference? e Write up a comparison of a small efficient car versus a large gas-guzzling car.

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Chapter 1 review

Ratio

Convert the ratio into two fractions.

U N SA C O M R PL R E EC PA T E G D ES

Dividing a quantity in a given ratio

A ratio is used to compare amounts of the same units in a definite order. Equivalent ratios are obtained by multiplying or dividing by the same number.

Review

Key ideas and chapter summary

Scale drawing

Multiply the quantity by both fractions.

Scale of a drawing = Drawing length : Actual length Scale is expressed in two ways: • using units, such as 1 cm to 1 m (or 1 cm = 1 m) • with no units, using a ratio such as 1 : 100.

Plans and elevations

A plan is a view of an object from the top. An elevation is a view of an object from one side such as a front elevation or side elevation.

Perimeter

When calculating the perimeter of the land, multiply side lengths on an aerial photograph by the scale.

Area

To estimate the area of land, divide it into square grids and count the number of squares. To calculate the area of land, use the appropriate formula. The trapezoidal rule for area of land with one irregular boundary gives an approximation.

Volume of water and the trapezoidal rule

Volume of a body of water of constant depth and a trapezoidal area, or volume of rainfall falling on a trapezoidal area, can be calculated using the trapezoidal rule for b area A ≈ d f + dl and the formula for the volume of a right prism V = Ah. 2

Rate

A rate is a comparison of quantities expressed in different units. E.g. km/h, L/100 km, $/hour.

Unitary method

Use division to go down to 1. Multiply to find the required amount.

Energy rate

Energy is the rate of power per unit of time (kilowatt hour - kWh) Running cost of the appliance is calculated by multiplying the energy consumption by the electricity charge.

Fuel consumption

A rate, expressed in litres/100 km or km/litre.

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Review

Chapter 1 Ratios and rates

Chapter checklist with success criteria A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

1 I can change a ratio to a simpler form, in three ways.

U N SA C O M R PL R E EC PA T E G D ES

e.g. Consider the ratio 4 : 6.

a Express the ratio 4 : 6 in simplest form.

b Express the ratio 4 : 6 in the form 1 : a. c Express the ratio 4 : 6 in the form b : 1.

2 I can use a ratio to find an unknown quantity.

e.g. If 4 : 3 is equal to a : 24, what is the value of a?

3 I can use the capture-recapture technique to estimate a population.

e.g. A biologist traps 50 carp from a dam, then tags and releases them. One month later he captures 40 carp, of which 16 are tagged. Estimate the number of carp in the dam.

4 I can divide a quantity in a given ratio. e.g. Divide $400 in the ratio: a 3:2

b 4:1

c 3 : 7.

5 I can use a scale drawing to calculate real distances.

e.g. On a map the scale is 1 cm = 50 m. Two points are 3.2 cm apart on the map. How far apart are they in real life?

6 I can use a scale on a building plan.

e.g. A building plan has been drawn using the scale 1 : 100. Therefore, every mm on the plan represents what distance, in centimetres?

7 I can calculate perimeter, area and volume from a plan and an aerial photo, using the trapezoidal rule when necessary.

e.g. The following shape represents the surface of a concrete pond used for cultivating fish. It has uniform depth of 1.5 metres. The scale is 1 : 500.

a Estimate the perimeter, in metres, by assuming the curve is a semicircle.

b Estimate the area, in square metres, using the two applications of the trapezoidal rule. c Estimate the number of litres of water required to fill the pond. d Judging by the shape, is this an underestimate or overestimate of the volume? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Chapter 1 review

8 I can use the unitary method to solve problems involving rates. A cake for 6 people requires 4 eggs. How many eggs will be required for a cake big enough to serve 15 people?

Review

9 I can determine which product is the best buy.

U N SA C O M R PL R E EC PA T E G D ES

e.g. A drink comes in two sizes: • 1.25 litres for $3.25

• 2 litres for $4.20.

Which is the best buy?

10 I can solve problems involving speed, distance and time.

e.g. Cathy Freeman’s best time for the 400-metre race was 48.63 seconds. A wombat can run at a speed of 40 km/h over short distances. If the wombat could keep up this speed for 400 metres, would it beat Cathy Freeman?

11 I can convert a rate to other units of measurement.

e.g. A dripping tap wastes 25 millilitres of water every minute. Convert this rate to litres per hour.

12 I can interpret and analyse a distance-time graph.

Distance from home (km)

e.g. Consider this distance-time graph. 70 60 50 40 30 20 10

1 2 3 4 5 6 Time (hours)

a What was the average speed on the first part of the journey?

b What was the average speed on the last part of the journey, correct to one decimal place?

c What was the average speed for the entire journey, including the rest break? d What was the average speed for the entire journey, without the rest break, correct to one decimal place?

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Chapter 1 Ratios and rates

13 I can calculate the cost of running electrical appliances. e.g. The power rating for electrical appliances is shown in the table.

Electrical appliance Power rating

Electric shaver 15 W a How much energy does a laptop Hair dryer 1000 W computer use for 12 hours? Answer Iron 900 W in kilowatt-hours. b How much energy does a microwave Laptop computer 50 W 1 use for hour? Answer in Microwave 1200 W 2 watt-hours. Vacuum cleaner 600 W c For how many hours was the vacuum cleaner used, if the cost of operating the vacuum cleaner was $6.36? The price of electricity is $0.2650 per kilowatt-hour.

U N SA C O M R PL R E EC PA T E G D ES

Review

14 I can calculate the fuel consumption rate of a motor vehicle. e.g. Rata’s car uses 11.26 litres per 100 km.

a How many litres of petrol will his car use on a trip of 155 km from Sydney to Newcastle? Answer correct to three decimal places. b The cost of petrol is $1.60 per litre. How much will the petrol cost for the 155 km trip?

c How many litres of petrol will his car use on a trip of 294 km from Armidale to Taree? Answer correct to three decimal places. d The cost of petrol is $1.35 per litre. How much will the petrol cost for the 294 km trip?

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Chapter 1 review

The ratio of adults to children in a park is 5 : 9. How many adults are in the park if there are 630 children? B 126 D 350

U N SA C O M R PL R E EC PA T E G D ES

A 70 C 280

Review

Multiple-choice questions

2 A 360-gram lolly bag is divided in the ratio 7 : 5. What is the mass of the smaller amount? A 150 g C 192 g

3 A scale drawing has a scale of 1 : 20. What is the actual length if the drawing length of an object is 20 mm? A 1 mm C 40 mm

B 100 m/s D 6000 m/h

How much energy does a 600-watt hair dryer use in 5 hours? A 1.2 kWh C 120 kWh

B 30 D 7200

7 Which of the following is the slowest speed? A 60 km/h C 10 000 m/min

B 11 D 13

6 A hose fills a 10 L bucket in 20 seconds. What is the rate of flow in litres per hour? A 0.0001 C 1800

B 0.25 mm D 40 cm

Youssef is a courier who delivers 1 parcel on average every 25 minutes. How many hours does it take to deliver 24 parcels? A 10 C 12

B 20 mm D 400 mm

4 The scale on a map is given as 1 mm = 150 m. If the distance between two points is 600 m, what is the map distance between these points? A 4 mm C 2.5 cm

B 168 g D 210 g

B 3 kWh D 3000 kWh

9 Flynn uses a 1.2-kilowatt dishwasher for 5 hours. He pays 25.72 cents per kilowatt-hour. What is the cost of this usage? A $0.31 C $1.54

1J 10

B $1.29 D $6.00

What is the fuel consumption for a vehicle that travelled 340 km using 51 litres of petrol? A 7 L/100 km C 15 L/100 km

B 9 L/100 km D 17 L/100 km

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Chapter 1 Ratios and rates

Short-answer questions 1A

Simplify the following ratios. a 500 : 100

b 20 : 30

c 28 : 7

e 12 : 9

f 56 : 88

g 4.8 : 1.6

d 10 : 15 : 30 3 1 h : 4 2

U N SA C O M R PL R E EC PA T E G D ES

Review

Gabriel mixes sand and cement in the ratio 5 : 2 by volume. If he uses 5 buckets of cement, how much sand should he use?

The ratio of Victoria’s height to Willow’s is 8 : 7. If Victoria is 176 cm tall, how tall is Willow?

Express each of the following scales as a ratio in the form 1 : a. a 1 cm to 3 m

b 1 mm to 6 cm

c 1 m to 2.5 km

5 A scientist catches 200 fish from a pond, tags them and releases them. One month later, they catch 50 fish, of which 20 have tags. Estimate the number of fish in the pond.

Daniel and Eddie own a business and share the profits in the ratio 3 : 4.

a The profit last week was $3437. How much does Daniel receive? b The profit this week is $2464. How much does Eddie receive?

7 The scale on a map is given as 1 cm = 5 km. If the distance between two points on the map is 46 mm, what is the actual distance between these points? Answer in kilometres.

A scale drawing has a scale of 1 : 50 000. What is the drawing length of these actual lengths? Express your answer in millimeters.

a 4 km c 5000 cm e 20 000 mm

b 1250 m d 6.5 km f 2125 m

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Chapter 1 review

The scale on a map is 1 : 400. Calculate the actual distances if the following are the distances on the map. Express your answer in metres. a Bike path 180 cm d Beach 210 mm

c Street 5 cm f Park 60 mm

A section of a floor plan is shown opposite. The longer dimension of the laundry is 3 metres.

Robe

U N SA C O M R PL R E EC PA T E G D ES

1E 10

b Town centre 20 cm e River 62 cm

Review

a Estimate a suitable scale for the floor plan. b What symbol is used for the bath? c What does ‘W.C.’ represent on the plan? d What are the dimensions of the laundry?

W.C.

LAUNDRY

1F 11

Sliding door

BATHROOM

A body of water has the following dimensions: b = 6 m, d f = 20 m, dl = 40 m and h = 2.5 m. Use the trapezoidal rule for area and the formula for the volume of a right prism to answer the following.

a Find the approximate volume of the body of water. b Find the approximate volume of water that falls on the same area when there is 9 mm of rain.

1G 12

Convert each rate to the units indicated.

a $15/kg to $/g c 120 cm/s to mm/min e 14 L/g to mL/kg

b 14 400 m/h to m/ min d 4800 kg/g to kg/mg f $3600/g to c/mg

1G 13

If 20 metres of curtain material costs $580, what would be the cost of 35 metres of the same material?

1G 14

A motor bike travels at a speed of 100 km/h. How far does it travel in 3.5 hours?

1G 15

A 5 kg bag of rice costs $9.20. What is the cost of the following amounts?

a 10 kg d 7 kg

1J 16

b 40 kg e 500 kg

c 3 kg f 250 kg

A car travels 960 km on 75 litres of petrol. How far does it travel on 50 litres?

Extended-response questions

2 Pump A can fill a tank in 5 minutes; pump B can fill the tank in 10 minutes. How long will it take to fill the tank if both pumps are working together?

There are 384 passengers and 144 crew on a cruise ship. If there are 22 lifeboats, enough for everyone, how many are allotted to the passengers and how many to the crew?

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U N SA C O M R PL R E EC PA T E G D ES

Relative frequency and probability

Birthday coincidences in football teams at the 2023 FIFA Women’s World Cup Every four years, certain national teams from all over the world manage to qualify for one of the biggest sporting events on the planet: the FIFA Women’s World Cup. They must choose 23 players to represent their country during this soccer competition. Thirty-two countries qualified for the 2023 Women’s World Cup, which was held in Australia. Would you expect that many of these teams had two or more players who celebrate their birthday on the same day every year? For example, think about the 23 players from Brazil. There are 365 days that they could have been born on, ignoring leap years, so what is the chance that two or more of them share a birthday? It seems quite low, but it happened. In fact, 17 of the teams had at least two players with the same birthday. So just over half of the teams experienced this coincidence, which seems extraordinary.

Four of the teams had two pairs of players with matching birthdays:

• • • •

Panama Brazil Columbia Denmark

Two of the teams, Morocco and Nigeria, even had three pairs of players with matching birthdays.

Mathematically, this is not surprising at all. It turns out that in any random group of 23 people, the chance that two or more people will share a birthday is approximately 50.73%. This chance, which is probably higher than you guessed, comes from the number of pairs that can be made with 23 people compared with the number of days in a year. The mathematics of this counter-intuitive situation is explained here: https://www.bbc.com/future/article/ 20230830-the-unexpected-maths-problemat-work-during-the-womens-world-cup

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Chapter contents Language of probability Definition of probability Tables and tree diagrams Range of probabilities Complementary events Relative frequency Expected frequency Venn diagrams and two-way tables

U N SA C O M R PL R E EC PA T E G D ES

2A 2B 2C 2D 2E 2F 2G 2H

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Models and solves problems involving the probabilities of multistage events (MST-12-S2-09). © NSW Education Standards Authority

Online resources

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Chapter 2 Relative frequency and probability

2A Language of probability Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand and use the language associated with probability • To assign words such as ‘likely’ in reference to the chance an event will occur • To be able to list the elements in a sample space • To understand that every event has a probability from 0 (impossible) to 1 (certain) Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Years 7 to 10 • Questions involving probability regularly appear in HSC Examinations

Probability is the chance that something will happen. We use fractions (from 0 to 1), decimals (from 0 to 1) and percentages (from 0% to 100%) to assign a numerical value to a probability. There are many words and phrases in the English language which are used to describe the chance that something will not happen, might happen or will happen. 0

1 4

1 2

3 4

0.25

0.5

0.75

25%

50%

75%

100%

Impossible No chance

Somewhat unlikely

Even chance 50-50

Quite likely

Certain Sure

Discussion

Which one of these is closest to 0: ‘slim chance’ or ‘fat chance’?

Where does ‘Buckley’s chance’ fit in? Who was Buckley? What is the origin of the phrase? Search the internet.

Probability

Probability is the chance that something will happen.

Fractions, decimals and percentages are used for probabilities.

All probabilities can be placed on a scale from 0 (impossible) to 1, or 100% (certain)

Example 1

Using the language of probability

Ying has a pear, an apple and an orange. She randomly selects one piece of fruit. Describe the chance of the following events using the words ‘certain’, ‘likely’, ‘even’, ‘unlikely’ or ‘impossible’. a Ying selects a banana. b Ying selects an orange. c Ying selects a piece of fruit. d Ying selects a pear or an apple.

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2A Language of probability

Explanation

a Impossible

No chance of selecting a banana.

b Unlikely

One chance out of three of selecting the orange.

c Certain

Selecting a piece of fruit must happen.

d Likely

Two chances out of three of selecting the pear or the apple.

U N SA C O M R PL R E EC PA T E G D ES

Solution

Now you try

Would you say the following phrases mean ‘more than 50% chance’ or ‘less than 50% chance’? a Sporting chance b Red hot chance c You don’t stand the chance of a cinder in snow d Long odds e Short odds f You are odds on to win

Sample space

Sample space is the set of all possible outcomes or possible results of a chance experiment. For example, if the experiment is tossing a coin, then the sample space is a head and a tail. When a die is tossed, the sample space is the numbers 1, 2, 3, 4, 5 and 6. Each outcome or data value is an element of the sample space. The sample space is usually listed between curly brackets {}, which are sometimes called braces.

Sample space

Sample space is the set of all possible outcomes of a chance experiment. Each outcome or data value is an element of the sample space.

Example 2

Identifying the sample space

a Daku is choosing a day of the week to start his holiday. List the sample space. b How many elements in the sample space? Solution

Explanation

a Sample space = {Mon, Tue, Wed, Thu, Fri, Sat, Sun}

Each day of the week is a possible outcome – Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday. List each element between the curly brackets.

b There are 7 elements in the sample space.

Count the elements in the sample space.

Now you try

How many elements in each sample space? a Months of the year c Tossing one coin

b Cards in a standard pack d Rolling one standard die

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Chapter 2 Relative frequency and probability

Example 3

Identifying the sample space

U N SA C O M R PL R E EC PA T E G D ES

Two unbiased coins are tossed.

a What is the sample space? b How many possible outcomes are there? Solution

Explanation

a First coin: H or T.

Each coin could land on a head or a tail.

Second coin: H or T.

If the first coin was a head, then the second coin could be a head or a tail.

Sample space = {HH, HT, TH, TT}

If the first coin was a tail, then the second coin could be a head or a tail.

b Number of possible outcomes is 4.

Count the elements in the sample space.

Now you try

Three cards are numbered 1, 2, and 3. A card is chosen then a coin is tossed, so one of the outcomes is 2H.

a List all the outcomes in the sample space b How many outcomes are there?

Exercise 2A 1

BUILDING

State whether the probability of each event is impossible or certain. a Christmas Day occurring on the 25th December b Winning a raffle without buying a ticket c Rolling the number 7 on a standard six-sided die d Friday being the day after Thursday e Selecting the letter ‘A’ from letters in the word ‘SCHOOL’ f The sum of 1 and 2 being 3 g Choosing an even number from the numbers 2, 4, 6 and 8 h Winning a car race without a car i Obtaining a head or a tail when a coin is tossed j Night following day

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2A Language of probability

Example 1

Describe the chance of the following events using the words ‘certain’, ‘likely’, ‘even’, ‘unlikely’ or ‘impossible’.

U N SA C O M R PL R E EC PA T E G D ES

a There being 31 days in January b Tossing a coin and getting a tail c Randomly choosing a person who has never watched television d Winning first division Lotto e Choosing a red ball from a bag containing an equal number of red and white balls f Snow on Mount Kosciuszko during a cold day in winter g Newborn baby being a girl h Drawing a heart from a normal pack of cards i Tossing a coin and turning up a head j Australia participating in the next cricket World Cup

Aliya has a better than even chance of winning the marathon.

a What word could you use to describe this probability? b Aliya is sick with a cold and running much slower. What word could be used to describe her chances now?

Two coins are tossed. Describe in words the probability of the following outcomes. a Two heads b Head and a tail c Two tails

A standard six-sided die is rolled. Describe in words the probability of the following outcomes.

a Odd number b Number between (and including) 1 and 6 c Number less than 10 d Number 3

Example 2

List the sample space for the following events.

a Tossing a $2 coin b Selecting a ball from a box containing 5 balls coloured red, green, white, blue and yellow c Choosing a letter from the word ASCOT d Drawing a card from cards labelled from 1 to 5 e The sex of a newborn kitten f Selecting the outcome of a football match

The arrow on this regular pentagon is spun once.

a What is the sample space? b How many elements are in the sample space? c Describe in words the probability of landing on red. d Describe in words the probability of not landing on blue.

Blue Red

White

Yellow Green

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Chapter 2 Relative frequency and probability

Example 3

A fair die is rolled. a List the sample space. b How many elements in the sample space?

There are 10 counters each labelled with the letters from A to J.

U N SA C O M R PL R E EC PA T E G D ES

a A counter is selected at random. List all the possible outcomes. b Describe in words the probability of selecting a vowel.

A Rugby Union match is being played between Australia and South Africa.

a List the sample space for the possible outcomes of the match. b How many elements in the sample space? c Is each outcome equally likely? Why?

Jacob is learning to play chess. He has a match against the Australian champion. There are three outcomes for the match – win, draw or lose.

a What is the sample space? b Are these events equally likely? Why?

DEVELOPING

Four cards are numbered 1, 2, 3 and 4. A card is chosen then a coin is tossed. How many outcomes are there in the sample space?

The probability of six outcomes is 0, 0.25, 0.5, 0.75, 0.9 and 1. Assign one of these numbers to the following word descriptions. a Even chance c Certain e Impossible

b Likely d Very likely f Unlikely

A bag contains one $10 note, one $20 note and one $50 note. One note is chosen from the bag. It is not replaced. Then another note is chosen.

a List the sample space for the first note. b How many distinct elements are in the sample space of the first note? c Assume the first note chosen was a $10 note. List the sample space of the second note. d Assume the first note chosen was a $20 note. List the sample space of the second note. e Assume the first note chosen was a $50 note. List the sample space of the second note.

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2A Language of probability

CHALLENGING 15

A fair coin and a standard die are thrown. Describe in words the probability of the following outcomes.

U N SA C O M R PL R E EC PA T E G D ES

a Head and a number less than 10 b Head and a number greater than 10 c Tail and a 1 d Tail or head and an even number

1 ‘Six students enter a swimming race. The chance of a particular student winning is .’ Is this 6 statement true or false? Give reasons to support your opinion.

Laura has an 80% chance of winning a tennis tournament. The other two players Maya and Emma are equally likely to win.

a What is the sample space? b Are the outcomes equally likely? Why? c What is the probability that Maya wins the tournament?

Communicate your reasoning, understanding and fluency 18

Tess says two things. Statement 1: “When you toss two coins together, there are three possibilities: 2 heads, 2 tails or one head and one tail.” Statement 2: “Therefore, the chance of tossing two heads is one-third.” Re-write Statement 1 and Statement 2 so that they are correct.

I rolled a fair standard die 3 times. I rolled a 4, three times in a row.

Rachel said: “There is a really good chance that the next one will be a 4.” Sophie said: “There is almost no chance that the next one will be a 4.” Aditi said: “If that is a fair die, the chance that the next one will be a 4 is one-sixth.” Write a paragraph in support of Aditi’s statement.

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Chapter 2 Relative frequency and probability

2B Definition of probability Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand and apply the definition of probability • To understand that the probability of an event can be expressed as a fraction of the form ‘7 out of 10’ • To understand that some outcomes are ‘equally likely’ and some are not • To express probabilities using fractions, decimals and percentages Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Years 7 to 10 • Questions involving probability regularly appear in HSC Examinations

There is quite a lot of probability terminology to unpack. For example, an event can be a result of a probability experiment, such as: • Rolling a 6

• Rolling a number less than 5

• Tossing a head

• Tossing a head then a tail

• Choosing a red ball

• Winning a tennis match

Two similar probability experiments, with an important difference

Bag A contains a red ball, a white ball and a green ball. One is chosen at random. The three possible outcomes R, W and G are ‘equally likely outcomes’. The sample space is {R, W, G}.

An event is ‘randomly choose a white ball’ What is the probability of choosing a white ball?

The answer is obvious. It is 1 out of 3 and can be written as a fraction, a decimal or a percentage. 1 P(white) = 3 There is a formula for this situation, using the letter P for probability: number of favourable outcomes P(event) = total number of outcomes

Bag B contains a red ball and two white balls. One ball is chosen at random. R and W are not equally likely outcomes. Is the sample space {R, W, W}? OR Is the sample space {R, W}? Are there two outcomes or three outcomes? This question is frequently discussed online.

An event is ‘randomly choose a white ball’ What is the probability of choosing a white ball?

There are 3 balls. Two of them are white. The probability is 2 out of 3 and can be written as a fraction, a decimal or a percentage. 2 P(white) = 3 This experiment shows the importance of being careful when calculating probability with outcomes that are not equally likely.

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2B Definition of probability

Probability that a certain event will occur If all the outcomes in the sample space are equally likely: number of favourable outcomes P(event) = total number of outcomes

U N SA C O M R PL R E EC PA T E G D ES

A ‘plain English’, more inclusive version: number of ways the event could occur P(event) = number of all the things which could happen

Example 4

Calculating the probability

A coin is chosen at random from 7 one dollar coins and 3 two dollar coins. Calculate the probability that the coin is a:

a one dollar coin b two dollar coin.

Solution

Explanation

7 a P($1) = 10 = 0.7

7 out of 10 coins are $1. Form a fraction. Express as a fraction, decimal or percentage.

= 70% 3 10 = 0.3

b P($2) =

3 out of 10 coins are $2. Form a fraction. Express as a fraction, decimal or percentage.

= 30%

Now you try

A bag contains three red balls, two white balls and one green ball. One ball is chosen at random. What is the probability that the ball is: a white?

b black?

c red, white or green?

A deck of playing cards A normal deck of playing cards has 52 cards.

• There are four suits called clubs, spades,

hearts and diamonds. • In each suit there are 13 cards from ace to king. • There are 3 picture cards in each suit (jack, queen and king). • Half the cards are red and half are black.

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Chapter 2 Relative frequency and probability

Example 5

Calculating the probability from playing cards

A single card is randomly chosen from the pack. What is the probability that it is a:

U N SA C O M R PL R E EC PA T E G D ES

a red four b diamond c picture card. Solution

Explanation

2 a P(Red 4) = 52 1 = 26

2 of the 52 cards are red fours. Reduce to simplest form.

13 52 1 = 4

b P(Diamond) =

12 52 3 = 13

c P(Picture) =

13 of the 52 cards are diamonds. Reduce to simplest form.

12 of the 52 cards are picture cards. Reduce to simplest form.

Now you try

The four aces are removed from the pack of cards. One card is then chosen at random. What is the probability that the card is a:

a red four? b diamond? c picture card?

Exercise 2B

Example 4

BUILDING

What is the probability of the following experiments?

a A card dealt from a normal deck of cards is a diamond. b A day selected at random from the week is on the weekend. c A head results when a coin is tossed. d A letter chosen randomly from the alphabet is a vowel. e A two results when a die is rolled. f A six is chosen randomly from {2, 4, 6, 8, 10}.

A bag contains 5 blue and 3 red balls. Find the probability of selecting the following at random.

a A blue ball b A red ball c Not a red ball.

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2B Definition of probability

Rangi chooses one ball at random from his golf bag. The table below shows the type and number of golf balls in his bag. Type of golf ball

Number

Titleist

Srixon

U N SA C O M R PL R E EC PA T E G D ES

Callaway

Find the probability of Rangi choosing: a a Srixon c a Titleist

b a Callaway d not a Srixon.

An unbiased coin is tossed three times. On the first two tosses the result is tails. What is the probability that the result of the third toss will be a tail?

In Amber Ave there are 3 high school students, 4 primary school students and 5 preschool students. One student from Amber Ave is chosen at random. What is the probability that a primary school student is chosen?

A box contains 3 blue, 4 green and 2 white counters. Find the probability of drawing at random a counter that is:

a blue c white or blue

Example 5

b green d not blue.

7 A card is chosen at random from a standard deck of 52 playing cards. Find the probability of choosing: a the seven of clubs d a red picture card g an even number

b a spade e a nine h a picture card

c a red card f the six of hearts i a black ace.

The weather on a particular day is described as either wet or dry. Therefore there is an even chance of a wet day. Do you agree with this statement? Give a reason.

DEVELOPING

A die with 16 faces marked 1 to 16 is rolled. Find the probability that the number is:

a an odd number c a multiple of 3 e less than or equal to 15

b neither a 1 nor a 2 d greater than 12 f a square number.

A wheel contains eight evenly spaced sectors numbered 1 to 8. The wheel is spun until it stops at a number. Given that the wheel is equally likely to stop at any number, find the probability that the wheel stops at:

a a7 c an odd number e a number less than 10

b a number greater than 5 d a 1 or 2 f a number divisible by 3.

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Chapter 2 Relative frequency and probability

Casey is dealt four cards from a normal deck: two aces and two kings. What is the probability that the next card is:

U N SA C O M R PL R E EC PA T E G D ES

a another ace? b another king? c not an ace? d not a king?

CHALLENGING

Two cards are drawn at random from a normal deck of cards. What is the probability that the second card is:

a a two if the first card was a two? b an ace if the first card was an ace? c the six of clubs if first card was a 10? d a two if the first card was a king? e a diamond if the first card was a diamond? f a picture card if the first card was a picture card?

A four-digit number is formed from the digits 2, 3, 4 and 5 without replacement. What is the probability that the number:

a starts with the digit 4? b is greater than 3000? c ends with a 2 or a 3? d is 2345?

Communicate your reasoning, understanding and fluency 14

1 If six students enter a swimming race, then the chance of any particular student winning must be . 6 Is this statement true or false? Give reasons to support your opinion.

The chance of rain on Saturday is 70%. The chance of rain on Sunday is 50%. Therefore, the chance of rain this weekend is 120%. Why does this not make any sense?

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2C Tables and tree diagrams

2C Tables and tree diagrams Learning intentions

• To use tables (arrays), tree diagrams and probability trees to list all the outcomes of multistage

U N SA C O M R PL R E EC PA T E G D ES

experiments • To determine the probability of multistage events Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Years 7 to 10 • Questions involving probability regularly appear in HSC Examinations

Some probability experiments are ‘multistage’. For example: • tossing a coin twice • tossing a coin then rolling a die • selecting three cards from a pack of cards.

Tables and tree diagrams are used to list all the possible results for a multistage experiment. If a coin is tossed then a die is rolled, one of the multistage events is H3. Tables, also known as arrays

A table is an arrangement of information in rows and columns. The table shows all the possible outcomes for tossing two coins. There are two events: tossing the first coin and tossing the second coin. The outcomes of the first event are listed down in the first column (Head or Tail). The outcomes of the second event are listed across the top row (Head or Tail). Each cell in the table is an outcome. There are four possible outcomes.

Head

Tail

Head

Tail

Sample space = {HH, HT, TH, TT}

Tables (arrays)

A table is an arrangement of information in rows and columns. Each cell in the table is a data value or an outcome.

Example 6

Using an array for a multistage event

Two red cards (R1, R2) and one black card (B1) are placed in a box. Two cards are selected at random with replacement. (i.e. the first card is put back into the box before the second one is chosen).

a Construct a table to list the sample space. b What is the probability of selecting R1 R1 or R2 R2? Solution a

Explanation

R1 R1

R1 R2

R1 B1

R2 R1

R2 R2

R2 B1

B1 R1

B1 R2

B1 B1

List the outcomes of the first stage (first card) down the first column. There are three outcomes: R1, R2 and B1. List the outcomes of the second stage (second card) across the top row. There are three outcomes: R1, R2 and B1. Write the outcome in each cell using the intersection of the row and column.

2 Form a fraction. 9 Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400 b P(R1 R1 or R2 R2) =

Chapter 2 Relative frequency and probability

Now you try Repeat Example 6, but this time the first card is not replaced before the second card is chosen.

Tree diagrams for equally likely outcomes

U N SA C O M R PL R E EC PA T E G D ES

A tree diagram is a diagram that details the outcomes of a multistage experiment with equally likely outcomes. It shows each event as a branch of the tree. The tree diagram below shows all the possible outcomes for tossing two coins. The outcomes of the first stage are listed (H or T) with two branches. The outcomes of the second stage are listed (H or T) with two branches on each of the outcomes from the first stage. The sample space is HH, HT, TH and TT. 1st

2nd H

Note:

1 4 1 P(TT) = 4

P(the coins are different) =

P(HH) =

2 4

Tree Diagrams

• Draw a tree diagram with each event as a new branch of the tree. • Always draw large clear tree diagrams and list the sample space on the right-hand side.

Example 7

Drawing a tree diagram

A coin is tossed and a die is rolled.

a Construct a tree diagram of these two stages to show the sample space. b What is the probability of a head and a 1? Solution a

Explanation

Coin

Die 1 H1 2

5 H5

Draw branches for first stage – tossing a coin. Tossing a coin has two outcomes (head or tail) so there are two branches. Draw branches for the second stage – rolling a die. Rolling a die has six outcomes (1, 2, 3, 4, 5 or 6) so there are six branches. Draw six branches for each of the two outcomes from the first stage. Use the branches of the tree to list the sample space. Write the outcomes down the right-hand side.

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2C Tables and tree diagrams

b P(H1) =

1 12

Number of favourable outcomes (H1) is 1. The total number of outcomes is 12. Form a fraction.

Now you try

U N SA C O M R PL R E EC PA T E G D ES

There are three cards in a pack, numbered 1, 2, and 3. A card is chosen at random. It is not replaced. A second card is chosen.

a Construct a tree diagram to show the sample space. b What is the probability that the sum of the two cards is less than 5?

Probability trees for outcomes which are not equally likely

Sometimes, a pair of branches in a tree diagram may not be equally likely. For example, in the year 2022 in Australia, 51% of babies born were male compared with 49% of babies born female. If two babies were randomly selected, the probabilities are written on the branches, as shown, using decimals or fractions. The tree diagram is then sometimes called a ‘probability tree’.

0.51

0.49

0.51

F M

0.49

The probabilities are then calculated using multiplication. P(MM) = 0.51 × 0.51 = 0.2601

0.49

P(FF) = 0.49 × 0.49 = 0.2401

If it is necessary to combine two or more outcomes, their probabilities are added together. P(same sex) = P(MM) + P(FF) = 0.2601 + 0.2401 = 0.5002

Example 8

Drawing a probability tree

A small pack of cards contains 5 red cards and 2 blue cards. One card is chosen and then replaced. A second card is then chosen.

a Draw a probability tree. b What is the probability that both cards are red? c What is the probability that the two cards are different colours? d What is the probability that one or more of the cards are red? Solution

Explanation

5 7

2 7

5 7

P(RR) = 25 49

2 7

5 7

P(RB) = 10 49 P(BR) = 10 49

2 7

The fractions are written on the branches. Multiply two fractions to get the probability of each outcome. Check that the sum of the last column is 1.

B P(BB) = 4 49

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Chapter 2 Relative frequency and probability

b P(RR) =

5 5 25 × = 7 7 49

c P(different colours) =

Multiply the two fractions. 10 10 20 + = 49 49 49

Add the two different outcomes. They are usually the same as each other.

10 10 25 45 + + = 49 49 49 49 49 4 45 Alternatively, − = 49 49 49

d P(one or two red) =

U N SA C O M R PL R E EC PA T E G D ES

Add the three different outcomes.

The four outcomes add to 1, so subtract P(BB) from 1, as that is the only outcome you do not need to include. This is a very useful shortcut.

Now you try

Repeat Example 8, but this time the first card is not replaced before the second card is chosen.

Exercise 2C 1

BUILDING

Riley and Bailey are planning to have two children. Male

Female

Male

Female

a Use a table to list the number of elements in the sample space. Consider the sex of each child as an event. b What is the total number of outcomes?

Two fair dice are thrown and their sum recorded.

a Use a table to list all possible outcomes. +

2 3

4 5 6

b How many elements are in the sample space?

Example 6

A menu has three entrees (E1, E2 and E3) and four mains (M1, M2, M3 and M4). Use a table to list all possible outcomes.

E1 E2 E3

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2C Tables and tree diagrams

Three people (A, B and C) applied for a manager’s position and two people (D and E) applied for an assistant manager’s position. a Use a table to list the all the possible outcomes. b What is the total number of outcomes? One bag contains two discs labelled ‘X’ and ‘Y’. A second bag contains four discs labelled ‘D’, ‘E’, ‘F’ and ‘G’. A disc is chosen from each bag at random. Use a table to list the sample space.

U N SA C O M R PL R E EC PA T E G D ES

DEVELOPING

Three yellow cards (Y1, Y2, Y3) and one green card (G1) are placed in a box. Two cards are selected at random with replacement.

a Use a table to list the number of elements in the sample space. b Find the probability of the following selections:

Y1 Y2 Y3

i Y1 Y2 ii two green iii two yellow.

Three cards (king, queen and jack) are placed face down on a table. One card is selected at random and the result recorded. This card is returned to the table. A second card is then selected at random.

a Use a table to list the elements in the sample space. b Find the probability of the following selections: i KJ ii two kings iii no kings.

Example 7

Two coins are tossed and the results recorded.

1st

a List the sample space by completing this tree diagram. b Find the probability of the following results:

2nd

i two tails ii head then a tail iii one head and one tail.

A survey has two questions whose answers are ‘Yes’ or ‘No’. Construct a tree diagram to list the sample space.

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Chapter 2 Relative frequency and probability

There are three questions in a True or False test.

1st

U N SA C O M R PL R E EC PA T E G D ES

a Fill in the remainder of the partially drawn tree diagram and list the sample space. b How many elements in the sample space? c Find the probability of the following random selections:

3rd

2nd

i all of the selections are false ii two true selections and one false selection iii at least one of the selections is false.

A two-digit number is formed using the digits 1, 2 and 3. The same number cannot be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit.

Tens

Units 2

a List the sample space from the tree diagram. b What is the probability of choosing 23?

A spinner has an equal number of red and green sections. This spinner is spun twice.

a Use a tree diagram to list the total possible outcomes. b What is the probability of the same colour when the spinner is spun twice?

Ebony tosses a coin and spins a spinner, which has a red, an amber and a green section. Each colour on the spinner is equally likely.

a Use a tree diagram to list the sample space. b What is the probability of a head on the coin and either a red or green section on the spinner?

Four cards (ace, king, queen and jack) are placed face down on a table. One card is selected at random and the result recorded. This card is not returned to the table. A second card is then selected at random. Use a tree diagram to list all the possible outcomes.

CHALLENGING

There are four candidates for the positions of leader and deputy leader. The four candidates are Mohammed, Bridget, Connor and Danielle.

a Construct a tree diagram with the leader as the first event and the deputy leader as the second event. Use a tree diagram to list the sample space. b What is the probability of Bridget being the leader or deputy leader?

A two-digit number is formed using the digits 3, 5 and 7. The same number can be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit.

a Construct a tree diagram to list the sample space. b What is the probability of forming a number less than 70?

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2C Tables and tree diagrams

The number of laptops sold in the past two weeks is shown below. Brand X

Brand Y

Total

Week 1

240

320

Week 2

150

200

Total

130

390

520

U N SA C O M R PL R E EC PA T E G D ES

a Of all brand X laptops sold, what fraction was sold in week 2? b What percentage of all sales in week 1 was brand Y? c During the two weeks, what was the probability that any laptop sold would be brand X?

Example 8

Repeat Question 6, assuming the first card is not replaced before the second card is chosen.

Repeat Question 7, assuming the first card is not replaced before the second card is chosen.

Repeat Question 8, for a coin which is biased so that it lands on heads 60% of the time.

Communicate your reasoning, understanding and fluency 21

Your calculator has a button Write down the value of:

. What does it do? Search the internet.

a 1! d 4!

b 2! e 10!

c 3! f 52!

The last one is a 68 digit number. That is the number of different ways a pack of cards can be shuffled.

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Chapter 2 Relative frequency and probability

2D Range of probabilities Learning intentions

• To understand that every event has a probability from 0 to 1 • To understand that the sum of the probabilities of all the events in an experiment is 1

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• These concepts build upon prior learning from Years 7 to 10 • Students should already be familiar with the contents of this section • Questions involving probability regularly appear in HSC Examinations

The probability of an event that is impossible is 0 and the probability of an event that is certain is 1. Probability is always within this range, or from 0 to 1. It is not possible to have the probability of an event as 2. It is also important to realise the following, which is true for every probability experiment: When all the outcomes are listed, the sum of all their probabilities is 1.

Certain

0.75 0.5

Even chance

0.25 0

Impossible

Range of Probability

Probability of an event is between 0 and 1. P(A) + P(B) + . . . = 1

A, B, . . . are all the possible outcomes or events.

Example 9

Using the range of probability

3 A box contains red, yellow and blue cards. The probability of selecting a red card is and the probability 5 1 of selecting a yellow card is . What is the probability of selecting a blue card? 10 Solution

Explanation

P(R) + P(Y) + P(B) = 1 3 1 + + P(B) = 1 5 10

Write the formula for the range of probability. Substitute into the! formula the probabilities of the other ! 3 1 events P(R) = and P(Y) = . 5 10

3 1 P(B) = 1 − − 5 10 3 = 10

Probability of a blue card is

Solve the equation by making P(B) the subject of the equation. Simplify the fraction if possible.

3 . 10

Write the answer in words.

Now you try For a particular couple, the chance they will have a male child is 10%. They are planning to have two children. What is the chance that they will not have two male children? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

2D Range of probabilities

Exercise 2D 1

3 and the A hat contains tickets labelled as ‘A’, ‘B’ and ‘C’. The probability of selecting ticket A is 10 7 probability of selecting ticket B is . 15 a What is the value of P(A)? b What is the value of P(B)? c What is the probability of selecting a ticket with the letter ‘D’? d What is the probability of selecting tickets A, B or C? e What is the probability of selecting ticket C?

U N SA C O M R PL R E EC PA T E G D ES

Example 9

BUILDING

2 A bag contains black, yellow and white cards. The probability of drawing a black card is 57% and the probability of drawing a yellow card is 8%. What is the value of the following, expressed as a fraction in simplest form? a P(Black)

b P(Yellow)

c P(White)

3 In a particular event the probability of Jun winning a gold 3 1 medal is and a silver medal is . There is no bronze 8 4 medal. a What are Jun’s chances of winning a gold or a silver medal? b What are Jun’s chances of not winning any medals?

Some picture cards from a deck of cards are placed face down on the table. The probability of drawing a king is 0.25 and a queen is 0.60. What is the value of the following expressed as a decimal?

a P(King) b P(Jack) c P(Jack) + P(King) d P(King) + P(Queen) + P(Jack)

There are four outcomes of an experiment. Three of the outcomes have probabilities of 20%, 25% and 40% respectively. What is the probability of the fourth outcome?

A biased die is rolled. The probability of obtaining an even number is 0.4 and the probability of a 1 or a 3 is 0.3. Find the value of the following probabilities.

a P(1, 2, 3, 4, 5, 6) b P(2, 4, 5, 6) c P(Odd)

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Chapter 2 Relative frequency and probability

DEVELOPING 7

A disc is chosen at random from a bag containing five different colours: black, green, pink, red 1 2 2 1 and white. If P(B) = , P(G) = , P(P) = and P(R) = , find the probability of the following 5 13 9 6 outcomes. b Pink or red disc d Black, green or pink disc f Black, green, pink or red disc

U N SA C O M R PL R E EC PA T E G D ES

a Black or green disc c Black or red disc e Black, green or red disc

A card is chosen at random from some playing cards. The probability of a spade is 0.24, the probability of a club is 0.27 and the probability of a heart is 0.23. Find the probability of the following outcomes.

a Black card c Club or a heart e Diamond

b Red card d Spade or a heart f Diamond or a club

Julia and Natasha are playing a game in which a standard six-sided die is rolled. Julia wins if an even number is rolled. Natasha wins if a number greater than three is rolled. What is the probability that the number rolled is neither even nor greater than three?

2 A bag contains white, green and red marbles. The probability of selecting a white marble is and the 7 1 probability of selecting a green marble is . What is the probability of selecting a red marble? 8

CHALLENGING

The numbers 1 to 20 are written on separate cards. One card is chosen at random. What is the probability that the card chosen is a prime number or is divisible by 3?

One letter is selected at random from a word containing the letters T, A, M, P, R. It is given that 1 2 1 1 P(T ) = , P(A) = , P(M) = and P(P) = . 5 5 10 10 a Find the probability of the following outcomes. i Letters T or A iii Letters M or P v Letter T, A, M or P

ii Letters T or P iv Letters A, M or P vi Letter R

b The word contains 10 letters. From the letters T, A, M, P, R how many of the following letters are in the word? i T

ii A

iii M

iv P

v R

c What is the word? (Hint: an Australian place.)

Communicate your reasoning, understanding and fluency 13

Your calculator has a button Write down the value of:

. What does it do? Search the internet.

a 45 C1

c 45 C3

b 45 C2

d 45 C4

e 45 C5

f 45 C6

The last one has something to do with Lotto. Write down what it is. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

2E Complementary events

2E Complementary events Learning intentions

• To be able to determine the complement of an event • To use the complement to determine probabilities

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

Sometimes, the simplest way to find the probability that something will happen is to find the probability that it won’t happen, then subtract that from 1.

If E is some event that could occur in a probability experiment, like rolling a 5 with a die, then the complement of E is all the other events which could occur. Think of ‘complement’ as meaning ‘opposite’. The complement of E is written as E. If E is ‘rolling a 5 with a standard die’, E is ‘rolling a 1, 2, 3, 4 or 6’. 1 5 In this case, P(E) = and P(E) = 6 6

If E is ‘tossing a head’, E is ‘tossing a tail’. 1 1 In this case, P(E) = and P(E) = 2 2 The probability of an event and the probability of the complement always have a sum of 1.

If the probability of an event is 0.7, the probability of the complement is 1 − 0.7, which is 0.3. P(an event does not occur) = 1 − P(the event does occur) P(E) = 1 − P(E)

Complementary Events

P(E) + P(E) = 1 or P(E) = 1 − P(E)

E – Event or outcome

E – Complement of event E

Example 10

Using the complement

Lisa selects a card at random from a normal pack.

Find the probability of obtaining the following outcomes.

a Not a 10 b Not a black jack (i.e. not a jack of clubs or spades) c Not a picture card

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Chapter 2 Relative frequency and probability

Solution

Explanation

a P(10) = 1 − P(10) 4 =1− 52 48 = 52 12 = 13

Write the formula for the complement. Substitute into the ! formula the probability for a 10 4 or P(10) = . 52 Evaluate. Simplify the fraction.

b P(black jack) = 1 − P(black jack) 2 =1− 52 50 = 52 25 = 26

Write the formula for the complement. Substitute into the formula the probability for a ! 2 . black jack or P(black jack) = 52 Evaluate. Simplify the fraction.

c P(picture) = 1 − P(picture) 12 =1− 52 40 = 52 10 = 13

Write the formula for the complement. Substitute into the formula the probability for a ! 12 . picture card or P(picture) = 52 Evaluate. Simplify the fraction.

U N SA C O M R PL R E EC PA T E G D ES

Now you try

(For the purposes of this question, ignore leap years, so 1 year = 365 days)

Remy was born on 29 March.

a What is the probability that Abdullah was born on the same day? b What is the probability that Abdullah was not born on the same day? c What is the probability that Lana was not born on the same day as Remy or Abdullah?

Exercise 2E 1

BUILDING

What is the event that is the complement of the following events?

a Selecting a black card from a normal pack of cards b Winning first prize in Lotto c Throwing an even number when a die is rolled d Obtaining a tail when a coin is tossed e Drawing a spade from a normal pack of playing cards f Choosing a green ball from a bag containing a blue, a red and a green ball

Find the value of P(E), given the following information about event E. 1 a P(E) = b P(E) = 0.9 c P(E) = 62% 5 3 e P(E) = f P(E) = 0.45 g P(E) = 37.5% 11

1 4 3 h P(E) = 7 d P(E) =

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2E Complementary events

The chances of the Sydney Swans winning the premiership are given as 29%. What are the chances that the Sydney Swans will not win the premiership?

U N SA C O M R PL R E EC PA T E G D ES

a Express your answer as a decimal. b Express your answer as a fraction.

Example 10

The probability of obtaining a 3 on a biased die is 0.6. What is the probability of not obtaining a 3?

11 The probability of a rainy day in March is . What is the probability that a particular day in March 15 does not have rain?

5 The probability of drawing a red marble from a bag is . What is the probability of not drawing a red 8 marble? Express your answer as a: a fraction

b decimal

c percentage.

DEVELOPING

A ball is chosen at random from a bag containing four different colours: brown, orange, purple and 2 2 1 yellow. If P(O) = , P(P) = and P(Y) = , find the probability of the following outcomes. 11 9 4 a Not a yellow ball b Not an orange ball c Not a purple ball d Orange or a purple ball e Yellow or a purple ball f Not a brown ball g A brown ball h Not an orange or a yellow ball

Suhaim selects a card at random from a normal pack. Find the probability of obtaining the following outcomes.

a Not a queen

b Not a red ace

What is the probability that a person selected at random will:

a not be born on Saturday? b not be born on a weekend?

CHALLENGING

A 12-sided die has faces marked 1 to 12. The die is biased. If P(8) = 0.1, P(2) = 0.15 and P(3) = 0.91, find: a P(8)

b P(2)

c P(3)

d P(8) + P(8)

e P(2) + P(2)

f P(3) + P(3)

g P(2) + P(8)

h P(2 or 8)

One card is selected at random from a non-standard pack of playing cards. If P(ace) = 8%, P(king) = 7% and P(queen) = 10%, find the probability of the following outcomes. a Not an ace c Not a queen e Ace or a queen

b Not a king d King or a queen f Not an ace, king or queen

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Chapter 2 Relative frequency and probability

The probability of selecting a card labelled ‘T’ from 32 cards is given as P(T ) =

3 . 16

a What is the probability of not selecting a card labelled with a ‘T’? b How many of the 32 cards were labelled with a ‘T’?

U N SA C O M R PL R E EC PA T E G D ES

Communicate your reasoning, understanding and fluency 13

Read the first page of this chapter, regarding the birthday coincidences in World Cup teams. This question explains the maths behind it.

a You want to find the probability that two or more people share a birthday. That is, you want to know P(two or more people have the same birthday). What is the complementary event? b Read carefully: • forget about leap years. Assume that a year has 365 days • you want everybody to have a different birthday

• the first person arrives and says their birthday is 11 March. That date is now ‘taken’

• the second person arrives. What is the probability this person has a different birthday? • third person: What is the probability this person has a different birthday to the first

two people? • fourth person: What is the probability this person has a unique birthday? • fifth person, etc. Copy and complete: 1st person

2nd person

3rd person

4th person

5th person

Probability of unique birthday

c Multiply all the probabilities together, then subtract the answer from 1 and convert to a percentage correct to two decimal places. This is P(2 or more of these 5 people share a birthday). Very low probability. d Extend the pattern to 23 people. A spreadsheet like this will be very useful. e Show that the probability that two or more people share a birthday is 50.73%, correct to two decimal places. f Conclusion: By the time you reach the 23rd person, there is a better than 50% chance that two or more of them share a birthday. Try this experiment with 23 random people. g We have assumed that nobody is born on February 29 in a leap year. What is another assumption we have made?

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2F Relative frequency

2F Relative frequency Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand that the probability of an event may be approximated by the relative frequency • To estimate probabilities using relative frequencies • To recognise that increasing the number of trials produces relative frequencies closer in value to the theoretical probabilities

Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Years 7 to 10 • Questions involving probability regularly appear in HSC Examinations

Sometimes it can be difficult or impossible to calculate the probability of an event.

For example, an insurance company needs to know the probability that an 18 year-old male driver will have a car accident in the year ahead, so they can set the price of their insurance policies. Sometimes it is possible to conduct a probability experiment and use relative frequency as an estimate for the probability. For example, when a thumbtack falls to the ground it could land in two different ways, as shown in the photo. Haruki tried this 50 times. These are the results: Frequency

Relative frequency

Point up

28 50

Point down

22 50

28 Based on this trial, the relative frequency for the outcome ‘point up’ can be expressed as , which reduces 50 14 to , or 0.56 as a decimal, or 56% as a percentage. 25 These figures can be used as an estimate of the probability. P(point up) ≈ 56%

The questions which face scientists and researchers are: • Are 25 trials enough to draw that conclusion? • Is this information reliable and conclusive? • If not, how many trials would be enough?

Generally speaking, if the number of trials is increased, the relative frequencies become closer in value to the theoretical probability. Note: The authors conducted this experiment 200 times with the same thumbtack. After 200 trials, the relative frequency was 60.5%.

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Chapter 2 Relative frequency and probability

Relative Frequency Relative frequency is usually collected from a survey or a probability experiment Number of times a data value was observed Relative frequency = Total number of data values The result can be expressed as a fraction, decimal or percentage.

U N SA C O M R PL R E EC PA T E G D ES

If the number of data values is sufficient, relative frequency can be used as an estimate of the theoretical probability.

Example 11

Finding the relative frequency

An experiment of tossing two coins was completed and the number of heads recorded in the frequency table shown. Find the relative frequency of obtaining the following number of heads: a 0

b 1

c 2

Number of heads

Frequency

100

192

108

Relative frequency

Solution

Explanation

a 100 + 192 + 108 = 400

Add the frequency column to determine the total number of frequencies. Write the formula for relative frequency. Substitute the frequency and total number of frequencies into the formula. Simplify the fraction if possible or express as a decimal. Write the answer in words.

100 400 1 = 4 = 0.25 or 25% Relative frequency of 0 heads is 0.25.

Rel. freq. =

192 400 12 = or 0.48 or 48% 25 Relative frequency of 1 head is 0.48.

Write the formula for relative frequency. Substitute the frequency and total number of frequencies into the formula. Simplify the fraction if possible or express as a decimal. Write the answer in words.

108 400 27 = 100 = 0.27 or 27%

b Rel. freq. =

c Rel. freq. =

Relative frequency of 2 heads is 0.27.

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2F Relative frequency

Now you try A thumbtack was tossed 200 times. These were the results: Frequency 121

Point down

U N SA C O M R PL R E EC PA T E G D ES

Point up

Relative frequency

a Find the relative frequency, as a percentage, for ‘point up’. b Find the relative frequency, as a percentage, for ‘point down’.

Exercise 2F

Example 11

BUILDING

The frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Express as a fraction in simplest form. a A b B c C d D

Letter

Frequency

Outcome

Frequency

Colour

Frequency

Black

105

Yellow

210

Red

145

Blue

170

Green

215

White

155

The frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Answer as a percentage.

Score

Frequency

a 30 c 32 e 34

2 The frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Answer correct to three decimal places. a HH b HT c TH d TT 3 The frequency table shows the outcomes of an experiment. What is the relative frequency for the following outcomes? Answer as a percentage correct to one decimal place. a Black c Red e Green

b Yellow d Blue f White

b 31 d 33

Relative frequency

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Chapter 2 Relative frequency and probability

Calculate the relative frequency for each of these numbers if the total frequency is 48. Write your answer as a fraction in simplest terms. a 16

b 40

c 24

d 6

Calculate the relative frequency for each of these numbers if the total frequency is 40. Write your answer as a percentage. b 30

c 15

d 32

U N SA C O M R PL R E EC PA T E G D ES

a 4

DEVELOPING

A retail store sold 512 televisions last year, of which 32 were faulty and returned to the store. What is the relative frequency of a faulty television last year? Answer as a percentage, correct to two decimal places.

A pistol shooter at the Olympic Games hits the target 24 out of 25 attempts. What is the relative frequency of the shooter hitting the target? Give the answer as a decimal, correct to two decimal places.

The birth statistics in a local community were 142 females and 126 males. What is the relative frequency for a female? Answer as a fraction in lowest terms.

In an experiment a die was thrown 120 times. The frequency of each data value is shown in the frequency table.

Number

Frequency

a How many times would you expect to obtain each of the data values? b Do you think that the die is fair? Give a reason for your answer.

Relative frequency

Create the spreadsheet below using the frequency table in question 4.

a Cell B10 has a formula that adds cells B5 to B9. Enter this formula. b The formula for cell C5 is ‘=B5/$B$10’. It is the formula for relative frequency. Fill down the contents for C6 to C9 using this formula. c Cells D5 to D9 have the same formula as cells C5 to C9. Enter this formula and format the cells to a percentage.

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2F Relative frequency

CHALLENGING A frequency distribution table is shown below. Score

Frequency

Relative frequency

0.20

0.30

0.25

U N SA C O M R PL R E EC PA T E G D ES

a What is the value of x? b What is the value of y? c What is the total number of scores?

Perform an experiment by rolling a die 120 times.

a Use a frequency table to record the results of the experiment. b Calculate the relative frequency of each outcome. c What result would you have predicted for each outcome? d Compare your results to those of the other students in your class.

Roll two standard dice and work out the difference between the two scores. For example, if one die shows a 2 and the other die shows a 5, the difference is 3. Do this 36 times.

Perform an experiment by tossing two coins 80 times.

Communicate your reasoning, understanding and fluency 16

Ashvin said: “Last year it rained on 140 days out of 365 in Dubbo. Therefore, the chance it will rain tomorrow is about 38%.” Make two or more comments which cast doubt over Ashvin’s statement.

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Chapter 2 Relative frequency and probability

2G Expected frequency Learning intentions

• To understand the concept of expectation • To calculate the expected number of times an outcome will occur by multiplying the probability by the

U N SA C O M R PL R E EC PA T E G D ES

number of trials Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Years 7 to 10 • Questions involving probability regularly appear in HSC Examinations

The expected frequency is the number of times that a particular event should occur. It may not equal the 1 actual results. For example, when a coin is tossed the probability of getting a head is . Hence, if a coin is 2 1 tossed 100 times, the expected number of heads is 50 or × 100. Clearly, if a coin is tossed 100 times it 2 may not result in exactly 50 heads. However, the larger the number of trials the more accurate the expected frequency will be.

Expected Frequency

Expected frequency is the number of times that a particular event should occur. Expected frequency = n × p = np

n = number of times the experiment is repeated p = probability of the event

The expected frequency may not be a whole number. It is an estimate of what to expect. For example, when 1 a die is tossed, the probability of getting a six is . Hence, if a die is tossed 100 times the expected number 6 1 2 2 of sixes is × 100 or 16 . Clearly, it is not possible to have of a six. The expectation is a whole number 6 3 3 2 close to 16 . 3

Example 12

Finding the expected frequency

Two coins are tossed 120 times and the results recorded.

a What is the expected frequency for two heads? b What is the expected frequency for a head and a tail? Solution

Explanation

1 a P(HH) = 4 Expected frequency = np 1 = × 120 4 = 30

Calculate the probability of two heads. Number of favourable outcomes is 1 (HH). The total number of outcomes is 4 (HH, HT, TH, TT). Write the formula for expected frequency. Substitute into the formula. Evaluate.

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2G Expected frequency

2 1 = 4 2 Expected frequency = np

b P(HT or TH) =

1 × 120 2 = 60

U N SA C O M R PL R E EC PA T E G D ES

Calculate the probability for a head and a tail. Number of favourable outcomes is 2 (HT, TH). The total number of outcomes is 4 (HH, HT, TH, TT). Write the formula for expected frequency. Substitute into the formula. Evaluate.

Now you try

1 When three coins are tossed, the probability of HHH = . If this experiment is conducted 50 times, how 8 many times would you expect to toss HHH?

Exercise 2G

Example 12

BUILDING

1 The probability of a red traffic light at an intersection is . How many red traffic lights are expected 3 on a trip that passes through 54 intersections?

The probability of a person living in a certain community of developing melanoma is four out of nine. There are 1404 people living in this community. What is the expected number of people who will develop melanoma?

Soren and Caitlin are planning to have five children. A genetic counsellor has calculated they have a 40% chance of having a child with green eyes. How many of Soren and Caitlin’s children are expected to have green eyes?

Amira is a goal shooter for her netball team. She usually scores on 88% of her shots. This year she has had 225 attempts at goal.

a How many goals would you expect Amira to have scored this year? b How many goals would you expect Amira to have missed this year?

Akira is a professional golfer who has a 78% chance of breaking par. He plays 150 games of golf in a year. How many times would you expect Akira to break par in a year?

The probability of a worker in an industrial plant having an accident is 0.12. The industrial plant employs 175 workers. What is the expected number of accidents?

A die is rolled 480 times and the results recorded.

a What is the probability of throwing a 4? b How many 4s are expected? c What is the probability of throwing an odd number? d How many odd numbers are expected? e What is the probability of throwing a number greater than 2? f How many numbers greater than 2 are expected? g What is the probability of throwing a number divisible by 3? h How many numbers divisible by 3 are expected?

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Chapter 2 Relative frequency and probability

DEVELOPING Five cards (A, K, Q, J and 10) are placed face down on a table. One card is selected at random and replaced. A second card is then selected at random. This experiment is repeated 200 times.

U N SA C O M R PL R E EC PA T E G D ES

a What is the probability of selecting two aces? b How many double aces are expected? c What is the probability of selecting an ace followed by a king? d How many aces then kings are expected? e What is the probability of exactly one of the cards being a 10? f How many single 10s are expected?

A three-digit number is selected from cards labelled 3, 4 and 5. The first card selected is the hundreds digit, the second card is the tens digit and the third card is the units digit. The cards are selected without replacement. This selection is repeated 30 times.

a What is the probability the number starts with the digit 3? b How many numbers starting with the digit 3 are expected? c What is the probability the number is 453? d How many times is the number 453 expected? e What is the probability the number ends with a 4 or a 5? f How many numbers ending with a 4 or 5 are expected?

A bag contains 6 yellow discs and 5 red discs. Two discs are drawn in succession from the bag. The first disc is not replaced before the second disc is drawn. This process is repeated 352 times.

a How many of the first discs are expected to be yellow discs? b How many of the first discs are expected to be red discs? c How many double yellow discs are expected? d How many double red discs are expected? e How many are expected to have a first disc yellow and a second disc red? f How many are expected to have a first disc red and a second disc yellow?

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2G Expected frequency

Create the spreadsheet below.

U N SA C O M R PL R E EC PA T E G D ES

a Cell C8 has a formula that multiplies cells B8 and B4. Enter this formula. b Fill down the contents of C8 to C11 using the formula in cell C8. c Change the number of trials from 230 to 800. Observe the change in C8:C11.

There are 240 families with three children. Assume P(male) = 50%.

a How many of these families are expected to have three male children? b How many of these families are expected to have exactly one male child? c How many of these families are expected to have exactly two male children? d How many of these families are expected to have no male children?

CHALLENGING

Two cards are selected at random from a normal playing pack with replacement. This experiment is repeated 2704 times with the cards being replaced each time.

a What is the expected number of times the result would be two spades? b What is the expected number of times the result would be two aces? c What is the expected number of times the result would be two picture cards? d What is the expected number of times the result would be two cards with a number less than 9?

Two dice are tossed one after the other onto a table. This event is repeated 144 times.

a On how many occasions would you expect the result to be a 6 then a 1? b On how many occasions would you expect the result to be two 3s? c On how many occasions would you expect the result to be two odd numbers?

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Chapter 2 Relative frequency and probability

Communicate your reasoning, understanding and fluency Mei rolls two standard dice, then works out the difference between the two scores. For example, if she rolls a 5 and a 2, the difference is 3. She pays $1 every time she plays this game. If her difference is 0, 1 or 2, she loses her dollar. If her difference is 3, 4 or 5, she gets her dollar back, plus another dollar.

U N SA C O M R PL R E EC PA T E G D ES

a Copy and complete these tables. 1

1 2

Outcome (difference)

Probability

6 36

5 6

b Use the tables to explain why this game is not fair. c If Mei plays this game 100 times, how much money would she expect to lose? d Change the rules to make the game fair.

The syllabus says that students should: ‘Examine the use of statistics and the associated probabilities in shaping decisions made by media, governments and companies’. Australia’s media, governments and some companies are heavily involved in the gambling industry. Use the internet to research the following:

a How much money did people in NSW lose on poker machines in a recent year? b What are some of the organisations which benefit from this loss? c The probability is clear: people who play poker machines will lose money in the long run. Some people are addicted to playing poker machines. Our governments are aware of this. What are some of the things that the NSW and Australian governments have done to address poker machine addiction? Suggestion: Watch the documentary ‘Ka-Ching! Pokie Nation’ to further explore poker machine addiction.

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2H Venn diagrams and two-way tables

2H Venn diagrams and two-way tables Learning intentions

• To understand that Venn diagrams with two attributes can be used to contain the information in a

U N SA C O M R PL R E EC PA T E G D ES

two-way table • To be able to transfer information between a Venn diagram and a two-way table and vice versa • To be able to solve problems using Venn diagrams and two-way tables Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Years 7 to 10 • Questions involving probability regularly appear in HSC Examinations, but Venn diagrams were not examined prior to 2027.

Consider this scenario:

Somebody asked you to choose a card randomly from a standard pack of 52 cards. They then asked you to answer these two questions: A: Was your card red? B: Was your card a picture card?

The diagram below shows the contents of the pack. Note: • 26 cards are red

• 12 cards are picture cards, of which 6 are red, so 6 cards are red picture cards • 20 cards are neither red cards nor picture cards.

There are four different responses to the two questions: Questions

Response 1

Response 2

Response 3

Response 4

A: Was your card red?

Yes

B: Was your card a picture card?

Yes

In this example, the four responses are not equally likely. The number of ways of achieving each of the four outcomes can be expressed in a two-way table and a Venn diagram (named after John Venn (1834 − 1923)). Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

100

Chapter 2 Relative frequency and probability

Two-way table

Venn diagram

Not A

Total

Not B

Neither A nor B

B 6

U N SA C O M R PL R E EC PA T E G D ES

Total

Both A and B

A not B

B not A

Two-way tables and Venn diagrams

These are displays of the responses to a survey or results of an experiment in which two questions were asked that have two answers (YES or NO). They can be used to estimate the probability that other people will answer the same questions in certain ways. Two-way table A

Not A

Venn diagram

Total

Not B Total

The number in the yellow cell is: • the sum of the bottom row and • the right column; or • the sum of the four orange cells A

Not A

The number in the yellow cell of the table is the sum of all four numbers (the boxes) in the Venn diagram.

Total

Not B Total

The shaded cell contains the number of people who answered ‘YES’ to both questions.

Venn diagram: The section in which the two circles overlap is called the ‘intersection’. In words: Both A and B are true. In symbols: A ∩ B Note: The syllabus does not mandate that set notation such as this must be used. This notation is unlikely to be used in HSC questions. However, it can be used by students in their solutions.

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2H Venn diagrams and two-way tables

Two-way table A

Not A

Venn diagram Total A

Not B

U N SA C O M R PL R E EC PA T E G D ES

Total

The shaded cell contains the number of people who answered ‘YES’ to one or both questions.

Not A

Venn diagram: All three sections inside the circles, called ‘the union’. In words: A or B are true. The word ‘or’ is inclusive. It means ‘A’ or ‘B’ or ‘both’. It is called the ‘union’. It does not mean ‘one or the other’. In symbols: A ∪ B

Total

Not B Total

The shaded cell contains the number of people who answered ‘NO’ to both questions.

Example 13

Venn diagram: The number in the purple cell is found in the section outside the circles but inside the rectangle. In words: Neither A nor B

Transferring information from a two-way table to a Venn diagram

a Complete the two-way table. A

Not A

Not B

Total

b Use the two-way table to draw a Venn diagram. Solution

Explanation

Not A

Total

Not B

Total

Find the sum of each row and each column.

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Chapter 2 Relative frequency and probability

b A

U N SA C O M R PL R E EC PA T E G D ES

Note: • the number 6 is inside the overlapping section • the positions of the numbers 6, 7, 8 and 9 in the table and the Venn diagram • the sum of all four numbers is 30 • the total in the A circle is 14 • the total in the B circle is 13.

Now you try

a Complete the two-way table. A

Not A

Not B

Total

b Use the two-way table to draw a Venn diagram.

Example 14

Transferring information from a Venn diagram to a two-way table

Use the Venn diagram to construct a two-way table.

A 5

Solution

Explanation

Not A

Total

Not B

Total

Note: • the number 10 is inside the overlapping section • the positions of the numbers 5, 8, 10 and 20 in the Venn diagram and the table • the sum of all four numbers is 43 • the total in the A column 15 • the total in the B row is 18.

Now you try

Use the Venn diagram to construct a two-way table.

40 10

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2H Venn diagrams and two-way tables

Example 15

Using a two-way table to calculate probabilities

A group of people were asked these two questions: A: Have you been to the North Island of New Zealand? B: Have you been to the South Island of New Zealand? Their responses are listed in the two-way table.

Not A

Total

U N SA C O M R PL R E EC PA T E G D ES

a Complete the two-way table. b If one person is chosen at random from this group, what is the probability that:

Not B

Total i they have been to both islands? ii they have been to the South Island? iii they have been to the South Island but not the North Island?

Solution

Explanation

Not A

Total

Not B

Total

100

20 1 = 100 5 32 8 ii P(South Island) = = 100 25 12 3 iii P(South, not North) = = 100 25 i P(both islands) =

Find the sum of each row and each column.

20 of the 100 people have been to both islands.

32 of the 100 people have been to the South Island. 12 of the 100 people have been to the South Island but not the North Island.

Now you try

Using Example 15, if one person is chosen at random from this group, what is the probability that:

a they have been to neither island? b they have been to the North Island? c they have been to the North Island but not the South Island?

Example 16

Using a Venn diagram to calculate probabilities

In Year 12 there are 120 students. Of these, 70 play a musical instrument (I), 60 play a sport (S), and 20 do neither.

a Using this information, complete the Venn diagram. b A Year 12 student is selected at random. What is the probability that the student plays an instrument and a sport? c Two students are chosen at random. What is the probability that they both play an instrument only?

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Chapter 2 Relative frequency and probability

Solution

Explanation

Start with 120 students, then subtract the 20 who do neither, leaving 100 students. 60 + 70 = 130 and 130 − 100 = 30, so 30 students do both. 70 − 30 = 40 students who play an instrument only. 60 − 30 = 30 students who play sport only.

I 30

U N SA C O M R PL R E EC PA T E G D ES

b P(instrument and sport) =

30 1 = 120 4

30 of the 120 students play both.

When the first student is chosen, 39 of the remaining 119 play an instrument.

c P(two students play instrument only) 40 39 13 = × = 120 119 119 Now you try

Using Example 16:

a A Year 12 student is selected at random. What is the probability that the student plays a sport but not an instrument? b Two students are chosen at random. What is the probability that they both play sport only?

Exercise 2H 1

BUILDING

Match the diagrams A, B, C, D, E with the correct description. a P and Q d P

b P or Q e Not P

c P but not Q f Neither P nor Q

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2H Venn diagrams and two-way tables

Customers at a restaurant were given the following choices to add to their dessert: A: Would you like cream? B: Would you like ice cream?

Not A

Total

U N SA C O M R PL R E EC PA T E G D ES

The results from 100 customers are shown in this table. a Copy the table and fill in the five missing totals. b How many people: i chose both cream and ice cream? ii chose neither cream nor ice cream? iii chose cream? iv chose ice cream? vi chose ice cream without cream? viii did not have cream?

Example 13

Not B

v chose cream without ice cream? vii did not have ice cream? ix chose cream or ice cream?

Draw this Venn diagram for each of the following, then shade in the region which shows: a A and B c A but not B e not A

Total

3 Convert the table in Question 2 into a Venn diagram by writing a number in each of the four boxes.

b A or B d A f neither A nor B.

5 Students in Year 12 were asked:

A: Would you like to represent the school in soccer? B: Would you like to represent the school in AFL?

The results from 100 students were as follows: A

How many people:

a chose both soccer and AFL? c chose soccer? e chose soccer but not AFL? g did not choose soccer? i chose soccer or AFL?

b chose neither soccer nor AFL? d chose AFL? f chose AFL but not soccer? h did not choose AFL?

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Chapter 2 Relative frequency and probability

Example 14

6 Convert the Venn diagram in Question 5 into a two-way table.

Not A

Total

B Not B Total Copy and complete the following two-way tables.

U N SA C O M R PL R E EC PA T E G D ES

Not A

Total

Not B Total

Not A

Not B

Total

100

In the following Venn diagrams, some of the numbers are missing, as indicated by empty boxes. What are the missing numbers?

a Overall total is 30.

A 5

b Overall total is 50.

c Total of A is 20. Total of B is 30. Grand total is 100.

d Grand total is 100. Total of A is 60. Total of B is 50.

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2H Venn diagrams and two-way tables

DEVELOPING 9

For each two-way table, fill in the missing numbers then draw a Venn diagram. a

A B

Not A

Total

Not A

Total

Not B

Total

120

130

U N SA C O M R PL R E EC PA T E G D ES

Not B

Total

120

A group of people were asked these two questions: A: Do you drink coffee every week? B: Do you drink tea every week?

• 50 people answered YES to both questions • 10 people answered NO to both questions • 75 people answered YES to question A • 65 people answered YES to question B

a Copy and complete this two-way table. A

Not A

Total

Not B Total

b Copy and complete this Venn diagram.

Example 15

11 a Copy and complete this two-way table: A

Not A

Total

Not B Total

b Calculate the following probabilities: i P(A and B) iii P(neither A nor B) v P(A) vii P(not B)

ii P(A or B) iv P(A but not B) vi P(B)

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Repeat Question 11 with this two-way table. A B Not B

Not A

Total

U N SA C O M R PL R E EC PA T E G D ES

Example 16

13 Repeat Question 11 with this Venn diagram.

CHALLENGING

In Year 12 there are 100 students. Of these, 40 play soccer (S), 50 play netball (N) and 15 do neither.

a Using this information, complete the Venn diagram.

b A Year 12 student is selected at random. What is the probability that the student plays a soccer and netball? c Two students are chosen at random. What is the probability that they both play soccer?

In a group of 30 students, it is found that 10 play both cricket and soccer, 5 play cricket only and 7 play soccer only.

a Represent this information in a Venn diagram. b How many students play neither cricket nor soccer? c How many students play cricket? d How many students play cricket or soccer or both? e If a student is selected at random from the group, what is the probability that the student plays cricket or soccer but not both? f If another student is chosen, what is the probability that both of them play cricket or soccer but not both?

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2H Venn diagrams and two-way tables

Consider this Venn diagram, in which the letters r, s, t and u represent the number of elements in each section. A t

U N SA C O M R PL R E EC PA T E G D ES

Write an algebraic fraction for the following probabilities.

a P(A and B) b P(A or B) c P(neither A nor B) d P(A but not B) e P(A) f P(B) g P(not B)

Communicate your reasoning, understanding and fluency

3 17 a Construct a two-way table and a Venn diagram in which P(A or B) = . This question has many 4 correct answers. Explain how you know that yours is one of them. 3 b Construct a two-way table and a Venn diagram in which P(A and B) = . This question has many 4 correct answers. Explain how you know that yours is one of them. 3 c Construct a two-way table and a Venn diagram in which P(A but not B) = . This question has 4 many correct answers. Explain how you know that yours is one of them.

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Chapter 2 Relative frequency and probability

Key ideas and chapter summary Language of probability

Probability is the chance of something happening. Sample space is the set of all possible outcomes. Each outcome is an element of the sample space. Equally likely outcomes have an equal chance of occurring. Multistage event consists of two or more events.

U N SA C O M R PL R E EC PA T E G D ES

Review

110

Definition of probability

If all the outcomes are equally likely, Number of favourable outcomes or Probability (Event) = Total number of outcomes n(E) P(E) = (also known as an array) n(S )

Arrays and tree diagrams

A table (also known as an array) is an arrangement of information in rows and columns. A tree diagram is another way to list all the possible results of a multistage experiment.

Range of probabilities

Probability of an event that is impossible is 0 and certain is 1. Probability of an event ranges from 0 to 1. P(A) + P(B) + . . . = 1 A, B, . . . are all the possible outcomes or events.

Complementary events

Complement of an event E is the event not including E. Probability of an event and its complement will sum to 1. P(E) + P(E) = 1 or P(E) = 1 − P(E) E – Event or outcome E – Complement of event E or the outcomes not including event E

Relative frequency

Relative frequency is an estimate for the probability of an event. Number of times a data value was observed Relative frequency = Total number of data values

Expected frequency Expected frequency is the number of times that a particular event should occur. of an event Expected frequency = n × p = np

n = number of times the experiment is repeated p = probability of the event

Venn diagrams and two-way tables

A 2

Not A

Not B

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111

Chapter 2 review

A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

Review

Chapter checklist with success criteria

1 I can use the terminology associated with probability.

U N SA C O M R PL R E EC PA T E G D ES

e.g. I am going to randomly choose one of the 5 vowels we use in the English alphabet. What is the sample space?

2 I can use fractions, decimals and percentages to assign probabilities to outcomes. e.g. I am going to randomly choose a letter from the alphabet. What is the probability that it is a vowel?

3 I can use arrays and tree diagrams to list the outcomes of a probability experiment and solve problems.

e.g. A coin is tossed and a 4-sided die is rolled.

List the outcomes in an array.

Draw a tree diagram.

The coin is biased and shows up as heads 60% of the time. The die is biased and rolls a ‘1’ 40% of the time. What is the probability of the outcome H1?

4 I can use tree diagrams to solve problems with replacement and without replacement. e.g. A bag contains 5 red balls and 3 green balls.

One ball is chosen, noted and then replaced. Then another ball is chosen. What is the probability that the two balls are the same colour?

Repeat part a, given that the first ball is not replaced.

5 I can use the range of probabilities.

2 1 e.g. In a bag containing 20 coloured blocks, P(red) = and P(black) = . How 5 5 many blocks are red or black?

6 I can use complementary events to solve problems.

e.g. In Question 5, what is the probability of choosing a block which is neither red nor black?

7 I can use relative frequencies to estimate probabilities.

e.g. I chose a card from a bag, then replaced it. This was done 50 times. 30 of the cards picked were red. What is the relative frequency of choosing a red, as a percentage?

8 I can determine an expected frequency based on the probability of an event. e.g. Using Question 7, if I conducted the experiment 250 times, how many cards would you expect to be red?

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Chapter 2 Relative frequency and probability

9 I can transfer between two-way tables and Venn diagrams and vice versa. e.g. a

Copy and complete this two-way table, then convert it into a Venn diagram. A

Not A

Not B

Total

U N SA C O M R PL R E EC PA T E G D ES

Review

112

Total

Use the Venn diagram to complete the two-way table. A 6

Not A

Total

Not B Total

10 I can solve problems using a two-way table. e.g. Using Question 9b, find the following: a

P(not B)

b P(both A and B)

number of A only

number of either A or B or both A and B.

11 I can solve problems using a Venn diagram.

At a sausage sizzle, customers can add some onion (O) or an egg (E) to the sausage in their roll. From the first 45 customers, 25 people chose onion, 30 chose egg and 15 had both. a

Construct a Venn diagram to represent this information.

How many people had neither onion nor egg?

How many people chose onion but not egg?

If one of these people is chosen at random, what is the probability that they chose egg but not onion?

Find P(did not choose onion).

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Chapter 2 review

An event has the probability of occurring equal to 0. What word can you use to describe this probability? B Impossible

C Likely

D Unlikely

U N SA C O M R PL R E EC PA T E G D ES

A Certain

Review

Multiple-choice questions

2 How many elements are there in the sample space when two cards are selected without replacement from cards labelled 1 to 7? A 7

C 42

D 49

3 A three-digit number is formed from the digits 5, 7, 8 and 9. What is the probability that the number will be odd? A 0.25

B 14

B 0.50

C 0.75

D 0.80

4 One card is selected from cards labelled 1, 2, 3, 4 and 5. What is the probability of an even number or a number divisible by 5? A 10%

B 50%

C 60%

D 100%

5 A letter is chosen at random from the word ‘NEWCASTLE’. What is the probability that the letter will not be a vowel? 1 2 1 2 A B C D 9 9 3 3

When two dice are rolled, how many outcomes are there? A 6

D 36

B 0.36

C 0.63

D 0.9

8 The frequency of an event is 6 and the total number of frequencies is 20. What is the relative frequency? A 14%

C 21

7 A bag contains black, white and grey balls. The probability of selecting a black ball is 0.3 and a grey ball is 0.6. What is the probability of selecting a white ball? A 0.1

B 12

B 26%

C 30%

D 70%

9 Two unbiased coins are tossed 100 times. Which calculation illustrates the expected number of times you would get a tail and a head? 1 1 1 1 A × 100 B × 100 C × 100 D × 200 4 2 3 4

2G 10

A card is selected at random from a normal pack of playing cards. The card is then replaced. What is the expected number of times you would select a king from 260 trials? A 5

B 20

C 52

D 65

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Chapter 2 Relative frequency and probability

Short-answer questions 2A

Describe each of these events as either impossible, likely, unlikely or certain. a Throwing an 8 when a standard die is thrown. b A student travelling to school on the wrong bus. c A piece of gold being found in your backyard.

U N SA C O M R PL R E EC PA T E G D ES

Review

114

2 List the sample space of the following events.

a Selecting a card from cards labelled with an even number from 1 to 10. b A spinner with each section labelled with a vowel. c Choosing at random a ball from a bag containing three white and four blue balls.

3 A raffle ticket is drawn from a box containing 50 raffle tickets numbered from 1 to 50. Find the probability of the following outcomes. a The number 50 c A number less than 20 e A number divisible by 5

b An even number d A number greater than 30 f A square number

What is the probability of choosing a black card from a standard deck of cards?

Four kings are taken from a standard deck of cards and placed face down on a table. One card is selected at random. What is the probability of selecting: a the king of clubs?

c is a girl or a boy?

b Either a 3 or a 5 d A number divisible by 3 f A prime number

8 There are five students in a group and their names are Adam, Sarah, Max, Hayley and David. If one name is chosen at random, what is the probability of selecting a name: a with five letters?

b is a boy?

7 An eight-sided die has the numbers 1 to 8 on it. What is the probability of rolling the following outcomes? a The number 2 c The number 9 e An odd number

c a picture card?

6 There are four girls, three boys and two adults in a house. If one person is chosen at random, what is the probability that the person: a is a girl?

b a black king?

b with the letter ‘a’?

c with one vowel?

9 A paper bag contains three green, four brown and five yellow beads. To win a game, Zoe needs to draw two green beads from the bag in two draws, without replacement. How many elements are in the sample space?

2E 10 A fair coin is tossed three times. The probability of throwing three tails is 0.125, two tails is 0.375 and one tail is 0.375. What is the probability of the following outcomes? a No tails c At least one tail e Not throwing two tails

b Three or two tails d Not throwing a head f Throwing one head

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Review

2E 11 There are three outcomes of a rugby league game: 5 1 win, lose or draw. If P(Win) = and P(Lose) = , 7 5 find the probability of the following.

U N SA C O M R PL R E EC PA T E G D ES

a Winning or losing the match b Drawing the match c Not winning the match d Not losing the match

2E 12 Abeer selects a card at random from a standard pack of cards. Find the probability of obtaining the following outcomes. a Not an ace

2F 13

b Not a heart

c Not a red six

A class frequency table shows the scores in a test. Score

Frequency

50−59

60−69

70−79

80−89

Relative frequency

What is the relative frequency for the following outcomes? Answer correct to two decimal places. a 50–59

2F 14

b 60–69

c 70–79

d 80–89

Last year Oscar bought a packet of biscuits every week and found 30 of these packets contained broken biscuits. What is the relative frequency of this event? Answer as a decimal correct to two decimal places.

1 2G 15 The probability of a couple having a baby with red hair is 33 %. If the couple have six children, how 3 many children with red hair are expected? 3 2G 16 A probability of a dog catching heartworm is . If there are 896 dogs in the local community, how 8 many of them would you expect to catch heartworm?

Extended-response question

A lie detector was used to indicate the guilt or innocence of 200 suspects. Accurate

Not accurate

Total

True statements

105

False statements

Total

165

200

a Find the probability a person selected at random, with an accurate test, made a true statement? b Find the probability a person selected at random, with a false statement, has an accurate test? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

U N SA C O M R PL R E EC PA T E G D ES

Investments and loans

How has inflation affected prices in the last 50 years? A basket of groceries worth $50 dollars in 1970 would cost approximately $729 in 2025. This is an increase of 1358% over 55 years at an average inflation rate of 5% . The Reserve Bank of Australia has a fantastic inflation calculator at https://www.rba.gov.au/ calculator/annualDecimal.html. How many Australians own shares?

Share prices go up and down daily. Over a long period of time shares on the Australian Stock Exchange (ASX) have risen steadily. In just the past 5 years, shares have risen approximately 40%. You might think that this does not affect many people. However, in Australia 7.7 million people own investments listed on the ASX. Ten million Australians hold investments other than their superannuation.

Credit cards

Credit cards are used by millions of people around the world as a simple way to use a line of credit. As long as you pay off your monthly spend by the due date, you should not have to pay any interest. However, if you do not pay off the entire amount the compound interest will add up quickly and your credit card can become a debt trap. Buy Now Pay Later (BNPL)

The rise of BNPL schemes in Australia has been driven by people wanting access to money in other ways than borrowing it from a bank. BNPL schemes work to allow people to get access to items without having to save up the money beforehand. The charges and fees can become a serious issue for people that do not keep track of how much they need to pay back each month or fortnight.

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Chapter contents

U N SA C O M R PL R E EC PA T E G D ES

3A Simple interest on investments 3B Compound interest on investments 3C Graphs of investments: Simple interest and compound interest 3D Appreciation and inflation 3E Shares and dividends 3F Straight-line depreciation 3G Declining-balance depreciation 3H Costs of borrowing money 3I Reducing-balance loans 3J Credit cards 3K Credit card statements 3L Fees and charges for credit card usage

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Models financial situations and solves problems involving interest, depreciation and borrowing money (MST-12-S2-02). © NSW Education Standards Authority

Online resources

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3A Simple interest on investments Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand simple interest on investments • To use the simple interest formula to calculate the interest and the future value • To solve practical problems involving simple interest Past, present and future learning

• Students may already be familiar with the contents of this section • Questions involving simple interest regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The formula for simple interest does not appear on the 2027 version.

Interest is the amount earned for lending money or investing. There are different ways of calculating interest. Simple interest (or flat interest) is a fixed percentage of the amount invested and is calculated on the original amount. For example, if we invest $100 in a bank account that pays interest at the rate of 5% per annum (per year) we would receive $5 each year. That is, 5 Interest = $100 × = $5 100

This amount of interest would be paid each year. Simple interest is always calculated on the initial amount, or the principal. An investment in the context of interest is money that you put into a bank, or other financial institution, in return for interest payments.

Formula for calculating simple interest I = Prn, where

I = Interest (simple or flat) earned for the use of money, paid by borrowers to lenders P = Principal is the initial amount of money borrowed, lent or invested

r = Rate of simple interest per time period expressed as a decimal, e.g. 5% = 0.05

n = Number of time periods (days, weeks, months or years)

Example 1

Finding the simple interest earned on an investment

Calculate the amount of simple interest paid on an investment of $12 000 at 10% simple interest per annum for 3 years.

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3A Simple interest on investments

Solution

Explanation

P = 12 000, r = 10% = 0.10, n = 3

Write the values for P, r and n.

I = Prn

Write the simple interest formula. Substitute the values for P, r and n.

= 12 000 × 0.10 × 3

U N SA C O M R PL R E EC PA T E G D ES

= 3600 Simple interest is $3600.

Now you try

Calculate the amount of simple interest paid on an investment of $15 000 at 9% simple interest per annum for 3 years.

Future value

The interest is added to the principal to determine the future value of an investment.

Formula for future value A= P+I

A = Future value

I = Interest (simple or flat) earned

P = Principal is the initial quantity of money borrowed, loaned or invested

Example 2

Calculating value of an investment

1 Joel plans to make an investment of $200 000 at 7 % p.a. simple interest 2 for 2 years. What is the total value of his investment at the end of 2 years?

Solution

Explanation

1 P = 200 000, r = 7 % = 0.075, n = 2 2

Write the values for P, r and n.

I = Prn

= 200 000 × 0.075 × 2

Write the simple interest formula. Substitute the values for P, r and n.

= 30 000

A = P + I = 200 000 + 30 000 = 230 000

Write the formula for future value. Substitute the values for P and I.

Total value of the investment is $230 000. Now you try Josephine plans to make an investment of $150 000 at 9% p.a. simple interest for 3 years. Calculate the total value of her investment after 3 years. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Example 3

Determine the value of an investment calculated periodically

Jackson plans to make an investment of $150 000 at 9% p.a. simple interest with the interest calculated monthly for 3 years. Calculate the total interest earned on his investment after 3 years. Explanation

P = 150 000

Write the value for P.

U N SA C O M R PL R E EC PA T E G D ES

Solution

9% 0.09 = = 0.0075 12 12

Calculate r as a monthly interest rate.

n = 3 × 12 = 36

Calculate n as the number of months in 3 years.

I = Prn

Write the simple interest formula. Substitute the values for P, r and n.

= 150 000 × 0.0075 × 36

= 40 500

Total value of the investment is $40 500. Now you try

June plans to make an investment of $17 500 at 8% p.a. simple interest with the interest calculated monthly for 4 years. Calculate the total interest earned on her investment after 4 years.

Exercise 3A

Example 1

BUILDING

Calculate the amount of simple interest for each of the following.

a Principal = $15 000, Interest rate = 13% p.a., Time period = 3 years 1 b Principal = $2000, Interest rate = 6 % p.a., Time period = 7 years 2 1 c Principal = $200 000, Interest rate = 9 % p.a., Time period = 2 years 4 1 d Principal = $3600, Interest rate = 9% p.a., Time period = 3 years 2 1 e Principal = $40 000, Interest rate = 7.25% p.a., Time period = 5 years 4

Example 2

Fiona invested $30 000 at a simple interest rate of 10% for 3 years. How much interest did she earn?

A sum of $100 000 was invested in a fixed–term account for 4 years.

a Calculate the simple interest earned if the rate of interest is 5.5% per annum. b Find the value of the investment at the end of 4 years.

Larry invested $2000 at 5% pa. What was the value of the investment after 3 years?

The simple interest rate is given as 4.8% per annum.

a What is the interest rate per quarter? b What is the interest rate per month? c What is the interest rate per six months? d What is the interest rate per nine months?

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3A Simple interest on investments

Example 3

Calculate the amount of simple interest for each of the following.

U N SA C O M R PL R E EC PA T E G D ES

a Principal = $800, Interest rate = 12% p.a., Time period = 1 month b Principal = $1600, Interest rate = 18% p.a., Time period = 6 months c Principal = $60 000, Interest rate = 9.6% p.a., Time period = 3 months d Principal = $20 000, Interest rate = 6% p.a., Time period = 9 months 7

Jodi invested $3000 at 7.5% p.a. How much interest did she earn in the first 3 months?

DEVELOPING

Create the spreadsheet shown.

a Cell D5 has a formula that calculates the simple interest. Enter this formula. b The formula for cell E5 is ‘=A5 + D5’. Fill down the contents of E6 to E12 using this formula.

Riley wants to earn $4000 a year in interest. How much must he invest if the simple interest rate is 10% p.a.?

Aisha invests $7000 at 5% for 2 years. She then reinvests the new amount for 4 years at 6%. How much is Aisha’s investment worth after 6 years?

CHALLENGING

Catherine wants to buy a car worth $22 000. How long will she need to invest $15 000 at 8.5% interest to be able to afford to buy the car? Answer correct to one decimal place.

1 Samira invests $16 000 for 2 years. What is the minimum rate of simple interest needed for her to 2 earn $3000?

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Communicate your reasoning, understanding and fluency 13

Jack invests $20 000 at 6% p.a. simple interest for 5 years. Arlo invests $20 000 at 0.5% per month simple interest for 5 years.

U N SA C O M R PL R E EC PA T E G D ES

a How much is Jack’s investment worth after 5 years? b How much is Arlo’s investment worth after 5 years? c Explain why Jack and Arlo earn the same amount of interest.

Find the value of A and B in the receipt below. FISHMARKET Receipt

Item Crab Salmon Whiting Perch *Soft drink

Value $ 12.00 $ 32.00 A $ 12.20 B

Total

85.20

GST included in total

0.50

FPO

* GST of 10% is included in the price of the item

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3B Compound interest on investments

3B Compound interest on investments Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand compound interest on investments • To use the future value formula to perform calculations involving compound interest • To solve practical problems involving compound interest Past, present and future learning

• Students may already be familiar with the contents of this section • Questions involving compound interest regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet

published by NESA. The formula for the future value of a compound interest investment appears on the 2027 version.

Compound interest is calculated on the principal at the start, and then at each time period, on the principal plus interest. It calculates interest on the interest. For example, if $100 is invested in a savings account at a compound interest rate of 10% per annum: First year:

Interest = $100 × 0.10 × 1 = $10

Amount owed = $100 + $10 = $110

Second year:

Interest = $110 × 0.10 × 1 = $11

Amount owed = $110 + $11 = $121

Third year:

Interest = $121 × 0.10 × 1 = $12.10

Amount owed = $121 + $12.10 = $133.10

These calculations show that the interest earned increased each year. In the first year it was $10, the second year $11 and the third year $12.10.

Compound interest formula A = P(1 + r)n

A = Amount (final balance) or future value of the loan

P = Principal is the initial amount of money borrowed or present value of the loan

r = Rate of interest per compounding time period expressed as a decimal n = Number of compounding time periods

Total compound interest earned or paid The total compound interest is calculated by subtracting the principal from the final balance i.e. the amount of money at the end.

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Interest earned or owed I = A−P A = Amount of money or final balance I = Interest (compound) earned or paid

U N SA C O M R PL R E EC PA T E G D ES

P = Principal is the initial amount of money

Example 4

Finding the compound interest earned

Paige invests $5000 over 5 years at a compound interest rate of 6.5% p.a. Calculate:

a the amount of the investment after 5 years, correct to the nearest cent b the interest earned after 5 years, correct to the nearest cent. Solution

Explanation

a P = 5000, r = 6.5% = 0.065, n = 5

Write the values for P, r and n.

A = P(1 + r)n

= 5000(1 + 0.065)5

Write the compound interest formula. Substitute the values for P, r and n.

= 6850.433317 . . . ≈ $6850.43

Amount of the investment after 5 years is $6850.43.

b I =A−P

= 6850.43 − 5000

Write the formula for future value. Substitute the values for A and P.

= $1850.43 Interest earned after 5 years is $1850.43.

Now you try

Peta invests $7000 over 4 years at a compound interest rate of 7.25% p.a. Calculate:

a the amount of the investment after 4 years, correct to the nearest cent b the interest earned after 4 years, correct to the nearest cent.

Future value

Recall that compound interest is calculated on the initial amount borrowed or invested plus any interest that has been charged or earned. It calculates interest on the interest. The future value formula FV = PV(1 + r)n and the compound interest formula A = P(1 + r)n are commonly used to calculate the compound interest. They are the same formula. The future value (FV) is the same as the amount (A) and the present value (PV) is the same as the principal (P).

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3B Compound interest on investments

Future value and present value formula A = P(1 + r)n

becomes

FV = PV(1 + r)n

PV =

FV = Future value of the loan or amount (final balance)

FV (1 + r)n

PV = Present value of the loan or principal (initial quantity of money)

U N SA C O M R PL R E EC PA T E G D ES

r = Rate of interest per compounding time period expressed as a decimal (or a fraction)

n = Number of compounding time periods A = Amount

Example 5

Calculating future value and present value of an investment

a Blake invests $7000 over 5 years at a compound interest rate of 4.5% p.a. Calculate the future value after 5 years. Answer correct to the nearest cent. b Calculate the present value of an investment that has a future value of $500 000 over 8 years with an interest rate of 8.5% per annum compounded monthly. Answer correct to the nearest cent. Solution

Explanation

a PV = 7000, r = 4.5% = 0.045, n = 5

Write the values for PV, r and n.

FV = PV(1 + r)n

= 7000(1 + 0.045)

Write the future value formula. Substitute the values for PV, r and n.

≈ $8723.27

The future value is $8723.27.

b FV = 500 000, r =

8.5% 0.085 = , 12 12

Write the values for PV, r and n.

n = 8 × 12 = 96 PV =

FV (1 + r)n 500 000 0.085 1+ 12

Write the formula for present value.

!96

Substitute the values for FV, r and n.

≈ $253 916.41

The present value is $253 916.41.

Now you try

a Zohyr invests $8500 over 6 years at a compound interest rate of 6.5% p.a. Calculate the future value after 6 years, correct to the nearest cent. b Calculate the present value of an investment that has a future value of $250 000 over 10 years at 5.5% p.a. compounded monthly. Answer correct to the nearest cent.

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Calculating compound interest Compound interest is calculated by subtracting the principal from the amount borrowed or invested. Alternatively, finance companies provide an investment calculator as an estimate of the value of an investment.

Compound interest earned or owed

U N SA C O M R PL R E EC PA T E G D ES

I = A − P becomes I = FV − PV

FV = Future value of the loan or amount (final balance)

PV = Present value of the loan or principal (initial quantity of money)

I = Interest (compounded) earned

Example 6

Finding the compound interest compounded quarterly

1 Tyson invests $10 000 over 10 years at a compound interest rate of 7 % p.a. compounded 2 quarterly. Calculate:

a the amount of the investment after 10 years, correct to the nearest cent b the interest earned after 10 years, correct to the nearest cent. Solution

Explanation

a PV = 10 000, r =

7.5% = 0.01875, n = 10 × 4 = 40 4

FV = PV(1 + r)n

= 10 000(1 + 0.01875)40

Write the values for PV, r and n.

Write the future value formula. Substitute the values for PV, r and n.

= 21 023.492772 . . . ≈ $21 023.49

Amount of investment after 10 years is $21 023.49.

b I = FV − PV

= 21 023.49 − 10 000

Write the formula for interest earned. Substitute the values for FV and PV.

= $11 023.49 Interest earned after 10 years is $11 023.49.

Now you try

Jordan invests $7000 over 4 years at a compound interest rate of 7.25% p.a. compounded quarterly. Calculate:

a the amount of the investment after 4 years, correct to the nearest cent b the interest earned after 4 years, correct to the nearest cent.

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3B Compound interest on investments

Exercise 3B Example 4

BUILDING

Calculate the amount owed for each of the following loans, correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

a Principal = $800, Compound interest rate = 4% p.a., Time period = 3 years 1 b Principal = $9000, Compound interest rate = 6 % p.a., Time period = 4 years 2 1 c Principal = $12 000, Compound interest rate = 11% p.a., Time period = 2 years 2 1 d Principal = $22 000, Compound interest rate = 5.5% p.a., Time period = 4 years 4

Example 5

Calculate the future value of each of the following, correct to the nearest cent.

a Present value = $400, Compound interest rate = 3% p.a., Time period = 2 years b Present value = $26 000, Compound interest rate = 8% p.a., Time period = 4 years c Present value = $48 000, Compound interest rate = 3.95% p.a., Time period = 10 years 1 d Present value = $3000, Compound interest rate = 5 % p.a., Time period = 5 years 2 1 e Present value = $18 000, Compound interest rate = 10% p.a., Time period = 2 years 2

Example 6

Use the formula A = P(1 + r)n to calculate the value of an investment of $10 000 over a period of 2 years with an interest rate of 0.8% compounding monthly. Answer correct to the nearest cent.

Use the formula FV = PV (1 + r)n to calculate the value of an investment of $16 000, over a period of 2 years with an interest rate of 5% compounding annually. Answer correct to the nearest cent.

Karim invested $13 500 over 7 years at 6.2% p.a. interest compounding quarterly. Calculate:

a the value of the investment after 7 years, correct to the nearest cent b the compound interest earned, correct to the nearest cent.

Jett sold his car for $35 600. He invested this amount at 7.2% p.a. with interest compounded annually. What is the value of his investment in 15 years? Answer correct to the nearest cent.

Bonnie wishes to invest $5000 for a period of 8 years. The following investment strategies are suggested to her. How much interest is earned on each investment strategy? Answer correct to the nearest dollar.

a Simple interest at 7% p.a. b Compound interest at 7% p.a. compounded annually. c Simple interest at 14% p.a. d Compound interest at 14% p.a. compounded annually.

Which of the following is the best investment over 10 years? Justify your answer. Investment A: Simple interest at 5% p.a. with $70 000. Investment B: Compound interest at 5% p.a. compounded annually with $70 000.

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Calculate the present value for each of the following, correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

a Future value = $34 000, Interest rate = 4% p.a., Time period = 4 years b Future value = $87 000, Interest rate = 5% p.a., Time period = 12 years c Future value = $190 000, Interest rate = 3% p.a., Time period = 15 years 1 d Future value = $200 000, Interest rate = 12 % p.a., Time period = 5 years 4 10

Calculate the future value for each of the following, correct to the nearest cent.

a Present value of $680 invested for 4 years at 5% p.a. compounded fortnightly b Present value of $5000 invested for 6 years at 6% p.a. compounded quarterly c Present value of $1400 invested for 3 years at 4.2% p.a. compounded monthly d Present value of $780 invested for 5 years at 9.8% p.a. compounded weekly

DEVELOPING

Create the spreadsheet below.

a Cell D5 has a formula that calculates the compound interest. Enter this formula. b The formula for cell E5 is ‘= D5 − A5’. Fill down the contents of E6 to E12 using this formula.

Which of the following is the best investment over 25 years? Justify your answer.

Investment A: Simple interest at 4% p.a. with $100 000 Investment B: Compound interest at 4% p.a. compounded annually with $100 000 Investment C: Compound interest at 4% p.a. compounded biannually with $100 000 Investment D: Compound interest at 4% p.a. compounded quarterly with $100 000 Investment E: Compound interest at 4% p.a. compounded monthly with $100 000

Calculate the present value, correct to the nearest dollar, for each of the following:

a future value of $1243, interest rate at 6% p.a. compounded biannually for 5 years b future value of $8200, interest rate at 4% p.a. compounded quarterly for 8 years c future value of $1580, interest rate at 4.8% p.a. compounded monthly for 4 years d future value of $19 600, interest rate at 8% p.a. compounded weekly for 3 years e future value of $3800, interest rate at 5% p.a. compounded fortnightly for 7 years.

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3B Compound interest on investments

What sum of money would Max need to invest to accumulate a total of $100 000 at the end of 7 years at 8% p.a. interest compounding biannually? Answer correct to the nearest cent.

What sum of money needs to be invested to accumulate a total of $40 000 after 10 years at 9.25% p.a. interest compounding monthly? Answer correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

CHALLENGING 16

How much more interest is earned on a $40 000 investment if the interest at 6% p.a. is compounded annually over 6 years, compared with the simple interest at 6% p.a. earned over the same time? Answer correct to the nearest cent.

How much more interest is earned on $60 000 if interest of 8% p.a. is compounded quarterly over 6 years than if simple interest of 8% is earned over the same time? Answer correct to the nearest cent.

Communicate your reasoning, understanding and fluency 18

Mikayla invests $200 000 for 10 years at 6% p.a. interest compounded quarterly. Abby also invests $200 000 for 10 years, but her interest rate is 6% p.a. compounded monthly.

a Calculate the value of Mikayla’s investment at maturity. Answer correct to the nearest cent. b Show that the compounded value of Abby’s investment is greater than the value of Mikayla’s investment. c Explain why Abby’s investment is worth more than Mikayla’s investment.

Explain why you would always choose to have an investment that has compound interest paid monthly over an investment that has compound interest paid annually.

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Graphs of investments: Simple interest and 3C compound interest Learning intentions

• To understand the difference between simple interest graphs, which are straight lines, and compound

U N SA C O M R PL R E EC PA T E G D ES

interest graphs, which are exponential • To compare the growth of investments which involve simple interest and compound interest for different rates and periods • To compare and contrast different investment strategies Past, present and future learning

• Some aspects of this section may be new to students • Questions involving graphs of investments regularly appear in HSC Examinations

Simple interest graphs

Simple interest graphs are used to model investments. When graphing simple interest, put the number of time periods (n) on the horizontal axis and the interest earned (I) on the vertical axis. Simple interest will increase by a constant amount each time period. This will always result in a straight-line graph.

Simple interest graphs

1 Construct a table of values for I and n using the simple interest formula. 2 Draw a number plane with n on the horizontal axis and I on the vertical axis. Plot the points. 3 Join the points to make a straight line.

Example 7

Constructing a simple interest graph

Draw a graph showing the amount of simple interest earned over a period of 10 years if $1000 is invested at 8% p.a. Use the graph to estimate the interest earned after 7.5 years.

Solution

Explanation

P = 1000, r = 8% = 0.08

Write the values for P and r.

I = Prn

Write the simple interest formula. Substitute the values for P and r to generate the equation I = 80n.

= 1000 × 0.08 × n

I = 80n n

160

320

480

640

800

To construct a table of values:

1 create your own values for n 2 use the equation I = 80n to fill in the table for each value of n.

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3C Graphs of investments: Simple interest and compound interest

Simple interest on $1000 at 8% p.a. I

Draw a number plane with: 1 vertical axis labelled as I 2 horizontal axis labelled as n.

800

Plot the points and use a ruler to draw a straight line.

U N SA C O M R PL R E EC PA T E G D ES

600

Use a ruler to draw a dotted vertical line at n = 7.5 to meet the straight line. From this point draw a dotted horizontal line to the I-axis.

400

Write the approximate value where the horizontal line meets the I-axis.

200

9 10

Interest after 7.5 years is approximately $600. Now you try

Draw a graph showing the amount of simple interest earned over a period of 10 years if $1500 was invested at 5% p.a. Use the graph to estimate the interest earned after 8.5 years.

Compound interest graphs

When graphing compound interest put the number of time periods (n) on the horizontal axis and the interest earned (I) on the vertical axis, like with simple interest. Compound interest will increase by a different amount each time period. This will result in an exponential curve.

Compound interest graphs

1 Construct a table of values for I and n using the compound interest formula. 2 Draw a number plane with n on the horizontal axis and I on the vertical axis. Plot the points. 3 Join the points to make an exponential curve.

Example 8

Constructing a compound interest graph

Draw a graph showing the interest earned over a period of 10 years if $1000 is invested at a compound interest rate of 8% p.a. Use the graph to estimate the interest earned after 7.5 years.

Solution

Explanation

PV = 1000, r = 0.08

Write the values for PV and r.

FV = PV(1 + r)n

Write the future value formula. Substitute the values for PV and r to generate the equation FV = 1000(1.08)n

= 1000 × (1.08)n

Continued on next page

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1000

1166

1360

1587

1851

2159

166

360

587

851

1159

U N SA C O M R PL R E EC PA T E G D ES

Compound interest earned on $1000 at 8% p.a.

To construct a table of values: 1 create your own values for n 2 use the equation FV = 1000(1.08)n to fill in the table for each value of n 3 subtract PV (1000) from each value of FV to fill in the Interest (I) column. Draw a number plane with: 1 vertical axis labelled as I 2 horizontal axis labelled as n.

1200

1000

Plot the points. Join the points to create a curved line.

800

Use a ruler to draw a dotted vertical line at n = 7.5 to meet the curved line. From this point draw a dotted horizontal line to the I-axis.

600

400

Write the approximate value where the horizontal line meets the I-axis.

200

9 10

Interest on the loan after 7.5 years is about $780. Now you try

Draw a graph showing the amount of compound interest earned over a period of 10 years if $1500 was invested at 7% p.a. Use the graph to estimate the interest earned after 8.5 years.

Example 9

Comparing the growth of simple interest and compound interest investments graphically

An investment of $10 000 earns simple interest at 5% per annum, while another investment of $10 000 earns compound interest at 5% per annum, compounded annually. Use a spreadsheet or graphing software to answer the following questions. a Create a table showing the value of each investment for each year over a 10-year period. b Use technology to graph both investments on the same set of axes. c Describe the differences in the growth patterns shown in your graph.

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3C Graphs of investments: Simple interest and compound interest

Explanation

U N SA C O M R PL R E EC PA T E G D ES

Solution a

Use formulas to calculate the amount the investment is worth at the end of each year.

Select cells A1:C12 and insert the recommended graph.

b Graph both investments on the same axes.

Highlight the vertical axis and edit it to start at $9000. Label the axis. Add a title.

c • Simple interest grows linearly at a constant rate. • Compound interest grows exponentially, increasing faster over time. • After 10 years, compound interest is $1, 288.95 higher than simple interest (using the table).

Use the cells in the spreadsheet. C12 − B12 = $1 288.95.

Now you try

Repeat Example 9 for an investment of $18 000 that earns simple interest at 4.25% per annum, and another investment of $18 000 that earns compound interest at 4.25% per annum, compounded annually.

Exercise 3C

Example 7

BUILDING

Harvey invested $2000 at 6% per annum simple interest for 5 years.

a Simplify the simple interest formula (I = Prn) by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. n

I c Draw a number plane with n as the horizontal axis and I as the vertical axis. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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d Plot the points from the table of values. Join the points to make a straight line. 1 e Use the graph to find the interest after 2 years. 2 f Extend the graph to find the interest after 6 years. g Estimate the interest earned after 6 years using the graph. Olive invested $800 at 7% per annum simple interest for 6 years.

U N SA C O M R PL R E EC PA T E G D ES

a Simplify the simple interest formula (I = Prn) by substituting values for the principal and the interest rate. b Use this formula to complete the following table of values. n

c Draw a number plane with n as the horizontal axis and I as the vertical axis. d Plot the points from the table of values. Join the points to make a straight line. 1 e Use the graph to find the interest after 2 years. 2 f Extend the graph to find the interest after 7 years. g Estimate the interest earned after 7 years using the graph.

Ellie is comparing three different interest rates for a possible investment.

a On the same number plane, draw a graph to represent the interest earned over 5 years on: i $1000 invested at 4% per annum simple interest ii $1000 invested at 6% per annum simple interest iii $1000 invested at 8% per annum simple interest.

b How much does each investment earn after 5 years? 1 c Use the graph to estimate the interest earned after 3 years. 2 d Find the time for each investment to earn $200 in interest.

Example 8

Ava invested $2000 at 6% per annum interest compounding annually for 5 years.

a Substitute the present value and the interest rate into the formula FV = PV(1 + r)n to obtain an expression for the future value. b Substitute the future value expression and the present value into the formula I = FV − PV. c Use these formulas to complete the following table of values. Answer correct to the nearest dollar. n

FV I

d Draw a number plane with n as the horizontal axis and I as the vertical axis. e Plot the points from the table of values. Join the points to make a curve. f Extend the graph to find the interest after 6 years. g Compare the interest earned after 5 years by Ava with Harvey in Question 1. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

3C Graphs of investments: Simple interest and compound interest

Raj invested $800 at 7% p.a. compound interest, paid annually, for 6 years. a Substitute the present value and the interest rate into the formula FV = PV(1 + r)n to obtain an expression for the future value. b Substitute the future value expression and the present value into the formula I = FV − PV. c Use these formulas to complete the following table of values. Answer correct to nearest dollar. 0

U N SA C O M R PL R E EC PA T E G D ES

FV I

d Draw a number plane with n as the horizontal axis and I as the vertical axis. e Plot the points from the table of values. Join the points to make a curve. 1 f Use the graph to find the interest earned after 2 years. 2 g Compare the interest earned after 6 years by Raj with Olive in Question 2.

Frankie is comparing three different interest rates for a possible investment.

a Draw on the same number plane the graph to represent the interest earned over 5 years on: i $1000 invested at 4% per annum interest compounding annually ii $1000 invested at 6% per annum interest compounding annually iii $1000 invested at 8% per annum interest compounding annually.

b What is the approximate value of the interest on each investment after 5 years? Answer correct to the nearest dollar. 1 c What is the approximate value of the interest on each investment after 3 years? Answer correct 2 to the nearest dollar. d Find the approximate time for each investment to earn $200 in interest. Answer correct to one decimal place. e Compare the interest earned after 5 years by Frankie with Ellie in Question 3 using the 8% interest rate for each.

DEVELOPING

Draw a graph showing the amount of interest paid over a period of 6 years if $1000 is invested at a 1 compound interest rate of 10% p.a. Use the graph to estimate the interest after 5 years. 2

Mick is comparing three different interest rates for a possible investment.

a Draw a graph to represent the interest earned for 6 months on: i $100 000 invested at 6% p.a. simple interest ii $100 000 invested at 9% p.a. simple interest iii $100 000 invested at 12% p.a. simple interest.

b How much does each investment earn after 2 months? c How much does each investment earn after 6 months? d Find the time for each investment to earn $3000 in interest.

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Tahnee is comparing three different interest rates for a possible investment. a Use Excel to draw these graphs on the same axes. A graph to represent the amount earned over 6 months on:

U N SA C O M R PL R E EC PA T E G D ES

i $100 000 invested at 6% p.a. interest compounding monthly ii $100 000 invested at 9% p.a. interest compounding monthly iii $100 000 invested at 12% p.a. interest compounding monthly. b What is the approximate value of each investment after 2 months, correct to the nearest dollar? c How much does each investment earn after 6 months, correct to the nearest dollar? d Find the time for each investment to earn $3000 in interest, correct to the nearest month.

CHALLENGING

The table below gives details for an investment product. The compound interest earned is paid quarterly. Investment

Rate of compound interest

4% p.a.

6% p.a.

8% p.a.

10% p.a.

Ethan is prepared to invest $50 000 in the above product.

a Draw a graph to represent the interest earned of these investments after 3 years. b What is the approximate value of investment B after 2 years? c What is the approximate value of investment C after 18 months? d Find the approximate time for investment D to earn $10 000 in interest.

Communicate your reasoning, understanding and fluency 11

Davis has the following options for an investment of $10 000.

A 10% simple interest, compounded quarterly. B 8.5% compound interest, compounded monthly.

a How much interest would each option earn after 3 years? Which option should Davis choose? b Draw one graph to represent the amount of interest earned on each investment over 5 years. c Draw a line on the graph to show how long it takes for option B to earn more interest. d What is the effect of compounding the simple interest quarterly? e Explain why option B with a lower interest rate earns more interest over time than option A.

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3C Graphs of investments: Simple interest and compound interest

An investment of $5000 earns simple interest at 6% per annum, while another investment of $5000 earns compound interest at 6% per annum, compounded annually. a Calculate the value of each investment after 5 years. b Compare the total interest earned by each investment. c Explain why the compound interest investment grows faster than the simple interest investment.

An investment of $8000 earns simple interest at 4.5% per annum, while another investment of $8000 earns compound interest at 4.5% per annum, compounded annually.

U N SA C O M R PL R E EC PA T E G D ES

Example 9

a Create a table showing the value of each investment for each year over a 10-year period. b Use technology to graph both investments on the same set of axes. c Describe the differences in the growth patterns shown in your graph.

Repeat Question 13 but this time consider the investment over a 12-year period.

Amber needs to choose between these three options when investing $25 000 for 5 years. Use Excel to model the growth of each option.

A 5% p.a. compounded annually. B 4.8% p.a. compounded monthly. C 4.75% p.a. compounded weekly.

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3D Appreciation and inflation Learning intentions

• To apply the compound interest formula to the appreciating value of assets and the inflating value of

U N SA C O M R PL R E EC PA T E G D ES

goods and services • To solve problems involving appreciation and inflation Past, present and future learning

• Some aspects of this section may be new to students • Questions involving appreciation and inflation regularly appear in HSC Examinations. • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The formula for the future value of a compound interest investment appears on the 2027 version. It may be used for appreciation and inflation.

Appreciation

Appreciation is the increase in value of items such as art, gold or land. This increase in value is often expressed as the rate of appreciation. Calculating the appreciation is similar to calculating the compound interest. For example, a painting worth $100 000 that has an annual rate of appreciation of 10% will be worth $110 000 after one year (an increase of $10 000). In the second year its value will increase by $11 000. The amount of appreciation has increased.

Appreciation

FV = PV(1 + r)n or A = P(1 + r)n FV = Future value of the item

PV = Present value of the item

r = Rate of appreciation per compounding time period expressed as a decimal

n = Number of compounding time periods

Example 10

Finding the appreciated value

Omar bought a unit for $690 000. If the unit appreciates at 9% p.a., what is its value after 7 years? Answer correct to the nearest dollar.

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3D Appreciation and inflation

Solution

Explanation

PV = 690 000, r = 0.09, n = 7; FV = PV(1 + r) = 690 000(1 + 0.09)

Write the values for PV, r and n. 7

= 1 261 346.993...

Write the formula for appreciation. Substitute the values for PV, r and n.

≈ $1 261 347

U N SA C O M R PL R E EC PA T E G D ES

Unit is valued at $1 261 347. Now you try

Manav bought a unit for $725 000. If the unit appreciates at 8% p.a., what is the value of the unit after 10 years? Answer correct to the nearest dollar.

Inflation

Inflation is a rise in the price of goods and services or Consumer Price Index (CPI). It is measured by comparing the prices of a fixed basket of goods and services. If inflation rises then a person’s spending power decreases. The inflation rate is the annual percentage change in the CPI. In Australia, the Reserve Bank aims to keep the inflation rate in a 2% to 3% band. Calculating inflation is similar to calculating appreciation or compound interest.

Inflation

Inflation rate is the annual percentage change in the CPI.

Use the formula FV = PV(1 + r)n to calculate the future value of an item following inflation.

Example 11

Finding the price of goods following inflation

What is the price of a $400 clothes dryer after three years if the inflation rate was 3.2% p.a.? Answer correct to the nearest cent.

Solution

Explanation

PV = 400, r = 0.032, n = 3

Write the values for PV, r and n

FV = PV(1 + r)n = 400(1 + 0.032)3

Write the formula for FV to calculate inflation. Substitute the values for PV, r and n.

≈ $439.64 Clothes dryer will cost $439.64. Now you try

What is the price of a $850 fridge after 4 years if the inflation rate was 2.8% p.a.? Answer correct to the nearest cent. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Exercise 3D 1

A vintage car was bought for $70 000 and appreciated at the rate of 6% p.a. What will be the value of the car after 4 years? Answer correct to the nearest cent.

The price of a house has increased by 4.5% for each of the last two years. It was bought for $490 000 two years ago. What is the new current value? Answer correct to the nearest dollar.

U N SA C O M R PL R E EC PA T E G D ES

Example 10

BUILDING

Rohan bought the following antiques.

a Tall boy valued at $4450. Each year its value appreciated by 5%. Calculate the value of the tall boy after 3 years. Answer correct to the nearest cent. b Table valued at $6200. Each year its value appreciated by 4%. Calculate the value of the table after 5 years. Answer correct to the nearest cent. c Chair valued at $1250. Each year its value appreciated by 9%. Calculate the value of the chair after 4 years. Answer correct to the nearest cent.

4 A collection of dolls was valued at $1500 four years ago. If it appreciated at 12% p.a., find its current value. Answer correct to the nearest cent.

Example 11

The price of a diamond ring has increased from $3400 to $5300 during the past five years due to inflation. What is the rise in the price of the ring?

The average inflation for the next five years is predicted to be 3% p.a.. Calculate the price of the following goods in five years’ time. Answer correct to the nearest cent. a 3 L of milk for $3.57 b Loaf of bread for $3.30 c 250 g honey for $4.50 d 800 g of eggs for $5.20

If the inflation rate is 5% p.a., what would you expect to pay, correct to the nearest dollar, in four years’ time for a house that costs: a $280 000? b $760 000? c $324 000? d $580 000? e $1 260 000? f $956 000?

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3D Appreciation and inflation

DEVELOPING 8

The graph on the right shows the value of cricket memorabilia for the past 6 years.

300 200

U N SA C O M R PL R E EC PA T E G D ES

a What was the value of the memorabilia after 5 years? b What was the initial value? c How much did the memorabilia appreciate each year? d Find the equation of the straight-line graph in terms of V and n.

Cricket memorabilia

V ($)

100

The following blocks of land have increased in value this year. What is the rate of appreciation? Answer correct to two decimal places. a $328 000 to $352 000 b $256 000 to $278 000

Jeva invested $700 on rugby memorabilia. It appreciated over 7 years at 8.2% p.a. interest compounding annually.

a What is the value of the investment after 7 years? Answer correct to the nearest cent. b Calculate the amount of appreciation.

CHALLENGING

An investment is appreciating at a rate of 8% of its value each year. Kumar decides to invest $20 000.

a What will be the investment’s value after 5 years? Answer correct to the nearest cent. b How much does the investment increase during the first 5 years? c When will the investment at least double in value? Answer to the nearest year.

The cost of a certain sculpture has increased during the past two years from $45 200 to $49 833 following inflation. What was the annual inflation rate? Answer correct to two decimal places.

Communicate your reasoning, understanding and fluency 13

Explain what happens to the price of goods if the rate of appreciation is negative.

Bianca buys a house for $800 000. She sells it after 10 years. The house appreciates at a rate of 10% p.a. for the first 6 years. After this the inflation rate dropped dramatically and became −8% p.a. How much is Bianca’s house worth, correct to the nearest thousand dollars: a 6 years after she purchased it? b 10 years after she purchased it?

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3E Shares and dividends Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To calculate the costs involved with buying and selling shares • To calculate the dividend paid on a portfolio of shares and the dividend yield • To solve problems involving shares and dividends Past, present and future learning

• The concepts addressed in this section may be new to students • Questions involving shares and dividends regularly appear in HSC Examinations

A share or stock is a part ownership in a company. Shares are bought and sold on the stock market or stock exchange, such as the ASX (Australian Securities Exchange). The ASX provides current information about share prices, market data, tools and resources for investment. When a company is first listed on the ASX the initial price is called the face value. The price of a share will change according to the performance of the company. The current price of a share is called the market price. The amount paid when a share is bought is called the cost price or issued price. The amount paid when a share is sold is called the selling price.

The share price can change each day, or even minute-by-minute. The changes are shown on screens and also on the ASX website. As well as tables of share prices over time, the website provides charts (graphs) of prices and the volume of shares traded. Shares are bought and sold using a broker. A broker receives a brokerage fee when shares are traded or a percentage value of the transaction. There are many different types of brokers, such as CommSec. Goods and Services Tax (GST) of 10% is charged for buying and selling shares.

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3E Shares and dividends

Example 12

Calculating the cost of shares

Leila bought 500 shares at a market value of $6.80 each. Brokerage fees incurred were $33 including GST. What is the total cost of purchasing the shares? Explanation

No. of shares = 500 Price = $6.80 Brokerage fee = $33

Write the values for the ‘No. of shares’, ‘price’ and ‘brokerage fee’.

Cost = (No. of shares × price) + brokerage fee Cost = (500 × 6.80) + $33

Write the formula for cost of shares. Substitute the values for the ‘No. of shares’, ‘price’ and ‘brokerage fee’.

U N SA C O M R PL R E EC PA T E G D ES

Solution

= $3433 Cost of the shares is $3433. Now you try

Wyatt purchased 800 shares at a market value of $14.20 each. The brokerage fees were $44 including GST. What was the total cost of purchasing the shares?

Dividends

The owners of a share are entitled to a proportion of the company’s profits. Profits are given to shareholders as a dividend. A dividend is a payment given as an amount per share or as a percentage of the issued price. A dividend is issued twice per year: interim dividend and final dividend. The dividend yield is the annual rate of return written as a percentage. It is calculated by dividing the annual dividend by the share’s market price. Dividend History

Type

Cents per share

Frank %

Ex Dividend Date

Dividend Pay Date

Interim

62.00

100

8/5/2018

1/7/2018

Final

74.00

100

8/11/2017

21/12/2017

Interim

62.00

100

14/5/2017

2/7/2017

Final

69.00

100

9/11/2016

15/12/2016

Interim

56.00

100

15/5/2016

3/7/2016

Dividends

A dividend is a payment given as an amount per share or a percentage of the issued price. Annual dividend Dividend yield (as a %) = × 100 Market price Dividend = Dividend yield × Market price

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Example 13

Calculating the dividend and dividend yield

The share price of a company is $28.42.

U N SA C O M R PL R E EC PA T E G D ES

a The predicted dividend yield is 3.5%. What would be the dividend, correct to the nearest cent? b The company decides to pay a dividend of $1.12. What is the dividend yield, correct to two decimal places? Solution

Explanation

a Dividend yield = 3.5% = 0.035 Market price = $28.42

Write the values for Dividend yield and Market price.

Dividend = Dividend yield × Market price = 3.5% × $28.42

= 0.035 × $28.42

Write the Dividend formula. Substitute the values for Dividend yield and Market price.

= $0.9947

Dividend is $0.99

b Annual dividend = $1.12 Market price = $28.42 Dividend yield =

Annual dividend × 100 Market price 1.12 × 100 28.42

Write the values for Annual dividend and Market price.

Write the Dividend yield formula. Substitute the values for Annual dividend and Market price.

≈ 3.94% The dividend yield is 3.94%.

Now you try

The share price of a company is $42.27.

a The predicted dividend yield is 2.9%. What would be the dividend, correct to the nearest cent? b The company decides to pay a dividend of $2.22. What is the dividend yield, correct to two decimal places?

Exercise 3E

Example 12

BUILDING

A broker charges a fee of $33 to trade shares. What is the total cost for these shares? a 150 shares, market price of $19.70 b 340 shares, market price of $2.41 c 60 shares, market price of $92.35 d 2000 shares, market price of $1.68 e 208 shares, market price of $49.61 f 3900 shares, market price of $56.23

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3E Shares and dividends

Example 13

Saanvi buys 3000 shares with a market price of $6.20. She pays a brokerage fee of $22.50. How much does she pay altogether?

Calculate the dividend yield, correct to two decimal places, on the following shares.

U N SA C O M R PL R E EC PA T E G D ES

a Market price of $33.70 and a dividend of $0.84 per share b Market price of $22.08 and a dividend of $1.63 per share c Market price of $20.58 and a dividend of $1.00 per share

d Market price of $37.72 and a dividend of $2.11 per share e Market price of $45.43 and a dividend of $2.86 per share f Market price of $4.00 and a dividend of 10 cents per share

A company with a share price of $8.40 declares a dividend of 56 cents. What is the dividend yield, correct to two decimal places?

Calculate the total dividend received on the following shares. Answer correct to two decimal places.

a 500 shares with a market price of $4.80 and a dividend yield of 5.2% b 80 shares with a market price of $88.10 and a dividend yield of 4.4% c 2200 shares with a market price of $9.56 and a dividend yield of 6.1% d 890 shares with a market price of $22.30 and a dividend yield of 1.9% e 3400 shares with a market price of $56.30 and a dividend yield of 4.6% f 780 shares with a market price of $12.58 and a dividend yield of 8.1%

A share with a market price of $1.30 has a dividend yield of 9.2%. What is the dividend per share, correct to two decimal places?

DEVELOPING

Tipeni bought 100 shares in a bank for $35.60 each. He sold them two years later for $48.90 each and paid a brokerage fee of needs to be more clear that 32.95 is subtracted twice.

a What is the profit made on these shares? b What is the profit as a percentage of the cost of the shares?

Cooper bought 2500 shares for $5.60 each.

a Cooper is charged a brokerage fee of 6.2 cents per share. What is the total cost of purchasing the shares? b Two months later a dividend of 36 cents per share was paid. What was the total dividend Cooper received? c Cooper sold the shares after receiving the dividend for $5.75 each. He was charged a brokerage fee of 6.2 cents per share. What was the profit on these shares?

The dividend yield on a company was 5%. How much is the dividend if you owned 500 shares with a market value of $4.80? Answer correct to the nearest cent.

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A company started trading on the ASX with a face value of $8.40. This graph shows the share price over time.

8.5 8.0 7.5 7.0

U N SA C O M R PL R E EC PA T E G D ES

a What was the share price at the beginning of June? b How much has the share price decreased during the 3 months? c What is the percentage decrease in the market value over the three months? Answer correct to two decimal places.

9.0

6.5

April

May

Jun

Jul

11 A spreadsheet can be used to record and graph the price of a share over time. The Excel file in the Interactive Textbook shows you how. Choose a company listed on the ASX and record its share price over a week and create a graph of the price. You will need to choose a company that is actively traded, with frequent changes of price.

You have $30 000 to invest for 5 years. Compare these options:

Option 1 Savings account (compound interest): 4% per annum, compounded annually. Option 2 Term deposit (simple interest): 4.5% per annum.

Option 3 Term deposit (compound interest): 3.8% per annum, compounded annually.

Option 4 Shares: Assume an average annual growth of 6.5% (compounded). Brokerage fees were $252 including GST. Option 5 Property: Assume an average annual growth of 4.8% (compounded).

Use the graph in the spreadsheet provided to answer the following questions.

a Which investment option provides the highest return after 5 years? b After 3 years, which investment has grown the most, and by how much compared to the lowest-performing option? c Why was the Option 4 investment valued at −$252.00 after 0 years? d Under what conditions would Option 2 have higher gains than Option 3 over the five years? e Research the positives and negatives of these investment types and complete a table to summarise your results.

CHALLENGING

A company has an after tax profit of $128 million. There are 200 million shares in the company. What dividend per share will the company declare if all the profits are distributed to shareholders?

Nyrie owns 400 $3 ordinary shares and 300 $2 preference shares. The current prices of the ordinary shares and preference shares are $4.20 and $3.60 respectively. The dividend on the ordinary shares is 55c and on the preference shares is 3%. Calculate Nyrie’s total dividend.

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3E Shares and dividends

Communicate your reasoning, understanding and fluency 15 a Explain what a dividend is in your own words. How is it calculated? b Explain what a dividend yield is in your own words. How is it calculated? c Research the ASX200. Find out the following:

U N SA C O M R PL R E EC PA T E G D ES

i the meaning of S&P/ASX200 ii the percentage increase or decrease in the value of the ASX200 in the past 5 days iii the percentage increase or decrease in the value of the ASX200 in the past month iv the percentage increase or decrease in the value of the ASX200 in the past 6 months v the percentage increase or decrease in the value of the ASX200 in the past 5 years.

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3F Straight-line depreciation Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To calculate the depreciation of an asset using the straight-line method • To use the straight-line depreciation formula • To solve problems involving straight-line depreciation Past, present and future learning

• Some aspects of this section may be new to students • Questions involving depreciation regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The formula for straight-line depreciation appears on the 2027 version.

Straight-line depreciation occurs when the value of the item decreases by the same amount each period. For example, if you buy a car for $20 000 and it depreciates by $2000 each year, the value of the car after one year is $20 000 − $2000, or $18 000. After the second year the value of the car is $20 000 − $2000 − $2000, or $16 000.

Straight-line depreciation S = V0 − Dn

S = Salvage value or current value of an item; also referred to as the book value

V0 = Purchase price of the item; value of the item when n = 0 D = Depreciated amount per time period

n = Number of time periods

Example 14

Calculating the salvage value

Yasmine pays $14 500 for a used car. It depreciates $1100 each year. How much will it be worth after three years?

Solution

Explanation

V0 = 14 500, D = 1100, n = 3

Write the values for V0 , D and n.

S = V0 − Dn

Write the salvage value formula. Substitute the values for V0 , D and n.

= 14 500 − 1100 × 3

= $11 200 The value of the car is $11 200. Now you try

Bakari pays $15 000 for a used car. It depreciates at $1250 each year. How much will the car be worth after 4 years?

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3F Straight-line depreciation

Example 15

Calculating the straight-line depreciation

A new car is purchased for $25 800. After 4 years its salvage value is $15 160. What is the annual amount of depreciation, if the amount of depreciation is constant? Explanation

S = 15 160, V0 = 25 800, n = 4

Write the values for S, V0 and n.

U N SA C O M R PL R E EC PA T E G D ES

Solution

S = V0 − Dn

Write the salvage value formula. Substitute the values for S, V0 and n.

15 160 = 25 800 − D × 4

D × 4 = 25 800 − 15 160

25 800 − 15 160 4 = $2660 Annual depreciation is $2660. D=

Now you try

Sally purchases a new car for $75 000. After 3 years its salvage value is $54 000. What is the annual amount of depreciation, if the amount of depreciation is a constant?

Exercise 3F 1

BUILDING

Farrah bought a used car for $15 200. She estimates that her car will depreciate in value by $3040 each year. a What is the loss in value (depreciation) during the first year? b What is the value of the car at the end of the first year? c What is the loss in value (depreciation) during the second year? d What is the value of the car at the end of the second year? e What is the loss in value (depreciation) during the third year? f What is the value of the car at the end of the third year?

Example 14

Harrison pays $9500 for a motor bike. It depreciates $850 each year. What will be the value of the bike after: a three years?

b five years?

c seven years?

d nine years?

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Faiz buys a car for $55 500 and it is depreciated at a rate of 10% of its purchase price each year. What is the salvage value of the car after four years?

The graph shows the depreciation of a car over four years.

12 000

Value ($)

U N SA C O M R PL R E EC PA T E G D ES

a What is the initial value? b How much did the car depreciate each year? c What is the value of the car after 3 years? d When was the car worth $8000? 1 e What is the value of the car after 3 years? 2 f What is the value of the car after 6 months?

16 000

8000

4000

2 Years

DEVELOPING

Ryan bought a commercial van three years ago. It has a salvage value of $36 000 and depreciated $4650 each year. How much did Ryan pay for the van?

Jamie bought a used car four years ago. It has a salvage value of $16 400 and depreciated $1250 each year. How much did Jamie pay for the used car?

Matteo has a car worth $9220. It depreciates by $420 each year.

a When will the car be worth $5440? b When will the car be worth $3340?

A ute is purchased for $18 600. After two years it has depreciated to $14 800 using the straight line method of depreciation.

a When will the ute be worth $9600? b When will the ute be worth $1500?

Example 15

A truck is purchased new for $64 000. After 3 years its market value is $44 800.

a What is the annual amount of depreciation, if the amount of depreciation is constant? b Determine the value of the truck after 7 years.

Grace bought an SUV costing $38 000. It is expected that the SUV will have an effective life of 10 years and then be sold for $14 000. Assume the SUV depreciated by the same amount each year. What is the annual depreciation?

11 A utility van is purchased new for $24 000. After 3 years its book value is $15 000. What is the annual amount of depreciation, if the amount of depreciation is constant?

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3F Straight-line depreciation

CHALLENGING 12

A caravan is bought for $82 000. It is expected to be used for 4 years and then sold for $50 000. Assume the caravan depreciates by the same amount each year.

U N SA C O M R PL R E EC PA T E G D ES

a How much does the caravan depreciate each year? b What is the total amount of depreciation for 4 years? c Copy and complete the following depreciation table for the first four years. Year

Current value

Depreciation

Depreciated value

1 2 3 4

d Graph the value in dollars against the age in years.

Communicate your reasoning, understanding and fluency 13

Calvin purchased a computer for $5000. He sold it for $2500 five years later.

a Calculate the annual amount of depreciation, assuming the amount of depreciation is a constant. b Use Excel to create a table that shows how many years it will take for the computer to have no value. c Use Excel to a graph to represent the depreciation over 10 years.

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3G Declining-balance depreciation Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the difference between straight-line depreciation and declining balance depreciation • To calculate the depreciation of an asset using the declining balance method • To use the declining balance depreciation formula • To solve problems involving declining balance depreciation Past, present and future learning

• Some aspects of this section may be new to students • Questions involving depreciation regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The formula for declining balance depreciation appears on the 2027 version.

Declining-balance depreciation occurs when the value of the item decreases by a fixed percentage each time period. For example, if you buy a car for $20 000 and it depreciates by 10% each year then the value of the car after one year is $20 000 − $2000, or $18 000. After the second year the value of the car is $18 000 − $1800, or $16 200. Notice that the amount of depreciation has decreased in the second year. Depreciation calculations have similarities with compound interest, except that the depreciation is subtracted from the value not added to it.

Declining-balance depreciation S = V0 (1 − r)n

S = Salvage value or current value of an item. Also referred to as the book value.

V0 = Purchase price of the item. Value of the item when n = 0.

r = Rate of depreciation per time period expressed as a decimal. n = Number of time periods.

Example 16

Calculating the current value using declining-balance depreciation

Arya purchased a new car five years ago for $32 000. During the first 2 years it had depreciated by 25% and during the next 3 years it had depreciated 20% p.a. What is the current value of the car?

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3G Declining-balance depreciation

Solution

Explanation

V0 = $32 000, r = 0.25, n = 2

Write the values for V0 , r and n.

S = V0 (1 − r)n

Write the declining-balance depreciation formula.

S = 32 000(1 − 0.25)2

Substitute the values for V0 , r and n.

U N SA C O M R PL R E EC PA T E G D ES

= $18 000 V0 = $18 000, r = 0.20, n = 3

Write the values for V0 , r and n.

S = V0 (1 − r)

Write the declining-balance depreciation formula.

S = 18 000(1 − 0.20)3

Substitute the values for V0 , r and n.

= $9216 The current value of the car is $9216. Now you try

The purchase value of a new car six years ago was $72 000. During the first 2 years it depreciated at 22%. During the next 4 years it depreciated at 18%. What is the current value of the car correct to the nearest dollar?

Example 17

Calculating the purchase price

Bodhi buys a car that depreciates at the rate of 26% per annum. After five years the car has a salvage value of $17 420. How much did Bodhi pay for the car, correct to the nearest dollar?

Solution

Explanation

S = 17 420, r = 0.26, n = 5

Write the values for S, r and n.

S = V0 (1 − r)n

Write the declining-balance depreciation formula.

17 420 = V0 × (1 − 0.26)5 V0 =

Substitute the values for S, r and n.

Make V0 the subject of the equation.

17 420

(1 − 0.26)

= $78 503.596 21...

≈ $78 504 Bodhi paid $78 504 for the car.

Now you try

Greg buys a car that depreciates at the rate of 22% p.a. After four years the value of the car is $20 200. How much did Greg pay for his car, correct to the nearest dollar?

Example 18

Calculating the percentage rate of depreciation

Madison bought a delivery van four years ago for $27 500. Using the declining-balance method for depreciation, she estimates its present value to be $8107. What annual percentage rate of depreciation did she use? Answer to the nearest whole number.

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Solution

Explanation

S = 8107, V0 = 27 500, n = 4 S = V0 (1 − r)

Write the values for S, V0 and n. Write the declining-balance depreciation formula.

8107 = 27 500 × (1 − r)

Substitute the values for S, V0 and n.

8107 27 500 r 8107 4 1−r = 27 500 r 8107 4 r =1− 27 500 = 0.263 145 28...

Make (1 − r)4 the subject of the equation.

U N SA C O M R PL R E EC PA T E G D ES

(1 − r)4 =

Take the fourth root of both sides. Make r the subject and solve.

≈ 26% Rate of depreciation is 26%.

Now you try

Morgan bought a car five years ago for $32 300. Using the declining-balance method for depreciation, she estimates its present value to be $9200. What annual percentage rate of depreciation did she use? Answer correct to the nearest whole number.

Exercise 3G

Example 16

BUILDING

A motor vehicle is bought for $22 000. It depreciates at 16% per annum and is expected to be used for 5 years. What is the salvage value of the motor vehicle after the following time periods? Answer correct to the nearest cent.

a One year b Two years c Three years

2 Quinn purchased a used car for $6560 two years ago. Use the declining-balance method to determine the salvage value of the used car if the depreciation rate is 15% per annum. Answer correct to the nearest dollar.

Kennedy purchased a motor cycle for $17 500. It depreciates at 28% per year. Answer correct to the nearest dollar.

a What is the book value of the motor cycle after 3 years? b How much has the motor cycle depreciated over the 3 years?

A new car is bought for $52 000. It depreciates at 22% per annum and is expected to be used for 4 years. How much has the car depreciated over the 4 years? Answer correct to the nearest dollar.

Chloe purchased a car for $19 900. It depreciates at 24% per year. Answer correct to the nearest dollar. a What is the salvage value of the car after 5 years? b How much has the car depreciated over the 5 years?

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3G Declining-balance depreciation

The depreciation of a used car over four years is shown in the graph below. 4000

U N SA C O M R PL R E EC PA T E G D ES

Value ($)

3000

2000

1000

2 Years

a What is the initial value of the used car? b How much did the used car depreciate during the first year? c When is the value of the used car $2000? d When is the value of the used car $1500? e What is the value of the used car after 4 years? 1 f What is the value of the used car after 1 years? 2

DEVELOPING

Example 17

Ali buys a car that depreciates at the rate of 20% per annum. After six years the car has a salvage value of $6400. How much did Ali pay for the car, correct to the nearest dollar?

Example 18

A hatchback vehicle was purchased for $16 980 three years ago. By using the declining-balance method of depreciation, the current value of the vehicle is $9614. What is the annual percentage rate of depreciation, correct to two decimal places?

A new car is valued at $35 000. After one year using the decliningbalance method, it is valued at $25 500.

a Determine the annual percentage rate of depreciation. Answer correct to three decimal places. b What is the value of the new car after three years? Answer correct to the nearest dollar.

Philip and Amy spent $200 000 on a luxury car 7 years ago. Its current value is $104 350. Using the declining-balance method, find the percentage depreciation rate over this period. Answer correct to one decimal place.

Jessica invested $18 820 to buy a new car for her business. How many years would it take for this car to depreciate to $4520? Assume declining-balance method of depreciation with a rate of depreciation of 30% per annum. Answer correct to the nearest year.

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CHALLENGING 12

A motor vehicle is bought for $32 000. It depreciates at 16% per annum and is expected to be used for 5 years.

U N SA C O M R PL R E EC PA T E G D ES

a How much does the motor vehicle depreciate in the first year? b Copy and complete the following depreciation table for the first five years. Answer correct to the nearest dollar. Year

Current value

Depreciation

Depreciated value

1 2 3 4 5

c Graph the value in dollars against the age in years.

Communicate your reasoning, understanding and fluency 13

Laurina purchased a car for $66 000. The car depreciated over time. Laurina wanted to compare the straight-line depreciation to declining balance depreciation.

a Using any spreadsheet program, create and complete the spreadsheet below.

b What will the formula need to be in cell D8? c What will the formula need to be in cell E8? d What will the formula need to be in cell G8? e What will the formula need to be in cell H8? f Using the completed spreadsheet, find the value of Laurina’s car after: i 4 years using declining-balance depreciation ii 4 years using straight-line depreciation.

g Draw one graph to compare the straight-line depreciation to declining balance depreciation. h Explain the factors that Laurina will need to consider when choosing which method of depreciation to use. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

3H Costs of borrowing money

3H Costs of borrowing money Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the various costs involved with borrowing money • To calculate the cost of different loans • To solve problems involving various forms of credit and loans Past, present and future learning

• Some aspects of this section may be new to students • Questions involving loans regularly appear in HSC Examinations

Buy now pay later (BNPL)

In Section 8C of the Year 11 textbook, we looked at how BNPL schemes work. In this section we will look more closely at the costs associated with BNPL schemes. BNPL schemes usually require the first monthly payment upfront – some services like Afterpay don’t require an initial payment. We will analyse the advantages and risks. Below is a table outlining key advantages and risks of BNPL. Advantages of BNPL

Risks of BNPL

Convenience and flexibility

Credit laws do not apply

Interest-free options

Difficulty tracking spending

Faster approval and lower requirements

Unaffordable debt

Budget management

Fees add up quickly

Access to higher-value Items

Credit rating can be affected

Example 19

Calculating the costs of borrowing using buy now pay later

Hugh made the following purchases using a BNPL scheme:

January 15th : Computer $700, four monthly payments of $175 February 15th : Clothes $200, four monthly payments of $50

March 15th : Car service $500, four monthly payments of $125

Each BNPL agreement charges a $25 late payment fee for any instalment that is not paid on time.

a How much did Hugh have to pay in: i February?

ii March?

iii April?

iv May?

b In April, Hugh was only able to pay $300 on time and paid the remaining April instalments one week late. In May, he delayed his entire May payment schedule by one week. How much did Hugh pay in late fees?

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Solution i Computer $175 + Clothes $50 = $200 ii Computer $175 + Clothes $50 + Car service $125 = $350 iii Computer $175 + Clothes $50 + Car service $125 = $350 iv Clothes $50 + Car service $125 = $175

In February only the computer and clothes BNPL are active. In March all three BNPL schemes are active.

U N SA C O M R PL R E EC PA T E G D ES

Explanation

b April = $25

In April all three BNPL schemes are active.

In May only the clothes and car service BNPL are active. Hugh is able to pay on time for the computer and the car service but not the clothes.

May = $50

Hugh has paid the Computer off so has two schemes he is late paying for. $25 + $25 = $50

Total penalty = $25 + $50

Total of three penalty payments.

= $75 Hugh pays a total of $75 in penalties.

Now you try

Jayden made the following purchases using a BNPL scheme: January 20th : Phone $800, four monthly payments of $200

February 20th : Clothes $150, four monthly payments of $37.50

March 20th : Birthday present $200, four monthly payments of $50

Each BNPL agreement charges a $25 late payment fee for any instalment that is not paid on time.

a How much did Jayden have to pay in: i February?

ii March?

iii April?

iv May?

b In April, Jayden was only able to pay $250 on time and paid the remaining April instalments one week late. In May, he delayed his entire May payment schedule by one day. How much did Jayden pay in late fees?

Short-term loans

A short-term loan can be taken out with a bank or with other short-term lenders. Recently, more people have been using non-bank lenders for short-term loans. A list of advantages and risks of short-term financing using non-bank lenders can be seen below. Advantages of short-term loans

Risks of short-term loans

Fast application process

High fees and interest rates

Almost immediate access to funds

The amount you borrow is limited

Easy repayment process

Borrowing can become a cycle

More flexibility

Debt always has some risk

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3H Costs of borrowing money

Example 20

Calculating the costs of a short-term loan

Leanne borrowed $1500 using a short-term finance company. There were no set up fees. The terms were no deposit and 5 monthly instalments of $300. If she does not pay on time, she will incur a 3% penalty on the outstanding balance and a $2 processing fee.

U N SA C O M R PL R E EC PA T E G D ES

a Leanne pays her third payment two days late. How much are her penalties? b Leanne pays her fourth payment five days late. How much are her penalties? c What is the total of all the penalties paid by Leanne? Solution

Explanation

a Outstanding balance = $900 Penalties = 900 × 0.03 + 2 = $29

Two payments made so outstanding balance is 1500 − 300 − 300 = $900.

b Outstanding balance = $600 Penalties = 600 × 0.03 + 2 = $20

Three payments made so outstanding balance is 1500 − 300 − 300 − 300 = $600.

c Total penalties = 29 + 20 = $49 Now you try

Elizabeth borrowed $1600 using a short-term finance company. There were no set up fees. The terms were no deposit and 3 monthly instalments of $400. If she does not pay on time, she will incur a 5% penalty on the outstanding balance and a $2.50 processing fee.

a Elizabeth pays her second payment two days late. How much are her penalties? b Elizabeth pays her third payment five days late. How much are her penalties? c What is the total of all the penalties paid by Elizabeth?

Long-term loans

Long-term loans enable the borrowing of larger amounts of money. There are many fees that banks charge to set up long-term loans. Long-term loans are used most frequently for buying a house or a car. Below is a list of common fees associated with a long-term loan. • Establishment fees cover the lender’s costs of processing the loan application. They can be a fixed • •

• •

amount or a percentage of the loan. Origination fees cover the cost of setting up the loan. They are a percentage of the loan. Valuation fees cover the cost charged to assess the property’s value. They are a fixed amount. Conveyancing fees are paid to a conveyancer or solicitor to transfer legal title of the property. They are a fixed amount. Mortgage registration fees are paid to the government to register the mortgage. They can be a fixed amount or a percentage of the loan. Lenders mortgage insurance must be paid if the deposit is less than 20% of the property value. This is a percentage of the loan. Stamp duty is paid to the government and is based on value and location of property.

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Example 21

Calculating the costs of a long-term loan

Angel plans to take out a loan for $600 000 for a house valued at $700 000. He has to pay the following fees which will need to be added to the loan: • Establishment fees: 0.1% of the loan • Origination fees: 0.05% of the loan

U N SA C O M R PL R E EC PA T E G D ES

• Valuation fees: $250

• Conveyancing fees: $350

• Mortgage registration fees: $171.70 • Stamp duty: $343

Calculate the actual amount the loan will need to be to cover the fees.

Solution

Explanation

Fees = 600 000 × 0.001 + 600 000 × 0.0005

Add all the costs for each fee.

+ 250 + 350 + 171.70 + 343

= $2014.70

Loan = 600 000 + 2014.70

Add fees to original loan amount.

= $602 014.70

Now you try

Burt plans to take out a loan for $700 000 for a house valued at $800 000. He has to pay the following fees which will need to be added to the loan: • Establishment fees: 0.1% of the loan • Origination fees: 0.05% of the loan • Valuation fees: $275

• Conveyancing fees: $375

• Mortgage registration fees: $171.70 • Stamp duty: $348.

Calculate the actual amount the loan will need to be to cover the fees, correct to the nearest dollar.

Fees and charges for a loan

Banks and financial institutions charge their customers for borrowing money. A loan account is created and an account service fee is charged per month. In addition to this fee there are a number of other loan fees and charges, depending on the financial institution. Many of these fees are negotiable and customers are advised to compare the fees and charges with the interest rate charged. Fees and charges for a loan may include: • loan application fee – costs in setting up the loan. • loan establishment fee – initial costs in processing the loan application. • account service fee – ongoing account-keeping fee. • valuation fee – assessment of the market value of a property. • legal fee – legal processing of a property.

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3H Costs of borrowing money

Exercise 3H 1

Lucinda purchased a washing machine for $1500 using a BNPL scheme. She paid a 10% deposit and then four equal monthly repayments. How much was each repayment?

Holly made the following purchases using a BNPL company. All arrangements had no deposit.

U N SA C O M R PL R E EC PA T E G D ES

Example 19

BUILDING

Item

Jul

Aug

Sep

Bed linen

$50

$175

Jewellery Presents

Oct

Nov

$175

$60

a How much did Holly spend in: i September? ii October? iii November?

3 Sajaad borrowed $1320 using a short-term finance company. A $30 set up fee was added. The terms were no deposit and 5 monthly instalments. He paid 4 instalments of $275. How much was left to be paid in the final instalment?

Example 20

Example 21

Faith borrowed $1740 using a short-term finance company. There were no set up fees. The terms were no deposit and 6 monthly instalments of $290. If she does not pay on time, she will incur a 5% penalty on the outstanding balance and a $2.50 processing fee. Faith forgets to make her second payment. What are her total fees?

Brandon wants to purchase a house for $900 000. He has a $200 000 deposit. Lenders mortgage insurance is payable if Brandon has less than a 20% deposit. Show, using working out, and explain why Brandon does not need to pay mortgage insurance.

Walter plans to take out a loan for $500 000 for a house valued at $610 000. The fee schedule is set out in the table below. Fee

Cost

Establishment

0.15% of the loan amount

Origination

0.075% of the loan amount

Conveyancing

$500

Mortgage registration

$171.70

Stamp duty

$418

Calculate the actual amount the loan will need to be to cover the fees.

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DEVELOPING 7

Roger purchased a lounge for $2500 using a BNPL scheme. He paid a 25% deposit and 5 equal monthly repayments on his credit card.

U N SA C O M R PL R E EC PA T E G D ES

a How much was each repayment? b The BNPL company charged 1% on all credit card payments. How much was added to each repayment for using a credit card?

Xavier borrowed $1000 using a short-term finance company. The set-up fees were 5% which were added to the amount borrowed. The terms were 10% deposit and 5 equal monthly instalments. If he does not pay on time, he will incur a 3.5% penalty on the outstanding balance and a $2 processing fee.

a How much is each monthly payment? b Xavier pays his third payment four days late. What are his penalties? c How much will his fifth payment need to be to cover the penalties?

Bree wants to purchase a house for $759 000. She has a $150 000 deposit. Lenders mortgage insurance is payable if Bree has less than a 20% deposit.

a Does Bree need to pay mortgage insurance? b Mortgage insurance is calculated using the formula: size of loan Mortgage Insurance = × 1000 value of property How much mortgage insurance does Bree need to pay?

CHALLENGING

Niaz wants to buy a unit valued at $725 000. He does not want to pay mortgage insurance which is payable if he has less than a 20% deposit. He has $105 000 saved. He can save $5000 a month.

a How long will it be until he can afford to buy the unit without paying mortgage insurance? b Using the formula in Question 9, calculate the mortgage insurance Niaz would pay right now. c Make a recommendation to Niaz as to whether he should buy the unit now or wait. (Remember the unit may sell if he waits too long.)

Communicate your reasoning, understanding and fluency 11

Kody wants to borrow $1000. Sometimes he earns $50 a week and sometimes he earns $100 a week.

a Explain to Kody the advantages and risks involved in using a BNPL scheme. b Explain to Kody the advantages and risks involved in using a short-term loan.

Modelling activity: Comparing financing options

This activity explores four different financing options—credit cards, personal loans, Buy Now Pay Later (BNPL), and revolving credit—by using a spreadsheet to analyse costs, interest, fees, and the impact of late payments across several real-world purchasing scenarios (purchasing a laptop, bike, car, and a holiday). The activity can be downloaded from your Interactive Textbook. It is recommended that this activity be done after you have completed sections 3H-3L. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

3I Reducing-balance loans

3I Reducing-balance loans Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand how reducing balance loans work • To perform calculations involving fees and charges associated with loans • To solve practical problems involving reducing balance loans Past, present and future learning

• Some aspects of this section may be new to students • Questions involving loans regularly appear in HSC Examinations

Reducing-balance loans are calculated on the balance owing. As payments are made, the balance owing is reduced and therefore the interest charged is reduced. This can save thousands of dollars on the cost of a loan. The calculations for reducing-balance loans are complicated and financial institutions publish tables related to loans.

Loan repayments

Total to be paid = Monthly repayment × Number of repayments Total to be paid = Principal + Interest

Example 22

Using a table for a reducing-balance loan to find interest paid

The table below shows the monthly repayments for a reducing-balance loan. Calculate the amount of interest to be paid on a loan of $200 000 over 13 years. Amount of the loan

Term

$100 000

$150 000

$200 000

$250 000

12 years

$1664

$2096

$2794

$3493

13 years

$1700

$2150

$2856

$3569

14 years

$1726

$2218

$2898

$3622

Solution

Explanation

Total to be paid

Loan is $200 000 and time period is 13 years. Find the intersection value from the table $2856 .

= Monthly repayment × Number of repayments = 2856 × 13 × 12 = $445 536

Multiply the intersection value by the number of years and 12 (number of months in a year) to determine the total to be paid. Continued on next page

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Substitute the total to be paid ($445 536) and principal ($200 000) into the formula.

Total to be paid for the loan is $445 536. Total to be paid = Principal + Interest 445 536 = 200 000 + I

U N SA C O M R PL R E EC PA T E G D ES

I = $245 536 Interest paid is $245 536. Now you try

Use the table from Example 22 to calculate the amount of interest paid on a loan of $250 000 over 14 years.

Example 23

Using a table for a reducing-balance loan to find amount saved

The table shows the monthly payments for each $1000 borrowed. Zainab is planning to borrow $280 000 to buy a house at 8% per annum over a period of 20 years. Period of loan

Interest rate

10 years

15 years

20 years

6% p.a.

$11.10

$8.44

$7.10

7% p.a.

$11.61

$9.00

$7.75

8% p.a.

$12.13

$9.56

$8.36

a What is Zainab’s monthly payment on this loan? b How much would Zainab pay in total to repay this loan? c How much would Zainab save if she repaid the loan over 15 years? Solution

Explanation

a $8.36

Find the intersection value from the table for interest rate 8% p.a. and time period 20 years.

Monthly repayment = $8.36 × 280 = $2340.80

b Total to be paid

= Monthly repayment × Number of repayments

= 2340.80 × 20 × 12

Multiply the intersection value by the number of thousands borrowed (280).

Multiply the monthly repayment by the number of years and 12 (number of months in a year) to determine the total to be paid.

= $561 792 Total to be paid for the loan is $561 792.

c 15 years Monthly repayment = $9.56 × 280 = $2676.80

Repeat the above calculations using 15 years instead of 20 years.

Total to be paid

= Loan repayment × Number of repayments = 2676.80 × 15 × 12 = $481 824 Amount saved = $561 792 − $481 824

Subtract the total to be paid for 15 years from the total to be paid for 20 years.

= $79 968 The amount saved 968. University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400 Uncorrected 3rd sample pagesis• $79 Cambridge

3I Reducing-balance loans

Now you try Logan is planning to borrow $460 000 to buy a house at 7% p.a. over 15 years. Using the table in Example 23:

U N SA C O M R PL R E EC PA T E G D ES

a What is Logan’s monthly payment for this loan? b How much would Logan pay in total for this loan? c How much would Logan save if he repaid the loan over 10 years?

Graph of a reducing-balance loan

The graph below shows the amount owed after each month on a reducing-balance loan. The amount borrowed is $50 000 at an interest rate of 10% p.a. It illustrates the difference between making repayments of $500 per month and making repayments of $1000 per month. When paying $500 a month, it takes 215 months to pay off the loan, and the interest charged is $57 500. However, when paying $1000 a month, it only takes 65 months to pay off the loan, and the interest charged is $15 000. Each graph is a gradual curve as each payment reduces the amount owed and slowly decreases the interest charged. Reducing-balance loan

$60 000

$50 000

Amount owed

$40 000

$500 per month

$30 000

$20 000

$1000 per month

$10 000

100 120 Month

140

160

180

200

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Exercise 3I Example 22

BUILDING

Tyler is considering an investment loan from the bank at a reducible interest rate of 9.9% p.a. (i.e a reducing-balance loan). The table below shows the monthly repayment for an investment loan.

U N SA C O M R PL R E EC PA T E G D ES

Investment loan Term

$50 000

$75 000

$100 000

$125 000

$150 000

5 years

$1060

$1590

$2120

$2650

$3180

10 years

$658

$987

$1316

$1645

$1974

15 years

$534

$801

$1068

$1336

$1603

a What is the monthly repayment for a loan of $75 000 over 15 years? b What is the monthly repayment for a loan of $150 000 over 10 years? c What is the monthly repayment for a loan of $100 000 over 5 years? d What is the monthly repayment for a loan of $50 000 over 15 years? e What is the monthly repayment for a loan of $125 000 over 5 years?

Kevin is applying for an investment loan from a bank of $75 000 over 5 years using the table in Question 1.

a What is the monthly repayment? b What is the total amount paid for this loan? c What is the interest paid on this loan?

The table below shows the monthly repayments per $1000 on a bank loan. Term

7.00%

7.25%

7.50%

7.75%

10 years

$16.39

$16.78

$17.18

$17.58

15 years

$15.33

$15.87

$16.44

$17.02

Calculate the monthly repayment on the following loans. a $310 000 at 7.50% p.a. for 15 years b $120 000 at 7.00% p.a. for 10 years c $450 000 at 7.75% p.a. for 10 years d $180 000 at 7.25% p.a. for 15 years

Binh is borrowing $35 000 at 7% p.a. for 10 years. Use the table in Question 3 to answer these questions. a What is the monthly repayment? b How much interest will he pay?

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3I Reducing-balance loans

Example 23

This table shows the monthly payments for a loan of $1000 for varying interest rates. Jack is planning to borrow $340 000 to buy a house at 10% p.a. over a period of 15 years. Period of loan 10 years

15 years

20 years

7% p.a.

$11.61

$9.00

$7.75

U N SA C O M R PL R E EC PA T E G D ES

Interest rate

8% p.a.

$12.13

$9.56

$8.36

9% p.a.

$12.67

$10.14

$9.00

10% p.a.

$13.22

$10.75

$9.65

a Calculate Jack’s monthly payment on this loan. b How much does Jack pay in total to repay this loan? c How much interest does Jack pay on this loan? d How much would Jack save if he repaid the loan over 10 years?

Hannah and Mitchell borrow $180 000 over 20 years at a reducible interest rate of 8.5% p.a. They pay $1754 per month.

a Calculate the total amount to be paid on this loan. b How much interest do they pay on the loan?

The graph opposite shows the amount owed each month on a reducing-balance loan. Use the graph to estimate the answers to these questions.

a How much was borrowed? b How much is owed after 20 months? c How much is owed after 40 months? d How much is owed after 60 months? e When is the amount owing $20 000? f When is the amount owing $60 000? g When is the loan paid?

Home loan

$120 000 $100 000 $80 000

Amount

$60 000

$40 000 $20 000

60 Month

100

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DEVELOPING 8

Mr and Mrs Smith intend to borrow $400 000 at 8% p.a. reducible interest for a term of 20 years. Use the table in Question 5 for the following calculations.

U N SA C O M R PL R E EC PA T E G D ES

a What is their monthly repayment? b Find the total amount to be repaid. c What is the interest paid for this loan? d How much would they save if they repaid the loan over 15 years? e How much would they save if they repaid the loan over 10 years? f What is the equivalent flat interest rate charged per annum on this loan? Answer correct to two decimal places.

CHALLENGING

Alinta borrows $160 000 over 8 years at a reducible interest rate of 21.2% p.a. She pays $1600 per fortnight.

a How much will Alinta pay back altogether? b What is the interest paid for this loan? c What is the equivalent flat interest rate charged per annum on this loan? Answer correct to one decimal place.

The formula below converts a flat rate of interest to the approximate equivalent reducible interest rate. 2NR N+1 E = Reducible rate of interest per annum as a decimal

R = Flat rate of interest per annum expressed as a decimal N = Number of repayments in the loan

Use the formula to calculate the approximate reducible interest rate for these loans. Answer correct to two decimal places.

a Loan of $20 000 over 3 years at 12% p.a. flat interest with monthly repayments b Loan of $50 000 over 5 years at 8.5% p.a. flat interest with monthly repayments c Loan of $58 000 over 4 years at 9% p.a. flat interest with quarterly repayments d Loan of $7000 over 12 months at 15% p.a. flat interest with biannual repayments e Loan of $62 000 over 6 years at 11% p.a. flat interest with quarterly repayments f Loan of $25 000 over 2 years at 19% p.a. flat interest with biannual repayments g Loan of $70 000 over 24 months at 15% p.a. flat interest with monthly repayments

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3I Reducing-balance loans

Communicate your reasoning, understanding and fluency 11

The table below simulates the first 6 payments on a $440 000 loan at 7% p.a. with interest compounded monthly. A

U N SA C O M R PL R E EC PA T E G D ES

1 2

3 4 5 6 7 8 9 10 11 12 13 14 15 16

Month

Balance at start of month

1 2 3 4 5 6

$4 40 000 $4 37 456.67 $4 34 898.50 $4 32 325.41 $4 29 737.30 $4 27 134.10

7% p.a. interest Balance compounded plus interest monthly $2 566.67 $2 551.83 $2 536.91 $2 521.90 $2 506.80 $2 461.62

$4 42 566.67 $4 40 008.50 $4 37 435.41 $4 34 847.30 $4 32 244.10 $4 29 625.72

Monthly Repayment

Balance at end of month

$5 110 $5 110 $5 110 $5 110 $5 110 $5 110

$4 37 456.67 $4 34 898.50 $4 32 325.41 $4 29 737.30 $4 27 134.10 $4 24 515.72

7 8 9 10 11 12

a What will the formula need to be in cell D4? b What will the formula need to be in cell E4? c What will the formula need to be in cell G4? d What will the formula need to be in cell C5? e Using any spreadsheet program to create and complete the above spreadsheet. f Using the completed spreadsheet, find: i the amount of interest paid in month 10 ii the balance at the end of 12 months.

g How much was paid off the loan in 12 months? h What is the total amount the loan has reduced by in 12 months? i Experiment with what changes occur when you increase or decrease the monthly repayments. j Experiment with what changes if a lump sum of $20 000 is paid off the loan after 3, 6 or 9 years.

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12 Open the provided spreadsheet model for reducing-balance loans. The current settings are: • Principal = $150 000 • Annual interest rate = 4% • Payments per year = 12

U N SA C O M R PL R E EC PA T E G D ES

• Term = 15 years

The regular repayment shown is $1 109.53.

a Use the sliders to increase the annual interest rate to 8% while keeping the term at 15 years. i What is the new regular repayment? ii How much more will you pay each month compared to the original setting?

b Now reduce the term to 10 years while keeping the interest rate at 8%. i What is the new repayment amount? ii Explain why the repayment changes when the term is shortened.

c Finally, increase the principal to $200 000 with the term at 10 years and interest rate at 8%. i What is the repayment now? ii Compare this to the repayment when the principal was $150 000.

d Based on your observations, explain how each factor (interest rate, term, and principal) affects the repayment amount.

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3J Credit cards

3J Credit cards Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand how credit cards work • To calculate the interest, fees and other charges on purchases and payments made with credit cards • To solve problems involving credit cards Past, present and future learning

• Some aspects of this section may be new to students • Questions involving credit cards regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The formula for the future value of a compound interest investment appears on the 2027 version. It may be used for credit cards.

Credit cards are used to buy goods and services and pay for them later. The time when interest is not charged on your purchases is called the interest-free period. If payment is not received when the statement is due then interest is charged from the date of purchase. Interest on credit cards is usually calculated daily on the outstanding balance using compound interest. The interest rate is usually much higher than for other kinds of loans and credit facilities.

Credit cards

Daily interest rate = FV = PV(1 + r)n

Annual interest rate 365 I = FV − PV

FV = Amount owing on the credit card

PV = Principal is the purchases made on the credit card plus the outstanding balance r = Rate of interest per compounding time period expressed as a decimal n = Number of compounding time periods I = Interest (compound) charged on the outstanding balance Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Example 24

Calculating the cost of using a credit card

U N SA C O M R PL R E EC PA T E G D ES

Luna has a credit card with a compound interest rate of 18% p.a. and no interest-free period. Luna used her credit card to pay for clothing costing $280. She paid the credit card account 14 days later. What is the total amount she paid for the clothing, including the interest charged?

Solution

Explanation

P = 280, r = 0.18 ÷ 365, n = 14

Write the values for P, r and n.

FV = PV(1 + r)n

Write the future value formula. Substitute the values for P, r and n.

= 280 1 +

0.18 365

!14

= $281.94 Total amount paid is $281.94.

Now you try

Carter has a credit card with a compound interest rate of 22% p.a. and no interest-free period. Carter used their credit card to pay for groceries which cost $196.50. They paid the credit card account 21 days later. What was the total they paid for their groceries including the interest charges?

Exercise 3J 1

BUILDING

A credit card has a daily interest rate of 0.05% per day. Find the interest charged on these outstanding balances. Answer correct to the nearest cent.

a $840 for 12 days c $5680 for 30 days e $240 for 6 days

Example 24

b $742.40 for 20 days d $128 for 18 days f $1450 for 15 days

Emir has a credit card with an interest rate of 0.04% compounding per day and no interest-free period. He uses his credit card to pay for a mobile phone costing $980. Calculate the total amount paid for the mobile phone if Emir paid the credit card account in the following time period. Answer correct to the nearest cent. a 10 days later c 30 days later e 50 days later

b 20 days later d 40 days later f 60 days later

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3J Credit cards

Calculate the compound interest charged on these outstanding balances. Answer correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

a Balance = $6820, Daily interest rate of 0.08%, Time period 70 days b Balance = $23 648, Daily interest rate of 0.06%, Time period 35 days c Balance = $1550, Daily interest rate of 0.05%, Time period 20 days d Balance = $35 800, Daily interest rate of 0.09%, Time period 100 days e Balance = $4500, Daily interest rate of 0.05%, Time period 27 days f Balance = $7680, Daily interest rate of 0.04%, Time period 180 days

4 Dasula’s credit card charges 0.045% compound interest per day on any outstanding balances. Calculate how much interest Dasula is charged on an amount of $450, which is outstanding on his credit card for 35 days. Answer correct to the nearest cent.

Cheyanne received a new credit card with no interest-free period and a daily compound interest rate of 0.05%. She used her credit card to purchase food for $320 and petrol for $50 on 18 July. This amount stayed on the credit card for 24 days. What is the total interest charged? Answer correct to the nearest cent.

Jhye used his credit card to buy a holiday to New Zealand. The cost of the package was $6500. The charge on the credit card is 1% interest per month on the unpaid balance. How much does Jhye owe for his holiday after six months? Answer correct to the nearest cent.

Calculate the amount owed, to the nearest cent, for each of the following credit card transactions. The credit card has no interest-free period.

a Transactions = $540, Compound interest rate = 14% p.a., Time period = 15 days b Transactions = $270, Compound interest rate = 11% p.a., Time period = 9 days c Transactions = $1400, Compound interest rate = 18% p.a., Time period = 22 days d Transactions = $480, Compound interest rate = 16% p.a., Time period = 18 days e Transactions = $680, Compound interest rate = 10% p.a., Time period = 9 days

Calculate the interest charged for each of the following credit card transactions. The credit card has no interest-free period. Answer correct to the nearest cent.

a Transactions = $680, Compound interest rate = 15% p.a., Time period = 20 days b Transactions = $740, Compound interest rate = 12% p.a., Time period = 13 days c Transactions = $1960, Compound interest rate = 17% p.a., Time period = 30 days d Transactions = $820, Compound interest rate = 21% p.a., Time period = 35 days e Transactions = $1700, Compound interest rate = 19% p.a., Time period = 32 days

Pedro has a credit card with a compound interest rate of 18.25% per annum.

a What is the daily percentage interest rate, correct to two decimal places? b Pedro has an outstanding balance of $4890 for a period of 30 days. How much interest, to the nearest cent, will he be charged?

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Alyssa uses a credit card with a no interest-free period and a compound interest rate of 15.5% p.a. from the purchase date. During April she makes the following transactions. Transaction details IGA Supermarket

$85.00

09 April

Kmart

$115.00

12 April

David Jones

$340.00

27 April

General Pants

$80.00

28 April

JB Hi-Fi

$30.00

U N SA C O M R PL R E EC PA T E G D ES

04 April

a What is the daily compound interest rate, correct to three decimal places? b Alyssa’s account is due on 30 April. What is the total amount due if you disregard the amount of interest to be paid? c How much interest has Alyssa paid on the IGA transaction during the month? Answer correct to the nearest cent. d How much interest has Alyssa paid on the Kmart transaction during the month? Answer correct to the nearest cent.

DEVELOPING

Vihaan’s credit card has up to 55 days interest free and is due on the 22nd of each month. The interest rate is 15.4% p.a. compounding daily. Vihaan buys furniture costing $5160 on 25 October. How much interest is he charged if he pays the balance on 22 December? Interest is charged from the date of purchase if the total amount owing has not been paid by the due date.

Harry buys two Blu-ray DVDs for $29.90 each and a shirt for $84.95 on his credit card. This amount stays on his credit card for 75 days. There is a 45-day interestfree period and a daily interest of 0.05% compound on his credit card.

a How much did Harry spend on his credit card? b Calculate the amount Harry owes on the credit card. c What was the interest charged on these purchases? d What would be the interest charged if Harry’s credit card did not have a 45-day interest-free period?

CHALLENGING

Amina and Joshua each use their credit cards to buy holiday packages to Adelaide. The cost of the package is $1700 for each person.

a The charge on Amina’s credit card is 0.9% compound interest per month on the unpaid balance. It has no interest-free period. Amina pays $800 after one month and another $500 the next month. How much does she still owe on her credit card? b The charge on Joshua’s credit card is interest-free in the first month, and 1.4% compound interest per month on any unpaid balance. Joshua pays $800 after one month and another $500 the next month. How much does he still owe on his credit card? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

3J Credit cards

Avery’s August credit card statement shows an opening balance of $1850, a purchase of $2450 on 5 August, and another purchase of $55 on 14 August. The minimum payment is 3% of the closing balance. The initial credit charge is 1.6% compounding per month of any amount outstanding.

U N SA C O M R PL R E EC PA T E G D ES

a What is the closing balance on this credit card for August? b Calculate the amount of interest charged for the month of August. c What is the minimum payment, to the nearest cent, required for August? d What is the opening balance for October if Emily paid the minimum payment in September for interest charged in August and made no purchases in September?

Communicate your reasoning, understanding and fluency 15

Open the provided spreadsheet Credit Card Interest Calculator, available within your Interactive Textbook.

The spreadsheet shows an outstanding balance of $2400, an annual interest rate of 18%, and the number of compounding periods per year as 365. The time in days until the card debt is paid off is currently set to 45 days, and the amount owing is displayed as $2453.84. a Change the time in days to 60.

i What is the new amount of interest owing? ii How much interest has been added compared to the original 45-day scenario?

b Change the time in days to 90.

i What is the amount of interest owing now? ii How does the interest compare to the 60-day scenario?

c Explain why the amount owing grows faster as the number of days increases. Use the concept of compound interest in your explanation. d Experiment by changing the annual interest rate to 20%. i What happens to the amount of interest owing after 45 days? ii Why does increasing the interest rate have such a large effect?

16 Research current credit card interest rates, home loan interest rates and short-term interest rates at four banks of your choice. a Complete the table below.

Interest rate comparison

Bank

Credit card

Home loan

Short-term loan

A B C

b Explain why there is such a difference between each of the interest rates.

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3K Credit card statements Learning intentions

• To understand and interpret credit card statements • To perform calculations based on credit card statements

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • Questions involving credit cards regularly appear in HSC Examinations

Credit card statements are issued each month and contain information such as account number, opening balance, new charges, payments, refunds, reward points, payment due data, minimum payment and closing balance. The credit card statement includes the date and cost of each purchase and could be regarded as a ledger. A ledger documents your spending. If the minimum payment is not made by the due date, the consequences can be expensive. You may be charged a late payment fee and, of course, you will be charged interest on it.

Example 25

Reading a credit card statement

Your Bank

Your Bank of Australia ABN 12 345 678 901

1 of 2 5 Oct 5 Nov

Page number Statement begins Statement ends Enquiries Credit Card

MR JOHN CITIZEN 123 SAMPLE STREET SUBURBIA NSW 2000

132 221

(24 hours a day, 7 days a week)

Your Bank Awards

131 661

(8am to 8pm Mon-Fri)

MasterCard 5353 1801 0001 0001

Payment due date

Opening balance

$207.72

30 November

New charges

$460.14

Minimum payment

Payments/refund

–$207.72

$25.00

Closing balance $460.14

Your Bank Awards 1000123456 Opening points balance Total points earned Points redeemed

Total Points Balance 34 910

50 500

460

–15 600

Answer the following questions using the above credit card statement. a What is the credit card account number? c What is the payment due date?

b What is the opening balance? d What is the minimum payment?

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3K Credit card statements

Explanation

a 5353 1801 0001 0001

1 Read the number after ‘MasterCard’.

b Opening balance is $207.72.

2 Read ‘Opening balance’.

c Payment due date is 30 Nov.

3 Read the box ‘Payment due date’.

d Minimum payment is $25.00.

4 Read the box ‘Minimum payment’.

U N SA C O M R PL R E EC PA T E G D ES

Solution

Now you try

Answer the following questions using the credit card statement from Example 25.

a What is the total points balance? b What is the closing balance? c What are the dates that the statement begins and ends? d What was the cost of new charges?

Exercise 3K

Example 25

BUILDING

Use the credit card statement below to answer these questions. Account summary Opening balance

$743.42

Payments and other credits

$743.42

Purchases

$172.91

Cash advances

$0.00

Interest and other charges

$0.00

Closing account balance

$172.91

Cardholder credit balances

4511.88

Payment summary

Card balances renewal Monthly payment Due date

Minimum amount due

$4684.79 $10.00

21/04/2020 $10.00

a What is the due date? b What is the cost of the purchases? c What is the closing account balance? d What is the minimum amount due? e What payment was made last month? f How much interest was charged? g What was the opening balance? h What is the cardholder’s credit balance?

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The transactions on a credit card are shown below. Account summary Account balance

Credit limit

$15 549.18

$3950.82

$19 500.00

Payment due date

Minimum payment due

7 Dec

$57.00

U N SA C O M R PL R E EC PA T E G D ES

Available credit

Last 5 transactions View more Date

Transaction description

Debit

30 Nov

WW Petrol

$24.38

29 Nov

Coles

$55.03

29 Nov

Woolworths

$34.63

28 Nov

Myer

$49.13

28 Nov

David Jones

$23.40

a What is the credit limit? c How many transactions are shown? e How much was spent on 29 November? g Where was $49.13 spent on 28 November? i What is the payment due date?

Credit

b What is the account balance? d What is the available credit? f How much was spent on 28 November? h Where was $24.38 spent on 30 November? j What is the minimum amount due?

DEVELOPING

Create the spreadsheet below.

a How many transactions are shown on the ledger? b How much has been spent at Manly Vale Pharmacy? c If the account begins on 15 November and ends on 14 December, how many days does it account for? d If the card has a $5000 credit limit, what is the available credit on 14 December? e If the minimum payment is $10 and is paid on the due date, what is the balance owing? f This credit card charges 0.06% per day compound interest on the unpaid balance. What is the interest charged per day on the closing balance? Answer correct to the nearest cent.

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3K Credit card statements

Consider the credit card statement shown below. Visa Account number

4557 0756 0833 1234

Credit limit

$12 000

Available credit

$6 361

U N SA C O M R PL R E EC PA T E G D ES

Account summary − Opening balance

$5821.31 DR

+ Payment & other credits received

$781.25 CR

− Purchases, cash advances & special promo debits

$511.93 DR

− Interest & other charges

$86.26 DR

= Closing balance

$5638.25 DR

a What is the opening balance? b What is the credit limit? c What is the available credit? d What is the closing balance? e How much has been spent on purchases, cash advances and special promo debits this month? f How much interest and other charges were paid last month? g This credit card charges 0.05% per day compound interest on the unpaid balances.

i What is the interest charged per day on the closing balance if no payment is made? Answer correct to the nearest cent. ii How much interest would be paid on the closing balance for a year? Answer correct to the nearest cent.

CHALLENGING

The transactions on a credit card with an opening balance of $5246.84 are shown below.

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a What is the opening balance? b What is the total of new charges and credits made on this account? c If the card has a $30 000 credit limit, what is the available credit on 1 December? d How many transactions were completed using card 6360? e How many transactions were completed using card 6378? f Which transaction was a credit? g What was the transaction date for Coles? h What is the closing balance? i This credit card charges 21.9% p.a. compound interest on the unpaid balances. What is the interest charge for 120 days on the closing balance? Answer correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

180

6 Payment details for a credit card are shown opposite.

a What is the closing balance if the minimum payment was paid on the 30 September? b This credit card charges 18.75% p.a. compound interest on the unpaid balance. What is the interest charged for the next 60 days if the minimum payment was made? Answer correct to the nearest cent.

Communicate your reasoning, understanding and fluency 7

Answer parts a–e below from the credit card statement shown.

a What is the name of the bank? b What is the closing balance? c Explain why the closing balance is the same as the new charges. d Explain why Mr A B Sea has paid no interest this month. e Mr A B Sea does not get paid from his job until 15th January. He made a payment of $30 on 10th January and then paid the balance on 15th January. How much interest was added to his next credit card statement as a result of his late payment?

Standard Bank

Standard Bank of Australia ABN 98 765 432 101

1 of 2 17-Nov 16-Dec

Page number Statement Start Statement End Enquiries Credit Card

MR A B Sea 321 Somewhere Place Overhere NSW 2001

132 224 (24 hours, 7days)

Standard Bank Rewards 131 624 (8am-8pm Mon-Fri)

MasterCard 4564 0011 1100 0220 (interest rate 22.5% p.a. compounded daily on overdue amounts only after the payment date)

Opening balance

$432.91

New charges/interest

$885.12

Payments/refunds

$432.91

Payment Due date 10th January

Minimum payment $30

Closing balance $855.12

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3L Fees and charges for credit card usage

3L Fees and charges for credit card usage Learning intentions

• To understand and interpret the fees and charges associated with credit card usage • To perform calculations of fees and charges when using a credit card

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • Questions involving credit cards regularly appear in HSC Examinations

Banks and financial institutions charge their customers an annual card fee for maintaining a credit card account. In addition to this fee, customers may be charged fees for late payment, cash advances and balance transfers. The late payment fee applies if the minimum payment has not been received by the due date. Interest is charged for retail purchases and the amount still owing from the previous month.

Fees and charges for credit card usage

• Annual card fee – maintaining credit card account • Interest charge – interest charged for retail purchases • Late payment fee – when minimum payment has not been received by the due date • Cash advances – withdrawing cash from the credit card account • Balance transfers – moving balance to another account, often held at another institution

Example 26

Calculating fees and charges when using a credit card

Hillary has a debit of $6000 on a credit card with an interest rate of 14.75% p.a. that compounds daily. She decided to transfer the balance to a new card with a 0% balance transfer for 6 months. However, after 6 months the new card reverted to an interest rate of 19.75% p.a that compounds daily. Show by how much Hillary is better off after 12 months.

Solution

Old card P = 6000, r = 0.1475 ÷ 365, n = 365 FV = PV(1 + r)n !365 0.1475 = 6000 1 + 365 ≈ $6953.39

Explanation

Write the values for P, r and n. Write the future value formula.

Substitute the values for P, r and n.

Continued on next page

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New card P = 6000, r = 0.1975 ÷ 365, n = 184 FV = PV(1 + r)n !182.5 0.1975 = 6000 1 + 365 ≈ $6627.94

Write the values for P, r and n. Write the future value formula. Substitute the values for P, r and n.

U N SA C O M R PL R E EC PA T E G D ES

182

Savings = Old card − new card.

Savings = $6953.39 − $6627.94

= $325.45 Hillary is $325.45 better off with the new card. Now you try

Grayson has a debt of $7500 on a credit card with an interest rate of 13.25% p.a. compounded daily. He decides to transfer his balance to a new card with a 0% balance transfer for 6 months. However, after 6 months the new card reverts to an interest rate of 21% p.a. compounded daily. Show by how much he is better off after 12 months.

Exercise 3L 1

BUILDING

Alicia’s bank charged an annual credit card fee of $350, a cash advance fee of $2.50 and a late payment fee of $20. Calculate Alicia’s banking costs for the year if she made:

a 11 cash advances and 4 late payments b 20 cash advances and 12 late payments c 50 cash advances and 6 late payments d 0 cash advances and 12 late payments e 100 cash advances and 0 late payments f 0 cash advances and 0 late payments.

The table below shows the credit card usage charges for four banks. Bank

Annual fee

Cash advance

Late payment

$225

$2.00

$15

$200

$2.20

$20

$250

$1.80

$12

$240

$1.90

$16

a What is the cost of the cash advance fee at bank B? b What is the cost of the late payment fee at bank D? c Which bank has the lowest annual fee? d Which bank has the highest cash advance fee? e Calculate the difference between the late payment fees at bank C and bank D. f Calculate the difference between the cash advance fees at bank B and bank C. g What is the average annual fee for these banks?

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3L Fees and charges for credit card usage

h What is the average late payment fee for these banks? i What are the annual banking costs for 30 cash advances and 1 late payment at: i Bank A? iii Bank C?

ii Bank B? iv Bank D?

j What are the annual banking costs for 100 cash advances and 6 late payments at: ii Bank B? iv Bank D?

U N SA C O M R PL R E EC PA T E G D ES

i Bank A? iii Bank C?

3 Elijah’s bank charged an annual credit card fee of $320, cash advance fee of $2.30 and late payment fee of $18. What are Elijah’s banking costs for the year if he made 80 cash advances and had 1 late payment fee?

DEVELOPING

The table below shows the banking charges for credit card usage. Fee

Charge

Cash advance

Late payment

$20

Balance transfer

Find the cost of the following activities. Answer correct to the nearest cent. a 1 cash advance of $400. b 10 cash advances of $50. c 100 cash advances of $100. d 1 balance transfer of $3580. e 1 balance transfer of $8620. f 1 balance transfer of $10 000. g 2 late payments and 3 cash advances of $300. h 10 late payments and 20 cash advances of $200. i 5 late payments and 50 cash advances of $80.

Example 26

5 George purchased a laptop on his credit card for $2500. Interest is charged at 0.04% per day. He decided to transfer the balance to a new card with a 0% interest? for 90 days. How much does he save in the first 180 days on the following new cards? Answer correct to the nearest cent. a Interest rate of 0.03% per day b Interest rate 0.04% per day c Interest rate of 0.05% per day d Interest rate 0.06% per day e Interest rate of 0.07% per day f Interest rate 0.08% per day

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Annabelle has a debit of $8000 on a credit card with an interest rate of 12% p.a. She decided to transfer the balance to a new card with a 0% balance transfer for 6 months. How much does she save in the first 12 months on the following new cards whose interest rate compounds daily? Answer correct to the nearest cent. b Interest rate 14% p.a. d Interest rate 18% p.a. f Interest rate 22% p.a.

U N SA C O M R PL R E EC PA T E G D ES

a Interest rate of 12% p.a. c Interest rate of 16% p.a. e Interest rate of 20% p.a.

CHALLENGING

Prisha August credit card statement shows an opening balance of $1850, a purchase of $2450 on 5 August, and another purchase of $1055 on 14 August. The minimum payment is 3% on the closing balance. The credit charge is 0.05% per day compound interest on the outstanding balance. Answer the following questions correct to the nearest cent.

a What was the total cost of the purchases in August on the credit card? b What is the closing balance on this credit card for August? c What is the minimum payment to the nearest cent required for August? d Calculate the amount of interest charged for the month of September if Prisha paid the minimum payment in August.

Communicate your reasoning, understanding and fluency 8

Trevor opens a new credit card account on 10 December with the interest and fees shown below. • Interest charged at 17.25% p.a. compounded daily • No interest-free period • $0 online payments • $3.50 cash payments • $10 establishment fee payable the day the account is opened.

Trevor purchases a flight valued at $1767 with the credit card on 10 December.

a Trevor worked out the balance owing on the credit card after 29 days in the following two ways. Explain which way is correct and why. Option 1 FV = PV(1 + r)n = 1767 1 +

0.1725 365

!29

+ 10

= $1801.38

Option 2 FV = PV(1 + r)n

0.1725 = 1777 1 + 365 = $1801.52

!29

b Trevor made a cash repayment of $800, 29 days after they bought the flight. Explain why the new balance owing is $1005.02.

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Chapter 3 review

FV or I = FV − PV (1 + r)n FV = Future value of the loan or amount (final balance) PV = Present value of the loan or principal (initial quantity of money) r = Rate of interest per compounding time period as a decimal (i.e. convert percentage rates to decimal) n = Number of compounding time periods I = Interest (compounded) earned. FV = PV(1 + r)n or PV =

U N SA C O M R PL R E EC PA T E G D ES

Future value formula

Review

Key ideas and chapter summary

Investment graphs

Construct a table of values for A and n using the simple interest formula or the future value formula.

Draw a number plane: n horizontal axis, A the vertical axis. Plot and join the points.

Appreciation and inflation

Use the formula FV = PV(1 + r)n for appreciation and inflation. Inflation rate is the annual percentage change in the CPI.

Shares and dividends

Dividend is a payment given as an amount per share or a percentage of the issued price. Annual dividend × 100% Dividend yield = Market price

Declining-balance depreciation

S = V0 (1 − r)n S = Salvage value or current value V0 = Purchase price of the item r = Rate of interest per time period (decimal) n = Number of time periods.

Reducing-balance loans

Total to be paid = Loan repayment × Number of repayments Total to be paid = Principal + Interest

Credit cards

Fees and charges for credit card usage

Annual interest rate Daily interest rate = 365 FV = PV(1 + r)n FV = Future value or the amount owing on the credit card PV = Present value or the purchases made on the credit card r = Rate of interest per compounding time period as a decimal n = Number of compounding time periods.

Annual card fee, interest charge, late payment fee, cash advances, balance transfers.

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Chapter checklist with success criteria A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

1 I can find the simple interest earned on an investment.

U N SA C O M R PL R E EC PA T E G D ES

e.g. Calculate the amount of simple interest paid on an investment of $22 000 at 5% simple interest per annum for 7 years.

2 I can calculate the value of an investment.

e.g. Albert plans to make an investment of $250 000 at 7% p.a. simple interest for 6 years. Calculate the total value of the investment after 6 years.

3 I can calculate the value of an investment calculated periodically.

e.g. Indigo plans to make an investment of $150 000 at 5.5% p.a. simple interest with the interest calculated monthly for 5 years. Calculate the total interest earned on the investment after 5 years.

4 I can find the compound interest earned on an investment.

e.g. Aahil invests $17 000 over 3 years at a compound interest rate of 8.25% p.a. Calculate: a the amount of the investment after 3 years, correct to the nearest cent b the interest earned after 3 years, correct to the nearest cent.

5 I can calculate the future value and present value of an investment. e.g.

a Ahmed invests $7500 over 7 years at a compound interest rate of 7.5% p.a. Calculate the future value after 7 years, correct to the nearest cent. b Calculate the present value of an investment that has a future value of $150 000 over 5 years at 9.5% p.a. compounded monthly.

6 I can find the compound interest compounded quarterly.

e.g. Winston invests $5500 over 5 years at a compound interest rate of 6% p.a. compounded quarterly. Calculate: a the amount of the investment after 5 years, correct to the nearest cent b the interest earned over the 5 years, correct to the nearest cent.

7 I can construct a simple interest graph.

e.g. Draw a graph showing the amount of simple interest earned over a period of 12 years if $1800 was invested at 5% p.a. Use the graph to estimate the interest earned after 9.5 years.

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Chapter 3 review

8 I can construct a compound interest graph. e.g. Draw a graph showing the amount of compound interest earned over a period of 10 years if $1500 was invested at 7% p.a. Use the graph to estimate the interest earned after 9.5 years. 9 I can find the appreciated value.

U N SA C O M R PL R E EC PA T E G D ES

Review

e.g. Avi bought a unit for $825 000. If the unit appreciates at 9.5% p.a., what is its value after 5 years? Answer correct to the nearest dollar.

10 I can find the price of goods following inflation.

e.g. What is the price of a $1050 fridge after five years if the inflation rate was 1.8% p.a.?

11 I can calculate the cost of purchasing shares.

e.g. Ernie purchased 200 shares at a market value of $4.20 each. The brokerage fees were $33 including GST. What was the total cost of purchasing the shares?

12 I can calculate the dividend and dividend yield. e.g. The share price of a company is $27.72.

a The predicted dividend yield is 4.25%. What would be the dividend? b The company decides to pay a dividend of $1.95. What is the dividend yield?

13 I can calculate the salvage value.

e.g. Koby pays $11 000 for a used car. It depreciates at $950 each year. How much will the car be worth after 3 years?

14 I can calculate the straight-line depreciation.

e.g. Naomi purchases a new car for $25 000. After 5 years its salvage value is $10 000. What is the annual amount of depreciation, if the amount of depreciation is a constant?

15 I can calculate the declining-balance depreciation.

e.g. The purchase value of a new car four years ago was $62 000. During the first 2 years it depreciated at 25% p.a. During the next 2 years it depreciated at 15% p.a. What is the current value of the car correct to the nearest $100?

16 I can calculate the purchase price.

e.g. Gabriella buys a car that depreciates at the rate of 24% p.a. After 6 years the value of the car is $12 150. How much did Gabriella pay for her car, correct to the nearest dollar?

17 I can calculate the percentage rate of depreciation. e.g. Morticia bought a car three years ago for $30 000. Using the declining-balance method for depreciation, she estimates its present value to be $11 100. What annual percentage rate of depreciation did she use? Answer correct to the nearest whole number.

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18 I can calculate the costs of borrowing using buy now pay later. e.g. Hayden made the following purchases using a BNPL scheme: January 10th : Clothes $200, four monthly payments of $50 February 10th : Phone $750, four monthly payments of $187.50

U N SA C O M R PL R E EC PA T E G D ES

March 10th : Bike $400, four monthly payments of $100 a

How much did Hayden have to pay in:

i February? ii March? iii April? iv May?

Hayden was only able to afford to pay $200 in April. As a result, he paid penalty fees of 17.5% of his outstanding debt and a $5 processing fee. How much were his total penalties in April?

19 I can calculate the costs of a short-term loan.

e.g. Yung Wei borrowed $1400 using a short-term finance company. There were no set up fees. The terms were no deposit and 4 monthly instalments of $350. If she does not pay on time, she will incur a 4% penalty on the outstanding balance and a $2.50 processing fee. a Yung Wei pays her third payment two days late. How much are her penalties? b Yung Wei pays her fourth payment five days late. How much are her penalties? c What is the total of all the penalties paid by Yung Wei?

20 I can calculate the costs of a long-term loan.

e.g. Bimal plans to take out a loan for $900 000 for a house valued at $955 000. He is required to pay the following fees which will need to be added to the loan:

• Establishment fees: 0.1% of the loan • Origination fees: 0.05% of the loan • Valuation fees: $222 • Conveyancing fees: $350 • Mortgage registration fees: $171.70 • Stamp duty: $284

Calculate the actual amount the loan will need to be to cover the fees.

21 I can use a table for a reducing-balance loan to find interest paid.

e.g. Use the table from Example 22 to calculate the amount of interest paid on a loan of $150 000 over 12 years.

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Chapter 3 review

22 I can use a table for a reducing-balance loan to find amount saved. e.g. Charlotte is planning to borrow $560 000 to buy a house at 6% p.a. over 20 years. Use the table in Example 23 showing the monthly repayments for each $1000 borrowed to answer the following questions.

U N SA C O M R PL R E EC PA T E G D ES

a What is Charlotte’s monthly payment for this loan? b How much would Charlotte pay in total for this loan? c How much would Charlotte save if she repaid the loan over 15 years?

Review

23 I can calculate the cost of using a credit card.

e.g. Duke has a credit card with a compound interest rate of 21% p.a. and no interest free period. Duke used his credit card to pay for headphones which cost $386. He paid the credit card account 10 days later. What was the total he paid for the headphones including the interest charges?

24 I can read a credit card statement.

e.g. Using the credit card statement from Exercise 3K Question 7 answer the following questions. a What is the credit card enquiries phone number? b What is the MasterCard number? c When is the credit card due? d What is the interest payable on overdue amounts?

25 I can calculate the fees and charges when using a credit card.

e.g. Betty has a debt of $9000 on a credit card with an interest rate of 14.25% p.a. compounded daily. She decides to transfer her balance to a new card with a 0% balance transfer for 90 days. However, after 90 days the new card reverts to an interest rate of 18.75% p.a. compounded daily. Show by how much she is better off after 12 months.

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Multiple-choice questions 3A

Find the simple interest earned on $3214 at 12% p.a for 6 years. A $2314.08

C $231

D $1950

2 $22 000 is invested for 5 years at 11% p.a. simple interest. How much is the investment worth in 5 years?

U N SA C O M R PL R E EC PA T E G D ES

B $2314.07

A $34 100

B $31 400

B $2648

B $60

C $588 800

D $602 966

B $8160

C $8198

D $30 400

7 A car depreciates in value from $39 000 to $12 250 in 4 years using the declining-balance method for depreciation. What is the annual rate of depreciation correct to the nearest whole number? B 18%

C 25%

D 26%

8 The table shows the monthly repayment of $1000 on a reducing-balance loan. What is the monthly repayment on $290 000 at 8.75% for 20 years? Term

8.00%

8.25%

8.50%

8.75%

20 years

$6.38

$6.77

$7.17

$7.57

A $1850.20

D $630

6 Ella bought 800 shares at a market value of $10.20 each. Brokerage costs incurred were $38 including GST. What is the total cost of purchasing the shares?

A 17%

C $600

B $492 200

A $48.20

D $10 648

5 A painting was bought for $460 000 and appreciated at the rate of 7% p.a. What will be the value of the painting after 4 years? Answer correct to the nearest dollar. A $473 016

C $8242

4 Clara invests $3000 at 10% p.a. interest compounding annually. What is the interest earned after 2 years? Answer correct to the nearest dollar. A $30

D $12 100

3 Bella invests $8000 at 10% p.a. interest compounding annually. What is the future value after 3 years? Answer correct to the nearest dollar. A $242

C $30 000

B $1963.30

C $2079.30

D $2195.30

9 Lachlan borrows $245 000 over 20 years at a reducible interest rate of 6.5% p.a. He pays $1856 per month. What is the total paid on this loan? A $200 440

B $318 500

C $445 440

D $563 500

3J 10 A credit card has a daily interest rate of 0.05% per day (no interest free period). Find the interest charged on $1530 for 14 days. Answer correct to the nearest cent. A $0.77

B $10.74

C $76.50

D $1540.74

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Find the simple interest earned if $1100 is invested for 3 years at 5% p.a.

Calculate the future value, correct to the nearest cent, for each of the following.

U N SA C O M R PL R E EC PA T E G D ES

a Present value = $920, Compound interest rate = 5% p.a., Time period = 4 years b Present value of $2100 invested for 3 years at 6.1% p.a. compounded monthly

Review

Short-answer questions

3 Calculate the present value, correct to the nearest cent, for each of the following.

a Future value = $26 000, Interest rate = 4.9% p.a., Time period = 3 years b Future value of $10 400, Interest rate at 9% p.a. compounded quarterly for 5 years

4 What sum of money would Amelia need to invest to accumulate a total of $200 000 at the end of 10 years at 12% p.a. interest compounding biannually? Answer correct to the nearest cent.

5 An investment is appreciating at a rate of 4% of its value each year. Alkira decides to invest $480 000.

a What will be the value of the investment after 10 years? Answer correct to the nearest dollar. b How much does the investment increase during the first ten years?

The average inflation for the next three years is predicted to be 2.5% p.a. Calculate the price of the following goods in three years’ time. Answer correct to the nearest cent. a 2 L of soft drink for $2.80 b Apple pie for $4.60 c Hamburger for $6.00 d Bottle of water for $1.60 e Punnet of strawberries for $4.50 f 500 g of chicken breast for $8.90

Dakota bought 50 shares in a bank for $82.64 each. She sold them two years later for $68.90. The brokerage fee paid for buying and selling the shares was $35.95. What was the loss on these shares?

A company with a share price of $2.42 declares a dividend of 12 cents. What is the dividend yield correct to two decimal places?

Esther purchases 1000 shares at $16.90 per share. Her broker charges $20 plus 1.5% of the purchase price. Calculate the brokerage for this purchase.

3F 10

Zhang Wei purchased a car for $19 900. It depreciates at 24% per year.

a What is the salvage value of the car after 5 years? Answer correct to the nearest dollar. b How much has the car depreciated over the 5 years?

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Chapter 3 Investments and loans

3G 11

Li Na takes out a loan of $21 000 over 36 months. The repayment rate is $753.42 per month. a How much will Li Na pay back altogether? Answer correct to the nearest dollar. b What is the equivalent simple interest rate per annum for the loan, correct to one decimal place?

3G 12

Jabari borrows $280 000 and repays the loan in equal fortnightly repayments of $1250 over 20 years. What is the flat rate of interest per annum on Jabari’s loan, correct to two decimal places?

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Review

192

3H 13

Derek purchased an air fryer for $275 using a BNPL scheme. He paid a 15% deposit and then 5 equal monthly repayments. How much was each repayment?

3H 14

Jason borrowed $470 using a short-term finance company. A $30 set up fee was then added. The terms were no deposit and 5 monthly instalments. He paid 4 instalments of $100. How much was left to be paid in the final instalment?

3I 15 The table below shows the monthly repayments for a loan of $1000 for varying interest rates. Jill borrows $527 000 at 9% over 15 years. Period of Loan

Interest Rate

10 years

15 years

20 years

$12.16

$9.88

$8.44

$12.70

$10.22

$9.09

10%

$13.35

$10.78

$9.71

a What were Jill’s monthly repayments? b How much does Jill pay in total for the loan? c How much interest does Jill pay on this loan? d How much would she save if she paid off the loan in 10 years?

3J 16

Cara has a credit card with an interest rate of 17% p.a. compounding daily and no interest-free period. Cara used her credit card to pay for shoes costing $170. She paid the credit card account 26 days later. What is the total amount she paid for the shoes including the interest charged? Answer correct to the nearest cent.

3J 17

Zemira’s bank charged an annual credit card fee of $300, a cash advance fee of $4.00 and a late payment fee of $20. Calculate Zemira’s banking costs for the year if she made: a 9 cash advances and 5 late payments b 15 cash advances and 7 late payments.

3K 18

Using the credit card statement in Exercise 3K Question 7, answer the follow questions.

a What is the name of the person who owns the credit card? b What is the ABN of the Standard Bank? c What is the opening balance? d What is the minimum payment?

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Chapter 3 review

Benjamin uses a credit card with a no interest-free period and an interest rate of 18.5% p.a. compounding daily from and including the purchase date and due date. Benjamin’s account is due on February 28. During February he makes the following transactions.

U N SA C O M R PL R E EC PA T E G D ES

Transaction Details

Review

Extended-response question

06 February

Coles

$278.00

07 February

Myer

$87.00

18 February

Big W

$259.00

18 February

JAG

$120.00

20 February

Bunnings

$460.00

21 February

Woolworths

$300.00

How much interest will Benjamin pay during the month on the following transactions? Answer correct to the nearest cent. a Coles transaction b Myer transaction c Big W transaction d JAG transaction e Bunnings transaction f Woolworths transaction

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U N SA C O M R PL R E EC PA T E G D ES

Trigonometry

Some uses of trigonometry

• Trigonometry makes GPS and Google Maps work: Satellites, TV, Wi-Fi and GPS use trigonometry to pinpoint your exact location on Earth by triangulating signals. Every ‘You are here’ dot is powered by trigonometry. • The sine and cosine graphs are infinite and repeating: These are periodic functions, meaning their patterns repeat forever. Great for modelling cycles — like tides, seasons, or even your sleep rhythm. • Music has trigonometry hidden inside it: All music/sound waves can be modelled using trigonometric functions and their graphs. An individual note can be modelled with a

sine wave while a chord can be modelled with multiple sine waves. • Trigonometry was a secret weapon in World War II: Navigation, artillery targeting and radar systems relied heavily on trigonometry to calculate distances, angles and bearings. • Trigonometry makes roller coasters possible: Engineers use sine and cosine functions to design and analyse roller coaster tracks, especially in calculating angles, distances and the shape of curves and loops, ensuring safe and thrilling rides.

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Chapter contents Trigonometry in right-angled triangles Angles of elevation and depression True bearings and compass bearings The sine rule The cosine rule The area of a triangle Problem solving using trigonometry Compass radial surveys

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4A 4B 4C 4D 4E 4F 4G 4H

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Applies trigonometry to solve problems involving right-angled and non-right-angled triangles (MST-12-S2-04). © NSW Education Standards Authority

Online resources

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Chapter 4 Trigonometry

4A Trigonometry in right-angled triangles Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To use Pythagoras’ theorem to find unknown sides in right-angled triangles • To use trigonometric ratios to find unknown sides and angles in right-angled triangles • To solve problems using trigonometric ratios in one or more right-angled triangle Past, present and future learning

• Students should already be familiar with the contents of this section • Questions involving right-angled triangles regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The formula for Pythagoras’ theorem does not appear on the 2027 version, but the three trigonometric ratios are provided.

Trigonometric ratios are defined using the sides of a right-angled triangle. The hypotenuse is opposite the right angle, the opposite side is opposite the angle θ and the adjacent side is the side next to θ.

Hypotenuse

Opposite

Adjacent

The trigonometric ratios sin θ, cos θ and tan θ are defined using the sides of a right-angled triangle. opposite adjacent opposite sin θ = cos θ = tan θ = hypotenuse hypotenuse adjacent

O A O (SOH) cos θ = (CAH) tan θ = (TOA) H H A In trigonometry an angle is usually measured in degrees, minutes and seconds such as 31◦ 05′ 51′′ (31 degrees 5 minutes 51 seconds). It is important to remember that one degree is equal to 60 minutes (1◦ = 60′ ) and one minute is equal to 60 seconds (1′ = 60′′ ). Calculators may also measure angles in decimal degrees such as 31.0975, and a button like or DMS is provided to convert between decimal degrees and degrees/minutes/seconds. Make sure that the degree mode (not radian mode) is selected on your calculator. sin θ =

Right-angled trigonometry

The mnemonic ‘SOH CAH TOA’ is pronounced as a single word. SOH: Sine – Opposite – Hypotenuse

CAH: Cosine – Adjacent – Hypotenuse

TOA: Tangent – Opposite – Adjacent

The order of the letters matches the ratio of the sides.

Pythagoras’ theorem

Pythagoras’ theorem links the sides of a right-angled triangle. In a right-angled triangle the side opposite the right angle is called the hypotenuse. The hypotenuse is always the longest side.

Hypotenuse (longest side)

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4A Trigonometry in right-angled triangles

Pythagoras’ theorem Pythagoras’ theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. (Hypotenuse) = (side) + (other side) 2

h =a +b 2

U N SA C O M R PL R E EC PA T E G D ES

Pythagoras’ theorem is used to find a missing side of a right-angled triangle if two of the sides are given. It can also be used to prove that a triangle is right angled.

Rounding angles to the nearest minute

In trigonometry, an angle is usually measured in degrees, minutes and seconds. Make sure the calculator is set up to accept angles in degrees. D or DEG should be displayed on your calculator.

Degrees

Minutes

1 degree = 60 minutes

1 minute = 60 seconds

1◦ = 60′

1′ = 60′′

Convert decimal degrees to degrees and minutes using

or DMS on your calculator.

Write the answer correct to the nearest minute.

Note that when rounding angles to the nearest minute, since there are 60 seconds in a minute, 30 seconds or more will round up. For example, 34◦ 12′ 32′′ is rounded to 34◦ 13′ .

Example 1

Finding an unknown side in a right-angled triangle (numerator)

Find the length of the unknown side x in the triangle shown. Answer correct to three decimal places.

35° 20

Solution

Explanation

The opposite side is the unknown, the adjacent side is 20. O tan θ = A x tan 35◦ = 20

Identify the sides of the right-angled triangle that are relevant to the question. Determine the ratio that uses these sides (TOA).

20 × tan 35◦ = x

To make x the subject, multiply both sides of the equation by 20.

x = 20 × tan 35◦

Substitute the known values and x for O.

= 14.00415076 . . .

Press 20 tan 35 exe or = .

≈ 14.004

Write the answer correct to three decimal places.

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Chapter 4 Trigonometry

Now you try Find the length of the unknown side x in the triangles shown. Answer correct to one decimal place. a

b 25

34°

U N SA C O M R PL R E EC PA T E G D ES

42°

Example 2

Finding an unknown side in the denominator

Find the length of the unknown side x in the triangle shown. Answer correct to two decimal places.

40°

Solution

Explanation

Adjacent = 12, hypotenuse = x

Identify the relevant sides of the right-angled triangle.

cos θ =

A H

cos 40◦ =

12 x

Determine the ratio that uses these sides. Substitute the known values.

Multiply both sides of the equation by x.

x × cos 40◦ = 12 x=

12 cos 40◦

Divide both sides by cos 40◦ .

Press 12 ÷ cos 40 = or exe .

x = 15.66488747 . . .

Write the answer correct to two decimal places.

≈ 15.66

Now you try

Find the length of the unknown side x in the triangles shown. Answer correct to one decimal place. a

63°

12.5°

Example 3

Finding an unknown angle in a right-angled triangle

Find the angle θ in the triangle shown. Answer correct to the nearest minute.

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4A Trigonometry in right-angled triangles

Solution

Explanation

Opposite = 13, hypotenuse = 22 O sin θ = H 13 sin θ = 22 θ = 36.22154662 . . .

Identify the relevant sides of the right-angled triangle. Determine the ratio that uses these sides. Substitute the known values.

U N SA C O M R PL R E EC PA T E G D ES

Look for the sin−1 !button on your calculator. 13 Evaluate sin−1 on your calculator. 22

θ ≈ 36◦ 13′

Convert decimal degrees to degrees and minutes using

DMS or your calculator. Write the answer correct to the nearest minute.

Now you try

Find the angle θ in the triangle shown. Answer correct to the nearest minute.

Example 4

Use Pythagoras’ theorem to find the length of a side in a right-angled triangle

Use Pythagoras’ theorem to find the value of x. a h cm 5 cm 9 cm

x mm

5 mm

12 mm

Solution

Explanation

a h2 = a2 + b2

Write Pythagoras’ theorem.

= 92 + 52 √ h = 92 + 52

≈ 10.30 cm

Substitute the length of the sides. Take the square root to find h.

Express the answer to correct to one decimal place.

h2 = a2 + b2

Write Pythagoras’ theorem.

122 = x2 + 52

Substitute the length of the sides.

x2 = 122 − 52 √ x = 122 − 52

Make x2 the subject.

≈ 10.9 mm

Take the square root to find x.

Express the answer to correct one decimal place.

Now you try Use Pythagoras’ theorem to find the value of x. Answer correct to two decimal places. a b x mm h mm

13 mm

20 mm

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Chapter 4 Trigonometry

Example 5

Application requiring the length of a side

63° 2.5 m

Peg

U N SA C O M R PL R E EC PA T E G D ES

A vertical tent pole is supported by a rope tied to the top of the pole and to a peg on the level ground. The peg is 2.5 m from the base of the pole and the rope makes an angle of 63◦ to the horizontal. What is the length of the rope between the peg and the top of the tent pole? Answer correct to two decimal places.

Solution

Explanation

Adjacent is 2.5 m, hypotenuse is the unknown x. A cos θ = H 2.5 cos 63◦ = x

Identify the unknown side and call it x, and identify the relevant sides of the right-angled triangle. Determine the ratio that uses these sides (CAH). Substitute the known values, and x for h.

x cos 63◦ = 2.5 2.5 x= cos 63◦ = 5.506723161 . . .

Multiply both sides of the equation by x. Divide both sides of the equation by cos 63◦ , so x is now the subject of the equation. Press 2.5 ÷ cos 63 exe or = . Write the answer correct to two decimal places.

≈ 5.51

Write the answer in words.

Length of the rope is 5.51 m.

Now you try

A pole is supported by a wire that runs from the top of the pole to a point on the level ground 5 m from the base of the pole. The wire makes angle of 42◦ with the ground. Find the height of the pole, correct to two decimal places.

42° 5m

Exercise 4A 1

Find the value of the following trigonometric ratios, correct to two decimal places. Your calculator must be in degree mode and needs to have a degrees/minutes/seconds button which you press after entering the degrees, so you can then enter the minutes. a sin 33◦ e cos 42◦ 15′

BUILDING

b cos 70◦ f tan 55◦ 28′

c 2 tan 51◦ g 2 cos 62◦ 59′

d 7 sin 24◦ h 8 sin 15◦ 42′

Given the following trigonometric ratios, find the value of θ to the nearest minute. Remember the sin−1 , cos−1 and tan−1 buttons give angles from the trigonometric ratio. Use the DMS button on the calculator to round to the nearest minute.

a sin θ = 0.4712 5 d tan θ = 8 97 g sin θ = 150

b cos θ = 0.3412 1 e sin θ = 2 61 h cos θ = 83

c tan θ = 0.3682 1 f cos θ = 4

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4A Trigonometry in right-angled triangles

Example 1

Find the length of the unknown side x in each triangle, correct to two decimal places. a

14 28°

c x

43°

28°

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35°12′

Example 2

75 33°58′

24°5′ 40

Find the length of the unknown side x in each triangle, correct to two decimal places.

21 47°

66°

39° x

47°

70°

44°

Example 3

Find the unknown angle θ in each triangle. Answer correct to the nearest minute.

7.8

4.2

5.2

65.1

θ 38.9

6.8

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Chapter 4 Trigonometry

Example 4

Using Pythagoras’ theorem, find the length of the unknown side. Answer correct to two decimal places. a

b 6 cm

5 cm

h cm 24 mm

12 cm

h mm

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h cm

8 cm

10 mm

2.3 cm

14.1 cm

x cm

24 mm

32 mm

9.5 cm

4.8 cm

x mm

DEVELOPING

7 a A rectangular block of land measures 25 m by 50 m. John wants to put a fence along the diagonal. How long will the fence be? Answer correct to three decimal places.

50 m

25 m

b The base of a ladder leaning against a wall is 1.5 m from the base of the wall. If the ladder is 4.5 m long, find how high the top of the ladder is from the ground, correct to one decimal place.

4.5 m

1.5 m

c The length of the diagonal of a square table is 150 cm. What is the side length of the table? Answer correct to one decimal place.

150 cm

Example 5

A balloon is tied to a string that is 25 m long. The other end of the string is secured by a peg to the surface of a level sports field. The wind blows so that the string forms a straight line making an angle of 37◦ with the ground. Find the height of the balloon above the ground. Answer correct to one decimal place.

25 m 37°

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4A Trigonometry in right-angled triangles

Ann noticed a tree was directly opposite her on the far bank of the river. After she walked 50 m along the side of the river, she found her line of sight to the tree made an angle of 39◦ with the river bank. Find the breadth of the river, to the nearest metre.

Tree

39° 50 m

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Ann

A ship at anchor requires 70 m of anchor chain. If the chain is inclined at 35◦ to the horizontal, find the depth of the water, correct to one decimal place.

Ship

70 m 35°

A pole casts a shadow of 5.4 m long. The sun’s rays make an angle of 36◦ with the level ground. Find the height of the pole to the nearest metre.

Sun

36° 5.4 m

A vertical tent pole is supported by a rope of length 3.6 m tied to the top of the pole and to a peg on the ground. The pole is 2 m in height. Find the angle the rope makes to the horizontal. Answer correct to the nearest degree.

3.6 m

A 3 m ladder has its foot 1.5 m out from the base of a wall. What angle does the ladder make with the ground? Answer correct to the nearest degree.

An archer, 80 m from a target and level with it, aims 2 m above the bullseye and hits it. What is the angle, to the nearest minute, that the arrow is inclined to the line of sight from the archer’s eye to the target?

CHALLENGING

A rope needs to be fixed with one end attached to the top of a 6 m vertical pole and the other end pegged at an angle of 65◦ with the level ground. Find the required length of rope. Answer correct to one decimal place.

A wheelchair ramp is being provided to allow access to first floor shops. The first floor is 3 m above the ground floor. The ramp requires an angle of 20◦ with the horizontal. How long will the ramp be? Answer correct to two decimal places.

A plane maintains a flight path of 19◦ with the horizontal after it takes off. It travels for 4 km along the flight path. Find, correct to one decimal place:

a the horizontal distance of the plane from its take-off point b the height of the plane above ground level.

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Two ladders are the same distance up a wall. The shorter ladder is 5 m long and makes an angle of 50◦ with the ground. The longer ladder is 7 m long. Find:

a the distance, correct to two decimal places, from the ground to where the ladders are on the wall b the angle, to the nearest degree, the longer ladder makes with the ground.

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50°

A pole is supported by a wire that runs from the top of the pole to a point on the level ground 7.2 m from the base of the pole. The height of the pole is 5.6 m. Find the angle, to the nearest degree, that the wire makes with the ground.

Communicate your reasoning, understanding and fluency

20 a Prove that ∆ABC is a right-angled triangle. b Calculate the size of the angle ABC correct to the nearest minute.

40 cm

41 cm

9 cm

Find θ to the nearest minute.

18 m

27 m

12 m

An aerial AB is tied down with two supporting wires CF and DE. Point D is halfway between C and B. BE is 25 metres and the smaller supporting wire, DE, makes an angle of 58◦ with the ground, as shown. Calculate the length of the supporting wire CF, correct to one decimal place.

58° F

E 25 m

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4B Angles of elevation and depression

4B Angles of elevation and depression Learning intentions

• To understand the difference between angles of elevation and depression • To solve problems involving angles of elevation and depression

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Students should already be familiar with the contents of this section • Questions involving angles of elevation and depression regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The three trigonometric ratios are provided on the 2027 version.

The angle of elevation is the angle measured upwards from the horizontal. The angle of depression is the angle measured downwards from the horizontal.

Angle of elevation

Angle of depression

Angle of elevation

Horizontal θ Angle of depression

θ Horizontal

Angle of elevation = angle of depression

They form alternate angles between two parallel lines. This information is useful to solve some problems. Angle of depression θ

Example 6

Angle of elevation

A park ranger measured the top of a plume of volcanic ash to be at an angle of elevation of 41◦ . From her map she noted that the volcano was 7 km away. Calculate the height of the plume of volcanic ash. Answer correct to two decimal places.

Volcanic plume

41° 7 km

Continued on next page

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Chapter 4 Trigonometry

Solution

Explanation Draw a diagram and label the required height as x. x

U N SA C O M R PL R E EC PA T E G D ES

41° 7 km

O A

Determine the ratio (TOA).

x 7

Substitute the known values.

tan θ =

tan 41◦ =

7 × tan 41◦ = x

Multiply both sides of the equation by 7.

x = 7 × tan 41

◦

x = 6.085007165 . . .

Write the answer correct to two decimal places.

≈ 6.09

The height of the volcanic plume was 6.09 km.

Write the answer in words.

Now you try

a Ti stands 25 m away from the base of a tall building and measures the angle of elevation to the top of the building as 72◦ . Find the height of the building, correct to the nearest metre.

72°

25 m

b A 22 m high tree casts a long shadow. Find the length of the shadow if the angle of elevation from the end of the shadow to the top of the tree is 15◦ . Round your answer to the nearest metre.

Example 7

20 m

15°

Finding a distance using angle of depression

The top of a cliff is 85 m above sea level. Minh saw a tall ship. He estimated the angle of depression to be 17◦ .

a How far was the ship from the base of the cliff? Answer to the nearest metre. b How far is the ship in a straight line from the top of the cliff? Answer to the nearest metre.

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4B Angles of elevation and depression

Solution

Explanation

17° y

85 m

Draw a diagram and label the distance to the base of the cliff as x and the distance to the top of the cliff as y.

17°

U N SA C O M R PL R E EC PA T E G D ES

tan θ =

O A

Determine the ratio (TOA).

85 x x × tan 17◦ = 85 85 x= tan 17◦ 85 x= tan 17◦ tan 17◦ =

Substitute the known values.

Multiply both sides of the equation by x. Divide both sides by tan 17◦ .

= 278.022 . . .

≈ 278 m ∴ The ship is 278 metres from the base of the cliff.

Write the answer correct to nearest metre. Write the answer in words.

O H

Determine the ratio (SOH).

85 y ◦ y × sin 17 = 85 85 y= sin 17◦

Substitute the known values.

b sin θ =

sin 17◦ =

Multiply both sides of the equation by y. Divide both sides by sin 17◦ .

= 290.7258 . . .

≈ 291 m ∴ The ship is 291 metres from the top of the cliff.

Write the answer correct to nearest metre. Write the answer in words.

Now you try

A iron ore seam of 120 m slopes down at an angle of depression from the horizontal of 38◦ . The mine engineer wishes to sink a vertical shaft, x, as shown. What is the depth of the required vertical shaft? Answer correct to the nearest metre. x

38°

120 m

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Chapter 4 Trigonometry

Example 8

Finding angles of elevation and depression

Shobna stands on the observation deck of the Sydney Tower Eye, which is 250 metres above the ground. From this vantage point, she spots the Sydney Opera House, located 1.6 km away. θ

FPO

U N SA C O M R PL R E EC PA T E G D ES

250 m

1.6 km

What is the angle of depression from the observation deck to the Opera House, to the nearest minute? Solution

Explanation

250 m

θ 1600 m

O A 250 tan θ = 1600

tan θ =

θ = 8.880 . . .

θ ≈ 8◦ 53′

Angle of depression = angle of elevation (alternate angles in parallel lines). Therefore, find the angle of elevation instead. Convert 1.6 km to 1600 m. Determine the ratio (TOA).

Substitute the known values, using angle of elevation as the unknown angle.

Look for the tan−1 button ! on your calculator. 250 Evaluate tan−1 on your calculator. 1600 Convert decimal degrees to degrees and minutes using or DMS on your calculator. Write the answer correct to the nearest minute.

Now you try

A helicopter is flying 850 m above sea level. It is also 1162 m in a straight line to a ship at sea level. What is the angle of depression from the helicopter to the ship? Answer correct to the nearest degree.

850 m

Exercise 4B

Example 6

1162 m

BUILDING

Arjun walked 400 m away from the base of a tall building, on level ground. He measured the angle of elevation to the top of the building to be 62◦ . Find the height of the building. Answer correct to the nearest metre.

62° 400 m

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4B Angles of elevation and depression

The angle of depression from the top of a TV tower to a satellite dish near its base is 59◦ . The dish is 70 m from the centre of the tower’s base on flat land. Find the height of the tower. Answer correct to one decimal place.

59° x 70 m

When Tasneem looked from the top of a cliff 50 m high, she noticed a boat at an angle of depression of 25◦ . How far was the boat from the base of the cliff? Answer correct to two decimal places.

25°

U N SA C O M R PL R E EC PA T E G D ES

Example 7

The pilot of an aeroplane saw an airport at sea level at an angle of depression of 13◦ . His altimeter showed that the aeroplane was at a height of 4000 m. Find the horizontal distance of the aeroplane from the airport. Answer correct to the nearest metre.

50 m

13°

4000 m

The angle of elevation to the top of a tree is 51◦ at a distance of 45 m from the point on level ground directly below the top of the tree. What is the height of the tree? Answer correct to one decimal place.

51° 45 m

Jack measures the angle of elevation to the top of a tree from a point on level ground as 35◦ . What is the height of the tree if Jack is 50 m from the base of the tree? Answer to the nearest metre.

35° 50 m

Example 8

A tourist viewing Sydney Harbour from a building 130 m above sea level observes a ferry 800 m from the base of the building. Find the angle of depression. Answer correct to the nearest degree.

130 m

800 m

What would be the angle of elevation to the top of a radio transmitting tower 130 m tall and 300 m from the observer? Answer correct to the nearest degree.

130 m

300 m

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Jarli observes the top of a tree at a distance of 60 m from the base of the tree. The tree is 40 m high. What is the angle of elevation to the top of the tree? Answer correct to the nearest degree.

40 m θ 60 m

A town is 12 km from the base of a mountain. The town is also a distance of 12.011 km in a straight line to the top of the mountain. What is the angle of depression from the top of a mountain to the town? Answer correct to the nearest tenth of a degree.

U N SA C O M R PL R E EC PA T E G D ES

12.011 km

12 km

Find, to the nearest degree, the angle of elevation of a railway line that rises 7 m for every 150 m along the track.

150 m

The distance from the base of the tree is 42 m. The tree is 28 m in height. What is the angle of elevation measured from ground level to the top of a tree? Answer correct to the nearest degree.

28 m

42 m

The angle of elevation to the top of a tree from a point A on the ground is 25◦ . The point A is 22 m from the base of the tree. Find the height of the tree. Answer correct to nearest metre.

A plane is 460 m directly above one end of a 1200 m runway. Find the angle of depression to the far end of the runway. Answer correct to the nearest minute.

DEVELOPING

A rocket launching pad casts a shadow 35 m long when the angle of elevation of the sun is 52◦ . How high is the top of the launching pad? Answer correct to nearest metre.

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4B Angles of elevation and depression

Ishaan is on the top of a 65 m high cliff. He observes a man swimming out to sea at an angle of depression of 51◦ . Ishaan also sees a boat out to sea at an angle of depression of 30◦ . Find, to the nearest metre, the distance:

51°

30°

65 m

x y

U N SA C O M R PL R E EC PA T E G D ES

a x of the man from the base of the cliff b y of the boat from the base of the cliff c from the man to the boat.

CHALLENGING

A lighthouse stands on the top of a cliff. A sailing boat is located 75 m from the base of the cliff. The angle of elevation to the top of the cliff and the top of the lighthouse is 42◦ and 50◦ respectively. Find the height of the lighthouse, correct to the nearest metre.

75 m

A plane is flying at an altitude of 880 m. Kayla is standing on the ground. She observes the angle of elevation to the plane as 67◦ 40′ and 25 seconds later the angle of elevation had changed to 24◦ 30′ .

880 m

a How far had the plane flown in that time? Answer to the nearest metre. b What is the speed (to the nearest km/h) of the plane?

Communicate your reasoning, understanding and fluency 19

A communication tower is located on the top of a hill. The angle of elevation to the top of the hill from an observer 2 km away from the base of the hill is 6◦ . The angle of elevation to the top of the tower from the observer is 8◦ . Find, to the nearest metre, the height of the:

a hill b hill and the tower c tower.

The angle of elevation from a boat out to sea to the top of a 350 m cliff is 13◦ . After the boat travels directly towards the cliff, the angle of elevation from the boat is 19◦ . How far did the boat travel towards the cliff? Answer correct to the nearest metre.

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4C True bearings and compass bearings Learning intentions

• To understand true bearings and compass bearings • To solve problems involving true bearings and compass bearings

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • Questions involving bearings regularly appear in HSC Examinations.

A bearing is the direction one object is from another object or an observer or a fixed point. There are two types of bearings: compass bearings and true bearings.

Compass bearings

Compass bearings use the four directions of the compass: north, east, south and west (N, E, S and W). The NS line is vertical and the EW line is horizontal. In between these directions are another four directions: north-east, south-east, south-west and north-west (NE, SE, SW and NW). Each of these directions makes an angle of 45◦ with the NS and EW lines.

A direction is given using a compass bearing by stating the angle either side of north or south. For example, a compass bearing of S50◦ W is found by measuring an angle of 50◦ from the south direction towards the west side.

Example 9

Understanding a compass bearing

a Find the compass bearing of: i A from O

30°

b Draw the compass bearing: i S40◦ W iii S10◦ E

ii B from O.

ii N40◦ W iv N52◦ E.

120°

Solution

Explanation

a i The compass bearing of A from O is N30◦ E

A compass bearing is written as starting from north or south, angle less than 90◦ , going towards the east or west. A from O is in the NE quadrant so it is north 30◦ going east.

ii The compass bearing of B from O is S60◦ E

B from O is in the SE quadrant so it is south 180◦ − 120◦ = 60◦ going east.

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4C True bearings and compass bearings

i S40◦ W is in the SW quadrant, south 40◦ towards the west. ii N40◦ W is in the NW quadrant, north 40◦ towards the west. iii S10◦ E is in the SE quadrant, south 10◦ towards the east. iv N52◦ E is in the NE quadrant, north 52◦ towards the east.

N N40°W

N52°E 40° 52° E

W 10°

U N SA C O M R PL R E EC PA T E G D ES

40°

S40°W

S S10°E

Now you try

a Find the compass bearing of:

i A from O ii B from O.

30°

b Draw the compass bearing:

i S65◦ W ii N65◦ W iii S35◦ E iv N27◦ E.

15° A

True bearings

A true bearing is the angle measured clockwise from north around to the required direction, and it is written with the letter T after the degree or minutes or seconds symbol. True bearings are sometimes called three-figure bearings because they are written using three numbers or figures. For example, 120◦ T is the direction measured 120◦ clockwise from north. It is the same bearing as S60◦ E.

Clockwise 120° from north

The smallest true bearing is 000◦ T and the largest true bearing is 360◦ T. The eight directions of the compass have the following true bearings:

315°T NW

360°T 000°T N

W 270°T

SW 225°T

045°T NE

E 090°T

S 180°T

SE 135°T

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The bearings in the following diagrams are given using both methods. N

N P 30° 170°

Compass bearing E S10°E True bearing 170°T

Compass bearing E N30°W True bearing 330°T

W 330°

U N SA C O M R PL R E EC PA T E G D ES

10°

Compass bearing

True bearing

A direction given by stating the angle either side of north or south, such as S60◦ E.

A direction given by measuring the angle clockwise from north to the required direction, such as 120◦ T.

Example 10

Understanding a true bearing

a Find the true bearing of:

i B from O ii C from O iii D from O.

50°

60°

b Draw each true bearing on a separate diagram. i 025◦ T ii 140◦ T iii 290◦ T

210°

Solution

Explanation

a i The bearing of B from O is 050◦ T

Measure the angle clockwise from north.

ii The bearing of C from O is 210◦ T

Measure the angle clockwise from north.

iii The bearing of D from O is 330◦ T

Measure the angle clockwise from north. 270◦ is due west + 60◦ further = 330◦

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4C True bearings and compass bearings

Draw angles clockwise from north.

N 025°T

U N SA C O M R PL R E EC PA T E G D ES

25°

ii N

140°

140°T

iii

290°T

290°

Now you try

a Find the true bearing of: i B from O

ii C from O

iii D from O.

b Draw each true bearing on a separate diagram. i 065◦ T

ii 170◦ T

iii 310◦ T

70°

25°

230°

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Solving problems using bearings When solving bearing problems involving two or more points, we can draw imaginary vertical lines extending north from each point. These north running lines are always parallel. Using the rules of angles formed by parallel lines, we can determine other angles within the diagram. N

U N SA C O M R PL R E EC PA T E G D ES

Cointerior angles in parallel lines add to 180◦ .

Example 11

Alternate angles in parallel lines are equal.

Corresponding angles in parallel lines are equal.

Solving problems using bearings

a If the bearing of P from O is 110◦ , what is the bearing of O from P? N

110°

b If the bearing of Q from O is 230◦ , what is the bearing of O from Q? N

230°

Solution

Explanation

Draw the north/south line at point P.

Use cointerior angles: 180◦ − 110◦ = 70◦

110°

70°

To find the bearing of O from P : 360◦ − 70◦ = 290◦

The bearing of O from P is 290◦ T

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4C True bearings and compass bearings

Draw the north/south line at Point Q. Use alternate angles: 230◦ − 180◦ south bearing = 50◦ Alternate angles are equal.

N O

230°

50°

U N SA C O M R PL R E EC PA T E G D ES

50° Q

The bearing of O from Q is 050◦ T

Now you try

Draw a diagram to answer the following.

a If the bearing of P from O is 120◦ , what is the bearing of O from P? b If the bearing of Q from O is 250◦ , what is the bearing of O from Q?

Example 12

Using bearings and trigonometry to calculate distance

Cassie runs 12 km on a bearing of 210◦ . How far south has she run? Answer correct to one decimal place.

Solution

Explanation

210°

Draw a diagram and form a right-angled triangle. s is for how far south she ran. The angle is found 210◦ − 180◦ S bearing = 30◦

12 km

30° s

cos θ =

A H

cos 30◦ =

Determine the ratio that uses these sides.

s 12

s = 12 × cos 30◦ s = 10.3923 . . . s ≈ 10.4 She ran 10.4 km south.

Substitute the known values.

Multiply both sides by 12. Round answer to one decimal place.

Now you try Finn runs 15 km on a bearing of 310◦ . How far north has he run? Answer correct to one decimal place.

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Exercise 4C 1

BUILDING

State the compass bearing and the true bearing of each of the following directions. a NE

Example 9, 10

b NW

c SE

d SW

State the compass bearing and the true bearing of each of the following directions. a

U N SA C O M R PL R E EC PA T E G D ES

52°

63°

55°

33°

20°

35°

g N

i N

59°

78°

98°

240°

283°

250°

Sketch each of these bearings on a separate diagram. a N10◦ E e 300◦ T

Example 11

b S25◦ W f 105◦ T

c N60◦ W g 219◦ T

d S42◦ E h 050◦ T

For each of these questions, draw a diagram and hence find the bearing.

a If the bearing of P from O is 065◦ , what is the bearing of O from P? b If the bearing of Q from O is 190◦ , what is the bearing of O from Q? c If the bearing of R from O is 155◦ , what is the bearing of O from R? d If the bearing of S from O is 310◦ , what is the bearing of O from S ?

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4C True bearings and compass bearings

DEVELOPING 5

A plane is travelling on a true bearing of 030◦ from A to B.

U N SA C O M R PL R E EC PA T E G D ES

a What is the compass bearing of A to B? b What is the true bearing of B to A? c What is the compass bearing of B to A?

The diagram shows the position of P, Q and R relative to S . In the diagram, R is NE of S , Q is NW of S and ∠PSR is 155◦ .

a What is the true bearing of R from S? b What is the true bearing of Q from S? c What is the true bearing of P from S? d What is the true bearing of S from R? e What is the true bearing of S from P? f What is the true bearing of S from Q?

155°

The bearing of E from D is N38◦ E, F is east of D and ∠DEF is 87◦ .

a Find the values of x and y. b What is the compass bearing of E from F? c What is the true bearing of E from F?

38° x°

87°

y°

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Hunter travels from X to Y for 125 km on a bearing of N32◦ W.

N Y

a How far did Hunter travel due north, to the nearest kilometre? b How far did Hunter travel due west, to the nearest kilometre?

32° W

U N SA C O M R PL R E EC PA T E G D ES

Diya cycled for 15 km west and then 24 km south.

a What is the value of θ correct to the nearest degree? b What is Diya’s true bearing from her starting point? c What is Diya’s compass bearing from her starting point?

24 km

Example 12

15 km θ

10 A boat sails 137 km from Port Stephens on a bearing of 065◦ T.

a How far east has the boat sailed? Answer correct to one decimal place. b How far north has the boat sailed? Answer correct to one decimal place.

CHALLENGING

A ship sails 5 kilometres west, then 5 kilometres south.

a What is the compass bearing of the ship from its original position? b What is the true bearing of the ship from its original position?

Lewis travelled for 8.5 km on a bearing of S30◦ W from his home.

a How far west is Lewis from home? Answer correct to two decimal places. b How far south is Lewis from home? Answer correct to two decimal places. c What is the compass bearing of his home from his current position?

A cyclist rides 15 km on a bearing of 290◦ .

a How far north is the cyclist? Answer correct to one decimal place. b How far west is the cyclist? Answer correct to one decimal place. c What is the true bearing of the cyclist’s starting position from the cyclist’s current position?

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4C True bearings and compass bearings

Isla drove 6 km east and then drove a further 13 km on a bearing of 070◦ T. How far east is Isla from her starting position?

Siya walked for 3 hours and 45 minutes at 6 km/h on a bearing of 145◦ to get to her friend Radha’s house.

U N SA C O M R PL R E EC PA T E G D ES

a How far south has she walked? b What is the true bearing of Siya’s starting point from Radha’s house?

Communicate your reasoning, understanding and fluency 16

Bradlee drives at a speed of 80 km/h on a bearing of 125◦ T for 2.5 hours.

a How far east is Bradlee from his starting position? Answer correct to the nearest kilometre. b What is the true bearing of Bradlee’s starting position from his current position?

A plane left O and travelled 350 km in the direction 225◦ T to reach P. It then changed direction and travelled 315◦ T to reach point Q which is due east of O. N

O 225°

P 315°

a Explain why angle QPO is 90◦ . b How far did the plane travel from P to Q?

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4D The sine rule Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the sine rule for non-right-angled triangles • To use the sine rule to find unknown sides and angles in non-right-angled triangles • To solve problems using the sine rule in one or more non-right-angled triangles • To understand why an unknown angle may be obtuse and learn how to determine the size of such an angle

Past, present and future learning

• The concepts in this section may be new to students • Questions involving non-right-angled triangles regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The sine rule is provided on the 2027 version.

Trigonometry is also applied to non-right-angled triangles. The sides of the triangle are named according to the opposite angle. • Side a is opposite angle A.

• Side b is opposite angle B.

• Side c is opposite angle C.

The sine rule relates the sides and angles in a triangle. It states that ‘Side a divided by the sine of angle A is equal to side b divided by the sine of angle B and is equal to side c divided by the sine of angle C’. This result is derived by constructing a perpendicular line of length h from C to D. h h In ∠ACD, sin A = In ∠BCD, sin B = a b h = a sin B h = b sin A The values of h must be equal to each other.

D c

a sin B = b sin A a b = sin A sin B

Similarly, by constructing a perpendicular line from B it can be shown that a c = sin A sin C

We can combine these two rules to obtain the result below.

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4D The sine rule

The sine rule Sine rule relates the sides and angles in a triangle.

a b c = = sin A sin B sin C sin A sin B sin C = = • To find an angle, use a b c

• To find a side, use

b c

U N SA C O M R PL R E EC PA T E G D ES

Sine rule is used in a non-right-angled triangle given:

• two sides and an angle opposite one of the given sides, or • two angles and one side.

Example 13

Using the sine rule to find an unknown side

Find the value of x, correct to one decimal place.

4.1

50°

37°

Solution x

Explanation

Check that the sides and angles are opposite each other.

4.1

50°

37°

a b = sin A sin B x 4.1 = ◦ sin 50 sin 37◦ 4.1 × sin 50◦ x= sin 37◦ = 5.218849806 . . .

Write the sine rule to find a side.

Substitute the known values (a = x, A = 50, b = 4.1 and B = 37). Multiply both sides of the equation by sin 50◦ . Evaluate.

Write the answer correct to one decimal place.

x ≈ 5.2

Now you try

Find the value of x correct to one decimal place. a

38°

6.2

55°

12.2

99° 50°

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Example 14

Using the sine rule to find an unknown angle

Zoe is standing 4.5 m from the base of a 3 m sloping wall. The angle of elevation to the top of the wall is 36◦ . Find the angle θ at the top of the wall, correct to the nearest minute. θ

U N SA C O M R PL R E EC PA T E G D ES

36°

4.5 m

Solution

Explanation

sin A sin B = a b

Check that the sides and angles are opposite each other.

sin θ sin 36◦ = 4.5 3 4.5 × sin 36◦ sin θ = 3

Write the sine rule to find an angle. Substitute the known values (a = 4.5, A = θ, b = 3 and B = 36). Multiply both sides of the equation by 4.5.

= 0.8816778784 . . .

Evaluate, using sin−1 on the calculator.

θ ≈ 61 51 ◦

′

Write the answer correct to the nearest minute.

The angle at the top of the wall is 61◦ 51′ .

Write the answer in words.

Now you try

Find the unknown angle θ, correct to the nearest minute. a

10 m

12.3 m

89°

47°

8.7 m

25 m

The ambiguous case for the sine rule

Investigation

Using a ruler, pencil and protractor, draw these triangles as accurately as you can. Triangle 1: ∆ABC with AB = 10 cm, ∠BAC = 30◦ , BC = 12 cm.

Did everybody draw the same triangle? Measure ∠ACB.

Triangles 2 and 3: ∆ABC with AB = 10 cm, ∠BAC = 30◦ , BC = 6 cm. This time, there are two different triangles with the same information.

Draw both triangles and measure ∠ACB in each. If you do that accurately you may notice something interesting about the two angles you just measured.

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4D The sine rule

The ambiguous case Since the sine of an angle is the same for both an acute angle and its supplementary obtuse angle, we may sometimes get two possible solutions for the angle.

10 cm

30° 124° 6 cm

6 cm

As in Triangle 2 and 3 in the investigation above, we can calculate angle C as θ = 56◦ , but another possible solution is 180◦ − 56◦ = 124◦ .

U N SA C O M R PL R E EC PA T E G D ES

56°

So sin 56◦ = sin 124◦ = 0.8290 (correct to 4 decimal places).

When you use the sine rule to find an unknown angle, your calculator will only give you the acute answer. If you need the obtuse answer, subtract it from 180◦ , then check that the angle sum of the triangle is less than 180◦ , because sometimes there is only one triangle with the given information (see Triangle 1 in the investigation).

Example 15

Using the sine rule to find the obtuse angle

Calculate the size of the angle θ given that it is an obtuse angle. Answer correct to the nearest minute.

42 m

12°

125 m

Solution

Explanation

sin B sin A = a b sin θ sin 12◦ = 125 42

Check that the sides and angles are opposite each other. Write the sine rule to find an angle. Substitute the known values.

sin θ =

sin 12◦ × 125 42

Multiply both sides by 125. Look for the sin−1 button on your calculator. ! sin 12 × 125 on your calculator. Evaluate sin−1 42 Convert decimal degrees to degrees and minutes using or DMS on your calculator. Write the answer correct to the nearest minute. For obtuse angles, sin θ = sin(180◦ − θ)

= 38.22744…

θ ≈ 38◦ 14′

But θ is obtuse: θ = 180◦ − 38◦ 14′ θ = 141◦ 46′ The obtuse angle θ is 141◦ 46′ . Now you try

Calculate the size of the angle θ given that it is an obtuse angle. Answer correct to the nearest minute. a

10°

38 m

7.5 cm

15° 25 cm 12 m

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Exercise 4D 1

BUILDING

For each triangle, state the lengths of sides a, b and c. a

U N SA C O M R PL R E EC PA T E G D ES

Example 13

Find the value of x, correct to two decimal places. 5 × sin 65◦ 10 × sin 76◦ a x= b x = sin 46◦ sin 25◦ x 8 x 4 d = e = sin 40◦ sin 62◦ sin 58◦ sin 78◦

3 × sin 100◦ sin 18◦ x 29 f = sin 21◦ sin 120◦

c x=

For each triangle, find the length of the unknown side x, correct to two decimal places. a

94°

48°

115°

45°

2.2

55°

27°

73°

29°

110°

49°

54°

Find the length of the unknown side x in each triangle, correct to two decimal places. a

7 60°23′

49°25′ x

94°5′

46°4′

64°

10 71°14′ 62°49′

Given the following trigonometric ratios, find the value of A to the nearest minute. 25 × sin 65◦ 12 × sin 71◦ a sin A = b sin A = 18 32 ◦ 24 × sin 103 sin A sin 53◦ 20′ c sin A = d = 40 50 65 ◦ ′ sin A sin 73 30 sin A sin 61◦ 46′ e = f = 187 236 6 7

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4D The sine rule

Example 14

Find the unknown angle θ. Answer correct to the nearest degree. a

71° 40

12 89°

U N SA C O M R PL R E EC PA T E G D ES

53°

56°

75°

12.7 37°

4.6

9.8

2.9

DEVELOPING

Find the unknown angle θ. Answer correct to the nearest minute. a

75°13′

54°2′ 19

Example 15

124°3′

8 Calculate the size of the angle θ, correct to the nearest minute, given that it is an obtuse angle. a A

28°

13 cm

14°

40 cm

7 cm

22 cm

12 cm

14°

9.3 cm

10°

38 cm

2.4 cm

18°

53 cm

125 cm

8°

32 cm

18 cm

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Find the value of the pronumeral in following questions, correct to two decimal places. a In triangle ABC, A = 56◦ , B = 75◦ and a = 15. Find b. b In triangle XYZ, X = 107◦ , Y = 45◦ and x = 30. Find y. c In triangle PQR, P = 118◦ , Q = 34◦ and p = 59.5. Find q.

10 Find the value of the pronumeral in following questions, correct to the nearest degree.

U N SA C O M R PL R E EC PA T E G D ES

a In triangle DEF, D = 46◦ , d = 10 and f = 5. Find angle F. b In triangle ABC, B = 102◦ , b = 18 and c = 13. Find angle C. c In triangle RS T , R = 68◦ , r = 15 and t = 8. Find angle T .

11 Find the length of the unknown side, x, in each triangle, correct to two decimal places. a

55° 4.8

84°

105°

51°

33°

57°

59°21′

1.2

85°21′

65°3′

40°59′

27°40′

32°29′

Sebastian is planning to build a triangular garden for his daughter. The vertices of the triangle are named PQR. He measured PQ as 2.7 m, QR as 3.1 m and ∠PRQ as 57◦ . Use the sine rule to find the size of ∠RPQ, correct to the nearest degree.

Triangle ABC has ∠ABC = 71◦ , ∠BCA = 69◦ and ∠CAB = 40◦ . The length of AB is 10.

a What is the value of a, correct to one decimal place? b What is the value of b, correct to one decimal place?

Triangle XYZ has sides YZ = 30 cm, XY = 24 cm and ∠Y XZ = 54◦ . Use the sine rule to find:

a the size of angle XYZ. Give your answer to the nearest degree b the size of y. Give your answer to the nearest centimetre.

24 cm

54°

30 cm

Find the length of the longest side of a triangle with angles of 42◦ , 55◦ and 83◦ , given that the length of the shortest side is 8.6 cm. Answer correct to one decimal place.

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4D The sine rule

CHALLENGING 16

Sienna was located at X and saw a fire in the direction N15◦ E. Seven kilometres to the east of X at Z, Dylan saw the fire in the direction N50◦ W.

a How far is X from the fire? Answer in kilometres, correct to one decimal place. b How far is Z from the fire? Answer in kilometres, correct to one decimal place.

Fire

U N SA C O M R PL R E EC PA T E G D ES

A man at R measures the angle of elevation to a plane at P as 45◦ 15′ . The plane travels 15 km from P to Q. The man then measures the angle of elevation to the plane at Q as 35◦ 42′ .

a What are the sizes of ∠PRQ, ∠PQR and ∠QPR? b What is the distance from R to P? Answer in kilometres, correct to one decimal place. c What is the distance from R to Q? Answer in kilometres, correct to one decimal place.

Oliver measured the angle of elevation to the top of the mountain as 28◦ . He moved 140 m closer to the mountain and measured the angle of elevation to the top of the mountain as 43◦ . How far in a straight line is Oliver from the top of the mountain at his new position? Answer correct to the nearest metre.

43°

28° 140 m

Communicate your reasoning, understanding and fluency 19

The captain of a ship, A, sighted the smoke of a volcanic island in the direction N53◦ E. A captain on another ship, B, 30 km due east of the first ship saw the smoke in the direction N42◦ W.

a How far is ship A from the volcano? Answer correct to one decimal place. b How far is ship B from the volcano? Answer correct to one decimal place. c The ship closer to the volcano is travelling at 20 km/h. How long will it take for this ship to reach the volcano? Answer correct to the nearest minute.

Smoke

42°

53°

30 km

Use the given diagram to answer the following questions. 120 sin 32◦ a Show that BC can be given by the expression . sin 43◦ b Show that h can be found by the expression 120 sin 32◦ × sin 75◦ . sin 43◦ c Calculate h correct to one decimal place.

75°

32° 120 m

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4E The cosine rule Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the cosine rule for non-right-angled triangles • To use the cosine rule to find unknown sides and angles in non-right-angled triangles • To solve problems using the cosine rule in one or more non-right-angled triangles Past, present and future learning

The cosine rule is another formula that relates the sides and angles in a triangle. It is applied to problems where three sides and one angle are involved. The cosine rule can be found by constructing a perpendicular line of length h from B to D. Let the distance from A to D be x and hence the distance from C to D is (b − x). In ∆ABD using Pythagoras’ theorem:

c2 = h2 + x2 ⇒ h2 = c2 − x2 (1)

x c x = c cos A . . . (2)

In ∆ABD, cos A =

b−x

Using Pythagoras’ Theorem in ∠BCD: a2 = h2 + (b − x)2

= h2 + b2 − 2bx + x2

= h2 + x2 + b2 − 2bx = c2 + b2 − 2bx

by substituting (1)

a2 = b2 + c2 − 2bc cos A

by substituting (2)

Similarly, by constructing perpendiculars from C and A, we can obtain the other two results shown below.

The cosine rule

To find the third side given two sides and the included angle in ∆ABC, a = b + c − 2bc cos A b2 = a2 + c2 − 2ac cos B c2 = a2 + b2 − 2ab cos C 2

(a is opposite ∠A) (b is opposite ∠B) (c is opposite ∠C)

To find an angle given three sides (rearrangements of the above formulas): cos A =

b2 + c2 − a2 2bc

cos B =

a2 + c2 − b2 2ac

cos C =

a2 + b2 − c2 2ab

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4E The cosine rule

Example 16

Using the cosine rule to find an unknown side

Find the value of x, correct to two decimal places.

A x

6 76° C

U N SA C O M R PL R E EC PA T E G D ES

Solution

Explanation

c2 = a2 + b2 − 2ab cos C

Write the cosine formula to find a side.

x2 = 122 + 62 − 2 × 12 × 6 × cos 76◦

Substitute the values for a, b, c and C.

x = 145.163247 x ≈ 12.05

Calculate the value of x2 . Take the square root of both sides. Write your answer to two decimal places.

Now you try

23 cm 58°

Find the value of x, correct to two decimal places.

27 cm

Example 17

Using the cosine rule to find an unknown angle

Find the value of the angle θ. Answer in degrees, correct to one decimal place.

Solution

cos X = cos θ =

y2 + z2 − x2 2yz 2 12 + 112 − 82 (2 × 12 × 11)

cos θ = 0.7613636364

θ = 40.41543902 . . .

Explanation

Write the cosine formula to find an angle. Substitute the values for x, y, z and X (x = 8, y = 12, z = 11 and X = θ).

Evaluate, using cos−1 on your calculator. Write the answer correct to one decimal place.

≈ 40.4◦

Now you try

Find the value of the angle θ, correct to the nearest minute.

Y 10 cm

15 cm

θ 13 cm

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Example 18

Solving a problem using the cosine rule

U N SA C O M R PL R E EC PA T E G D ES

Samuel shoots for goal when he is 2.5 m from one post and 3.2 m from the other post. The goal is 2 m wide. What is the size of the angle θ for Samuel to score a goal? Answer correct to the nearest minute. 2m

2.5 m

3.2 m

Solution

cos A = cos θ =

Explanation

b2 + c2 − a2 2bc 2 2.5 + 3.22 − 22

Write the cosine formula to find an angle.

Substitute the values for a, b, c and A (a = 2, b = 2.5, c = 3.2 and A = θ).

(2 × 2.5 × 3.2)

cos θ = 0.780625

Evaluate, using cos−1 on your calculator.

θ = 38.68216452 . . . ≈ 38◦ 41′

The size of the angle for Samuel to score a goal is 38◦ 41′ .

Use your calculator to convert decimal degrees to degrees and minutes. Write the answer in words.

Now you try

Theresa shoots for goal when she is 3.8 m from one post and 4 m from the other post. The goal is 2 m wide. What is the size of angle θ for Theresa to score a goal? Answer correct to the nearest minute.

3.8 m

Exercise 4E 1

BUILDING

In each triangle, write the cosine rule with x2 as the subject of the equation. a

70°

50°

45°

A x

17 B

62°

4 59°

40°

x C

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4E The cosine rule

Find the value of the x, correct to one decimal place. a x2 = 32 + 42 − 2 × 3 × 4 cos 39◦ c x2 = 22 + 52 − 2 × 2 × 5 cos 45◦ 15′

Example 16

b x2 = 72 + 102 − 2 × 7 × 10 cos 135◦ d x2 = 132 + 92 − 2 × 13 × 9 cos 101◦ 34′

Find the length of the unknown side x in each triangle, correct to two decimal places. a

B 44

U N SA C O M R PL R E EC PA T E G D ES

69°

50°

105°

113° 98

46°

58°

Find the length of the unknown side x in each triangle, correct to three decimal places. a

1.9

x 91°53′

78°18′

43°29′ 14

2.6

In the triangle ABC, a = 24, b = 22 and C = 42◦ . What is the value of side c? Answer correct to three significant figures.

Find the value of the angle. Answer in degrees correct to two decimal places. 72 + 92 − 62 162 + 202 − 242 a cos A = b cos A = 2×7×9 2 × 16 × 20 c cos A =

Example 17

4.92 + 3.42 − 5.72 2 × 4.9 × 3.4

d cos A =

32 + 42 − 62 2×3×4

Find the size of the unknown angle θ. Answer correct to the nearest degree. a

d 74

8.1

8.7 9.3

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In the triangle ABC, a = 10, b = 11 and c = 12. What are the sizes of the following angles? Answer correct to the nearest minute. a Angle A

c Angle C

In triangle XYZ, XY = 8 cm, YZ = 12 cm and XZ = 16 cm. What is the size of angle Y, correct to the nearest minute? Find the size of the unknown angle θ. Answer correct to the nearest minute.

U N SA C O M R PL R E EC PA T E G D ES

b Angle B

3.2

2.5

3.9

In triangle FGH, the length of GH is 14 cm, FH is 9 cm and FG is 8 cm. Find ∠HFG, correct to the nearest minute.

DEVELOPING

Example 18

DEF is a triangle for which DF = 37 cm, EF = 46 cm and ∠DFE = 44◦ . Use the cosine rule to find the length of DE, correct to the nearest millimetre.

A triangle RS T has ∠RS T = 51◦ , ∠S T R = 63◦ , RT = 40 and RS = 48.

a What is the size of ∠T RS ? b Find the length of S T using the cosine rule. Answer correct to three significant figures.

Imogen drives a four-wheel drive along a track from point A due west to a point B, a distance of 14 km. She then turns and travels 19 km to point C. Use the cosine rule to calculate the distance Imogen is from her starting point. Answer correct to one decimal place.

103°

Passengers in a car travelling east, along a road that runs east–west, see a castle 10 km away in the direction N65◦ E. When they have travelled a further 4 km east along the road, what will be the distance to the castle? Answer correct to two decimal places.

Castle

10 km

65°

E 4 km

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4E The cosine rule

An A-frame stepladder has legs of length 120 cm and the angle between them is 15◦ . Calculate the distance between the legs on the ground, correct to the nearest centimetre.

15°

U N SA C O M R PL R E EC PA T E G D ES

CHALLENGING 17

A triangle has sides measuring 4 m, 5 m and 7 m. What is the size of the largest angle in this triangle? Answer correct to the nearest minute.

Use the given diagram to answer these questions.

12 cm

50°

a Find the length of WY, correct to one decimal place. b Hence, find the length of WZ, correct to one decimal place.

48°

18 cm

Consider this diagram of 3 towns A, B and C and the distances between them. B is due south of A. What is the bearing of Town C from Town A correct to the nearest degree?

92 km

85 km

55 km

Communicate your reasoning, understanding and fluency 20

In the diagram, Condobolin (C), Parkes (P) and Dubbo (D) are locations of 3 New South Wales towns and the distances between them are shown. Parkes is due east of Condobolin and the bearing of Dubbo from Parkes is 030◦ .

120 km

a Show that the size of ∠CPD is 120◦ .

102 km

b Find the distance from Condobolin to Dubbo correct to the nearest km. c What is the bearing of Parkes from Dubbo?

A plane is flying 120 km from X to Y on a bearing of 070◦ . It then turns and flies 150 km to Z on a bearing of 145◦ .

a Show that the size of ∠XYZ is 105◦ . b How far is the plane from its starting point to the nearest km? c Calculate ∠Y XZ correct to the nearest degree. d What is the bearing of the plane from its starting point?

145°

120 km 70°

150 km

X Z

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4F The area of a triangle Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the area rule for non-right-angled triangles • To use the rule to find the area of triangles, given two sides and their included angle • To solve problems using the area rule in one or more non-right-angled triangles Past, present and future learning

The area of a triangle can be calculated in a triangle without a right angle. It requires that two sides and the included angle are known. The area of a triangle is half the product of two sides multiplied by the sine of the angle between the two sides (the included angle). This result can be demonstrated by constructing a perpendicular line of length h from B to D. Area of a triangle is calculated using the formula: 1 A = bh 2

h a h = a sin C

In ∆BCD, sin C =

D b

Substituting a sin C for h into A =

1 bh 2

1 ba sin C 2 1 = ab sin C 2

Similarly, by constructing perpendiculars from C and A, we can obtain the other two results shown below.

Area of a triangle

1 bc sin A C 2 Side 1 a Angle b A = ac sin B 2 Side B A 1 c A = ab sin C 2 Area of a triangle is half the product of two sides multiplied by the sine of the angle between the two sides (the included angle). A=

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4F The area of a triangle

Example 19

Finding the area of a triangle

Find the area of the triangle, correct to the nearest square centimetre.

B 20 cm C

36° A

U N SA C O M R PL R E EC PA T E G D ES

32 cm

Solution

Explanation

1 A = bc sin A 2 1 = × 32 × 20 × sin 36◦ 2

Write the formula. We are given two sides b, c and the angle A between these sides (the included angle). Substitute values for b, c and A. Evaluate. Write correct to the nearest square centimetre.

= 188.0912807 . . . ≈ 188 cm2

Now you try

Find the area of the triangle, correct to the nearest square centimetre.

12 cm

100°

10 cm

Exercise 4F

Example 19

BUILDING

Find the area of each triangle, correct to one decimal place. a

46° 19

53°

109°

41 cm

13 mm

31°

65°

69°

28 cm

16 mm

In triangle ABC, side a is 36 cm, side b is 48 cm and angle C is 68◦ . Find the area of the triangle. Answer correct to two decimal places.

In triangle XYZ, side x is 4 m, side y is 7 m and angle Z is 34◦ . Find the area of the triangle, correct to two decimal places.

Find the area of triangle DEF if DF = 5 cm, EF = 6 cm and ∠DFE = 40◦ . Answer correct to the nearest square centimetre.

D 5 cm E

40° 6 cm

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Find the area of each triangle, correct to one decimal place. a

b 19

27°

8.2 5.1

128°

56 64°

4.3

U N SA C O M R PL R E EC PA T E G D ES

60 cm

50°

73°

19 mm

23 mm

32 cm

2.5 m

2.4 m

101°

36 cm

1.6 m

25 mm

DEVELOPING

In triangle DEF, the length of DF is 21 cm, EF is 28 cm, ∠FDE is 64◦ and ∠DEF is 43◦ . Find the area of triangle DEF correct to the nearest square centimetre.

64°

21 cm

43° 28 cm

A parallelogram PQRS has PS = 4 cm, S R = 5 cm and ∠PSR = 40◦ .

a What is the area of triangle PRS ? Answer correct to one decimal place. b What is the area of parallelogram PQRS ? Answer correct to one decimal place.

A drawing of a farmer’s property is shown here. W

8 km

114°

5 km

84°

7 km

7.5 km

a What is the area of triangle WXZ? Answer correct to one decimal place. b What is the area of triangle XYZ? Answer correct to one decimal place.

c Find the total area of the property in square kilometres. Answer correct to one decimal place.

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4F The area of a triangle

CHALLENGING 9

The triangle ABC has an area of 243 m2 , BC = 30.1 m and ∠BCA = 54◦ 21′ . What is the length of AC in metres, correct to one decimal place?

30.1 m 54°21′

U N SA C O M R PL R E EC PA T E G D ES

A rhombus ABCD has a side length of 13 cm and ADC = 64◦ 35′ . What is the area of the rhombus? Answer correct to the nearest square centimetre.

A triangle has an angle of 129◦ 56′ with one arm of this angle 15 cm long. What is the length of the other arm of this angle if the area of the triangle is 69.94 cm2 ? Answer correct to the nearest centimetre.

Communicate your reasoning, understanding and fluency 12

Find the area of a regular hexagon with side length 10.5 cm. Answer correct to one decimal place.

∆XYZ has an area of 300 m2 . Find the value of θ correct to one decimal place.

10.5 cm

35 cm θ

Janet calculated the area of ∆ABC. Her working out is shown. 1 A = ab sin C 2 1 = × 15 × 18 × sin 73◦ 2 = 129.1 cm2

25.6 cm

19.8 cm

73°

18 cm

15 cm

Describe the error Janet made in her working and find the correct area of the triangle.

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4G Problem solving using trigonometry Learning intentions

• To use trigonometry to solve a variety of practical problems Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• The concepts in this section may be new to students • Questions involving trigonometry regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA.

Trigonometry is used to solve many practical problems. Use the steps below to solve them.

Solving a trigonometric word problem

1 Read the question and underline the key terms. 2 Draw a diagram and label the information from the question. 3 If the triangle is right-angled, use SOH CAH TOA. 4 If the triangle does not have a right angle: a use the sine rule if given two sides and two angles b use the cosine rule if given three sides and one angle. 5 Check that the answer is reasonable and units are correct.

Example 20

Solving problems involving non-right-angled triangles

The dimensions of a block of land are shown opposite.

a What is the length of x, correct to the nearest metre? b What is the length of y, correct to the nearest metre?

320 m

70°

210 m

38°

82° R y

Solution

Explanation

a Use the cosine rule in ∆PQS to find x.

Looking at ∆PQS , three sides (including x) and one angle are given and the side we want to find (x) is opposite the known angle. Determine which rule fits this situation.

p2 = q2 + s2 − 2qs cos P

Write the cosine rule to find a side, changing ABC to PQS to match the triangle in question.

x2 = 2102 + 3202 − 2 × 210 × 320 × cos 70◦

Substitute the values for p, q, s and P.

x2 = 100532.4927 . . . x = 317.068 x ≈ 317 m

Take the square root of both sides. Evaluate correct to the nearest metre.

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4G Problem solving using trigonometry

b Use the sine rule to find a side in ∆QRS to find y.

x y = sin Y sin X

Looking at ∆QRS , two sides (including y, and x is known from part a) and two angles are given, and the unknown side is opposite a known angle. Write out the rule that fits this situation.

U N SA C O M R PL R E EC PA T E G D ES

y 317.068 = ◦ sin 38 sin 82◦ y=

Substitute the known values.

317.068 × sin 38◦ sin 82◦

= 197.125328 . . . ≈ 197 m

Multiply both sides by sin 38◦ .

Evaluate correct to the nearest metre. (Note: Do not use the approximate value from part a as it may result in an error.)

Now you try

The dimensions of a block of land are shown in the diagram.

150 m

a What is the length of x, correct to the nearest metre? b What is the length of y, correct to the nearest metre?

25°

80°

92°

120 m

Example 21

Solving problems involving a triangle without a right angle

Sia measures the angle of elevation to the top of a wall as 32◦ . She walks 10 m horizontally towards the wall and measures the angle of elevation as 51◦ . Find the height of the wall. Answer correct to the nearest metre.

32° 10 m

51°

Solution

Explanation

51° 32° 10 m C C

129°

Draw a diagram consisting of two triangles. Label each diagram. To find the height of the wall h we need to calculate the value of a.

Continued on next page

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Use the sine rule in ∆ABC to find a. ∠ABC = 180◦ − (32◦ + 129◦ ) = 19◦

Write the sine rule.

U N SA C O M R PL R E EC PA T E G D ES

b a = sin A sin B a 10 = ◦ sin 32 sin 19◦ 10 × sin 32◦ a= sin 19◦ ≈ 16.276753 ≈ 16 m

Looking at ∆ABC there are two sides (including a) and two angles. Determine which rule to use. To calculate ∠ABC, use the angle sum of a triangle is 180◦ .

In ∆BCD, D is a right angle. Use a trigonometric ratio SOH to find h. O sin C = H h sin 51◦ = a ◦ a × sin 51 = h h = a × sin 51◦

= 12.64941286 . . .

≈ 13 m The wall is 13 m high.

Substitute the known values for b, A and B.

Multiply both sides of the equation by sin 32◦ . Evaluate.

Determine whether to use a trigonometric ratio or a rule to find h, and which one. Write the ratio for SOH. Substitute the known value of C.

Multiply both sides of the equation by a. Evaluate using the exact value of a.

Write the answer correct to the nearest metre. Write the answer in words.

Now you try

Mason measures the angle of elevation to the top of the wall as 25◦ .

He walks 15 m towards the wall and measures the angle of elevation as 48◦ . Find the height of the wall correct to the nearest metre.

48°

25° 15 m

Exercise 4G 1

BUILDING

The bearing of Y from X is 240◦ and the distance of Y from X is 20 km.

a What is the value of a? b If Z is 18 km due north of X, calculate the distance of Y from Z, correct to the nearest kilometre. c What is the value of b correct to the nearest degree?

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4G Problem solving using trigonometry

Seth plans to build a triangular flower bed in the lower part of a circular garden. The length of T S is 24 m, ∠T S R is 45◦ and ∆RT S is 40◦ . a What is the size of ∠S RT ? b Use the sine rule to find the length of T R. Answer correct to the nearest metre.

24 m

40°

45°

U N SA C O M R PL R E EC PA T E G D ES

Example 20

The diagram shows information about the locations of towns P, Q and R. Amber takes 2 hours and 30 minutes to walk directly from Town P to Town Q.

North

a What is Amber’s walking speed, correct to the nearest km/h? b What is the distance from P to R? Answer correct to the nearest kilometre. c How long would it take Amber to walk from P to R? Answer correct to the nearest minute.

108°

61°

XYZ represents a triangular area of land. The bearing of X from Y is 041◦ and the bearing of Z from Y is 305◦ . The distance XY is 27 m and the distance YZ is 37 m.

37 m

A ship is located at B and two radar stations 75 km apart are located at A and C (C is east of A). The bearing of the ship from one radar station is 038◦ and from the other radar station it is 306◦ .

North

38°

306°

A triangle DEG has a right angle at E and DG is 8 m in length. A line is drawn from D to F with ∠DFG = 144◦ and ∠GDF = 7◦ .

a Find the length of DF. Answer correct to two decimal places. b Find the size of ∠DFE. c Find the length of FE. Answer correct to two decimal places.

27 m

a How far is the ship from the radar station at A, correct to the nearest kilometre? b How far is the ship from the radar station at C, correct to the nearest kilometre? c What is the bearing of the radar station at C from the ship at B?

Example 21

R 17 km

M is 50 km north of O. The bearing of N from M is 108◦ and from O it is 061◦ . Answer the following questions correct to the nearest kilometre.

a What is the size of angle XYZ? Answer correct to the nearest degree. b Calculate the area of the land to the nearest square metre. c What is the length of the boundary XZ, to the nearest metre?

111°

19 km

a What is the distance between M and N? b What is the distance between N and O? c What is the bearing of M from N? d What is the bearing of O from N?

North

8m 7° 144° G

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Two ladders are the same distance up the wall. The shorter ladder is 4.9 m long and makes an angle of 48◦ with the ground. The longer ladder is 6.7 m long. a How far up the wall do the ladders reach? Answer correct to one decimal place. b What is the angle, correct to the nearest degree, that the longer ladder makes with the ground? c Find the distance between the ladders on the ground. Answer correct to one decimal place.

6.7 m

4.9 m

U N SA C O M R PL R E EC PA T E G D ES

48°

The angle of depression from A to D is 65◦ . The length of AB is 18 m and the length of BD is 16 m.

65°

18 m

a What is the size of ∠ADB? Answer correct to the nearest minute. b Calculate the angle of elevation from D to B. Give your answer correct to the nearest minute. c Find the length of AC, correct to the nearest metre.

16 m

DEVELOPING

The diagram shows five roads: DE, DG, EF, GE and GF. The bearing of FG is 304◦ , ED is 292◦ and DG is 040◦ . The distance from E to F is 8 km and E is due west of F.

a Find the size of ∠GFE. b What is the distance GE? Answer correct to one decimal place. c What is the size of ∠DGE, ∠GED and ∠GDE? Answer correct to the nearest degree. d What is the distance DE? Answer correct to one decimal place. e What is the distance DG? Answer correct to one decimal place.

40°

292°

304°

A survey of a park is shown in the diagram. A path is proposed from B to D and a fence is required from B to C to D.

a What is the length of the path? Answer correct to the nearest metre. b Calculate the required length of fencing. Answer correct to the nearest metre. c What is the area of the park? Answer correct to the nearest square metre.

37°

41 m

99°

35 m

74°

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4G Problem solving using trigonometry

CHALLENGING 12

Chris and Dylan are 55 m apart. They both observe a bird in the sky at B.

Bird B

U N SA C O M R PL R E EC PA T E G D ES

a Prove that the distance between Dylan and the bird (BD) is given 55 sin 35◦ by BD = sin 16◦ b What is the distance between Chris and the bird? Answer correct to three significant figures. c How high is the bird above the ground? Answer correct to three significant figures.

C Chris

A boat travels 90 km from point X to point Y on a bearing of 065◦ . It then turns and travels on a bearing of 140◦ from point Y to point Z. Point Z is on a horizontal line from its starting point (X).

The diagram below shows a plan for a triangular park GHI. Fence posts have been located at points G, H and I. GH is 75 m and HI is 108 m. The bearing H from G is 045◦ and the bearing of I from H is 125◦ .

a Show that ∠GHI is 100◦ . b Calculate the area of the triangular park GHI. c What length of fence would be required to go around the perimeter of the park, correct to the nearest metre? d What is the bearing of G from I, correct to the nearest degree?

A ferry sails from Island A, towards Island B on a bearing of 058◦ for 18 km. After dropping off passengers, it sails to Island C at a bearing of 312◦ for 27 km. Then it returns to Island A.

a Show this information on a diagram. b What is the distance from Island C to Island A, correct to the nearest kilometre? c Find ∠ABC and ∠CAB. Answer correct to the nearest degree. d In what direction does the ferry sails from Island C to Island A, correct to the nearest degree?

D Dylan

a Copy the diagram and show the given information on the diagram. b What are the sizes of ∠XYZ and ∠Y XZ? c How far is the boat from its starting point (X to Z)? Answer correct to two decimal places.

51°

35°

125°

75 m

108 m

45°

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Chapter 4 Trigonometry

Two groups of hikers decide to leave campsite A to visit some natural landmarks. The first group hikes to Empress Falls, F which is 14 km on a bearing of 035◦ . The second group hikes to Lincoln’s Rock, R which is 24 km on a bearing of 317◦ .

U N SA C O M R PL R E EC PA T E G D ES

a Copy the given diagram and add the information above. b How far is Empress Falls from Lincoln’s Rock, correct to one decimal place? c Find the bearing of Empress Falls from Lincoln’s Rock.

Communicate your reasoning, understanding and fluency 17

The National Park bike track starts at Beetle Bench, B. The riders ride for 12 km on a bearing of 150◦ to Panda Playground, P. At P, the track changes course to a bearing of 040◦ until the riders reach Dolphin Dam, D. From D, it is a 14 km ride back to B.

150°

a Show that ∠BPD is 70◦ . b Calculate ∠PDB correct to the nearest degree. c What is the bearing of B from D? d How far is it once around the bike track, correct to the nearest kilometre?

Suzanna cycles around a park starting at A, passing through B, C, D and back to A as shown in the diagram below.

14 km N

12 km

40°

10 km

a Find the total distance around the park, correct to the nearest kilometre. b Find the area of the park, correct to the nearest square kilometre.

110°

20 km

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4H Compass radial surveys

4H Compass radial surveys Learning intentions

• To construct and interpret compass radial surveys • To use trigonometry to solve a variety of practical problems

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

A compass radial survey is used to create a diagram of a piece of land that is in the shape of a polygon. It involves drawing lines from a central point radiating out to the corners of the polygon. The length of the lines and the angles between them at the central point (usually labelled O for origin) are measured.

Paddock

8m 11 m 95˚ 55˚ 52˚ 9m O 158˚ 12 m

Paper

Table

Conducting a compass radial survey

1 A ‘plane table’ (a surveyor’s table) and large sheet of paper are placed in the centre of a field. 2 A point is marked near the middle of the paper and labelled O. 3 A line is drawn on the paper from O to reflect the line of sight to each corner (the radial line). 4 The distance from the O on the paper to the first corner on the land boundary is measured. 5 The length of the line is marked off on the line drawn on the paper using a suitable scale (e.g. 1 m on the ground is 1 cm on the paper). 6 Steps 3–5 are repeated to draw lines from O to every corner of the piece of land. 7 Boundary lines are drawn connecting the ends of the radial lines, representing the land boundary. 8 The angles at O between the radial lines are measured using a compass.

Example 22

Calculating area using a radial survey

The diagram shows a compass radial survey of a triangular piece of land.

What is the area of ∆ABC?

Answer in square metres correct to one decimal place.

136°

82°

4 m 142° 7 m

Continued on next page

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Chapter 4 Trigonometry

Solution 1 × a × b × sin C 2

The three smaller triangles each have known angles between known side lengths. Use the sine rule for area of a triangle. (Round this step to 3 decimal places before your final answer.)

U N SA C O M R PL R E EC PA T E G D ES

Explanation

1 × 5 × 7 × sin 82◦ ≈ 17.330 m2 2 1 In ∆BCO : A = × 4 × 7 × sin 142◦ ≈ 8.619 m2 2 1 In ∆ACO : A = × 4 × 5 × sin 136◦ ≈ 6.947 m2 2 In ∆ABO : A =

Total area ≈ 17.330 + 8.619 + 6.947 ≈ 32.9 m2

For each triangle, substitute the two sides and the included angle into the area formula, and evaluate.

Add up the areas correct to one decimal place.

Now you try

The diagram shows a compass radial survey of a triangular piece of land. Calculate the area of ∆ABC.

Answer in square metres correct to one decimal place.

150°

95°

115°

Example 23

Calculating distance using a compass radial survey

A compass radial survey for a field is shown opposite.

P 072°T

R 290°T

a What is the size of ∠POQ? b What is the size of ∠POR? c Calculate the length of PQ, correct to two decimal places. d Calculate the length of PR, correct to two decimal places. e What is the perimeter of the field? Answer correct to three significant figures.

4.8 m

4.7 m

4.5 m

Q 145°T

Solution

Explanation

a ∠POQ = 145 − 72

Subtract the bearing of P from the bearing of Q.

= 73◦

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4H Compass radial surveys

North P 072°T

R 290°T 4.7 m

70° 72°

4.8 m

Calculate the angle between R and the north direction. Identify the angle between P and the north direction from the bearing of P. Add the two angles formed with the north direction (70◦ and 72◦ ).

U N SA C O M R PL R E EC PA T E G D ES

∠POR = 70 + 72 = 142◦

c c2 = a2 + b2 − 2ab cos C

x2 = 4.82 + 4.52 − 2 × 4.8 × 4.5 × cos 73◦ x2 = 30.65954236 . . .

x ≈ 5.54 m Length PQ is 5.54 metres

d c2 = a2 + b2 − 2ab cos C

x2 = 4.72 + 4.82 − 2 × 4.7 × 4.8 × cos 142◦ x2 = 80.6850452 . . .

x ≈ 8.98 m Length PR is 8.98 metres

e Perimeter ≈ 5.54 + 8.98 + 4.5 + 4.7 ≈ 23.72

≈ 23.7 m

In ∆POQ, ∠POQ and two sides are known, so use the cosine formula to find a side (PQ is c). Substitute the values for a, b, c and C. Calculate the value of x2 . Take the square root of both sides. Write the answer correct to two decimal places. Write the cosine formula to find a side. Substitute the values for a, b, c and C. Calculate the value of x2 . Take the square root of both sides. Write the answer correct to two decimal places.

The field has four sides. Perimeter is the sum of the lengths of each side. Write the answer to three significant figures. Note: Point O, the origin, may be on the boundary of the piece of land, or even outside it.

Now you try

A compass radial survey for a field is shown.

a Find the size of ∠POQ. b Calculate the size of ∠POR. c Calculate the length of PQ, correct to two decimal places. d Calculate the length of PR, correct to two decimal places. e What is the perimeter of the field? Answer correct to three significant figures.

P 050°T

R 300°T

8.9

5.2

6.1

Q 140°T

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Chapter 4 Trigonometry

Exercise 4H Example 22

BUILDING

The diagram below shows a compass radial survey of a piece of land. All length measurements are in metres.

U N SA C O M R PL R E EC PA T E G D ES

56° 43° O 47°

a What is the area of ∆AOB? Answer correct to two decimal places. b What is the area of ∆BOC? Answer correct to two decimal places. c What is the area of ∆COD? Answer correct to two decimal places. d What is the total area of the piece of land? Answer correct to the nearest square metre. e Find the length of AB, correct to two decimal places. f Find the length of BC, correct to two decimal places. g Find the length of CD, correct to two decimal places. h Calculate the perimeter of the piece of land. Answer correct to the nearest metre.

Example 23

A compass radial survey of a triangular field is shown.

a What is the size of ∠AOB? b What is the size of ∠BOC? c What is the size of ∠AOC? d What is the area of ∆AOB? Answer correct to two decimal places. e What is the area of ∆BOC? Answer correct to two decimal places. f What is the area of ∆AOC? Answer correct to two decimal places.

C 316°

58 m N 82 m

A 075°

O 65 m

B 214°

g What is the total area of the triangular field? Answer correct to the nearest square metre. h What is the length of AB? Answer correct to two decimal places. i What is the length of BC? Answer correct to two decimal places. j What is the length of AC? Answer correct to two decimal places. k Calculate the perimeter of the piece of land. Answer correct to the nearest metre.

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4H Compass radial surveys

DEVELOPING 3

A compass radial survey of land DEFG is shown opposite.

E 036°

D 344° 66 m

52 m

U N SA C O M R PL R E EC PA T E G D ES

a What is the size of ∠FOG? b Find the area of triangle FOG correct to the nearest square metre. c What is the length of FG, correct to the nearest metre?

The diagram opposite is a compass radial survey of field VWXYZ. All distances are in metres. Answer the following questions, correct to one decimal place.

a What is the length of XY? b What is the length of ZY? c What is the length of VZ? d What is the length of VW? e What is the length of XW? f Calculate the perimeter of the field.

A compass radial survey of land ABCD is shown opposite. Answer the following questions correct to the nearest metre.

a What is the area of ∆AOB? b What is the area of ∆BOC? c What is the area of ∆COD? d What is the area of ∆AOD? e What is the area of the land?

30 m

G 260°

45 m

F 157°

W 300°

V 270°

Y 090°

Z 135°

A 324°

B 027°

33 m

50 m

24 m

18 m

C 080°

224° D

CHALLENGING

Consider the compass radial survey shown opposite.

a If the area of ∆AOB is 70 m , find the bearing of B from O. Answer correct to the nearest degree b Find the area of ∆DOC. Answer correct to one decimal place.

A 320°

13 m

11 m

12 m

D 260°

14 m

C 160°

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Chapter 4 Trigonometry

A compass radial survey of a field is shown.

B 035° A

a A is north-west of O. What is the size of ∠AOB? b Find the length of AB, correct to one decimal place. c Find the area of ∆AOB, correct to one decimal place. d What is the bearing of D from O?

25 m

22 m O

22 m

72°

21 m

U N SA C O M R PL R E EC PA T E G D ES

C 140°

Consider the compass radial survey shown opposite.

P 345°

a Calculate ∠POQ, correct to the nearest degree. b Hence, find the bearing of Q from O. c Calculate the area of ∆QOR, correct to the nearest square metre.

36 m

29 m

32 m

25 m

R 168°

A compass radial survey of a paddock ABCD is shown. A farmer wishes to use the shaded region DOBC for his horses to graze. Find the area of the shaded region DOBC correct to the nearest square metre.

A 050°

295° D

96 m

82 m

FPO O

85 m

80 m

B 155°

115 m

Communicate your reasoning, understanding and fluency 10

Archie, Brodie and CJ are standing in a field. CJ (C) is 15 metres away from Brodie (B) on a bearing of 032◦ . Archie (A) is 20 metres away from Brodie on a bearing of 315◦ .

a Draw a diagram to represent the positions A, B and C. Mark the information from the question on the diagram. b Show that ∠ABC is 77◦ . c What is the area of triangle ABC? Answer correct to the nearest square metre. d How far is Alex from Connor? Answer correct to the nearest metre.

A compass radial survey of a parkland is shown in the diagram.

a Show that the size of ∠AOB is 109◦ . b Calculate the length of AB, correct to the nearest metre. c Find the area of ∆AOB, in hectares correct to three significant figures.

315° A

B 064°

304 m

310 m O

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Chapter 4 review

Review

Key ideas and chapter summary Use the mnemonic ‘SOH CAH TOA’. SOH: Sine-Opposite-Hypotenuse CAH: Cosine-Adjacent-Hypotenuse TOA: Tangent-Opposite-Adjacent

U N SA C O M R PL R E EC PA T E G D ES

Right-angled trigonometry

Degrees and minutes

One degree is equal to 60 minutes (1◦ = 60′ ). One minute is equal to 60 seconds (1′ = 60′′ ).

Angles of elevation and depression

θ Horizontal

Bearings

Angle of elevation

Horizontal θ

Angle of depression

Compass bearing

True bearing

A direction given by stating the angle either side of north or south, such as S60◦ E

A direction given by measuring the angle clockwise from north, such as 120◦ T

Area of a triangle

Area of a triangle is half the product of two sides multiplied by the sine of the angle between the two sides (included angle).

The sine rule

Sine rule is used in a non-right-angled triangle given information about two sides and two angles. a b c • To find a side, use = = . sin A sin B sin C

1 A = bc sin A 2

sin A sin B sin C = = . a b c • To find an obtuse angle using the sine rule, subtract the acute angle from 180◦ . Always check that the angle sum of a triangle does not exceed 180◦ .

• To find an angle, use

The cosine rule

Cosine rule is used in a non-right-angled triangle given information about three sides and one angle.

• To find a side, use a2 = b2 + c2 − 2bc cos A. b2 + c2 − a2 • To find an angle, use cos A = . 2bc

Compass radial surveys

Compass radial surveys involve measuring the angles between radial lines drawn from a central point in the direction of the corners of the land, and measuring the lengths of the radial lines.

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Review

258

Chapter 4 Trigonometry

Chapter checklist with success criteria ✓

A printable version of this checklist is available in the Interactive Textbook. Checklist

1 I can apply trigonometric ratios to right-angled triangles to find the lengths of

U N SA C O M R PL R E EC PA T E G D ES

unknown sides.

e.g. Find the lengths of the unknown sides, correct to one decimal place. a

63°

42°

2 I can apply trigonometric ratios to right-angled triangles to find the size of unknown angles in both degrees, and degrees and minutes.

e.g. Find the angle θ in the triangle shown. Answer correct to the nearest minute.

3 I can use Pythagoras’ theorem to find the missing side in a right-angled triangle. e.g. The base of a ladder leaning against a wall is 1.5 m from the base of the wall. If the ladder is 4.5 m long, find how high the top of the ladder is from the ground, correct to one decimal place.

4.5 m

1.5 m

4 I can solve practical problems involving angles of elevation and depression. e.g. An iron ore seam of 120 m slopes down at an angle of depression from the horizontal of 38◦ . The mine engineer wishes to sink a vertical shaft, x, as shown. What is the depth of the required vertical shaft?

38°

120 m

Answer correct to the nearest metre.

5 I can identify and interpret true bearings and compass bearings and convert between them.

e.g. State the compass bearing and true bearing of B from O.

30°

120°

B S

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Chapter 4 review

6 I can solve practical problems involving true bearings and compass bearings. e.g. Simran runs 12 km on a bearing of 210◦ . How far south has she run? Answer correct to one decimal place.

7 I can apply the sine rule

U N SA C O M R PL R E EC PA T E G D ES

the unknown side.

a b c = = in a given triangle ABC to find the value of sin A sin B sin C

Review

e.g. Find the value of x, correct to one decimal place.

4.1

50°

37°

sin C sin A sin B = = in a given triangle ABC to find the value a b c of the unknown angle, excluding the ambiguous case.

8 I can apply the sine rule

e.g. Find the unknown angle θ. Answer correct to the nearest degree.

4.6

75°

2.9

9 I can apply the sine rule in a given triangle to find the value of the unknown angle, given that the unknown angle is obtuse.

e.g. Calculate the size of the angle θ given that it is an obtuse angle. Answer correct to the nearest degree.

42 m

12°

125 m

10 I can apply the cosine rule c2 = a2 + b2 − 2ab cos C in a given triangle ABC to find the value of an unknown side.

e.g. Find the value of x, correct to two decimal places.

76°

11 I can apply the cosine rule cos C = of an unknown angle.

a2 + b2 − c2 in a given triangle ABC to find the value 2ab

e.g. Find the value of the angle θ. Answer in degrees, correct to one decimal place.

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Review

260

Chapter 4 Trigonometry

1 ab sin C, where a and b are the sides that form the 2 included angle C in a given triangle ABC, to solve problems in a range of contexts.

12 I can apply the formula A =

e.g. Find the area of the triangle correct to the nearest square centimetre.

D 5 cm 40°

U N SA C O M R PL R E EC PA T E G D ES

6 cm

13 I can solve practical problems involving Pythagoras’ theorem, the trigonometry of

right-angled and non-right-angled triangles, angles of elevation and depression, and true bearings and compass bearings.

e.g. A triangle DEG has a right angle at E and DG is 8 m in length. A line is drawn from D to F with ∠DFG = 144◦ and ∠GDF = 7◦ . a Find the length of DF. Answer correct to two decimal places. b Find the size of ∠DFE. c Find the length of FE. Answer correct to two decimal places.

7°

144°

14 I can construct and interpret compass radial surveys and solve related problems. e.g. A compass radial survey of land DEFG is shown opposite.

a What is the size of ∠FOG? b Find the area of triangle FOG correct to the nearest square metre. c What is the length of FG, correct to the nearest metre?

E 036°

D 344°

66 m

G 260°

30 m

52 m

45 m

F 157°

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Chapter 4 review

Review

Multiple-choice questions 4A

What is the length of a? A 14 cos 63◦

B 14 sin 63◦

a 63°

14 cos 63◦

14 sin 63◦

U N SA C O M R PL R E EC PA T E G D ES

C 4D

What is the correct expression for x? 5 × sin 29◦ 5 × sin 39◦ A B sin 39◦ sin 29◦ C

5 8 sin 64◦

5 × sin 112◦ sin 29◦

64°

7 8 sin 64◦

What is the correct formula for the cosine rule to find angle A? a2 + c2 − b2 a2 + b2 − c2 A cos A = B cos A = 2ac 2ab C cos A =

29°

How would sin θ be calculated? 7 sin 64◦ 5 sin 64◦ A B 8 8 C

5 × sin 112◦ sin 39◦

39°

b2 + c2 − a2 2ab

D cos A =

b2 + c2 − a2 2bc

What is the length of x? Answer correct to two decimal places. A 6.93

B 6.94

C 48.14

D 48.15

50°

The largest angle in the triangle is θ. What is the value of cos θ? 62 + 72 − 52 52 + 72 − 62 A cos θ = B cos θ = 2×6×7 2×5×7 C cos θ =

52 + 72 − 62 2×5×7

D cos θ =

52 + 62 − 72 2×5×6

What is the area of the triangle correct to the nearest square metre? A 115 m2 C 205 m2

B 185 m2 D 220 m2

32 m

20 m

40°

18 m

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Chapter 4 Trigonometry

Short-answer questions 4A

Find the value of the following trigonometric ratios, correct to two decimal places. a tan 123◦ 41′ e 5 sin 134◦

b cos 93◦ 26′ f 7 tan 128◦

c sin 130◦ 59′ g 11 cos 149◦

d cos 167◦ 35′ h 4 tan 137◦

2 Find the value of x, correct to two decimal places.

U N SA C O M R PL R E EC PA T E G D ES

Review

262

25 m

56°

34°

35 m

17 mm

3 Find the unknown angle θ in each triangle. Answer correct to the nearest minute. a

4 Susan looked from the top of a cliff, 62 m high, and noticed a ship at an angle of depression of 31◦ . How far was the ship from the base of the cliff? Answer correct to one decimal place.

31°

62 m

Gurvinder rode for 8.5 km on a bearing of N43◦ W from her home.

a How far north is Gurvinder from home? Answer correct to one decimal place. b How far west is Gurvinder from home? Answer correct to one decimal place. c What is the compass bearing of her home from her current position?

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263

Chapter 4 review

Find the value of θ correct to the nearest minute. a 15 cm

60 cm 52 cm

17 cm

104 cm θ

55°

108 cm

90 cm

18 cm

7 Find the length of the unknown side x in each triangle, correct to two decimal places.

U N SA C O M R PL R E EC PA T E G D ES

Review

4.9

6.1

51°

83°

109°

32°

58°

50°

8 Find the length of the unknown side x in each triangle, correct to two decimal places. a

63°

3.1

96°

2.5

49°

Extended-response questions

A triangle BCE has a right angle at C, ∠BEC = 24◦ and BE is 20 m in length. A line is drawn from B to D with ∠DBE = 29◦ .

a Find the length of BD. Answer correct to two decimal places. b Find the size of angle BDC. Answer correct to the nearest minute. c Find the length of CD. Answer correct to two decimal places.

20 m

29°

24°

A compass radial survey of land ABC is shown.

C 315°T

a What is the size of ∠AOB? b Find the area of triangle AOB correct to the nearest square metre. c What is the length of AB, correct to the nearest metre?

256°T

A 108°T

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U N SA C O M R PL R E EC PA T E G D ES

Simultaneous linear equations

Using algebra to make business decisions Olivia runs a market stall every Saturday. She sells bracelets which she makes herself.

In this chapter we will use algebra to unpack situations like these.

Relevant information:

Her costs (also known as expenditure) form a straight line on the Cartesian plane.

• how much does the stall cost? • how much does it cost to make the bracelets? • how much might people be willing to pay for them? • how much should she ask for? • how many might she sell every Saturday? • how many will she need to sell just to break even? • if she sells 50 every Saturday, how much profit or loss will she make?

Her income (also known as revenue) forms a different straight line. The point at which the lines meet is very important to the success or failure of this business venture.

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Chapter contents

U N SA C O M R PL R E EC PA T E G D ES

5A Straight lines (CONSOLIDATING) 5B Linear relationships (CONSOLIDATING) 5C Solving simultaneous equations graphically 5D Solving simultaneous equations algebraically 5E Solving practical problems using simultaneous equations 5F Break-even analysis

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Represents the relationships between quantities algebraically and graphically to solve problems and make predictions in practical contexts (MST-12-S2-01). © NSW Education Standards Authority

Online resources

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Chapter 5 Simultaneous linear equations

5A Straight lines

CONSOLIDATING

Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To construct straight-line graphs on the Cartesian plane • To understand the gradient and the vertical intercept • To use the gradient–intercept formula to draw straight-line graphs • To determine the equation of a line from a graph Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Year 11 • Questions involving straight lines regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The gradient–intercept formula is not provided on the 2027 version.

Linear relationships: The big picture

This topic should be familiar to you from previous years of schooling.

In this chapter, the following concepts will be consolidated and extended, with emphasis on real-life scenarios which are linear relationships. A linear relationship can be viewed and represented in several different ways, as follows. A real-life scenario

Asra has $4 in her money box at present.

This is a single, one-off, once-only amount, at the beginning of a time period. The time is 0. She decides to add $3 every week. This is a repetitive increase.

This is a linear relationship, because she is always adding the same amount every week. A number pattern 4, 7, 10, 13, . . .

The numbers in the pattern are sometimes called terms. The number 4 is the present value.

The numbers 7, 10, 13, . . . are the future values of this investment. A table of values

The numbers 4, 7, 10, 13, . . . are placed in the bottom row of a table. The top row tracks the weeks, starting from 0. Weeks (W)

Dollars (D)

Variables ‘Weeks’ and ‘Dollars’ are called variables. The top row is the independent variable. The bottom row is the dependent variable. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

5A Straight lines

An equation, sometimes called a ‘rule’ or a ‘function’ You may recall an activity called ‘guess the rule’, but guessing is really not required. You just need to look in the correct places: • Asra is adding $3, repeatedly. This is known as the gradient, slope or rate of change. • She began with $4. This is known as the constant, or the vertical intercept, or the y-intercept.

U N SA C O M R PL R E EC PA T E G D ES

Choose any number in the top row. Multiply by 3 then add 4. That will give you the number in the bottom row. W ×3+4= D 3W + 4 = D

The equation is usually written ‘backwards’. D = 3W + 4

Note:

• the gradient (3) has a letter attached to it

• the vertical intercept (4) is the constant term.

Every straight line can be written in the form: y = mx + c, where:

m is the gradient, sometimes called the slope

c is the y-intercept, also known as the vertical intercept.

A series of points on the Cartesian plane

Every column in the table is a point: (0, 4) then (1, 7), (2, 10), (3, 13), . . . A graph on the Cartesian plane

The points can be joined with one straight line.

The relationship between weeks and dollars is linear. This is an example of a linear relationship.

Note:

• the top row of the table is the horizontal axis

• the bottom row of the table is the vertical axis

• the scale on the horizontal axis can be different to the scale on the

vertical axis.

Amount saved ($)

Asra’s savings are increasing at a constant rate.

D 16 14 12 10 8 6 4 2 O

Asra’s savings

1 2 3 4 5 6 7 8 Number of weeks

Summary

Every linear relationship can be viewed and represented as a: • worded real-life scenario • number pattern • table of values containing a series of points • rule called an equation • straight-line graph on the Cartesian plane.

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In this chapter, you will: • become more familiar with each of these • learn how to move efficiently from one representation to the others • solve problems involving real-life scenarios.

U N SA C O M R PL R E EC PA T E G D ES

Graphing a straight line using a table of values 1 Construct a table of values with the independent variable as the first row and the dependent variable as the second row. 2 Draw a number plane with the independent variable on the horizontal axis and the dependent variable as the vertical axis. Plot the points. 3 Join the points to make a straight line.

Example 1

Drawing a straight line graph

Draw the graph of y = −2x − 1.

Solution

Explanation

−2

−1

−3

−5

Draw a table of values for x and y. Let x = −2, −1, 0, 1 and 2. Find y using the linear function y = −2x − 1.

3 2 1

–5

–4

–3

–2

–1

O –1 –2 –3 –4 –5

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points (−2, 3), (−1, 1), (0, −1), (1, −3) and (2, −5). Join the points to make a straight line.

Now you try

Draw the graph of y = 2x − 1.

Gradient

The gradient of a line is the slope or steepness of the line. It is calculated by dividing the vertical rise by the horizontal run. The larger the gradient, the steeper the slope. The letter m is used to indicate gradient.

rise run

Gradient of a straight line m =

vertical rise horizontal run

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5A Straight lines

Like reading a book, it is best to view a line from left to right, considering the run first, then the rise (or fall). A fraction can be formed with rise over run. Positive gradient

Negative gradient

U N SA C O M R PL R E EC PA T E G D ES

run

rise

run

Run is positive. Rise is positive. The line goes ‘uphill’.

Run is positive. Rise is negative. The line goes ‘downhill’.

The y-intercept

Lines can cross the x-axis and the y-axis at ‘intercepts’. The intercept on the vertical axis is called the y-intercept and is denoted by the letter c.

Gradient

Intercept

Gradient of a line is the slope of the line. rise m= run

The intercept of a line is where the line cuts the axis.

Vertical intercept is often denoted by c. In the table of values, the vertical intercept is always the number in the bottom row, under the 0.

When the equation of a line is written as y = mx + c, it is said to be in ‘gradient–intercept form’. The gradient is m, the coefficient of x, and the y-intercept is c, the constant term. The independent variable in the formula is x and the dependent variable in the formula is y. Gradient–intercept form is useful in modelling relationships in many practical situations. However, the variables are often changed to reflect the situation. For example, the equation c = 25n + 100 has c as the cost of the event ($) and n as the number of guests. These letters are the dependent and independent variables.

Using the gradient–intercept formula y = mx + c

m = gradient of the line (vertical rise over the horizontal run) c = y−intercept where the line cuts the y-axis or vertical axis

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Example 2

Draw a graph from a table of values, find gradient and y-intercept

Draw the graph of y = 3x − 2 from a table of values. Find the gradient and y-intercept of this line. Solution

Explanation

−2

−1

Construct a table of values for x and y.

−8

−5

−2

Let x = −2, −1, 0, 1 and 2. Find y using the linear function y = 3x − 2.

U N SA C O M R PL R E EC PA T E G D ES

4 3 2 1

Draw a number plane with x as the horizontal axis and y as the vertical axis.

3 Vertical rise

1 Horizontal run x O −3 −2 −1 1 2 3 −1 −2 y-intercept −3 −4

Plot the points (−1, −5), (0, −2), (1, 1) and (2, 4). The point (−2, −8) has not been plotted as it does not fit the scale of the number plane.

Join the points to make a straight line. Calculate the gradient of the line using the ‘rise over run formula’ and read the value of the y-intecept where the line crosses the vertical axis. Write the values of the gradient and y-intercept.

The gradient is the coefficient of x and the y-intercept is −2.

Now you try

Draw the graph of y = 2 − 3x. Find the gradient and y-intercept of this line.

Sketching a straight line requires at least two points. When an equation is written in gradient–intercept form, one point on the graph is immediately available: the y-intercept. A second point can be quickly calculated using the gradient.

Example 3

Sketching a linear function using the gradient and y-intercept

Sketch the graph of 3y + 6x = 9. Solution

Explanation

3y + 6x = 9

Rearrange the equation into gradient form y = mx + c. Subtract 6x from both sides.

3y = 9 − 6x 9 − 6x y= 3 y = 3 − 2x

y = −2x + 3 Gradient is −2 and y-intercept is 3.

Divide both sides by 3 and simplify.

The equation is now written in the form y = mx + c. Therefore m = −2 and c = 3.

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5A Straight lines

A gradient of −2 means that for every unit across in the positive x-axis direction, you go down 2 in the negative y-axis direction.

y y-intercept 3 1 Horizontal run

Plot the y-intercept (0, 3), then move across 1 (horizontal run) and down 2 (vertical rise) to plot the point (1, 1).

−2 Vertical rise

U N SA C O M R PL R E EC PA T E G D ES

2 1

−2

−1

Join the points (0, 3) and (1, 1) to make a straight line.

−1

Alternatively, Let x = 0:

Let x = 0 to find the y-intercept. Solve for y. Plot the point (0, 3).

3y + 6x = 9 3y = 9

y=3 y-intercept is (0, 3).

Let y = 0 to find the x-intercept. Solve for x. Plot the point (1.5, 0).

Let y = 0:

3y + 6x = 9 6x = 9

x = 1.5 x-intercept is (1.5, 0).

Plot these two points. Now you try

Sketch the graph of 2x + 3y = 6.

Exercise 5A 1

BUILDING

Plot the following points on a number plane and join them to form a straight line. a

−2

−1

−2

−1

−3

−1

2 Complete the following table of values and graph each straight line. a y= x−1 x

b y = −2x

c y = 2x + 3

d y = −x + 2

x y

−2

−1

−2

−1

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Example 1

Example 2

3 Draw the graphs of these lines by first completing a table of values.

2 x−1 3

a y = 2x + 2

b y = −x + 3

c y=

1 d y=− x+1 2

e y = 3x − 1

f y = −2x + 3

Find the gradient of the following straight lines. b y= x−2 2 e y=− x+3 3

c y = −2x 3 1 f y= x+ 4 2

U N SA C O M R PL R E EC PA T E G D ES

a y = 5x + 1 1 d y= x+4 2

Find the y-intercept of the following straight lines.

a y = 3x − 5 1 d y=− x−1 3

b y = −x + 2 2 e y=− x+7 5

c y = −4x 1 3 f y= x+ 4 5

Find the equation of the following straight lines defined by: a gradient 2, passing through (0, 1) b gradient −3, passing through (0, 4) c gradient 0.5, passing through (0, −2) d gradient 0, passing through (0, 6) 2 e gradient − , passing through (0, 4) 5 1 f gradient , passing through (0, 0). 3

What is the equation of each of the following lines? a

2 1

3 2 1

O −1 −1

1 2 3

4 2

O −6 −4 −2 −2 −4

5 10

2 1

4 2 1 2

−10−5−5O −10

3 6

2 1

1 2

10 5

−6

−2

O −6 −3 −3

2 4 6

−4

O −2 −1 −1

−2 −1 O −1 −2

6 3

O –6 −4 −2 −2

1 2

3 2 1

O −1 −1

x 2 4

O −2 −1 −1

x 1 2 3

−2 −3

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5A Straight lines

Write an equation for each line. a m = 5, c = −1 c m = 5, c = 1

b m = −5, c = 1 d m = −5, c = −1

Plot the pair of points on a Cartesian plane, then find the equation of the line. b (−2, 0), (0, 4) d (1, 0), (0, −1) f (0, 4), (4, 2)

U N SA C O M R PL R E EC PA T E G D ES

a (0, 3), (3, 0) c (2, 0), (0, 2) e (−1, 0), (0, 3)

Express the following linear equations in gradient–intercept form (y = mx + c). a y + 2 = 3x 1 c y− x=1 2 1 e x−y=1 3

b x+y−4=0

d 4x − y + 2 = 0 f 4 − y = 3x

Draw the graph of each line in question 10.

DEVELOPING

Example 3

12 Draw the graphs of these lines using the gradient and y-intercept. a y= x+2 1 c y= x+2 3 e y+2= x

b y = −2x + 1 1 d y=− x−1 2 f 3x − y = 0

Consider the equation 2x − y + 5 = 0.

a What is the value of y when x = 3? b What is the value of x when y = 21? c Does the point (6, 17) lie on the line? d What is the gradient and the y-intercept? e What is the equation of a line parallel to 2x − y + 5 = 0 passing through (0, −2)? f Draw a graph of 2x − y + 5 = 0.

CHALLENGING

Answer true or false to the following questions.

a The point (3, 0) lies on the line y = 3x. b The line y = x − 7 passes through the point (1, −6). c The point (1, 4) lies on the line x + 2y = 9. d The line 2x − 3y = 0 passes through the point (0, 0). e The point (1, −1) lies on the line 4x − y + 1 = 0. f The line 3x − 4y + 1 = 0 passes through the point (2, 0).

It is known that y varies directly with x. When x = 3, y = 12. a Write a linear equation in the form y = mx to describe this situation. b Draw the graph of y against x.

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It is known that y varies directly with x. When x = 6, y = 2. a Write a linear equation in the form y = mx to describe this situation. b Draw the graph of y against x.

A motorcyclist is travelling at constant speed. He travels 350 km in 7 hours.

U N SA C O M R PL R E EC PA T E G D ES

a Write a linear equation in the form d = mt to describe this situation. b Draw the graph of d against t.

Communicate your reasoning, understanding and fluency 18

Draw these three straight lines on the same Cartesian plane: y = 3x y = 3x + 1 y = 3x − 2 What do you notice about these three lines?

Complete this sentence: If two or more lines have the same gradient, ________. What is the equation of the line parallel to y = 3x which passes through the point (1, −4)?

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5B Linear relationships

CONSOLIDATING

Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To apply linear relationships to practical situations, including direct variation • To understand, use and interpret linear models Past, present and future learning

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Year 11 • Questions involving linear relationships regularly appear in HSC Examinations

Linear modelling occurs when a practical situation is a linear relationship. For example, the gradient– intercept form of a straight-line graph can sometimes be used to model catering costs. A catering company charges a base amount of $100 plus a rate of $20 per guest. Using this information, we can write down a linear equation to model the cost of the event. Let c be the cost of the event ($) and n be the number of guests. We can then write c = 20n + 100.

Note: The number of guests (n) must be greater than zero and a whole number.

The graph of this linear model has been drawn. There are two important features of this linear model: 1 gradient is the rate per guest ($20) 2 the vertical-intercept is the base amount ($100).

Catering cost

5 10 15 20 25 30 35 40 45 50

1500

1000 500

Some linear models have limitations and restrictions In the catering cost scenario:

• the number of guests (n) must be a positive number, not negative • the number of guests must be a whole number

• the venue for the event may have a limit of 100 people

• the catering company might only be able to cater for a maximum of 100 people.

Other examples of restrictions:

• if a water tank is being filled, there is a maximum amount which can fit in the tank, so the model only

applies up to that moment • if a water tank is being emptied, the model stops when the tank is empty • a teenager may be growing taller by 12 cm per year, but eventually they will stop growing taller, so the model only applies up to that time • if you are spending the money you saved up for your holiday, the model stops when the money runs out.

Linear models A section of a straight line is used to model a linear relationship.

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Example 4

Using linear models for direct variation

The graph opposite is used to convert Australian dollars (AUD) to New Zealand dollars (NZD). Use the graph to convert:

Australian dollars to New Zealand dollars NZD 60 50 40 30 20 10

U N SA C O M R PL R E EC PA T E G D ES

a 40 AUD to NZD b 25 NZD to AUD.

AUD

10 20 30 40 50

Solution

Explanation

a 50 NZD

Read from the graph (when AUD = 40, NZD = 50). Read from the graph (when NZD = 25, AUD = 20).

b 20 AUD

Now you try

Brad can make 4 surfboards every 5 days. Draw a graph to represent this situation.

Example 5

Interpreting linear models

Tala sells insurance. She earns a base salary and a commission on each new insurance policy she sells. The graph shows Tala’s weekly income (I) plotted against the number of new policies (n) she sells in that week. The relationship between I and n is linear.

a What is Tala’s base salary? b What is Tala’s salary in a week in which she sells 8 new policies? c How many policies does Tala need to sell to earn $700 for the week?

1000 800 600 400 200

Income ($)

d Find the equation of the straight line in terms of I and n. e Use the equation to calculate the weekly income when Tala sells 3 new policies. f How much does Tala earn for each new policy?

O 2 4 6 8 10 Number of new policies

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5B Linear relationships

Solution

Explanation

a $400 b $800 c 6 new policies d c = 400

Read from the graph (when n = 0, I = 400). Read from the graph (when n = 8, I = 800). Read from the graph (when I = 700, n = 6). Determine the vertical intercept (400).

rise run 400 m= 8 m = 50

U N SA C O M R PL R E EC PA T E G D ES

Find the gradient by choosing two suitable points. (0, 400) and (8, 800). Calculate the gradient (m) between these points using the gradient formula.

I = mn + c

Substitute the gradient and y-intercept into the gradient–intercept form y = mx + c. Use the appropriate variables (I for y, n for x).

I = 50n + 400

e I = 50 × 3 + 400

Substitute n = 3 into the equation. Evaluate. Check the answer using the graph. The gradient of the graph is the commission for each new policy.

= $550

f $50

Now you try

Qing has $10 in her money box. She is going to add $6 every week.

a What is the equation of the linear relationship, using D for dollars and w for weeks? b How much will she have 2 years from now? c How long will it take her to reach $1000?

Exercise 5B 1

BUILDING

Complete the following tables of values and graph each line. a c=d+5 c

b I = −3n

2 Draw the graphs of these lines using a table of values from 1 to 10. a C = 2n + 10

Example 4

b V = −5t + 30

3 a The conversion graph opposite is used to convert Australian dollars to British pounds. Use the graph to calculate these exchanges. i 30 Australian dollars to pounds ii 50 Australian dollars to pounds iii 20 pounds to Australian dollars iv 10 pounds to Australian dollars

b What is the gradient of the conversion graph? c What is the equation of this graph?

1 c M = n − 10 2

Australian dollars to British pounds GBP

50 40 30 20 10

AUD 10 20 30 40 50

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The relationship of the age of machinery (a) in years to its value (v) in $1000 is v = −4a + 20.

U N SA C O M R PL R E EC PA T E G D ES

a Construct a table of values for age against value. Use values of a from 0 to 4. b Draw the graph of age (a) against value (v). c What is the initial cost of the machinery? d What is the age of the machinery if its current value is $15 000 ? e What is the value of the machinery after 1 3 years? 2 f What will be the value of the machinery after 2 years? g When will the machinery be worth half its initial cost?

An ultrasound machine was purchased by a medical centre for $150 000. Its value is depreciated each month as shown in the graph.

a What was the value of the machine after 40 months? b What was the value of the machine after five years? c When does the line predict the machine will have no value? d Find the equation of the straight line in terms of v and t. e Use the equation to predict the value of the machine after 6 months.

Value of ultrasound machine

10 20 30 40 50 60 70 80 90 100 Time (months)

160 140

Value ($1000)

Example 5

120

100 80 60

40 20

A car is travelling at constant speed. It travels 360 km in 9 hours.

a Write a linear equation in the form d = mt to describe this situation. b Draw the graph of d against t.

7 The cost ($C) of hiring a taxi consists of two elements: a fixed flagfall and a figure that varies with the number (n) of kilometres travelled. If the flagfall is $4 and the cost per kilometre is $2, determine a rule that gives C in terms of n. 8 The weekly wage, $w, of a vacuum cleaner salesperson consists of a fixed sum of $350 plus $20 for each cleaner sold. If n cleaners are sold per week, construct a rule that describes the weekly wage of the salesperson.

A gardening shop will deliver mulch for $55 per scoop plus $45 delivery.

If c is the cost and s is the number of scoops, what is the equation of this linear relationship?

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5B Linear relationships

Wei converted 100 Australian dollars (AUD) to 60 euros (EUR).

U N SA C O M R PL R E EC PA T E G D ES

a Draw a conversion graph with Australian dollars on the horizontal axis and euros on the vertical axis. b How many euros is 25 Australian dollars? Use the conversion graph. c How many Australian dollars is 45 euros? Use the conversion graph. d Find the gradient and vertical intercept for the conversion graph. e Write an equation that relates Australian dollars (AUD) to euros (EUR).

DEVELOPING

A car rental company charges $85, plus an additional 24 cents per kilometre.

a Write a rule to determine the total charge, $C, for hiring a car and travelling x kilometres. b What would be the cost to travel 250 kilometres?

An electrician charges $50 to call and $80 per hour. Assume that the relationship between charge and time is linear.

a Find a formula for $C, the charge, in terms of the time spent (t hours). b Draw a graph of C against t. c Calculate the cost of a job that takes 2 hours 45 minutes to complete.

The cost of hiring a taxi is $5 flagfall and $1.50 per kilometre travelled.

a Construct a table of values using 0, 10, 20, 30 and 40 as values for the kilometres travelled (d). Calculate the cost of the taxi hire (C) for each value of d. b Draw the graph of the kilometres travelled (d) against cost of the taxi hire (C). c Use the graph to find the cost of the taxi hire after travelling 15 kilometres. d Use the graph to find the distance travelled if the cost of the taxi hire was $35. e Find the gradient and vertical intercept for this graph. f Find the equation of the straight line in terms of d and C.

CHALLENGING

An electronic bank teller registered $775 after it had counted 120 notes and $975 after it had counted 160 notes.

a Find a formula for the sum registered ($C) in terms of the number of notes (n) counted. b Was there a sum already on the register when counting began? If so, how much?

A printing firm charges $35 for printing 600 sheets of headed notepaper and $47 for printing 800 sheets.

a Find a formula, assuming the relationship is linear, for the charge, $C, in terms of number of sheets printed, n. b How much would they charge for printing 1000 sheets?

The tyres on a racing car had lost 3 mm of tread after completing 250 km of a race and 4 mm of tread after completing 1000 km. Assuming that a linear relationship exists between loss of tread and distance covered, find the total loss of tread, d mm, after s km from the start of the race. What would be the tread loss by the end of a 2000 km race? Give your answer correct to one decimal place.

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Communicate your reasoning, understanding and fluency 17

Tala from Example 5 has been offered a new pay deal by her employer: • base salary of $800 • commission of $25 on each new insurance policy she sells.

U N SA C O M R PL R E EC PA T E G D ES

Using this new model, compared to the current model in Example 5: a under what circumstances will Tala earn the same income? b under what circumstances will Tala earn more income? c under what circumstances will Tala earn less income?

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5C Solving simultaneous equations graphically

5C Solving simultaneous equations graphically Learning intentions

• To find the point of intersection of two lines by drawing their graphs Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• Students should already be familiar with the contents of this section • These concepts build upon prior learning from Year 11 • Questions involving simultaneous equations regularly appear in HSC Examinations

Two straight lines will intersect (unless they are parallel). The point at which two straight lines intersect can be found by sketching the two graphs on a Cartesian plane and reading off the coordinates of the point of intersection. Finding the point of intersection is said to be ‘solving the equations simultaneously’. In addition to graphing the straight lines, the point of intersection could be determined by looking at the table of values. If the same value for x and y occurs in both tables it is the point of intersection.

Solving a pair of simultaneous equations graphically 1 Draw a number plane. 2 Graph both linear equations on the number plane. 3 Read the point of intersection of the two straight lines. 4 Interpret the point of intersection for practical applications.

Example 6

Graphical solution of simultaneous linear equations

Find the simultaneous solution of y = x + 3 and y = −2x.

Solution

Explanation

y= x+3 Gradient is +1, y-intercept is 3. y = −2x Gradient is −2, y-intercept is 0.

Use the gradient–intercept form to determine the gradient (coefficient of x) and y-intercept (constant term) for each line. Draw a number plane.

y 4 3 2 1

O −3 −2 −1 −1

Sketch y = x + 3 using the y-intercept of 3 and a gradient of 1 (blue line).

1 2

The lines intersect at the point (−1, 2). Simultaneous solution is x = −1 and y = 2.

Sketch y = −2x using the y-intercept of 0 and a gradient of −2 (red line).

Find the point of intersection of the two lines. The simultaneous solution is the point of intersection.

Continued on next page

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−2

−1

−2

−1

−2

−4

Alternatively, construct a table of values for x and y. Let x = −2, −1, 0, 1 and 2. Find y using the linear function y = x + 3. Repeat to find y using the linear function y = −2x. The same value of x and y occurs in both tables when x = −1 and y = 2.

U N SA C O M R PL R E EC PA T E G D ES

282

Simultaneous solution is x = −1 and y = 2. Now you try

Find the point of intersection of y = 2x + 1 and y = 3 − 2x.

Exercise 5C 1

BUILDING

What is the point of intersection for each of these pairs of straight lines? a

3 y=x+3

–1 O –1

−3 −2 −1 O −1

−2

–2

y = 0.5x + 1

y=x–2

–3

y = −2x

y = –x + 2

y = –2x + 1

–3 −2 −1 O −1 −2

−3

y = −1x − 2

–3

–2

–1 O –1

–2

y = 4x + 1 –3

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5C Solving simultaneous equations graphically

Example 6

Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines?

y 8

6 5 4

U N SA C O M R PL R E EC PA T E G D ES x

2 1

Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines? x

−2

−1

−6

−5

−4

−3

−2

−1

−3

−6

9 8 7 6 5 4 3 2 1

Plot the following points on a number plane and join them to form two straight lines. What is the point of intersection of these straight lines? x

6 5 4 3 2 1

–3 –2 –1 O –1 –2 –3 –4 –5 –6

1 2 3

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Complete the following table of values and plot the points on a number plane. Find the simultaneous solution of these pairs of equations. All solutions are whole numbers. a y = 2x + 3 x

−2

y = −x −1

x y

b y= x+4

y = 2x

−1

U N SA C O M R PL R E EC PA T E G D ES

−2

−1

−2

−1

c y = 3x + 1 x

−2

y = 5x − 3

−1

−2

DEVELOPING

Complete the following table of values and plot the points on a number plane. What is the solution to each pair of simultaneous equations? Some of the solutions are not whole numbers. a y = 3x − 3 x

−2

y= x+1

−1

−2

−1

−2

−1

c y=x

y = 4x + 3

−2

−1

y = −x + 4

−2

b y = 3x − 2 x

−2

Draw the graphs of the following pairs of equations to find their simultaneous solution. Some of the solutions are not whole numbers. a y = −x and y = 4 − 2x c y = x + 1 and y = −2x e y = x + 1 and y = −3x + 2

b y = 5x + 1 and y = 3x − 7 d y = 2x − 4 and y = −x + 5 f y = 8x − 10 and y = 4x

CHALLENGING

Estimate the solution to each of the following pairs of simultaneous equations by graphing them on the same number plane. Answer correct to one decimal place. a 4y = x and x + y = 1 c x + y = 3 and y = 4x e x − 2y = 1 and 4x + 3y = 3

b x − y = 2 and 1 + y = 3x d x = 2 + y and 8x + 4y = 7 f 4x + 3y = 4 and 4x − 6y = 1

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5C Solving simultaneous equations graphically

Communicate your reasoning, understanding and fluency 9

Here is a very interesting party trick:

U N SA C O M R PL R E EC PA T E G D ES

1 Start with 4 then add 3, then add 3 again to make this pattern: 4, 7, 10 2 Now make that into a straight line: 4x + 7y = 10 3 Now you try it with your own numbers: think of any number, then add a number, then add it again. Examples include: • 2, 5, 8 • 1, 3, 5 • 3, 4, 5 • 7, 6, 5

4 Choose one of the above examples or make your own pattern, then choose another. Then make them into a straight line. For example: x + 3y = 5 and 7x + 6y = 5 Draw your chosen straight lines and find the point of intersection of the two lines. Compare your result with others. You are in for a big surprise!

Modelling activity: Finding the break-even point for a small business

This task guides you through building a spreadsheet a model to conduct a break-even analysis for a small business. This modelling task should be done after you have covered sections 5C–5F. The activity can be downloaded from your Interactive Textbook.

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5D Solving simultaneous equations algebraically Learning intentions

• To find the point of intersection of two lines by using the substitution method • To find the point of intersection of two lines by using the elimination method

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Students may already be familiar with the contents of this section • These concepts build upon prior learning from Year 11 • Questions involving simultaneous equations regularly appear in HSC Examinations

Drawing two straight lines is time-consuming and error-prone. There are two algebraic methods of solving simultaneous equations: • the substitution method

• the elimination method.

Method 1: Substitution

When solving simultaneous equations by substitution, the process is to substitute a variable from one equation into the other equation.

Simultaneous equations: substitution method

1 Make one pronumeral the subject in one of the equations. 2 Substitute the expression for this subject into the other equation. 3 Solve this new equation to find the value of one pronumeral. 4 Substitute this value into one of the equations to find the value of the second pronumeral.

Example 7

The substitution method

Solve simultaneously the equations y = 2x − 5 and y = 3x + 2.

Solution

Explanation

(A) y = 2x − 5 (B) y = 3x + 2 Substitute (A) into (B):

Name the equations (A) and (B).

2x − 5 = 3x + 2

Note: It is also possible to substitute (B) into (A).

3x + 2 = 2x − 5

Swap LHS and RHS, for convenience.

Substitute (A) into (B) so that the y in (B) becomes 2x − 5.

3x = 2x − 7

Subtract 2 from both sides.

x = −7

Subtract 2x from both sides.

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5D Solving simultaneous equations algebraically

When x = −7, y = 2 × −7 − 5

These two numbers satisfy both equations: (A) y = 2x − 5 −19 = 2 × −7 − 5 (B) y = 3x + 2 −19 = 3 × −7 + 2

U N SA C O M R PL R E EC PA T E G D ES

y = −19 Solution: x = −7, y = −19 The point of intersection is (−7, −19).

Substitute the x-value into (A) or (B). Choose the simplest equation.

Now you try

Solve simultaneously the equations y = 2x and 3x + 2y = 7.

Note: The substitution method is very useful when one equation, or both equations, begins with ‘y =’ or ‘x =’

See Question 1 in Exercise 5D. Every part contains an equation like y = 2x − 5, with just one letter on the LHS.

Method 2: Elimination

Sometimes, adding the two equations together causes x or y to ‘vanish’.

For example:

(A) y = 2x − 5

(B) y = −2x + 9

(A) + (B) eliminates x and gives: 2y = 4, so y = 2

Sometimes, subtraction is more useful. For example:

(A) 2x + 3y = 14 (B) 2x + y = 6

(A) − (B) eliminates x and gives: 2y = 8, so y = 4

Note: The elimination method is very useful when both equations have the same coefficient of x, or y.

See Question 3 in Exercise 5D. The letter x, or y, can be eliminated by adding the equations. In Question 4, subtraction is required.

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Example 8

The elimination method

Solve the simultaneous equations 2x − 3y = 5 and x + 3y = 7. Explanation

(A) 2x − 3y = 5 (B) x + 3y = 7 (A) + (B): 3x = 12

Name the equations (A) and (B).

U N SA C O M R PL R E EC PA T E G D ES

Solution

x=4 Substitute x = 4 into (B) x + 3y = 7 4 + 3y = 7

Adding the equations together will eliminate the −3y and +3y.

Substitute the x-value into (A) or (B). Choose the simplest equation. Solve the equation.

3y = 3

y=1 Solution: x = 4, y = 1 The point of intersection is (4, 1)

These two numbers satisfy both equations: (A) 2x − 3y = 5 2×4−3×1=5 (B) x + 3y = 7 4+3×1=7

Now you try

Solve the simultaneous equations 2x + 3y = 5 and x + 3y = 7.

Exercise 5D

Example 7

BUILDING

Solve the following pairs of simultaneous equations using the substitution method.

a x+y=7 c

b x + y = −3

y= x+3

y= x+1

y = 2x

d x + 3y = −30

x + 5y = 22

y = 3x

y = 10 − 3x

f 2x + y = 7

2x − y = 10

x=y−4

Make x or y the subject of one equation and then use the substitution method to solve the equations. Check all solutions. x−y=7

b 2x − y = 2

x + 2y = 4

x + 2y = 11

c 2x + y = 0

d 2x + y = 7

2x − y = −8

x+y=5

e x + 2y = 1

3x + y = 13

2x + y = 6

2x − 3y = 4

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5D Solving simultaneous equations algebraically

Example 8

Use the elimination method to solve these equations simultaneously by adding the equations. a 2x − y = 1

b x − 2y = 7

x + y = 14

x + 2y = 3

c 3x + 2y = 7

d −x + y = 6

5x − 2y = 1

x + 3y = 10

x + 2y = −4

f 4x + 2y = 9

−x + 2y = 12

5x − 2y = 0

U N SA C O M R PL R E EC PA T E G D ES

Use the elimination method to solve these equations simultaneously by subtracting the equations. a 3x + y = 1

7x + 2y = −1

5x + y = 7

10x + 2y = 2

c 2x − y = 2

d 5x − 3y = 8

5x − y = 14

5x − y = 16

e 4x − 2y = 4

f 3x + 5y = 17

7x − 2y = 1

2x + 5y = 20

DEVELOPING

Solve the following pairs of simultaneous equations using the elimination method. a 4x − 3y = −1

b 8x + 4y = 4

2x + 3y = 13

8x − 2y = 34

c 3x − 2y = 7

d x + y = 10

5x + 2y = 1

x−y=4

e 3x − 2y = 15

f x + 2y = 8

2x + 2y = 20

x − 2y = 4

Which is easier: substitution or elimination? 3x + 2y = 4 3x − y = 10 What is the solution?

Multiply one of the equations by a constant and then use the elimination method to solve these pairs of simultaneous equations.

a 2x + 3y = 17

b 2a + b = 7

c 3x + 2y = 10

x+y=7

a + 2b = 11

5x + y = 12

d x + 3y = 9

4x − y = 10

g 2x − 7y = 19 x + 2y = 4

4m − n = 6

3m + 2n = −1

x + 3y = 7

5x − 2y = −16

2x + y = 8

i x + 3y = 8

5x + 2y = −3

3x − y = 9

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CHALLENGING In this question, make sure that the coefficients of x or y are the same in both equations of the pair. This may require multiplying or dividing one or both equations by a number. So for a, you could multiply the first equation by 3 to get 3 × 4x + 3 × 2y = 3 × 0, which becomes 12x + 6y = 0. (When both sides of an equation are multiplied or divided by the same number, the equation remains true.) So now the pair of equations both have 12 as the coefficient of x. As an alternative you could multiply the first equation by 3 and the second equation by 2 so that the coefficient of y is 6 in both equations. Then use the elimination method to solve these pairs of simultaneous equations.

U N SA C O M R PL R E EC PA T E G D ES

4x + 2y = 0

b 4x + 3y = 15

12x − 3y = 18

5x + 2y = 10

c 3a − 7b = 2

d 5m + 2n = 3

9a + 5b = 32

7m + 3n = 4

e 5x + 2y = 28

f 4x + 3y = −28

3x + 5y = 51

5x − 6y = −35

Communicate your reasoning, understanding and fluency 9

Write a pair of simultaneous equations to represent the following problems. Find a solution to the simultaneous equations to solve the problem.

a The sum of two numbers is 38 and their difference is 24. What are the numbers? b The sum of two numbers is 12 and one of the numbers is three times the other. Find the numbers. c The sum of x and y is −4. When y is subtracted from 3x the result is 16. Find x and y. d One apple and one orange cost $0.75. Two apples and three oranges cost $1.85. What is the cost of each fruit? e Hamza has twice as much money as Kylie. If I give Kylie $25, she will have three times as much as Hamza. How much do Hamza and Kylie have altogether?

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5E Solving practical problems using simultaneous equations

5E Solving practical problems using simultaneous equations Learning intentions

• To solve practical problems by finding the point of intersection of two lines Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

When two practical situations are described using simultaneous equations, the point of intersection has an important and often different meaning depending on the situation. For example, when income is graphed against costs the point of intersection represents the point where a business changes from a loss to a profit.

Simultaneous equations as models for real life scenarios

When simultaneous equations of real life scenarios are solved, the point of intersection is often the solution to a problem.

Example 9

Using simultaneous equations as models

Zaina buys and sells books. Income received by selling a book is calculated using the formula I = 16n. Costs associated in selling a book are calculated using the formula C = 8n + 24.

a What is the income when 6 books are sold? b What are the costs when 6 books are sold? c Draw the graph of I = 16n and C = 8n + 24 on same number plane. d Use the graph to determine the number of books needed to be sold for the costs to equal the income.

Solution

Explanation

a I = 16n = 16 × 6 = 96 ∴ Income for six books is $96

Substitute 6 for n into the formula for income I = 16n.

b C = 8n + 24 = 8 × 6 + 24 = $72 ∴ Costs for six books are $72

Substitute 6 for n into the formula for costs C = 8n + 24.

Continued on next page

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c Amount ($)

100 Income 80 60 Costs

U N SA C O M R PL R E EC PA T E G D ES

Draw a number plane. Use the gradient–intercept form to determine the gradient and vertical intercept for each line. Gradient is the coefficient of n. Vertical intercept is the constant term. Sketch I = 16n using the vertical intercept of 0 and gradient of 16. Sketch C = 8n + 24 using the vertical intercept of 24 and gradient of 8.

1 2 3 4 5 6 Number of books sold

d Income is equal to costs when n = 3 ∴ 3 books

Find the point of intersection of the two lines (3, 48).

Now you try

The cost of running a dinner is given by C = 200 + 50n, where n is the number of people who attend. Tickets are sold for $75 each, so the revenue is given by R = 75n.

a Draw the two lines on the same number plane. b Use the graph to determine the number of people required for the revenue to equal the cost.

Example 10

Solving problems using intersecting graphs

Isabella’s Mathematics mark exceeded her English mark by 15. She scored a total of 145 for both tests. Find Isabella’s marks in each subject by plotting intersecting graphs.

Solution

Explanation

Let the Mathematics mark be m. Let the English mark be e. m = e + 15 Gradient is 1, vertical intercept is 15.

Express the relationship between the Mathematics and the English mark as a linear equation.

m + e = 145

m = −e + 145 Gradient is 1, vertical intercept is 145.

Use the gradient–intercept form to determine the gradient and vertical intercept for the line. Gradient is the coefficient of e. Vertical intercept is the constant term. Express the total of the two marks as a linear equation. Use the gradient–intercept form to determine the gradient and vertical intercept for the line.

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5E Solving practical problems using simultaneous equations

Draw a number plane. Sketch m = e + 15 using the vertical intercept of 15 and gradient of 1 (blue line). Sketch m = −e + 145 using the vertical intercept of 145 and a gradient of −1 (red line). The simultaneous solution is the point of intersection.

140 120 100 80

U N SA C O M R PL R E EC PA T E G D ES

60 40 20

10 20 30 40 50 60 70 80 90 100

Intersection is (65, 80) so e = 65 and m = 80 Isabella scored 65 in English and 80 in Mathematics.

Find the point of intersection of the two lines. Write the solution in words using the context of the question.

Now you try

Anna has $1000 in her money box. She begins spending $50 per week.

a Draw a graph of this model. b What is the equation of this line?

Jasmine has no money in her money box for the first 8 weeks, then she begins to add $50 per week to her money box.

c Draw a graph of this model. d At what time, in weeks, do Anna and Jasmine have the same amount of money in their money boxes?

Exercise 5E 1

BUILDING

Matilda and Nate have wages m and n respectively.

a Matilda earns $100 more than Nate. Write an equation to describe this information. b The total of Matilda’s wages and Nate’s wages is $1200. Write an equation to describe this information. c Draw a graph of the two equations on the same number plane. Use n as the horizontal axis and m as the vertical axis. d Use the intersection of the two graphs to find Matilda’s and Nate’s wages.

Let one number be represented by a and the other number by b.

a The sum of the two numbers is 42. Write an equation to describe this information. b The difference of the two numbers is 6. Write an equation to describe this information. c Draw a graph of the two equations on the same number plane. Use a as the horizontal axis and b as the vertical axis. d Use the intersection of the two graphs to find the two numbers.

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Let one number be represented by p and another number by q.

U N SA C O M R PL R E EC PA T E G D ES

a The sum of the two numbers is 15. Write an equation to describe this information. b One of the numbers is twice the other number. Write an equation to describe this information. c Draw a graph of the two equations on the same number plane. Use p as the horizontal axis and q as the vertical axis. d Use the intersection of the two graphs to find the two numbers. 4

Sana and Nghi work for the same company and their wages are a and b respectively.

a Sana earns $100 more than Nghi. Write an equation to describe this information. b The total of Sana’s and Nghi’s wages is $1500. Write an equation to describe this information. c Draw a graph of the above two equations on the same number plane. Use a as the horizontal axis and b as the vertical axis. d Use the intersection of the two graphs to find Sana’s and Nghi’s wages.

Example 9

5 A factory produces items whose costs are $1000 plus $10 for every item. The factory receives $60 for every item sold. a Write an equation to describe the relationship between the: i costs (C) and number of items (n) ii income (I) and number of items (n).

b Draw a graph and find the number of items when income equals costs.

DEVELOPING

Let the cost of an apple be a and the cost of a banana be b.

a One apple and one banana cost $1.25. Write an equation (using cents, not dollars) to describe this information. b Two apples and three bananas cost $3.25. Write an equation (using cents, not dollars) to describe this information. c Draw a graph of the two equations on the same number plane. Use a as the horizontal axis and b as the vertical axis. d Use the intersection of the two graphs to find the cost (in cents) of each fruit.

Example 10

Let Zara’s hourly wage rate be a and Bayani’s hourly wage rate be b.

a The sum of Zara’s hourly wage rate and Bayani’s hourly wage rate equals $70. Write an equation to describe this information. b Twice Zara’s wage rate is the same as three times Bayani’s wage rate. Write an equation to describe this information. c Draw a graph of the two equations on the same number plane. Use a as the horizontal axis and b as the vertical axis. d Use the intersection of the two graphs to find the wage rates.

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5E Solving practical problems using simultaneous equations

CHALLENGING A business has five times as many junior workers as it does senior workers. If the weekly wage for a senior is $800 and for a junior is $600, find how many of each are employed if the total weekly wage bill is $57 000.

A cinema has 2340 seats. All of the rows of seats in the cinema have either 50 seats or 45 seats. If there are three times as many rows with 50 seats as those with 45 seats, find the total number of rows in the cinema.

U N SA C O M R PL R E EC PA T E G D ES

At a birthday party for 20 people, each person could order chocolate or strawberry ice-cream. If there were four times as many orders for chocolate as for strawberry, how many orders were taken for chocolate ice-cream? How many people ordered strawberry?

Communicate your reasoning, understanding and fluency 11

A factory makes watches and clocks. In any week:

• the total number of watches and clocks made is 100 • the maximum number of clocks made is 70

• the maximum number of watches made is 80.

The graph below shows the number of clocks (x) and watches (y) that can be made. y A 100

Number of watches

Number of clocks

100

a What is the equation of the line AD? b Why is this line only applicable between the points B and C for this factory? c The profit per week, $P, can be found using the equation P = 40x + 50y. Compare the profits at B and C.

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5F Break-even analysis Learning intentions

• To use straight lines to model income (revenue) and costs (expenses) • To find the break-even point at which income and costs are equal

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Students may already be familiar with the contents of this section • Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving break-even analysis regularly appear in HSC Examinations

The break-even point is reached when costs (or expenses) and income are equal. There is no profit or loss at the break-even point. For example, if the break-even point for a business is 100 items per month, the business will make a loss if it sells fewer than 100 items each month; if it sells more than 100 items per month, it will make a profit. A profit (or loss) is calculated by subtracting the costs from the income (Profit = Income – Costs). Income is a linear function of the form I = mx, where x is the number of items sold and m is the selling price of each item. Cost is a linear function of the form C = mx + c, where x is the number of items sold, m is the cost price per item manufactured and c is the fixed costs of production.

Break-even analysis

Break-even point occurs when costs equal income. Profit = Income − Costs

Income: I = mx

Costs: C = mx + c

Interpreting the point of intersection of two graphs

Noor buys and sells wallets. Income received by selling wallets is calculated using the formula I = 30x. Costs associated with selling wallets are calculated using the formula C = 20x + 30.

a Use the graph to determine the number of wallets that Noor needs to sell to break even. b How much profit or loss does she make when four wallets are sold?

160

Income

140

Amount ($)

Example 11

120

Loss zone

Costs

100 80 60

Proﬁt zone

3 4 2 Number of wallets sold

Solution

Explanation

a Break-even point occurs when the income equals the costs. Intersection is at (3, 90). So x = 3. Number of wallets = 3

Consider when the break-even point occurs.

Read the point of intersection of the two linear graphs.

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5F Break-even analysis

b Profit = Income − Cost

Profit is determined by subtracting the costs from the income. Read from the graph the values of I and C when x = 4. Evaluate.

I = 120 and C = 110 Profit = 120 − 110

U N SA C O M R PL R E EC PA T E G D ES

= $10

Write the answer in words.

Profit for selling 4 wallets is $10.

Now you try

Using Example 11:

a what is the loss if Noor sells no wallets? b what is the profit if she sells 10 wallets?

Example 12

Break-even analysis

A firm sells its product at $20 per unit. The cost of production $C is given by the rule C = 4x + 48, where x is the number of units produced.

a Find the value of x for which the cost of the production of x units is equal to the income or revenue received by the firm for selling x units. b Check your answer algebraically. Explanation

a Let the income or revenue for producing x units be $I. Formula is: I = 20x Gradient is 20, vertical intercept is 0. Cost of production $C is given by: C = 4x + 48 Gradient is 4, vertical intercept is 48

Set up the income equation and determine the gradient and vertical intercept.

Amount ($)

Solution

200 180 160 140 120 100 80 60 40 20

Income

I = 20x

Break-even point

Costs

C = 4x + 48

Set up the cost of production equation and determine the gradient and vertical intercept.

Draw a number plane. Use x as the horizontal axis. Use I and C as the vertical axis. Sketch I = 20x using the vertical intercept of 0 and gradient of 20. Check this line using some valid points such as (1, 20). Sketch C = 4x + 48 using the vertical intercept of 48 and gradient of 4. Check this line using some valid points such as (1, 52).

1 2 3 4 5 6 7 8 9 10 Number of units (x)

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Read the value of x at the point of intersection of the two linear graphs.

b Check algebraically.

Substitute x = 3 into the formula I = 20x.

U N SA C O M R PL R E EC PA T E G D ES

The point of intersection of the two linear graphs occurs when x = 3. This is the break-even point, the value of x for which cost of production is equal to income.

Income

Costs

I = 20x

C = 4x + 48

= 20 × 3

= 4 × 3 + 48

= 60

Substitute x = 3 into the formula C = 4x + 48.

Income equals costs, so answer to a is correct.

Check that I is equal to C.

Now you try

Using Example 12, how many units needs to be sold to make a profit $112?

Exercise 5F

Example 11

BUILDING

What is the break-even point for the following graphs?

300

Income

Costs

Amount ($)

Costs

200

100

Income

2 3 4 Number of shirts (n)

2 The graph on the right shows the cost of producing boxes of chocolates and the income received from their sale.

6 8 4 Number of printers (n)

Income

Amount ($)

a Use the graph to determine the number of boxes that need to be sold to break even. b How much profit or loss is made when 3 boxes are sold? c How much profit or loss is made when 1 box is sold? d What are the initial costs?

Costs

2 3 4 Number of boxes

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5F Break-even analysis

The graph on the right shows the cost of producing cartons of soft drinks and the income received from their sale.

40 30

Costs

U N SA C O M R PL R E EC PA T E G D ES

a Use the graph to determine the number of cartons that need to be sold to break even. b How much profit or loss is made when five cartons are sold? c How much profit or loss is made when two cartons are sold? d What is the initial cost? e What is the gradient of the straight line that represents income? f What is the vertical intercept of the straight line that represents income?

Amount ($)

Income

3 4 2 Number of cartons

g Write an equation to describe the relationship between income and the number of cartons. h What is the gradient of the straight line that represents costs? i What is the vertical intercept of the straight line that represents costs? j Write an equation to describe the relationship between costs and the number of cartons.

DEVELOPING

The graph on the right shows the cost of producing bottles of wine and the income received from their sale.

a Use the graph to determine the number of bottles that need to be sold to break even. b Write an equation to describe the relationship between income and the number of bottles. c Write an equation to describe the relationship between costs and the number of bottles.

200

Amount ($)

150

Costs

100

Income

6 8 4 Number of bottles

Levi and Atama earn a salary of L and A respectively.

a Levi earns $10 000 more than Atama. Write an equation to describe this information. b The total of Levi’s salary and Atama’s salary is $150 000. Write an equation to describe this information. c Draw a graph of the above two equations on the same number plane. Use A($1000) as the horizontal axis and L($1000) as the vertical axis. d Use the intersection of the two graphs to find Levi’s and Atama’s salaries.

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CHALLENGING Example 12

An industrial plant produces items whose costs are $250 plus $50 for every item. The plant receives $75 for every item sold. a Write an equation to describe the relationship between the:

U N SA C O M R PL R E EC PA T E G D ES

i costs (C) and number of items (x) ii income (I) and number of items (x).

b Draw a graph to represent the costs and income for producing the item. c How many items need to be sold to break even? d Check your answer to part c algebraically.

Jun owns a nursery that grows native plants. It costs Jun $4000 per month to cover costs such as rent, water, electricity and wages. Each plant costs $5 and is sold for $17.50.

a Write an equation to describe the relationship between: i the costs (C) and the number of plants (x) ii income (I) and the number of plants (x).

b Draw a graph to represent the costs and income for producing the plants. c How many plants need to be sold each month to break even? d Check your answer to part c algebraically.

Communicate your reasoning, understanding and fluency 8

Sakura makes and sells birdhouses for $40 each. The cost of doing this is $500 plus $15 per bird house. She wants to make $1900 profit. How many bird houses will she need to make?

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Chapter 5 review

Construct a table of values with the independent variable as the first row and the dependent variable as the second row.

Draw a number plane with the independent variable on the horizontal axis and the dependent variable as the vertical axis. Plot the points.

U N SA C O M R PL R E EC PA T E G D ES

Graphing lines

Review

Key ideas and chapter summary

Join the points to make a straight line.

Gradient–intercept formula

Linear equation: y = mx + c. m = Slope or gradient of the line (vertical rise over the horizontal run). c = y-intercept Where the line cuts the y-axis or vertical axis.

Linear models

Linear models describe a practical situation mathematically using a straight line.

Simultaneous equations – graphically

Simultaneous equation – algebraically

Draw a number plane.

Graph both linear equations on the number plane.

Read the point of intersection of the two straight lines.

Interpret the point of intersection for practical applications (break-even point).

Substitution method

Make one pronumeral the subject and substitute the expression for this subject into the other equation.

Elimination method

Eliminate one pronumeral by adding or subtracting the two equations.

Simultaneous equation as models

Simultaneous equation models use two straight lines to describe a practical situation and the point of intersection is often the solution to a problem.

Break-even analysis

Break-even point occurs when costs equals income. Profit = Income – Costs Income: I = mx Costs: C = mx + c

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Review

302

Chapter 5 Simultaneous linear equations

Chapter checklist with success criteria A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

1 I can use the equation of a straight line to graph it on the Cartesian plane.

U N SA C O M R PL R E EC PA T E G D ES

e.g. Use a table of values to draw the straight line y = 2x − 4.

2 I can use the y-intercept and gradient to graph a straight line on the Cartesian plane. e.g.

Make y the subject of the subject of the equation 2x + y − 6 = 0.

Use the gradient and y-intercept to draw the line.

3 I can determine the equation of a straight line from a graph. e.g. What is the equation of this straight line? y

3 2 1

–3 –2 –1 O –1 –2 –3 –4 –5

1 2 3

4 I can use straight lines as models for linear relationships.

e.g. A taxi driver charges $4 base fee and $2 per kilometre. What is the equation for the cost of a taxi fare (C) for n kilometres?

5 I can solve problems involving linear relationships.

e.g. Using the equation in Question 4, how many kilometres can be travelled for $50?

6 I can solve simultaneous equations graphically.

e.g. Draw the lines y = x + 1 and y = 5 − x. What is their point of intersection?

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Chapter 5 review

7 I can solve simultaneous equations algebraically. e.g. a

Use the substitution method to solve simultaneously. 2x + y = 5

Review

U N SA C O M R PL R E EC PA T E G D ES

y=2+x b

Use the elimination method to solve simultaneously. 2x + y = 7

2x − y = −3

8 I can solve practical problems using simultaneous equations.

e.g. The sum of two numbers is 114. One number is twice as big as the other. What are the numbers?

9 I can solve problems involving profit, loss and the break-even point. e.g. The cost of making gift packs is $200 plus $40 per pack. They sell for $60 each.

How many need to be sold to break even?

How many need to be sold to make a profit of $1000?

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Chapter 5 Simultaneous linear equations

Multiple-choice questions 5A

2 What is the y-intercept of the line shown at right?

y 3 2 1

What is the gradient of the line shown at right? 3 2 2 3 A − B − C D 2 3 3 2

O −3 −2 −1 −1

U N SA C O M R PL R E EC PA T E G D ES

Review

304

A −2

C +1

B −1

1 2 3

−2 −3

D +3

B $150

C $190

D $230

5 A car is travelling at a constant speed. It travels 80 km in 4 hours. This situation is described by the linear equation d = mt. What is the value of m? A 0.05

D 2

The cost of manufacturing bags (C) is given by the formula C = 40x + 150, where x is the number of bags sold. What is the cost of manufacturing two bags? A $40

C 1

3 A straight line has the equation of y = −x − 3. What is the y-intercept? A −3

B −1

B 3

C 20

D 60

6 What is the point of intersection of the lines y = x + 2 and y = −x + 2? A (2, 0) B (0, 2) C (0, −2) D (1, 1)

Use the graph below to answer questions 7–9.

Amount ($)

300

200

Costs

100

Income

B $20

C $220

D $240

C 20 items

D 80 items

C $40

D $50

8 What is the break-even point? A 10 items

4 6 8 Number of items

What is the profit for selling 12 items? A $10

B 12 items

9 What is the loss for selling 5 items? A $20

B $30

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Chapter 5 review

Review

Short-answer questions 5A

Draw the graph of these linear functions.

U N SA C O M R PL R E EC PA T E G D ES

a y= x+2 b y = −3x + 1 c y = 2x − 2

2 Find the equation of the following straight-line graphs. y 4 3 2 1

O −4 −3 −2 −1 −1

1 2 3 4 x

−2 −3 −4

y 4 3 2 1

1 2 3 4 x

−2 −3 −4

3 The table below shows the speed v (in km/s) of a plane at time t seconds. Time (t)

Speed (v)

2.5

5.5

8.5

a Draw a number plane with t as the horizontal axis and v as the vertical axis. Plot the points and join them to make a straight line. b Determine a linear model in the form y = mx + c to describe this situation. c What does the model predict will be the plane’s speed when t = 2.5 seconds? d What does the model predict will be the plane’s speed when t = 6 seconds? e What does the model predict will be the plane’s speed when t = 7 seconds? f What does the model predict will be the plane’s speed when t = 10 seconds?

An internet access plan charges an excess fee of $12 per GB. Data (d)

Cost (c)

a Draw a graph of data against cost. b Use the graph to find d if c is 30. c Use the graph to find c if d is 3.5. d Estimate the cost of 7 GB of data. e Estimate the cost of 10 GB of data. f Estimate the cost of 8.5 GB of data.

What is the point of intersection of the lines y = 3x + 3 and = x − 2 ?

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Chapter 5 Simultaneous linear equations

This graph shows the cost of making picture frames and the income received from their sale. a Use the graph to determine the number of picture frames that need to be sold to break even. b How much profit or loss is made when one picture frame is sold? c How much profit or loss is made when four picture frames are sold? d What is the initial cost?

U N SA C O M R PL R E EC PA T E G D ES

Review

306

100

Amount ($)

80 60

Costs

40 20

Income

2 1 3 Number of frames

This graph shows the cost of manufacturing tables and the income received from their sale.

a Use the graph to determine the number of tables which need to be sold to break even. b Write an equation to describe the relationship between income and the number of tables. c Write an equation to describe the relationship between costs and the number of tables. d How much profit or loss is made when 10 tables are sold? e How much profit or loss is made when 4 tables are sold? f How much profit or loss is made when 16 tables are sold? 500

Income

Amount ($)

400 300

Costs

200 100

2 4 6 8 10 12 14 16 Number of tables

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Chapter 5 review

Solve these simultaneous equations algebraically. a

y = 2x − 1

b x+y=6

x+y=5 x−y=0

d 2x + 3y = 4

2x − y = 5

2x + 5y = 8

U N SA C O M R PL R E EC PA T E G D ES

y = 2x − 9

Review

Extended-response questions

Let the cost of item A be x and the cost of item B be y.

a The total cost of both items is $1.80. Write an equation (using cents, not dollars) to describe this information. b One item A and two item B’s cost $2.75. Write an equation (using cents, not dollars) to describe this information. c Draw a graph of the above two equations on the same number plane. Use x as the horizontal axis and y as the vertical axis. d Use the intersection of the two graphs to find the cost (in cents) of item A.

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U N SA C O M R PL R E EC PA T E G D ES

Bivariate data analysis

Bivariate data analysis in sports Coaches use bivariate data analysis to inform critical decisions that impact player selection, rest schedules and how to maximise potential on the field. By examining relationships between different variables, they can refine team strategies, optimise individual contributions and ultimately improve their chances of winning games. Variables analysed include:

• • • •

fielding errors vs game pressure possession time vs points scored years of experience vs game performance player height vs shooting accuracy (basketball, netball) • batting average vs home runs (baseball) • distance run vs ball possession time (soccer) • player weight vs position (NRL).

• training hours vs injury rate • weather conditions vs win rate

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Chapter contents 6A 6B 6C 6D 6E 6F 6G

Constructing a bivariate scatter plot Using a bivariate scatter plot Pearson’s correlation coefficient Line of best fit Least-squares regression line Interpolation and extrapolation Statistical investigations

U N SA C O M R PL R E EC PA T E G D ES

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Analyses bivariate datasets using statistical processes (MST-12-S2-08). © NSW Education Standards Authority

Online resources

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Chapter 6 Bivariate data analysis

6A Constructing a bivariate scatter plot Learning intentions

• To understand and read bivariate scatter plots to draw conclusions • To construct bivariate scatter plots to identify patterns in data

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving bivariate scatter plots regularly appear in HSC Examinations

Bivariate data is data that has two variables.

A scatter plot is used to determine if there is a relationship between two numerical variables. Data is collected on the two variables and often displayed in a table of ordered pairs. A scatter plot is a graph of the ordered pairs of numbers. Each ordered pair is a dot on the graph. To illustrate this process, a scatter plot has been constructed to determine the relationship between the height and arm span. The data collected on these variables is shown below in the table of ordered pairs. Height (cm)

172 159 178 162 156 174 151 162 165 185 186 176 166 180 158

Arm span (cm) 172 162 182 164 159 180 151 165 168 189 188 184 167 184 161 200 190

Arm span (cm)

Each person has two numerical variables, height and arm span. To construct a scatter plot, draw a number plane with the height on the horizontal axis and arm span on the vertical axis. Plot each ordered pair as a dot. The scatter plot shows there is a relationship between these variables.

180 170 160

150 O150

160

170 180 Height (cm)

190

200

Constructing a scatter plot 1 Draw a number plane. 2 Determine a scale and a title for the horizontal or x-axis. 3 Determine a scale and a title for the vertical or y-axis. 4 Plot each ordered pair of numbers with a dot. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

6A Constructing a bivariate scatter plot

Example 1

Reading a scatter plot

The average numeracy score for year 6 students and their general rate of internet use (%) for 10 countries are displayed in this scatter plot.

Numeracy score

U N SA C O M R PL R E EC PA T E G D ES

200 150 100 50

30 40 50 Internet use (%)

a What is the scale for the vertical axis? b What is the average numeracy score for the country which has an internet use rate of 24%? c What is the internet use (%) for the country which has an average numeracy score 160? d How many countries have internet use of less than 50%? e How many countries have a numeracy score greater than 100? f Is there a relationship between these two variables? Solution

Explanation

a 1 unit = 10

Count the number of divisions between 0 and 50 (5). Therefore 1 unit is 50 divided by 5 (10).

b 120

Read from the scatter plot (when internet use is 24% the numeracy score is 120).

c 50%

Read from the scatter plot (when the numeracy score is 160 the internet use is 50%).

d 6 countries

Count the number of dots less than 50% (left-hand side).

e 6 countries

Count the number of dots greater than 100 (top-half).

f Yes, there is a relationship. When the internet use is greater than 20%, both the variables are increasing.

Look for any pattern in the dots. In this scatter plot when the internet use is greater than 20%, there is a clear increase in the numeracy score. However, this relationship does not exist when the internet use is less than 20%.

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Chapter 6 Bivariate data analysis

Now you try

220 200 180 160 140 120 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 Fuel Price ($/L)

U N SA C O M R PL R E EC PA T E G D ES

Train commuters (Thousands)

The scatter plot below shows the cost of fuel (in cost per litre) and the average number of people on trains per day (per 1000 people).

a What is the scale for the vertical axis? b How many train commuters will there be if the cost of fuel is $1.70/L? c If there are 170 000 commuters, what would be the price per litre of fuel? d Is there a relationship between these two variables?

Exercise 6A

The scatter plot shows the results for 15 students in two tests. a What is the highest mark in test 2? b What is the lowest mark in test 1? c What is the range for test 1? d What is the mode for test 2? e How many students scored greater than 6 in test 1?

10 8

6 4

The scatter plot shows the head height to age for 10 people.

a What is the head height for the person who is 21 years old? b What is the age of the person who has a head height of 22 cm? c What is the largest head height? d What is the age of the youngest person? e How many people have a head height greater than 23 cm? f Is there a clear relationship between these two variables?

10 12 Test 1

Head height (cm)

Test 2

Example 1

BUILDING

24 23 22 21

22 24 Age (years)

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6A Constructing a bivariate scatter plot

The table below shows the height (in cm) of a eucalypt tree seedling as it grows. Time (weeks) 0 Height (cm)

0 6.6 8.8 9.0 10.5 12.0 13.5 15.2 15.4 15.8 15.9

a Copy the number plane opposite to construct a scatter plot using the above table. b What is the increase in the height of the seedling during the first week? c What is the increase in the height of the seedling during the last week? d How many weeks does it take for the seedling to increase in height from 9 cm to 12 cm? e Estimate the height of the seedling after 4.5 weeks. f Estimate the time taken for the seedling to grow to a height of 14 cm.

U N SA C O M R PL R E EC PA T E G D ES

Height (cm)

12 10 8

6 4

6 8 4 Time (weeks)

DEVELOPING

Haruto is a political commentator who has been studying the effects of television exposure time on the approval ratings of nine politicians. The data is shown below. Time (minutes)

Approval rating (%)

a Construct a scatter plot of the data given in the table. b Are there any conclusions to be drawn from the scatter plot?

The table shows the number of runs scored and the number of balls faced by batsmen in a one-day cricket match. Balls faced

Runs scored

a Construct a scatter plot of the data given in the table. b Are there any conclusions to be drawn from the scatter plot?

The maximum wind speed and maximum temperature in a location were recorded for 2 weeks. The data is displayed in the table below. Wind speed (km/h)

Temperature (◦ C)

28 26 23 22 21 22 19 16 20 24 25 24 19 26

12 15 19 20 22 25 17 14

11 24 13

a Construct a scatter plot of the data given in the table. b Are there any conclusions to be drawn from the scatter plot? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Chapter 6 Bivariate data analysis

CHALLENGING The data below shows the litres of petrol used over a certain distance travelled in kilometres. Petrol used (L)

Distance travelled (km)

100

120

145

180

210

U N SA C O M R PL R E EC PA T E G D ES

a Construct a scatter plot using the data above. b Approximately how far would you travel on 14 litres of petrol? c If you travel 200 km, how many litres of petrol will you need? d Describe the relationship between the two variables.

Communicate your reasoning, understanding and fluency 8

The table below shows the age (in months) of six cars, and the minimum stopping distance (in metres) when the car is travelling at 60 km/h. Age of car (months)

Stopping distance (metres)

a Construct a scatter plot using the above data. b Calculate the mean age of the car and the mean stopping distance. Answer correct to one decimal place. c Do you agree with the following statement? Justify your answer. ‘As a car gets older, its stopping distance increases.’ d Are there any limitations when making predictions from this data?

Investigation: Investigating how the presentation of data can influence decision-making in a government funding meeting

In this assignment, you will examine how statistics and visual communication influence decisionmaking by reviewing and critiquing a real-world style infographic used in a government funding meeting. You’ll then work with bivariate data to come up with your own infographic. This investigation should be done after you have covered sections 6A-6G. The activity can be downloaded from your Interactive Textbook.

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6B Using a bivariate scatter plot

6B Using a bivariate scatter plot Learning intentions

• To use bivariate scatter plots to describe the patterns, features and associations of bivariate datasets • To identify the independent variable and the dependent variable

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving bivariate scatter plots regularly appear in HSC Examinations

What are the features in a scatter plot that will identify and describe any relationship? First look for a clear pattern. In the scatter plot opposite, there is no clear pattern in the points: they are just randomly spread on the scatter plot. There is no relationship or association between the variables.

For the three examples below, there is a clear (but different) pattern in each set of points, so we conclude that there is an association in each case.

Having found a clear pattern, we need to be able to describe these associations clearly, as they are obviously quite different. There are three things we look for in the pattern of points: form, direction and strength.

Form of an association

If an association exists between the variables then the points in a scatter plot tend to follow a linear pattern or a curved pattern. This is called the form of an association. Linear form

Non-linear form

If the points seem to approximate a straight line, the association is a linear form. y

If the points seem to approximate a curve, the association is a non-linear form. y

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Chapter 6 Bivariate data analysis

Form of an Association Linear form – when the points tend to follow a straight line. Non-linear form – when the points tend to follow a curved line.

U N SA C O M R PL R E EC PA T E G D ES

Direction of an association There are two types of direction if the association is in linear form. Positive

Negative

Positive association exists between the variables if the gradient of the line is positive. That is, the dots on the scatter plot tend to go up as we go from left to right.

Negative association exists between the variables if the gradient of the line is negative. That is, the dots on the scatter plot tend to go down as we go from left to right.

Direction of an Association

Positive – gradient of the line is positive.

Negative – gradient of the line is negative.

Strength of an association

The strength of an association is a measure of how much scatter there is in the scatter plot. Strong

Moderate

Weak

In a strong association, the dots will tend to resemble the shape of a line or smooth curve. A pattern is clearly seen. There is only a small amount of scatter in the plot.

In a moderate association, the amount of scatter in the plot increases and the pattern becomes less clear. This indicates that the association is less strong.

In a weak association, the amount of scatter increases further and the pattern becomes even less clear. Linear form is less evident.

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6B Using a bivariate scatter plot

Strength of an Association Strong – small amount of scatter in the plot. Moderate – modest amount of scatter in the plot. Weak – large amount of scatter in the plot.

U N SA C O M R PL R E EC PA T E G D ES

Example 2

Describing bivariate datasets

The table below shows the height (cm) and arm length (cm) of nine people. Height (cm)

163

165

170

175

178

180

182

186

190

Arm length (cm)

a Construct a scatter plot using the above table. b Describe the form of the association. c Describe the direction of the association. d Describe the strength of the association. e Predict the arm length of a person who is 173 cm tall using the scatter plot. f Predict the height of a person who has a arm length of 75 cm using the scatter plot.

Solution l a

Explanation

Draw a number plane with height as the horizontal axis and arm length as the vertical axis.

Arm length (cm)

80 75

Determine a scale for the horizontal axis. Let each unit represent 1 cm.

Determine a scale for the vertical axis. Let each unit represent 1 cm.

Write titles for the horizontal and vertical axes.

Plot the points from the table.

50 160

165

170 175 180 Height (cm)

185

190

b Linear form

Look for a pattern. The points approximate a straight line.

c Positive

Gradient of the line is positive. The dots tend to go up as you move from left to right.

d Strong

There is a small amount of scatter in the scatter plot. Draw an imaginary vertical line from 173 cm.

e The person has an arm length of approximately 65 cm.

Try to maintain the linear relationship and guess the arm length. Draw an imaginary horizontal line from 75 cm.

f The person’s height is approximately 187 cm.

Try to maintain the linear relationship and guess the height.

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Chapter 6 Bivariate data analysis

Now you try This table shows the heights (in cm) and masses (in kg) of six kangaroos. Height (cm)

100

120

140

160

180

200

Mass (kg)

U N SA C O M R PL R E EC PA T E G D ES

a Draw a scatter plot using the table. b Is the form linear or non-linear? c Describe the direction of the association. d Describe the strength of the association. e Predict the mass of a kangaroo who is 180 cm tall using the scatter plot. f Predict the height of a kangaroo who has a mass of 60 kg using the scatter plot.

Independent and dependent variables

Bivariate data has two variables that are often identified as the independent and dependent variables. The independent variable is the input. It is not affected by the other variable and is represented on the horizontal axis of the scatter plot. The dependent variable is the output and is ‘dependent’ on the independent variable. It is represented on the vertical axis of a scatter plot.

Example 3

Identifying independent and dependent variables

The table below shows the time taken (hours) relative to the distance travelled (km). Distance d (km)

Time t (hours)

0.25

0.38

0.59

0.82

1.00

a Draw a scatter plot using the above table. b Which are the independent and dependent variables? c Describe the strength and direction of the association. Solution

Explanation

Draw a number plane with d as the horizontal axis and t as the vertical axis. Determine a scale for the horizontal axis. Let each unit represent 10 km. Determine a scale for the vertical axis. Let each unit represent 0.2 hours. Write titles for the horizontal and vertical axes. Plot the points from the table.

Time (hours)

0.8 0.6 0.4 0.2

d 10

20 30 40 Distance (km)

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6B Using a bivariate scatter plot

b Independent variable is distance (d).

The independent variable is the input and represented on the horizontal axis of the scatter plot.

Dependent variable is time (t).

The dependent variable is the output and represented on the vertical axis of the scatter plot. Points lie in a line with a positive gradient. Clear pattern, strong association.

U N SA C O M R PL R E EC PA T E G D ES

c strong and positive Now you try

The table shows satisfaction levels in comparison to overtime hours worked. Overtime hours worked

Satisfaction levels

90%

85%

80%

60%

45%

15%

a Draw a scatter plot using the table. b Identify the dependent and independent variables. c Describe the strength and direction of the scatter plot.

Exercise 6B

Example 2

BUILDING

Describe the association in the following scatter plots as: i linear or non-linear ii positive or negative iii strong, moderate or weak.

50 40 30 20 10

90 80 70 60

6 5 4 3 2 1

5 10 15 20 25 30 35 40 Time (hours)

Performance level

180

Daughter’s height (cm)

9 8 7

Price (S’000)

Mark (%)

170 160

4 6 Age (Years)

5 4 3 2 1

150

1 2 3 4 5 6 7 8 9 10 Time spent practising

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Chapter 6 Bivariate data analysis

f Wife’s age (years)

10 5

40 35 30 25 20 15 10

U N SA C O M R PL R E EC PA T E G D ES

Score on test

25 30 35 Temperature (°C)

10 15 20 25 30 35 40 45 Husband’s age (years)

For each of the following pairs of variables, indicate whether you expect an association to exist and, if so, whether you would expect the association to be positive or negative. a Independent variable: Distance travelled

Dependent variable: Time taken

b Independent variable: Amount of daily exercise

Dependent variable: Fitness level

c Independent variable: Foot length of an adult

Dependent variable: Intelligence

d Independent variable: Number of pages

Dependent variable: Book price

e Independent variable: Temperature above 30◦ C

Dependent variable: Comfort level

DEVELOPING

Example 3

Chocolates are sold for $12 per kg. The following table shows weight against cost. Weight w (kg)

Cost c ($)

a Which is the independent variable? b Which is the dependent variable? c Draw a scatter plot of weight against cost. d Is the form linear or non-linear? e Is the direction positive or negative? f Is the strength strong, moderate or weak?

The following table shows the drug dosage against the response time for the drug to start taking effect. Dosage d (mg/kg)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0

Response time t (days)

a Which is the independent variable? b Which is the dependent variable? c Draw a scatter plot of dosage against reaction time. d Is the form linear or non-linear? e Is the direction positive or negative? f Is the strength strong, moderate or weak?

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6B Using a bivariate scatter plot

Salah conducted a science experiment and presented the results in a table. Mass m (grams)

Time t (seconds)

8.2

6.7

5.2

3.7

2.2

U N SA C O M R PL R E EC PA T E G D ES

a Which is the independent variable? b Which is the dependent variable? c Draw a scatter plot of mass against time. d Is the form linear or non-linear? e Is the direction positive or negative? f Is the strength strong, moderate or weak?

CHALLENGING

The following table shows leg length compared to height. Leg length (cm)

Height (cm)

166

167

170

174

179

178

183

185

188

a Draw a scatter plot using the above table. b Describe the association between the leg length and height.

The following table shows forearm length and hand length. Forearm (cm)

Hand (cm)

18.5

19.5

17.5

18.5

a Draw a scatter plot using the above table. b Describe the association between the forearm length and hand length.

Communicate your reasoning, understanding and fluency 8

The following table demonstrates the effects of sleep against test scores for a group of 10 students. Sleep (hours)

Test scores (%)

20%

40%

30%

40%

35%

50%

60%

50%

60%

a Draw a scatter plot using the table. b Is there an association between the two variables? If so, what type of form does the association have? c Which are the dependent and independent variables? d Describe the strength and direction of the association. e Discuss the limitations of this scatter plot.

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6C Pearson’s correlation coefficient Learning intentions

• To calculate and interpret Pearson’s correlation coefficient to quantify the strength of a linear association

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving correlation regularly appear in HSC Examinations

The strength of a linear association is an indication of how closely the points in the scatter plot fit a straight line. If the points in the scatter plot lie exactly on a straight line, we say there is a strong linear association. If there is no fit at all we say there is no association. To measure the strength of a linear association we use Pearson’s correlation coefficient (r), which has the following properties: No linear association

Positive linear association

Negative linear association

r=0

r = +1

r = −1

Pearson’s correlation coefficient r has a value between −1 and +1. The closer it is to −1 or +1, the stronger the association. The scatter plots below show the approximate values of r for positive linear associations of varying strengths. Weak positive linear

Moderate positive linear

Strong positive linear

association r = +0.3

association r = +0.6

association r = +0.9

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6C Pearson’s correlation coefficient

Pearson’s Correlation Coefficient Pearson’s correlation coefficient (r) measures the strength of a linear association (−1 ≤ r ≤ +1).

U N SA C O M R PL R E EC PA T E G D ES

• Positive correlation (0 to + 1) – Both quantities increase or decrease at the same time. • Zero or no correlation (0) – No association between the quantities. • Negative correlation (−1 to 0) – One quantity increases, the other quantity decreases. Note: High correlation between two variables does not imply that one causes the other.

Strength of an association using Pearson’s correlation coefficient

The table below provides a guideline for classifying the strength of a linear association using Pearson’s correlation coefficient. Association

Pearson’s correlation coefficient

Perfect negative

r = −1

Strong negative

r = −0.75 to − 0.99

Moderate negative

r = −0.50 to − 0.74

Weak negative

r = −0.25 to − 0.49

No linear association

r = −0.24 to + 0.24

Weak positive

r = +0.25 to + 0.49

Moderate positive

r = +0.50 to + 0.74

Strong positive

r = +0.75 to + 0.99

Perfect positive

r = +1

Summary of Pearson’s correlation coefficient

−1.0

−0.99 to −0.75

−0.74 to −0.50

Perfect negative

Strong negative

Moderate negative

−0.49 to −0.25

−0.24 to +0.24

+0.25 to +0.49

Weak negative

No correlation

Weak positive

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U N SA C O M R PL R E EC PA T E G D ES

+0.50 to +0.74

+0.75 to +0.99

1.0

Moderate positive

Strong positive

Perfect positive

Example 4

Calculating Pearson’s correlation coefficient

The table below shows foot length and height in centimetres for a group of students. Foot length (cm)

24.5

25.6

26.1

25.9

26.2

27.0

27.4

27.5

28.1

Height (cm)

167

168

170

174

176

180

181

185

187

a Calculate Pearson’s correlation coefficient, correct to three decimal places. b Classify the strength and direction of the association using Pearson’s correlation coefficient. Explanation

a r = 0.94781831 . . . r ≈ 0.948

Enter the statistics mode of the scientific calculator. Select 2 : A + BX and enter the data. Enter the foot length data into List X. Enter the height data into List Y.

UUUU

U U U UUUUUUUUUUUU UUUU UU UU UUUU U UU UUUU U

U U U

Solution

U UUU UU UUU U UUU U

UUUU

Once all the data has been entered, calculate the correlation coefficient: Press AC-SHIFT – 1 –

Select Reg (regression) and r to view the results.

b Strong positive linear association

The value of Pearson’s correlation coefficient (r) is between +0.75 to +0.99, which indicates a strong positive linear association.

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6C Pearson’s correlation coefficient

Now you try The data below shows the litres of petrol used over a certain distance travelled in kilometres. Petrol used (L)

Distance travelled (km)

100

120

145

180

210

U N SA C O M R PL R E EC PA T E G D ES

a Calculate Pearson’s correlation coefficient, correct to four decimal places. b Classify the strength and direction of the association using Pearson’s correlation coefficient.

Exercise 6C 1

BUILDING

State whether Pearson’s correlation coefficient is positive, negative or zero. a y

b y

c y

d y

e y

f y

2 Draw a scatter plot that shows: a high positive correlation c zero correlation

b low positive correlation d low negative correlation.

Describe the strength of the linear associations with the following Pearson’s correlation coefficients. a r = 0.3 d r = −0.9

b r = −1 e r = −0.3

c r = 0.9 f r=1

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Chapter 6 Bivariate data analysis

The age of a child versus head circumference is shown in the scatter plot.

Head circumference (cm)

U N SA C O M R PL R E EC PA T E G D ES

a State whether the association is positive or negative. b Describe the strength of the association as strong, moderate or weak. c Estimate the value of Pearson’s correlation coefficient.

Example 4

1.0 2.0 Age (years)

3.0

Calculate Pearson’s correlation coefficient to three decimal places for each table.

30 0.0

Calculate Pearson’s correlation coefficient correct to three decimal places for each scatter plot. a

10 9 8 7 6 5 4 3 2 1

1 2 3 4 5 6 7 8 9 10

10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10 y

10 9 8 7 6 5 4 3 2 1

x 1 2 3 4 5 6 7 8 9 10

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6C Pearson’s correlation coefficient

The mass (in kg) and body mass index (BMI) of nine students is shown in the scatter plot.

28 Body mass index (BMI)

27 26

U N SA C O M R PL R E EC PA T E G D ES

a State whether the association is positive or negative. b Describe the strength of the association as strong, moderate or weak. c Calculate the value of Pearson’s correlation coefficient, correct to three decimal places. d Do you think BMI is related to mass?

25 24

64 66 Mass (kg)

The table below depicts mass (kg) in comparison to height (cm). Height h (cm)

100

120

140

150

155

160

180

200

Mass m (kg)

a Calculate Pearson’s correlation coefficient, correct to four decimal places. b Classify the strength and direction of the association using Pearson’s correlation coefficient.

The table below shows satisfaction levels in comparison to overtime hours worked. Overtime hours worked

Satisfaction levels

90%

85%

80%

60%

45%

15%

a Calculate Pearson’s correlation coefficient, correct to four decimal places. b Classify the strength and direction of the association using Pearson’s correlation coefficient.

DEVELOPING

The table below shows the heart rate (beats per minute) and the score out of 20 in a test. Heart rate h (BPM)

Score on test s

a Draw a scatter plot using the data in the table. b State whether the association is positive or negative. c Describe the strength of the association as strong, moderate or weak. d Calculate the value of Pearson’s correlation coefficient, correct to four decimal places.

CHALLENGING

Rose believes tall women tend to taller male partners than do short women. Do you agree? Explain your view using the data in the table. Height women (cm)

168

163

168

165

178

165

170

175

169

174

Height men (cm)

183

173

178

173

180

165

168

181

179

175

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Chapter 6 Bivariate data analysis

A report found a high positive correlation between the shoe size of primary school students and their reading ability.

U N SA C O M R PL R E EC PA T E G D ES

a Does this mean people with big feet are better at reading? Explain your answer. b What other factors might explain this relationship?

Communicate your reasoning, understanding and fluency 13

There is a high positive correlation between health and income.

a This correlation is commonly thought to reflect a causal link running from income to health. Explain how higher income can cause better health. b Some experts believe the correlation is a causal link running from health to income. Explain how better health can cause higher income.

A developing nation compared body mass to income. Pearson’s correlation coefficient for these quantities was −0.2. What is the meaning of this correlation?

Each member of a group of males had his height and foot length measured and recorded. Pearson’s correlation coefficient between these variables was 0.9. Describe the association between height and foot length.

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6D Line of best fit

6D Line of best fit Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the purpose of a line of best fit • To draw lines of best fit by eye • To find the equation of lines of best fit • To use the line to make predictions and expectations Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving lines of best fit regularly appear in HSC Examinations

If the points on a scatter plot tend to lie in a straight line, then we can draw a straight line on the scatter plot. This line is known as the line of best fit, which is also called the least-squares regression line of best fit or a trendline.

The simplest method is to draw a line of best fit by eye that seems to be a balance of points above and below the line. The more mathematically accurate line is the least-squares regression line, which is calculated using the data’s coordinates. The line of best fit does not need to pass through any of the data points, although it may pass through some of the points.

Distance travelled (km)

Petrol used vs Distance travelled

220 200 180 160 140 120 100 80 60 40

Data points Trendline

4 6 8 10 12 14 16 18 Petrol used (Litres)

The aim of the line of best fit is to model the association between two numerical variables by using the equation of a straight line. The equation of a straight line is found:

• using a calculator to find the equation of the least-squares regression line (this is examined in detail in

Section 6E) • using the gradient-intercept formula: y = mx + c, where m is the gradient and c is the y-intercept.

Line of best fit A line of best fit is a straight line that approximates a linear association between points. The equation of the line of best fit is found using the gradient-intercept formula: y = mx + c. The line of best fit is used to make a prediction about one of the variables. When it is used to make a prediction within the data range it is called interpolation. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Chapter 6 Bivariate data analysis

Extrapolation is a prediction outside the data range and must be used carefully, as the line of best fit may not apply (for example, predicting an adult’s height based on their increasing height as a child). Interpolation and extrapolation will be examined in detail in Section 6F.

Example 5

Drawing a line of best fit by eye

Draw a line of best fit on the scatter plot.

Price (S’000)

U N SA C O M R PL R E EC PA T E G D ES

9 8 7 6 5 4 3 2 1

Solution

Price (S’000)

4 6 Age (Years)

Explanation

The line does not need to pass through any of the points but should balance the points above and below it.

9 8 7 6 5 4 3 2 1

6 4 Age (Years)

Now you try

Draw a line of best fit on the scatter plot. 80

Mass (kg)

60 55 50 160

165

170 175 180 Height (cm)

185

190

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6D Line of best fit

Finding the equation of a line of best fit using y = mx + c From the Year 11 topic of Linear relationships, we know that the equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept. When given the line of best fit or the least-squares regression line, we can calculate the gradient and the y-intercept and hence find the equation of the trendline.

Finding the equation of the line of best fit using y = mx + c

U N SA C O M R PL R E EC PA T E G D ES

Example 6

The scatter plot shows a mother’s height (in cm) and her daughter’s height (in cm). 180

Daughter’s height (cm)

(174, 176)

170

160

(152, 160)

150 150

170 160 Mother’s height (cm)

180

Use the two points given on the line of best fit to: a determine the gradient of the line b determine the y-intercept c form an equation of the line of best fit. Solution

176 − 160 174 − 152 8 m= 11

a m=

8 × 152 + c 11 544 c= 11

b 160 =

8 544 x+ 11 11 or to 2 decimal places: y = 0.73x + 49.45

c y=

Explanation

Gradient = m =

rise y2 − y1 = run x2 − x1

Substitute one of the points on the line (152, 160) 8 and gradient m = into y = mx + c to solve for c. 11

Substitute the m found in part a and c found in part b into y = mx + c to get the equation of a straight line.

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Now you try The heights and shoe size of 10 people were recorded and plotted in the scatter plot below. A line of best fit is already drawn. Use the two points given on the line of best fit to:

y 200 190 180 170 160 150 140 130

Height (cm)

U N SA C O M R PL R E EC PA T E G D ES

a determine the gradient of the line b determine the y-intercept c form an equation of the line of best fit.

(11, 180)

(10, 170)

6 7 8 9 10 11 12 13 Shoe size

Exercise 6D 1

Use a ruler to draw a line of best fit on each of the scatter plots below.

Daily temperature ˚C

40 35 30 25 20 15

140 120

Speed (km/h)

Example 5

BUILDING

10 5

100 80 60

40 20

81 80 79 78 77 76 75 74 73 72 71

Sales ($000)

Pulse rate (bpm)

20 40 60 80 100120140 Ice cream consumption

20 40 60 80 100120 Time taken (minutes)

120 100 80 60 40 20

5 10 15 20 25 30 Proﬁt ($000)

0 20 40 60 80 100 120 Weight (kg)

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6D Line of best fit

Draw a scatter plot and a line of best fit by eye for the following points. a (0, 0) (10, 30) (20, 67) (30, 93) (40, 126) (50, 158) (60, 178) b (5, 20) (10, 42) (15, 73) (20, 94) (25, 122) (30, 150) (35, 165) c (0, 6) (2, 24) (3, 39) (4, 44) (5, 59) (6, 64) (7, 79) (8, 84) d (10, 55) (12, 45) (14, 20) (16, 40) (18, 30) (20, 28) (22, 25)

U N SA C O M R PL R E EC PA T E G D ES

DEVELOPING Example 6

Given two points that lie on the line of best fit, determine:

i the gradient of the line ii the y-intercept iii the equation of the line of best fit drawn through the given points. a (10, 40) and (60, 190) b (3, 8) and (1, 12) c (14, 20) and (7, −1) d (10, 12) and (4, 9)

The equation relating a person’s weight (in kg) to shoe size is given below: weight = 2.2 × shoe size + 48.1 (only for shoe sizes between 6 and 12). a What is the weight of a person with a shoe size of 7? b What is the weight of a person with a shoe size of 5? c What is the weight of a person with a shoe size of 11? d What is the shoe size of someone who weighs 70 kg?

CHALLENGING

The examination score (%) and hours of study of 12 students who sat the same test were recorded and plotted in the scatter plot below. The least-squares regression line is also shown.

Exam score (%)

100 90

(9, 90)

80 70

60 50

(3, 55)

0 2 4 6 8 10 12 Hours of study

a Determine the gradient of the line. b Determine the y-intercept. c Hence, find the equation of the least-squares regression line drawn through the given points. d Using the equation or otherwise, how many hours of study did a person do if they scored 70% on the test? Answer correct to one decimal place. e Describe the strength and direction of the relationship. f Interpret the gradient of the least-squares regression line in the context of the data. g Interpret the y-intercept. Does this value have meaning in this context?

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The heights and weights of small dog breeds are measured and recorded below. Height h (cm)

Mass m (kg)

2.0

3.5

4.5

6.0

7.0

8.5

10.0

10.5

U N SA C O M R PL R E EC PA T E G D ES

a Draw a scatter plot using the table. b Draw a line of best fit by eye. c Determine the gradient of the line of best fit. Interpret the gradient in the context of the data. d Determine the y-intercept and explain why interpreting the y-intercept does not make sense in the context of the data. e Find the equation of the line of best fit drawn through the given points.

The art assessment results for a group of students and their creativity score have been recorded in the scatter plot below. A line of best fit has been drawn.

Creativity score

a Determine the gradient of the line of best fit. b Determine the y-intercept. c Find the equation of the line of best fit drawn through the given points. d Describe the strength and direction of the relationship. 20 19 18 17 16 15 14 13 12

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6D Line of best fit

Communicate your reasoning, understanding and fluency The rainfall (in mm) and the temperature (in ◦ C) during the last month was recorded and plotted in the scatter plot below. The least-squares regression line is also shown. 14 12 10 8 6 4

(18, 12)

Rainfall (mm)

U N SA C O M R PL R E EC PA T E G D ES

(26, 5)

17.5 22.5 27.5 32.5 Temperature (ºC)

a By finding the gradient and y-intercept, determine the equation of the least-squares regression line drawn through the given points. b Using the equation or otherwise, how much rain would there be if the temperature was 21◦ C? Answer correct to one decimal place. c If 5 mm of rain fell, what would the maximum daily temperature be? d Describe the strength and direction of the relationship. e Interpret the gradient of the least-squares regression line in the context of the data. f Interpret the y-intercept. Does this value have meaning in this situation?

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Chapter 6 Bivariate data analysis

6E Least-squares regression line Learning intentions

• To find the equation of the least-squares regression line • Use the equation to make predictions and expectations

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving lines of best fit regularly appear in HSC Examinations

Method of least squares

Drawing a line of best fit that seems to balance out the points around the line is not reliable and everyone is likely to come up with a slightly different line. One method of overcoming this problem is to use the method of least squares. First a line of best fit is drawn on the scatter plot. Then the vertical distance (the residual) between each data point and the line is calculated. If the line is a perfect fit, this distance is zero for all the data points. The worse the fit, the larger the differences. Finally, each vertical distance is squared and the squared values are added together. The least-squares line of best fit is the line that minimises the sum of the squares of the vertical distances.

(6, 5)

(3, 5)

5 4 3 2

(2, 1)

1 0

−1

Least-squares line of best fit

A line of best fit minimises the sum of the squares of the vertical distances (or residuals).

The equation of least-squares line of best fit

The equation of the least-squares line of best fit is found using the gradient–intercept formula, y = mx + c. By using the statistics mode on your scientific calculator, select A + BX to enter your x and y values into a table: A is the y-intercept, c B is the gradient, m

Substitute these values into y = mx + c and you have an equation of the least-squares regression line of best fit (see Example 7).

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6E Least-squares regression line

Example 7

Using the calculator to find the least-squares line of best fit

The table below shows the heights (cm) and masses (kg) of nine people. 163

165

170

175

178

180

182

186

190

Mass m (kg)

U N SA C O M R PL R E EC PA T E G D ES

Height h (cm)

a Find Pearson’s correlation coefficient, correct to 3 decimal places. b Use a calculator to determine the equation of the least-squares line of best fit. Give your values of A and B correct to 2 decimal places. c Draw a scatter plot showing the least-squares line of best fit. d Describe the association between height and mass. e Interpret the gradient of the least-squares regression line in the context of the data. f Interpret the y-intercept. Does this value have meaning in this context? Solution

Explanation

a r = 0.9743297568 . . . r ≈ 0.974

Enter the statistics mode of the scientific calculator. Select 2 : A + BX and enter the data. Select Reg (regression) r to view the results.

b A = −65.63424252 . . . y − interept B = 0.7512323365... gradient

Select Reg (regression) and A and B to view the results.

y = mx + c

Write the gradient–intercept formula.

= Bx + A

Substitute into the formula.

= 0.75x − 65.63

Evaluate.

Draw a number plane with height on the horizontal axis and mass on the vertical axis. Plot the points from the table. Select two heights in the given domain such as 170 and 180. Use y = 0.75x − 65.63 and x = 170 to calculate a value for y (about 62). Plot (170, 62). Use y = 0.75x − 65.63 and x = 180 to calculate a value for y (about 69). Plot (180, 69). Draw a line between (170, 62) and (180, 69).

y = 0.75x − 65.63

Mass (kg)

75 70 65 60 55

50 160

165

170 175 180 Height (cm)

d Strong positive linear association.

185

190

The value of Pearson’s correlation coefficient (r) is between +0.75 and +0.99. Continued on next page

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Chapter 6 Bivariate data analysis

3 Gradient is approximately 0.75 or . 4

e For approximately every 3 kg increase in mass, height increases by 4 cm. f The y-intercept is a mass of −65 kg when the height is 0. A persons’ high cannot be 0 and weights cannot be negative. This is outside the range of data and does not make sense in context of the data.

U N SA C O M R PL R E EC PA T E G D ES

The y-intercept is −65.

Now you try

The table below shows the number of tourists visiting a country and the average maximum temperature of the country. Temperature (◦ C)

Number of tourists (000’s)

a Find Pearson’s correlation coefficient, correct to 3 decimal places. b Comment on the association between temperature and number of tourists. c Use a calculator to determine the equation of the least-squares regression line of best fit. Give the values of A and B correct to three decimal places. d Construct a scatter plot for the table with temperature as the horizontal axis. Show the line of best fit on the scatter plot. e Interpret the gradient and y-intercept of the least-squares regression line in the context of the data.

Exercise 6E

Example 7

BUILDING

For the data below:

i using the statistics mode on your scientific calculator and the A + BX function, determine the equation for the least-squares regression line of best fit. Answer correct to 3 decimal places where necessary ii calculate Pearson’s correlation coefficient correct to 3 decimal places and describe the direction and strength of the data.

110

180

200

190

1.5

2.5

3.5

4.5

6.3

4.5

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6E Least-squares regression line

DEVELOPING 2

Create the spreadsheet and scatter plot below.

U N SA C O M R PL R E EC PA T E G D ES

a Use the trendline tool to insert the least-squares line of best fit onto the scatter plot. b Display the equation of the leastsquares line of best fit onto the scatter plot. c Use the equation to predict the pulse rate when the weight is 65 kg. d Use the equation to predict the weight when the pulse rate is 60 beats per minute. e Comment on the reliability of the predictions using the line of best fit.

The heights and masses of young children are measured and recorded below. Height h (cm)

Mass m (kg)

1.5

3.1

3.6

5.5

6.0

6.9

7.6

8.6

10.0

11.2

a Determine the equation of the least-squares line of best fit. b What is the expected mass of a child given their height is 73 cm? c What is the expected mass of a child given their height is 48 cm? d What is the expected height of a child given their mass is 4.8 kg? e What is the expected height of a child given their mass is 9.0 kg? f Find the correlation coefficient and comment on its strength and direction.

CHALLENGING

The table below shows the amount of energy (in megajoules, MJ) used per day for 12 people of various mass (in kg). Energy (MJ)

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

Mass (kg)

101

110

115

119

125

a Draw a scatter plot using energy for the horizontal axis and mass for the vertical axis. b Calculate Pearson’s correlation coefficient. Answer correct to three decimal places. c Find the equation of the least-squares line of best fit in terms of energy (e) and mass (m).

d Use the equation to predict the mass when 1.55 MJ of energy is used. e Use the equation to predict the mass when 2.15 MJ of energy is used. f Use the equation to predict the energy used when the mass is 100 kg. g Use the equation to predict the energy used when the mass is 80 kg.

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Chapter 6 Bivariate data analysis

The table compares a mother’s height in centimetres with her son’s height in centimetres. Mother’s height (cm)

155

160

165

170

175

180

185

Son’s height (cm)

161

167

171

174

179

185

189

U N SA C O M R PL R E EC PA T E G D ES

a Draw a scatter plot using mother’s (m) height as the horizontal axis and son’s height (s) as the vertical axis. b Determine the equation of the least-squares line of best fit. c Use the equation to predict the son’s height if the mother’s height is 163 cm. d Use the equation to predict the mother’s height if the son’s height is 178 cm. e Comment on the reliability of the predictions using the line of best fit.

Communicate your reasoning, understanding and fluency 6

The table below shows the number of rebounds per game for 10 people who play different amounts of minutes in a basketball game. Minutes played

Rebounds grabbed

a Draw a scatter plot using ‘Minutes played’ as the horizontal axis and ‘Rebounds grabbed ‘as the vertical axis. b Find the equation of the least-squares line of best fit in terms of minutes played and rebounds grabbed. c Find Pearson’s correlation coefficient, correct to 3 decimal places. Describe the correlation in terms of strength and direction. d Use this equation to predict the number of rebounds grabbed when a player plays for 38 minutes. e Use the equation to predict the minutes played if a player grabs 11 rebounds.

The table below depicts the test scores out of 40 for a student and the hours of study they have completed. Hours of study

Test Score

a Which variable is the independent variable? b Draw a scatter plot using the ‘Hours of study’ for the horizontal axis and ‘Test scores’ for the vertical axis. c Calculate Pearson’s correlation coefficient and describe the correlation in terms of strength and direction. Answer correct to three decimal places. d Use a calculator to find the equation of least-squares line of best fit in terms of test score and hours of study. Answer correct to two decimal places. e Use this equation to predict the test score of a student who studied for 7 hours. f Interpret the gradient and y-intercept of the least-squares regression line in the context of the data.

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6F Interpolation and extrapolation

6F Interpolation and extrapolation Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand the meaning and purpose of interpolation and extrapolation • To use lines of best fit to make predictions and expectations • To understand the limitations of interpolation and extrapolation Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions involving lines of best fit regularly appear in HSC Examinations

The equation of the line of best fit or regression line can provide important information and be used to make predictions. The gradient (m) indicates the change in dependent variable as the independent variable increases by 1 unit. The vertical intercept (b) indicates the value of the dependent variable when the independent variable is zero. In addition to this information, the equation of best fit is used for interpolation and extrapolation.

Interpolation

Interpolation is the use of the linear regression line to predict values within the range of the dataset. If the data has a strong linear association then we can be confident our predictions are accurate. However, if the data has a weak linear association, we are less confident with our predictions.

Example 8

Making predictions using interpolation

Life expectancy at birth for females and males is shown below. Year

1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Female

Male

Life expectancy

Female Male

Linear (Female)

Linear (Male)

50 1900

1920

1940

1960 Year

1980

2000

2020

a What was the life expectancy in 1935 for females? b What was the life expectancy in 1995 for males? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Solution

Explanation

a Life expectancy for females in 1935 was approximately 67 years.

Draw a vertical line from 1935 until it intersects the blue line. At this point draw a horizontal line until it reaches the vertical axis. Read the value. Draw a vertical line from 1995 until it intersects the red line. At this point draw a horizontal line until it reaches the vertical axis. Read the value.

U N SA C O M R PL R E EC PA T E G D ES

b Life expectancy for males in 1995 was approximately 74 years. Now you try

The table below depicts the findings of a study which investigated the effects of wages (in thousands $) on satisfaction levels. Wages (000’s)

Satisfaction level (%)

Wages vs Satisfaction level

80 70 60 50 40 30 20

30 35 40 45 50 55 60 65 70 Wages (000’s)

a What is the satisfaction level for someone whose wage is $45 000? b If your satisfaction level is 70%, what wage could you be earning?

Extrapolation

Extrapolation is the use of the linear regression line to predict values outside the range of the dataset. Predicted values are either smaller or larger than the dataset. The accuracy of predictions using extrapolation depends on the strength of the linear association similar to interpolation. It may not be reasonable to extrapolate too far as this example shows.

Example 9

Making predications using extrapolation

The table below shows the age of a student and their height in centimetres. Age (years)

Height (cm)

133

139

144

149

156

163

170

174

177

181

a Construct a scatter plot from the table using age from 0 to 30 and height from 100 to 300. b Draw the line of best fit and describe the association between age and height. c Predict the height of the student when they are aged 19 years. d What are the limitations of this linear model? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

6F Interpolation and extrapolation

Solution

Explanation

Draw a number plane with age as the horizontal axis and height as the vertical axis. Determine a scale for the horizontal axis. Let each unit represent 1 year. Determine a scale for the vertical axis. Let each unit represent 10 cm. Write a title for the horizontal and vertical axes. Plot the points from the table.

300

Extrapolation

U N SA C O M R PL R E EC PA T E G D ES

Height (cm)

250 200

Interpolation

150

Extrapolation

100

15 20 Age (years)

There is a small amount of scatter in the scatter plot. Read the height from the scatter plot when age is 19.

b Strong positive linear association.

c Height of the student is 200 cm when they are 19 years old.

Extrapolation too far from the dataset needs to be done carefully.

d Adult height does not grow at the same rate as a child. Using the model to extrapolate is flawed, e.g., the prediction is the height will be 260 cm at age 30.

Now you try

The table below depicts the test score out of 50 for nine students and the hours of study they have completed. Hours of study

Test score

a Draw a scatter plot using the hours of study for the horizontal axis and test scores for the vertical axis. b Calculate Pearson’s correlation coefficient and describe the association between hours of study and test scores. c Draw the line of best fit. d Use a calculator to find the equation of least-squares line of best fit in terms of test score and hours of study. e Use this equation to predict the test score of a student who studied for 7 hours. f What would be the score of a student that studied for 40 hours? What are the limitations of this linear model?

Interpolation

Extrapolation

Predicting values within the dataset range

Predicting values outside the dataset range

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Exercise 6F Example 8

BUILDING

The scatter plot opposite shows the rainfall (in mm) and the percentage of clear days for each month in 2018.

120 100 80

U N SA C O M R PL R E EC PA T E G D ES

Rainfall (mm)

a How many months had 10% of clear days? b What was the percentage of clear days when the rainfall was 70 mm? c Predict the rainfall in the month given the following percentage of clear days: i 4%

ii 20%

iii 26%.

The scatter plot shows the average annual female income plotted against average annual male income for 15 countries. a What was the average annual female income for a country whose average annual male income was $450 00? b How many countries had an average annual female income of $250 00? c Predict the female income given the following male income: i $20 000 ii $40 000 iii $60 000.

10 20 30 Clear days (%)

35 30 25 20 15 10 5

d Predict the male income given the following female income: i $25 000 ii $30 000 iii $35 000.

iii 100 mm.

Female ($1000)

ii 90 mm

40 20

d Predict the percentage of clear days in the month given the following rainfall: i 80 mm

30 40 50 Male ($1000)

In a study of errors made in a test, for times ranging from 5 to 25 seconds, the equation relating the number of errors to time is: errors = −0.53 × time + 15

Use this equation to predict the number of errors (correct to nearest whole number) with the following times. Are you interpolating or extrapolating? a 10 seconds

b 20 seconds

c 30 seconds

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6F Interpolation and extrapolation

For minimum temperatures from 5◦ C to 20◦ C, the equation relating the maximum and minimum temperature (in◦ C) at a weather station is shown below: maximum = 0.67 × minimum + 13 Use this equation to predict the maximum temperature given the following minimum temperatures. Are you interpolating or extrapolating? b 20◦ C

c 30◦ C

U N SA C O M R PL R E EC PA T E G D ES

a 10◦ C

The equation relating life expectancy at birth from 1900 to the current year for a particular country is given below: life expectancy = 0.21 × year − 353.78

Use this equation to predict a life expectancy in the following years. Are you interpolating or extrapolating?

a 1900

b 1950

c 1870

d 2000

e 2040

f 1970

A student-run business sells homemade cookies. For production levels between 160 and 190 cookies, the equation relating profit (in $) and the number of cookies (n) is shown below: profit = 0.75 × n − 65.63

Use this equation to predict the profit for the following production levels. Are you interpolating or extrapolating? a 150

b 175

c 200

When a worker’s average pay rate is between $5 and $25, the equation relating a country’s development index (%) to the average pay rate (in dollars per hour) is shown below: development index = 0.272 × pay rate + 81.3

Use this equation to predict a country’s development index with the following average pay rates. Are you interpolating or extrapolating? a $40 per hour b $20 per hour c $10 per hour

When the area in a large city is between 1 km2 and 8 km2 , the equation relating a population to area (in square kilometres) is shown below: population = 2680 × area + 5330

Use this equation to predict the population with the following areas. Are you interpolating or extrapolating? a 2.5 km2 b 5.0 km2 c 7.5 km2

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DEVELOPING 9

A factory produces video game cartridges. The least-squares regression line for cost of production (C) as a function of the number of cartridges (n) produced is given by: C = 7.2n + 500

U N SA C O M R PL R E EC PA T E G D ES

Furthermore, this function is deemed accurate when producing between 100 and 1000 cartridges. a What is the cost to produce 400 cartridges? b How many cartridges can be produced for $5900? c What is the cost to produce 10000 cartridges? d What are the ‘fixed’ costs for this production? e Which of the parts a to c is an extrapolation?

Example 9

The table below compares the length (in centimetres) for 10 fish and their weight (in grams). Length (cm)

Weight (g)

250 310 340 355 445 450 500 520 550 565

a Draw a scatter plot using length as the horizontal axis and weight as the vertical axis. b Determine the equation of the least-squares line of best fit. c What is the rate at which the weight is changing relative to the length? d What is the median weight of the 10 fish? e Use interpolation to predict a fish’s weight if its length is 35.5 cm. f Use extrapolation to predict a fish’s weight if its length is 45.5 cm. g What is the length of a fish given it weighs 300 g?

CHALLENGING

The table below compares the literacy score for 10 countries and their internet use (%) as a percentage of the population. Internet use (%)

Literacy score

90 102 131 148 282 295 350 375 390 400

a Draw a scatter plot using internet use as the horizontal axis and literacy as the vertical axis. b Calculate Pearson’s correlation coefficient. Answer correct to four decimal places. c Describe the association between internet use and literacy score. d Find the least-squares regression equation, correct to two decimal places. e Using the least-squares equation, if a country has an internet use of 30% predict the country’s literacy score. Answer correct to the nearest whole number. f If a country has a literacy score of 300, predict the country’s internet use, correct to the nearest whole number. g Comment on the internet use for a country with a literacy score of 600.

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6F Interpolation and extrapolation

Communicate your reasoning, understanding and fluency 12

The population of a town over a 10-year period was recorded starting from the year 2010 and is shown in the table. 1

U N SA C O M R PL R E EC PA T E G D ES

Year after 2010 Population

9400 9300 9250 8500 8400 7000 7300 6300 5000 4900

a Draw a scatter plot with ‘Year after 2010’ on the horizontal axis and ‘Population’ on the vertical axis. b Calculate Pearson’s correlation coefficient, correct to three decimal places. c Describe the association between population each year after 2010. d Find the equation of the least squares regression line of best fit, correct to the nearest whole number. e Interpret the gradient and y-intercept of the least-squares regression line in the context of the data. f Using this equation, what year after 2010 did the population reach 8000? g Comment on the population of the town in the year 2030.

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6G Statistical investigations Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To know and understand the statistical investigation process • To view and undertake statistical investigations involving bivariate datasets • To explain the difference between variables that show correlation and those that have a causal relationship

Past, present and future learning

• Some aspects of this section may be new to students • These concepts build upon prior learning from Year 11 • Questions about statistical investigations regularly appear in HSC Examinations

Statistical investigation is the process of gathering statistics. The information gained from a statistical investigation is a vital part of our society. A statistical investigation involves four steps.

Four steps in a statistical investigation

1 Collect the data Collecting data involves deciding what to collect, locating it and collecting it. The gathering of statistical data may take the form of a: • census – data is collected from the whole population • survey – data is collected from a smaller group of the population. It is important that procedures are in place to ensure the collection of data is accurate, up-to-date, relevant and secure. If the data collected comes from unreliable sources or is inaccurate, the information gained from it will be incorrect. When taking a sample, the data gathered must be representative of the entire population otherwise the information collected may be biased towards a particular outcome. 2 Organise the data Organising data is the process of arranging, representing and formatting data. It is carried out after the data is collected. The organisation of the data depends on the purpose of the statistical investigation. For example, to store and search a large amount of data, the data needs to be categorised. Organising gives structure to the data. 3 Summarise and display the data Displaying data is the presentation of the data and information. Information must be well organised, readable, attractively presented and easy to understand. Information is often displayed using graphs such as scatter plots, dot plots, histograms, line graphs, stem-and-leaf plots and box plots. Data is summarised using statistics such as the mean, median, mode and standard deviation.

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6G Statistical investigations

U N SA C O M R PL R E EC PA T E G D ES

4 Analyse the data Analysing data is the process of interpreting data and transforming it into information. It involves examining the data and giving meaning to it. When analysing bivariate data, the form, direction and strength of the association is determined. Scatter plots and lines of best fit are commonly used to analyse the data. They make it easy to interpret data by making instant comparisons and revealing trends. Predictions and conclusions are completed by interpolating and extrapolating the data.

Statistical Investigation

A statistical investigation involves four steps: collecting data, organising data, summarising and displaying data and analysing data.

Example 10

Case study of a statistical investigation

Coen has been asked to complete a statistical investigation on whether the blood glucose level (in mg/100 mL) of an adult can be predicted from their weight (in kg).

Weight

Glucose

Weight

Glucose

60.2

88.1

71.6

94.9

61.3

91.5

72.0

95.7

Coen performed the following steps.

63.5

88.7

72.0

95.1

1 Collecting the data – Coen accessed the medical data on 20 adults. 2 Organising the data – Coen categorised the data into blood glucose levels and weight. 3 Summarising and displaying the data – Coen presented the bivariate data into the table shown opposite and the scatter plot shown below.

63.7

90.6

72.5

96.4

65.0

90.9

73.7

96.6

66.4

94.0

73.8

97.0

66.9

92.1

74.8

96.3

70.1

96.5

75.9

99.1

70.2

93.9

76.3

98.2

70.5

95.2

78.9

99.9

110

Glucose (mg/100 mL)

105 100 95 90 85 80

70 75 Weight (kg)

100

Continued on next page

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r = 0.9507479097 . . . r ≈ 0.951 This indicates a strong positive linear association between weight and blood glucose levels.

U N SA C O M R PL R E EC PA T E G D ES

4 Analysing the data: a Coen calculated Pearson’s coefficient to measure the strength of a linear association.

b Coen calculated the equation and graphed the least-squares regression line on the scatter plot.

A = 52.78718161...

B = 0.5966957534 . . .

y = mx + b

= Bx + A

= 0.60x + 52.79

glucose = 0.60 × weight + 53

glucose = 0.60 × 75 + 53 = 98

c Coen applied the results of his statistical investigation to predict the glucose level of a person who weighs 75 kg.

∴ A person weighing 75 kg has a blood glucose level of 98 mg/100 mL.

Now you try

Shreya has been asked to complete a statistical investigation on whether height above sea level can be predicted by temperature. a What association would you expect Shreya to find? b What other factors might affect these findings?

Issues in a statistical investigations

A statistical investigation raises a number of ethical issues such as bias, accuracy, copyright and privacy.

• Data needs to be free from bias. Bias means that the data is unfairly skewed or gives too much weight to a

particular result. For example, if a survey about favourite music was only completed by teenagers, and the results were generalised to the entire population, it would have a bias. Several checks should be made to limit the impact of bias. • The accuracy of the collected data is a vital ingredient of a statistical investigation. It depends on the source of the data and whether the data has been recorded correctly. The accuracy of the data is often difficult to check in a reasonable time. It is often necessary to compare data from a number of different sources and determine which data is accurate. • Copyright is the right to use, copy or control the work of authors and artists. It is against the law to infringe copyright. You are not allowed to use or copy the work of another person without their permission. If data is collected from the internet, it should be assumed to be protected by copyright. • Privacy is the ability of an individual to control personal data. Data collected on individuals is not always accurate. Inaccuracies can be caused by mistakes in gathering or entering the data, by mismatch of the data and the person or by information being out-of-date. Most people give information about themselves to selected parts of the outside world. Often people are quite willing to tell A something but would be shocked if B knew. But what prevents A telling B?

Issues in a statistical investigation A statistical investigation raises a number of ethical issues such as bias, accuracy, copyright and privacy. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

6G Statistical investigations

Causation

U N SA C O M R PL R E EC PA T E G D ES

Causation indicates that one event is the result of the occurrence of another event (or variable). This is often referred to as the cause and effect. That is, one event is the cause of another event happening. For example, the bell at the end of the period is an event that causes students to leave for the next period. When completing a statistical investigation it is important to be aware that two events (or variables) may have a high correlation but be unrelated. That is, high correlation does not imply causation. For example, the increase in the use of mobile phones has a strong correlation to the increase in life expectancy. However, the use of mobile phones does not cause the increase in life expectancy.

Causation

Causation indicates that one event is the result of the occurrence of another event (or variable).

Exercise 6G 1

BUILDING

Copy and complete the following sentences:

a A statistical ________________ involves four steps: collecting data, organising data, summarising and displaying data, and analysing data. b Census data is collected from the whole ________________. c When taking a ________________ the data gathered must be representative of the entire population. d Displaying data is the ________________of the data and information. e When analysing ________________ data the form, direction and strength of the association is determined. f Bias means that the data is unfairly ________________ or gives too much weight to a particular result.

True or false?

a A survey is when data is collected from a smaller group of the population. b Data collected from unreliable sources results in incorrect information. c Data is often displayed using graphs such as scatter plots, dot plots, histograms, line graphs, stem-and-leaf plots and box plots. d Analysing data is the process that interprets data, transforms it into information. e You are allowed to use or copy the work of another person without their permission. f Data collected on individuals is always accurate.

The Australian Bureau of Statistics collects data for our society. Collecting data is one step in a statistical investigation. List all the four steps involved in a statistical investigation.

Explain the difference between a census and a survey.

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Chapter 6 Bivariate data analysis

How can you limit the impact of biased data?

There is a strong positive correlation between number of car accidents and the number of teachers in cities around the world. Can we conclude from this that teachers are causing car accidents? Give a possible explanation.

There is a strong positive correlation between the number of churches in a town and the amount of alcohol consumed by its inhabitants. Does this mean that religion is encouraging people to drink? What common cause might counter this conclusion?

U N SA C O M R PL R E EC PA T E G D ES

352

DEVELOPING

Implement a statistical investigation to explore the link between a person’s weight and the number of kilometres they run each week. The data below was collected from 15 young adults. Distance run (km) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Weight (kg)

86.5 87.5 88.0 85.5 86.0 86.5 87.0 85.0 86.5 86.0 85.5 90.0 84.5 84.0 84.0

a Draw a scatter plot to display the data. b Describe the association between distance run and weight. c Calculate Pearson’s correlation coefficient. Answer correct to four decimal places. d Find the least-squares regression equation. e The line graph below was constructed by omitting the value for 6 km. What value for weight would you read from this graph for a person running 6 km? 90.0 89.0

Weight

88.0 87.0 86.0 85.0 84.0 83.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Kilometres run per week

f The graph above was constructed to encourage people to run more kilometres to reduce their weight. Comment on the misrepresentation of the data.

The issue of privacy has been a concern for decades. Discuss this statement. What is your view on privacy? Use the internet to find information about privacy. Compare Australian and overseas data.

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6G Statistical investigations

CHALLENGING Example 10

10 Implement a statistical investigation to solve these problems.

U N SA C O M R PL R E EC PA T E G D ES

a Students in Year 12 have decided to have a party after the HSC at a local venue. You have been given the responsibility of choosing the venue. Investigate a number of different venues and collect data such as the name and address of the venue; the cost per head; the initial deposit required; details of meals; and available dates. Design a survey for Year 12 students that will decide on the venue. Students will vote on the venue and if they are committed to attend. The survey should also decide on possible dates and meal arrangements. b The principal would like to promote the positive features of your school outside the local community. He is presenting these features at a conference centre. Develop a solution for the principal. Design a survey for Year 12 students that will decide on the best five features of your school and use the most appropriate method of presentation.

Communicate your reasoning, understanding and fluency 11

Discuss whether a cause-and-effect relationship exists between these variables when a strong positive correlation is found.

a The heights of children and their reading speeds. b A company expenditure on advertising and sales. c Daily temperature and the consumption of ice cream. d Ice cream sales and drowning accidents. e Cheese consumption and the number of movies Nicole Kidman appears in. f The distance travelled and the litres of fuel a car consumes. g Hours of study and exam results. h Revenue generated by gambling and electric car sales.

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Chapter 6 Bivariate data analysis

Key ideas and chapter summary Scatter plot

Constructing a scatter plot: 1

draw a number plane

determine a scale and a title for the horizontal or x-axis

determine a scale and a title for the vertical or y-axis

plot each ordered pair of numbers with a dot.

U N SA C O M R PL R E EC PA T E G D ES

Review

354

Using a bivariate scatter plot

Pearson’s correlation coefficient

Form of an association

Linear form – a straight line Non-linear form – a curved line

Direction of an association

Positive – gradient of the line is positive Negative – gradient of the line is negative

Strength of an association

Strong – small amount of scatter Moderate – modest amount of scatter Weak – large amount of scatter

Pearson’s correlation coefficient (r) measures the strength of a linear association (−1 ≤ r ≤ +1).

• Positive correlation

• Zero or no correlation • Negative correlation

– Both quantities increase or decrease at the same time. – No association between quantities. – One quantity increases, the other quantity decreases.

Line of best fit

Line of best fit is a straight line that approximates a linear association between points.

Least-squares line of best fit

The equation is given by y = mx + c or A + Bx function on the calculator. Gradient (or slope) is m or B and the y-intercept is c or A.

Interpolation

Predicting values within the range of the dataset.

Extrapolation

Predicting values outside the range of the dataset.

Statistical investigations

Four steps: collecting data, organising data, summarising and displaying data, and analysing data. A statistical investigation raises a number of ethical issues such as bias, accuracy, copyright and privacy.

Causation

Indicates that one event is the result of the occurrence of another event (or variable).

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Chapter 6 review

✓

A printable version of this checklist is available in the Interactive Textbook. Checklist

Review

Chapter checklist with success criteria

1 I can represent a bivariate dataset using a scatter plot.

U N SA C O M R PL R E EC PA T E G D ES

e.g. The table below shows the height (in cm) of a eucalypt tree seedling as it grows. Time (weeks) 0

a b c

d e f

Copy the number plane shown to construct a scatter plot using the given table. What is the increase in the height of the seedling during the first week? What is the increase in the height of the seedling during the last week? How many weeks does it take for the seedling to increase in height from 9 cm to 12 cm? Estimate the height of the seedling after 4.5 weeks. Estimate the time taken for the seedling to grow to a height of 14 cm.

16 14 12 10 8

6 4

6 8 4 Time (weeks)

2 I can read bivariate data in a scatter plot. e.g. The average numeracy score for year 6 students and their general rate of internet use (%) for 10 countries are displayed in the given scatter plot.

200

Numeracy score

0 6.6 8.8 9.0 10.5 12.0 13.5 15.2 15.4 15.8 15.9

Height (cm)

150 100 50

a b c d e f

30 40 50 Internet use (%)

What is the scale for the vertical axis? What is the average numeracy score for the country which has an internet use rate of 24%? What is the internet use (%) for the country which has an average numeracy score 160? How many countries have internet use of less than 50%? How many countries have a numeracy score greater than 100? Is there a relationship between these two variables?

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Chapter 6 Bivariate data analysis

3 I can describe the strength of a linear relationship between two variables as strong, moderate or weak, and its direction as positive or negative.

Head circumference (cm)

e.g. 50

U N SA C O M R PL R E EC PA T E G D ES

Review

356

a b

30 0.0

1.0 2.0 Age (years)

3.0

State whether the association is positive or negative. Describe the strength of the association as strong, moderate or weak.

4 I can identify the independent and dependent variable within a bivariate dataset where appropriate.

Head circumference (cm)

e.g. Determine the independent variable in this scatter plot.

30 0.0

1.0 2.0 Age (years)

3.0

5 I can describe the form of a dataset as linear or non-linear based on the association between two variables.

Head circumference (cm)

e.g. Is the dataset linear or non-linear? 50

30 0.0

1.0 2.0 Age (years)

3.0

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Chapter 6 review

6 I can calculate and interpret Pearson’s correlation coefficient (r) for a bivariate dataset using a scientific calculator and describe its strength and direction of the linear association between the two variables.

e.g. The table below shows the heights (cm) and masses kg of nine people.

Review

Height h (cm) 163 165 170 175 178 180 182 186 190 55

U N SA C O M R PL R E EC PA T E G D ES

Mass m (kg)

a b

Calculate Pearson’s correlation coefficient, correct to three decimal places. Describe the strength and direction of the linear association between the two variables.

7 I can create a line of best fit on a scatter plot for a bivariate dataset, by eye.

Price (S’000)

e.g. Draw the line of best by eye on the scatter plot. 9 8 7 6 5 4 3 2 1

4 6 Age (years)

8 I can determine and interpret the intercept and gradient of the line of best fit from a given graph to form an equation of the line.

Use the two points given on the line of best fit to:

a determine the gradient of the line b determine the y-intercept c form an equation of the line of best fit.

200 190

Height (cm)

e.g. The heights and shoe sizes of 10 people were recorded and plotted in the scatter plot shown. A line of best fit is already drawn.

(11, 180)

180 170

(10, 170)

160 150 140 130

9 10 11 12 13 Shoe size

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Chapter 6 Bivariate data analysis

9 I can determine the equation of the least-squares regression line for a bivariate dataset using a scientific calculator.

e.g. The table below shows the heights (cm) and masses (kg) of nine people. Height h (cm) 163 165 170 175 178 180 182 186 190 55

U N SA C O M R PL R E EC PA T E G D ES

Mass m (kg)

Use a calculator to determine the equation of the least-squares regression line of best fit.

10 I can examine lines of best fit to make predictions and recognise limitations of

interpolation and extrapolation for bivariate datasets within a variety of contexts.

e.g. For minimum temperatures from 5◦ C to 20◦ C, the equation relating the maximum and minimum temperature (in ◦ C) at a weather station is given as: maximum = 0.67 × minimum + 13

Use this equation to predict the maximum temperature given the following minimum temperatures. Are you interpolating or extrapolating? a 10◦ C

b 20◦ C

c 30◦ C

11 I can explain the difference between variables that show correlation and those that have a causal relationship.

e.g. There is a strong positive correlation between number of car accidents and the number of teachers in cities around the world. Can we conclude from this that teachers are causing car accidents? Give a possible explanation.

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Chapter 6 review

Review

Multiple-choice questions 6B

Blood pressure levels for women increase as they get older. What is the best description for the association between blood pressure levels and a woman’s age? B Zero correlation D Constant correlation

U N SA C O M R PL R E EC PA T E G D ES

A Positive correlation C Negative correlation 6B

2 What is the correlation between the variables in the scatter plot? A Strong positive C Strong negative

B Weak positive D Weak negative

Which of the following scatter plots shows weak negative correlation? A y

B y

C y

D y

4 The masses of Great Danes at 1 month and 1 year are given in the table below. 1 month

1.9

2.4

2.6

2.7

2.9

3.2

3.4

3.6

1 year

47.6

53.1

52.2

56.2

57.6

59.9

55.3

56.7

What is the value of Pearson’s correlation coefficient? A 0.5360

B 0.6182

C 0.7863

D 0.8232

5 Which of the following would most likely be due to causation?

A The number of people unemployed and the cost of bread. B Family income and the number of family pets. C Daily temperature and ice-cream consumption. D Wine consumption and the number of movies made in Australia.

The equation of a regression line that enables weekly amount spent on entertainment (in dollars) to be predicted from weekly income is given by: amount = 0.10 × income + 40. What is the predicted amount spent on entertainment with a weekly income of $600? A $40

B $46

C $100

D $240

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Chapter 6 Bivariate data analysis

Short-answer questions 1

The scatter plot shows the navel height and the body height for nine students. a Which has been plotted as the independent variable? b Which has been plotted as the dependent variable? c Is the association between these two variables linear or non-linear? d Describe the association as strong, moderate or weak. e What is the body height for the student with a navel height of 112 cm? f What is the navel height for the student with a body height of 166 cm? g Use the scatter plot to predict the body height of a student with a navel height of 110 cm.

190 Body height (cm)

180

U N SA C O M R PL R E EC PA T E G D ES

Review

360

a What is the age for a student when their height is 120 cm? b What is the height for the student whose age is 11 years? c State whether the association is positive or negative. d Describe the strength of the association as strong, moderate or weak. e Estimate the value of Pearson’s correlation coefficient. Answer correct to two decimal places.

The scatter plot shows a student’s height (in cm) and their age (in years).

Age (years)

170 160

100 110 120 Navel height (cm)

11 10 9 8 7 6 5 4 3 2 1 0

100

110 120 Height (cm)

130

140

150

The table below shows the length of the right foot (in cm) and body height (in cm). Length of right foot (cm)

27.5

Body height (cm)

174.4 156.0 155.3 160.5 170.7 169.3 163.3 164.9

24.0

22.6

23.7

26.4

27.1

25.5

26.1

a Draw a scatter plot using the above table. b Draw a line of best fit by eye. c Calculate Pearson’s correlation coefficient and describe the strength and direction of the relationship. d Using your calculator, determine the equation of the least-squares regression line of best fit.

A strong positive linear association exists between the hours spent studying for an exam and the mark achieved. The equation for this association is mark = 4.5 × study hours + 2. a Predict the exam mark if the student studied for 20 hours a week. b Predict the amount of study hours required to achieve a mark of 56.

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Review

Chapter 6 review

The table below shows the age (in months) and the height (in cm) of a young plant. Age (months)

Height (cm)

114

128

131

159

169

188

U N SA C O M R PL R E EC PA T E G D ES

a Find the equation of the least-squares line of best fit. Answer correct to two decimal places. b What is the predicted height of the plant after 19 months? Answer correct to the nearest centimetre. c What is the predicted height of the plant after 3 years? Answer correct to the nearest centimetre. d Which of the above questions involves interpolation?

6C 6E

The table shows the birth rate (live births per 1000) and the life expectancy (in years). Birth rate

Life expectancy

a Draw a scatter plot using the above table. b State whether the association is positive or negative. c Describe the strength of the association as strong, moderate or weak. d Calculate the value of Pearson’s correlation coefficient. Answer correct to four decimal places. e Determine the equation of the least-squares line of best fit, correct to two decimal places. f Use the equation to predict the life expectancy when the birth rate is 35, correct to the nearest whole number. g Use the equation to predict the birth rate when the life expectancy is 60, correct to the nearest whole number.

Extended-response questions

A marketing firm wanted to investigate the relationship between the number of times a song was played on the radio (played) and the number of downloads sold the following week (weekly sales). The following data was collected from a random sample of ten songs. Played

Weekly sales 3950 2500 3700 2800 2900 3750 2300 4400 2200 3400

a Construct a scatter plot using the above table. b Calculate the value of Pearson’s correlation coefficient. Answer correct to four decimal places. c Describe the relationship between weekly sales and played in terms of direction, strength and form. d Determine the equation of the least-squares line of best fit. e Interpret the slope and intercept of the regression line in the context of the problem. f Use the equation to predict the number of downloads of a song when it was played on the radio 100 times in the previous week. Are you interpolating or extrapolating?

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U N SA C O M R PL R E EC PA T E G D ES

Annuities

We all use annuities without even knowing it

As Australia’s population ages, we are left with the challenge of how to maintain our lifestyle, infrastructure and services with a larger proportion of the population in retirement and out of the workforce. This has been a problem recognised by governments for many years. In 1992, the Treasurer at the time introduced a system known as the superannuation guarantee. This made it compulsory for employers to make superannuation contributions on behalf of employees. Over time the mandatory contribution rate made by employers has changed as shown below.

Year

Percentage paid by employer into superannuation

1992

3% (4% for larger employers)

2002

2013

9.25%

2014

9.5%

2021

10%

2022

10.5%

2023

11%

2024

11.5%

2025

12%

It has long been a policy of the Australian government to improve retirement savings, taking pressure off the age pension. While these changes have helped retirees in Australia, many people still rely on the government to fund their retirement through the age pension.

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Chapter contents

U N SA C O M R PL R E EC PA T E G D ES

7A What are annuities? 7B Tracking annuities every period 7C Using tables for the future value of an annuity 7D Using tables for the present value of an annuity 7E Solving problems involving superannuation and loans

NSW Syllabus

In this chapter, a student:

Online resources

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Chapter 7 Annuities

7A What are annuities? Learning intentions

• To understand that an annuity is a series of repeated investments • To solve problems involving financial decisions

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• The concepts in this section will be new to students • These concepts build upon prior learning from Chapter 3 • Questions involving annuities regularly appear in HSC Examinations

Annuity

Annuity is a form of investment that involves the regular contribution of money. Investments into superannuation or a monthly loan repayment are examples of annuities. The future value of an annuity is the sum of the money contributed, plus the compound interest earned. It is the total value of the investment at the end of a specified term. For example, if $1000 is invested at the end of each year for 4 years at 10% per annum compound interest, then the value of the investment is calculated as follows: End of first year: Interest = $0 (payment at end of the year)

FV = PV(1 + r)n + 1000

= 0 × (1 + 0.10)1 + 1000 = $1000

End of second year: Interest = ($1000) × 0.10 × 1 = $100

FV = PV(1 + r)n + 1000

= 1000 × (1 + 0.10)1 + 1000 = $2100

End of third year: Interest = ($2100) × 0.10 × 1 = $210

FV = PV(1 + r)n + 1000

= 2100 × (1 + 0.10)1 + 1000 = $3310

End of fourth year: Interest = ($3310) × 0.10 × 1 = $331

FV = PV(1 + r)n + 1000

= 3310 × (1 + 0.10)1 + 1000 = $4641

These calculations show that the future value of an annuity is $4641 after 4 years. It is the total value of the investment at the end of the fourth year. After 4 years the money contributed was $4000 and the compound interest earned was $641. Be aware that interest received each year has been increasing exponentially, or compounding. It has not just been increasing by $100 each year.

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7A What are annuities?

Modelling an annuity using a table

U N SA C O M R PL R E EC PA T E G D ES

The above calculation to find the future value of an annuity is an example of a recurrence relation. It uses the previous result to obtain the next result. For example, the future value at the end of the first year ($1000) is the present value for the second year and the future value at the end of the second year ($2100) is the present value for the third year. This can be represented in a table as shown:

Year

Value at start of Year

Deposit

Interest

Value at end of Year

$1000

$100

$1100

$1000

$110

$2100

$1000

$210

$3310

$1000

$331

$4641

Example 1

Modelling an annuity on a loan using a recurrence relation

Freya borrows $1000 at an interest rate of 15% per annum, compounding monthly. She will repay the loan by making 4 monthly payments of $257.85.

a Construct a table to model this loan. b Use the table to find the balance of the loan after Freya has made each of the four payments. c What is the balance of the loan (the amount she still owes) after she has made two payments? Give your answer correct to the nearest cent. d Is the loan fully paid out after four payments have been made? If not, how much will the last payment have to be to ensure that the loan is fully repaid after four payments? Solution 0.15 a r= = 0.0125 12

Explanation

D = $257.85

Value at Balance at start of end of the Month Month Payment Interest Month 1

$1000

$257.85

$12.50

$754.65

$754.65 $257.85

$9.43

$506.23

$506.23 $257.85

$6.33

$254.71

$254.71 $257.85

$3.18

$0.04

Write the values for r and D. Create a table with columns as shown. Interest = value at start of month × 0.15 ÷ 12. To use the calculator recursively (i.e. repeatedly) to find the balance of the loan: i Type ‘1000 =’ ii Type ‘× 1.0125 − 257.85’ iii Press ‘=’ for each month and record the result each time.

b Balance of the loan is $0.04

Balance at the end of the 4th month is read from the table.

c After two months Freya owes $506.23.

Balance at the end of the 2nd month is read from table.

d No. Freya would still owe 4 cents. Last payment = 257.85 + 0.04 = $257.89

Add 0.04 to the repayment amount (D) to find the amount that will fully repay the loan.

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Chapter 7 Annuities

Now you try Jedda borrows $1500 at an interest rate of 13.5% p.a., compounded monthly. She will repay the loan by making 5 monthly repayments of $308.95.

U N SA C O M R PL R E EC PA T E G D ES

a Construct a table to model this loan. b Use the table to find the balance of the loan after Jedda has made each of the five payments. c What is the balance of the loan (the amount she still owes) after she has made 3 payments? Give your answer correct to the nearest cent. d Is the loan fully paid out after five payments have been made? If not, how much will the last payment have to be to ensure the loan is fully repaid after four payments?

Example 2

Modelling an annuity as an investment using a recurrence relation

Spencer makes an initial deposit of $1000 on an investment taken out over one year at a rate of 7.2% per annum compounded quarterly and an additional deposit of $100 is made each quarter.

a Create a table to model this investment. b How much interest is earned by this investment? c What is the total value of this investment at the end of the year? Solution

Explanation

0.072 a V0 = 1000; r = = 0.018; D = 100 4

Write the values for V0 , r and D. Create a table with columns as shown. Interest = value at start of quarter × 0.072 ÷ 12.

Quarter

Value at start of quarter

Balance at end of the Payment Interest quarter

$1000

$100.00

$18.00

$1118.00

$100.00

$20.12

$1238.12

$100.00

$22.29

$1360.41

$100.00

$24.49

$1484.90

To use the calculator recursively (i.e. repeatedly) to find the balance of the loan:

i Type ‘1000 =’ ii Type ‘× 1.018 + 100’ iii Press ‘=’ for each month and record the result each time.

b Interest = $18.00 + $20.12 + $22.29 + $24.49 = 84.90 ∴ The interest earned in one year is $84.90.

c Total value of investment is $1484.90

Interest earned is the sum of the Interest column.

Value of the investment is the number in the right-hand bottom corner of the table.

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7A What are annuities?

Now you try Tao makes an initial investment of $1250 on an investment taken out over one year at a rate of 9.6% p.a. compounded quarterly, and an additional deposit of $125 is made each quarter.

U N SA C O M R PL R E EC PA T E G D ES

a Create a table to model this investment. b How much interest is earned by this investment? c What is the total value of this investment at the end of the year?

Exercise 7A

Example 1

A loan of $2500 is charged interest at a rate of 9% p.a. Payments of $600 are made at the end of each year. Use your calculator to determine recursively the balance of the loan (correct to the nearest cent) after: a one payment

d four payments.

b two payments

c three payments

d four payments.

b two payments

c three payments

d four payments.

An investment of $60 000 earns interest at a rate of 0.84% per month. Additional deposits of $2000 are made at the end of each month. Use your calculator to determine the balance of the investment (correct to the nearest cent) after: a one payment

Example 2

c three payments

An investment of $5000 earns interest at a rate of 6% p.a. Additional deposits of $800 are made at the end of each month. Use your calculator to determine the balance of the investment (correct to the nearest cent) after: a one payment

b two payments

A loan of $100 000 is charged interest at a rate of 0.55% per month. Payments of $1500 are made at the end of each month. Use your calculator to determine recursively the balance of the loan (correct to the nearest cent) after: a one payment

BUILDING

b two payments

c three payments

d four payments.

Kiara makes an initial deposit of $12 000 on an investment taken out over four years at a rate of 7.8% compounding per annum and an additional deposit of $500 is made each year. Complete the given table for the first four years.

Month

Value at start of month

Deposit

Interest

Balance at end of month

1 2 3 4

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Chapter 7 Annuities

DEVELOPING Datu borrows $250 000 at a rate of 4.8% p.a. compounding monthly and makes a repayment of $2000 per month. Complete the given table for the first four months. Answer correct to the nearest dollar.

Month

Value at start of month

Payment

Interest

Balance at end of month

U N SA C O M R PL R E EC PA T E G D ES

2 3 4

Blair makes an initial deposit of $30 000 on an investment taken out over 6 years at a rate of 8.4% p.a. compounding biannually and an additional deposit of $1500 is made every six months. a Complete the given table for the first three years. Answer correct to the nearest dollar. b How much interest has been earned over this time?

Month

Value at start of month

Deposit

Interest

Balance at end of month

1 2 3 4 5 6

CHALLENGING

Petelo deposits $5000 into an investment account at the beginning of the year. He received interest of $150 after the first quarter.

a What is the quarterly interest rate? b Anthony made an extra deposit of $600 at the end of each quarter for a year. Use Excel to create a spreadsheet to model this investment? c Calculate the balance of Petelo’s investment account at the end of the year.

Indiana takes out a reducing-balance loan of $32 000 whose interest rate is 12% p.a. compounded monthly.

a What is the monthly interest rate? b Indiana makes a monthly repayment of $5500. Create a spreadsheet to model Indiana’s investment for the first six months. c Calculate the balance of the loan after six months. Answer correct to the nearest dollar. d How many months does it take for Indiana to repay the loan? e What is the final payment in the last month? Answer correct to the nearest dollar.

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7A What are annuities?

Althea has $3000 in a savings account with an interest rate of 6.5% p.a. compounded annually. She wants to calculate the amount she will receive in 4 years when she plans to buy a car.

U N SA C O M R PL R E EC PA T E G D ES

a What amount is available for her car if she deposits $6000 each year into the savings account? Answer correct to the nearest dollar. b Althea decides to transfer her $3000 into a different savings account with an interest rate of 7.5% p.a. compounded annually. What amount is available for her car if she deposits $6000 each year into this savings account? Answer correct to the nearest dollar. c What is the extra amount saved by investing $6000 each year at 7.5% p.a. compared with $6000 each year at 6.5% p.a.?

Communicate your reasoning, understanding and fluency 11

Rory and Gloria each invest $40 000 with an interest rate of 6% p.a. Rory’s interest is compounded monthly whereas Gloria’s interest is compounded yearly. They both make monthly deposits of $500.

a Show how much each investment is worth in 5 years, correct to the nearest dollar. b Explain the reasons why there is a difference between their investments.

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Chapter 7 Annuities

7B Tracking annuities every period Learning intentions

• To use technology to model an annuity • To investigate the outcome of an annuity

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• The concepts in this section will be new to students • These concepts build upon prior learning from Chapter 3 • Questions involving annuities regularly appear in HSC Examinations

Spreadsheets have several built in functions for working with annuities. Let’s consider an annuity where $1000 is invested at the end of each year for 4 years at 10% per annum compound interest. We require the value of the investment after 4 years. The spreadsheet function for this calculation is ‘= FV(0.10, 4, 1000)’ where 0.10 is the rate (10%), 4 is the total number of payment periods and 1000 is the payment per period. The format of the functions in Excel is shown in the table below. The spreadsheet in the Interactive Textbook shows them in action.

Function

Formula

Example

Future value

= FV(rate, nper, pmt, [pv], [type])

= FV(0.10, 4, 1000)

Present value

= PV(rate, nper, pmt, [fv], [type])

= PV(0.10, 4, 1000)

Number of periods

= NPER(rate, pmt, [pv], [fv], [type])

= NPER(0.10, 4, 1000)

Payment

= PMT(rate, nper, pv, [fv], [type])

= PMT(0.10, 4, 1000)

Rate

= RATE(nper, pmt, pv, [fv], [type], [guess])

= RATE(0.10, 4, 1000)

Arguments

Rate: interest rate per period expressed as decimal such as 0.10 for 10%. Nper: the total number of payment periods in an annuity. Pmt: payment made each period and cannot change over the life of the annuity. Pv: present value, or amount of money if invested now would equal the future value. Fv: future value, or sum of the money contributed plus the compound interest earned. Type: 0 if payments are due at the end of the period or 1 at the beginning of the period. Guess: guess for what the rate will be. Note: if parts of the formula are left blank, Excel will assume that their value is zero.

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7B Tracking annuities every period

Example 3

Using Excel to calculate the future value

U N SA C O M R PL R E EC PA T E G D ES

Yara has decided to save for a cruise to be taken when she finishes her TAFE course. She decides to invest $250 per month, at the end of each month, placing it into an account earning 6.4% per annum compounding monthly. Yara will do this for three years. How much will Yara have for her cruise? Answer correct to the nearest cent.

Solution

Explanation

Enter the values into the spreadsheet as follows:

Select the “Future Value” tab.

rate = 6.4

nper = 3 × 12 = 36 pmt = −250

Write the values for ‘rate’, ‘nper’ and ‘pmt’. The Excel formula requires the regular payment per month to be expressed as a negative.

The future value is $9893.09.

The future value will be calculated in cell C5.

Now you try

Farida wants to take a cruise when she finishes university. She invests $200 per month, at the end of each month, by depositing the money into an account earning 7.5% p.a. compounding monthly. Farida will do this for 3 years. How much will Farida have saved for her cruise?

Example 4

Using a spreadsheet to calculate the payment

Khaled is planning to invest into an annuity at the end of each year that pays 10% per annum compound interest. Use a spreadsheet to calculate the payment Khaled needs to make to have $10 000 at the end of 4 years. Answer correct to the nearest cent. The spreadsheet below shows this example, allocating values to cells.

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Continued on next page

377

Chapter 7 Annuities

Solution

Explanation

Cell A1 = PMT(rate, nper, pv, [fv], [type])

Select the “Payment” tab in the excel document.

B11 = annual rate = 10 B7 = nper = 4 B4 = FV = −10 000

Write the values for ‘rate’, ‘nper’ and ‘FV’. The Excel formula requires the FV to be expressed as a negative.

Cell C8 PMT(0.10, 4, 0, −10 000)’

After entering the values into spreadsheet, the rate per payment will be calculated in cell C9 and the payment will be calculated in cell C8.

U N SA C O M R PL R E EC PA T E G D ES

378

The future value is $2154.71.

Now you try

Beni is planning to invest into an annuity at the end of each year that pays 11% p.a. compound interest. Use a spreadsheet to calculate the payment Beni needs to make to have $15 000 in 5 years. Answer correct to the nearest cent.

Example 5

Using Excel to calculate the present value

Find the present value of an annuity in which $1300 was invested for 6 years at a rate of 5.8% p.a. compounded annually at the end of every year. Answer correct to the nearest cent. The spreadsheet below shows this example, allocating values to cells.

Solution

Explanation

Present value = PV(rate, nper, pmt)

Select the “Present Value” tab in the excel document. Write the values for ‘rate’, ‘nper’ and ‘pmt’. The Excel formula requires the regular payment per year to be expressed as a negative.

B11 rate = 5.8 B7 = nper = 6 B8 = pmt = −1300

Cell C6 ‘= PV(0.058, 6, −1300)’ The present value is $6432.89.

After entering the values into the spreadsheet, the rate per payment will be calculated in cell C9 and the payment will be calculated in cell C6.

Now you try Using a spreadsheet, find the present value of an annuity in which $1500 was invested for 5 years at 6.25% p.a. compounded annually at the end of each year. Answer correct to the nearest cent. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

7B Tracking annuities every period

Exercise 7B Example 3

BUILDING

Use Excel to calculate the future value of following investments, correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

a RATE = 0.05, NPER = 3 and PMT = −700 b RATE = 0.06, NPER = 5 and PMT = −2000 c RATE = 10%, NPER = 4 and PMT = −1100 d RATE = 9%, NPER = 2 and PMT = −5000

Use Excel to find the future value correct to the nearest cent for each of the following annuities.

a $23 000 at the end of every year for 3 years with an interest rate of 8.5% p.a. b $1850 at the end of every year for 5 years with an interest rate of 7.25% p.a. c $15 000 at the end of every year for 4 years with an interest rate of 4 12 % p.a. d $7450 at the end of every year for 9 years with an interest rate of 4 23 % p.a.

Linh is planning to invest $2000 at the end of each year at 8% p.a. interest. What will the future value of this annuity be worth after: a 5 years?

Example 4

b 10 years?

c 15 years?

d 20 years?

e 25 years?

f 30 years?

Molly’s grandparents invest $450 for her at the end of each year. They started the annuity on Molly’s first birthday with an interest rate of 8.2% per annum compounded annually. What is the value of the annuity at the end of the year when Molly turns 15? Answer correct to the nearest cent.

Create the spreadsheet shown opposite. A template is available to download from the Interactive Textbook.

a Cell B8 has a formula that calculates the future value. Enter this formula. b Fill down the contents of B8 to B17 using the formula in cell B8. Change the annual interest rate to: c 10%

d 15%

e 20%.

Change the payment to: f $3000

Example 5

g $4000

h $5000.

Use a spreadsheet to calculate the present value of following investments, correct to the nearest cent.

a RATE = 0.05, NPER = 3 and PMT = −700 b RATE = 0.06, NPER = 5 and PMT = −2000 c RATE = 10%, NPER = 4 and PMT = −1100 d RATE = 9%, NPER = 2 and PMT = −5000

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Use a spreadsheet to find the present value correct to the nearest cent for each of the following annuities.

U N SA C O M R PL R E EC PA T E G D ES

a $23 000 at the end of every year for 3 years with an interest rate of 8.5% p.a. b $1850 at the end of every year for 5 years with an interest rate of 7.25% p.a. c $15 000 at the end of every year for 4 years with an interest rate of 4 21 % p.a. d $7450 at the end of every year for 9 years with an interest rate of 4 23 % p.a. 8

Yunchen pays $5680 into a managed fund at the end of each year. The fund pays 12% per annum compounding yearly. What is the present value of the annuity if it has had 10 years to mature? Answer correct to the nearest dollar.

Mrs Zhang aims to give Jenny $200 for her birthday for the next 12 years. What lump sum could she invest now, in an account paying 8% per annum, to achieve this goal? Answer correct to the nearest cent.

Create the spreadsheet shown opposite. A template is available to download from the Interactive Textbook.

a Cell B8 has a formula that calculates the future value. Enter this formula. b Fill down the contents of B8 to B17 using the formula in cell B8. Change the interest rate to: c 10%

d 15%

e 20%.

g $4000

h $5000.

Change the payment to: f $3000

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7B Tracking annuities every period

DEVELOPING 11

Use a spreadsheet to calculate the future value of following investments, correct to the nearest cent. a $420 at the end of every six months for 3 years at 6% p.a. compounded ambiguous b $960 at the end of every six months for 5 years at 9% p.a. compounded biannually

U N SA C O M R PL R E EC PA T E G D ES

c $2000 at the end of every quarter for 9 years at 8% p.a. compounded quarterly

d $1300 at the end of every quarter for 6 years at 6.4% p.a. compounded quarterly e $1200 at the end of every month for 4 years at 4.8% p.a. compounded monthly f $360 at the end of every month for 7 years at 7.2% p.a. compounded monthly

g $680 at the end of every week for 2 years at 5.2% p.a. compounded weekly h $840 at the end of every week for 6 years at 7.8% p.a. compounded weekly

i $320 at the end of every fortnight for 3 years at 4% p.a. compounded fortnightly

j $560 at the end of every fortnight for 4 years at 10% p.a. compounded fortnightly

Janisha’s mother invested in an annuity for Janisha when she was born. She invested $230 per month earning 7.8% per annum, compounded monthly.

a What is the future value of the annuity on Janisha’s tenth birthday?

b On Janisha’s tenth birthday they stopped regular payments. However, she left the money in this account earning the same rate of interest. What is the value of the investment on Janisha’s eighteenth birthday?

Mitch decides to invest $720 per quarter into his superannuation fund. The interest rate quoted is 6.8% per annum, compounded quarterly.

a What is the future value of his superannuation at the end of 20 years?

b Calculate the payment needed per quarter to have $40 000 after 10 years.

Use a spreadsheet to calculate the present value of investments shown in question 11 if you were withdrawing rather than depositing payments. Answer correct to the nearest cent.

Harpreet contributes $860 at the end of each quarter for 15 years into a superannuation fund that earns 9.6% p.a. compounding quarterly.

a What is the future value of this investment? (Answer correct to the nearest dollar.) b What is the present value of this investment? (Answer to the nearest dollar.)

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Qiang is planning to save for a motorbike in four years’ time. Each fortnight he will invest $150 into an annuity earning 5.6% per annum, compounded fortnightly.

U N SA C O M R PL R E EC PA T E G D ES

a How much will Qiang’s investment be worth in four years’ time? Answer correct to the nearest dollar. b Calculate the contribution needed per fortnight to have $20 000 after 4 years. Answer correct to the nearest dollar.

CHALLENGING

Ash aims to have a deposit for a house in 4 years’ time by investing in an annuity. The deposit for the house is $80 000. The annuity has an interest rate of 8% p.a. compounded monthly. How much money per month must Ash invest to reach their target? Answer correct to the nearest cent.

Seo-yun plans to buy a car in 3 years’ time by investing in an annuity. The annuity has an interest rate of 6.6% p.a. compounded fortnightly. She estimates that the car will cost $42 000. Calculate the amount of money Seo-yun must invest at the end of each fortnight to reach her goal. Answer correct to the nearest cent.

Ezekial is saving for a holiday, costing $18 000. He has already saved $10 000. He has deposited the $10 000 into a term deposit at 0.75% per month compounded monthly. In addition he is investing $150 per month into an annuity, paid at the end of each month, earning 5.9% p.a., compounding monthly. Give your answers correct to the nearest cent.

a What is the value of the term deposit after two years? b Calculate the future value of the annuity at the end of two years. c Does Ezekial have enough money for his holiday at the end of two years? Justify.

Tom invests $7500 at 7% per annum compounded monthly for one year. Jaspreet invests $620 at the end of each month at 8% per annum compounded monthly for one year. Which is worth more at the end of the year? Justify your answer.

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7B Tracking annuities every period

Communicate your reasoning, understanding and fluency 21

Shazia will need at least $240 000 to upgrade her hairdressing salon in five years’ time. She plans to make regular monthly payments of $4200 into an investment account that pays a rate of interest of 8.4% p.a. compounding monthly.

U N SA C O M R PL R E EC PA T E G D ES

a Will Shazia have enough money to upgrade her salon? Explain your answer by calculation. b How much would Shazia have needed to invest in a lump sum now to have the required $240 000 in five years’ time? Assume the same rate of interest.

Somchai plans to buy a new car whose present value is $45 000. He decides to take out a loan with an interest rate of 1% per month. The interest on this loan is compounding monthly. What is the monthly contribution required to payout the loan in three years? Answer correct to the nearest cent.

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7C Using tables for the future value of an annuity Learning intentions

• To use a table to determine the future value of an annuity

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• The concepts in this section will be new to students • These concepts build upon prior learning from Chapter 3 • Questions involving annuities regularly appear in HSC Examinations

Investment problems are made easier by using tables. The table below shows the future value of an annuity when $1 is invested at the end of the period at the given interest rate for the given number of periods. The interest is compounded per period. For example, the value of $1 after 6 periods at an interest rate of 6% per period is $6.9753. Future value of $1

Period

1.0000

2.0100

2.0200

2.0300

2.0400

2.0500

2.0600

3.0301

3.0604

3.0909

3.1216

3.1525

3.1836

4.0604

4.1216

4.1836

4.2465

4.3101

4.3746

5.1010

5.2040

5.3091

5.4163

5.5256

5.6371

6.1520

6.3081

6.4684

6.6330

6.8019

6.9753

Using a future value table

1 Determine the time period and rate of interest. 2 Find the intersection of the time period and rate of interest in the table. 3 Multiply the number in the intersection with the payment (FV = intersection × PMT ).

Example 6

Using tables of future values to find the future value of an investment

Use the above table to calculate the future value of:

a $34 000 per year for 3 years at 5% p.a. compounded annually b $5000 per half-year for 2 years at 6% p.a. compounded six-monthly c $300 per month for 5 months at 12% p.a. compounded monthly.

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7C Using tables for the future value of an annuity

Solution

Explanation

a Period = 3, Rate = 5%, PMT = 34 000

Write the values for period, rate and PMT. Find the intersection value in the table for period = 3 and rate = 5%. Write the future value formula used with a future value table. Substitute in the intersection value and the PMT.

Intersection value = 3.1525

U N SA C O M R PL R E EC PA T E G D ES

FV = intersection value × PMT FV = 3.1525 × 34 000 = 107 185

The future value is $107 185.

b Period = 2 × 2 = 4, Rate = 6% ÷ 2 = 3%, PMT = 5000

Write the values for period, rate and PMT. Find the intersection value in the table for period = 4 and rate = 3%. Write the future value formula used with a future value table. Substitute in the intersection value and the PMT.

Intersection value = 4.1836

FV = intersection value × PMT FV = 4.1836 × 5000 = 20 918

The future value is $20 918.

c Period = 5, Rate = 12% ÷ 12 = 1%, PMT = 300

Write the values for period, rate and PMT. Find the intersection value in the table for period = 5 and rate = 1%. Write the future value formula used with a future value table. Substitute in the intersection value and the PMT.

Intersection value = 5.1010

FV = intersection value × PMT FV = 5.1010 × 300 = 1530.3

The future value is $1530.30.

Now you try

Use the table above to calculate the future value of:

a $12 875 per year for 4 years at 6% p.a. compounded annually b $125 per half-year for 3 years at 4% p.a. compounded six-monthly c $750 per month for 6 months at 24% p.a. compounded monthly.

Example 7

Using tables of future values to find the payment when investing

The table shows the future value of an annuity with a contribution of $1. Use the table to answer the following questions. Future value of $1

Period

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

1.000

2.010

2.015

2.020

2.025

2.030

2.035

2.040

3.030

3.045

3.060

3.076

3.091

3.106

3.122

4.060

4.091

4.122

4.153

4.184

4.215

4.246

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Chapter 7 Annuities

a Find the payment or contribution per period of an annuity with a future value of $76 713 at 3.5% p.a. compounded annually for 4 years. b Find the payment or contribution per period of an annuity with a future value of $6105 at 7% p.a. compounded biannually for one year. Explanation

a Period = 4, Rate = 3.5%, FV = 76 713

Write the values for period, rate and FV. Find the intersection value in the table for period = 4 and rate = 3.5%. Write the version of the future value formula used with a future value table that has PMT as the subject of the formula. Substitute in the FV and the intersection value.

U N SA C O M R PL R E EC PA T E G D ES

Solution Intersection value = 4.215 FV PMT = Intersection value 76 713 PMT = 4.215 = 18 200

Payment per month is $18 200.

b Period = 1 × 2 = 2, Rate = 7% ÷ 2 = 3.5%, FV = 6105

Write the values for period, rate and FV. Find the intersection value in the table for period = 2 and rate = 3.5%. Write the version of the future value formula used with a future value table that has PMT as the subject of the formula. Substitute in the FV and the intersection value.

Intersection value = 2.035 FV PMT = Intersection value 6105 PMT = 2.035 = 3000

Payment per month is $3000.

Now you try

Use the table above to find the payment or contribution per period of an annuity with a future value of:

a $15 610 at 4% p.a. compounded annually for 3 years b $5439.20 at 6% p.a. compounded six-monthly for 2 years.

Exercise 7C

Example 6

BUILDING

The table below shows the future value of an annuity with a contribution of $1. Future value of $1

Period

10%

12%

1.00

2.02

2.04

2.06

2.08

2.10

2.12

3.06

3.12

3.18

3.25

3.31

3.37

4.12

4.25

4.37

4.51

4.64

4.78

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7C Using tables for the future value of an annuity

Use the table to calculate the future value of the following annuities.

U N SA C O M R PL R E EC PA T E G D ES

a $2000 per year for 2 years at 12% p.a. compounded annually b $26 000 per year for 3 years at 4% p.a. compounded annually c $32 000 per year for 4 years at 8% p.a. compounded annually d $7000 per year for 2 years at 6% p.a. compounded annually e $10 000 per year for 3 years at 10% p.a. compounded annually f $1600 per year for 4 years at 12% p.a. compounded annually

Caleb saves $2650 by the end of each year and invests it into an account earning 12% p.a. compound interest. Use the table in question 1 to calculate the future value of the investment after 3 years.

The given table shows the future value of an annuity with a contribution of $1. Future value of $1

Period

1.00

2.01

2.02

2.03

2.04

2.05

2.06

2.07

3.03

3.06

3.09

3.12

3.15

3.18

3.21

4.06

4.12

4.18

4.25

4.31

4.37

4.44

5.10

5.20

5.31

5.42

5.53

5.64

5.75

Use the table to calculate the future value of the following annuities. a $34 000 per year for 5 years at 4% p.a. compounded annually b $18 000 per year for 2 years at 6% p.a. compounded annually c $45 000 per year for 4 years at 1% p.a. compounded annually d $8600 per year for 3 years at 7% p.a. compounded annually e $14 200 per year for 5 years at 5% p.a. compounded annually f $90 000 per year for 2 years at 3% p.a. compounded annually

4 Isaac is saving to buy a car. He deposits $5500 into an account at the end of every year for 4 years. The account pays interest of 6% p.a., compounding annually. Future value of $1

End of year

1.0000

2.0500

2.0600

2.0700

2.0800

3.1525

3.1836

3.2149

3.2464

4.3101

4.3746

4.4399

4.5061

Use the given table to calculate the following amounts. a What is the value of Isaac’s investment at the end of the following years? i Two

ii Three

iii Four

b How much interest does Isaac earn on his investment over the following years? i Two

ii Three

iii Four

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Example 7

5 The given table shows the future value of an annuity with a contribution of $1. Future value of $1 1%

12%

16%

20%

1.0000

2.0100

2.0400

2.0800

2.1200

2.1600

2.2000

U N SA C O M R PL R E EC PA T E G D ES

Period

3.0301

3.1216

3.2464

3.3744

3.5056

3.6400

4.0604

4.2465

4.5061

4.7793

5.0665

5.3680

5.1010

5.4163

5.8666

6.3528

6.8771

7.4416

6.1520

6.6330

7.3359

8.1152

8.9775

9.9299

Use the table to calculate the future value of the following annuities.

a $7000 per year for 2 years at 1% p.a. compounded annually b $16 000 per year for 6 years at 12% p.a. compounded annually c $58 000 per year for 6 years at 4% p.a. compounded annually d $120 000 per year for 6 years at 1% p.a. compounded annually e $950 per quarter for 1 year at 16% p.a. compounded quarterly f $1200 per quarter for 1 year at 4% p.a. compounded quarterly g $8400 per quarter for 1 year at 16% p.a. compounded quarterly h $37 000 per half-year for 3 years at 16% p.a. compounded biannually i $5400 per half-year for 2 years at 8% p.a. compounded biannually j $12 100 per half-year for 1 year at 8% p.a. compounded biannually

DEVELOPING

The given table shows the future value of an annuity with a contribution of $1. Future value of $1

Period

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

1.000

2.010

2.015

2.020

2.025

2.030

2.035

2.040

3.030

3.045

3.060

3.076

3.091

3.106

3.122

4.060

4.091

4.122

4.153

4.184

4.215

4.246

5.101

5.152

5.204

5.256

5.309

5.362

5.416

6.152

6.230

6.308

6.388

6.468

6.550

6.633

7.214

7.323

7.434

7.547

7.662

7.779

7.898

8.286

8.433

8.583

8.736

8.892

9.052

9.214

Use the table to calculate the future value of the following annuities. a $600 per month for 4 months at 18% p.a. compounded monthly b $1280 per month for 5 months at 12% p.a. compounded monthly c $970 per month for 7 months at 42% p.a. compounded monthly d $11 000 per quarter for 9 months at 4% p.a. compounded quarterly Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

7C Using tables for the future value of an annuity

e $23 000 per quarter for 1 year at 8% p.a. compounded quarterly f $5000 per quarter for 2 years at 12% p.a. compounded quarterly g $32 000 per half-year for 18 months at 7% p.a. compounded six-monthly h $67 000 per half-year for 2 years at 2% p.a. compounded six-monthly i $14 000 per half-year for 4 years at 8% p.a. compounded six-monthly Create the spreadsheet shown on the right. A template is available to download from the Interactive Textbook.

U N SA C O M R PL R E EC PA T E G D ES

a Cell B6 has a formula = FV($B$5, A6, −1) that calculates the future value. Enter this formula. b Fill down the contents of B6 to B13 using the formula in cell B6. c Cells C6 : F6 have similar formulas to B6. Enter these formulas to calculate the future value. d Fill down the contents of C6 to F13 using the formulas in cells C6 : F6.

CHALLENGING

The table below shows the future value of an annuity with a payment or contribution of $1. Future value of $1

End of year

1.0000

2.0300

2.0400

2.0500

2.0600

2.0700

2.0800

3.0909

3.1216

3.1525

3.1836

3.2149

3.2464

4.1836

4.2465

4.3101

4.3746

4.4399

4.5061

5.3091

5.4163

5.5256

5.6371

5.7507

5.8666

6.4684

6.6330

6.8019

6.9753

7.1533

7.3359

Using the table find the payment per period of an annuity whose future value is:

a $8242 at 7% p.a. compounded annually for 3 years b $8780 at 5% p.a. compounded annually for 2 years c $13 784 at 6% p.a. compounded annually for 4 years d $20 800 at 8% p.a. compounded annually for 3 years e $165 000 at 5% p.a. compounded annually for 4 years.

Use the table in question 8 to find the annual compound interest rate of an annuity whose future value is:

a $2972.50 when $1450 is invested at the end of each year for 2 years b $4884.37 when $920 is invested at the end of each year for 5 years c $12 861.32 when $2940 is invested at the end of each year for 4 years d $63 743.07 when $20 420 is invested at the end of each year for 3 years e $8085.50 when $1250 is invested at the end of each year for 6 years. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Chapter 7 Annuities

Thomas aims to have a deposit for an apartment of at least $56 000 in 3 years’ time by investing in an annuity. The annuity has an interest rate of 8% p.a. compounded biannually. Use the table in question 8 to calculate Thomas’s six-monthly contribution to achieve his deposit. Answer correct to the nearest dollar.

Use the table in question 6 to answer the following questions, correct to the nearest dollar. a What would be the future value of an $820 per quarter annuity at 8% p.a. for 2 years, with interest compounding quarterly? b What is the value of an annuity that would provide a future value of $98 375 after 6 years at 3% p.a. compound interest? c An annuity of $3800 per half-year is invested at 4% p.a. compounded six-monthly for 4 years. What will be the amount of interest earned?

U N SA C O M R PL R E EC PA T E G D ES

390

Communicate your reasoning, understanding and fluency 12

Hui is choosing between two investment opportunities. Opportunity 1: Deposit $10 000 now and earn 1.2% p.a. compounded monthly. Opportunity 2: Deposit $2000 at the end of each quarter at 8% p.a. compounded quarterly. Future value of $1

Period

1.0000

2.0100

2.0200

2.0300

2.0400

2.0500

2.0600

3.0301

3.0604

3.0909

3.1216

3.1525

3.1836

4.0604

4.1216

4.1836

4.2465

4.3101

4.3746

5.1010

5.2040

5.3091

5.4163

5.5256

5.6371

6.1520

6.3081

6.4684

6.6330

6.8019

6.9753

Use the future value table to answer the following questions.

a What is the value of Opportunity 1 after 5 years correct to the nearest dollar? b What is the value of Opportunity 2 after 5 years? c How much interest does Hui earn with Opportunity 1? d How much interest does Hui earn with Opportunity 2? e If you were 20 years old and given the same opportunities, explain which option you would choose and why.

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7D Using tables for the present value of an annuity

7D Using tables for the present value of an annuity Learning intentions

• To use a table of present values to calculate the present value of an annuity

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• The concepts in this section will be new to students • These concepts build upon prior learning from Chapter 3 • Questions involving annuities regularly appear in HSC Examinations

The present value of an annuity is the amount of money that, if invested now, would become the future value of the annuity. The present value is the principal (P) in the compound interest formula and the future value is the amount (A) or FV = PV(1 + r)n . If the present value is made the subject of this formula, then the present FV value is the future value divided by (1 + r)n or PV = . (1 + r)n Calculating the present value is made easy using tables and provides an alternative to using the formula.

Using a present value table

1 Determine the time period and rate of interest. 2 Find the intersection of the time period and rate of interest in the table. 3 Multiply the number in the intersection by the amount of money contributed.

Example 8

Using tables of present values to find the present value of an investment

Use the table to calculate the present value, if:

a $9600 was contributed per year for 7 years at 4% p.a. compounded annually b $7000 was contributed per quarter for 2 years at 16% p.a. compounded quarterly. Present value of $1

End of period

5.4172

5.2421

5.0757

4.9173

6.2303

6.0021

5.7864

5.5824

7.0197

6.7327

6.4632

6.2098

7.7861

7.4353

7.1078

6.8017

Solution

Explanation

a Period = 7, Rate = 4%, Contribution = 9600

Write the values for period, rate and contribution.

Intersection value = 6.0021

PV = Intersection value × Contribution PV = 6.0021 × 9600 = 57 620.16 The present value is $57 620.16.

Find the intersection value in the table for period = 7 and rate = 4%. Write the present value formula used with a present value table. Substitute in the intersection value and the contribution.

Continued• (03) on next Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 8671 page 1400

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Chapter 7 Annuities

b Period = 2 × 4 = 8, Rate = 16% ÷ 4 = 4%, Contribution = 7000

Write the values for period, rate and contribution.

Intersection value = 6.7327

Find the intersection value in the table for period = 8 and rate = 4%. Write the present value formula used with a present value table.

U N SA C O M R PL R E EC PA T E G D ES

PV = Intersection value × Contribution PV = 6.7327 × 7000 = 47 128.90

The present value is $47 128.90.

Substitute in the intersection value and the contribution.

Now you try

Use the table above to calculate the present value if:

a $1275 is contributed per year for 9 years at 6% p.a. compounded annually b $625 is contributed per half-year for 3 years at 10% p.a. compounded six-monthly.

Example 9

Using tables of present values to find the payment when investing

The table shows the present value of an annuity with a contribution of $1. Use the table to answer the following questions, to the nearest dollar. Present value of $1

Period

10%

12%

0.99

0.98

0.97

0.96

0.94

0.93

0.91

0.89

1.97

1.94

1.91

1.89

1.83

1.78

1.74

1.69

2.94

2.88

2.83

2.78

2.67

2.58

2.49

2.40

3.90

3.81

3.71

3.63

3.47

3.31

3.17

3.04

a Find the payment or contribution per period of an annuity with a present value of $33 240 at 6% p.a. compounded annually for 3 years. b Find the payment or contribution per period of an annuity with a present value of $10 000 at 8% p.a. compounded biannually for 1 year. Solution

Explanation

a Period = 3, Rate = 6%, PV = 33 240

Write the values for period, rate and PV. Find the intersection value in the table for period = 3 and rate = 6%. Write the version of the present value formula used with a present value table that has payment (PMT ) as the subject of the formula. Substitute in the PV and the intersection value.

Intersection value = 2.67 PV PMT = Intersection value 33 240 PMT = 2.67 ≈ 12 499 Payment is $12 499

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7D Using tables for the present value of an annuity

b Period = 1 × 2 = 2, Rate = 8% ÷ 2 = 4%, PV = 10 000

Write the values for period, rate and PV.

Intersection value = 1.89

Find the intersection value in the table for period = 2 and rate = 4%.

PV Intersection value 10 000 PMT = 1.89 ≈ 5291

PMT =

U N SA C O M R PL R E EC PA T E G D ES

Write the version of the present value formula used with a present value table that has payment (PMT) as the subject of the formula. Substitute in the PV and the intersection value.

Payment is $5291

Now you try

Use the table above to find the payment or contribution per period (to the nearest dollar) of an annuity with a present value of:

a $64 125 at 8% p.a. compounded annually for 4 years b $5555 at 12% p.a. compounded six-monthly for 2 years.

Exercise 7D

Example 8

BUILDING

The table on the right side shows the present value of an annuity with a contribution of $1. Use the table to calculate the present value of the following annuities.

Present value of $1

a $8000 per year for 2 years at 10% p.a. compounded annually b $35 000 per year for 4 years at 2% p.a. compounded annually

Period

10%

12%

0.98

0.96

0.94

0.93

0.91

0.89

1.94

1.89

1.83

1.78

1.74

1.69

2.88

2.78

2.67

2.58

2.49

2.40

3.81

3.63

3.47

3.31

3.17

3.04

c $22 000 per year for 2 years at 6% p.a. compounded annually d $43 000 per year for 3 years at 12% p.a. compounded annually e $6000 per year for 4 years at 8% p.a. compounded annually.

Nala pays $5600 into a managed fund at the end of each year. The fund pays 10% p.a. interest compounding yearly. Use the table in question 1 to calculate the present value of the annuity if it has 3 years to mature.

The table on the right side shows the present value of an annuity with a contribution of $1. Use the table to calculate the present value of the following annuities.

Present value of $1

End of year

5.4172

5.2421

5.0757

4.9173

6.2303

6.0021

5.7864

5.5824

7.0197

6.7327

6.4632

6.2098

7.7861

7.4353

7.1078

6.8017

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a $5000 per year for 6 years at 3% p.a. compounded annually b $21 000 per year for 9 years at 5% p.a. compounded annually c $14 000 per year for 7 years at 6% p.a. compounded annually d $1000 per year for 9 years at 4% p.a. compounded annually e $8000 per year for 8 years at 6% p.a. compounded annually Use the present value table in question 3 to calculate the present value when $3200 is invested at the end of each year for 9 years at 5% p.a. compound interest. Answer correct to the nearest cent.

Farooq would like a new kitchen. He can afford to pay $8400 into an account at the end of every year. The account has an interest rate of 9% p.a. compounded annually. The table on the right side shows the present value of the annuity with a payment of $1.

U N SA C O M R PL R E EC PA T E G D ES

Present value of $1

Period

0.9524

0.9434

0.9346

0.9259

0.9174

1.8594

1.8334

1.8080

1.7833

1.7591

2.7232

2.6730

2.6243

2.5771

2.5313

3.5460

3.4651

3.3872

3.3121

3.2397

Use the table to do the following calculations. Answer correct to the nearest dollar.

a What is the present value when the payments are made at the end of the following years? i Two

ii Three

iii Four

b Farooq found another account whose interest rate is of 6% p.a. compounded annually. What is the present value when the same payment is made into this account? i Two

Example 9

ii Three

iii Four

6 The table below shows the present value of an annuity with a contribution of $1. Present value of $1

Period

10%

12%

0.9901

0.9804

0.9615

0.9434

0.9259

0.9091

0.8929

1.9704

1.9416

1.8861

1.8334

1.7833

1.7355

1.6901

2.9410

2.8839

2.7751

2.6730

2.5771

2.4869

2.4018

3.9020

3.8077

3.6299

3.4651

3.3121

3.1699

3.0373

4.8534

4.7135

4.4518

4.2124

3.9927

3.7908

3.6048

5.7955

5.6014

5.2421

4.9173

4.6229

4.3553

4.1114

Use the table to calculate the present value of the following annuities. Answer to the nearest dollar. a $8000 per year for 4 years at 6% p.a. compounded annually b $11 000 per year for 5 years at 10% p.a. compounded annually c $13 000 per year for 6 years at 12% p.a. compounded annually d $7000 per year for 2 years at 1% p.a. compounded annually e $51 000 per year for 6 years at 4% p.a. compounded annually f $100 000 per year for 6 years at 1% p.a. compounded annually g $3500 per quarter for 1 year at 8% p.a. compounded quarterly

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7D Using tables for the present value of an annuity

h $8000 per quarter for 1 year at 4% p.a. compounded quarterly i $29 000 per half-year for 2 years at 2% p.a. compounded biannually j $6100 per half-year for 3 years at 8% p.a. compounded biannually

DEVELOPING The table below shows the future value of an annuity with a contribution of $1.

U N SA C O M R PL R E EC PA T E G D ES

Present value of $1

Period

12%

16%

20%

0.9901 0.9804 0.9615 0.9434 0.9259 0.8929 0.8621 0.8333

1.9704 1.9416 1.8861 1.8334 1.7833 1.6901 1.6052 1.5278

2.9410 2.8839 2.7751 2.6730 2.5771 2.4018 2.2459 2.1065

3.9020 3.8077 3.6299 3.4651 3.3121 3.0373 2.7982 2.5887

Use the table to find the present value (to nearest dollar) of the following annuities. a $8400 per quarter for 1 year at 16% p.a. compounded quarterly b $950 per quarter for 1 year at 16% p.a. compounded quarterly c $1200 per quarter for 1 year at 4% p.a. compounded quarterly d $12 100 per half-year for 1 year at 8% p.a. compounded biannually e $37 000 per half-year for 1.5 years at 16% p.a. compounded biannually f $5400 per half-year for 2 years at 8% p.a. compounded biannually

8 The table below shows the value of an annuity with a contribution of $1. Present value of $1

Period

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%

0.9901

0.9852

0.9804

0.9756

0.9709

0.9662

0.9615

0.9569

0.9524

1.9704

1.9559

1.9416

1.9274

1.9135

1.8997

1.8861

1.8727

1.8594

2.9410

2.9122

2.8839

2.8560

2.8286

2.8016

2.7751

2.7490

2.7232

3.9020

3.8544

3.8077

3.7620

3.7171

3.6731

3.6299

3.5875

3.5460

4.8534

4.7826

4.7135

4.6458

4.5797

4.5151

4.4518

4.3900

4.3295

5.7955

5.6972

5.6014

5.5081

5.4172

5.3286

5.2421

5.1579

5.0757

6.7282

6.5982

6.4720

6.3494

6.2303

6.1145

6.0021

5.8927

5.7864

7.6517

7.4859

7.3255

7.1701

7.0197

6.8740

6.7327

6.5959

6.4632

Use the table to find the present value (to the nearest dollar) of the following annuities. a $800 per month for 6 months at 30% p.a. compounded monthly b $630 per month for 7 months at 42% p.a. compounded monthly c $1520 per month for 5 months at 12% p.a. compounded monthly d $28 000 per quarter for 1 year at 8% p.a. compounded quarterly e $11 000 per quarter for 9 months at 4% p.a. compounded quarterly f $8000 per quarter for 2 years at 16% p.a. compounded quarterly

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g $30 000 per half-year for 18 months at 7% p.a. compounded six-monthly h $19 500 per half-year for 4 years at 8% p.a. compounded six-monthly i $61 000 per half-year for 2 years at 7% p.a. compounded six-monthly Create the spreadsheet below. A template is available to download from the Interactive Textbook.

U N SA C O M R PL R E EC PA T E G D ES

a Cell B6 has a formula = PV($B $5, A6, −1) that calculates the present value. Enter this formula. b Fill down the contents of B6 to B13 using the formula in cell B6. c Cells C6 : K6 have similar formulas to B6. Enter these formulas to find the present value. d Fill down the contents of C6 to K13 using the formulas in cells C6 : K6.

CHALLENGING

Use the above spreadsheet table in question 9 to find the payment per period, correct to the nearest dollar, of an annuity whose present value is:

a $10 684 at 5% p.a. compounded annually for 2 years b $5005 at 7% p.a. compounded annually for 3 years c $139 420 at 5% p.a. compounded annually for 4 years d $16 860 at 6% p.a. compounded annually for 4 years e $22 100 at 10% p.a. compounded annually for 3 years.

Use the above spreadsheet table in question 9 to find the annual compound interest rate of an annuity whose present value is:

a $25 431.65 when $7850 is invested at the end of each year for 4 years b $6318.60 when $1500 is invested at the end of each year for 5 years c $4468.56 when $610 is invested at the end of each year for 8 years d $70 709.55 when $25 480 is invested at the end of each year for 3 years e $12 405.39 when $2290 is invested at the end of each year for 6 years.

Dao plans a holiday that has a present value of $3200. She decides to take out a loan with an interest rate of 12% p.a. compounded monthly. What is the monthly contribution required to pay out the loan in 6 months? Use the above spreadsheet table in question 9 and answer to the nearest dollar.

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7D Using tables for the present value of an annuity

Communicate your reasoning, understanding and fluency 13

In twenty years, the future value of an investment will be $200 000. The interest rate is 6% p.a., compounded half-yearly. Which equation will give the present value (PV) of the investment? 200 000 (1 + 0.06)20

U N SA C O M R PL R E EC PA T E G D ES

A PV =

B PV =

200 000 (1 + 0.06)40

C PV =

200 000 (1 + 0.03)20

D PV =

200 000 (1 + 0.03)40

Eric made a deposit into a new bank account which earns 2.5% p.a. compound interest. Present value of $1

Years

2.5%

9.471

9.222

8.983

8.752

18.046

17.169

16.351

15.589

25.808

24.016

22.396

20.930

Use the present value table above to calculate the minimum deposit Eric will have needed to make given that he made the following withdrawals: a $750 at the end of each year for 10 years starting one year after the account was opened b $5000 each year for 10 years starting 11 years after the account opened.

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7E Solving problems involving superannuation and loans Learning intentions

• To solve problems involving superannuation and loans Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• The concepts in this section will be new to students • These concepts build upon prior learning from Chapter 3 • Questions involving annuities regularly appear in HSC Examinations

Loans

The regular loan repayment is an example of an annuity. In a reducing-balance loan, as the payments are made, the balance owing is reduced and therefore the interest charged is reduced. A table is often used to summarise the key properties of an annuity. It shows the payment number, the payment received, the interest earned, the principal reduction and the balance of the annuity after each payment has been received.

Example 10

Reading and interpreting an annuity table for a loan

Jessica takes out a car loan for $24 000 with an interest rate of 10% p.a. compound interest. She agrees to a regular yearly payment of $5500 over six years. A table for this annuity is shown. Payment number

Payment paid

5500.00

Interest paid

Principal reduction

Balance of annuity

24 000.00

2400.00

3100.00

20 900.00

5500.00

2090.00

3410.00

17 490.00

5500.00

1749.00

3751.00

13 739.00

5500.00

1373.90

4126.10

9612.90

5500.00

961.29

4538.71

5074.19

5500.00

507.42

4992.58

81.61

0.00

Determine, to the nearest cent:

a the interest paid when payment 1 is made b the principal reduction when payment 3 is made c the balance of the annuity after payment 4 has been made d the value of the last payment if the balance of the annuity is to be zero after the 6th payment is received e the total amount of interest paid. Solution

Explanation

a Interest paid is $2400.00 b Principal reduction is $3751.00 c Balance of the annuity is $9612.90 d Payment = $5500 + $81.61 = $5581.61 e Interest = $2400 + $2090 + . . .

Read the value in the table. Read the value in the table. Read the value in the table. Add the regular payment and the balance of the annuity. Sum the interest paid column.

= $9081.61 Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

7E Solving problems involving superannuation and loans

Now you try Wajma takes out a loan for a new family car for $34 750 with an interest rate of 8% compound interest. She agrees to regular yearly payments of $7500 over 6 years. A table for this annuity is shown. Payment number

Payment paid 0.00

Principal reduction

0.00

Balance of annuity 34 750.00

U N SA C O M R PL R E EC PA T E G D ES

Interest paid

7500.00

2780.00

4720.00

30 030.00

7500.00

2402.40

5097.60

24 932.40

7500.00

1994.59

5505.41

19 426.99

7500.00

1554.16

5945.84

13 481.15

7500.00

1078.49

6421.51

7059.64

7500.00

564.77

6935.23

124.41

Determine to the nearest cent:

a the interest paid when payment 2 is made b the principal reduction when payment 3 is made c the balance of the annuity after payment 4 is made d the value of the last payment if the balance of the annuity is to be zero after the last payment e the total amount of interest paid.

Superannuation

The regular payment into a superannuation fund is an example of an annuity. Superannuation is money invested while you are working, so you can enjoy a regular income later in life when you retire. In Australia your employer must pay 12% of your income into a super account. You can also add extra money to increase the amount you have in retirement.

Example 11

Reading and interpreting an annuity table for superannuation

Michael is aged 60 and is planning to retire at 65 years of age. His current superannuation fund has a balance of $500 000 and is delivering 7% p.a. compound interest. Michael is making a payment of $20 000 each year into his superannuation fund. A table for this annuity is shown.

Payment number Payment received Interest earned Principal increase Balance of annuity 0

20 000.00

0.00

500 000.00

35 000.00

55 000.00

555 000.00

20 000.00

38 850.00

58 850.00

613 850.00

20 000.00

47 377.37

67 377.37

744 196.87

20 000.00

52 093.78

72 093.78

816 290.65

What is the value of A, B and C? Answer correct to the nearest cent. Continued on next page Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Chapter 7 Annuities

Solution

Explanation

For A PV = 613 850, Rate = 7% = 0.07, n = 1 FV = PV(1 + r)n

Write the values for PV, rate and n. Write the future value formula. Substitute in the values PV, rate and n.

FV = 613 850(1.07)1

U N SA C O M R PL R E EC PA T E G D ES

= $656 819.50 I = FV − PV

Write the interest formula. Substitute in the values for FV and PV.

= 656 819.50 − 613 850 = 42 969.50

A is $42 969.50

For B Increase = 42 969.50 + 20 000

Add the interest earned to the payment.

= 62 969.50

B is $62 969.50

For C Balance = 62 969.50 + 613 850

Add the principal increase to the previous balance of the annuity.

= 676 819.50

C is $676 819.50

Now you try

Bethany is aged 62 and is planning to retire at age 67. She has a current balance in her superannuation fund of $650 000. The fund earns 7.5% interest. Bethany is making a payment of $25 000 each year into her superannuation fund. A table for this annuity is shown. Payment number

Payment received

Interest paid

Principal increase

Balance of annuity

0.00

650 000.00

25 000.00

48 750.00

73 750.00

723 750.00

25 000.00

54 281.25

79 281.25

803 031.25

25 000.00

66 619.39

91 619.39

979 877.99

25 000.00

73 490.85

98 490.85

1 078 368.84

What is the value of A, B and C? Answer correct to the nearest cent.

Annuity table

A table is used to summarise the key properties of an annuity. It shows the payment number, the payment received, the interest earned, the principal reduction and the balance of the annuity after each payment has been received.

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7E Solving problems involving superannuation and loans

Exercise 7E 1

Sheila is a university student who received a scholarship of $6000 for her first year of study. She invests the money in an annuity paying an interest rate of 3% per annum, compounding monthly. From this annuity, she receives a monthly payment of $508. A table for this annuity is shown.

Payment number

Payment received

Interest earned

Principal reduction

Balance of annuity

0.00

6000.00

508.00

15.00

493.00

5507.00

508.00

13.77

494.23

5012.77

508.00

12.53

495.47

4517.30

508.00

11.29

496.71

4020.59

508.00

10.05

497.95

3522.64

508.00

7.56

500.04

2523.01

508.00

6.31

501.69

2021.32

508.00

5.05

502.95

1518.37

508.00

3.80

504.20

1014.17

508.00

2.54

505.46

508.70

508.00

1.27

506.73

1.97

U N SA C O M R PL R E EC PA T E G D ES

Example 10

BUILDING

Determine, correct to the nearest cent:

a the monthly interest rate b the interest earned when payment 1 is received c the principal reduction when payment 3 is received d the balance of the annuity after payment 5 has been received e the values of A, B and C f the value of the last payment if the balance of the annuity is to be zero after the 12th payment is received g the total payments received from the annuity h the total amount of interest paid.

The interest rate on the annuity has increased to 6% p.a. Determine, correct to the nearest cent:

i the monthly interest rate j the interest earned when payment 1 is received k the principal reduction when payment 1 is received l the balance of the annuity after payment 1 has been received m the difference in the balance of the annuity after 1 payment with 6% interest compared to the balance with 3% interest.

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Chapter 7 Annuities

Example 11

2 Jennifer is aged 50 and is planning to retire at 60 years of age. Her current superannuation fund has a balance of $450 000 and is delivering 6% p.a. compound interest. Jennifer is making a payment of $18 000 each year into her superannuation fund. A table for this annuity is shown.

Payment number

Payment received

Interest earned

Principal increase

Balance of annuity

450 000.00

0.00

18 000.00

27 000.00

45 000.00

495 000.00

18 000.00

29 700.00

47 700.00

542 700.00

18 000.00

32 562.00

50 562.00

593 262.00

18 000.00

35 595.72

53 595.72

646 857.72

18 000.00

38 811.46

56 811.46

703 669.18

18 000.00

42 220.15

60 220.15

763 889.33

18 000.00

45 833.36

63 833.36

827 722.69

18 000.00

53 723.16

71 723.16

967 09.22

18 000.00

58 026.55

76 026.55

1 043 135.77

U N SA C O M R PL R E EC PA T E G D ES

11 12 13

Determine, correct to the nearest cent:

a the interest earned when payment 1 is received b the principal increase when payment 3 is received c the balance of the annuity after payment 5 has been received d the values of A, B and C e the final balance of the annuity after payment 10 has been received f the total payments received by the annuity g the total amount of interest earned h the extra interest earned if Jennifer delays retirement for one year (after 11 payments) i the principal increase if Jennifer delays retirement for one year (after 11 payments) j the balance of the annuity if Jennifer delays retirement for one year (after 11 payments) k the interest earned in the twelfth year if Jennifer delays retirement for two years (after 12 payments) l the principal increase in the twelfth year if Jennifer delays retirement for two years (after 12 payments) m the balance of the annuity if Jennifer delays retirement for two years (after 12 payments) n the interest earned in the thirteenth year if Jennifer delays retirement for three years (after 13 payments) o the principal increase in the thirteenth year if Jennifer delays retirement for three years (after 13 payments) p the balance of the annuity if Jennifer delays retirement for three years (after 13 payments).

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7E Solving problems involving superannuation and loans

DEVELOPING Adrian borrowed $500 000 for an apartment. The interest rate was 5% p.a. compounding monthly and the monthly repayment is $2500. Copy and complete the annuity table shown for the first 12 months. Use your calculator answer to complete each cell of the table, not the approximation. Answer correct to the nearest cent.

U N SA C O M R PL R E EC PA T E G D ES

Payment number

Payment received

0.00

2500.00

Interest charged

Principal reduction

Balance of annuity

0.00

500 000.00

2 3 4 5 6 7 8 9

10 11 12

CHALLENGING

Stephen is aged 55 and is planning to retire at 65 years of age. His current superannuation fund has a balance of $700 000 and is delivering 4% p.a. compound interest. Stephen is making a payment of $22 000 each year into his superannuation fund. a Construct a table to summarise the key properties of this annuity for the next ten years. b By what amount has the annuity grown if Stephen retired at 57 years of age? c By what amount has the annuity growth if Stephen retired at 62 years of age? d By what amount has the annuity growth if Stephen retired at 67 years of age?

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Chapter 7 Annuities

Communicate your reasoning, understanding and fluency Rhys is 60 years old and would like to retire. When he was 40 years old he had $150 000 in his superannuation account. For the first ten years Rhys received 8% p.a. compound interest and deposited $12 000 each year. For the next ten years he received 9.5% p.a. compound interest and was able to deposit $15 000 each year.

U N SA C O M R PL R E EC PA T E G D ES

a Model this annuity in a spreadsheet. b How much will Rhys have in his superannuation account when he is 50? c How much interest does he earn in the 20 years of the model? d Rhys decides (when he is 50) that he would like 1.5 million dollars in his super when he retires. Correct to the nearest $100, how much extra will he need to pay each year for the last ten years to make this happen?

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405

Chapter 7 review

An annuity is a form of investment that involves the regular contribution of money. Investments into superannuation or a monthly loan repayment are examples of annuities.

Annuity table

An annuity table uses the previous result to obtain the next result. For example, the future value at the end of the first year ($1000) is the present value for the second year and the future value at the end of the second year ($2100) is the present value for the third year. This can be represented using a table.

U N SA C O M R PL R E EC PA T E G D ES

Annuity

Review

Key ideas and chapter summary

Value at start

Value at end

Year

of Year

Deposit

Interest

of Year

$1000

$100

$1100

$1000

$110

$2100

$1000

$210

$3310

$1000

$331

$4641

Excel has several built in functions for working with annuities.

= FV(rate, nper, pmt, [pv], [type]) = PV(rate, nper, pmt, [fv], [type]) = NPER(rate, nper, pmt, [pv], [type]) = PMT(rate, nper, pv, [fv], [type]) = RATE(nper, pmt, pv, [fv], [type])

Future value of an annuity

The future value of an annuity is the sum of the money contributed plus the compound interest earned. It is the total value of the investment at the end of a specified term.

Present value of an annuity

The present value of an annuity is the amount of money if invested now would become the future value of the annuity.

Using a future value or present value table with factors

1 Determine the time period and rate of interest.

2 Find the intersection of the time period and rate of interest.

3 Multiply the number in the intersection with the money contributed.

Use of annuities

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Chapter 7 Annuities

Chapter checklist with success criteria A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

1 I can model an annuity on a loan using a table.

U N SA C O M R PL R E EC PA T E G D ES

e.g. Hando borrows $1750 at an interest rate of 14.25% p.a., compounded monthly. She will repay the loan by making 5 monthly repayments of $361.75. a Construct a table to model this loan. b Use the table to find the balance of the loan after Hando has made each of the four payments. c What is the balance of the loan (the amount she still owes) after she has made 3 payments? Give your answer correct to the nearest cent. d Is the loan fully paid out after four payments have been made? If not, how much will the last payment need to be to ensure the loan is fully repaid after four payments?

2 I can model an annuity as an investment using a table.

e.g. Jiemba makes an initial investment of $1750 on an investment taken out over one year at a rate of 8.25% p.a. compounded quarterly and an additional deposit of $120 is made each quarter. a Create a table to model this investment. b How much interest is earned by this investment? c What is the total value of this investment at the end of the year?

3 I can use a spreadsheet to calculate future value.

e.g. Aarav wants to take a holiday when he finishes university. He invests $150 per month, at the end of each month, by depositing the money into an account earning 5.5% p.a. compounded monthly. Aarav will do this for 4 years. How much will Aarav have saved for his holiday?

4 I can use a spreadsheet to calculate the payment.

e.g. Effie is planning to invest into an annuity at the end of each year that pays 9.75% p.a. compound interest. Use a spreadsheet to calculate the payment Effie needs to make to have $12 000 in 5 years.

5 I can use a spreadsheet to calculate the present value.

e.g. Using a spreadsheet, find the present value that will need to be invested so as to match the value of an annuity in which $1800 is invested at 7.7% p.a. compounded annually at the end of each year for 6 years.

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Chapter 7 review

6 I can use a table of future values to find the future value of an investment. e.g. Use the table below to calculate the future value of:

U N SA C O M R PL R E EC PA T E G D ES

a $12 000 per year for 2 years at 5% p.a. compounded annually b $725 per half-year for 3 years at 6% p.a. compounded six-monthly c $1750 per month for 4 months at 12% p.a. compounded monthly.

Review

Future value of $1

Period

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2.0100 2.0200 2.0300 2.0400 2.0500 2.0600

3.3010 3.6040 3.0909 3.1216 3.1525 3.1836

4.0604 4.1216 4.1836 4.2465 4.3101 4.3746

5.1010 5.2040 5.3091 5.4163 5.5256 5.6371

6.1520 6.3081 6.4684 6.6330 6.8019 6.9753

7 I can use a table of future values to find the payment when investing.

e.g. Use the table in Question 6 to find the payment or contribution per period of an annuity with a future value of: a $6907 at 5% p.a. compounded annually for 5 years b $7874.28 at 12% p.a. compounded six-monthly for 2 years.

8 I can use a table of present values to find the present value of an investment. e.g. Use the given table to calculate the present value needed to be invested in order to match an annuity if:

End of Period

5.4172 5.21421 5.0757 4.9173

6.2303

6.0021

5.7864 5.5824

7.0197

6.7327

6.4632 6.2098

9 7.7861 7.4353 b $400 is contributed per half-year for 3 years at 12% p.a. compounded six-monthly.

7.1078 6.8017

a $1500 is contributed per year for 8 years at 3% p.a. compounded annually

Present value of $1

9 I can use a table of present values to find the payment when investing.

e.g. Use the table in Question 8 to find the payment or contribution per period correct (to the nearest dollar) of an annuity with a present value of: a $16 900 at 6% p.a. compounded annually for 8 years b $8000 at 10% p.a. compounded six-monthly for 4 years.

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Chapter 7 Annuities

10 I can read and interpret an annuity table for a loan. e.g. Inaya takes out a loan for a new family car for $27 000 with an interest rate of 7% compound interest. She agrees to regular yearly payments of $5100 over 7 years. A table for this annuity is shown. Balance Payment number Payment paid Interest paid Principal reduction of annuity

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0.00

27 000.00

5100.00

1890.00

3210.00

23 790.00

5100.00

1903.20

3196.80

20 593.20

5100.00

1647.46

3452.54

17 140.66

5100.00

1371.25

3728.75

13 411.91

5100.00

1072.95

4027.05

9384.86

5100.00

750.79

4349.21

5035.65

5100.00

402.85

4697.15

338.50

Determine, correct to the nearest cent:

a the interest paid when payment 1 is made b the principal reduction when payment 3 is made c the balance of the annuity after payment 5 is made d the value of the last payment if the balance of the annuity is to be zero after the last payment e the total amount of interest paid.

11 I can read and interpret an annuity table for superannuation.

e.g. Iman is aged 60 and is planning to retire at age 65. She has a current balance in her superannuation fund of $450 000. The fund earns 6% interest. Iman is making a payment of $17 500 each year into her superannuation fund. A table for this annuity is shown.

Balance Payment number Payment received Interest paid Principal increase of annuity 0

0.00

450 000.00

17 500.00

27 000.00

44 500.00

494 500.00

17 500.00

29 670.00

47 170.00

541 670.00

17 500.00

32 500.20

50 000.20

591 670.20

17 500.00

38 680.22

56 180.22

700 850.63

What are the values of A, B and C?

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Peel borrows $25 000 at an interest rate of 8% p.a. compounding monthly. She will repay the loan by making monthly repayments of $500. What is the balance of the loan after four payments? Answer correct to the nearest whole number. B $23 653

C $23 993

D $24 331

U N SA C O M R PL R E EC PA T E G D ES

A $24 667

Review

Multiple-choice questions

2 An annuity has $250 invested at the end of each month for 4 years. The interest rate is 12% p.a. compounding monthly. What is the future value of the annuity? Use Excel to find the answer correct to the nearest dollar. A $1015

B $12 238

C $15 306

D $477 897

3 Calculate the present value, to the nearest dollar, needed to match an annuity which consists of quarterly deposits of $400 that are invested at 12% p.a. with interest compounding quarterly. The annuity will mature in 16 years. A $3331

B $5024

C $11 323

D $14 369

Use the table below to answer questions 4 to 7.

Future value of $1

Period

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.060

4.091

4.122

4.153

4.184

4.215

4.246

6.152

6.230

6.308

6.388

6.468

6.550

6.633

8.286

8.433

8.583

8.736

8.892

9.052

9.214

Find the future value of an annuity of $3000 per year for 6 years at 4% p.a. compounded annually. A $6633

C $18 924

D $19 899

5 Find the future value of an annuity of $800 per month for 4 months at 12% p.a. compounded monthly. A $3248

B $12 738

B $3347

C $4060

D $38 976

6 Use this table to answer this question.

Present value of $1

End of Period

5.4172

5.21421

5.0757

4.9173

6.2303

6.0021

5.7864

5.5824

7.0197

6.7327

6.4632

6.2098

7.7861

7.4353

7.1078

6.8017

The payment per year of an annuity with a present value of $72 100 at 5% p.a. compound interest for 7 years, to the nearest dollar, is: A $12 461

B $12 460

C $10 600

D $417 199

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Chapter 7 Annuities

Use this table to find the balance of the annuity after payment 4 is made: Payment number

Payment paid

Interest paid

Principal reduction

Balance of annuity

0.00

27 000.00

5100.00

1890.00

3210.00

23 790.00

5100.00

1903.20

3196.80

20 593.20

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5100.00

1647.46

3452.54

17 140.66

5100.00

1371.25

3728.75

13 411.91

5100.00

1072.95

4027.05

9384.86

5100.00

750.79

4349.21

5035.65

5100.00

402.85

4697.15

338.50

A $5100.00

B $3728.75

C $13 411.91

D $17 140.66

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Chapter 7 review

Arjit makes an initial deposit of $5000 on an investment taken out over one year at a rate of 8.5% p.a. compounding quarterly, and an additional deposit of $1000 is made quarterly.

U N SA C O M R PL R E EC PA T E G D ES

a Create a table to model the investment. b How much interest is earned by this investment? c What is the total value of the investment at the end of the year?

Review

Short-answer questions

Use Excel to find the future value correct to the nearest cent for each of the following annuities.

a $15 000 at the end of every year for 4 years with an interest rate of 10% p.a. b $3950 at the end of every year for 6 years with an interest rate of 9.25% p.a.

Liam decides to invest $980 per quarter into his superannuation fund. The interest rate quoted is 9.2% per annum, compounded quarterly. Use Excel for the following questions.

a What is the future value of his superannuation at the end of 8 years? b Calculate the contribution needed per quarter to have $50 000 after 8 years.

Use Excel to find the present value correct to the nearest cent using the following future values.

a FV = $400 000, r = 6% p.a. compounded biannually and n = 10 years b FV = $30 000, r = 8% p.a. compounded quarterly and n = 12 years

Monaro invests $1450 per year at 9.8% p.a. compound interest into a 15-year annuity. What lump sum does Jose need to deposit today to have the same amount as Monaro after 15 years? Jose’s account also pays 9.8% p.a. compounding annually. Use Excel to find the answer correct to the nearest cent.

The table below shows the future value of an annuity with a contribution of $1. Future value of $1

Period

10%

12%

14%

6.31

6.63

6.98

7.34

7.72

8.12

8.54

13.41

15.03

16.87

18.98

21.38

24.13

27.27

21.41

25.65

30.91

37.45

45.60

55.75

68.39

30.42

39.08

50.82

66.76

88.50

118.16

158.66

Use the table to calculate the future value of the following annuities. Answer correct to the nearest dollar.

a $6000 per year for 18 years at 10% p.a. compounded annually b $4920 per year for 6 years at 14% p.a. compounded annually c $2800 per quarter for 6 years at 8% p.a. compounded quarterly d $11 700 per half-year for 12 years at 12% p.a. compounded biannually e $700 per month for 6 months at 24% p.a. compounded monthly

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Chapter 7 Annuities

The table below shows the present value of an annuity with a contribution of $1. Present value of $1 Period

10.58

9.95

9.39

8.86

8.38

7.94

7.54

18.91

16.94

15.25

13.80

12.55

11.47

10.53

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25.49

21.83

18.91

16.55

14.62

13.04

11.72

30.67

25.27

21.20

18.08

15.65

13.73

12.19

Use the table to calculate the present value of the following annuities.

a $25 000 per year for 36 years at 3% p.a. compounded annually b $12 600 per year for 12 years at 8% p.a. compounded annually c $5700 per year for 24 years at 5% p.a. compounded annually d $9500 per quarter for 12 years at 8% p.a. compounded quarterly e $5200 per half-year for 24 years at 6% p.a. compounded biannually

8 Use the table in question 6 to find the contribution per period (correct to the nearest dollar) of an annuity that has these future values. a $60 300 at 4% p.a. compounded annually for 6 years b $342 300 at 10% p.a. compounded annually for 12 years c $165 200 at 8% p.a. compounded quarterly for 6 years d $15 600 at 24% p.a. compounded monthly for 1 year e $19 500 at 12% p.a. compounded biannually for 12 years

9 Use the table in question 7 to find the contribution per period (correct to the nearest dollar) of an annuity that has these present values. a $64 600 at 8% p.a. compounded annually for 36 years b $31 800 at 3% p.a. compounded annually for 12 years c $87 000 at 12% p.a. compounded quarterly for 6 years d $150 000 at 8% p.a. compounded biannually for 12 years e $72 000 at 36% p.a. compounded monthly for 2 years

7E 10

Karan is 62 years old and would like to retire in 5 years’ time. He currently has $500 000 in his superannuation account. For the next five years he received 9.25% p.a. compound interest and was able to deposit $20 000 each year. The annuity is modelled in this table.

Payment number Payment received Interest paid Principal increase Balance of annuity 0

0.00

500 000.00

21 000.00

46 250.00

67 250.00

567 250.00

21 000.00

73 470.63

640 720.63

21 000.00

59 266.66

720 987.29

21 000.00

66 691.32

87 691.32

21 000.00

74 802.77

95 802.77

904 481.38

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Chapter 7 review

Cameron would like $16 000 to go overseas for a holiday in four years’ time. The best investment Cameron can find pays 10.5% p.a. interest, compounded quarterly.

U N SA C O M R PL R E EC PA T E G D ES

a Calculate the present value of the investment needed for the holiday. Answer correct to the nearest cent. b Cameron plans to save for the trip by depositing $80 per week into an annuity at 10.5% p.a. interest, compounded weekly. Will Cameron have enough money for the holiday in four years’ time? Justify your answer with the appropriate calculations. Answer correct to the nearest cent.

Review

Extended-response question

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U N SA C O M R PL R E EC PA T E G D ES

Exponential, quadratic and reciprocal relationships

One city, two very large parabolas The City of Sydney in NSW has one of the finest harbours in the world. As the city expanded it became necessary to build bridges connecting the north and south. Two of these are the Sydney Harbour Bridge and Gladesville Bridge. In both cases, it was deemed essential for safety reasons to have high bridges without supporting pylons in the water, which would have been a danger to shipping. The two solutions had one thing in common: they both have a very long arch, shaped very much like a parabola.

The weight of road and rail traffic pulls down on the arch and the forces are directed along the arch, down to the land, where the bridge is connected to massive underground concrete blocks.

It is the tenth-longest spanning-arch bridge in the world and the tallest steel arch bridge, measuring 134 m from top to water level. It was also the world’s widest long-span bridge, at 48.8 m wide.

In 1964, the new Gladesville Bridge replaced the inadequate original bridge, which had to open to allow ships through. A scaffold was built, concrete blocks were pushed up to the top of the scaffold, and then the scaffold was removed. It is easily strong enough to support 6 lanes of traffic. When it was built, it was the longest single span concrete arch ever constructed.

The two arches of the Sydney Harbour Bridge were built from both sides of the harbour, supported by cables buried deep in the sandstone. People were nervous about the arches falling down. When the arches were joined the cables were removed, and the road was then hung from the arch. It has stood Uncorrected 3rd sample pages •opened Cambridge strong since the bridge inUniversity 1932. Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Chapter contents

U N SA C O M R PL R E EC PA T E G D ES

8A Graphs of exponential relationships 8B Exponential models in practical contexts 8C Graphs of quadratic relationships 8D Quadratic models in practical contexts 8E Graphs of reciprocal relationships 8F Reciprocal models in practical contexts 8G Inverse variation 8H Problems involving exponential, quadratic and reciprocal relationships

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Represents the relationships between quantities algebraically and graphically to solve problems and make predictions in practical contexts (MST-12-S2-01). © NSW Education Standards Authority

Online resources

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Chapter 8 Exponential, quadratic and reciprocal relationships

8A Graphs of exponential relationships Learning intentions

• To recognise exponential models • To use equations and worded scenarios to draw exponential graphs

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• The concepts in this section will be new to many students • Questions involving exponential relationships regularly appear in HSC Examinations

In Chapter 5, we encountered linear relationships. We used straight lines on the Cartesian plane as models for practical situations. In this chapter, we examine non-linear relationships, with emphasis on real-life scenarios. Every non-linear relationship can be viewed and represented as a: • worded real-life scenario • number pattern

• table of values containing a series of points • an equation, also known as a rule

• curved graph on the Cartesian plane.

In this chapter, you will:

• become more familiar with each of these

• learn how to move efficiently from one representation to the others • solve problems involving real-life scenarios.

Exponential relationships: the big picture

An exponential relationship can be viewed and represented in several different ways, as follows. A real-life scenario

In previous chapters we encountered investments which earn compound interest. For example, $1 is invested at 5% per annum, compounding annually. Every year, the amount in the account: • increases by 5%, so

• it grows from 100% to 105%, which is 1.05, so

• to go from one year to the next, the amount is multiplied by 1.05.

This is an exponential relationship, because we are always multiplying by the same amount every year. A number pattern

We can also see exponential relationships in certain number patterns. Try this on your calculator: Type the number 1, then press =

Multiply by 1.05, then press = = = = = 1, 1.05, 1.1025, 1.157625, . . .

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8A Graphs of exponential relationships

At first, the numbers don’t seem to be changing by much. It takes a long time for $1 to become $2. But if you keep on pressing = , the numbers start to increase dramatically. They are ‘growing exponentially’, which is a commonly used phrase. Try it again with 12% rather than 5%. It only takes 6 or 7 years to double your money.

U N SA C O M R PL R E EC PA T E G D ES

Below are some common examples of exponential number patterns.

Exponential growth by a factor of 2, starting from 1 1, 2, 4, 8, 16, . . .

The equation is y = 2 x

1 Exponential decay by a factor of , starting from 1 2 1 1 1 1 1, , , , , . . . 2 4 8 16 The equation is y = 2−x

Exponential growth by a factor of 3, starting from 1 1, 3, 9, 27, . . .

The equation is y = 3 x

Exponential growth by a factor of 4, starting from 2 2, 8, 32, 128, . . .

The equation is y = 2 × 4 x

1 Exponential decay by a factor of , which is 2−1 , starting from 256 2 256, 128, 64, 32, 16, 8, . . .

The equation is y = 256 × 2−x

We can also see exponential relationships in certain number patterns. A table of values

The numbers 1, 1.05, 1.1025, 1.157625, . . . that we looked at earlier that we looked at earlier are placed in the bottom row of a table. The top row tracks the years, starting from 0. Time x (years)

Money y ($)

1.05

1.1025

1.157625

‘Time’ and ‘Money’ are called variables. The top row in the table above is the independent variable. Time is usually in the top row. The bottom row is the dependent variable.

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Chapter 8 Exponential, quadratic and reciprocal relationships

An equation, sometimes called a ‘rule’ or a ‘function’ The numbers in the bottom row are calculated as follows: 0

Money y ($)

1 × 1.05

1 × 1.052

1 × 1.053

1 × 1.054

U N SA C O M R PL R E EC PA T E G D ES

Time x (years)

This can be generalised to: y = 1 × 1.05 x where:

• y is the term in the pattern • x is number of years.

More generally: x

k × a0

k × a1

k × a2

k × a3

k × a4

Equations of exponential relationships

Exponential relationships can be modelled by y = k × a x or y = ka x , where:

• y is the term in the pattern, which is the y-value • k is the value when x is zero • a is the number by which we multiply to get from one term to the next • x is number in the top row of the table. (Note that when we we’re talking about time as the independent variable, we often use n or t instead of x.)

A graph on the Cartesian plane

Every column in the tables above contain two numbers which make a point on the Cartesian plane. The points can be joined with a curve, not a straight line. It is non-linear. If y represents an investment, we say that the investment is increasing at an increasing rate. Notes:

• the top row of the table is the horizontal axis

• the bottom row of the table is the vertical axis • the y-intercept is 1, because we invested $1

• the scale on the x-axis can be different to the scale on the y-axis.

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8A Graphs of exponential relationships

y 50 40 30

U N SA C O M R PL R E EC PA T E G D ES

20 10

10 20 30 40 50 60 70 80

Graphs of exponential functions y = kax

y = ka–x

Key features of exponential graphs

• The graph lies wholly above the x-axis because a x is always positive, regardless of the value of x. It is •

• • • •

•

impossible for the y values to be zero or negative. The graph always passes through the point (0, k) because when x = 0 then y = a0 = 1, regardless of the value of a. The x-axis is an asymptote. That is, it is a line that the curve approaches by getting closer and closer to it but never reaching it. The graph y = a−x is the reflection of the graph y = a x about the y-axis. Increasing the value of a such as changing y = 2 x to y = 3 x affects the steepness of the graph. The y values increase at a greater rate when the x values increase. The exponential function y = a x when a > 1 is often referred to as exponential growth because, as the x values increase, the y values increase. The exponential function y = a−x when a > 1 is often referred to as exponential decay because, as the x values increase, the y values decrease.

To graph an exponential function:

1 construct a table of values 2 draw a number plane 3 plot the points 4 join the points to make a curve.

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Chapter 8 Exponential, quadratic and reciprocal relationships

Example 1

Exponential growth

Draw the graph of y = 3 x . Solution −2

−1

Construct a table of values for x and y. Let x = −2, −1, 0, 1, and 2. Find y using the exponential function y = 3 x .

U N SA C O M R PL R E EC PA T E G D ES

Explanation

1 9

1 3 y

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points from the table. Join the points to make a curve. Note: Sometimes it is necessary to rescale the axes to plot the points as some! points are 1 impractical to plot, such as –2, . 9

10 8 6 4 2

−3 −2 −1 O

Now you try

Draw the graph of y = 2 x .

Example 2

Exponential decay

Draw the graph of y = 3−x .

Solution

Explanation

−2

−1

1 3

1 9

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points from the table. Join the points to make a curve. Note: The exponential curve y = 3−x is the reflection of y = 3 x about the y-axis. Both curves pass through (0, 1) and have the x-axis as an asymptote.

10 8 6 4 2

−3 −2 −1 O

Construct a table of values for x and y. Let x = −2, −1, 0, 1, and 2. Find y using the exponential function y = 3−x .

Now you try Draw the graph of y = 2−x . Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

8A Graphs of exponential relationships

Desmos widget 8A Graphing exponential functions with technology Spreadsheet activity: Graphing exponential functions

Exercise 8A

U N SA C O M R PL R E EC PA T E G D ES

BUILDING

What are the next three terms?

a 3, 6, 12, . . . 1 1 b 1, , , . . . 2 4

Use a table and a graph of y = 2 x to answer these questions.

a Is it possible for y to have negative values? b Is it possible for x to have negative values? c Is it possible to calculate y when x = 0? If so, what is it? d Is it possible to calculate x when y = 0? If so, what is it? e What is the approximate value of y when x = 0.5? f What is the approximate value of y when x = 1.5?

Example 1

Example 2

Complete the following table of values by expressing the y values, correct to two decimal places. Graph each exponential curve.

a y = 4x x

−3

0.02

−2

−1

b y = 4−x x

−3

−2

−1

0.06

Use the graph of y = 4 x in Question 3a to answer these questions.

a Is it possible for y to have negative values? b Is it possible for x to have negative values? c Is it possible to calculate y when x = 0? If so, what is it? d Is it possible to calculate x when y = 0? If so, what is it? e What is the value of y when x = 1.5? f What is the value of y when x = 2.5?

DEVELOPING

Sketch these graphs on the same set of axes. a y = 2x

b y = 3x

c y = 4x

Sketch these graphs on the same set of axes. a y = 2−x

b y = 3−x

c y = 4−x

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What is the effect on the graph of changing the value of a in y = a x ? Hint: Use your graphs in Questions 5 and 6.

Complete the following table of values and graph each exponential curve. a y = 0.5 x −2

−1

U N SA C O M R PL R E EC PA T E G D ES

1 4

b y = 0.5−x x y

−2

−1 1 2

Use the graphs in Questions 5a, 6a and 8 to answer these questions.

a What are the coordinates on the graph of y = 0.5 x where x = 1? b What are the coordinates on the graph of y = 2−x where x = −1? c What are the coordinates on the graph of y = 0.5−x where x = 2? d What are the coordinates on the graph of y = 2 x where x = −2? e Is the graph of y = 0.5 x the same as y = 2−x ? Explain your answer. f Is the graph of y = 0.5−x the same as y = 2 x ? Explain your answer.

CHALLENGING

Sketch these graphs on the same set of axes. Label the y-intercept.

a y = 2x b y = 2 × 2x 1 c y = × 2x 2 d How does multiplying the graph y = a x by a number affect the graph?

11 What does the graph of y = 1 x look like?

Communicate your reasoning, understanding and fluency 12

Consider the equation y = 100 × 3 x . Answer the following in words, without drawing a table or a graph.

a What does the number 100 tell you about this curve? b What does the number 3 tell you about the numbers in the bottom row of the table of values? c Will this graph model exponential growth or decay?

Consider the equation y = 4000 × 4−x . Answer the following in words, without drawing a table or a graph. a What does the number 4000 tell you about this curve? b What does the number 4 tell you about the numbers in the bottom row of the table of values? c Will this graph model exponential growth or decay?

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8B Exponential models in practical contexts

8B Exponential models in practical contexts Learning intentions

• To construct and analyse exponential models to solve problems in practical contexts Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• The concepts in this section will be new to many students • Questions involving exponential relationships regularly appear in HSC Examinations

Exponential modelling occurs when a practical situation is described mathematically using an exponential equation, table and graph. The quantity usually experiences fast growth or decay.

Exponential model

Exponential growth: Quantity increases rapidly according to the function y = ka x where a > 1.

Exponential decay: Quantity decreases rapidly according to the function y = ka−x where a > 1.

Limitations of exponential models in practical contexts When we use t for time:

• the top row of the table only contains 0 and positive numbers • the graph can be drawn in the first quadrant.

Example 3

Using an exponential model

A fish population is modelled using the formula, N = 500 × 1.5t where N is the number of fish and t is the time in years.

a Construct a table of values for t and N. Use values for t from 0 to 4. Approximate the number of fish to the nearest whole number. b Draw the graph of N = 500 × 1.5t . c How many fish were present after 2 years? d How many extra fish will be present after 4 years compared to 2 years? e Estimate the number of fish after 18 months.

Continued on next page

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Solution Time (t years)

Fish (N)

500

750

1125

1688

2531

Construct a table of values for t and N. Let t = 0, 1, 2, 3 and 4. Find N using N = 500 × 1.5t . Express the values for N as a whole number. Draw a number plane with t as the horizontal axis and N as the vertical axis. Plot the points from the table. Join the points to make the curve.

U N SA C O M R PL R E EC PA T E G D ES

Explanation

4000

3500 3000 2500 2000 1500 1000 500

Look up t = 2 in the table and find N. Subtract the number of fish for 2 years from 4 years. Read the approximate value of N from the graph when t = 1.5.

c 1125 fish d Extra = 2531 − 1125 = 1406 e Approximately 900 fish. Now you try

a In Example 3, what are the values of N when t = 10 and t = 20? b Do you think the number of fish can grow in this way in a real population? What factors might limit this model?

Exercise 8B 1

BUILDING

The equation N = 6t is used to model the growth in the number of insects (N) after t days.

a Copy and complete the table of values for t and N. t

b Copy the given number plane and draw the graph of N = 6t . c What was the initial number of insects? d How many insects were present after 3 days? e How many extra insects will be present after 4 days compared to 2 days? f Estimate how many days it took for the number of insects to exceed 1000.

1400 1200 1000 800 600 400 200

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8B Exponential models in practical contexts

Example 3

The population of an endangered reptile is decreasing exponentially according to the formula, P = 100 × 1.5−t where P is the population of reptiles after t years. a Copy and complete the table of values for t and P. Express the population of reptiles to the nearest whole number. t

U N SA C O M R PL R E EC PA T E G D ES

b Graph P = 100 × 1.5−t using the table of ordered pairs in part a. c What is the initial population of reptiles? d Estimate the population of reptiles after 3 years. e Estimate the population of reptiles after 7 years. f What is the difference in the population of reptiles after 2 years compared to 6 years? g Estimate the time taken (to the nearest year) for the population of reptiles to be below 1.

Note: This highlights another limitation of exponential models in practical concepts. If P is a population of animals experiencing exponential decay, eventually they will become extinct, but the model keeps on producing decimals which are close to zero.

DEVELOPING

The size of a flock of birds, F, after t years is decaying exponentially according to F = 200 × 0.5t .

a Make a table of values for t and F. Use values for t from 0 to 5. Express F correct to the nearest whole number. b Draw the graph of F = 200 × 0.5t . c What was the size of initial flock of birds? d How many birds were present after 3 years? e How many fewer birds will be present after 4 years compared to 2 years? f Estimate how years it will take for the number of birds to fall to less than one bird.

The number of bacteria grows exponentially according to the formula b = 30 × 1.2t where b is the number of bacteria after t hours.

a Construct a table of ordered pairs using 0, 5, 10, 15 and 20 as values for t. Express the number of bacteria to the nearest whole number. b Graph b = 30 × 1.2t using the table of ordered pairs in part a. c What is the initial number of bacteria? d What is the number of bacteria after 4 hours? e Estimate the time taken for the number of bacteria to reach 120.

Modelling activity: Modelling real-world scenarios with non-linear relationships This task explores the limitations of three common non-linear mathematical models—exponential, quadratic and reciprocal—by applying each to a real-world scenario and analysing when the model is valid and when it breaks down. This modelling activity should be done after you have covered sections 8B–8H. The activity can be downloaded from your Interactive Textbook.

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CHALLENGING Owen invested $1000 into a managed fund that appreciated in value for 5 years. The amount of money (A) in the fund for each year (t) is shown below. t

$1000

$1300

$1690

$2197

$2856

$3713

U N SA C O M R PL R E EC PA T E G D ES

a Draw a number plane with t as the horizontal axis and A as the vertical axis. b Plot the points from the table of values. Join the points to make a curve.

An exponential growth model in the form y = 1000 × a x describes this situation.

c What is the model if x and y are replaced with t and A? d Determine the value of a in the model by using t = 1. e What is the exponential model using t and A?

1 f Use the model to find the value (to the nearest dollar) of the fund after 2 years. 2 g Use the model to find the value (to the nearest dollar) of the fund after 7 years. h Estimate the time when the value of the fund is approximately $1480.

Communicate your reasoning, understanding and fluency 6

A population of Animal A lives on a small island. Another population of Animal B was introduced to the same island. This caused the population of Animal A to decline. The populations have been modelled using these equations, in which: • N is the number of animals • t is the number of years

Animal A: N = 5000 × 1.2−t Animal B: N = 200 × 1.2t

Note: This activity may be completed using a spreadsheet.

a Is the population of Animal A increasing or decreasing? How can you tell from the equation? b Is the population of Animal B increasing or decreasing? How can you tell from the equation? c Complete the following table, using whole numbers for A and B. Time

A B

d How many Animal B’s were introduced onto the island? e Approximately when will the B population outnumber the A population? f What will happen to Animal A if this trend continues?

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8C Graphs of quadratic relationships

8C Graphs of quadratic relationships Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To recognise quadratic models • To know the key features of a parabola • To use equations to draw parabolas Past, present and future learning

• The concepts in this section will be new to many students • Questions involving quadratic relationships regularly appear in HSC Examinations

Quadratic relationships: The big picture

Another type of non-linear relationship is called ‘quadratic’. This relationship produces graphs called parabolas. A quadratic relationship can be viewed and represented in several different ways, as follows. A real-life scenario

If the sides of a square are x centimetres long, its area is the square of x, written as x2 . A table of values Side of square, x

Area of square, y

This is a quadratic relationship, because in order to get from top row to bottom row, a power of 2 is used. Variables

‘Side of square’ and ‘Area of square’ are called variables. The top row is the independent variable. The bottom row is the dependent variable. An equation, sometimes called a ‘rule’ or a ‘function’ The numbers in the bottom row are calculated as follows:

Choose a number in the top row then multiply it by itself. The equation of this parabola is y = x2 .

More generally, parabolas have equations like these: y = x2 − 4

y = x2 + 3x y = 2x2

y = 2x2 + 3x − 4

The general form for every quadratic relationship is: y = ax2 + bx + c where a, b and c are numbers which are called coefficients. A series of points on the Cartesian plane Every column in the table contains two numbers which make a point on the Cartesian plane.

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A graph on the Cartesian plane The points can be joined with a curve, not a straight line. It is non-linear. • The top row of the table is the horizontal axis. • The bottom row of the table is the vertical axis. • The y-intercept is 0, because we have 02 = 0.

U N SA C O M R PL R E EC PA T E G D ES

• The scale on the x-axis can be different to the scale on the y-axis. y 10 9

Area of square

8 7 6 5

4 3

2 1

2 Side of square

Key features of a parabola

The basic parabola has the equation y = x2 .

• The vertex (or turning point) is (0, 0).

• Axis of symmetry is x = 0 (the y-axis). • y-intercept is 0 and x-intercept is 0.

• The graph y = −x2 is the reflection of the graph y = x2 about the x-axis.

1 2 x affects the shape of the parabola. 2 • Adding or subtracting a number to the equation such as y = x2 + 1 or y = x2 − 1 does not change the shape but moves the parabola up or down. • Changing the coefficient of the equation such as y = 2x2 or y =

y 10 8

Axis of symmetry

y = x2

4 2 –3

–2

–1

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8C Graphs of quadratic relationships

Parabolas The parabola y = x2 is concave up

The parabola y = −x2 is concave down y

y 10 9 8 7 6 5 4 3 2 1

1 2 3

U N SA C O M R PL R E EC PA T E G D ES

−3 −2 −1 O −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

−3 −2 −1 O

1 2 3

To graph a parabola:

1 construct a table of values 2 draw a number plane 3 plot the points 4 join the points to make a parabola.

Example 4

Graphing a parabola

Draw the graph of y = x2 + 1.

Solution

Explanation

−3

−2

−1

y 10 8 6 4 2

O −6 −4 −2

Construct a table of values for x and y using x = −3, −2, −1, 0, 1, 2 and 3. Find y by substituting into y = x2 + 1.

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points from the table. Join the points to make a curve in the shape of a parabola.

2 4 6

Now you try

Draw the graph of y = x2 − 1.

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Example 5

Determining the features of a parabola

Draw the graph of y = x2 − 4x + 3 (use x = −1, 0, 1, 2, 3, 4, 5) and find the following features: b axis of symmetry d x-intercepts

U N SA C O M R PL R E EC PA T E G D ES

a vertex c y-intercept e minimum y-value Solution

Explanation

−1

y Axis of symmetry 5 4

y-intercept

3 2 1

−1 O −1 −2

Construct a table of values for x and y using x = −1, 0, 1, 2, 3, 4 and 5. Find y by substituting into y = x2 − 4x + 3. Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points from the table. Join the points to make a curve in the shape of a parabola.

x-intercepts

Vertex

a vertex is (2, −1)

Find where the graph changes direction.

b Axis of symmetry is x = 2

Find the line that splits the graph into two.

c y-intercept is 3

Find the points where the graph cuts the y-axis.

d x-intercepts are 1 and 3

Find the point where the graph cuts the x-axis.

e Minimum y-value is −1

Determine the smallest value of y.

Now you try

Repeat Example 5, using y = x2 − 4x.

Desmos widget 8C

Graphing a parabola function with technology

Spreadsheet activity: Graphing a parabola with a spreadsheet

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8C Graphs of quadratic relationships

Exercise 8C 1

BUILDING

Write the missing features for this graph.

a The coordinates of the vertex are _________. b The y-intercept is _________. c The x-intercepts are _________ and _________. d The axis of symmetry is _________. e The maximum value is _________.

U N SA C O M R PL R E EC PA T E G D ES

3 2 1

−4

−3

−2

−1

Write the missing features for this graph.

−1

a The coordinates of the vertex are _________. b The y-intercept is _________. c The x-intercepts are _________ and _________. d The axis of symmetry is _________. e The minimum value is _________.

−3

−2

−1

−2 −3 −4

Complete the following table of values and graph each parabola on the same number plane.

a y = x2 x

−3

−2

−1

−3

−2

−1

b y = 2x2 x y

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c y = 3x2 x

−3

−2

−1

y 1 2 x 2

U N SA C O M R PL R E EC PA T E G D ES

d y= x

−3

−2

−1

e What is the axis of symmetry for each of the above parabola? f Is the vertex for each of the above parabolas a maximum or minimum? g What is the effect of changing the coefficient of x2 in the quadratic equation y = x2 ?

4 Complete the following table of values and graph each parabola on the same number plane. a y = −x2 x

−3

−2

−1

−2

−1

−2

−1

−2

−1

b y = −2x2 x

−3

c y = −3x2 x

−3

1 d y = − x2 2 x

−3

e What is the axis of symmetry for each of the above parabolas? f Is the vertex for each of the above parabolas a maximum or minimum? g What is the effect of changing the coefficient of x2 in the quadratic equation y = −x2 ?

DEVELOPING

Example 4

Complete the following table of values and graph each parabola on the same number plane.

a y = x2 + 2 x

−3

−2

−1

−2

−1

y b y = x2 − 2 x

−3

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8C Graphs of quadratic relationships

c What is the axis of symmetry for each of the above parabolas? d Is the vertex for each of the above parabolas a maximum or minimum? e What is the effect of adding or subtracting a number to the parabola y = x2 ? Example 5

6 Complete the following table of values and graph each parabola on the same number plane.

U N SA C O M R PL R E EC PA T E G D ES

a y = x2 + 2x + 1 x

−6

−4

−2

−4

−2

−4

−2

−4

−2

b y = x2 + 4x + 4 x

−6

c y = x2 − 2x + 1 x

−6

d y = x2 − 4x + 4 x

−6

e What do all of the above parabolas have in common?

CHALLENGING

Complete the following table of values and graph each parabola on the same number plane.

a y = x2 − 4x + 3 x

−1

b y = −x2 + 4x − 3 x

−1

c y = 2x2 − 8x + 6 x

−1

d y = −2x2 + 8x − 6 x

−1

y e What do all of the above parabolas have in common?

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What is the equation for each of these tables of values which form parabolas? a

−3

−2

−1

−3

−2

−1

−3

−2

−1

−9

−4

−1

−4

−9

−3

−2

−1

−3

−2

−1

−3

−2

−1

U N SA C O M R PL R E EC PA T E G D ES

Communicate your reasoning, understanding and fluency

Note: Questions 9 and 10 may be enhanced with the use of graphing software such as Desmos.

9 For each question part, draw the graphs of the parabolas on the same set of axes, from x = −5 to x = 5. a y = (x + 1)2 , y = (x + 2)2 and y = (x + 3)2 b y = (x − 1)2 , y = (x − 2)2 and y = (x − 3)2 c Explain how the constant k in y = (x + k)2 transforms the graph y = x2 .

10 For each question part, draw the graphs of the parabolas on the same set of axes, from x = −5 to x = 5. a y = (x + 1)2 − 2, y = (x + 2)2 − 3 and y = (x + 3)2 − 5 b y = (x − 1)2 + 2, y = (x − 2)2 + 3 and y = (x − 3)2 + 5 c Explain how the constant h in y = (x + k)2 + h transforms the graph y = (x + k)2 .

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8D Quadratic models in practical contexts

8D Quadratic models in practical contexts Learning intentions

• To construct and analyse quadratic models to solve problems in practical contexts Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• The concepts in this section will be new to many students • Questions involving quadratic relationships regularly appear in HSC Examinations

Quadratic modelling occurs when a practical situation is described mathematically using a quadratic equation and a parabola.

Quadratic model

A quadratic model describes a practical situation using an equation in the form y = ax2 + bx + c where a, b and c are numbers. These are graphed to make a curve in the shape of a parabola.

Limitations

Sometimes it is not possible to use all values of x in the model. In the following example:

• x must be a positive number

• x must be greater than 0 and less than 6.

Example 6

Using a quadratic model

Tina is going to make a rectangular garden next to two walls. She has enough material for 6 metres of wall. If one side is x, the other side is 6 − x.

The area is A = x(6 − x), so A = 6x − x2 . 6−x

Wall

a Complete the table of values. Side (x)

Area ( A)

b Draw the graph of A = 6x − x2 using the table. c Use the equation to calculate the area of the garden when the length of the garden is 4.5 m. d What is the maximum area of the garden? e What is the side length in order to have maximum area?

Continued on next page

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Solution a

Explanation

Let x = 0, 1, 2, 3, 4, 5 and 6 and find A by substituting these values into the parabola A = 6x − x2 . Draw a number plane with x as the horizontal axis and A as the vertical axis. Plot the points from the table. Join the points to make a parabolic curve.

U N SA C O M R PL R E EC PA T E G D ES

A 10 8 6

4 2

c Let x = 4.5

A = 6x − x2

= 6 × 4.5 − 4.52 = 6.75 m2

Draw a vertical line from x = 4.5 on the horizontal axis until it intersects the parabola. At this point draw a horizontal line until it connects with the vertical axis. Read this value.

d Maximum area of the garden is 9.

Read the largest value for A.

e Maximum area occurs when x = 3.

Read the value on the x-axis when the A is largest.

Now you try

Repeat Example 6 with 20 metres of fence and using one wall. What is the maximum area? x

Wall

Stopping distance

The stopping distance is the distance a vehicle travels from the time a driver sees an event occurring to the time the vehicle is brought to a stop. It is calculated by adding the reaction distance and the braking distance. Reaction distance (or thinking distance) is the distance travelled by the vehicle, from the time when a driver decides to brake to when the driver first commences braking. The reaction time averages 0.75 second for a fit and alert driver. The braking distance is affected by the road surface (wet, slippery, uneven or unsealed), slope of the road (uphill or downhill), mass of the vehicle and the condition of the brakes.

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8D Quadratic models in practical contexts

A formula for stopping distance Stopping distance = Reaction distance + Braking distance d=

v2 5vt + 18 170

(formula is an approximation using average conditions)

d = Stopping distance in metres

U N SA C O M R PL R E EC PA T E G D ES

v = Velocity or speed of the motor vehicle in km/h

t = Time reaction in seconds

Note: There are several other formulas for stopping distance.

Example 7

Using a quadratic model

Pratik was driving at a speed of 60 km/h and reaction time of 0.80 seconds. Calculate the stopping distance using the formula d=

v2 5vt + 18 170

Answer correct to the nearest whole metre.

Solution

Explanation

5vt v2 + 18 170

Write the stopping distance formula.

5 × 60 × 0.80 602 + 18 170

Substitute v = 60 and t = 0.80 into the formula.

= 34.50980 . . .

≈ 35 m

∴ Stopping distance is 35 m.

Evaluate. Express the answer correct to the nearest whole number. Write the answer in words.

Now you try

By what distance is the stopping distance increased if the car is travelling at 80 km/h?

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Exercise 8D Example 6

BUILDING

The area (A) of a rectangular enclosure of length x metres is given by the formula A = x(7 − x). The graph of this formula is shown on the right. a What is the area of the enclosure when x is 1 metre? b What is the area of the enclosure when x is 5 metres? c What is the enclosure’s length in order to have maximum area? d What is the maximum area of the enclosure?

A 12

U N SA C O M R PL R E EC PA T E G D ES

1 2 3 4 5 6 7

The movement of an object with a velocity v (in metres/second) at time t (seconds) is given by the formula v = 15t − 5t2 . The graph of this formula is shown on the right. v

a What was the initial velocity of the object? b What was the greatest velocity reached by the object? c How many seconds did it take for the object to reach maximum velocity? d Estimate the number of seconds when the velocity is greater than 6 metres per second.

12 10 8 6 4 2

A marine supply company sells rope that is stored by winding onto large circular coils. The price ($P) of rope depends on the diameter (d), in metres, of the coiled rope. The quadratic equation P = 3d2 is used to model this situation.

a Complete the following table of values, correct to th nearest whole number. d

b Draw the graph of P = 3d2 using the number plane shown. c What is the price of the rope when the diameter of the coiled rope is 12 metres? d What is the price of the rope when the diameter of the coiled rope is 23 metres? e What is the difference between the price of the rope when the diameter of the coiled rope is 5 metres compared to a diameter of 25 metres?

1800 1600 1400 1200 1000 800 600 400 200

d 4 8 12 16 20 24

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8D Quadratic models in practical contexts

A stone falls from rest down a mine shaft. The distance it falls, d metres, at time t seconds is given by the quadratic equation d = 4.9 t2 . a Complete the following table of values, correct to the nearest whole number. t

d 250

5 200

U N SA C O M R PL R E EC PA T E G D ES

b Draw the graph of d = 4.9 t2 using the number plane shown. c What is the distance travelled by the stone after 1 second? d What is the distance travelled by the stone after 5 seconds? e What is the distance travelled by the stone after 2.5 seconds? f How long did it take for the stone to travel 100 metres? Estimate from the graph.

150 100 50

g How long did it take for the stone to travel 200 metres? Estimate from the graph.

The equation d = 0.005s(s − 1) is used to model the stopping distance for a train where d is the stopping distance in metres and s is the train’s speed in km/h.

a Complete the following table of values, correct to the nearest whole number. s

100

Distance (m)

b Draw the graph of d = 0.005s(s − 1) using the number plane shown. c What is the stopping distance when the train is travelling at 20 km/h? d What is the stopping distance when the train is travelling 75 km/h? e Estimate the maximum speed (km/h) a train could be travelling to stop within 15 m. f Estimate the maximum speed (km/h) a train could be travelling to stop within 30 m.

50 40 30 20 10

10 20 30 40 50 60 70 80 90 100 Speed (km/h)

g What is the difference between the stopping distances when a train is travelling at a speed of 40 km/h compared to travelling at a speed of 80 km/h? h What is the difference between the stopping distances when a train is travelling at a speed of 15 km/h compared to travelling at a speed of 95 km/h?

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DEVELOPING Example 7

Amihan was driving at a velocity of v km/h and has a reaction time (t) of 0.50 seconds. An approximation between the stopping distance, d (in metres), and the velocity is given using the quadratic equation shown below. v2 5vt + 18 170

5v × 0.50 v2 + 18 170

2.5v v2 + 18 170

U N SA C O M R PL R E EC PA T E G D ES

Answer the following questions, correct to the nearest whole number.

a Copy and complete a table of values for v and d. v

100

2.5v v2 b Draw the graph of d = + using the number plane 18 170 shown. c What is the stopping distance when the velocity is 5 km/h? d What is the stopping distance when the velocity is 95 km/h? e What is the stopping distance when the velocity is 125 km/h? f What is the difference in the stopping distance between 20 km/h and 80 km/h? g What is the difference in the stopping distance between 20 km/h and 140 km/h? Answer correct to one decimal place.

80 70 60 50 40 30 20 10

20 40 60 80 100

Min-jun is driving with a reaction time of t = 0.70 seconds. Calculate the stopping distance using the 5vt v2 formula d = + for each of the following speeds. 18 170 a 25 km/h b 50 km/h c 75 km/h d 100 km/h e 125 km/h f 150 km/h

Ling was driving her car at 40 km/h through a school zone. A school student ran onto the road 12 metres in front of her. Ling has a reaction time of 0.80 seconds.

a Do you think Ling was able to stop without running over the child? Give a reason for your answer. b What would have happened if Ling was driving her car at 60 km/h? Explain your answer.

Emilio uses the freeway to travel to work. Emilio has a reaction time of 0.60 seconds and usually drives at the speed limit of 110 km/h. 5vt v2 a What is the stopping distance on the freeway using the formula d = + ? 18 170 b If Emilio is driving at 110 km/h at night, how far ahead does he need to be able to see in order to avoid hitting a stopped car in the road?

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8D Quadratic models in practical contexts

A ball was dropped from the top of a multistorey building to the ground below. Ignoring air resistance, the relationship between the height, h, above the ground (in metres) and the time, t (in seconds), for it to fall from that height is given by the formula h = 100 − 4t2 .

U N SA C O M R PL R E EC PA T E G D ES

a Construct a table of values for t and h. Use values for t from 0 to 5. b Draw the graph of h = 100 − 4t2 . c What is the height of the building? d How long did it take for the ball to hit the ground? e How high above the ground was the ball after 2 seconds? f Estimate how long it took for the ball to fall 64 metres. g During which second did the ball fall the greatest distance? Estimate from the graph.

A child’s circular swimming pool has a depth of 1 metre. The relationship between the volume (V, in m3 ) and the radius (r, in m) is given by the formula V = πr2 h (where h = 1 m).

a Complete a table of values for r and V. Use r from 0 to 5 and express V correct to one decimal place. b Draw the graph of V = πr2 using the table of values. c What is the maximum volume of water if the radius is between 0 and 5 metres (correct to one decimal place)?

CHALLENGING

The petrol consumption (P, litres per 100 km) and the speed of a car (s, km/h) are modelled by the formula P = 0.02s2 − 2s + 55.

a Draw the graph of P = 0.02s2 − 2s + 55. Use s from 0 to 100 km/h. b What is the petrol consumption at 40 km/h? c Estimate the speed of the car which gives the lowest petrol consumption.

A rectangular patio is constructed so that the length is 8 metres less than the breadth.

a Give an example of a length and breadth that would be possible for this patio. b Write an equation for the area (A) of the patio in terms of its length (x). c Construct a graph to compare the area of the patio with its length. d Estimate the dimensions of the patio if it had an area of 65 m2 .

Communicate your reasoning, understanding and fluency 14

Using graphing software such as Desmos, find the equation of any parabola which:

a passes through (0, 0) and (6, 0) and is concave up b passes through (0, 0) and (6, 0) and is concave up c passes through (−3, 0), (3, 0) and (0, 9) d has vertex at (2, 0) and passes through (0, 4).

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8E Graphs of reciprocal relationships Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To recognise reciprocal relationships • To know the key features of a hyperbola • To use equations to draw hyperbolas Past, present and future learning

• The concepts in this section will be new to many students • Questions involving reciprocal relationships regularly appear in HSC Examinations

Reciprocal relationships: the big picture

Another type of non-linear relationship is a ‘reciprocal’. This relationship produces graphs called hyperbolas. These are produced by a situation called inverse variation.

A reciprocal relationship can be viewed and represented in several different ways, as follows. A real-life scenario

The two sides of a rectangle have lengths x and y.

Consider a rectangle which has an area of 1. What could the sides be? A table of values

Length of rectangle

1 5

1 4

1 3

1 2

Breadth of rectangle

1 2

1 3

1 4

1 5

This is a reciprocal relationship, as every column of the table contains two numbers which are the reciprocal of each other. Their product is 1. Variables

‘Length of rectangle’ and ‘Area of rectangle’ are called variables. The top row is the independent variable. The bottom row is the dependent variable.

In this situation, as x increases in value from left to right in the table, y decreases. One is the inverse of the other. Limitations

In this scenario, the variables are lengths, so the values are limited to positive numbers only. An equation, sometimes called a ‘rule’ or a ‘function’ The numbers in the bottom row are calculated as follows: Pick a number in the top row then find the reciprocal of that number. 1 The equation of this hyperbola is xy = 1, which can also be written as y = . x

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8E Graphs of reciprocal relationships

More generally, hyperbolas have equations like these: 3 y= x xy = 40

U N SA C O M R PL R E EC PA T E G D ES

Every reciprocal relationship is a hyperbola of the form: k y = or xy = k x Where k is called the constant of variation. A series of points on the Cartesian plane

Every column in the table can be represented as a point on the Cartesian plane. A graph on the Cartesian plane

The points can be joined with a curve, not a straight line. The graph is non-linear. • The top row of the table is the horizontal axis.

• The bottom row of the table is the vertical axis. • It is impossible for x or y to be equal to 0.

4 3 2 1

Key features of a hyperbola

1 The basic hyperbola has the equation y = . x • No value exists for y when x = 0. • The curve has two parts called branches. Each branch is the same shape and size; they are symmetrical and are in opposite quadrants. • The x-axis and the y-axis are asymptotes of the curve. That is, the curve approaches the x-axis and the y-axis but never touches them.

4 3

Asymptotes

Branch

2 1

−4 −3 −2 −1 O −1 Branch

−2 −3 −4

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Hyperbolas A hyperbola has the form y = Hyperbola : y =

k where k is a number. x

1 x

y 4 3

U N SA C O M R PL R E EC PA T E G D ES

2 1

−4 −3 −2 −1 O −1

−2 −3 −4

To graph a hyperbola:

1 construct a table of values 2 draw a number plane 3 plot the points 4 join the points to make a hyperbola.

Example 8

Graphing a reciprocal function

2 Draw the graph of y = . x

Solution

Explanation

−4

−0.5 −1 −2

−2 −1 −0.5 0.5 1 2 −4

4 3 2 1

O −4 −3 −2 −1 −1 −2 −3 −4

1 2 3 4

2 1 0.5

Construct a table of values for x and y. Let x = −4, −2, −1, −0.5, 0.5, 1, 2 and 4. Find y using the reciprocal function.

Draw a number plane with x as the horizontal axis and y as the vertical axis. Plot the points from the table. No value exists for y when x = 0. This results in the curve having two branches. Join the points to make a curve in the shape of a hyperbola.

Now you try Draw a table of value for the equation xy = 10. Use the same x-values as Example 8.

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8E Graphs of reciprocal relationships

Example 9

Graphing a reciprocal function

2 a Draw the graph of y = − . x b What are the asymptotes for this graph? Explanation

U N SA C O M R PL R E EC PA T E G D ES

Solution a

−4

y 0.5

−2 −1 −0.5 0.5 1

−4

−2 −1 −0.5

Construct a table of values for x and y. Let x = −4, −2, −1, −0.5, 0.5, 1, 2 and 4. Find y using the reciprocal function.

4 3 2 1

O −4 −3 −2 −1 −1

1 2 3 4

−2 −3 −4

b Asymptotes are x = 0 and y = 0.

The curve approaches the x-axis and the y-axis but never touches them.

Now you try

Draw a table of values for xy = −12. Use the same x-values as Example 9.

Exercise 8E

Example 8

BUILDING

1 Using the equation y = : x a complete the following table of values x −4 −2 −1 −

1 2

b graph the hyperbola using the number plane shown.

4 3 2 1

O −4 −3 −2 −1 −1

1 2 3 4

−2 −3 −4

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Example 9

1 2 Using the equation y = − : x a complete the following table of values x −4 −2 −1 −

1 2

y 4 3 2 1

U N SA C O M R PL R E EC PA T E G D ES

O −4 −3 −2 −1 −1

b graph the hyperbola using the number plane shown.

1 2 3 4

−2 −3 −4

DEVELOPING

Complete the following table of values and graph each hyperbola on a separate number plane. 3 3 a y= b y=− x x x −9 −3 −1 −

1 3

c y=

x −9 −3 −1 −

1 3

1 2

4 x

x −8 −4 −1 −

d y=−

1 2

4 Draw the graph of the following hyperbolas. 6 a y= x 5 c y=− x

4 x

x −8 −4 −1 − y

b y=

8 x

d y=−

7 x

2 5 Find the coordinates on the graph of y = , where: x a x=2 b x=4 c x = −1 d x = −8

5 Find the coordinates on the graph of y = − , where: x a x = 10 b x=7 c x = −5 d x = −2

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8E Graphs of reciprocal relationships

CHALLENGING 7

What is the equation of the following hyperbolas? y

U N SA C O M R PL R E EC PA T E G D ES

9 6

(1, 5)

6 (−1, 5) 3

4 2

−10 −8 −6 −4 −2 O −2

8 10

−4 (−1, −5) −6

−12 −9 −6 −3 O −3 −6

9 12

(1, −5)

−9

−8

−12

−10

Communicate your reasoning, understanding and fluency 8

4 Do the following points lie on the hyperbola y = ? Justify your answer. x a (1, 4) b (−2, −2) c (4, 16) d (8, 0.5)

9 Do the following points lie on the hyperbola y = −

10 ? Justify your answer. x

a (10, 1) b (−5, −2) c (−20, −0.5) d (5, −2)

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8F Reciprocal models in practical contexts Learning intentions

• To construct and analyse reciprocal models to solve problems in practical contexts Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• The concepts in this section will be new to many students • Questions involving reciprocal relationships regularly appear in HSC Examinations

Reciprocal modelling occurs when a practical situation is described mathematically using a hyperbola.

Reciprocal models

k A reciprocal model describes a practical situation using a function in the form y = or xy = k, where k x is a number. When graphed they make a curve in the shape of a hyperbola.

Example 10

Using a reciprocal model for speed and time taken

On a road trip of 2000 km, the time taken varies inversely with speed.

The time taken (t), in hours, for a road trip, at speed (s), in 2000 km/h, is given by the equation t = . s a Construct a table of values for s and t. 2000 . b Draw the graph of t = s c How long did the road trip take at a speed of 70 km/h?

d Why is it practically impossible to complete the road trip in 10 hours? Solution

Explanation

Speed (s) 10

20 40 50 60 80 100

Construct a table of values for s and t.

Time (t) 200 100 50 40 33 25 20

Choose appropriate values for s, the speed of the car. Let s = 10, 20, 40, 50, 60, 80 and 100.

Note: The product of every column is 2000.

2000 . s Express the values for t as a whole number.

Find t using the reciprocal function t =

200

Draw a number plane with s as the horizontal axis and t as the vertical axis.

150

Plot the points from the table. Join the points to make a branch of a hyperbola.

100 50

s 20

100

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8F Reciprocal models in practical contexts

c Approximately 30 hours

Read the approximate value of t from the graph when s = 70.

d Speed required to complete the trip in 10 h is 200 km/h, which is above the speed limit on Australian roads.

Read the value of s from the table. Make sense of the result.

U N SA C O M R PL R E EC PA T E G D ES

Now you try

The minutes taken (M) for an ice cube to melt varies inversely with temperature (T ). Use the equation MT = 180 to fill in the table below. Temperature

Minutes

Example 11

Using a reciprocal model for sharing food

The cost per person of sharing a $24 pizza ($C) is dependent on the number of people (n) eating the pizza. 24 The reciprocal equation C = is used to model this n situation.

a Describe the possible values for n. b Construct a table of values for n and C. 24 c Draw the graph of C = . n d What is the cost per person if six people are sharing a pizza? e How many people shared a pizza if the cost was $2.40 per person? Solution

Explanation

a n is a positive whole number and likely to be less than 10.

The variable n represents the number of people sharing a pizza.

Construct a table of values for n and C. Choose appropriate values for n. Let n = 1, 2, 3, 4, 5, 6, 7 and 8. 24 Find C using C = . n

3 4

C 24 12 8 6 4.8 4 3.4 3

Draw a number plane with n as the horizontal axis and C as the vertical axis. Plot the points from the table. Join the points to make a branch of a hyperbola.

32 28 24 20 16 12 8 4

n 1 2 3 4 5 6 7 8

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Read the value of C from the table or graph when n = 6.

d Cost per person is $4.

e 2.4 =

Substitute 2.4 for C into the reciprocal equation. Solve the equation for n by rearranging the formula and evaluate. Check that the answer is reasonable. Write the answer in words.

24 = 10 2.4

U N SA C O M R PL R E EC PA T E G D ES

24 n

∴ number of people sharing the pizza was 10.

Now you try

The members of a class are going to buy a present for their teacher, provided 10 or more of them are willing to share the cost equally. The gift costs $300. N is the number of students willing to pay.

C is the cost, per student.

Use the equation NC = 300 to fill in the table below. Students

Cost

Exercise 8F

Example 10

BUILDING

The time taken (t), in hours, for a road trip, at an average speed (s), in km/h, is given by the formula 1500 t= . The graph of this relationship is shown. s a How long would the road trip take at 50 km/h? b How long would the road trip take at 75 km/h? c What is the speed required to complete the road trip in 25 hours? d What is the speed required to complete the road trip in 100 hours? e Why is it practically impossible to complete the road trip in 5 hours? t

250 200 150 100 50

s 20

100

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8F Reciprocal models in practical contexts

The cost per person of hiring a house ($C) is dependent on the number of people (n) sharing the total cost. The 320 reciprocal equation C = is used to model this situation. n a What is the cost per person of hiring the house if 2 people share the total cost? b What is the cost per person of hiring the house if 8 people share the total cost? c How many people are required to share the cost of hiring a house for $80? d How many people are required to share the cost of hiring a house for $320? e Is it possible for the cost per person to be $1?

c 400 350 300 250

U N SA C O M R PL R E EC PA T E G D ES

Example 11

200 150 100 50

The time taken (t in minutes) to type an essay depends on the typing speed (s in words per minute). 150 The reciprocal equation t = is used to model this situation. s t

50 40 30

20 10

a Complete the following table of values, correct to the nearest whole number. s

b Draw the graph of t =

150 using the number plane shown. s

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DEVELOPING 4

The time taken (t in hours) to dig a hole is dependent on the number of people (n) digging the hole. 6 This relationship is modelled using the formula t = . n a Complete the following table of values, correct to the nearest whole number.

U N SA C O M R PL R E EC PA T E G D ES

n 1 2 3 4 5 6 t

6 b Draw the graph of t = using the number n plane shown. c What is the time taken by one person to dig a hole? d What is the time taken by 3 people to dig a hole? e What is the time taken to by 6 people to dig a hole? f How many people could dig the hole in two hours? g How many people could dig the hole in 30 minutes? h How long would it take for 360 people to dig the hole? Is this possible?

10 8 6 4 2

The maximum number of people (n in 1000’s) attending an outdoor concert is dependent on the area 1.2 models this practical situation. (A in m2 ) allowed per person. The reciprocal equation n = A a Complete the following table of values, correct to the nearest whole number. A 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0 n

1.2 using the number plane shown. A c How many people can attend this concert if the area allowed per person is 0.5 m2 ? d How many people can attend this concert if the area allowed per person is 0.25 m2 ? e What is the area allowed per person if the maximum number of people attending the concert is 2000? f What is the area allowed per person if the maximum number of people attending the concert is 5000? g Is it possible for 12 000 people to attend this concert? Justify your answer. b Draw the graph of n =

12 10 8 6 4 2

A 0.2

0.4

0.6

0.8

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8F Reciprocal models in practical contexts

CHALLENGING 60 where t is the time (hours). t a Construct a table of ordered pairs using 1, 5, 10, 15, 20 and 60 as values for t. 60 b Draw the graph of s = . t c What is the speed of the truck if the time taken was 30 hours? d What is the time taken if the speed of the truck was 25 km/h? The speed in km/h (s) of a truck is given by the formula s =

U N SA C O M R PL R E EC PA T E G D ES

The number of chairs (n) in a row is dependent on the distance (d, in metres) between the chairs. 48 This relationship is modelled using the reciprocal equation n = . d a Construct a table of ordered pairs using 2, 4, 8, 12, 16, 20 and 24 as values for d. 48 b Draw the graph of n = . d c How many chairs can be placed in a row if the distance between them is 3 metres? d What is the problem with the model if the distance between the chairs is 50 metres?

Communicate your reasoning, understanding and fluency 8

900 Felix is planning to subdivide his land into rectangular blocks using the formula l = , where l is b the length in metres and b is the breadth in metres.

a Make a table of values for b and l. Use 5, 10, 30, 50 and 100 for b. 900 b Draw the graph of l = . b c What is the length of the block if the breadth is 2.5 metres? d What is the breadth of the block if the length is 150 metres? e If the block were to be a square, what would be its length? f Which would cost more to fence, a block that was 60 m wide, or one that was 25 m wide? Justify your answer using relevant calculations.

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8G Inverse variation Learning intentions

• To recognise reciprocal relationships involving inverse variation Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

• The concepts in this section will be new to many students • Questions involving inverse variation regularly appear in HSC Examinations

Inverse variation (or inverse proportion) occurs when one variable increases while the other variable decreases. The variables are dependent on each other but change in opposite directions. For example, the time taken to paint a house depends inversely on the number of people available to paint. The more painters available the less time it would take to paint the house.

Solving an inverse variation problem

k 1 Write an equation relating the two variables. y is inversely proportional to x so the equation is y = x or xy = k. (k is the constant of variation). 2 Solve the equation for k by substituting values for x and y. 3 Write the equation with the solution for k and solve the problem by substituting a value for either x or y.

Example 12

Solving an inverse variation problem

The cost per person (c) to hire a reception centre is inversely proportional to the number of people attending (n). If there are 50 people, the cost per person is $36.

a What is the cost per person when there are 20 people attending? b How many people are required for the cost per person to be $25? Solution

Explanation

k n k 36 = 50 k = 1800 1800 ∴ c= n 1800 = 20 = $90 Cost is $90 per person. 1800 b c= n 1800 25 = n n = 72 72 people required.

Cost is inversely proportional to the number of people. k Use the formula y = by replacing the y with c and the x with n. x Substitute 36 for c and 50 for n into the formula.

Evaluate. Write the formula using the value for the proportionality constant (k = 1800). Substitute 20 for n into the formula. Evaluate. Write the answer in words.

Write the formula using the value for the proportionality constant (k = 1800). Substitute 25 for c into the formula. Evaluate. Write the answer in words.

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8G Inverse variation

Now you try P varies inversely with Q, so that PQ = k.

U N SA C O M R PL R E EC PA T E G D ES

a When P = 50, Q = 10. Find k. b Find P when Q = 20. c Find Q when P = 100.

Example 13

Solving an inverse variation problem

The number of people (n) waiting at Circular Quay for a ferry is inversely proportional to the time (t, in minutes) until the next ferry arrives. If there are 64 people waiting, the ferry will 1 arrive in 3 minutes. 2 a Find the constant of variation. b Write the inverse variation equation. c How many people are waiting at Circular Quay 4 minutes before the ferry arrives? d How long before the ferry arrives if there are 28 people waiting? Solution a

k t k 64 = 3.5 n=

k = 224

224 t 224 c n= t 224 = 4 = 56 56 people are waiting.

b n=

224 d n= t 224 28 = t t=8 Ferry will arrive in 8 minutes.

Explanation

Number of people is inversely proportional to the time. k Use the formula y = by replacing the y with n and the x x with t. Substitute 64 for n and 3.5 for t into the formula. Evaluate. Substitute 224 for k in the inverse variation equation. Write the inverse variation equation. Substitute 4 for t into the formula.

Evaluate.

Write the answer in words.

Write the inverse variation equation. Substitute 28 for n into the formula. Evaluate.

Write the answer in words.

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Now you try k or xy = k: x a find k when x = 30 and y = 40. b find x when y = 60. c find y when x = 10.

U N SA C O M R PL R E EC PA T E G D ES

Using y =

Exercise 8G 1

BUILDING

y is inversely proportional to x and y = 4 when x = 3. a Write an equation connecting y and x. c What is y when x is 6?

b Calculate the proportionality constant. d What is x when y is 12?

2 y is inversely proportional to x and y = 18 when x = 2. a Write an equation connecting y and x. c What is y when x is 9?

b Calculate the proportionality constant. d What is x when y is 2?

3 It is known that m varies inversely with n. If m = 6 then n = 8. k a Write an equation in the form m = to describe this situation. n b Find the value of m if the value of n is 24. c Find the value of n if the value of m is 4. 4

2 It is known that c varies inversely with d. If c = 108 then d = . 3 k a Write an equation in the form c = to describe this situation. d b Find the value of c if the value of d is 48. c Find the value of d if the value of c is 8.

DEVELOPING

Example 12

The time taken (t), in days, to paint a house varies inversely with the number of painters (p) employed. It takes six painters 5 days to paint a house.

a Find the constant of variation. b Write an inverse variation equation to describe this situation. c How long does it take two painters to paint the house? d How many painters does it take to paint the house in 3 days?

Example 13

6 Time taken in hours for a trip varies inversely with average speed. It takes five hours to complete a trip travelling at 68 km/h. a Find the constant of variation. b Write an inverse variation equation to describe this situation. c What is the time taken for a trip with an average speed of 34 km/h? d What speed is required to complete the trip in 4 hours?

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8G Inverse variation

The time taken (T ), in days, to assemble a shed varies inversely with the number, N, of people employed. It takes three people 2 days to assemble a shed.

U N SA C O M R PL R E EC PA T E G D ES

a How long does it take one person to assemble the shed? b How many people are required to assemble the shed in 3 days? c Is this statement correct? ‘If the number of people is doubled then the time taken to assemble the shed is halved.’ Give a reason for your answer. 8

The cost ($c) per passenger of hiring a bus is inversely proportional to the number of passengers (n) on the bus. If there were 20 passengers, the cost per passenger is $30.

a What is the cost per passenger when there are 32 passengers? b How many passengers are required for the cost per passenger to be $20?

The cost per person (c), in dollars, to attend a formal dance varies inversely with the number of people (n) attending. When 90 people attend, the cost is $60 per person.

a If only 80 people attend the formal dance, how much will each person have to pay?

b The organising committee has suggested a cost per ticket of $40. How many people need to attend the formal dance for the tickets to be $40?

CHALLENGING

The number of people (N) who attend a show varies inversely with the amount of floor space (in cm2 ) allowed per person (A). A venue can hold 480 people if each person is allowed 2000 cm2 .

a How many people could the venue hold if each person was allowed 2400 cm2 ? b What is the space allowed per person to allow 500 to attend the show?

Connor uses a microwave oven to reheat a meal. The time taken (in minutes) for heating is inversely proportional to the power setting. The meal takes three minutes at a power setting of 60%. How long would it take at a power setting of 80%?

Communicate your reasoning, understanding and fluency 12

Explain the difference between direct variation and inverse variation. For each type of variation give: • a clear practical example • an equation • a graph

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Problems involving exponential, quadratic and 8H reciprocal relationships Learning intentions

• To solve problems involving non-linear relationships

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Past, present and future learning

• The concepts in this section will be new to many students • Questions involving non-linear relationships regularly appear in HSC Examinations

Algebraic modelling occurs when a practical situation is described mathematically using an equation and a graph. This involves gathering data and analysing the data to determine the best equation. Determining the equation is made easier using technology.

Algebraic models

• Algebraic models are used to describe practical situations. • Algebraic models may have limitations that restrict their use.

Example 14

Modelling physical phenomena

This table shows the mass (M kg) of a baby orangutan and its age after x months. Age

Mass

1.5

1.8

2.2

2.6

3.1

3.7

4.5

a Plot the points from the table onto a number plane. b The formula M = 1.5(1.2) x models the data in the table. Graph M = 1.5(1.2) x on the same number plane. c Use the model to determine the mass of the orangutan after 2.5 months, correct to 1 decimal place. d This model only applies when x is less than or equal to 6. Why? Solution

Explanation

a, b

Draw a number plane with x as the horizontal axis and M as the vertical axis. Plot the points from the table. The formula M = 1.5(1.2) x has the same table of values. Join the points to make a curve.

5 4 3

2 1

c M = 1.5(1.2)2.5 = 2.4 kg

Substitute 2.5 for x into the formula. Evaluate, correct to one decimal place.

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8H Problems involving exponential, quadratic and reciprocal relationships

d M = 1.5(1.2)48

Use the model for x greater than 6. Let x be 48 months or 4 years. Substitute 48 for x into the formula. Evaluate. Write the answer in words.

= 9479.6 kg Orangutans are less than 100 kg in general so the answer of 9479.6 kg is unreasonable.

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Now you try

The distance (d metres) that an object falls in t seconds is shown in this table. Time

Distance

125

180

a Plot the points from the table of values on the number plane. Join the points to make a curve. b A quadratic model in the form y = ax2 describes this situation.

200

i What is the model if x and y are replaced with t and d? ii Determine the value of a in the model by using t = 1. iii What is the quadratic model using t and d? 1 c Use the model to find the distance fallen after 3 seconds. 2 d Use the model to find the distance fallen after 10 seconds. e What is the time taken for an object to fall 320 metres? f Earth’s atmosphere is approximately 1000 km thick. What limitations would you place on this model?

150

100

Exercise 8H

Example 14

BUILDING

A new piece of equipment is purchased by a business for $150 000. The value of the equipment (v in thousands of dollars) depreciates each year (t) as shown in the table. Years

Value

150

a Draw a number plane as shown. b Plot the points from the table of values. Join the points. c An exponential model in the form y = 2−x × a describes this situation.

i What is the model if x and y are replaced with t and v? ii Determine the value of a in the model by using t = 0. iii What is the exponential model using t and v?

160 140 120 100 80 60 40 20

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d Use the model to predict the value of the equipment after 2.5 years. Answer correct to the nearest thousand. e Use the model to predict the value of the equipment after 6 months. Answer correct to the nearest thousand. f When will the value of the equipment be $75 000? g Use the model to predict the value of the equipment after 20 years. Explain your answer. 2

The number of tadpoles (N) in a pond after t months is shown in this table. Time

Number

216

384

600

1176

a Draw a number plane with t as the horizontal axis and N as the vertical axis. b Plot the points from the table of values. Join the points to make a curve. c A quadratic model in the form y = ax2 describes this situation.

i What is the model if x and y are replaced with t and N? ii Determine the value of a in the model by using t = 2. iii What is the quadratic model using t and N?

d Use the model to find the number of tadpoles after 11 months.

e Use the model to find the time taken for the number of tadpoles to reach 2400. f Use the model to predict the number of tadpoles after 4.5 months. What limitations would you place on this model?

DEVELOPING

The speed of a car (s, in km/h) and the time taken (t, in hours) is shown in this table. Time

Speed

120

a Draw a number plane with t as the horizontal axis and s as the vertical axis. b Plot the points from the table of values. Join the points to make a curve. a c A hyperbolic model in the form y = describes this situation. x i What is the model if x and y are replaced with t and s? ii Determine the value of a in the model by using t = 1. iii What is the hyperbolic model using t and s?

d Use the model to find the speed of the car after 8 hours. e What is the time taken for a car travelling at a speed of 48 km/h? f Use the model to predict the speed of the car after 30 minutes. Is this possible? Explain your answer.

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8H Problems involving exponential, quadratic and reciprocal relationships

CHALLENGING Filomena invested $100 into a managed fund that appreciated in value for 5 years. The amount of money (A) in the fund for each year (t) is shown in this table. Years

Amount

$100

$150

$225

$338

$506

$759

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a Draw a number plane with t as the horizontal axis and A as the vertical axis. b Plot the points from the table of values. Join the points to make a curve. c An exponential model in the form y = 100 × (a) x describes this situation. i What is the model if x and y are replaced with t and A? ii Determine the value of a in the model by using t = 1. iii What is the exponential model using t and A?

1 d Use the model to find the value (correct to the nearest dollar) of the fund after 2 years. 2 e Use the model to find the value (correct to the nearest dollar) of the fund after 7 years. f When is the value of the fund approximately $5767?

Communicate your reasoning, understanding and fluency 5

The cost of using a new motorway was initially free and 4500 vehicles used it each day. A toll was introduced that gradually increased. Let the toll charge be c (in dollars) and the number of vehicles using the toll be n.

a An exponential model in the form y = b−x × a describes this situation.

i What is the model if x and y are replaced with c and n? ii Determine the value of a in the model by using c = 0. iii Determine the value of b in the model if there were 3000 vehicles using the toll for a charge of $1. iv Determine an exponential model using C and n to describe this situation.

b Construct a table of values by using c (dollars) to represent the value of the toll and (n) for the number of vehicles using the motorway each day. c Plot the table of values on a number plane. d How many vehicles use the toll if the charge was $3? e What is the income received when the toll charge was $3? f ‘A higher toll always means a higher total daily income’. Do you agree with this statement? Explain your answer.

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Key ideas and chapter summary Exponential models

y = ka x , a > 0 (k is constant)

y = ka−x , a > 0 (k is constant) y

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Review

462

Quadratic models

Exponential growth

Exponential decay

Quantity increases rapidly using y = a x

Quantity decreases rapidly using y = a−x

Parabola has the form y = ax2 + bx + c where a, b and c are numbers.

• Parabola (y = x )

Concave up • Parabola (y = −x2 ) Concave down

A quadratic model describes a practical situation using a function in the form y = ax2 + bx + c where a, b and c are numbers.

Reciprocal models

–3

–2

–1

k situation using a function in the form y = x where k is a number.

Vertex or turning point

1 • Hyperbola : y = x

1 x A reciprocal model describes a practical

y=x

k A reciprocal function has the form y = x where k is a number.

• Hyperbola : y = −

Axis of symmetry

Asymptotes

Branch

2 1

−4 −3 −2 −1 O −1 Branch

−2

−3

−4

Inverse variation

Write an equation relating the two variables. (k is the constant). y is inversely k proportional to x so the equation is y = or xy = k. x

Solve the equation for k by substituting values for x and y.

Write the equation with the solution for k and solve the problem by substituting a value for either x or y.

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Chapter 8 review

A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

Review

Chapter checklist with success criteria

1 I can recognise the equation and shape of an exponential relationship and draw

U N SA C O M R PL R E EC PA T E G D ES

the graph.

e.g. On the same Cartesian plane draw the following two graphs from x = −3 to x = 3.

a y = 2x b y = 3 × 2x

2 I can use exponential models in practical contexts.

e.g. A town of 1000 people is growing by 10% per annum. How many years will it take for the population to double?

3 I can recognise the equation and shape of a quadratic relationship and draw the graph of a parabola.

e.g. Draw the graph of y = 25 − x2 from x = 0 to x = 5.

4 I can use quadratic models in practical contexts.

e.g. A rectangle with a perimeter of 20 metres has sides x and y.

a Show that the side labelled y is equal to 10 − x. b What is the formula for the area of the rectangle, in terms of x? c Draw a graph with the base length on the horizontal axis and the area on the vertical axis. d What is the maximum area of the rectangle?

5 I can recognise the equation and shape of a reciprocal relationship and draw the graph of a hyperbola.

e.g. Draw the graph of the hyperbola xy = 20.

6 I can use reciprocal models in practical contexts

e.g. Long rents a holiday house for a surfing weekend with his friends. It costs $1000. They will divide the cost evenly. If N is the number of people and C is the cost per person:

a Write a formula involving N and C. b Draw a graph of this model. c How many people will be required to bring the cost below $125 per person?

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7 I can solve problems involving inverse variation. e.g. y varies inversely to x. When x = 20, y = 8. Find the value of x when y = 50.

8 I can choose an appropriate model relevant to a problem and use it to solve the problem.

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e.g. a What are the missing numbers in this table? x

135

b What is the equation of the relationship between x and y?

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Chapter 8 review

Review

Multiple-choice questions 8A

What is the y-intercept of the exponential function y = 2−x ? A −1 C 1

Which point lies on the parabola y = 2x2 − x + 1?

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B 0 D 2

A (−1, 0) C (1, 2)

B (0, −1) D (2, 16)

What is the y-intercept of the parabola y = −x2 + 4x − 3? A −3 C 2

B 1 D 4

4 The equation d = 0.4(s2 + s) is used to model the stopping distance for a bicycle where d is the stopping distance in metres and s is the bicycle’s speed in m/s. What is the stopping distance, given a speed of 5 metres per second? A 5m C 12 m

B 10 m D 15 m

8 5 Which of the following points lies on the reciprocal function y = ? x A (−2, 8) B (−1, 8) C (0, 8) D (2, 4)

200 The speed in km/h (s) of a vehicle is given by the formula s = where t is the time in hours. t What is the time taken if the average speed was 80 km/h?

A 0.4 h C 120 h

B 2.5 h D 280 h

If the temperature (t) varies inversely with the cost (c), which formula correctly expresses t in terms of c and k, where k is the constant of variation? k c A t= B t= c k C t = kc D t = kc2

This graph shows the insect population (N) plotted against the time (t) in days. t

30 20

What type of function would model this data?

A Exponential B Hyperbolic C Quadratic D Reciprocal

t 1

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Short-answer questions 8A

Complete the following tables of values by expressing the y-values correct to one decimal place. Graph each exponential function. a y = 1.5 x −3

−2

−1

−3

−2

−1

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b y = 0.5 x

2 The height, h cm, of a plant and its age after x months is given in this table. x

1.1

2.4

5.3

11.7

25.8

56.7

124.7

a Plot the points from the above table onto a number plane. b The formula h = 1.1 × 2.2 x models the data in the table. Draw this function. c Use the model to determine the height of a plant after 1.5 months. Answer correct to one decimal place. d Use the model to determine the height of a plant after 3.5 months. Answer correct to one decimal place.

The population of earthworms grows exponentially according to the formula w = 25 × 1.1t , where w is the number of earthworms after t days.

a Construct a table of ordered pairs using 0, 5, 10, 15 and 20 as values for t. Express the number of earthworms correct to the nearest whole number. b Graph w = 25 × 1.1t using the table of ordered pairs in part a. c What is the initial number of earthworms? d What is the number of earthworms after 3 days? e Estimate the time taken for the earthworms to reach a population of 75.

Complete the following tables of values and graph each parabola. 1 a y = 3x2 b y = − x2 3 x −3 −2 −1 0 1 2 3 x −9 −3 y

d y = x2 − 5x − 4

c y = x2 + 3 x

−1

−3

−2

−1

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Chapter 8 review

Ramandeep throws a rock and it takes 6 seconds to reach the ground. The height it reaches is given by the formula h = −t2 + 6t, where h is the height (in metres) and t is the number of seconds after it has been thrown. a Complete the following table of values. t

Review

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b Draw the graph of h = −t2 + 6t. c What was the maximum height reached by the rock? d When was the maximum height reached?

6 Complete the following tables of values and graph each reciprocal function on the same number plane. 7 7 a y= b y=− x x x

−7

−1

−

1 7

−7

−1

−

1 7

7 The number of chairs (c) in a row varies inversely with the distance (d in metres) between them. When the chairs are 2 m apart the row can accommodate 60 chairs. a How many chairs can be placed in a row if the distance between them is 1.5 m? b What is the distance between the chairs if the number of chairs is 40?

8 The cost per person (c) to use a conference centre is inversely proportional to the number of people attending (n) the conference. If there were 20 people, the cost per person is $60. How many people are required for the cost per person to be $40?

Extended-response questions

A rectangular patio has a length of x metres and a breadth of (4 − x) metres.

Patio

a Find a quadratic equation for the area (A) of the patio in terms of the length (x). b Complete the table of values and draw the graph of this quadratic equation. x

0.5

1.5

2.5

3.5

4−x

c What is the maximum area of the patio? d What is the patio length in order to have maximum area?

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The normal distribution

What is a normal distribution?

Normal distributions in nature?

A normal distribution is a dataset with the majority of data situated toward the middle of the range of the dataset. It is symmetrical about the mean. The mean, mode and median are all equal in a normal distribution. When working with a normal distribution we use µ to represent the mean instead of x̄. We also use σ for standard deviation instead of SD. The shape of a normal distribution resembles a bell curve.

Normal distributions occur where we don’t even look for them. The sizes of snowflakes, IQ score, arm length, hand span, the birthweight of babies and blood pressure are all examples of normal distributions that occur in nature. In fact, we can look almost anywhere in nature to find a normally distributed dataset.

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Chapter contents Normally distributed datasets The empirical rule z-scores Converting z-scores into actual scores 9E Using z-scores to compare data 9F Further applications of normal distributions

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9A 9B 9C 9D

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Analyses normally distributed datasets using statistical processes (MST-12-S2-10). © NSW Education Standards Authority

Online resources

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Chapter 9 The normal distribution

9A Normally distributed datasets Learning intentions

• To recognise the characteristics of datasets which are normally distributed • To understand that the mean, median and mode are equal in a normal distribution

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Past, present and future learning

A normal distribution has the same mean, mode and median. It is symmetrical about the mean. The graph of a normal distribution is often called a ‘bell curve’ (or ‘normal curve’) due to its shape, as shown to the right.

‘bell curve’

50 40 30 20 10 0

100

120

140

160

180

200

Mean and median in a normal distribution

The mean is a measure of the centre. It is calculated by summing all the scores and dividing by the number of scores. The mean of a dataset is what most people call the ‘average’. The median is the middle score or value. To find the median, list all the scores in increasing order and select the middle one. The mean and median are exactly equal for a perfect normally distributed dataset. For example, consider the set of data: 2, 3, 3, 4. The mean of this set of data is given by: X x 2+3+3+4 mean µ = = n 4 12 =3 = 4

M 3

mean

Data is ordered: 2, 3, 3, 4. Median = 3

This small dataset is normally distributed. Both the mean and median are equal to 3. The balance point of the distribution is also the point that splits the distribution in half; that is, there are two data points to the left of the mean and two to the right. This is a general characteristic of a normal distribution.

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9A Normally distributed datasets

However, consider the set of data: 2, 3, 3, 8.

The mean of this dataset is given by: X x 2+3+3+8 mean = = n 4 16 =4 = 4

8 4

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mean

Data is ordered: 2, 3, 3, 8. Median = 3

This dataset is skewed. The mean is 4 and is not equal to median of 3. The mean is affected by changing the largest data value but the median is not.

Characteristics of a normal distribution

The normal distribution is arguably the most important distribution in statistics. There are many datasets that arise in practice that are roughly symmetrical and have an approximate bell shape, as shown here.

170

180 175 Blood pressure

185

5 6 7 8 9 10 11 12 13 14 15 16 Hand span measurements (centimetres)

55 60 65 70 75 80 85 90 95 Weight (kilograms)

2.0

2.5 3.0 3.5 4.0 Sepal width of flower (centimetres)

4.5

Normal distributions can differ in their means and in their standard deviations. The diagram opposite shows three normal distributions. The blue distribution has a mean of −3 and a standard deviation of 0.5, the distribution in red has a mean of 0 and a standard deviation of 1, and the distribution in black has a mean of 2 and a standard deviation of 3. All of these line graphs represent a normal distribution. Notice that all three graphs are symmetrical about the centre and most of the data is at the centre.

There are five important characteristics of a normal distribution: • a normal distribution is symmetrical about the mean

• the mean, median and mode (the score that occurs the most) are equal • the majority of the data is located at the centre with less data at the tails (or edges) • the graph of a normal distribution has an asymptote on both ends of the x-axis • the mean and standard deviation of the dataset are used to define the normal distribution.

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Normal distribution A normal distribution has the same mean, mode and median. It is symmetrical about the mean. The graph of a normal distribution is called a ‘bell curve’.

Example 1

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Drawing a ‘bell-shaped’ curve to represent a normally distributed dataset

The heights (in cm) of 11 women are shown below.

163, 164, 165, 165, 166, 166, 166, 167, 167, 168, 169

a Construct a frequency table b Find the mean and median. c What is the standard deviation, correct to two decimal places? d Draw a dot plot and bell-shaped curve to show this distribution. Solution

Explanation

a Frequency table

Create a frequency table to represent the dataset.

Score

Tally

Frequency

163

164

165

166

|||

167

168

169

b Mean = 166

163, 164, 165, 165, 166, 166, 166, 167, 167, 168, 169 Median = 166

Find the middle score. Keep the data from part b in your calculator. Calculate the standard deviation.

c Standard deviation σ x ≈ 1.65

Create a dot plot. Draw a ‘bell-shaped’ curve to match the dots.

d Dot plot and bell-shaped curve

163

164

165

166

167

Place your calculator in STATISTICS mode. Enter the dataset. Calculate the mean. Write the scores in increasing order.

168

169

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9A Normally distributed datasets

Now you try The shoe sizes of 15 students are shown. 6, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 14

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a Construct a frequency table. b Find the mean and the median. c Find the standard deviation, correct to one decimal place. d Draw a dot plot and a bell-shaped curve to show the distribution.

Exercise 9A 1

BUILDING

Copy and complete the following sentences.

a A normal distribution has the same mean, mode and _________. b A normal distribution is symmetrical about the _________. c The graph of a _________ distribution is often called a ‘bell-shaped’ curve. d Mean is calculated by summing all the _________ and dividing by the number of scores. e To find the median, list all the scores in increasing order and select the _________ one. f Normal distributions can differ in their means and in their _________ deviations. g The majority of the data in a normal distribution is located at the _________. h A normal distribution has an _________ on both ends of the x-axis.

2 Find the mean and median for each normally distributed dataset. Answer correct to the nearest whole number.

a 5, 8, 9, 9, 10, 11, 11, 13. b 37, 38, 38, 39, 40, 40, 40, 40, 40, 40, 41, 41, 41, 41, 42, 42. c 65, 70, 71, 71, 72, 73, 75, 75, 77, 77, 78. d 33, 23, 31, 24, 30, 32, 26, 32, 29, 30, 40. e 117, 127, 116, 119, 122, 123, 120, 125, 120, 106, 125. f 95, 93, 94, 95, 94, 98, 95, 93, 96, 101, 93, 94, 100, 92, 98, 87, 91, 88, 89.

Example 1

The time (in minutes) it takes Layla to travel to work is shown below. 60, 61, 62, 62, 63, 63, 63, 64, 64, 64, 64, 65, 65, 65, 66, 66, 67, 68

a Construct a frequency table. b Find the mean and median. c What is the standard deviation? d Draw a dot plot and ‘bell-shaped’ curve to show this distribution.

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The weight (in grams) of Henry’s oranges is shown below. 149, 150, 146, 148, 150, 149, 150, 147, 148, 152, 153, 154, 151, 152, 151

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a Construct a frequency table. b Find the mean and median. Answer to the nearest whole number. c What is the standard deviation? Answer correct to two decimal places. d Draw a dot plot and ‘bell-shaped’ curve to show this distribution.

DEVELOPING

Create the spreadsheet below.

a In Cell B14, use a formula to calculate the total of the frequency values. b In Cell C14, use a formula to calculate the total of the f x column. c Create a column chart using horizontal (x-values) A5:A13 and y-values B5:B13. d Change the scores by adding 2 to each score. e Change the frequencies by adding these scores: 79, 82, 83, 84 and 87.

A sample of 11 students were given a general knowledge test with the following results: 20, 20, 19, 21, 21, 18, 20, 22, 23, 17, 19

a calculate the mean and standard deviation of the test scores, correct to one decimal place b the median test score is 20, which is similar in value to the mean. What does this tell you about the distribution of test scores? c draw a column graph to show the distribution of test scores.

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9A Normally distributed datasets

CHALLENGING 7

The stem plot shows the distribution of weights (in kg) of 23 footballers.

Weight (kg) 6

6678

0012334

55569

U N SA C O M R PL R E EC PA T E G D ES

a What is the mean and median for this distribution? Answer correct to 1 decimal place where necessary. b Is this a normal distribution? Give a reason for your answer. c Which measure of centre, the mean or the median, do you think would best indicate the typical weight of these footballers?

Communicate your reasoning, understanding and fluency 8

Consider this frequency histogram.

Coins collected

Frequency

20 15 10

5 0

a Is the dataset normally distributed? b Explain the features of the dataset that support your answer to part a.

Modelling activity: Predicting viral videos

In this modelling task, you’ll analyse a normally distributed dataset of video view counts and then use ‘newcomer’ z-scores to judge which creators show signs of unusually strong performance and may have ‘viral potential’. The activity can be downloaded from your Interactive Textbook. It is recommended that this activity be done after you have completed sections 9A–9F.

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Chapter 9 The normal distribution

9B The empirical rule Learning intentions

• To understand the empirical rule • To apply the empirical rule to problems involving normal distributions

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Past, present and future learning

• The concepts in this section will be new to many students • These concepts build upon prior learning from Year 11 • Questions involving normal distributions regularly appear in HSC Examinations • Students need to be familiar with the formulas which appear on the HSC Reference Sheet published by NESA. The empirical rule is provided on the 2027 version.

A normal distribution has the same mean, mode and median. It is symmetrical about the mean. The percentage of data that lie within a certain number of standard deviations (or z-scores) is always the same for a normal distribution. That is, 68% are within one standard deviation, 95% are within two standard deviations and 99.7% are within three standard deviations. This gives rise to what is known as the 68–95–99.7% rule, or the empirical rule.

The empirical rule

For a normal distribution, approximately:

• 68% of data lie within one standard deviation of the mean • 95% of data lie within two standard deviations of the mean • 99.7% of data lie within three standard deviations of the mean.

Note: 50% of data will lie above the mean and 50% of data will lie below the mean.

Graphical form of the empirical rule If a data distribution is approximately normal, then:

• around 68% of data will lie within one standard deviation

(SD) of the mean (µ ± σ). This also means that 32% of data lie outside this region. As the distribution is symmetric, we can also say that around 16% of data lie in each of the tails (shaded blue, as shown in the diagram).

16% mean − 1 SD

68% mean

16% mean + 1 SD

• around 95% of data will lie within two standard deviations

(SD) of the mean (µ ± 2σ). This also means that 5% of data lie outside this region. As the distribution is symmetric, we can also say that around 2.5% of data lie in each of the tails (shaded blue, as shown in the diagram).

95%

2.5%

mean − 2 SD

2.5%

mean

mean + 2 SD

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9B The empirical rule

• around 99.7% of data will lie within three standard

deviations (SD) of the mean (µ ± 3σ). This also means that 0.3% of data lie outside this region. As the distribution is 0.15% symmetric, we can also say that around 0.15% of data lie in each of the tails (shaded blue, as shown in the mean − 3 SD diagram).

99.7% 0.15% mean + 3 SD

U N SA C O M R PL R E EC PA T E G D ES

mean

Finally, because the normal distribution is symmetric, the mean and the median coincide. Therefore, 50% of data will lie above the mean and 50% of data will lie below the mean.

50% 50% mean

Example 2

Introducing the empirical rule

A normal distribution of scores has a mean of 75 and standard deviation (SD) of 5. Complete the information about these percentages of the scores:

a 50% lie above ____ b 68% lie between ____ and ___ c 95% lie between ____ and ____ d 99.7% lie between ____ and ____. Solution

Explanation

a 75 ∴ 50% lie above 75 b 75 − 1 × 5 = 70

A normal distribution is symmetrical about the mean. 50% of the scores are above the mean of 75. 68% of the scores lie within one standard deviation of the mean.

75 + 1 × 5 = 80 ∴ 68% lie between 70 and 80 c 75 − 2 × 5 = 65

75 + 2 × 5 = 85 ∴ 95% lie between 65 and 85 d 75 = 75 − 3 × 5 = 60

75 = 75 + 3 × 5 = 90 ∴ 99.7% lie between 60 and 90

95% of the scores lie within two standard deviations of the mean.

99.7% of the scores lie within three standard deviations of the mean.

Now you try

A normal distribution has scores with a mean of 60 and a standard deviation of 10. Complete the following.

a 50% of scores lie below ____________. b 68% of scores lie between _______ and ______. c 95% of scores lie between _______ and ______. d 99.7% of scores lie between _______ and ______.

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Example 3

Applying the empirical rule

In a normal distribution, the mean recovery time after contracting a certain disease is 12 days with a standard deviation (SD) of 2 days.

U N SA C O M R PL R E EC PA T E G D ES

a How long would it take for half the patients to recover? b What percentage of patients would recover within 8 to 16 days? c What percentage of patients would recover within 10 days? d If 400 patients contracted the disease last year, how many recovered within 16 days? Solution

Explanation

a 12 days

A normal distribution is symmetrical about the mean. 50% of the scores are above the mean of 12. 8 days is 2 standard deviations below the mean. 16 days is 2 standard deviations above the mean. 95% of the scores lie within two standard deviations of the mean.

b 12 − 2 × 2 = 8 days

12 + 2 × 2 = 16 days ∴ 95% recover within 8 to 16 days c 12 − 1 × 2 = 10 days ∴ 16% recover within 10 days d 12 + 2 × 2 = 16 days

97.5 × 400 100 = 390 ∴ 390 patients recover within 16 days. Patients =

10 days is 1 standard deviation below the mean. 32% of scores lie outside 1 standard deviation away from the mean. Therefore, half of this is less than 10 days, which is 16%. 16 days is 2 standard deviations above the mean. The percentage outside 2 standard deviations is 5%. Half of this is 2.5%. So, 97.5% of people recovered within 16 days.

Now you try

In a normal distribution, the mean recovery time after contracting a virus is 15 days with a standard deviation of 3 days.

a How long would it take for half the patients to recover? b What percentage of patients will recover in 12 to 18 days? c What percentage of patients will recover in 6 days? d If 500 patients contract the virus, what percentage will recover in 21 days?

Example 4

Applying the empirical rule using diagrams

The distribution of delivery times for pizzas made by House of Pizza is approximately normal, with a mean of 25 minutes and a standard deviation (SD) of 5 minutes.

a What percentage of pizzas have delivery times of between 15 and 35 minutes? b What percentage of pizzas have delivery times of greater than 30 minutes? c In 1 month, House of Pizza delivers 2000 pizzas. How many of these pizzas are delivered in less than 10 minutes?

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9B The empirical rule

Solution

Explanation mean = 25 SD = 5

10 15 20 25 30 35 40 Delivery time

Shade the region under the normal curve representing delivery times of between 15 and 35 minutes. Note that delivery times of between 15 and 35 minutes lie within two SDs of the mean. (15 = 25 − 2 × 5 and 35 = 25 + 2 × 5) 95% of values are within two SDs of the mean.

U N SA C O M R PL R E EC PA T E G D ES

mean = 25 SD = 5

Draw, scale and label a normal distribution curve with a mean of 25 and an SD of 5.

95%

10 15 20 25 30 35 40 Delivery time ∴ 95% of pizzas will have delivery times of between 15 and 35 minutes. mean = 25 SD = 5 16%

10 15 20 25 30 35 40 Delivery time ∴ 16% of pizzas will have delivery times of greater than 30 minutes.

c Number = 2000 Percentage delivered in less than 10 minutes = 0.15% Number of pizzas delivered in less than 10 minutes = 2000 × 0.15% 0.15 = × 2000 100 =3

Draw, scale and label a normal distribution curve with a mean of 25 and an SD of 5. Shade the region under the normal curve representing delivery times of greater than 30 minutes. Delivery times of greater than 30 minutes are more than one SD above the mean. (30 = 25 + 1 × 5) 16% of values are more than one SD above the mean.

Write down the number of pizzas delivered. Delivery times of less than 10 minutes are more than three SDs below the mean. (10 = 25 − 3 × 5). 0.15% of values are more than three SDs below the mean. Multiply total number of pizzas delivered by 0.15% and evaluate.

∴ Number of pizzas delivered in less than 10 minutes is 3.

Now you try

The distribution of delivery times for meals from Meals in a Flash is approximately normal, with a mean of 26 minutes and a standard deviation of 4 minutes.

a What percentage of meals are delivered in between 18 and 34 minutes? b What percentage of meals have delivery times greater than 30 minutes? c Meals in a Flash delivers 675 meals in one week. How many of these meals take less than 14 minutes to deliver?

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Exercise 9B 1

BUILDING

What percentage of data in a normal distribution lie:

U N SA C O M R PL R E EC PA T E G D ES

a above the mean? b within one standard deviation of the mean? c outside one standard deviation of the mean? d within two standard deviations of the mean? e outside two standard deviations of the mean? f within three standard deviations of the mean? g outside three standard deviations of the mean?

Example 2

Scores in a class test are normally distributed with a mean of 65 and standard deviation of 10. Given this information it can be concluded that:

a 50% of the scores lie above _________ b 68% of the scores lie between _________ and _________ c 95% of the scores lie between _________ and _________ d 99.7% of the scores lie between _________ and _________.

The lifetimes of batteries used in a toy have a mean of 12 hours and a standard deviation of 3 hours. The battery lifetimes are normally distributed. Given this information it can be concluded that:

a 50% of the batteries have lifetimes above _________ b 68% of the batteries have lifetimes between _________ and _________ c 95% of the batteries have lifetimes between _________ and _________ d 99.7% of the batteries have lifetimes between _________ and _________.

The monthly rainfall at Georgetown follows a ‘bell-shaped’ curve with a mean of 55 mm and a standard deviation of 2.5 mm. Given this information it can be concluded that:

a 50% of the monthly rainfall measurements lie above _________ b 68% of the monthly rainfall measurements lie between _________ and _________ c 95% of the monthly rainfall measurements lie between _________ and _________ d 99.7% of the monthly rainfall measurements lie between _________ and _________.

The weight of coffee in a glass jar is normally distributed with a mean of 101.5 g and a standard deviation of 0.7 g. Given this information it can be concluded that:

a 50% of the weights lie above _________ b 68% of the weights lie between _________ and _________ c 95% of the weights lie between_________ and _________ d 99.7% of the weights lie between _________ and _________.

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9B The empirical rule

Example 3

The blood pressure readings for executives are normally distributed with a mean systolic blood pressure of 134 and a standard deviation of 20. Given this information it can be concluded that:

U N SA C O M R PL R E EC PA T E G D ES

a 68% of the executives have blood pressures between _________ and _________ b 95% of the executives have blood pressures between _________ and _________ c 99.7% of the executives have blood pressures between _________ and _________ d 16% of the executives have blood pressures above _________ e 2.5% of the executives have blood pressures below _________ f 0.15% of the executives have blood pressures below _________ g 50% of the executives have blood pressures above _________.

The mean length of 1000 garden stakes is 180 cm. The standard deviation of the lengths is 3 cm. The length of the garden stakes is normally distributed. What percentage of the garden stakes lie between: a 177 cm and 183 cm? b 174 cm and 186 cm? c 171 cm and 189 cm?

Example 4

8 The weights of packets of potato chips are normally distributed. The mean is 200 grams and the standard deviation is 8 grams.

a What percentage of packets will have a weight between 184 grams and 216 grams? b What percentage of packets will have a weight between 192 grams and 208 grams? c What percentage of packets will have a weight between 176 grams and 224 grams? d What percentage of packets will have a weight greater than 200 grams?

The time taken for 2000 bus trips between two towns is normally distributed with a mean of 78 minutes and a standard deviation of 4 minutes. What percentage of bus trips take: a 78 minutes or more? c between 66 and 90 minutes? e between 70 and 86 minutes? g less than 82 minutes?

b less than 78 minutes? d between 74 and 82 minutes? f more than 82 minutes? h more than 86 minutes?

An assessment task has a normal distribution with a mean of 67 and a standard deviation of 7. What percentage of scores lie between: a 46 and 88? d 60 and 67?

b 53 and 81? e 67 and 74?

c 60 and 74? f 67 and 81?

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DEVELOPING 11

The average weight of a bag of 10 blood plums picked at U-Pick Orchard is normally distributed with a mean of 1.88 kg and a standard deviation of 0.20 kg. What percentage of the bags of 10 plums weigh: b between 1.28 and 2.48 kg? d between 1.88 and 2.08 kg? f more than 2.28 kg? h less than 1.88 kg?

U N SA C O M R PL R E EC PA T E G D ES

a between 1.68 and 2.08 kg? c between 1.48 and 2.28 kg? e more than 2.08 kg? g less than 1.28 kg?

The distribution of times taken for walkers to complete a circuit in a park is normal, with a mean time of 14 minutes and a standard deviation of 3 minutes. What percentage of walkers complete the circuit in:

a between 11 and 14 minutes? c between 14 and 20 minutes? e more than 17 minutes? g less than 8 minutes?

b between 14 and 23 minutes? d between 5 and 14 minutes? f more than 23 minutes? h less than 5 minutes?

The distribution of heights of 19-year-old women is approximately normal, with a mean of 170 cm and a standard deviation of 5 cm.

a What percentage of these women have heights: i between 155 and 185 cm? ii greater than 180 cm? iii between 160 and 180 cm?

b In a sample of 5000 of these women, how many have heights: i between 155 and 185 cm? ii greater than 180 cm? iii between 160 and 180 cm?

The distribution of resting pulse rates of 20-year-old men is approximately normal, with a mean of 66 beats/minute and a standard deviation of 4 beats/minute.

a What percentage of these men have pulse rates of: i higher than 66? ii between 66 and 70? iii between 58 and 74?

b In a sample of 2000 of these men, how many have pulse rates: i higher than 66? ii between 66 and 70? iii between 58 and 74?

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9B The empirical rule

CHALLENGING The heights (to the nearest 5 cm) of a number of children are given below. Heights (m)

1.30

1.35

1.40

1.45

1.50

1.55

1.60

1.65

Frequency

U N SA C O M R PL R E EC PA T E G D ES

a How many children were measured? b What is the mean for this data? Answer correct to two decimal places. c What is the median for this data? d Find the standard deviation. Answer correct to two decimal places. e Is this data normally distributed or skewed? Justify your answer. f Find the range of values (correct to two decimal places) for: i µ±σ

ii µ ± 2σ

iii µ ± 3σ

g Ignoring rounding and assuming all children have exactly the heights shown in the table, use the table to find how many children have a height that corresponds to: i at least one standard deviation below the mean? ii at least one standard deviation above the mean?

16 The results of a numeracy test are normally distributed and shown below. Score

Frequency

125

200

250

200

125

a How many scores are there? b What is the mean for this data? c What is the median for this data? d Find the standard deviation? Answer correct to one decimal place. e Find the range of values (correct to one decimal place) for: i µ±σ

ii µ ± 2σ

iii µ ± 3σ

f How many scores correspond to:

i at least two standard deviations above the mean? ii at least two standard deviations below the mean?

Communicate your reasoning, understanding and fluency 17

A cake mix should contain 65 mg of sodium per 100 g of the product with a standard deviation of 4.3 mg. However, Madison analysed a sample cake mix and found it contained 77.9 mg of sodium. What conclusions could Madison draw assuming the data is normally distributed?

The life expectancy of car headlights is normally distributed with a standard deviation of 1000 hours. a What is the mean life, in hours, if 99.7% of the car headlights last up to 10 000 hours? b Jade had two car headlights die out last month. Explain why this is extremely unlikely to ever happen again.

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9C z-scores Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To understand z-scores • To describe a z-score in terms of the mean and standard deviation • To calculate z-scores of data values in normal distributions using the formula Past, present and future learning

A z-score is a statistical measure of how many standard deviations a raw score is above or below the mean. A z-score can be positive or negative, indicating whether it is above or below the mean, or zero. The z-score, or standardised score, is used to compare scores in a normal distribution. The z-score is calculated by subtracting the mean from the score and dividing the result by the standard deviation. The z-score is the number of standard deviations the score is from the mean.

z-score or standardised score

x−µ σ µ = Mean of a set of scores

z-score is the number of standard deviations the score is from the mean: z = z = z-score or standardised score

x = Score

σ = Standard deviation

A z-score of 0 indicates the score is equal to the mean. A positive z-score is above the mean and a negative z-score is below the mean. A z-score of 1 is one standard deviation above the mean and a z-score of −1 is one standard deviation below the mean. The larger the z-score (ignoring the positive or negative), the further away it is from the centre of the data.

Example 5

Calculating the z-score

For a normal distribution, the mean is 62 and the standard deviation is 11. Find the z-score for 84 and interpret this result.

Solution x−µ z= σ 84 − 62 = 11 =2

Explanation

Write the z-score formula, Substitute x = 84, µ = 62 and σ = 11. Interpret the meaning of the z-score.

The z-score of 2 tells us that 84 is two standard deviations above the mean.

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9C z-scores

Now you try For a normal distribution, the mean is 58 and the standard deviation is 8. Find the z-score for a score of 78 and interpret the results.

Calculating standard scores

U N SA C O M R PL R E EC PA T E G D ES

Example 6

The heights of a group of young women have a mean of µ = 160 cm and a standard deviation of σ = 8 cm. Determine the standard z-scores of a woman who is: a 172 cm tall

b 150 cm tall

Solution x−µ a z= σ =

c 160 cm tall.

Explanation

Write the z-score formula Substitute x = 172, µ = 160, σ = 8.

172 − 160 8

= 1.5

b z= =

x−µ σ

Write the z-score formula Substitute x = 150, µ = 160, σ = 8.

150 − 160 8

= −1.25

c z= =

x−µ σ

Write the z-score formula Substitute x = 160, µ = 160, σ = 8.

160 − 160 8

Now you try

The length of time a ball takes to roll down a hill has µ = 150 seconds and σ = 10 seconds. Determine the z-score if the time it takes the ball to roll down the hill is: a 120 seconds

Example 7

b 165 seconds

c 150 seconds.

Describing the z-score

What is the z-score that corresponds to a score that is:

a the mean? b one standard deviation above the mean? c one standard deviation below the mean?

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Explanation

a x=µ x−µ z= σ µ−µ = σ =0

The score is equal to the mean. Write the z-score formula Substitute x = µ.

b x=µ+σ x−µ z= σ (µ + σ) − µ = σ σ = σ =1

The score is equal to the mean plus the SD (σ). Write the z-score formula Substitute x = µ + σ.

c x=µ−σ

The score is equal to the mean minus the SD (σ). Write the z-score formula Substitute x = µ − σ.

U N SA C O M R PL R E EC PA T E G D ES

Solution

x−µ σ (µ − σ) − µ = σ −σ = σ = −1

Now you try

What is the z-score for a score that is:

a 2 standard deviations above the mean? b 1 standard deviation below the mean? c equal to the mean?

Exercise 9C 1

BUILDING

Calculate the z-score, correct to one decimal place, for each of the following:

a x = 60, µ = 75 and σ = 9 c x = 22, µ = 27 and σ = 2 e x = 10, µ = 5 and σ = 4

Example 5

b x = 73 , µ = 64 and σ = 5 d x = 134, µ = 120 and σ = 30 f x = 93, µ = 78 and σ = 6.

In a common test the mean was 50 and the standard deviation was 10. Find the z-scores for the following test scores. a 60 e 40 i 55

b 70 f 30 j 45

c 80 g 20 k 65

d 90 h 10 l 35

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9C z-scores

A set of scores has a mean of 100 and a standard deviation of 20. Standardise the following scores. a 120 d 100

c 80 f 110

The mean number of hours spent on the internet each week by students is 12 hours with a standard deviation of 3 hours. Complete the table, which shows the hours spent on the internet by five students.

Student

Hours

Amber

z-score

U N SA C O M R PL R E EC PA T E G D ES

Example 6

b 140 e 40

Example 7

Bailey

Chloe

Dylan

Ella

Packets of pasta are each labelled as having a mass of 500 g. The mass of these packets has a mean of 510 g with a standard deviation of 5 g. What are the z-scores for the packets of pasta with the following masses? a 500 g

b 505 g

c 510 g

d 515 g

The mean height for a group of students is 180 cm with a standard deviation of 6 cm. What is Julie’s standardised score (z-score) if her height is 171 cm?

7 What is the z-score that corresponds to a score that is two standard deviations: a above the mean?

b below the mean?

8 What is the z-score that corresponds to a score that is three standard deviations: a above the mean?

b below the mean?

DEVELOPING

Create the spreadsheet below.

a Cell D5 has a formula that calculates the mean. Enter this formula. b Cell D6 has a formula that calculates the standard deviation. Enter this formula. c Cell B6 has a formula that calculates the z-score. Enter this formula. d Fill down the contents of B6 to B14 using the formula in cell B6. e Change the scores by adding 2 to each score. f Delete the last 6 scores. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Shiloh scored 70 in a Science test. The test mean was 60 and the standard deviation was 4.

U N SA C O M R PL R E EC PA T E G D ES

a What was Shiloh’s z-score? b There was a mistake calculating the mean. It should have been a mean of 65. What is Shiloh’s new z-score? c There was a mistake calculating the mean and standard deviation. It should have been a mean of 65 and a standard deviation of 10. What is Shiloh’s new z-score? d What z-score corresponds to a score of 3 standard deviations above the updated mean and standard deviation? e What z-score corresponds to a score of 3 standard deviations below the updated mean and standard deviation?

CHALLENGING

Parker’s class achieved the following results in a Mathematics test: 29, 78, 55, 39, 96, 55, 74, 46, 63, 65, 49, 61, 57 and 73.

a What is the mean and standard deviation? Answer correct to one decimal place. b What is the z-score for the highest score? Answer correct to one decimal place. c What is the z-score for the lowest score? Answer correct to one decimal place. d Parker usually achieves a score two standard deviations above the mean. What is Parker’s z-score and expected mark?

Communicate your reasoning, understanding and fluency 12

Mehki’s class test had a mean of 65% with a standard deviation of 10%.

a What is the lowest mark that Mehki can achieve to be placed in the top 16% of students in his class? b Explain your answer using z-scores.

Explain why any complete set of z-scores for data arising from a random variable that is normally distributed has a mean of 0 and a standard deviation of 1.

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9D Converting z-scores into actual scores

9D Converting z-scores into actual scores Learning intentions

• To understand the relationship between z-scores and actual scores • To know how to determine an actual score from a z-score

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

Having learnt how to calculate z-scores, you also need to be able to convert z-scores back into actual scores. The rule for converting a z-score into an actual score involves making x the subject of the z-score formula. This is shown below. x−µ z= σ z×σ= x−µ x=µ+z×σ

In other words, the actual score can be calculated by adding the mean to the product of the z-score and the standard deviation.

Converting z-scores into actual scores

z-score is the number of standard deviations the score is from the mean. x−µ x = µ + z × σ or make x the subject of z = σ z = z-score or standardised score x = Score

µ = Mean of a set of scores

σ = Standard deviation

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Example 8

Converting a positive z-score into an actual score

In a normal distribution the mean was 30 and the standard deviation was 3.5. Quinn achieved a standardised score of 2. Calculate Quinn’s actual score by:

U N SA C O M R PL R E EC PA T E G D ES

a using the formula x = µ + z × σ x−µ b making x the subject of z = . σ Solution

Explanation

a x=µ+z×σ

Write the formula x = µ + z × σ. Substitute µ = 30, z = 2, σ = 3.5. Solve to find the score.

= 30 + 2 × 3.5

= 37

x−µ σ x − 30 2= 3.5 z=

x−µ . Write the formula z = σ Substitute µ = 30, z = 2, σ = 3.5. Solve to find the score.

2 × 3.5 = x − 30

x = (2 × 3.5) + 30

= 37 Quinn scored 37.

Now you try

In a normal distribution the mean was 50 and the standard deviation was 12. Norton achieved a standard score of 1.5. Calculate Norton’s score by:

a using the formula x = µ + z × σ x−µ b making x the subject of z = . σ

Example 9

Explaining z-scores and converting z-scores to actual scores

A class test had a mean of 34 and a standard deviation of 4. Jin had a z-score of −1.5 and Marli had a z-score of 2.5.

a Explain Jin’s z-score. b Explain Marli’s z-score. c Calculate Jin and Marli’s actual scores.

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9D Converting z-scores into actual scores

Explanation

a Jin’s z-score of −1.5 represents a low mark as it is 1.5 standard deviations below the mean.

A negative z-score represents a mark below the mean.

b Marli’s z-score of 2.5 represents an excellent mark as it is 2.5 standard deviations above the mean.

A positive z-score represents a mark above the mean.

c For Jin’s score x = µ + z × σ

Write the formula x = µ + z × σ. Substitute µ = 34, z = −1.5, σ = 4. Solve to find the score.

U N SA C O M R PL R E EC PA T E G D ES

Solution

x = 34 + (−1.5) × 4 x = 28

For Marli’s score x = 34 + 2.5 × 4 x = 34 + 2.5 × 4 x = 44

Write the formula x = µ + z × σ. Substitute µ = 34, z = 2.5, σ = 4. Solve to find the score.

Now you try

In a normal distribution the mean was 63 and the standard deviation was 7.5. Sun achieved a standard score of −1.

a Explain the meaning of Sun’s z-score of −1. b Calculate Sun’s actual score.

Exercise 9D

Example 8

BUILDING

Find the actual score by using the formula x = µ + z × σ.

a µ = 60, z = 2.0, and σ = 5 c µ = 50, z = −1.5, and σ = 8 e µ = 70, z = 3, and σ = 2 g µ = 80, z = −2.5, and σ = 6

The mean number of accidents at an intersection is 25 with a standard deviation of 8. Find the score that corresponds to each of the following z-scores. a z=1 e z=0 i z = −2.5

b µ = 60, z = −2.0, and σ = 5 d µ = 50, z = −1.5, and σ = 16 f µ = 70, z = −3, and σ = 2 h µ = 80, z = 0.5, and σ = 6

b z=3 f z = −2 j z = 0.5

c z = −1 g z = −3 k z = −0.25

d z=2 h z = 1.5 l z = 2.5

In a common test the mean was 60 and the standard deviation was 5. What score corresponds to the following standardised scores? a 1.4 e 0.9 i −2.7

b 2.8 f 2.1 j −2.5

c 3.1 g −0.6 k −0.8

d 0.2 h −1.3 l −1.8

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The concentration span on a reading task has a mean of 47 seconds and a standard deviation of 12 seconds. What is the concentration span on the following z-scores? b z=2 e z = 1.4 h z = −2 k z = −1.8

c z=3 f z = 2.6 i z = −3 l z = −2.5

U N SA C O M R PL R E EC PA T E G D ES

a z=1 d z = 0.8 g z = −1 j z = −0.3 5

Example 9

The mean time spent on the internet each week by students is 12 hours with standard deviation of 3 hours. The table shows the z-scores of hours spent on the internet by five students. Complete the table by finding the number of hours spent on the internet by each student. Answer correct to one decimal place.

Student

z-score

Alexis

1.5

Ben

2.2

Chris

0.9

Debbie

−1.6

Evan

−2.3

Hours

The monthly rainfall in Rock Waters has a mean of 55 mm and a standard deviation of 1.8 mm. What is the monthly rainfall if the standardised score was 3.2?

DEVELOPING

Issa gained a standardised score (z-score) of −3 for a fitness test.

a Describe Issa’s results in terms of the mean and standard deviation of the test. b The fitness test has a mean of 75 and a standard deviation of 7. What is the actual mark scored by Issa?

8 An assessment task is marked out of 20. The results of the test are shown below. 10 11 19 17 14 15 17 12

11 13

a Calculate the mean and standard deviation for these scores. Answer correct to one decimal place. b Panya scored the lowest mark in the task. What is his z-score? c Ren was absent for this task, but his mark is usually 1.5 standard deviations above the mean. What mark should his teacher give him?

An assessment task is marked out of 100. The results of the test are shown below. 75 80 65 66 79 83 61

a Calculate the mean and standard deviation for these scores. Answer correct to one decimal place. b Imara scored the highest mark in the task. What is her z-score? c Alma was absent for this task, but her mark is usually 1.9 standard deviations below the mean. What mark should her teacher give her?

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9D Converting z-scores into actual scores

CHALLENGING Ismael’s factory produces 200-gram packets of biscuits. In the past week the factory has produced packets with a mean weight of 201.5 grams and a standard deviation of 3 grams.

U N SA C O M R PL R E EC PA T E G D ES

a Quality control requires any packet with a z-score of less than −1 to be rejected. i What is the minimum weight that will be accepted? ii What is the probability that any given packet will be rejected for being too light?

b Packets of biscuits with a z-score greater than 3 are also rejected.

i What is the maximum weight that will be accepted? ii What is the probability that any given packet will be rejected for being too heavy?

c What is the probability that any packet chosen at random will be accepted?

Hugo was given his numeracy test result in the form of a z-score. He gained a z-score of 1.8.

a Explain the meaning of Hugo’s result in terms of the mean and standard deviation. b Calculate Hugo’s mark if the mean mark is 61 and the standard deviation is 7.2. Answer correct to the nearest whole mark.

Communicate your reasoning, understanding and fluency 12

All of the Year 10 cohort were given a test. The test results of one class are represented below as z-scores. −2.2, −2.1, −2.0, −1.8, −1.7, −1.5, −1.3, −1.3, −1.0, −1.0, −0.6, −0.4, 0, 0, 0.1, 0.2, 0.4, 0.6, 0.7, 0.9, 1.0

a Explain the meaning of a z-score of zero. b Did this class achieve well in this test? Explain. c If the mean was 55 and the standard deviation was 12, what was the highest and lowest mark (as a whole number)? d What was the median mark in this class?

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9E Using z-scores to compare data Learning intentions

• To be able to use z-scores to compare data values from different datasets Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

It is often difficult to compare two sets of data because they have different distributions. For example, if two tests have different means and standard deviations how do we compare a student’s marks in these tests? Using the z-score allows us to determine the number of standard deviations the mark is above or below the mean. The larger the z-score (ignoring the positive or negative) the further away it is from the centre of the data. It makes comparisons easier between two sets of data.

Using z-scores to compare data

z-scores are used to compare scores from different datasets.

1 Read the question carefully to determine whether a higher or lower z-score is better. 2 The larger the z-score (ignoring the positive or negative), the further away it is from the centre of the data.

Example 10

Comparing data using z-scores

David compares his swimming times in freestyle and butterfly. In freestyle, David swam 35 s and the club average was 43 s with a standard deviation of 4. In butterfly, David swam 37 s and the club average was 46 s with a standard deviation of 3. In which event did David do better, relative to other swimmers in his club? Justify your answer.

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9E Using z-scores to compare data

Solution

Explanation

Freestyle z = =

x−µ σ 35 − 43 4

U N SA C O M R PL R E EC PA T E G D ES

= −2 x−µ Butterfly z = σ

Calculate the z-score for freestyle. Write the z-score formula. Substitute x = 35, µ = 43 and σ = 4 into the formula.

37 − 46 3

= −3 Butterfly is the better result as it is further from the mean (z = −3) than the freestyle result (z = −2). Lower relative time.

Calculate the z-score for butterfly. Write the z-score formula. Substitute x = 37, µ = 46 and σ = 3 into the formula.

Justify the answer in words using the z-scores. The lower the relative time in swimming the better.

Now you try

Peter runs in the 100 m and 200 m races at his athletics club.

Peter ran 13.2 s for 100 m, with the club average being 14 s with a standard deviation of 0.4 s. Peter ran 22.8 s for 200 m, with the club average being 23.5 s with a standard deviation of 0.7 s. In which event did Peter do better in relation to other runners in his club? Justify your answer.

Example 11

Comparing data using z-scores

The mean and standard deviation of Algebra and Finance exams are shown in the table. Anisa scored 76 in Algebra and 74 in Finance.

a What is Anisa’s z-score for her result in Algebra? b What is Anisa’s z-score for her result in Finance? c Anisa was very pleased with her result in Algebra. Did she do better in Algebra or Finance, relative to the others who took the exams? Justify your answer. Solution

Topic

Mean

Standard deviation

Algebra

Finance

Explanation

x−µ σ 76 − 72 z= 8

a Algebra: z =

x−µ . σ Substitute x = 76, µ = 72, σ = 8. Solve to find the z-score. Write the formula z =

z = 0.5

Continued on next page

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x−µ σ 74 − 67 z= 10

b Finance z =

x−µ . σ Substitute x = 74, µ = 67, σ = 10. Write the formula z =

Solve to find the z-score.

z = 0.7 0.7 is further above the mean than 0.5. The Finance result is better than the Algebra result.

U N SA C O M R PL R E EC PA T E G D ES

c Anisa’s z-score of 0.7 is further above the mean than her z-score of 0.5. Anisa did better in Finance. Now you try

The mean and standard deviation of two tests are shown in the table below.

Topic

Standard deviation

Mean

Brooke scored 70 in both her Physics and Chemistry tests.

Physics 55 10 a What is Brooke’s z-score for her Physics test? Chemistry 65 5 b What is Brooke’s z-score for her Chemistry test? c Did Brooke do better relative to other students in Physics or Chemistry? Explain your answer.

Exercise 9E

Example 10

BUILDING

Tameeka’s z-scores obtained in her examinations for three subjects are shown below. English (z = 0.5) Maths (z = 1) Science (z = −2)

a Tameeka scored the mean mark in History. What is the z-score for History? b What is Tameeka’s best result in these examinations? c What is Tameeka’s worst result in these examinations? d Order Tameeka’s results from best to worst.

Example 11

2 The table shows the mean and standard deviation of the results in a Chinese and a Vietnamese exam. Hoang scored 81 in Chinese and 77 in Vietnamese. a What is Hoang’s z-score for his result in Chinese? b What is Hoang’s z-score for his result in Vietnamese? c In which of the two exams was his result better? Justify your answer.

The results of two class tests are normally distributed. The mean and standard deviation of the tests are displayed in the table. Diwa scored 71 in test 1 and 70 in test 2.

Subject

Mean

Standard deviation

Chinese

Vietnamese

Test 1

Test 2

Mean

Standard deviation

12.40

12.00

a What is Diwa’s z-score for her result in test 1? Answer correct to two decimal places. b What is Diwa’s z-score for her result in test 2? Answer correct to two decimal places. c Diwa thinks she has performed better in test 1. Do you agree? Justify your answer.

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9E Using z-scores to compare data

Carlos scored 82 in task 1 in which the mean was 72 and the standard deviation was 6. He scored 75 in task 2 in which the mean was 63 and the standard deviation was 9. In which task did Carlos perform better? Give reasons for your answer.

Harper scored 68% in her TAFE course for semesters 1 and 2. In semester 1, the class mean was 70% and the standard deviation was 4, and in semester 2 the class mean was 72% and the standard deviation was 6. Which semester was Harper’s better result? Justify your answer using appropriate calculations.

U N SA C O M R PL R E EC PA T E G D ES

Luca scored 75% in this Mathematics test (mean 57, standard deviation 11.5) while scoring 72% in Chemistry (mean 53, standard deviation 9.6). In which subject did he perform better? Justify your answer using appropriate calculations.

Himari has received her first two assessment results for this term. The details are shown below. Sports coaching: mark 66, mean 58.6, standard deviation 5.1 Modern history: mark 74, mean 67.1, standard deviation 8.1

a What is Himari’s z-score for her result in sports coaching? b What is Himari’s z-score for her result in modern history? c In which of the two assessment tasks was her result better? Justify your answer. d Himari’s best performing subject is usually economics. She achieved a mark of 82 in economics whose mean was 76 and standard deviation of 6.5. Is economics still the better performing subject compared to sports coaching and modern history?

The given table contains the scores a student obtained in the Trial HSC. Also shown is the mean and standard deviation for each subject.

Subject

Mark

Mean

Standard deviation

English

Mathematics

Biology 69 60 a Calculate their z-score for Business studies 75 60 each subject. Legal studies 55 44 b Use the z-score to rate their performance in each subject from the best performing subject to the worst performing subject.

4 5

c The Mathematics result was entered incorrectly. The mean was 62 not 82. What is the new z-score for Mathematics? d How does the above error affect the rating of Mathematics? Give reasons for your answer.

Theodore compared his recent marks in two subjects:

Construction: mark 77, mean 64.6, standard deviation 10.1 Legal studies: mark 74, mean

, standard deviation 12.5

a What is Theodore’s z-score (correct to two decimal places) for his mark in Construction? b Theodore achieved the same z-score in both Construction and Legal studies. What is the mean for Legal studies? Answer correct to one decimal place. c What mark does Theodore require in Construction to achieve a z-score of 2? d What mark does Theodore require in Legal studies to achieve a z-score of 2?

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DEVELOPING Livanshi’s assessment task results are recorded in the table below.

Topic

Mean

Standard deviation

Livanshi’s result

Task 1

a Standardise Livanshi’s results for each task. Answer correct to two decimal places.

Task 2

Task 3

Task 4

U N SA C O M R PL R E EC PA T E G D ES

b Rank her tasks in order of relative performance. c Describe Livanshi’s performance in these assessment tasks. d All the tasks are moderated so that the mean mark is 60 and the standard deviation is 12. What will be Livanshi’s moderated mark on these tasks if the z-score is maintained? Answer correct to the nearest whole number.

CHALLENGING

The scores of three students on their Multimedia and Technology projects are given below. The Multimedia mean was 63 and the standard deviation 15. The Technology mean was 58 and the standard deviation 11.

Student

Multimedia

Technology

Kushmeet

Irene

Katherine 88 66 a Convert Kushmeet’s Multimedia and Technology marks to z-scores. Which result was better in comparison to the other students in the course? b Convert Irene’s Multimedia and Technology marks to z-scores. Which result was better in comparison to the other students in the course? c Convert Katherine’s Multimedia and Technology marks to z-scores. Which result was better in comparison to the other students in the course? d Students on each assessment task are given a merit award if their result is one standard deviation above the mean. Which students would receive the award, and on which result would their award be based?

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9E Using z-scores to compare data

Communicate your reasoning, understanding and fluency Archer and his cousin Fran go to different schools. They both do Mathematics, History and Music. Archer tries to tell Fran he did better than her in all three subjects. However, Fran thinks that she did better than Archer in all three subjects even though her marks were lower. The given tables show all the marks.

U N SA C O M R PL R E EC PA T E G D ES

Topic

Mean

Standard deviation

Score in Exam

Archer

Mathematics

History

Music

Mean

Standard deviation

Score in Exam

Topic

Fran

Mathematics

History

Music

Show why Fran is correct and develop an explanation for each subject so she can convince Archer.

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9F Further applications of normal distributions Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To apply z-scores and the empirical rule • To use collected data from a normal distribution to make judgements • To use z-scores to identify probabilities of a given event Past, present and future learning

A normal distribution has the same mean, mode and median. It is symmetrical about the mean. The percentage of scores that lie within a certain number of standard deviations (or z-score) is always the same for a normal distribution. That is, 68% are within one standard deviation, 95% are within 2 standard deviations and 99.7% are within 3 standard deviations. z-score between 1 and −1

z-score between 2 and −2

68%

95%

99.7%

2.5%

16%

z-score between 3 and −3

2.5%

16%

0.15%

The empirical rule and z-scores In a normal distribution:

• 68% of scores have a z-score between 1 and −1 (data is mostly in this range) • 95% of scores have a z-score between 2 and −2 (data is very probably in this range) • 99.7% of scores have a z-score between 3 and −3 (data is almost certainly in this range). A bell-shaped curve represents a normal distribution. The z-scores on either side of the mean have the same percentage of the scores (data is symmetrical).

34%

0.15%

34%

13.5%

2.35%

−4

−3

−2

0.15%

13.5%

2.35%

−1

0 z-scores

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9F Further applications of normal distributions

Example 12

Applying the empirical rule with z-scores

The heights of children are normally distributed. The mean is 150 cm and the standard deviation is 5 cm. What percentage of children lie between 145 cm and 155 cm in height? Solution

Explanation For 155 cm: x−µ z= σ 155 − 150 = 5

= −1

Write the z-scores formula for each height.

U N SA C O M R PL R E EC PA T E G D ES

For 145 cm: x−µ z= σ 145 − 150 = 5

Substitute x = 145 (left) and x = 155 (right), µ = 150 and σ = 5 into the formulas.

Use the z-scores of −1 and 1 to determine the percentage of children within this range (68%).

The percentage of children in this range is 68%. Now you try

The length of pencils in a new pencil set is normally distributed. The mean is 17 cm and the standard deviation is 0.5 cm. What percentage of pencils lie between 16 cm and 18 cm?

Example 13

Applying the empirical rule using a bell curve

The mass of 700 jars of jam is normally distributed. The mean is 420 grams and the standard deviation is 3 grams.

a What percentage of the jars are: i more than 426 grams? ii less than 423 grams?

b What percentage of jars are predicted to have a weight between 414 and 429 grams? c If one jar is selected at random, what is the probability that it weighs more than 423 grams?

Solution

Explanation

a i

Draw a bell curve and add the data for this question. Highlight the sections that are required.

34%

0.15%

408

13.5%

34%

13.5%

2.35% 2.35% 411 414 417 420 423 426 429 z-scores of jam jars in grams

More than 426 g = 2.35 + 0.15

0.15%

432

Add the percentages for each section.

= 2.5% Continued on next page

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Draw a bell curve and add the data for this question. Highlight the sections that are required.

34% 34% 0.15%

13.5%

0.15%

2.35% 2.35% 411 414 417 420 423 426 429 z-scores of jam jars in grams

432

U N SA C O M R PL R E EC PA T E G D ES

408

Less than 423 g = 34 + 34 + 13.5 + 2.35 + 0.15

Add the percentages for each section.

= 84%

Draw a bell curve and add the data for this question.

34% 34%

0.15%

13.5%

2.35% 2.35% 411 414 417 420 423 426 429 z-scores of jam jars in grams

408

0.15%

432

Jars between 414 g and 429 g = 13.5 + 34 + 34 + 13.5 + 2.35 Add the percentages for each section. = 97.35%

Draw a bell curve and add the data for this question.

34%

0.15%

408

34%

13.5%

2.35% 2.35% 411 414 417 420 423 426 429 masses of jam jars in grams

Probability = 13.5% + 2.35% + 0.15% = 16%

0.15%

432

Treat the percentages as probabilities. Add the percentages for each section.

Now you try

The number of matches in a box is normally distributed with a mean of 100 and a standard deviation of 2.

a What percentage of boxes have: i more than 102 matches? ii less than 104 matches?

b What percentage of boxes have between 96 and 106 matches? c If one box is selected at random, what is the probability that it has more than 100 matches?

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9F Further applications of normal distributions

Exercise 9F 1

BUILDING

What percentage of scores in a normal distribution have a z-score: b between −2 and 2? e between −1 and 0? h between 0 and 3?

c between −3 and 3? f between 0 and 2? i between −3 and 0?

U N SA C O M R PL R E EC PA T E G D ES

a between −1 and 1? d between 0 and 1? g between −2 and 0? Example 12

A normal distribution of scores has a mean of 240 and a standard deviation of 20. What percentage of scores lie between: a 220 and 260?

c 180 and 300?

The lengths of timber are normally distributed with a mean of 1.80 m and a standard deviation of 0.05 m. What percentage of length of timber lies between: a 1.75 m and 1.85 m?

b 200 and 280?

b 1.70 m and 1.90 m?

c 1.65 m and 1.95 m?

The weights of boxes of chocolates are normally distributed. The mean is 748 grams and the standard deviation is 5 grams.

a A box of chocolates has a weight of 738 grams. What is the z-score? b What percentage of boxes will have a weight between 743 grams and 753 grams? c What percentage of boxes will have a weight between 738 grams and 758 grams?

The ages of the community members are normally distributed. The mean is 42 years and the standard deviation is 12 years. What percentage of the community members are aged between:

a 18 and 66 years? d 42 and 54 years? g 30 and 42 years?

b 30 and 54 years? e 42 and 66 years? h 18 and 42 years?

c 6 and 78 years? f 42 and 78 years? i 6 and 42 years?

An assessment task has a normal distribution with a mean of 72 and a standard deviation of 8. What percentage of scores lie between: a 48 and 96? d 48 and 72?

b 64 and 80? e 72 and 96?

c 56 and 88? f 64 and 72?

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Example 13

What percentage of scores in a normal distribution have a z-score: a greater than 1? d less than −1?

b greater than 2? e less than −2?

c greater than 3? f less than −3?

What percentage of scores in a normal distribution have a z-score: b between −1 and −2? e between −1 and 2?

c between 2 and 3? f between −1 and 3?

U N SA C O M R PL R E EC PA T E G D ES

a between 1 and 2? d between −3 and −1?

A set of scores with a normal distribution has a mean of 68 and a standard deviation of 8.

a Draw a diagram to represent this normal distribution including the appropriate percentages. b What percentage of scores lie: i above 68?

ii below 76?

iii above 84?

iv below 68?

c If a score was selected at random, what is the probability that it is: i below 84?

ii below 60?

iii below 52?

iv above 76?

Kumar is a company that produces packets of tea. The mass of the packets is normally distributed, with a mean of 200 grams and a standard deviation of 10 grams.

a In what range of mass should 68% of the packets lie? b In what range of mass should 95% of the packets lie? c If a packet of tea is randomly selected, what is the probability that it weighs more than 210 grams?

11 Nneka achieved a z-score of 2 in a general knowledge quiz. The results of the quiz are normally distributed. a Between which two z-scores will people sitting the quiz mostly lie? b Between which two z-scores will people sitting the quiz almost certainly lie? c Between which two z-scores will people sitting the quiz very probably lie?

Mahmoud sells a diet plan with a money-back guarantee if the customer does not lose 9 kg in the first three months. The weight loss after three months is normally distributed with a mean loss of 13.2 kg and a standard deviation of 1.4 kg.

a What is the z-score corresponding to a weight loss of 9 kg? b What percentage of customers will not reach the 9 kg in three months? c Mahmoud has sold 2000 diet plans. How many refunds are expected?

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9F Further applications of normal distributions

DEVELOPING 13

A CityRail train is timetabled to arrive at Central station at 8:44 am each day of the week. After 3 months the mean arrival time was 8:44 am with a standard deviation of 3 minutes. The data has been normally distributed.

U N SA C O M R PL R E EC PA T E G D ES

a Draw a diagram to represent this normal distribution including the appropriate percentages. b What percentage of times will the train arrive between 8:38 am and 8:50 am? c What percentage of times will the train arrive between 8:35 am and 8:47 am? d What percentage of times will the train arrive before 8:38 am? e What percentage of times will the train arrive after 8:47 am? f What percentage of times will the train arrive on time or earlier? g CityRail does not classify a train as late unless it arrives 6 minutes later than the scheduled time. What percentage of times will the train arrive late according to CityRail? h How often will this train run late according to CityRail, in 200 days? i How often will this train run late according to CityRail, if it runs every day of the year? j What is the probability that a person will arrive at Central station before 8:50 am?

The marks in an examination are expressed as a percentage and normally distributed with a mean of 60 and a standard deviation of 12.5.

a What are the boundaries between which you would mostly expect the marks to lie? b What are the boundaries between which you would very probably expect the marks to lie? c What are the boundaries between which you would almost certainly expect the marks to lie? d What percentage of students will get a mark of 85 or higher? e What percentage of students will get a mark of 35 or higher? f What percentage of students will get a mark of less than 72.5? g If there were 80 000 students who sat for this examination, how many students are expected to get a mark of 35 or higher? h If there were 60 000 students who sat for this examination, how many students are expected to get a mark of less than 72.5?

The mean height of an Irish wolfhound is 82 cm with a standard deviation of 1.5 cm. The data has been normally distributed.

a In what height range should 68% of Irish wolfhounds lie? b In what height range should 95% of Irish wolfhounds lie? c What height are 84% of Irish wolfhounds more than? d What height are 97.5% of Irish wolfhounds less than? e What percentage of Irish wolfhounds have a height greater than 86.5 cm? f What percentage of Irish wolfhounds have a height less than 80.5 cm? g If there are 800 Irish wolfhounds, how many Irish wolfhounds are expected to have a height of greater than 86.5 cm? h If there are 500 Irish wolfhounds, how many Irish wolfhounds are expected to have a height of less than 80.5 cm? i If you came across an Irish Wolfhound, what is the probability that they would be greater than 85 cm in height?

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CHALLENGING The normal distribution below represents the masses of 200 students. They have a mean of 70 and a standard deviation of 10.

U N SA C O M R PL R E EC PA T E G D ES

a What percentage of the students will have a mass that lies in the region marked with the letter A? b How many students will have a mass in the region marked with the letter A? c What percentage of the students will have a mass that lies in the region marked with the letter B? d How many students will have a mass in the region marked with the letter B? e How many students will have a mass between 50 and 60? f How many students will have a mass between 60 and 90? g How many students will have a mass greater than 50?

The maximum temperature for 600 summer days in Katoomba was recorded. The mean was 21.9◦ C and the standard deviation is 3.6◦ C. The maximum temperatures were normally distributed.

a What percentage of summer days in Katoomba would have maximum temperatures between: i 18.3◦ C and 25.5◦ C? iii 11.1◦ C and 29.1◦ C?

ii 29.1◦ C and 32.7◦ C? iv 14.7◦ C and 18.3◦ C?

b How many summer days will have a maximum temperature less than 18.3◦ C? c How many summer days will have a maximum temperature less than 14.7◦ C? d How many summer days will have a maximum temperature greater than 21.9◦ C? e How many summer days will have a maximum temperature greater than 29.1◦ C?

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Chapter 9 review

Normally distributed data

A normal distribution has the same mean, mode and median. The graph of a normal distribution is called a ‘bell curve’.

Review

Key ideas and chapter summary

• A normal distribution is symmetrical about the mean.

U N SA C O M R PL R E EC PA T E G D ES

• The mean, median and mode are equal. • The majority of the data is located at the centre.

• A normal distribution graph has the x-axis as an asymptote.

• Mean and standard deviation define a normal distribution.

The empirical rule

In a normal distribution:

• 68% of data lie within 1 standard deviation of the mean

• 95% of data lie within 2 standard deviations of the mean

• 99.7% of data lie within 3 standard deviations of the mean

• 50% of data lie above the mean and 50% of data lie below the mean.

z-score or standardised score

z-score is the number of standard deviations from the mean. x−µ µ: Mean of a set of scores z= z: z-score σ x: score σ: Standard deviation

Converting z-scores into actual scores

x = µ + z × σ or make x the subject of the above formula.

Using z-scores to compare data

z-scores are used to compare scores from different datasets.

• Read the question carefully to determine whether a higher or lower z-score

is better. • The larger the z-score (ignoring the positive or negative) the further away it is from the centre of the data.

Further applications of normal distributions

In a normal distribution:

• 68% of scores have a z-score mostly between 1 and −1

• 95% of scores have a z-score very probably between 2 and −2

• 99.7% of scores have a z-score almost certainly between 3 and −3.

A bell-shaped curve represents a normal distribution

−4

Probability using z-scores

34%

2.35%

0.15%

34%

2.35%

13.5%

−3

−2

−1

0.15%

13.5%

0 z-scores

In a normal distribution z-scores can be used to calculate the probability of an event occurring by using the empirical rule.

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Chapter checklist with success criteria A printable version of this checklist is available in the Interactive Textbook. Checklist

✓

1 I can draw a bell-shaped curve to represent a normally distributed dataset.

U N SA C O M R PL R E EC PA T E G D ES

e.g. The hand span measurements (in cm) of 13 people are given below. 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23

a Construct a frequency table. b Find the mean and the median. c Find the standard deviation, correct to one decimal place. d Draw a dot plot and a bell-shaped curve to show the distribution.

2 I understand the meaning of the empirical rule.

e.g. A normal distribution has scores with a mean of 70 and a standard deviation of 3. Complete the following. a 50% of scores lie below ____________. b 68% of scores lie between _______ and ______. c 95% of scores lie between _______ and ______. d 99.7% of scores lie between _______ and ______.

3 I can apply the empirical rule.

e.g. In a normal distribution, the mean recovery time after contracting a virus is 9 days with a standard deviation of 2 days.

a How long would it take for half the patients to recover? b What percentage of patients will recover in 5 to 13 days? c What percentage of patients will recover in 11 days? d If 1500 patients contract the virus, what percentage will recover in 3 days?

4 I can use diagrams to help me apply the empirical rule.

e.g. The distribution of delivery times for parcels from Quick Delivery Parcels is approximately normal, with a mean of 30 minutes and a standard deviation of 7 minutes.

a What percentage of parcels are delivered in between 23 and 37 minutes? b What percentage of parcels have delivery times less than 16 minutes? c Quick Delivery Parcels delivers 2500 parcels in one week. How many of these parcels take greater than 44 minutes to deliver?

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5 I can calculate the z-scores in a normal distribution. e.g. For a normal distribution, the mean is 72 and the standard deviation is 12. Find the z-score for a score of 66 and interpret the results.

Review

6 I can calculate standard scores in a normal distribution.

U N SA C O M R PL R E EC PA T E G D ES

e.g. The height of a group of giraffes has a mean of µ = 520 cm and σ = 12 cm.

Find the z-score for a giraffe that is: a 532 cm tall b 490 cm tall c 500 cm tall.

7 I can describe a z-score in a normal distribution. e.g. What is the z-score for a score that is: a 1 standard deviation above the mean? b 2 standard deviations below the mean? c equal to the mean?

8 I can convert a positive z-score to an actual score.

e.g. In a normal distribution the mean was 40 and the standard deviation was 7. Ahmed achieved a standard score of 1. Calculate Ahmed’s score by:

a using the formula x = µ + z × σ x−µ b making x the subject of z = . σ

9 I can explain z-scores and convert z-scores to actual scores.

e.g. In a normal distribution the mean was 80 and the standard deviation was 9. Sayid achieved a standard score of −2.

a Explain the meaning of Sayid’s z-score of −2. b Calculate Sayid’s score.

10 I can compare data using z-scores.

e.g. Yvette runs in the 400 m and 1500 m races at her athletics club.

Yvette ran 71 s for 400 m, with the club average being 77 s with a standard deviation of 4 s.

Yvette ran 5 min 30 s for 1500 m, with the club average being 6 min with a standard deviation of 30 s.

In which event did Yvette do better in relation to other runners in her club? Justify your answer.

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11 I can apply the empirical rule with z-scores. e.g. The height of bonsai trees in a shop is normally distributed. The mean is 30 cm and the standard deviation is 2 cm. What percentage of bonsai trees lie between 28 cm and 32 cm?

12 I can apply the empirical rule using a bell curve.

U N SA C O M R PL R E EC PA T E G D ES

e.g. The number of marbles in a bag is normally distributed with a mean of 125 and a standard deviation of 3. Model each question using a bell curve. a

What percentage of bags have: i more than 131 marbles? ii less than 128 marbles?

What percentage of marble bags have between 119 and 134 marbles?

13 I can calculate the probability of an event occurring using z-scores and the empirical rule.

e.g. The number of lollies in a jar is normally distributed with a mean of 600 and a standard deviation of 6. a Draw a bell curve to model this distribution. b If a jar is chosen at random, what is the probability that it will have more than 606 lollies?

c What is the probability that a jar chosen at random will have a z-score of −1 or less?

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Review

Multiple-choice questions 9A

A normal distribution is symmetrical about the: A mean C maximum

The ages of the citizens who live in Bishop are normally distributed. The mean age is 53 years and the standard deviation is 14. What percentage of the citizens are younger than 67?

U N SA C O M R PL R E EC PA T E G D ES

B standard deviation D z-score

A 34% C 84%

B 68% D 95%

3 Hua scored 76 in a science test. The science test had a mean of 60 and a standard deviation of 8. A recent history test had a mean of 56 and a standard deviation of 12. What mark in the history test would have been equivalent to Hua’s science mark? A 68 C 76

B 72 D 80

4 What is the z-score if x = 20, x̄ = 16 and s = 4? A −2 C 1

B −1 D 2

A student’s mark on a test is 50. The mean mark for their class is 55 and the standard deviation is 2.5. Their standard score is: A −2.0 C 2.0

B −1.5 D 2.5

Chase’s class achieved the following results in a topic test: 44, 91, 78, 55, 65, 49, 61, 55, 74, 46, 73, 63, 57 and 29. Using the standard deviation, how many scores have a z-score greater than 2? A 0 C 2

Merindah scored 80 in a maths test. The test mean was 70 and the standard deviation was 5. This test is being rescaled to a mean of 60 and a standard deviation of 8 without changing Merindah’s z-score. What is Merindah’s new score? A 64%

B 70%

C 76%

D 80%

Suresh achieved a z-score of 2.5 on his last assessment task. The class mean was 58% and standard deviation 6%. What was Suresh’s mark? A 60.5%

B 1 D 3

B 61%

C 67%

D 73%

9 What percentage of scores in a normal distribution has a z-score between 0 and 3? A 0.15%

9F 10

B 0.30%

C 49.85%

D 97%

The results of a literacy exam were normally distributed. Keeley gained a z-score of −1. What percentage of students scored better than Keeley? A 16%

B 34%

C 68%

D 84%

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Short-answer questions 9B

A normal distribution of scores has a mean of 136.5 and a standard deviation of 18.9. In what range of values would you expect to find: a 68% of the scores?

c 99.7% of the scores?

2 A normally distributed set of data has a mean of 9.9 and a standard deviation of 0.8. Find the percentage of data that lies between these ranges.

U N SA C O M R PL R E EC PA T E G D ES

b 95% of the scores?

a 9.1 and 10.7 d 11.5 and 12.3

b 7.5 and 12.3 e 8.3 and 9.1

c 8.3 and 11.5 f 9.9 and 12.3

4 In a university exam, the mean mark was 59%. Addison scored 68% in this exam and it gave her a z-score of 1.5. What is the standard deviation of this exam?

5 A set of scores has a mean of 20 and a standard deviation of 3.2. Find the z-score for each of the following scores.

Nick scored 72 in an assessment task. The mean for this task was 65 with a standard deviation of 14. What is Nick’s z-score?

a 26.4

b 15.2

c 27.2

d 16.8

6 Lenny’s mark on his exam paper was converted to a z-score of −1.5. The mean of this exam paper was 74 and the standard deviation was 4. What was Lenny’s exam mark?

7 A quiz has these results: 11, 15, 19, 12, 9, 16, 14, 10, 13, 7, 17, 13, 10, 18, 13 and 16.

a Calculate the mean and standard deviation for these marks. Answer correct to two decimal places. b Jade scored the lowest mark in the class. What is her z-score? Answer correct to one decimal place. c Lani was absent for this quiz. She usually scores 2.3 standard deviations above the mean. What mark would you expect Lani to achieve on this quiz?

Lily scored 78 in English, 75 in Mathematics and 77 in Science. The class mean and standard deviations are shown in the table below.

Subject

Mean

Standard deviation

English

Mathematics

a What was Lily’s z-score for English? Science 68 6 b What was Lily’s z-score for Maths? c What was Lily’s z-score for Science? d What was Lily’s best result? e The Science result is moderated so that the new mean is 72 and standard deviation is 8. What will be Lily’s mark for Science if she maintains the same z-score?

9 Kai scored a mark of 74 in a literacy test whose mean was 67.6 and standard deviation was 5.1. In a numeracy test with a mean of 74.1 and standard deviation of 8.1, he scored 80. On which test did Kai perform better? Justify your answer with appropriate calculations.

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Boxes of breakfast cereal are labelled as having a mass of 2 kg. The mass of these boxes is normally distributed with a mean of 2.02 kg and a standard deviation of 0.02 kg. What percentage of the boxes are:

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a greater than 2.06 kg? b less than 1.98 kg? c between 1.96 kg and 2.04 kg? d What is the probability that a box of breakfast cereal chosen at random is greater than 1.96 kg?

Review

9F 10

9F 11 The systolic blood pressures of 200 people are found to be normally distributed with a mean of 110 and a standard deviation of 5.

a Find the number of people whose systolic blood pressure is between 100 and 120. b Find the number of people whose systolic blood pressure is between 105 and 120. c How many people have a systolic blood pressure less than 100? d How many people have a systolic blood pressure more than 115? e What is the probability that a person chosen from this group of 200 has a systolic blood pressure greater than 125?

9F 12 The body weights of pets in Ramsey Street are approximately normally distributed and have a mean of 54 kg and a standard deviation of 10 kg. a Determine the standardised mass of a box that weighs 56 kg. b Another box a standardised weight of −0.75. Determine its actual mass.

Assuming the masses of the boxes are approximately normally distributed with a mean of 54 kg and a standard deviation of 10 kg, determine the: c percentage of these boxes that weigh more than 74 kg d percentage of these boxes that weigh between 54 and 64 kg. e A box is chosen at random. What is the probability that the box will weigh less than 54 kg?

Extended-response questions

A factory makes car pistons with an average diameter of 70 mm and standard deviation of 0.002 mm. The diameter of the pistons is normally distributed.

a In what range of diameters should 99.7% of the pistons lie? b What percentage of pistons will have a diameter greater than 70.002 mm? c What percentage of pistons will have a diameter less than 69.994 mm? d If the factory made 600 pistons, how many would you expect to be undersized by at least 0.004 mm? e What is the probability that a piston chosen at random has a diameter less than 69.998 mm? f What is the probability that a chosen at piston random has a diameter between 70.004 mm and 70.006 mm?

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U N SA C O M R PL R E EC PA T E G D ES

Network flow and critical path analysis

When is critical path analysis used?

Critical path analysis (CPA) is used where timing and coordination are crucial. Some examples include:

• Construction projects: in sequencing all phases of construction. • NASA for their Apollo Missions: CPA helped ensure that delays in any task on the critical path didn’t jeopardise the mission timeline. • Event planning: CPA assists in booking venues, sending out invitations and managing logistics.

• Film and media production: CPA helps manage the different stages of film production from financing, pre- and post-production, distribution & marketing. CPA helps to optimise shooting schedules and minimise potential delays. • Research and development in drug trials and product testing: CPA helps coordinate phases like lab testing, compliance checks and reporting.

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Chapter contents

U N SA C O M R PL R E EC PA T E G D ES

10A Network flow problems 10B Introduction to critical path analysis 10C Constructing a network diagram given a precedence table 10D Dummy activities 10E Forward and backward scanning 10F Float times and the critical path 10G Minimum completion time 10H Gantt charts

NSW Syllabus

In this chapter, a student:

• Develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly (MAO-WM-01) • Applies network flow concepts to solve problems and model decision-making in practical contexts (MST-12-S2-06) • Applies critical path analysis to solve problems and model decision-making in practical contexts (MST-12-S2-07). © NSW Education Standards Authority

Online resources

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10A Network flow problems Learning intentions

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• To define and use network flow terminology including source, sink, outflow, inflow and cut • To be able to convert information presented in a table to a weighted and directed network diagram • To determine the flow capacity of a network • To solve small-scale network flow problems • To understand and use the maximum-flow minimum-cut theorem to solve problems Past, present and future learning

• The concepts in this section will be new to many students • These concepts build upon prior learning from Year 11 • Questions involving network flow regularly appear in HSC Examinations

One of the applications of directed graphs to real-life situations is network flow problems. These involve the transfer or flow of material from one point, called the source, to another point called the sink. Examples of this include water flowing through pipes, or traffic flowing along one-way roads. source → flow through network → sink

The source is like the start, and the sink is like the finish, in the critical path analysis section.

Water flows through pipes in only one direction. In a directed graph representing water flow, the vertices are the origin and destination of the water and the edges represent the pipes connecting them. The weights on the edges would be the amount of water that can flow through the pipe in a given time. The weights of flow problem directed graphs are called capacities. The diagram on the right shows two pipes that are joined together, connecting the source of water to the sink. There is a small pipe with capacity 25 litres per minute joined to a large pipe with capacity 58 litres per minute. When an edge is at its maximum flow we say it is saturated.

58 L/min

25 L/min

Even though the large pipe has a capacity greater than 25 litres per minute, the small pipe will only allow 25 litres of water through each minute. The flow through the large pipe will never be more than 25 litres per minute. The large pipe will experience flow below its capacity. We say that it has an excess flow capacity. If we reverse the connection and direct water through the large capacity pipe into the smaller capacity pipe, there will be a ‘bottleneck’ of flow at the junction.

58 L/min

25 L/min

The large capacity pipe can deliver up to 58 litres of water every minute to the small pipe, but the small pipe will only allow up to 25 litres per minute to pass.

In both of these situations, the flow through the entire pipe system (both pipes from source to sink) is restricted to a maximum of 25 litres per minute. This is the capacity of the smallest pipe in the connection.

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10A Network flow problems

If we connect more pipes together, one after the other, we can calculate the overall flow capacity or maximum flow of the pipe system by looking for the smallest capacity pipe in that system.

U N SA C O M R PL R E EC PA T E G D ES

Maximum flow

If pipes of different capacities are connected one after the other, the maximum flow (or flow capacity) through the pipes is equal to the maximum capacity of the individual smallest pipe.

Saturated Edges An edge is saturated if:

• the flow through the edge equals the capacity of the edge • the edge is at its maximum flow.

The source is the starting vertex of a weighted and directed network. The sink is the finishing vertex of a weighted and directed network. A flow diagram is the weighted and directed network.

The maximum flow from source to sink is called the flow capacity. Inflow and Outflow

The inflow capacity is the total weight of edges arriving at a vertex. The outflow capacity is the total weight of edges leaving a vertex. The maximum outflow from a vertex cannot exceed the inflow, except at the source.

150

100

Vertex A has an inflow of 80 and a maximum outflow of 80

120

How to analyse and calculate the flow capacity of a network Consider this weighted and directed network:

200

150

100

150

C 60 Diagram 1

To calculate the flow through a network, you need to consider all paths from the source, S to the sink, T .

Consider path S–A–T from Diagram 1

• The capacity of this path is 150 because it is the minimum

weighting on this path. • Subtract 150 from S–A to show an excess flow capacity of 50. • Subtract 150 from A–T to show no excess flow capacity, it is saturated.

50 200 S

150 T

100 150

100 B

C 60 Diagram 2

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Consider path S–A–C–T from Diagram 2 • The capacity of this path is 50 because it is the minimum

150

200 50 100

S 100 B

C 60 Diagram 3

T 100 150

U N SA C O M R PL R E EC PA T E G D ES

weighting on this path. • Subtract 50 from S–A to show no excess flow capacity, it is now saturated. • Subtract 50 from A–C to show an excess flow capacity of 50. • Subtract 50 from C–T to show an excess flow capacity of 100.

Consider path S–B–C–T from Diagram 3

• The capacity of this path is 60 because it is the minimum

50 150 weighting on this path. 200 T • Subtract 60 from S–B to show an excess flow capacity of 40. 50 S 40 100 • Subtract 60 from B–C to show no excess flow capacity, it is 100 40 150 100 saturated. C 60 B • Subtract 60 from C–T to show an excess flow capacity of 40 0 There are no more paths to consider. Diagram 4 The total flow capacity through the network from S to T is 150 + 50 + 60 = 260. This is also the maximum flow from S to T .

Example 1

Calculating the maximum flow

A weighted and directed network diagram is shown.

Calculate the maximum flow through the network. The source is vertex S, and the sink is vertex T .

20 22

Solution

Explanation

Consider all paths from S to T and calculate the maximum flow through each path.

20 22

S –A–T is 12

Consider path S –A–T . The capacity of this path is 12. Subtract 12 from that path, to show an excess flow capacity S –A = 18, A–T = 0. A–T is now saturated.

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10A Network flow problems

0 12

30 18

15 B 5

20 22

Consider path S –A–B–T . The capacity of this path is 15. Subtract 15 from this path, to show an excess flow capacity S –A is now 18 − 15 = 3, A − B = 0. A–B is now saturated. B−T = 5

U N SA C O M R PL R E EC PA T E G D ES

S –A–B–T is 15

13 18

5 0

Consider path S –B–T . The capacity of this path is 5 (using B − T = 5 from above). Subtract 5 from this path, to show an excess flow capacity S − B = 13, B − T = 0. B–T is now saturated.

S –B–T is 5

13 18

0 18

Consider path S –C–T . The capacity of this path is 18. Subtract 18 from this path, to show an excess flow capacity C − T = 4, S − C = 0. This is saturated.

S –C–T is 18

Maximum flow = 12 + 15 + 5 + 18

Maximum flow is 50 Now you try

Calculate the maximum flow through the network. The source is vertex S, and the sink is vertex T . A

10 B

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Chapter 10 Network flow and critical path analysis

Cuts It is difficult to determine the maximum flow by inspection for directed networks that involve many vertices and edges. We can simplify the search for maximum flow by searching for cuts within the directed graph.

1000

700

400

U N SA C O M R PL R E EC PA T E G D ES

A cut divides the network into two parts, completely separating the source from the sink. It is helpful to think of cuts as imaginary breaks within the network that completely block the flow through that network. For the network or water pipes shown in this diagram, the dotted line is a cut. This cut completely blocks the flow of water from the source (S) to the sink (A).

1500

600

1200

Cut

Not a cut

The dotted line on the graph below is a cut because it separates the source and the sink completely. No material can flow from the source to the sink.

The dotted line on the graph below is not a cut because material can still flow from the source to the sink. Not all of the pathways from source to sink have been blocked by the cut.

source

sink

Cuts

• A cut is an imaginary line across a directed graph that completely separates the source (start of the flow) from the sink (destination of the flow). • Cut capacity is the sum of all the capacities of the edges that the cut passes through, taking into account the direction of flow. The capacity of an edge is only counted if it flows from the source side to the sink side of the cut. • A cut is like a snapshot across the system of the flow at that time.

In the simple network shown, the cut passes through three edges. The edge B to A is not counted in the capacity of the cut because the flow for that edge is from the sink side to the source side of the cut. A

source

sink

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10A Network flow problems

Example 2

Calculating the cut capacity

Calculate the capacity of the four cuts shown in the network on the right. The source is vertex S and the sink is vertex T.

15 B S

10 T

15 12

20 F C2

20 C3

U N SA C O M R PL R E EC PA T E G D ES

Solution

Explanation

The capacity of C1 = 15 + 20 = 35. The capacity of C2 = 14 + 20 = 34. The capacity of C3 = 14 + 15 + 20 = 49. The capacity of C4 = 20 + 10 = 30.

1 All edges in C1 are counted. 2 In C2 the edge from F to B is not counted. 3 All edges in C3 are counted. 4 In C4 the edge from D to C is not counted.

Now you try

Calculate the capacity of the four cuts shown in the network on the right. The source is vertex S and the sink is vertex T .

C3 C4

Maximum-flow minimum-cut theorem

The capacity of a cut is important to help determine the maximum flow through any directed graph. Look for the smallest, or minimum, cut capacity that exists in the graph. This will be the same as the maximum flow that is possible through that graph. This is known as the maximum-flow minimum-cut theorem. A

150

100

100 50

100 40

Minimum cut is 260

100

150

50 200

150 100 40

60 C 0 Maximum ﬂow is 260 B

The minimum cut coincides with edges that have a flow capacity equal to their maximum capacity. The minimum cut only passes through edges that are saturated, with no excess flow capacity.

Maximum-flow minimum-cut theorem The flow through a network cannot exceed the value of any cut in the network and the maximum flow equals the value of the minimum cut.

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Example 3

Drawing a weighted and directed network diagram to calculate its maximum flow

The tables show the flow capacities (FC) of the edges of a network. Determine the maximum flow from S to T. To

From

U N SA C O M R PL R E EC PA T E G D ES

From S

Solution

Explanation

Construct a network graph using the values in the cells as the flow capacity of the edges.

S–A–T

Calculate the flow through the network, consider all paths from S to T. Identify the minimum cut (red dotted line).

S–C–B–T 3 S–C–T

Minimum cut capacity = 3 + 3 + 1 = 7. Maximum flow = 3 + 3 + 1 = 7. Therefore the maximum flow is 7.

Find the minimum cut capacity. Find the maximum flow capacity. Write the answer in words. Note: The maximum flow must equal the minimum cut.

Now you try

The table shows the flow capacities (FC) of the edges of a network. Determine the maximum flow from S to T . From

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10A Network flow problems

Exercise 10A Example 1

BUILDING

Find the maximum flow for each of the following network diagrams. The source is vertex S, and the sink is vertex T. a

U N SA C O M R PL R E EC PA T E G D ES

9 T

Example 2

Find the capacity of each of the cuts in the following network diagrams. The source is vertex S and the sink is vertex T. a

C1 5

C1 2

C4 T

C2 C3

C C4 3 C5 6 T 7

5 7

3 Find the maximum flow for each of the following network diagrams. The source is vertex S and the sink is vertex T. Check your answer by finding the minimum cut. a

2 C

S 4

7 B

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10 S

4 8

12 3

U N SA C O M R PL R E EC PA T E G D ES

D 4

DEVELOPING

Example 3

The table given shows the flow capacities of the edges of a network.

a Construct a network diagram showing the direction and quantity of the flow. The source is vertex A and the sink is vertex E. b Find the maximum flow through the network. c Check part b by drawing the minimum cut.

From

Capacity

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10A Network flow problems

The table given shows the flow capacities of the edges of a network.

Capacity

U N SA C O M R PL R E EC PA T E G D ES

a Construct a network diagram showing the direction and quantity of the flow. The source is vertex C and the sink is vertex H. b Find the maximum flow through the network. c Check part b by drawing the minimum cut through the network.

From

The table given shows the flow capacities of the edges of a network.

a Construct a network diagram showing the direction and quantity of the flow. The source is vertex A and the sink is vertex H. b Find the maximum flow through the network. c Check the minimum flow from part b by finding the minimum cut through the network.

The table given shows the flow capacities of the edges of a network.

a Construct a network diagram showing the direction and quantity of the flow. The source is vertex P and the sink is vertex V. b Find the maximum flow through the network. c Check the maximum flow from part b by finding the minimum cut through the network.

From

Capacity

From

Capacity

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CHALLENGING A plane journey consists of a connected sequence of stages formed by edges on the following directed network from city A to city B. The number of available seats for each stage is indicated beside the corresponding edge, as shown on this diagram.

9 C1 3

C4 7 8

City A

1 8

4 City B

7 1 4

U N SA C O M R PL R E EC PA T E G D ES

4 C5

a What is the capacity of each of the five cuts shown? b Explain why cut 5 is not a valid cut when trying to find the minimum cut between city A and city B. c Find the maximum number of available seats for a plane journey from city A to city B. d Find one example of the maximum number of available seats for a plane journey from city A to city B.

In the directed graph given, the vertices A, B, C, D and E represent towns. The edges of the graph represent roads and the weights of these edges are the maximum number of cars that can travel on the road each hour. The roads allow only one-way travel.

600

800

300

500

150

a What is the outflow from vertex B? b Find the maximum traffic flow from A to E. c A new road is being built to allow traffic from town D to town C. This road can carry 500 cars per hour. i Add this road to the directed graph. ii Find the maximum traffic flow from A to E after this road is built and explain the impact of adding this new road to the network.

d During peak hour traffic, up to 600 cars per hour travel on these roads from A to E. What changes would you make, if any, to the original network above to meet the demand of traffic flow during peak hour? Explain your answer.

10 The network diagram given shows the number of litres of water that can flow from a water tank S, through different parts of a farm. C

150

600

300

400

200

100

150

a What is the outflow from vertex C? b Find the maximum flow of this network. c Draw the minimum cut. d Only one of the pipes in the network will be replaced with a pipe that can carry more water. Explain which pipe you recommend being upgraded to increase the maximum flow from the source S to the sink T , and by how much should its capacity be increased?

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10A Network flow problems

The network diagram given shows the walking tracks (edges) through a wildlife park. The walking tracks are one-way paths through breeding grounds showing the number of people permitted to walk along each track each day. The vertices (B, C, E, F, G) represent information hubs. A is the Entrance and D is the Exit to the wildlife park. C

U N SA C O M R PL R E EC PA T E G D ES

a What is the outflow from vertex C? b Find the maximum number of people permitted to walk through the park each day. c Draw the minimum cut. d Track E-B has been temporarily closed. Describe how this impacts the flow capacity. e One day, all available walking tracks will be used by students on a school excursion. If there are 50 students, which walking tracks could be increased to allow a maximum of 50 students through the wildlife park?

Communicate your reasoning, understanding and fluency 12

This directed graph shows vertices A, B, C and D representing intersections. The edges represent roads showing the number of cars per hour (cars/h) that can travel from S (source) to T (sink), through the intersections. The roads allow only one-way travel. 250

400

100

250

350

a What is the outflow from vertex B? b Determine the maximum number of cars per hour that can travel through the network. c Draw the minimum cut. d Choose one road that you would increase in capacity to maximise the flow from S to T . Explain the impact of your choice. e Road C–D was added to the above network, carrying 200 cars/h. Explain the impact this road will have on the network. f During peak hour traffic, up to 400 cars per hour travel on these roads from S to T . What two changes would you make to the original network above to meet the demand of traffic flow during peak hour? Explain the impact these changes will have on the traffic flow.

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Chapter 10 Network flow and critical path analysis

10B Introduction to critical path analysis Learning intentions

• To understand and use precedence tables (also known as activity charts) to represent projects • To be able to complete a precedence table

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

• The concepts in this section will be new to many students • These concepts build upon prior learning from Year 11 • Questions involving critical path analysis regularly appear in HSC Examinations

This chapter will develop your awareness of critical path analysis. Critical path analysis is a technique that identifies the activities required to complete a project or task, including identifying the time necessary to complete each activity and the relationships between the activities. Critical path analysis helps to predict if a project will finish on time. For example, building a house, manufacturing a product, organising a wedding and other similar projects all require many individual activities to be completed before the project is finished. The individual activities often rely upon each other and some can’t be performed until other activities are complete. In the organisation of a wedding, invitations would be sent out to guests, but a plan for seating people at the tables during the reception can’t be completed until the invitations are accepted. When building a house, the plastering of the walls can’t begin until the house is sealed from the weather.

Critical path analysis

Critical path analysis is a technique that identifies the activities required to complete a project, the time necessary to complete each activity and the relationships between the activities.

Precedence table

A precedence table or an activity chart is used to describe the activities in a project. It consists of a table containing the activities of a project, their duration and their immediate predecessors. Let’s consider some simple projects and their precedence table. Project: To make a phone call. Activity

Task

Duration (in min)

Immediate predecessors

Look up the number

−

Dial the number and wait for answer

Deliver the message

Hang up the phone

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10B Introduction to critical path analysis

Project: To make old-style lemonade. Activity

Task

Duration (in min)

Immediate predecessors

Fill a tall glass with cracked ice

−

Add lemon juice and sugar

Stir well

U N SA C O M R PL R E EC PA T E G D ES

Top with water

Add a slice of lemon

Serve with drinking straws

Precedence table

A table containing the activities of a project, their duration and their immediate predecessors.

Immediate predecessors and scheduling

For any project, if activity A must be completed before activity B can begin then activity A is said to be an immediate predecessor of activity B. The activities within a project can have multiple immediate predecessors. A time schedule indicates the length of time or duration of each activity. Allocating time to the completion of activities in a project is called scheduling. It shows the immediate predecessors and duration for each activity. Notice that checking email can only be completed after Sandra gets out of bed, has breakfast, and gets dressed. Activity

Example 4

Task

Duration (in min)

Immediate predecessors

Get out of bed

−

Cook breakfast

Eat breakfast

Brush teeth

Have shower

Get dressed

Check email

C, F

Constructing a precedence table

Project: To catch a train.

The tasks shown below are to catch a train. However, they are in the wrong sequence. Wait for train (8 minutes)

Go to the correct platform (2 minutes) Scan your Opal card (1 minute) Board the train (1 minute) Travel to the train station (5 minutes)

Correctly sort the above tasks and construct a precedence table for this project with the four headings: Activity, Task, Duration and Immediate predecessors. Continued on next page

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Solution

Explanation

Correct sequence of tasks: Travel to the train station, scan your Opal card, go to the correct platform, wait for train and board the train.

Analyse the tasks and sort into a logical sequence.

Duration Immediate in min predecessors

Draw a precedence table with 4 columns.

Activity Task

Label the 5 activities (A to E).

U N SA C O M R PL R E EC PA T E G D ES

530

Travel to train station

−

Write the sorted tasks in the task column.

Scan your Opal card

Record the duration for each activity.

Go to platform

Determine the predecessor for each activity.

Wait for train

Board the train

Now you try

Project: To bake a cake.

The tasks shown below are to bake a cake. However, they are in the wrong sequence. Mix the icing (4 minutes)

Cool the cake (30 minutes)

Bake the cake (35 minutes)

Put cake mixture in cake tin (3 minutes) Heat oven (10 minutes)

Mix ingredients (6 minutes) Ice the cake (8 minutes)

Correctly sort the above tasks and construct a precedence table for this project with the four headings: Activity, Task, Duration and Immediate predecessors.

Example 5

Reading a precedence table

Joshua wants to create a patio in his garden. The precedence table for this project is shown below. Activity

Task

Duration (in hours)

Immediate predecessors

Clean up garden

−

Measure area of patio

−

Plan the patio

Buy fencing

Buy plants and pots

A, C

Insert plants in pots

Buy paving

Make garden and patio

D, F, G

Note: Activity A (clean up garden) is an immediate predecessor of activity E (buy plants and pots) as it determines the number of plants required to fill the space. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

10B Introduction to critical path analysis

U N SA C O M R PL R E EC PA T E G D ES

a How many activities in the project? b What are the first activities? c Which activity has the longest time interval? d What is the total time for Joshua to complete this project? e What is the immediate predecessor for activity C? f What is the immediate predecessor for activity G? g Why are there three immediate predecessors for activity H? Solution

Explanation

a There are 8 activities. b A and B are the first activities.

Count the number of activities (A to H). List the activities with no immediate predecessor. Find the largest number in the duration column. Add the time listed in the duration column. List the activity in the predecessor column. List the activity in the predecessor column. Look for the 3 immediate predecessors of activity H.

c Making the garden and patio (H) takes 5 hours.

d Total time = 3 + 1 + 2 + 1 + · · · + 5 = 17 h e Immediate predecessor of C is B. f Immediate predecessor of G is C. g Activity H cannot be completed until Joshua buys the materials (fencing, plants, pots and paving). Now you try

Jasper wants to tile his balcony. The precedence table for this project is shown below. Activity

Task

Duration (in hours)

Immediate predecessors

Clean up balcony

−

Prepare the surface

Measure area of balcony

Plan the tiling

Buy waterproofing

Buy Tiles, glue and grout

C, D

Install waterproofing

Tile balcony

F, G

Grout tiles

Final Clean

a How many activities in the project? b What is the first activity? c Which activity has the longest time interval? d What is the total time for Jasper to complete this project? e What is the immediate predecessor for activity D? f What is the immediate predecessors for activity H?

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Exercise 10B 1

BUILDING

Copy and complete the following sentences.

U N SA C O M R PL R E EC PA T E G D ES

a Critical path analysis identifies the _________ required to complete a project. b Precedence table contains the activities, _________ and their immediate predecessors. c If activity A must be completed before activity B can begin then activity A is said to be an _________ of activity B. d Allocating time to the completion of activities in a project is called _________.

Example 5

An incomplete precedence table for washing your hands is shown. Activity

Task

Duration (in min)

Immediate predecessors

Turn on water

−

Wet hands

Clean hands with soap

Rinse off soap

Dry hands

a Copy and complete the precedence table shown. b How many activities in the project? c Which activity has the longest time interval? d What is the total time for this project? e What is the immediate predecessor for activity C? f What activities precede activity B?

3 An incomplete precedence table for preparing a meal is shown. Activity A

Task

Duration (in min)

Immediate predecessors

Find recipe

Prepare ingredients

Cook meal

Serve meal

Clean up

a Copy and complete the precedence table shown. b How many activities in the project? c Which activity has the shortest time interval? d What is the total time for this project?

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10B Introduction to critical path analysis

Example 4

The tasks for each of the following projects have errors in their sequence. Find these errors and construct a precedence table for each project with the four headings: Activity, Task, Duration and Immediate predecessor.

U N SA C O M R PL R E EC PA T E G D ES

a Project: To watch your favourite television show Turn television off (1 minute) Turn television on (1 minute) Find remote control (2 minutes) Watch television show (30 minutes) Select television show (1 minute) b Project: To listen to a music playlist Select play (1 minute) Turn off phone (1 minute) Unlock phone (1 minute) Select the music app (1 minute) Turn on phone (2 minutes) Select playlist (3 minutes) Listen to music (50 minutes)

c Project: To plant a tree in the garden Water the tree (5 minutes) Decide on the tree required (10 minutes) Purchase the tree (8 minutes) Dig the hole (12 minutes) Plant the tree (5 minutes) Mulch the tree (3 minutes)

d Project: To install a smoke alarm Select a section of the ceiling (2 minutes) Screw the mounting plate into the ceiling (5 minutes) Install the batteries into the alarm (3 minutes) Remove the mounting plate from the alarm (2 minutes) Choose a room (2 minutes) Connect the alarm body to the mounting plate (4 minutes)

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DEVELOPING Artem is required to organise a birthday party. He has estimated the following time for these tasks: buying the food – 2 hours; preparing the food – 3 hours; serving the meal – 1 hour; cleaning up – 2 hours. Construct a precedence table for this project.

Christina needs her new laptop operating in less than a day. She has estimated the following time for these tasks: buying the laptop – 1 hour; unpacking the laptop – 0.5 hours; installing the software – 4 hours; copying the data – 3 hours; testing the software with the data – 1 hour. Construct a precedence table for this project.

U N SA C O M R PL R E EC PA T E G D ES

Consider each of the following tasks as projects and construct a precedence table. Use the four headings: activity, task, duration and immediate predecessor.

a Make buttered toast b Pump up a bicycle tyre c Make a cheese sandwich

CHALLENGING

Veronica is preparing for exams by doing practice tests. She has organised this project into nine separate tasks. Prepare for English test (15 min) Work on English test (90 min) Review English test (30 min) Prepare for Maths test (10 min) Work on Maths test (90 min) Review Maths test (20 min) Prepare for Physics test (10 min) Work on Physics test (90 min) Review Physics test (25 min)

a Which tasks have no predecessors? b Construct a precedence table for this project.

Juan is cooking a pizza for dinner. He has divided this project into five tasks: Defrost the pizza base – 5 min; prepare the toppings – 10 min; place the sauce and topping on pizza – 2 min; heat oven – 10 min; cook pizza – 25 min.

a What tasks can be performed at the same time? b Which tasks are predecessor tasks for cooking the pizza? c Construct a precedence table for this project.

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10B Introduction to critical path analysis

Communicate your reasoning, understanding and fluency 10

Zari is renovating her lounge room. She has organised this project into six unordered tasks.

U N SA C O M R PL R E EC PA T E G D ES

Arrange new furniture – 1 day. Remove furniture and curtains – 1 day; Lay new carpet – 1 day; Hang new curtains – 1 day; Paint walls – 4 days; Prepare walls – 2 days;

a What is the first task? Give a reason for your answer. b What is the immediate predecessor to the following tasks? i Prepare walls ii Paint walls iii Lay new carpet iv Hang new curtains

c List the two immediate predecessors for ‘arranging new furniture’. d Construct a precedence table for this project.

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Chapter 10 Network flow and critical path analysis

10C Constructing a network diagram given a precedence table Learning intentions

• To be able to draw a network diagram to represent the activities to be completed during a project Past, present and future learning

U N SA C O M R PL R E EC PA T E G D ES

A network diagram is often used for critical path analysis. It can represent the activities required to complete a project, the time necessary to complete each activity and the relationships between the activities. In critical path analysis a network diagram does not have labelled vertices. The activities in the project are represented by the edges of the diagram and so it is the edges that must be labelled, not the vertices. Consider this precedence table. To draw a network diagram, we follow these steps.

Activity

Immediate predecessors

−

C, D

E, F

1 Activities A and B have no immediate predecessors. These activities can start immediately and can be completed at the same time.

start

2 Activity A is an immediate predecessor of activity C, so activity C must follow immediately after activity A. Activity C is an immediate predecessor of activity F, so activity F must follow immediately after activity C. 3 Activity D has immediate predecessor activity B so it follows immediately after activity B. Activity D is also an immediate predecessor of activity F so activity F must follow immediately after activity D. 4 Activity E has immediate predecessor activity B so it will follow immediately after activity B. Activity G has immediate predecessor activity F and activity E and so it must follow immediately after both of these activities.

start

finish

start

5 Activity G is not an immediate predecessor for any activity and so the project is finished after this activity is complete.

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10C Constructing a network diagram given a precedence table

Constructing a network diagram given a precedence table Activities that have no immediate predecessors follow from the start vertex. Activities that are not immediate predecessors for other activities lead to the finish vertex. For every other activity, look for:

U N SA C O M R PL R E EC PA T E G D ES

• the activities for which it is an immediate predecessor • the activities it has as immediate predecessors.

Example 6

Constructing a network diagram given a precedence table

Draw a network diagram from the chart shown. In this solution, the network diagram will be drawn from the finish back to the start.

Solution

Activity

Immediate predecessors

−

E, F, G

Explanation

H is not an immediate predecessor for any other activity so it will lead to the finish of the project.

finish

H has immediate predecessors E, F and G and so these three activities will lead into activity H.

finish

E F

finish

Activity D is an immediate predecessor of activity G and has immediate predecessor activity A. There will be a path through activity A, activity D and then activity G.

finish

Activity C is an immediate predecessor of activity F and has immediate predecessor activity A. There will be a path through activity A, activity C and then activity F.

Continued on next page

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Chapter 10 Network flow and critical path analysis

start A

E F

Activity B is an immediate predecessor of activity E and has immediate predecessor activity A. There will be a path through activity A, activity B and then activity E.

finish

Activity A has no immediate predecessors, so it is the start of the project.

U N SA C O M R PL R E EC PA T E G D ES

E F

start

finish

Now you try

Draw a network diagram from the precedence table shown. Activity

Immediate predecessors

−

C, E

D, F

Exercise 10C 1

BUILDING

Answer the following questions for each network diagram in parts a to d. i How many activities in the project? ii What is the beginning activity? iii What is the ending activity? iv Which activity is the immediate predecessor to activity D? v Which activities have no predecessors? a

G C

E B

F D

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10C Constructing a network diagram given a precedence table

Construct a precedence table for each of the following network diagrams. a

b A

E B C

G R

U N SA C O M R PL R E EC PA T E G D ES

F G

Example 6

Construct a network diagram for each of the precedence tables shown. a

Activity

Immediate predecessors

Activity

Immediate predecessors

−

Activity

Immediate predecessors

Activity

Immediate predecessors

−

D, E

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Chapter 10 Network flow and critical path analysis

Activity

Immediate predecessors

Activity

Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

Activity

Immediate predecessors

Activity

Immediate predecessors

−

C, D, E

Activity

Immediate predecessors

Activity

Immediate predecessors

−

R, S

DEVELOPING

Construct a precedence table for each of the following network diagrams. a

F G

H J

I L

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10C Constructing a network diagram given a precedence table

CHALLENGING 5

Construct a network diagram for each of the precedence tables below. Activity

Immediate predecessors

Activity

Immediate predecessors

−

V, W

G, I

H, J

Activity

Immediate predecessors

Activity

Immediate predecessors

−

N, L

A, D

E, C

R, Q

E, C

H, I

Activity

Immediate predecessors

U N SA C O M R PL R E EC PA T E G D ES

Activity

Immediate predecessors

−

E, D

I, G, J

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10C

Communicate your reasoning, understanding and fluency Consider this precedence table. Activity

Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

−

D, E

H, I, J

a Which activities follow from the start vertex? b Which activities lead to the finish vertex? c Draw a network diagram. d What are the immediate predecessors for activity M?

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10D Dummy activities

10D Dummy activities Learning intentions

• To know when dummy activities are required • To know how to include a dummy activity in a network diagram

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

Sometimes two activities will have some of the same immediate predecessors, but not all of them. In this precedence table, activity D and activity E share the immediate predecessor activity B, but they both have an immediate predecessor activity that the other does not. This overlap of predecessors presents some difficulty when constructing the network diagram. That is, you cannot have an edge that represents activity B being drawn to both the edge for activity D and the edge for activity E. This difficulty is easily overcome by introducing a dummy activity.

Activity

Immediate predecessors

−

A, B

B, C

Activity D and activity E are not immediate predecessors for any other activity, so they will lead directly to the finish vertex of the project. Activities A, B and C have no immediate predecessors, so they will be the starting vertex of the project.

We need to use the precedence information for activity D and activity E to join these two parts together. Activity D needs to follow directly from activity A and activity B, but we can only draw one edge for activity D. Activity E needs to follow directly from both activity B and activity C, but again we only have one edge for activity E, not two. The solution is to draw the diagram with activity D starting after one of its immediate predecessors, and using a dummy activity for the other. The dummy activities are represented by dotted edges and are, in effect, imaginary. They are not real activities, but they allow all of the predecessors from the table to be correctly represented.

The dummy activity for D allows activity D to directly follow both activity A and B. A dummy activity is also needed for activity E because it, too, has to start after two different activities, activity B and C.

dummy

dummy dummy

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Dummy activities

U N SA C O M R PL R E EC PA T E G D ES

• A dummy activity is required if two activities share some, but not all, of their immediate predecessors. • A dummy activity will be required from the end of each shared immediate predecessor to the start of the activity that has additional immediate predecessors. • A dummy activity is not real but imaginary, and so it takes no time (its duration is 0). • Dummy activities are represented in the network diagram using dotted lines so the label ‘dummy’ on the diagram is not actually needed, and won’t be used from now on, but you can include it on your own network diagrams for emphasis.

If two activities share the same immediate predecessor and end at the same node, a dummy activity is not required. Instead use multiple lines between the two nodes. D, 4

B, 6

A, 3

Example 7

C, 8

Using a dummy activity in a network diagram

Draw a network diagram from the precedence table shown below. Activity

Immediate predecessors

−

C, D

E, F

Solution

Explanation

start

end

A and B will lead from the start vertex. G will lead to the end vertex. A dummy will be required from the end of activity C (shared immediate predecessor) to the start of activity E (the activity with an additional immediate predecessor).

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10D Dummy activities

Now you try Draw a network diagram from the precedence table shown. Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

Activity

Example 8

B, C

D, E

Creating a precedence table from a network diagram

Construct a precedence table for the network diagram shown here.

start

Solution

end

Explanation

Activity

Immediate predecessors

−

D, C

E, F

G, H

Create a table with a row for each activity.

Look at the start of an activity. Write down all of the activities that lead directly to this activity in the immediate predecessor column. The dummy activity D1 makes activity C a predecessor of activities F and G as well as E. The dummy activity D2 makes activity G a predecessor of activity J as well as I.

Now you try

Construct a precedence table for this network diagram. C

end

start H B

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Chapter 10 Network flow and critical path analysis

Scheduling on a network diagram

U N SA C O M R PL R E EC PA T E G D ES

Precedence tables often include a duration to indicate the length of time of each activity. You will recall that allocating time to the completion of activities in a project is called scheduling. Knowing how long individual activities within a project are likely to take allows managers of such projects to hire staff, book equipment and also to estimate overall costs of the project. Scheduling on a network diagram is shown using a weighted edge. The number or weight on the edge indicates the duration of the activity. The weight of dummy activities is always zero.

Example 9

Scheduling on a network diagram

Draw a network diagram from the precedence table shown here.

Activity

Duration (in days)

Immediate predecessors

−

C, D

E, F

Solution

Explanation

C, 1

A, 8

E, 3

G, 2

B, 6

D, 2

F, 1

H, 1

A and B will lead from the start vertex. Indicate the time as a weight edge. H will lead to the end vertex. Indicate the time as a weighted edge. Both F and G have two predecessors. F shares one of its predecessors (C) with activity E, but F and E do not have exactly the same predecessors. Activity G’s immediate predecessors E and F are not shared with any other activity. So a dummy activity is required from the end of activity C to the start of activity F to show that E and F share an immediate predecessor, C.

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10D Dummy activities

Now you try Activity

Duration (in days)

Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

Draw a network diagram from the precedence table shown.

B, C

D, E

Scheduling on a network diagram

Scheduling on a network diagram is shown using a weighted edge. The number on the edge indicates the duration of the activity. The weight of dummy activities is always zero.

Exercise 10D

Example 7

BUILDING

Complete each network diagram by inserting a dummy activity for each precedence table. a

Activity

Immediate predecessors

−

H, G

Activity

Immediate predecessors

−

B, C

Example 9

C, 4

A, 4

E, 1 B, 5

D, 3

Activity

Duration (in days)

Immediate predecessors

−

A, B

C, D

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d C, 2

A, 5

D, 8 B, 6

Duration (in hrs)

Immediate predecessors

−

A, B

Activity

Duration (in mins)

Immediate predecessors

−

A, B

C, E

U N SA C O M R PL R E EC PA T E G D ES

E, 2

Activity

F, 12

C, 8

A, 6

D, 7

E, 10

B, 9

Example 8

Complete each precedence table for the network diagrams shown. a

Activity

start

−

Immediate predecessors

finish

D E

D, B

Activity

Immediate predecessors

−

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10D Dummy activities

D, 8 F, 3

C, 5

A, 3

Activity

Duration (in mins)

G, 7 B, 9

Immediate predecessors

B C

E, 2

U N SA C O M R PL R E EC PA T E G D ES

D E

G, 1

D, 5

Activity

Duration (in hrs)

H, 7

Immediate predecessors

B, 4

A, 2

F, 6

E, 8

C, 11

D E

DEVELOPING

Draw a precedence table for each network diagram. a

4 The following network diagram shows the activities in a project to repair a dent in a car panel. The activities are listed in the table shown. A

C B

a Which activity or activities are the immediate predecessors of the event ‘remove broken component’? b Which activities are the immediate predecessors of the activity ‘install new component’?

Activity

Description

Remove panel

Order component

Remove broken component

Pound out dent

Repaint

Install new component

Replace panel

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Chapter 10 Network flow and critical path analysis

CHALLENGING 5

Construct a network diagram for each of the precedence tables shown. Activity

Immediate predecessors

Activity

Immediate predecessors

−

B, C

R, S

R, S, T

F, G

U N SA C O M R PL R E EC PA T E G D ES

Activity

Duration (in days)

Immediate predecessors

Activity

Duration (in minutes)

Immediate predecessors

−

A, B

C, D

D, E, F

Activity

Duration (in days)

Immediate predecessors

Activity

Duration (in hours)

Immediate predecessors

−

A, B

B, C

A, D

B, C

E, F

D, E

E, F

G, H

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10D Dummy activities

Communicate your reasoning, understanding and fluency 6

This network shows the activities that are needed to complete a project.

D, 3 F, 7

A, 2

a List the immediate predecessors for start activity D. b List the immediate predecessors for activity E. c Are the following statements true or false?

E, 6

B, 3

finish

U N SA C O M R PL R E EC PA T E G D ES

G, 2 C, 4

Dummy D1

i Activity B is an immediate predecessor for activity D. ii Activity A must be complete before activity E can begin. iii The dummy line D1 has a duration of 4 days.

This precedence table shows the activities and their immediate predecessors required to complete a project. Activity

Duration (in days)

Immediate predecessors

−

B, C

D, E

F, G

a From the table:

i which activities will require a dummy activity? ii which activities are at the start? iii which activities meet the finish vertex?

b Create a network diagram for this project. c How many activities must be completed before activity E begins?

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10E Forward and backward scanning Learning intentions

• To use forward scanning to find the earliest starting time (EST) • To use backward scanning to find the latest starting time (LST)

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

Forward scanning

For a project to be completed in the shortest time possible, it is important to be aware of each activity’s earliest starting time (EST) after the start of the entire project. An EST of 8 means an activity cannot start until 8 hours (or whatever time unit is given) after the start of the project. The EST for each activity is found by a process called forward scanning, described below.

Example 10

Finding the earliest starting time using forward scanning

Find the earliest starting time (EST) for the project represented in the network diagram opposite. The duration of each activity is in days. C, 1

A, 8

E, 3

start

D, 2

B, 6

C, 1

E, 3

start

D, 2

B, 6

A 0 start B 0

G, 2

H, 1

finish

F, 1

C, 1

A, 8

E, 3

B, 6

finish

Explanation

A, 8

H, 1

F, 1

Solution

G, 2

D, 2

F, 1

G, 2

H, 1

finish

Draw a box, split into two cells, next to each vertex of the network diagram, as shown in the diagram opposite. If more than one activity begins at a vertex, draw a box for each of these activities.

Activity A and B start the project and have an EST of zero. Write zero in the left box (shaded yellow).

Activity C has an EST of 8 (add the EST of the immediate predecessor to the duration of the activity).

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10E Forward and backward scanning

8 A 0 start B 0

C, 1

A, 8

E, 3 0

H, 1

finish

12 F, 1

D, 2

B, 6

G, 2

Note:

Repeat step 3 for each activity. However, if an activity has more than one predecessor, calculate the EST using each of the predecessors and choose the largest value. EST of F = 6 + 2 = 8 or

U N SA C O M R PL R E EC PA T E G D ES

EST of F = 9 + 0 = 9. Use 9. EST of G = 9 + 1 = 10 or

EST of G = 9 + 3 = 12. Use 12.

A 0 start B 0

C, 1

A, 8

E, 3

G, 2

12 F, 1

D, 2

B, 6

The EST value at the finish of the project is the minimum time it takes to complete the project.

H, 1

finish 15

∴ EST of the project is 15 days.

Write the answer in words.

Now you try

Find the earliest starting time (EST) for the project presented in the network diagram. The duration of each activity is in days. C, 4

A, 6

E, 3

G, 5

start

H, 1

finish

F, 2

B, 4

D, 2

Backward scanning

Some activities have some flexibility around the time they can start. The latest start time, or LST, for each activity is the latest time after the start of the project that the individual activity can start for the project to still be completed in the minimum time. LSTs for each activity are calculated using the reverse of the process used to calculate the ESTs. This process is called backward scanning.

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Example 11

Finding the latest starting time using backward scanning

Find the latest starting time (LST) for each activity using the network diagram and completed forward scanning from the previous example. The duration of each activity is in days.

8 A 0 start B 0

C, 1 9

A, 8

E, 3 0

12 F, 1

D, 2

B, 6

G, 2

H, 1 14

finish 15

U N SA C O M R PL R E EC PA T E G D ES

Solution

Explanation

A 0 start B 0

A, 8

E, 3

H, 1

finish 1515

C, 1

A, 8

E, 3

B, 6

D, 2

8 8

A 0 0 start B 0 3

G, 2

12 F, 1

D, 2

B, 6

A 0 start B 0

Copy the minimum time to complete the project into the right cell shown shaded blue in the diagram. Calculate the LST for each activity by subtracting the duration of the activity from the LST of the following activity.

C, 1 9

G, 2

H, 1

1212 1414

finish 1515

If more than one activity has the same predecessor, calculate the LST using each of the activities that follow and choose the smallest value.

H, 1

The LST for each activity is shown in shaded blue cell in the diagram opposite.

F, 1 9 11

C, 1 9 9

E, 3

A, 8

G, 2

B, 6

12 12 14 14 F, 1

D, 2

6 9

finish 15 15

9 11

Now you try

Find the latest starting time (LST) for each activity using the network diagram and completed forward scanning from the Now you try question for Example 10. The duration of each activity is in days. 6

C, 4

A, 6

E, 3

A 0 start B 0

G, 5

B, 4

H, 1

finish 19

F, 2

D, 2

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10E Forward and backward scanning

Latest starting time (LST)

• The EST is the earliest possible time an activity can begin after all preceding activities have been completed. • Give the starting vertex a time of zero. Going forward, start calculating EST from here. • To calculate the EST of an activity, add the EST of the immediate predecessor to the duration of the activity. • If the activity has more than one predecessor, calculate the EST using each of the predecessors, and choose the largest value.

• The LST is the latest possible time an activity can begin before delaying the entire project after all preceding activities have been completed. • The LST = EST for the finishing vertex. Going backward, start calculating LST from here. • To calculate the LST of an activity, subtract the duration of the activity from the LST of the following activity. • If more than one activity has the same predecessor, calculate LST using each of the activities that follow and choose the smallest value.

U N SA C O M R PL R E EC PA T E G D ES

Earliest starting time (EST)

Exercise 10E

Example 10, 11

BUILDING

A section of a network diagram is shown here.

a What are the earliest start times for each of these activities? i A ii B

9 9

A 6 6

15 15

B 6 11

iii C iv D b What are the latest start times for each of these activities? i A ii B

8 13

iii C iv D c Find the duration of the four activities.

A section of a network diagram is shown here.

a What are the earliest start times for each of these activities? i A ii B iii C

iv D

v F vi G b What are the latest start times for each of these activities? i A ii B iii C

iv D

v F

vi G

start 0 0

B 2 2 A, 2

27 27 B, 25 D, 10

C 2 31

37 37

C, 6

G, 8

finish 45 45

F, 5

8 37

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Chapter 10 Network flow and critical path analysis

Write down the value of each pronumeral in the sections of network diagrams shown. a 8 8

b 4 8

p 12

w 14

P, 4

W, 6

c m n

12 12

A, 5 6 a

b 15

U N SA C O M R PL R E EC PA T E G D ES

M, 4

e Q 4 4

14 14

Q, q

p q

C, 4

B, 5

B 1 1

P 4 4

D, 2

A, 3

R, r

P, p

1010

r s

A 1 5

n 9

4 What are the values of f and g in the section of a network diagram shown? 3

F, 6

G, 7

Consider the section of a network diagram shown in the given diagram.

a What is the duration of activity B? b What is the latest start time for activity E? c What is the earliest time that activity E can start?

4 6

9 10 E, 2 12 12 F

D 0 1

B 0 0

C, 4

A 0 3

12 12

Find the earliest starting times (EST) and latest start times (LST) for all activities given the precedence tables and network diagrams provided.

Activity

Duration (in hrs)

Immediate predecessors

−

B, C

C, 1 A, 1 A B, 2 start B

D, 1

E, 2

finish

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10E Forward and backward scanning

Duration (in hrs)

Activity

Immediate predecessors −

−

B, 7 C, 4

−

D, 2 E, 3

F, 15 finish G, 8

D E

U N SA C O M R PL R E EC PA T E G D ES

A, 5

A start B

Activity

Duration (in days)

Immediate predecessors

−

start

A, 4

D, 1

B, 5

C, 2

E, 6 finish

DEVELOPING

Determine the earliest starting times (EST) and latest starting time (LST) for each activity in the network diagrams shown.

E, 7

A, 11

F, 2

H, 5

B, 2

C, 4

start

J, 12

D, 4

finish

I, 2

G, 7

F, 9

B, 5

start

A, 10

J, 3

C, 3

G, 6

D, 2

E, 5

K, 1

M, 2

finish

H, 4 L, 4 I, 3

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CHALLENGING 8

The precedence table for a project is shown. Activity

Duration (in min)

Immediate predecessors

−

B, C

D, E

Activity

Duration (in min)

Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

a Construct a network diagram using the given precedence table. b Show the earliest starting times (EST) for each activity on the network diagram. c Show the latest starting times (LST) for each activity on the network diagram.

The precedence table for a project is shown.

Complete each diagram to determine the earliest starting times (EST) and latest starting times (LST) for the activities in the network diagrams provided. The duration of each activity is in days.

D, 8

B, 7

EST LST

F, 3

A, 4

H, 2

C, 2

G, 1

E, 4

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10E Forward and backward scanning

D, 4 C, 2

E, 2

EST LST B, 3

U N SA C O M R PL R E EC PA T E G D ES

A, 3

B, 3

EST LST

C, 3

F, 4

E, 3

A, 5

I, 3

D, 4

J, 2

H, 9

G, 6

A, 6

EST LST A

D, 8

B, 8

E, 2

G, 4

C, 3

F, 4

A, 4

EST LST A

C, 3

E, 1

F, 6

B, 1

D, 5

A, 4

EST LST A C

B, 3

C, 5

E, 8

D, 6 G, 7

F, 10

H, 2

I, 4

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D, 8 A, 4 EST LST

F, 4 B, 6 I, 1

U N SA C O M R PL R E EC PA T E G D ES

H, 6 C, 5

G, 2

E, 3

D, 1

A, 8

E, 4

EST LST

G, 4

B, 2

H, 1

F, 6

C, 3

Communicate your reasoning, understanding and fluency 11

A network diagram for a project is shown below. A forward scan and a backward scan showing earliest starting times (EST) and latest starting times (LST) has been completed. The duration for the activities, in days, is missing from the diagram. 4

14 14

Give possible durations for each activity.

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10F Float times and the critical path

10F Float times and the critical path Learning intentions

U N SA C O M R PL R E EC PA T E G D ES

• To be able to calculate float times to identify critical and non-critical activities • To find the critical path • To locate the critical path to find the minimum time for a project to be completed Past, present and future learning

The diagram below shows a small section of a network diagram. There are three activities shown, with their individual durations, in hours. A, 5

B, 3

C, 6

Activity B and activity C are both immediate predecessors to the next activity, so the project cannot continue until both of these tasks are finished. Activity B cannot begin until activity A is finished.

Activity C can be completed at the same time as activity A and activity B. Activity A and B will take a total of 5 + 3 = 8 hours, while activity C only requires 6 hours. There is therefore some flexibility around when activity C needs to start. There are 8 − 6 = 2 hours spare for the completion of activity C. This value is called the float time (or slack time) for activity C. The flexibility around the starting time for activity C can be demonstrated with the following diagram. 1st

2nd

3rd

4th

5th

6th

7th

8th

Start at same time

Float

Delay C by 1 hour

Float

Delay C by 2 hours

Float

Hours:

• The five red squares represent the 5 hours it takes to complete activity A.

• The three green squares represent the 3 hours it takes to complete activity B. • The six yellow squares represent the 6 hours it takes to complete activity C.

Activity C does not have to start at the same time as activity A because it has some float time available (2 hours). Activity C should not be delayed by more than 2 hours because this would cause delays to the project and would affect the EST of the next activity. The next activity requires B and C to be complete before it can begin.

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Float times

U N SA C O M R PL R E EC PA T E G D ES

• Float time is the amount of time that a task in a project network can be delayed without causing a delay to subsequent tasks. • Float time is calculated by subtracting the earliest start time from the latest start time. Float time = LST − EST

Identifying float times

8 8

A 0 0 start B 0 3

9 9

C, 1

A, 8

E, 3

B, 6

D, 2

6 9

G, 2

12 12

H, 1

finish 15 15

14 14

F, 1

9 11

The boxes at the vertices in the network diagram above give the EST and LST for the activity that begins at that vertex. The EST for activity D is 6 and the LST for activity D is 9. This means activity D has a float time of 9 − 6 = 3 hours.

D, 2

6 9

Activity D can be delayed by 3 hours without delaying the rest of the project.

The EST for activity C is 8 and the LST for activity C is 8. This means activity C has a float time of 8 − 8 = 0 hours.

8 8

C, 1

Activity C has no flexibility around its starting time at all. Any delay to the start of this activity will delay the whole project and extend the minimum time for completion. This means that activity C is a critical activity.

Critical path

All activities that have no float time are critical activities for completion of the project. Tracking through the network diagram along the edges of critical activities gives the critical path for the project. The critical path for the previous project is highlighted in red on the diagram below.

We say the critical path is A → C → E → G → H, and that activities A, C, E, G and H are all critical activities.

The process for determining the critical path is called critical path analysis. The steps involved in critical path analysis are outlined below.

8 8

A 0 0 start B 0 3

9 9

C, 1

A, 8

E, 3

G, 2

B, 6

D, 2

6 9

12 12

H, 1

14 14

finish 15 15

F, 1

9 11

1 Draw a box with two cells next to the start of each edge of the network diagram and at the finish. 2 Calculate the EST for each activity by forward scanning: EST = EST of predecessor + duration of predecessor. 3 If an activity has more than one predecessor, the EST is the largest of the alternatives. 4 The minimum overall completion time of the project is the EST value at the finish. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

10F Float times and the critical path

U N SA C O M R PL R E EC PA T E G D ES

5 The LST of the finish is the same as the EST of the finish. Calculate the LST for each activity by backward scanning: LST = LST of following activity − duration of activity. 6 If an activity has more than one following activity, the LST is the smallest of all the alternative following activities. 7 Float time of an activity = LST − EST. 8 If the float time = 0, the activity is on the critical path.

Critical path

Minimum time

The critical path is the sequence of the critical activities. Critical activities are ones that cannot be delayed without affecting the overall completion time of the project.

• The minimum overall completion time of the project is the EST value at the finish. • The minimum time to complete the project is the same duration as the critical path in the network.

Example 12

Locating the critical path

What is the critical path in the network diagram shown below? F 15 16

start 0 0

B 10 10 B, 5 A, 10

J 24 25

F, 9

15 15

C, 3 H 15 15

D, 2

M, 2

G, 6

26 26

H, 4

E, 5

L, 4

Solution

22 22

Explanation

F 15 16

B 10 10 B, 5 A, 10

C 10 12

finish 28 28

I, 3

19 19

12 14

J, 3

K, 1

G 15 20

C 10 12

D 10 12

start 0 0

K 24 25

J 24 25

F, 9

K 24 25 J, 3

K, 1

M, 2

G, 6

C, 3 H 15 15

G 15 20

26 26

D, 2

D 10 12

H, 4

E, 5

12 14

L, 4

I, 3

19 19

The critical path of this project is A – B – H – I – L – M.

22 22

finish 28 28

The critical path has activities with a zero float time. Look for Float = LST − EST = 0. These instances are highlighted by the red boxes. The critical path is highlighted in red. The dummy activity is not included in the critical path. Write the answer in words.

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Now you try What is the critical path in the network diagram shown below? A 0 12 B 0

A, 4 C 4 16

G 11 23

C, 7

G, 2 H 25 25 H, 3

28 28

U N SA C O M R PL R E EC PA T E G D ES

D, 6

F, 10

B, 3

D 3 17 E 3

Example 13

E, 12

F 15 15

Finding the float time and the critical path

A project has six activities as shown in the precedence table.

a Draw a network diagram for this project. b Calculate the EST and LST for each activity. c What is the float time for each activity? d Write down the critical path of this project. e What is the minimum time required to complete the project?

Activity

Estimated completion time (days)

Immediate predecessors

−

A, B

C, E

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10F Float times and the critical path

Solution

Explanation

F, 12

C, 8

A and B have no predecessors and so can begin at the same time.

finish

A, 6 D, 7

start

E, 10

B, 9

Forward scan to determine the EST (yellow box) and backward scan to determine the LST (blue box).

U N SA C O M R PL R E EC PA T E G D ES b

A 0 3 start

A, 6

B 0 0

B, 9

C 9 18 D 9 9

26 26

C, 8

F, 12

D, 7

3838 finish

E, 10

16 16

c A: Float time = 3 − 0 = 3 days

Subtract the earliest start time from the latest start time for each activity to calculate their float time.

B: Float time = 0 − 0 = 0 days

C: Float time = 18 − 9 = 9 days

D: Float time = 9 − 9 = 0 days

E: Float time = 16 − 16 = 0 days F: Float time = 26 − 26 = 0 days

Look for activities with a zero float time to find the critical path. (Float = LST − EST = 0) The critical path is highlighted in red in answer b.

d Activities with a float time of zero are in the critical path.

The critical path of this project is B – D – E – F.

The minimum completion time is in EST of the end box.

e The minimum completion time of this project is 38 days. Now you try

A project has six activities as shown in the precedence table. Duration

Immediate

Activity

(in days)

predecessors

−

B, C

a Draw a network diagram for this project. b Calculate the EST and LST for each activity. c What is the float time for each activity? d Write down the critical path for this project. e What is the minimum time required to complete the project? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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Chapter 10 Network flow and critical path analysis

Exercise 10F 1

BUILDING

Find the float time for all activities (measured in minutes) in the network diagrams shown. a

finish 1616

D 8 9 E 8 8 D, 5 1314 B, 3 G, 4 B 5 5 E, 4 C, 6 F, 6 H, 3 A, 5

U N SA C O M R PL R E EC PA T E G D ES

start 0 0

9 9 D, 1 10 10 B 4 4 B, 5 E, 6 A, 4 C, 2

Example 12

C 4 14

start 0 0 C5 6

1818

1212

finish 2121

2 Identify the critical path in the network diagrams shown. a

5 5

A, 5

A 0 0

F, 15

B, 7

start

B 0 13

C, 4

C 0 5

D, 2

27 27 D, 10 B, 25 B 2 2 37 37 A, 2

finish 20 20

start

C, 6

0 0

G, 8

finish 45 45

F, 5

C 2 31

E, 3

D 4 18

G, 8

8 40

7 12

E 4 9

Example 13

3 The network diagram shows the time (in days) for all activities, EST and LST. a Calculate the float times for all the activities. b Write down the critical path for this project. c What is the minimum time for the project to be completed?

3 4

A 0 1

D, 12

A, 3

E, 10 F, 20

C, 7

start D 0 0

22 22

finish 42 42

C 0 15

The network diagram shows the time (in hours) for all activities, EST and LST.

a Calculate the float times for all the activities. b Write down the critical path for this project. c What is the minimum time for the project to be completed?

12 12

B, 8

F 13 13

G 13 14

A 0 1 A,10 start B 0 0

14 14

F,1

10 11 D,2 B,9

E,4

G,1 14 15

9 9

I,2

C 0 14

H,3

J,1

ﬁnish 18 18

C,3

17 17

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10F Float times and the critical path

DEVELOPING An incomplete precedence table and network diagram for a project are shown here.

Activity

Duration (in weeks)

Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

D E

a Complete the precedence table. b Find the float times for all activities. c What is the critical path of this project? d What is the minimum completion time for the project?

A 0 3

6 6 C, 6 12 12 E, 7 19 19

A, 3

start B 0 0

G, 3

finish 22 22

I, 2

B, 6

D 6 13 D, 5 I 6 20

H, 3

11 18 F, 1 12 19

CHALLENGING

The precedence table for a project is shown.

a Draw a network diagram for this project. b Complete the critical path analysis to calculate the EST and LST for each activity. c Write down the critical path of this project. d What is the minimum time required to complete the project?

Activity

Duration (weeks)

Immediate predecessors

−

R, T, U

W, X

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10F

Chapter 10 Network flow and critical path analysis

The precedence table for a project is shown. Activity

Duration (weeks)

Immediate predecessors

−

U N SA C O M R PL R E EC PA T E G D ES

J, N

L, M

J, N

Q, R

Communicate your reasoning, understanding and fluency 8

For the network diagrams shown, time is in days. Without doing a forward scan and backward scan (that is, do not calculate EST or LST), examine each diagram to determine: i the critical path in the networks ii the minimum time to complete the project iii the float time for the non-critical activities. a

D, 8

B, 7

F, 3

A, 4

H, 2

I, 1

G, 1

C, 2

E, 4

D, 4

C, 2

E, 2

A, 3

B, 3

C, 3

E, 3

F, 4

A, 5

I, 3

D, 4

H, 9

G, 6

A, 4

J, 2

C, 3 E, 1

F, 6

B, 1 D, 5

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10G Minimum completion time

10G Minimum completion time Learning intentions

• To examine the impact of increasing or reducing the duration of an activity on the completion time of

U N SA C O M R PL R E EC PA T E G D ES

the project or task being modelled • To be able to calculate shorter completion times when an activity is changed Past, present and future learning

The minimum time it takes to complete a project depends upon the time it takes to complete the individual activities of the project, and upon the predecessors for each activity. Critical path analysis can be completed to find the overall minimum completion time.

Completing activities in a shorter time

Sometimes, the managers of a project might arrange for one or more activities within the project to be completed in a shorter time than originally planned. Changing the conditions of an activity within a project, and recalculating the minimum completion time for the project, is sometimes called crashing. An individual activity could be crashed by employing more staff, sourcing alternative materials or simply because weather or other factors allow the activity to be completed in a shorter time than usual.

Example 14

Finding the minimum completion time

This network diagram shows the time (in hours) for all activities, EST and LST. The critical path is shown in red.

a What is the minimum completion time for the project? b Which of these activities, if reduced in time individually, would not result in an earlier completion of the project? c Activity D was reduced in time to 4 hours. What is the new minimum completion time?

2 6

B, 4

6 10

C, 3

A, 2 A 0 4

finish

start

D 0 0

13 13

E, 7

D, 6

6 6

Solution

Explanation

a The minimum completion time of this project is 13 hours.

The minimum completion time is the EST of the end box.

b Activities A, B or C

The activities not on the critical path will not shorten the time taken to complete the project.

Continued on next page

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2 4

B, 4

Decrease the weighted edge for activity D from 6 to 4 hours. Recalculate the EST and LST for each activity.

6 8 C, 3

A, 2 A 0 2

finish

start D 0 0

11 11 E, 7

U N SA C O M R PL R E EC PA T E G D ES

D, 4

4 4

The minimum completion time is now 11 hours.

The minimum completion time is the EST of the end box.

Now you try

This network diagram shows the time (in hours) for all activities, EST and LST. The critical path is shown in red. D 5

D, 4

E 9

C, 2

E, 2

11 11

A, 3

B, 3

C 3

B 3

Example 15

Finding the minimum completion time

A community centre is being built. Nine activities have been identified for a building project. This network diagram shows the activities, completion times in weeks, ESTs and LSTs. 4 9

A, 4

A 0 5 start B 0 0

D, 1

5 10

E 4 7

G, 3

C 4 4 E, 4 11 11

I, 6

H, 2 13 13 F, 4

B, 4

C, 3

finish 19 19

7 7

a What is the critical path and the minimum time, in weeks, to complete this project? b The builders of the community centre are able to speed up the project. Activities A, C, F and G can be reduced in time at an additional cost. Which of these activities, if reduced in time individually, would not result in an earlier completion of the project? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

10G Minimum completion time

c The owner of the estate is prepared to pay the additional cost to achieve early completion. The cost of reducing the time of each activity is $5000 per week. The maximum reduction in time for each one of the five activities, A, C, F and G is 2 weeks. Find the minimum time, in weeks, for the project to be completed given these activities are reduced in time. Explanation

a Critical path is B – C – F – H – I

Look for activities with a zero float time to find the critical path. The minimum completion time is the EST of the end box.

U N SA C O M R PL R E EC PA T E G D ES

Solution

∴ Minimum completion time is 19 weeks.

b Activities A and G are not on the critical path. These activities would not result in an earlier completion of the project.

Look for activities not on the critical path. Reducing their time will not affect the overall time for the project.

c C can now be completed in 1 week. F can now be completed in 2 weeks. Update the network diagram as shown.

Activities C and F are on the critical path. Reduce the time of each activity by 2 weeks.

4 6

D, 1 5 7

A, 4 E 4 4

A 0 2 start B 0 0

Recalculate the EST and LST.

G, 3

C 4 5 E, 4

8 8

I, 6

H, 2 10 10 F, 2

B, 4

C, 1

finish 16 16

5 6

New critical path is B – E – H – I ∴ Minimum completion time is now 16 weeks.

Check for change in the critical path. Look for activities with a zero float time to find the critical path.

Now you try

A community library is being built. Nine activities have been identified for a building project. This network diagram shows the activities, completion times in weeks, ESTs and LSTs. 4

21 21

A, 4

EST LST A 0 0

C 0

B, 3

E, 8

F, 10

E 13 13

31 31

D, 6

I, 4

35 35

C, 5

H, 2

G, 7

G 13 22

20 29

a What is the critical path and the minimum time, in weeks, to complete this project? b The builders of the community library can speed up the project. Activities A, C, F and G can be reduced in time at an additional cost. Which of these activities, if reduced in time individually, would not result in an earlier completion of the project? c The owner of the estate is prepared to pay the additional cost to achieve early completion. The maximum reduction in time for each one of the activities A, C, F and G is 2 weeks. Find the minimum time, in weeks, for the project to be completed given these activities are reduced in time.

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Chapter 10 Network flow and critical path analysis

Exercise 10G Example 14

BUILDING

This network diagram shows the time (in days) for all activities, EST and LST. The critical path is shown in red. A, 5 C 5 8 B 5 5

C, 6 B, 4

finish 14 14

U N SA C O M R PL R E EC PA T E G D ES

start 0 0

E, 4

D, 1

10 10

9 9

a What is the minimum completion time? b Reducing the time taken in which activities would not result in a reduction in the minimum completion time? c Activity A was reduced in time to 3 days. What is the new minimum completion time?

This network diagram shows the time (in weeks) for all activities, EST and LST. The critical path is shown in red. 9 9

5 6

A 0 1 A, 5 start D 0 0

B, 3

E, 5

D, 9

F, 3

C, 7

14 14

finish 17 17

C 0 7

a What is the minimum completion time for the project? b Reducing the time taken in which activities would not result in a reduction in the minimum completion time? c Activity E was reduced in time to 1 week. What is the new minimum completion time?

3 This network diagram shows the time (in hours) for all activities, EST and LST. The critical path is shown in red. 5 5

F, 13

A, 5

A 0 0 start B 0 11

C 0 4

B, 7

finish 18 18

D, 2

C, 3

D 3 16

E 3 7

G, 8

E, 3

6 10

a What is the minimum completion time for the project? b Reducing the time taken in which activities would result in a reduction in the minimum completion time? c Activity B was reduced in time to 5 hours. What is the new minimum completion time? Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

10G Minimum completion time

This network diagram shows the time (in minutes) for all activities, EST and LST. The critical path is shown in red. 8 8 A, 8

E, 3 0 F, 1

B, 6

G, 2 12 12

finish 14 14

U N SA C O M R PL R E EC PA T E G D ES

A 0 0 start B 0 3

9 9

C, 1

D, 2

6 9

9 11

a What is the minimum completion time? b Reducing the time taken in which activities would result in a reduction in the minimum completion time? c Both activities A and E are reduced by 2 minutes. What is the new minimum completion time?

DEVELOPING

Example 15

The network diagram for a project is shown. The duration for each activity is in hours. 3 7

A 0 4

B 0 0 start C 0 2

D, 4 11 11

H, 8

E, 5

A, 3

B, 6

J, 3

G, 3

6 6

I, 7

C, 2

19 19

finish 22 22

F, 4

2 4

G 6 8

I 6 12

a Identify the critical path for this project. b What is the minimum completion time for the project? c What is the maximum number of hours that the completion time for activity E can be reduced without changing the critical path of the project? d What is the maximum number of hours that the completion time for activity H can be reduced without affecting the critical path of the project? e If activity B is reduced from 6 hours to 4 hours, what is the minimum completion time for the project?

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CHALLENGING 6

The network diagram for a project is shown. The duration for each activity is in hours. G, 3 A, 5

U N SA C O M R PL R E EC PA T E G D ES

I, 8

start

finish

B, 1

F, 12

C, 2

H, 4

J, 1

E, 7

a Find the earliest starting times (EST) and latest start times (LST) for all activities. b Identify the critical path for this project. c How much time overall is saved if activity F is reduced in duration by 2 hours? d If only one activity can have its duration reduced, what is the maximum possible reduction in the completion time of the project?

The network diagram for a project is shown. The duration for each activity is in hours.

A, 6

start

B, 5

D, 4

E, 6

F, 2

finish

C, 2

a Find the earliest starting times (EST) and latest start times (LST) for all activities. b Determine the shortest time in which this project can be completed. c Write down the critical path for this project. d The time it takes to complete activity A can be reduced by one hour at a cost of $50. Explain why this will not affect the completion time of this project. e Activity B can be reduced in time at a cost of $100 per hour. Activity E can be reduced in time at a cost of $50 per hour. What is the cost of reducing the completion time of this project as much as possible?

The network diagram for a project is shown. The durations of activities are in hours.

a Find the EST and LST for all activities. b Find the minimum time required to complete the project. c Find the critical path for the project. d How many hours can activity C be reduced by to remain a critical activity but reduce the overall finishing time of the project?

A, 3

B, 6

C, 8

E, 5

D, 2

F, 7

G, 3

H, 6

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10G Minimum completion time

The network diagram for a project is shown. The durations of activities are in days. C, 8 D, 6

G, 11

E, 9

H, 10

U N SA C O M R PL R E EC PA T E G D ES

A, 7

B, 10

F, 6

a Find the EST and LST for all activities. b Find the minimum time required to complete the project. c Find the critical path for the project. d What impact will reducing activity E to 5 days have on the network diagram?

Communicate your reasoning, understanding and fluency 10

This network shows the activities for the completion of a project, where the time is in hours. D, 8

B, 5

F, 4

A, 4

I, 4

C, 6

H, 4

E, 5

G, 3

a Find the critical path and the minimum completion time. b Is the critical path unique? Give reasons. c What is the float time of the non-critical activities? d Which activities, if reduced in time, would not result in an earlier completion time of the project? e Activity H needs to be increased to 6 hours. What impact does this have on the finishing time? f How many hours can activity D be increased by before it becomes a critical activity? g The project time needs to be shortened by 4 hours. Which activity can be reduced to achieve this? Give reasons.

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10H Gantt charts Learning intentions

• To be able to construct a Gantt chart to represent a project from a network diagram indicating the critical path

U N SA C O M R PL R E EC PA T E G D ES

Past, present and future learning

What is a Gantt chart?

A Gantt chart is a visual tool used to represent a project’s schedule. It displays activities along a timeline, showing their earliest start time, durations and dependencies. Each activity is represented by a horizontal bar, with the length of the bar indicating the task’s duration. The horizontal bar for each activity is drawn at the earliest start time of that activity. Any float time is drawn afterwards, dotted. The critical path is typically displayed along the top. Non-critical tasks, along with their float times, appear below the critical path. The Gantt chart offers a clear and structured timeline, making it easier to:

• track project progress

• detect potential delays early

• manage resources efficiently.

Henry Laurence Gantt (1861–1919)

• An American mechanical engineer and management consultant who developed the chart between 1910

and 1915. He created it as a visual tool to enhance project planning, scheduling, resource allocation, and tracking the progress of the project. • Today, Gantt charts are commonly used across various industries, including construction, software development, manufacturing, IT and event planning. With a Gantt chart, teams can assign tasks, set deadlines, estimate costs and monitor real-time progress.

Example 16

Reading a Gantt Chart

The Gantt chart on the right shows the completion of 7 activities in a project. Time is in days.

a Calculate the duration of each activity. b Calculate the float time for non-critical activities. c Identify the critical path and minimum completion time. d What is the EST and LST for activity E? e What is the EST and LST for activity F? f What is the predecessor for activity G? g Construct a precedence table showing the duration and immediate predecessors of each activity.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 D

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10H Gantt charts

Explanation

a Activity A is 6 days; B is 8 days, C is 3 days, D is 8 days, E is 2 days, F is 4 days, G is 4 days

Calculate the length of each bar with a solid line.

b Activity B is 6 days, C is 3 days, E is 9 days, F is 3 days and G is 3 days

Calculate the length of the dotted bars.

U N SA C O M R PL R E EC PA T E G D ES

Solution

c Critical Path is A-D with a minimum completion time of 14 days

Critical path will be activities with zero float time. It should be easy to see. A series of bars (solid line) side-by-side that stretches from start to finish.

d EST for activity E is day 3 LST for activity E is day 12

Activity E starts on day 3. Slide Activity E across to the right to the end of its float time to read LST day 12.

e EST for activity F is day 3 LST for activity F is day 6

Activity F starts on day 3. Slide Activity F across to the right to the end of its float time to read LST day 6.

f Activity G predecessor is Activity F

Activity G starts at the end of activity F.

Activity E and F start straight after C. Therefore, C is an immediate predecessor to activities E and F.

Activity

Duration (days)

Immediate predecessors

−

Now you try

The Gantt chart on the right shows the completion of 5 activities in a project. Time is in days.

a Calculate the duration of each activity. b Calculate the float time for non-critical activities. c Identify the critical path and the minimum completion time. d What is the EST and LST for activity B? e What is the EST and LST for activity E? f What is the predecessor for activity D? g Construct a precedence table showing the duration and immediate predecessors of each activity.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C

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When reading Gantt charts:

U N SA C O M R PL R E EC PA T E G D ES

• The length of the solid horizontal bar is the duration of the activity (Activity D has a duration of 1 day) • The length of the dotted bar behind a solid bar is the float time (Activity D has float time of 2 days) • The EST is the start of the solid horizontal bar (Activity D EST is Day 9) • The LST is found by dragging the solid bar across to the end of the float time (Activity D LST is Day 11) 0

10 11 12 13 14

Example 17

Constructing a Gantt chart

Consider the network diagram on the right, where time is in weeks.

4 9

A, 4

a Locate the critical path and determine the minimum completion time. b Determine the float times for all non-critical activities. c Use the information in parts a and b to draw a Gantt chart. Draw the critical path first with non-critical activities underneath. 0

A 0 5 start B 0 0

D, 1

5 10

E 4 7

G, 3

C 4 4 E, 4 11 11

I, 6

H, 2 13 13 F, 4

B, 4

C, 3

finish 19 19

7 7

9 10 11 12 13 14 15 16 17 18 19

Solution

Explanation

a Critical path is B–C–F–H–I Minimum completion time is 19 weeks b Activity A is 5 weeks Activity D is 5 weeks Activity G is 5 weeks Activity E is 3 weeks

Activities on the critical path have EST = LST. Float times = LST−EST.

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10H Gantt charts

c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 B

Add the non-critical activities to start after their predecessor. Duration of the activity is shown with a solid line and the float times are shown dotted.

U N SA C O M R PL R E EC PA T E G D ES

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Draw the critical path first along the top row of the timeline. The length of each bar is the activity duration. Keep the order of the critical path.

Activity A EST of 0, duration 4 weeks, float time of 5 weeks. Activity D starts after activity A, EST of 4 weeks, duration 1 week, float time of 5 weeks. Activity G starts after activity D, EST of 5 weeks, duration 3 weeks, float time of 5 weeks. Activity E starts after activity B, EST of 4 weeks, duration 4 weeks, float time of 3 weeks.

Now you try

Consider the network diagram shown, where time is in weeks.

2 6

A 0 4

B, 4

A, 2

6 10

C, 3

finish

start

D 0 0

13 13

D, 6

E, 7

6 6

9 10 11 12 13

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Chapter 10 Network flow and critical path analysis

Exercise 10H Example 16

BUILDING

For the following Gantt chart, time is in days. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 E

U N SA C O M R PL R E EC PA T E G D ES

D A

a What is the minimum completion time of the project? b Which activities are on the critical path? State the duration of these activities. c What is the length of activity A and its float time? d What is the earliest start time and latest start time for activity B? e State the length and the float time for Activity B. f What is the earliest start time and latest start time for activity C? g What is the length of activity C and its float time? h The network diagram shown has been drawn from the Gantt chart. Complete the missing information on the network diagram. B, 3

A, 5

D, 9

F, 3

start

finish

i Complete a forward scan and backward scan of the network diagram to check your answers.

For the following Gantt chart, time is in weeks.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 A

a What is the minimum completion time of the project? b Which activities are on the critical path? State the duration of these activities. c What is the length of activity B and its float time? d What is the earliest start time and latest start time for activity C? e What is the length of activity D and its float time? f What is the EST and LST for activity D?

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10H Gantt charts

g Which activities start immediately after activity C? h What is the EST and LST for activity G? i What is the predecessor for activity G? j The network diagram shown has been drawn from the Gantt chart. Complete the missing information on the network diagram. F, 13

U N SA C O M R PL R E EC PA T E G D ES

A, 5

start

finish

C, 3

G, 8

k Complete a forward scan and backward scan of the network diagram to check your answers.

The Gantt chart for a project with nine activities is shown. Time is in weeks.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 B

a Identify the critical path and the minimum time to complete the project. b Find the earliest start time and the latest start time for activities D and E. c Identify the float times for the non-critical activities. d Construct a precedence table showing the duration and immediate predecessors of each activity.

DEVELOPING

Example 17

A forward and backward scan of a network has been completed, showing the duration and completion time of 5 activities in days. 5

D, 4

C, 2

E, 2

EST LST 0 0

11 11

A, 3

B, 3

a What is the minimum completion time? b Write down the critical path for this project. c Determine the float times for the non-critical activities. d Complete the Gantt chart for the network diagram on the timeline shown, showing the critical path, non-critical activities and their float times. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

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A forward and backward scan of a network has been completed, showing the duration and completion time of 8 activities in days. 1

9 10 11

U N SA C O M R PL R E EC PA T E G D ES

A 0

B 0

A, 4

C, 3

G, 2

10 10 H, 3

13 13

D, 2

F, 4

B, 3

D 3

E 3

E, 3

F 6

9 10 11 12 13

For the network diagram shown, time is in weeks.

A, 2

B, 1

E, 4

F, 3

EST LST

C, 5

D, 3

G, 1

H, 2

I, 5

a Complete a forward scan and backward scan to calculate the EST and LST for each activity. b Locate the critical path and the minimum completion time. Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

10H Gantt charts

c Determine the float times of the non-critical activities. d Using the information from the network diagram, compete the Gantt chart shown. Show the critical path and all non-critical activities and their float times. 1

9 10 11 12 13 14 15 16 17 18 19 20

U N SA C O M R PL R E EC PA T E G D ES

e List all activities not on the critical path which could be occurring on week 10.

For the network diagram shown, time is in days. D

H, 3

A, 2

B, 3

D, 3

E, 8

G, 2

C, 4

F, 3

a Complete a forward scan and backward scan to calculate the EST and LST for each activity. b Locate the critical path and the minimum time to complete the project. c Determine the float times of the non-critical activities. d Using the information from the network diagram, compete the Gantt chart. Show the critical path and all non-critical activities and their float times. 0

9 10 11 12 13 14

e List all activities not on the critical path which could be occurring on week 3.

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CHALLENGING 8

For the Gantt charts shown, time is in weeks. For each Gantt chart:

U N SA C O M R PL R E EC PA T E G D ES

i determine the critical path and the minimal time to complete the project ii find the earliest and latest start times for each activity iii draw the network diagram iv construct a precedence table showing the duration and immediate predecessors of each activity. a 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 C

b 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 D

c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 B

9 Using a spreadsheet to model a project, assist with resource allocation and determine the length of the project. A precedence table for a project that has five activities is shown. A Gantt chart has been drawn using a spreadsheet for the project showing the five activities, the sequence of the activities and the critical path. The cells of each activity have been merged and shaded to show an activity’s duration.

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10H Gantt charts

The table on the right shows the total duration of the project to be 8 days. The critical path contains activities B, D and E. There are two lines that need resourcing for this project: activities B-D-E and A-C. Once activity C is finished, then all resources go into the first line to complete the critical path. No float times are shown in the spreadsheet table.

U N SA C O M R PL R E EC PA T E G D ES

For the precedence tables shown, model the project on a spreadsheet, using a table similar to the one above. Merge and shade cells to show the activity duration. For each precedence table: i what is the minimum completion time of each project? ii what is the critical path? a

Activity

Predecessor

Duration (in days)

−

Activity

Predecessor

Duration (in days)

−

B, C

Communicate your reasoning, understanding and fluency 10

The precedence table for some minor home renovations involving nine activities is shown, including each activity’s duration and the immediate predecessors.

Activity

Duration (in min)

Immediate predecessors

−

From the precedence table, construct: a a network diagram b a Gantt chart c a spreadsheet representation (refer to Question 9).

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Chapter 10 Network flow and critical path analysis

Key ideas and chapter summary Maximum flow

If pipes of different capacities are connected one after the other, the maximum flow (or flow capacity) through the pipes is equal to the minimum capacity of the individual pipes.

Cuts

A cut is an imaginary line across a directed graph that completely separates the source (start of the flow) from the sink (destination of the flow).

U N SA C O M R PL R E EC PA T E G D ES

Review

586

Maximum-flow minimum-cut theorem

The flow through a network cannot exceed the value of any cut in the network and that the maximum flow equals the value of the minimum cut.

Critical path analysis

Critical path analysis is a technique that identifies the activities required to complete a project, the time necessary to complete each activity and the relationships between the activities.

Activity chart

A table containing the activities of a project, duration and their immediate predecessors.

Constructing a network diagram given a precedence table

Activities that have no immediate predecessors follow from the start vertex. Activities that are not immediate predecessors for other activities lead to the finish vertex. For every other activity, look for: • the activities for which it is an immediate predecessor • the activities it has as immediate predecessors.

Dummy activities

A dummy activity is required if two activities share some, but not all, of their immediate predecessors.

Scheduling on a network diagram

Scheduling on a network diagram is shown using a weighted edge. The number on the edge indicates the duration of the activity. The weight of dummy activities is always zero.

Earliest starting time (EST)

The earliest time any activity can be started after all prior activities have been completed.

Latest starting time (LST)

The latest time any activity can be started after all prior activities have been completed.

Float times

Float time is the amount of time that a task in a project network can be delayed without causing a delay to subsequent tasks. Float time = LST − EST

Critical path

The critical path is the sequence of activities that cannot be delayed without affecting the overall completion time of the project.

Minimum time

The minimum overall completion time of the project is the EST value at the end vertex.

Crashing

Crashing involves recalculating the minimum completion time for the project after the conditions of an activity within a project have changed.

Gantt charts

A Gantt chart is a visual tool used to represent a project’s schedule. Activities are represented by a horizontal bar displayed along a timeline, showing EST, durations, float times, dependencies and the critical path.

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Chapter 10 review

✓

A printable version of this checklist is available in the Interactive Textbook. Checklist

10A

Review

Chapter checklist with success criteria

1 I can draw a weighted and directed network diagram from information presented in a

U N SA C O M R PL R E EC PA T E G D ES

precedence table.

e.g. The table shows the flow capacities (FC) of the edges of a network. Construct a network diagram and determine the maximum flow from S to T .

10A

From

2 I can calculate the inflow and outflow of a vertex.

e.g. From the diagram shown, calculate the inflow and outflow of vertex B. 3

10A

3 I can calculate and analyse the flow capacity of a network.

e.g. By considering all paths from S to T , calculate the flow capacity of the network. 3

10A

4 I can calculate the cut capacity.

e.g. Find the capacity of each of the cuts in the network diagram. The source is S and the sink is T . C1

3 3

5 7

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10A

5 I can calculate the maximum flow and minimum cut from a network diagram. e.g. By considering all paths from S to T , calculate the flow capacity of the network. Hence, determine the maximum flow and draw the minimum cut of the network. A

C 50

U N SA C O M R PL R E EC PA T E G D ES

10A

6 I can solve network flow problems.

e.g. In the directed diagram shown, the vertices A, B, C, D and E represent towns. The edges of the graph represent roads, and the weights of these edges are the maximum number of cars that can travel on the road each hour. The roads allow only one-way travel. B

600

800

300

10B

500

150

Find the maximum traffic flow from A to E.

If one road was upgraded, which road would you upgrade to increase the maximum flow from A to E? Explain your answer.

If a road from D to C was constructed on this network diagram, what would be the minimum capacity required to increase the number of cars going from A to E? Explain your answer.

7 I can construct a precedence table from a sequence of tasks. e.g. Project: To catch a train.

The tasks shown below are to catch a train. However, they are in the wrong sequence. Wait for train (8 minutes)

Go to the correct platform (2 minutes) Scan your Opal card (1 minute) Board the train (1 minute)

Travel to the train station (5 minutes) Correctly sort the above tasks and construct a precedence table for this project with the four headings: Activity, Task, Duration and Immediate predecessors.

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8 I can read a precedence table. e.g. Mateo wants to tile his balcony. The precedence table for this project is shown. Duration Activity Task

Immediate

Review

10B

(in hours) predecessor

Clean up balcony

−

Prepare the surface

Measure area of balcony

Plan the tiling

Buy waterproofing

Buy Tiles, glue and grout

C, D

Install waterproofing

Tile balcony

F, G

Grout tiles

Final Clean

U N SA C O M R PL R E EC PA T E G D ES

10C

How many activities in the project?

What is the first activity?

Which activity has the longest time interval?

What is the total time for Mateo to complete this project?

What is the immediate predecessor for activity D?

What are the immediate predecessors for activity H?

9 I can construct a network diagram from a precedence table. e.g. Construct a network diagram from the precedence table. Activity Immediate predecessors T

−

V, W

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10C 10 I can construct a precedence table from a network diagram.

e.g. Construct a precedence table from the network diagram. L

J M N

U N SA C O M R PL R E EC PA T E G D ES

T P O

10D 11 I can construct a network diagram from a precedence table to include those involving dummy activities.

e.g. Draw a network diagram from the precedence table, showing the dummy activity. Activity Immediate predecessors A

−

B, C

10E 12 I can apply forward and backward scanning to determine the earliest start time (EST) and latest start time (LST) for each activity in a project.

e.g. Complete a forward and backward scan to find the earliest start times (EST) and the latest start times (LST) for all activities in the network diagram. D, 1

start

B, 5 A, 4

E, 6 finish

C, 2

10F 13 I can calculate float times to identify critical and non-critical activities.

e.g. List the non-critical activities and their float times for the network below, where time is in days. 8 8

A 0 0 start B 0 3

9 9

C, 1

A, 8

E, 3

B, 6

D, 2

6 9

G, 2

12 12

H, 1

14 14

finish 15 15

F, 1

9 11

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e.g. Write down the critical path for this project. 3 4 A 0 1

12 12

B, 8 D, 12

E, 10 F, 20

U N SA C O M R PL R E EC PA T E G D ES

A, 3

Review

10F 14 I can analyse EST and LST to locate the critical path for the project.

C, 7

start D 0 0

22 22

finish 42 42

C 0 15

10G 15 I can apply critical path analysis to determine the minimum completion time for a project.

e.g. What is the minimum time, in days, for the project to be completed? 3 4

A 0 1

12 12

B, 8

D, 12

A, 3

E, 10 F, 20

C, 7

start D 0 0

22 22

finish 42 42

C 0 15

10G 16 I can examine the impact of increasing or decreasing the duration of an activity on the critical path and completion time of the project.

e.g. The network diagram below shows the time (in minutes) for all activities, EST and LST. The critical path is shown in red. 8 8

A 0 0 start B 0 3

C, 1

9 9

E, 3

A, 8

G, 2

B, 6

12 12 14 14 F, 1

D, 2

6 9

H, 1

finish 15 15

9 11

What is the minimum completion time?

Reducing the time taken in which activities would result in a reduction in the minimum completion time?

Both activities A and E are reduced by 2 minutes. What is the new minimum completion time?

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10H 17 I can read a Gantt chart.

e.g. This Gantt chart shows the completion of 7 activities in a project. Time is in days. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 D

U N SA C O M R PL R E EC PA T E G D ES

A B

Calculate the duration of each activity.

Calculate the float time for non-critical activities.

Identify the critical path and minimum completion time.

What is the EST and LST for activity E?

What is the EST and LST for activity F?

What is the predecessor for activity G?

10H 18 I can construct a Gantt chart to represent a project from a network diagram, indicating the critical path.

e.g. Draw a Gantt chart from the network diagram provided. Determine the critical path and the minimum time to complete the project. Time is in days. F 13 13

G 13 14

A 0 1 A, 10 start B 0 0

14 14

F, 1

10 11 D, 2 B, 9

E, 4

G, 1 14 15

H, 3

9 9

I, 2

C 0 14

J, 1

ﬁnish 18 18

C, 3

17 17

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Duration Immediate Activity (in hrs) predecessors A

−

U N SA C O M R PL R E EC PA T E G D ES

e.g. Construct a Gantt chart from the precedence table provided. Determine the critical path and the minimum time to complete the project.

Review

10H 19 I can construct a Gantt chart to represent a project from a precedence table, indicating the critical path.

10H 20 I can construct a network diagram and a precedence table from a Gantt chart.

e.g. This Gantt chart shows the completion of 7 activities in a project. Time is in days. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 D

Construct a network diagram and a precedence table using the Gantt chart.

10H 21 I can use a spreadsheet to model a project, construct a table and merge and shade cells to indicate an activity’s length from a precedence table.

e.g. Construct a spreadsheet representation of this precedence table.

Activity

Duration Immediate (in min) predecessors

−

B, C

D, E

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Multiple-choice questions 10A

For the network shown below, the capacity of the cut is: 7 9 source

2 4

sink

U N SA C O M R PL R E EC PA T E G D ES

A 3

10A

B 7

C 10

D 15.

The maximum flow in the network diagram below is: B

A 10

10E

B 11

C 12

D 13.

The network diagram represents a manufacturing process with activities and their duration (in hours) listed on the edges of the graph. What is the earliest time (in hours) after the start that activity G can begin? E, 1

A, 2

G, 1

C, 4

F, 2

H, 4

B, 3

D, 2

A 3

10C

B 6

C 7

D 8

The table below lists the six activities in a project and the earliest start time, in hours, and the predecessor(s) of each task. The time taken for activity E is two hours. Without affecting the time taken for the entire project, how much could the time taken for activity C be increased by?

A 0 hours

Activity

Immediate predecessors

EST

−

D, E

B 8 hours

C 11 hours

D 22 hours

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The edges in this network diagram correspond to the tasks involved in the preparation of an examination. The numbers indicate the time, in weeks, needed for each task. What is the minimum completion time to prepare this examination? B, 6 A, 3

Review

10E

E, 2 H, 6

D, 5 G, 4

U N SA C O M R PL R E EC PA T E G D ES

C, 2

F, 3

A 14 B 15 C 16 D 17

10E

This network diagram is for a project where the component times are shown in days. The critical path for the network of this project is given by: E, 3

B, 4

A, 5

C, 3

D, 6

G, 2

H, 3

I, 3

K, 6

J, 1

F, 6

A A−B−E−I−K B A−D−H−I−K C A−C−G−H−I−K D A − C − F − J − K.

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Chapter 10 Network flow and critical path analysis

Short-answer questions 10A

In the network shown, the values on the edges give the maximum flow possible between each pair of vertices. The arrows show the direction of flow in the network. Also shown is a cut that separates the source (S) from the sink (T).

3 8 10

9 5

U N SA C O M R PL R E EC PA T E G D ES

Review

596

a Find the capacity of the cut shown. b Determine the maximum flow through this network.

10A

Determine the maximum flow from S to T for the network graph shown. Check your answer by finding the minimum cut.

Cut

160

110

135

100

120

205

10B

Camila is renovating her family room. She has organised this project into six tasks. Buy hardware – 1 day Prepare walls – 1 day Paint walls – 4 days Sand timber floor – 3 days Seal floor – 2 days Put furniture in the room – 1 day.

a Which task has no immediate predecessor? Give a reason for your answer. b What is the immediate predecessor to the following tasks? i Buy hardware ii Prepare walls iii Paint walls iv Sand timber floor v Seal floor vi Put furniture in the room

c Construct a precedence table for this project. Use the headings; Activity, Task, Duration (in days) and Immediate predecessors. Label the above tasks as activities A to F respectively. d Draw a network diagram from this precedence table.

10C

Draw a precedence table for the network diagram shown. C, 5

A, 9

D, 7

G, 6

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Jet 12 makes light aircraft for the civil aviation industry. They have identified 10 activities required for production of their new aircraft. These activities and their durations are given in this table.

Activity

Duration (weeks)

Immediate predecessors

−

E, F

D, G

H, I

U N SA C O M R PL R E EC PA T E G D ES

Review

10C

a Copy and complete the network diagram shown for this project by inserting and labelling activity D.

start

A, 4

F, 4

C, 7 B, 5

H, 4

E, 7

G, 2

J, 1

I, 2

finish

b Find the earliest starting times (EST) for all activities. c Find the latest starting times (LST) for all activities. d What is the critical path for this network?

10F

A network diagram for a project is shown. The duration for each activity is in hours.

a Calculate the float times for all the activities. b Identify the critical path for this project. c What is the minimum completion time for the project?

2 15

M, 7

I, 2

I 0 13 start

18 20

K, 5

M 11 13

J, 5

J 0 0

L, 6

O 11 15

N, 1

O, 6

finish

P 11 11 Q, 4

21 21

P, 6

5 5

17 17

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Extended-response questions 10C 1 10E

A development project involves completing a number of activities as shown in the table. Each activity is shown with its duration. Time is measured in days.

10F

Immediate predecessors

−

C, D

G, H

U N SA C O M R PL R E EC PA T E G D ES

Activity

Duration (days)

a Construct a network diagram for this project, labelling the activities on the edges with their associated shortest durations. b Determine the earliest start time for each activity from your graph. c Determine the latest start time for each activity from your graph. d State the critical path. e How long is the estimated project time under this set of activity durations? f If the final activity, K, had to be delayed, how many days could this delay take before the project schedule was disrupted?

10H 2

Consider the network diagram shown, where time is in weeks. D, 5

A, 3

G, 2

B, 4

start

E, 10

finish

H, 2

C, 6

F, 4

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U N SA C O M R PL R E EC PA T E G D ES

Review

a Complete a forward scan and backward scan to determine the earliest start time (EST) and latest start time (LST) for each activity. b What is the minimum completion time for this project? c Determine the critical path. d Determine the float time for the non-critical activities. e Complete the Gantt chart for the network diagram on the timeline below, showing the critical path, non-critical activities and their float times. 0

10 11 12 13 14 15 16

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Practice Paper 1 Section I Attempt Questions 1–15 (15 marks). Allow about 20 minutes for this section. A 4 L tin of paint is made using a mixture of blue, white and green paint in the ratio 3 : 5 : 2. How much blue paint is needed per tin?

U N SA C O M R PL R E EC PA T E G D ES

A 300 mL

D 20

B $27.61

C $69.04

D $276.14

B 1 : 15

C 15 : 10

D 15 : 1

B $27 000

C $703 023

D $853 023

A car uses on average 7 L per 100 km in fuel. How much fuel would be used on a trip of 382 km? A 26.74 L

C 10

What is the amount of interest paid on a $150 000 loan over 25 years if the interest rate is 7.2% p.a. compounding annually? Answer correct to the nearest dollar. A $10 800

B 3

The ratio of 1.5 m to 10 cm is: A 1.5 : 10

D 2000 mL

A bank charges 0.05753% compound interest per day on the amount owing on a credit card. What is the interest charged in four weeks on a balance of $1200? A $19.48

C 1800 mL

A car is travelling at a constant speed. It travels 60 km in 3 hours. This situation is described by the linear equation d = mt. What is the value of m?

A 0.05

B 1200 mL

B 34.72 L

C 38.20 L

D 54.57 L

When two coins are tossed together, what is the probability that only one of them shows heads? 1 1 1 B C D A 0 4 3 2 A fair coin is tossed 10 times and the result is 10 heads in a row. Which statement about the next toss is the most correct?

A The chance of tossing a head on the next toss is basically 0. B There is a very good chance of tossing a head on the next toss. C That experiment was rigged. D It is unusual to toss a head 10 times in a row, but the chance of tossing a head on the next toss is still 50%. 9 When three coins are tossed, what is the probability of tossing one or more heads? 1 1 7 A 0 B C D 8 2 8 10 Two standard six-sided dice are rolled. The two scores are added together to give a total score. What is the most likely total score? A 5

B 6

C 7

D 8

Sadie has planted red and white rose bushes in the ratio 2 : 3. How many white rose bushes are there if she planted a total of 30 rose bushes? A 6

B 12

C 18

D 20

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Practice Paper 1

Which symbol represents a toilet in a floor plan? A

Which of the following amounts would earn approximately $9 in interest in 29 days, if the interest is compounded at a daily rate of 0.037%? B $850

C $971

D $24 649

U N SA C O M R PL R E EC PA T E G D ES

A $246

Nicholas uses a 1.6−kilowatt dishwasher for a total of 2.5 hours. He is charged at a rate of 27.4 cents per kilowatt-hour. What is the cost of using the dishwasher? A $1.10

B $1.12

C $4.00

D $109.60

Suneil breathes about 15 times each minute. How many times would he breathe in 9 hours? A 90

B 216

C 8100

D 486 000

Section II

Attempt Questions 16–18 (45 marks). Allow about 70 minutes for this section. All necessary working should be shown in every question. Question 16 (15 marks) a

A group of three people share a sum of $1200 in the ratio 5 : 3 : 8. How much does each person receive? b The diagram shows a site plan. 32 m

(2 marks)

20 m

12 m

House

Garage

Not to scale

12 m

i What is the total area of land of the house and garage? ii What percentage of the site is taken up by the house and garage? Answer correct to one decimal place. iii There is a brick wall, 800 mm high, along the northern boundary of the site. What is the area of the brick wall in square metres? iv 600 bricks are needed to build 10 square metres of wall. How many bricks were used to build the wall in part iii? v The guttering around the perimeter of the house is to be replaced. Guttering costs $60 per metre. How much will the new guttering cost?

(1 mark)

(2 marks) (1 mark) (1 mark)

(2 marks)

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Practice Paper 1

Kane bought 1000 shares for $6.40 each. He received a dividend of 5.5% on his purchase price. Kane sold these shares for $7.20 each after 3 months. Calculate Kane’s profit from the dividend and the sale of these shares. d The table below shows the banking charges for credit card usage. Fee

(3 marks)

Charge 5%

Late payment

$30

Balance transfer

1.25%

U N SA C O M R PL R E EC PA T E G D ES

Cash advance

i Find the cost of 40 cash advances of $100. ii Find the cost of a balance transfer of $7000. iii Find the cost of 6 late payments and 5 cash advances of $200.

(1 mark) (1 mark) (1 mark)

Question 17 (15 marks) a A rectangular poster is to be made to advertise energy-efficient housing. It is to be 1.6 m long and 1 m wide. i Draw a scale drawing of the rectangle using the scale 1 : 20. ii Outline two features of an energy-efficient house that could be part of the poster.

(1 mark) (2 marks)

A painting was bought for $695 at the beginning of 2017. It is expected that the painting will appreciate in value by 8% each year. What will be the value of the painting at the end of 2027? Answer correct to the nearest dollar. c Light travels at a speed of 2.9979 × 108 metres per second. How many kilometres does light travel in one hour? Answer in scientific notation correct to three significant figures. d A card is chosen from a standard pack of 52 playing cards. What is the probability that the card is: i an ACE ii a red ACE iii a red card iv a red card or an ACE v red or black vi a red picture card vii neither red nor a picture card.

(2 marks)

(3 marks)

(1 mark) (1 mark) (1 mark) (1 mark) (1 mark) (1 mark) (1 mark)

Question 18 (15 marks) a Ezra plays cards with a normal deck and draws a card from the deck. i What are the chances of drawing a 3 or a black card? ii What are the chances of not drawing an ace?

(1 mark) (1 mark)

i Harriet has a debt of $15 890 with an interest rate of 17% p.a. compounded annually. How much interest is charged if she makes no payments for 2 years? (2 marks) ii Harriet could have taken the loan at 15% p.a. compounded monthly. Would she have paid more or less interest this way? (2 marks)

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Practice Paper 1

Rosa uses a hot water unit with a rating of 12 kW for 10 minutes per day. i How much electricity is used each day? ii Electricity is charged at 15.25 cents per kWh. What is the electricity cost of using the hot water unit for one week?

(1 mark)

Toni bought a new car for $45 000. When will the depreciated value of the car fall below $10 000 using the declining-balance method of depreciation if the rate of depreciation is 15% p.a.?

(2 marks)

Complete the two-way table and the Venn diagram for a standard pack of 52 playing cards.

(5 marks)

U N SA C O M R PL R E EC PA T E G D ES

(1 mark)

Red card

Not red card

Total

Picture card

Not picture card Total

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Practice Paper 2 Section I Attempt Questions 1–15 (15 marks). Allow about 20 minutes for this section. Which trignometric formula would be most useful in calculating the length of side BC?

U N SA C O M R PL R E EC PA T E G D ES

A c2 = a2 + b2 − 2ab cos C

10 m

1 B A = ab sin C 2 a2 + b2 − c2 C cos C = 2ab b c a D = = sin A sin B sin C

144°

The scatterplot shows the weights of Great Danes at 1 month old and at 1 year old. How is this association best described? A Weak positive non-linear B Weak negative non-linear C Strong positive linear D Strong negative linear

65 60 55 50

45 1.5

Weight at 1 year (kg)

7°

What is the value of sin θ in the triangle below? b a A B c b b a C D a c

2.5 3 3.5 4 Weight at 1 month (kg)

4.5

What is the value of y predicted by the least-squares regression line y = −0.78x − 5.6 when x = 10? A −13.4

B −7.2

C 2.2

D 7.8

5 Which of the following expressions would give the height (h), of the tree in the diagram? A 42 × tan 34◦

42 tan 34◦

34° 42 m

C 42 × cos 34◦ D

42 cos 34◦

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Practice Paper 2

Use the following graph to answer questions 6 to 8. y 4 3 2

U N SA C O M R PL R E EC PA T E G D ES

−4 −3 −2 −1 O −1

4 x

−2 −3 −4

line A

What is the gradient of line B? A −3

line B

B 3

C −

1 3

What is the equation of line A? A y − 2x = 3

B x−y=3

C y−x=3

D −2y − x = 6

8 What is the point of intersection of line A and line B? A (−1, −2)

B (−1, 2)

C (1, −2)

9 Which of the following equations is closest to this line of best fit? A y = −0.9x + 8.7 B y = −8.7x + 0.9 C y = 0.9x − 8.7 D y = 8.7x − 0.9

D (−3, 3)

10 9 8 7 6 5 4 3 2 1

1 2 3 4 5 6 7 8 9 10

What is the approximate length of the side marked with the letter x? A 10 B 31 C 39 D 67

127° 25.8 A

18° x C

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Practice Paper 2

The compass bearing of Y from X is S49◦ W. What is the compass bearing of X from Y? A S49◦ W B S41◦ W C N49◦ E D N41◦ E

U N SA C O M R PL R E EC PA T E G D ES

49° Y

For the graph opposite, R is the revenue, in dollars, earned by a mushroom farmer selling n kilograms of mushrooms, and C is the cost, in dollars, of producing n kilograms of mushrooms. To make a profit, how many kilograms of mushrooms should the farmer’s production exceed? A 25 B 50 C 100 D 250

R = 7n

C = 500 + 5n

Suppose that the least-squares regression line which would enable the amount of money spent at the shopping centre (in dollars) to be predicted from the number of hours spent at the shopping centre is given by: Shopping expenditure = 20 + 50 × Hours spent shopping

The slope of the line can be interpreted as follows.

A On average, for each extra 50 minutes spent shopping an extra $1 is spent. B On average, for each extra hour spent shopping an extra $50 is spent. C On average, customers spend $20 per hour at the shopping centre. D On average, customers spend $50 at the shopping centre.

A sailing boat travels due east from A to B. It then turns and sails on a bearing of 210◦ . What is the boat’s bearing from A when the boat is closest to A?

A 030◦ B 120◦ C 150◦ D 210◦

If Sue buys three packets of chips and two bottles of cola, it will cost her $13.20. However, if she buys two packets of chips and three bottles of cola it will cost her only $11.80. What is the price of each item?

A cola = $3.20, chips = $1.80 B cola = $2.10, chips = $3 C cola = $1.80, chips = $3.20 D cola = $2, chips = $2.90 Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Practice Paper 2

Section II Attempt Questions 16–18 (45 marks). Allow about 70 minutes for this section. All necessary working should be shown in every question.

U N SA C O M R PL R E EC PA T E G D ES

Question 16 (15 marks) a The cost ($C) of hiring a car is given by C = 2.5x + 65, where x is the number of kilometres travelled. A person is charged $750 for the hire of the car. What was the number of kilometres travelled? b Calculate Pearson’s correlation coefficient (to three decimal places) for the following table.

(2 marks) (1 mark)

A radial survey of a park is shown below.

P 040°

304° R

82 m

81 m

92 m

Q 142°

i What is the size of ∠POR? ii Use the cosine rule to calculate the length of PR, correct to the nearest metre.

(1 mark) (2 marks)

d What is the point of intersection of the lines x + 3y = −4 and x + y = 2? e A tower 110 m high stands on the top of a hill. From a point, A, at the foot of the hill, the angle of elevation of the bottom of the tower is 7◦ , and to the top of the tower it is 10◦ .

(3 marks)

110 m

10°

7°

i Find the size of angles T AB and AT B. ii Use the sine rule to find the length AB. Answer correct to the nearest metre. iii Find CB, the height of the hill. Answer correct to the nearest metre.

(2 marks) (2 marks) (2 marks)

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Practice Paper 2

Question 17 (15 marks) a The graphs show the cost of growing rose plants and the income received from selling them.

Income

60 Costs

U N SA C O M R PL R E EC PA T E G D ES

Dollars ($)

2 3 Roses

i Use the graph to determine the number of rose plants that need to be sold to break even. ii How much profit or loss is made when 1 rose is sold? iii How much profit or loss is made when 4 roses are sold? iv What is the initial cost?

(1 mark) (1 mark) (1 mark) (1 mark)

Calculate the size of the smallest angle in this triangle, correct to the nearest degree.

(2 marks)

34 cm

24 cm

24.3 cm

The cost of preparing meals in a school canteen is linearly related to the number of meals prepared. To help the caterers predict the costs, data were collected on the cost of preparing meals for different levels of demand. The data are shown below. Number of meals

Cost (dollars)

138

154

159

182

198

214

208

238

234

244

i Find the equation of the least-squares line of best fit. Answer correct to one decimal place. ii What is the predicted cost of producing 48 meals? iii What is the predicted number of meals if the cost is $249.50?

(2 marks) (1 mark) (1 mark)

A straight line has the equation of 2x + y = −2.

i What is the gradient of this line? ii What is the y-intercept? iii What is the point of intersection between 2x + y = −2 and y = 2?

e Every Year 12 student in a school had their height and armspan measured and recorded. Pearson’s correlation coefficient between these variables was 0.9. Describe the association between height and armspan.

(1 mark) (1 mark) (1 mark)

(2 marks)

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Practice Paper 2

Female income (dollars)

40 000

35 000

U N SA C O M R PL R E EC PA T E G D ES

Question 18 (15 marks) a The scatterplot shows the average annual female income plotted against average annual male income for 16 countries. A least-squares regression line is fitted to the data.

30 000

25 000

20 000 30 000

34 000 50 000 60 000 Male income (dollars)

i What was the female income for a country whose average annual male income was $60 000? ii How many countries had an average annual female income greater than $30 000? iii Find the gradient of the least-squares line of best fit. Answer correct to one decimal place.

70 000

(1 mark) (1 mark) (1 mark)

In the triangle ABC, AB = 8 cm, BC = 5 cm and CD = 4 cm. B

8 cm

5 cm

D 4 cm

i Find the size of angle BCD, correct to the nearest degree. ii What is the length of AD? Answer correct to two decimal places. iii Find the area of triangle ABC. Answer correct to two decimal places.

Lia sells pillows for $65 each.

i Write an equation for the revenue, R dollars, that Lia receives from the sale of x pillows. ii The cost, C dollars, of making pillows is given by C = 500 + 40x. Find the cost of making 30 pillows. iii Draw the graph of C = 500 + 40x and your equation in part i on the same number plane. iv Use the graph to estimate the number of pillows Lia will need to sell to break even.

(1 mark) (2 marks) (1 mark) (1 mark)

(1 mark) (2 marks) (1 mark)

For time ranging from 30 to 60 seconds, the equation relating the number of correct answers to time is: Correct answers = 0.72 × time + 30 Use this equation to predict the number of correct answers with the following times. Are you interpolating or extrapolating? i 25 seconds

ii 50 seconds

iii 75 seconds

(3 marks)

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Practice Paper 3 Section I Attempt Questions 1–15 (15 marks). Allow about 20 minutes for this section. The maximum flow in the network diagram is shown below:

U N SA C O M R PL R E EC PA T E G D ES

A 4

B 10

D 21

The amount of money in a fund is given by A = 800 (1.1)t where A is the amount of money and t is the time in years. What is the initial amount of money invested in the fund? A $800

C 14

B $880

C $1000

D $1100

In the fund described in Question 2, by what percentage does the fund increase each year?

A 1%

B 1.1%

C 10%

D 110%

4 In a normally distributed set of scores, the mean is 70 and the standard deviation is 6. Approximately what percentage of the scores will lie between 58 and 82? A 34%

B 68%

C 95%

D 99.7%

C (7, 0)

D (1, 0)

5 The graph of y = 3x2 − 6x + 7 meets the y-axis at: A (0, 1)

B (0, 7)

6 Which graph best represents y = 2 x ? A

6 5

4 3

2 1

x 1

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Practice Paper 3

U N SA C O M R PL R E EC PA T E G D ES

Use the table below to answer questions 7 and 8.

7 A reducing-balance loan of $8000 will be repaid with quarterly installments. The first three periods are shown below. Payment number

Payment paid

Interest paid

Principal reduction

Balance of loan

0.00

8000

2043.94

70.00

1973.94

6026.06

2043.94

52.73

1991.21

4034.85

The annual percentage interest rate on this loan is: A 1.2%

B $3965.15

C $4034.85

B around average

D $6016.06

C exactly average

D well above average

How many of these activities could be delayed without affecting the minimum completion time of the project? start

A, 5

B, 3

C, 3

G, 8

H, 2

D, 5 E, 4

A 2

D 8.75%

In an examination, Morgan’s z-score was z = 2.1. This result is:

A well below average

C 3.5%

What is the reduction in the above loan after 6 months? A $2043.94

B 3.42%

B 3

I, 2

finish

F, 2

C 4

D 5

For the network shown on the right, the capacity of the cut is 30. What is the value of x? 5

9 10

8 x

A 0

B 7

C 10

D 14

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602

Practice Paper 3

The time taken to make a bus trip between the two towns is normally distributed with a mean of 78 minutes and a standard deviation of 4 minutes. A standardised time for a trip is z = −0.25. The actual time (in minutes) is: A 77

C 77.75

D 78.25

The cost per person of hiring a yacht (c) varies inversely with the number of people sharing the total cost (n). Which formula correctly expresses c in terms of n and k, where k is the constant of variation? k n C c= D c= A c = kn B c = kn2 k n

U N SA C O M R PL R E EC PA T E G D ES

B 77.25

A student’s mark on a test is 64. The mean mark for their class is 73 and the standard deviation is 4.5. Their z-score is: A −2.0

B −1.5

C 2.0

D 2.5

The network diagram shows the activities and their duration (in hours) listed on the edges of the graph. The earliest time (in hours) after the start that activity H can begin is: E, 5

A, 5

B, 3 C

A 4

B 10

F, 4

C, 3

start

G, 2

D, 1

finish

H, 3

C 11

D 13

Section II

Attempt Questions 16–18 (45 marks). Allow about 70 minutes for this section. All necessary working should be shown in every question.

Question 16 (15 marks) a A factory produces bags of sultanas. The weights of the bags are normally distributed, with a mean of 900 g and a standard deviation of 50 g. What percentage of bags weigh more than 1000 g? b Asher borrows $14 000 to build fishing platforms and shelters around his lake. Interest will be charged at the rate of 9.6% per annum, compounding monthly. Asher will repay the loan with monthly payments of $760. i Find the value of the loan after 1 month. ii How much interest did Asher pay with the first payment of $760? iii By how much has the balance of the loan been reduced by the first payment of $760? iv Find the value of the loan after five months, correct to the nearest cent. v Show that Asher paid a total of $507.74 interest, correct to the nearest cent, after five months.

(2 marks)

(1 mark) (1 mark) (1 mark) (2 marks) (2 marks)

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

Practice Paper 3

The number of people (N) who attend a show varies inversely with the amount of floor space (in cm2 ) allowed per person (A). A venue can hold 4800 people if each person is allowed 200 cm2 . i How many people could the venue hold if a person was allowed 2400 cm2 ? ii What is the space allowed per person for 5000 people to attend the show? A network diagram with six activities is shown below.

U N SA C O M R PL R E EC PA T E G D ES

(2 marks) (1 mark)

i What activities have no predecessor? ii Construct a precedence table for this network diagram.

(1 mark) (2 marks)

Question 17 (15 marks) i Complete the table of values for the equation y = x2 − 4. a

(1 mark)

−3

−2

−1

ii Draw the graph of y = x2 − 4. iii Complete the table of values for the equation y = −x2 + 2x + 8. x

−2

−1

(1 mark) (1 mark)

iv Draw the graph of y = −x2 + 2x + 8.

(1 mark)

The table below shows the future value of an annuity with a contribution of $1 per period. XXX XXX Rate XXX Period XXX

1.0000

2.0100

2.0200

2.0300

2.0400

2.0500

2.0600

3.0301

3.0604

3.0909

3.1216

3.1525

3.1836

4.0604

4.1216

4.1836

4.2465

4.3101

4.3746

Use the above table to calculate the future value of the following annuities. i $12 000 per year for 3 years at 2% p.a. compounded annually. ii $8 900 per half-year for 2 years at 6% p.a. compounded six-monthly.

(1 mark) (1 mark)

Results for a class test are given as z-scores. In this test, Noah gained a z-score of −2.

i Interpret this z-score in terms of the mean and standard deviation of the test. ii If the test has a mean of 67 and a standard deviation of 7, calculate the actual mark scored by Noah.

(1 mark) (1 mark)

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © Powers, et al 2026 • 978-1-009-76285-4 • (03) 8671 1400

603

604

Practice Paper 3

The network diagram shows the time (in days) for all activities, EST and LST. D 9 B 3 3 B, 6 A, 3 C

E, 4 15 15

C, 3

3 12

finish

U N SA C O M R PL R E EC PA T E G D ES

start 0 0

9 D, 2 E 11 11

i Find the float times for all the activities. ii What is the critical path for this project?

(2 marks) (1 mark)

The weights of boxes of muesli are normally distributed. The mean is 756 grams and the standard deviation is 4 grams.

i A box of muesli has a weight of 748 grams. What is the z-score? ii What is the weight of a box that has a z-score of 3? iii What percentage of boxes will have a weight between 752 grams and 760 grams? iv If a box of muesli is chosen at random, what is the probability that it weighs more than 764 grams?

(1 mark) (1 mark) (1 mark) (1 mark)

Question 18 (15 marks) a The table shows the present value of a $1 annuity.

Present value of $1

Period

10%

0.9901

0.9804

0.9615

0.9434

0.9259

0.9091

1.9704

1.9416

1.8861

1.8334

1.7833

1.7355

2.9410

2.8839

2.7751

2.6730

2.5771

2.4869

3.9020

3.8077

3.6299

3.4651

3.3121

3.1699

4.8534

4.7135

4.4518

4.2124

3.9927

3.7908

5.7955

5.6014

5.2421

4.9173

4.6229

4.3553

i What would be the present value of an $11 000 per year annuity at 10% per annum for 5 years, with interest compounding annually? ii An annuity of $8000 each three months is invested at 4% per annum, compounded quarterly for 1 year. What is the present value of the annuity? iii What is the value of the payment that would provide a present value of $47 934 after 3 years at 8% per annum compound interest? Answer correct to the nearest dollar.

(1 mark) (1 mark)

(2 marks)

The ages of the residents in Green Hill are normally distributed. The mean age is 42 years and the standard deviation is 11 years.

i Determine the percentage of residents younger than 42 years. ii Determine the percentage of residents younger than 53 years. iii If a resident of Green Hill is chosen at random, what is the probability that they are older than 64 years?

(1 mark) (1 mark) (1 mark)

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Practice Paper 3

The number of bacteria grows exponentially according to the formula b = 30 (1.2)t , where b is the number of bacteria after t hours. i What is the initial number of bacteria? ii Copy and complete the table of ordered pairs. Express the number of bacteria correct to the nearest whole number. 0

(2 marks)

U N SA C O M R PL R E EC PA T E G D ES

(1 mark)

iii Draw the graph of b = 30 (1.2)t . iv Estimate the time taken for the number of bacteria to reach 120.

(1 mark) (1 mark)

Determine the maximum flow from S to T for the network graph shown below. Check your answer by finding the minimum cut.

(3 marks)

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