NSM9_51_53

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Enhanced

Mathematics

9

Enhanced

STAGE

5.1–5.3

Alan McSeveny

Rob Conway

Steve Wilkes

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Let the wise listen and add to their learning,

and let the discerning get guidance.

Proverbs 1:5

Pearson Australia

(a division of Pearson Australia Group Pty Ltd) 20 Thackray Road, Port Melbourne, Victoria 3207 PO Box 460, Port Melbourne, Victoria 3207 www.pearson.com.au

Other offices in Sydney, Melbourne, Brisbane, Perth, Adelaide and associated companies throughout the world.

10 9 8 7 6 5 4

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(www.copyright.com.au).

Reproduction and communication for other purposes

Except as permitted under the Act (for example any fair dealing for the purposes of study, research, criticism or review), no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All enquiries should be made to the publisher at the address above. This book is not to be treated as a blackline master; that is, any photocopying beyond fair dealing requires prior written permission.

Publisher: Leah Kelly Editor: Liz Waud Designer: Pierluigi Vido Typesetter: Nikki M Group

Cover Designers: Bob Mitchell and Ruth Comey Copyright & Pictures Editor: Michelle Jellett Project Editor: Carlie Jennings

Production Controller: Jem Wolfenden Cover art: Corbis Australia Pty Ltd

Illustrators: Michael Barter, Bruce Rankin and Wendy Gorton Printed in China

National Library of Australia Cataloguing-in-Publication entry McSeveny, A. (Alan)

New signpost mathematics enhanced 9 / Alan McSeveny, Rob Conway and Steve Wilkes.

9781442506978 (pbk. : Stage 5.1–5.3) Includes index.

For secondary school age. Mathematics--Textbooks. Other Authors/Contributors: Conway, Rob. Wilkes, Steve. 510

Pearson Australia Group Pty Ltd ABN 40 004 245 943 NSME 9 5-3 SB_00_ii-vii.fm Page ii Tuesday, November 30, 2010 11:22 AM

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

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B Exercises using triangles 125 C Exercises using quadrilaterals 127

5:02 Polygons 129

The angle sum of a polygon 130 The exterior angle sum of a convex polygon 131 Regular polygons and tessellations 133

Spreadsheet 134

The game of Hex 135

5:03 Deductive reasoning in non-numerical

exercises 136

5:04 Congruent triangles 139 5:05 Proving two triangles congruent 143 5:06 Using congruent triangles to find unknown

sides and angles 147 5:07 Deductive geometry and triangles 149 5:08 Deductive geometry and quadrilaterals 153

Theorems and their converses 158 What do you call a man with a shovel? 158

5:09 Pythagoras’ theorem and its converse 159

Proving Pythagoras’ theorem 159 Maths terms • Diagnostic test • Revision

Indices and Surds 167

6:01 Indices and the index laws 168

Exploring index notation 172

Family trees 172

6:02 Negative indices 173

Zero and negative indices 176

6:03 Fractional indices 177

Why is a room full of married people

always empty? 180

Reasoning with fractional indices 180

6:04 Scientific (or standard) notation 181

Multiplying and dividing by powers of 10 181

6:05 Scientific notation and the calculator 184

Using scientific notation 186

6:06 The real number system 187

Proof that is irrational 189

f-stops and 190

6:07 Surds 191

6:08 Addition and subtraction of surds 193 6:09 Multiplication and division of surds 195

Iteration to find square roots 197

6:10 Binomial products 198 6:11 Rationalising the denominator 200

What do Eskimos sing at birthday parties? 201 Rationalising binomial denominators 202 Maths terms • Diagnostic test • Revision

Measurement 208

7:01 Perimeter 209

Staggered starts 214

Skirting board and perimeter 215

Covering floors 223

7:03 Surface area of prisms and cylinders 224

How did the boy know that he had an

affinity with the sea? 229

7:04 Surface area of composite solids 230

Truncated cubes 232

7:05 Volume of prisms, cylinders and composite

solids 233

Perimeter, area and volume 237

7:06 Practical applications of measurement 238

Wallpapering rooms 242

Equations, Inequations and Formulae 248

8:01 Equivalent equations 249 8:02 Equations with grouping symbols 252

If I have 7 apples in one hand and 4 in the other, what have I got? 254 Solving equations using a spreadsheet 254

8:03 Equations with fractions (1) 255

Who holds up submarines? 257

8:04 Equations with fractions (2) 257

Equations with pronumerals in the

denominator 259

8:05 Solving problems using equations 260

Who ‘dunnit’? 265

8:06 Inequations 265

Operating on inequations 266 Read carefully (and think!) 269

8:07 Formulae: Evaluating the subject 270

Spreadsheet formulae 273

8:08 Formulae: Equations arising from

substitution 274

8:09 Solving literal equations (1) 277 8:10 Solving literal equations (2) 279 8:11 Solving problems with formulae 282

Why are cooks cruel? 285

Consumer Arithmetic 291

9:01 Working for others 292 9:02 Extra payments 296

Jobs in the papers 299

9:03 Wage deductions 300

9:04 Taxation 304

Income tax returns 306

What is brought to the table, cut,

but never eaten? 307

9:05 Budgeting 308

9:06 Best buy, shopping lists and change 310 9:07 Goods and services tax (GST) 314

Shopper dockets 316

9:08 Ways of paying and discounts 317

Chapter 6

2 2

Chapter 7

Chapter 8

Chapter 9

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Let’s plan a disco 325 Maths terms • Diagnostic test • Revision

Coordinate Geometry 330

10.01 The distance between two points 331 10.02 The midpoint of an interval 336 10.03 The gradient of a line 340

Gradients in building 345

10.04 Graphing straight lines 346

What is the easiest job in a watch factory? 351

10.05 The gradient–intercept form of a

straight line: y = mx + b 352

What does y = mx + b tell us? 352

10.06 The equation of a straight line, given point

and gradient 358

10.07 The equation of a straight line, given

two points 360

10.08 Parallel and perpendicular lines 363 10.09 Graphing inequalities on the number plane 367

Why did the banana go out with a fig? 371 Maths terms • Diagnostic test • Revision

Factorising Algebraic Expressions 377

11:01 Factorising using common factors 378 11:02 Factorising by grouping in pairs 380 11:03 Factorising using the difference of

two squares 382

The difference of two cubes 383

11:04 Factorising quadratic trinomials 384

How much logic do you have? 385

11:05 Factorising further quadratic trinomials 386

Another factorising method for harder

trinomials 389

11:06 Factorising: Miscellaneous types 390

What did the caterpillar say when it saw

the butterfly? 391

11:07 Simplifying algebraic fractions:

Multiplication and division 392 11:08 Addition and subtraction of algebraic

fractions 395

Statistics 402

12:01 Frequency and cumulative frequency 403 12:02 Analysing data (1) 410

Codebreaking and statistics 413

12:03 Analysing data (2) 414

Which hand should you use to stir tea? 421

Adding and averaging 422

12:04 Grouped data 423

The aging population 428

Simultaneous Equations 436

Solving problems by ‘guess and check’ 437

13:01 The graphical method of solution 438

Solving simultaneous equations using a

graphics calculator 442

What did the book say to the librarian 442

13:02 The algebraic method of solution 443 A Substitution method 443 B Elimination method 445 13:03 Using simultaneous equations to solve

problems 448

Breakfast time 451

Trigonometry 455

14:01 Right-angled triangles 456 14:02 Right-angled triangles: the ratio of sides 458 14:03 The trigonometric ratios 460 14:04 Trig. ratios and the calculator 466

The exact values for the trig. ratio of

30°, 60° and 45° 469

14:05 Finding an unknown side 470 14:06 Finding an unknown angle 476 14:07 Miscellaneous exercises 479 14:08 Problems involving two right triangles 484

What small rivers flow into the Nile? 487 Maths terms • Diagnostic test • Revision

Graphs of Physical Phenomena 492

15:01 Distance/time graphs 493 A Linear graphs 493

Graphing coins 502

Can you count around corners? 502

B Non-linear graphs 503

Rolling down an inclined plane 509

15:02 Relating graphs to physical phenomena 510

Spreadsheet graphs 519

Make words with your calculator 520 Curves and stopping distances 521 Maths terms • Diagnostic test • Revision

Answers 528 Answers to ID Cards 598 Index 599 Acknowledgements 604

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

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Interactive Student CD

1:02A Integers 2

Set A Addition and subtraction of integers 2 Set B Integers: Signs occurring side by side 2 Set C Multiplication and division of integers 3 Set D Order of operations 4

1:02B Fractions 5

Set A Improper fractions to mixed numerals 5 Set B Mixed numerals to improper fractions 5 Set C Simplifying fractions 6 Set D Equivalent fractions 7 Set E Addition and subtraction of fractions (1) 7 Set F Addition and subtraction of fractions (2) 8 Set G Addition and subtraction of mixed numerals 9 Set H Harder subtractions of mixed numerals 10 Set I Multiplication of fractions 11 Set J Multiplication of mixed numerals 11 Set K Division of fractions 12 Set L Division of mixed numerals 13

1:02C Decimals 15

Set A Arranging decimals in order of size 15 Set B Addition and subtraction of decimals 15 Set C Multiplication of decimals 16 Set D Multiplying by powers of ten 16 Set E Division of a decimal by a whole number 17 Set F Division involving repeating decimals 17 Set G Dividing by powers of ten 18 Set H Division of a decimal by a decimal 18 Set I Converting decimals to fractions 19 Set J Converting fractions to decimals 20

1:02D Percentages 22

Set A Converting percentages to fractions 22 Set B Converting fractions to percentages 23 Set C Converting percentages to decimals 24 Set D Converting decimals to percentages 24 Set E Finding a percentage of a quantity 25 Set F Finding a quantity when a part of it is known 26 Set G Percentage composition 28 Set H Increasing or decreasing by a percentage 29 Appendix Answers

1:05 Decimals 1

1:11 Angles review 4

1:12 Triangles and quadrilaterals 5 3:01 Generalised arithmetic 6

3:02 Substitution 7

3:04A Simplifying algebraic fractions 8 3:04B Simplifying algebraic fractions 9 3:05 Grouping symbols 10 4:02 Experimental probability 11 4:03 Theoretical probability 12 5:02 Formulae 13 5:03 Non-numerical proofs 14 5:05 Congruent triangles 15 5:09 Pythagoras’ theorem 16 6:01 The index laws 17 6:02 Negative indices 18 6:03 Fractional indices 19 6:04 Scientific notation 20

6:07 Surds 21

6:08 Addition and subtraction of surds 22 6:09 Multiplication and division of surds 23 6:10 Binomial products—surds 24

7:01 Perimeter 25

7:02 Area 26

7:03 Surface area of prisms 27 7:04 Surface area of composite solids 28

7:05 Volume 29

8:01 Equivalent equations 30 8:02 Equations with grouping symbols 31 8:03 Equations with fractions (1) 32 8:04 Equations with fractions (2) 33 8:05 Solving problems using equations 34 8:06 Solving inequations 35

8:07 Formulae 36

8:09 Solving literal equations 37 9:02 Extra payments 38

9:04 Taxation 39

9:06 Best buy, shopping lists, change 40 9:07 Goods and services tax 41 10:01 Distance between points 42

10:02 Midpoint 43

10:03 Gradients 44

10:04 Graphing lines 45 10:05 Gradient–intercept form 46 10:06 Point–gradient form 47 10:08 Parallel and perpendicular lines 48 10:09 Graphing inequalities 49 11:01 Common factors 50 11:02 Grouping in pairs 51 11:04 Factorising trinomials 52

Student Book

Appendixes

Foundation Worksheets

You can access this material by clicking on the links provided on the Interactive Student CD. Go to the Home Page for information about these links.

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12:01 Frequency and cumulative frequency 54 12:02 Mean, median and mode 55 12:03 Mean and median 56 13:01 Graphical method of solution 57 13:02A The substitution method 58 13:03 Using simultaneous equations to solve

problems 59

14:05 Using trigonometry to find side lengths 60 14:07 Angles of elevation and depression, and

bearings 61

14:08 Problems with more than one triangle 62 Worksheet Answers

3:05 Fractions and grouping symbols 1 5:02 Regular polygons and tessellations 2 6:03 Algebraic expressions and indices 3 12:04 Australia’s population 4 13:03 Solving three simultaneous equations 5 14:03 The range of values of the trig. ratios 6 14:06 Trigonometry and the limit of an area 7 14:08 Solving three-dimensional problems 8 Worksheet Answers

The material below is found in the Companion Website which is included on the Interactive Student CD as both an archived version and a fully featured live version. Activities and Investigations

2:01C Sharing the prize 3:02 Substitution 3:02 Magic squares Chapter 4 Probability 5:02 Spreadsheet 5:08 Quadrilaterals

6:01 Who wants to be a millionaire? 6:06 Golden ratio investigations 7:05 Greatest volume

8:03 Flowcharts

8:08–8:10 Substituting and transposing formulae 9:03 Wages

10:05 Equation grapher Chapter 12 Sunburnt country 13:01 Break-even analysis 14:06 Shooting for a goal 15:01 World record times 15:02 Filling tanks Drag and Drops

Chapter 1: Maths terms 1A,

Maths terms 1B,

Significant figures, Triangles and quadrilaterals, Angles

Chapter 3: Maths Terms 3, Addition and subtraction

of algebraic fractions, Multiplication and division of algebraic fractions, Grouping symbols, Binomial products, Special products

Chapter 4: Maths terms 4, Two dice, Pack of cards

Chapter 5: Maths terms 5, Angles and parallel lines,

Triangles, Quadrilaterals, Angle sum of polygons, Pythagoras’ theorem

Chapter 6: Maths terms 6, Index laws, Negative

indices, Fractional indices, Simplifying surds, Operations with surds

Chapter 7: Maths terms 7, Perimeter, Area of sectors

and composite figures, Surface area, Volume

Chapter 8: Maths terms 8, Equations with fractions,

Solving inequations, Formulae, Equations from formulae, Solving literal equations

Chapter 9: Maths terms 9, Find the weekly wage,

Going shopping, GST.

Chapter 10: Maths terms 10, x and y intercept and

graphs, Using y = mx + b to find the

gradient, General form of a line, Parallel and perpendicular lines, Inequalities and regions

Chapter 11: Maths terms 11, Factorising using

common factors, Grouping in pairs, Factorising trinomials 1, Factorising trinomials 2, Mixed factorisations

Chapter 12: Maths terms 12

Chapter 14: Maths terms 14, The trigonometric ratios,

Finding sides, Finding angles, Bearings 1, Bearings 2

Animations

Chapter 10: Linear graphs and equations

Chapter 14: Trigonometry ratios

Chapter Review Questions These can be used as a diagnostic tool or for revision. They include multiple choice, pattern-matching and fill-in-the-gaps style questions.

Destinations

Links to useful websites that relate directly to the chapter content.

Challenge Worksheets

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What does the package

consist of?

• Full-colour Student Book with free Student CD • Homework Book

• Pearson Places Website • Teacher Edition

• LiveText DVD

Student Book

• Improved full-colour design and layout makes the text more appealing for students and easier to navigate. • Original features that form

the backbone of the series

are retained to ensure this new edition meets the high standards set by earlier editions.

• Graded exercises are colour coded to indicate levels of diffi culty.

• Working Mathematically is fully integrated and also features as a separate section at the end of each chapter.

• Foundation worksheets provide alternative exercises for consolidation of earlier stages. • Challenge activities and worksheets provide more

diffi cult investigations.

• Enhanced technology is used extensively throughout, with fully integrated links to both the Student CD and the Pearson Places Website. The Student CD accompanies each

book and contains:

• a fully unlocked pdf of the Student Book than can be copied and pasted • a direct link to all the technology

components in the Student Book

• a cached version of the Companion Website • a link to the live Companion Website.

Homework Book

The Homework Book provides a complete homework program linked directly to the Student Book.

Enhanced

STAGE 5

Mathematics

S

T

A

G

E

5

9

9 Enhanced

M

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The latest edition of the best-selling mathematics series on the market! New Signpost Mathematics Enhanced features an updated, easier to navigate design, fantastic new technology and THE most comprehensive teacher support available in the form of a Teacher Edition. It is enhanced both in design, technology and teaching resources.

New Signpost Mathematics Enhanced 9 and 10 are designed to complete Stage 5 of the syllabus, but also to

assist students in achieving outcomes relevant to their stage of development. Working with this series, teachers will be able to select an appropriate program of work for all students.

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Teacher Edition

A Teacher Edition is available for each Student Book. These innovative resources allow any teacher to confi dently approach the teaching and learning of mathematics using the New Signpost Maths Enhanced package.

Each Teacher Edition book features:

• pages from the Student Book with ‘wrap- around’ notes

• lists of learning outcomes covered by activities and sections of the Student Book

• a wealth of teaching strategies and activities directly related to the Student Book

• additional examples and content

• Working mathematically and problem solving questions

• starter questions and extension activities • ICT strategies

• Teacher CD, including an electronic version of the Student Book.

ix

For more information on the New Signpost Mathematics Enhanced series,

visit www.pearsonplaces.com.au

LiveText DVD

LiveText is an electronic version of the Student Book, with additional features and resources, for whole-class teaching using any Interactive Whiteboard or data projector. Stimulating, fun and engaging, LiveText grabs students’ attention and provides a good platform for classroom teaching and discussion. • A Resource bank gives teachers everything

needed to deliver lessons: animations, quick quizzes, review questions, drag and drops, Excel spreadsheets, challenge worksheets, foundation worksheets and much more.

• Zoom functionality.

• Annotation tools to emphasise certain parts of the book and customise pages. • Print function that prints the displayed page

with any annotations made.

• Hotspots with multiple functions for zooming and linking to resources such as Flash activities and downloadable documents.

Pearson Places Website

The Pearson Places Website contains a wealth of support material for students and teachers:

• Chapter Review Questions for use as a diagnostic tool or for revision. These are auto-correcting and include multiple-choice, pattern-matching and fi ll-in-the-gaps style questions. Results can be emailed directly to the teacher or parents. • Technology Applications –

activities that apply concepts covered in each chapter and are designed for students to work independently:

– Activities and investigations using technology such as Excel spreadsheets and The Geometer’s Sketchpad. – Drag and Drop Interactives to

improve basic skills. – Animations to develop skills

by manipulating interactive demonstrations of key mathematical concepts. • Quick Quizzes for each chapter

Ch

ap

ter Review Quest ion

s

Technology

Drag-and-drop

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Student Book

Chapter-opening pages summarise the key content and present the syllabus outcomes addressed in each chapter.

Clear syllabus references are included throughout the text to make programming easier: in the chapter-opening pages, in each main section within each chapter and in the Foundation Worksheet references. For example, Outcome NS5·1.

Well-graded exercises where levels of diffi culty are indicated by the colour of the question number.

1 green foundation

4 blue Stage 5.3 level

9 red extension

1 Find the simple interest charged for a loan of: a 2× 3 b 5× 7 c 3× 11

3 a A straight line has a gradient of 2 and passes through the point (3, 2). Find the equation of the line.

4 Solve each literal equation for x: a a + x = b – x

b ax = px + q c x + a = ax + b

Worked examples are used extensively and are easy for students to identify.

Worked example

1 Express the following in scientifi c notation a 243 b 60 000 c 98 800 000

Important rules and concepts are clearly highlighted at regular intervals throughout the text.

Cartoons are used to give students friendly advice or tips.

Prep Quizzes review skills needed to complete a topic. These anticipate problems and save time in the long run. These quizzes offer an excellent way to start a lesson. Challenge activities and worksheets provide more diffi cult investigations and exercises. They can be used to extend more able students.

Fun Spots provide amusement and interest, while often reinforcing coursework. They encourage creativity and divergent thinking, and show that mathematics is enjoyable. Investigations encourage students to seek knowledge and develop research skills. They are an essential part of any mathematics course.

Diagnostic Tests at the end of each chapter assess students’ achievement of outcomes. More importantly, they indicate the weaknesses that need to be addressed and link back to the relevant section in the Student Book or CD.

How to use this book

The New Signpost Mathematics Enhanced 9 and 10 learning package gives complete coverage of the New South Wales Stage 5 Mathematics syllabus. The following features are integrated into the Student Book, Student CD and the Companion Website:

The table of

values looks

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Assignments are provided at the end of each chapter. Where there are two assignments, the fi rst revises the content of the chapter, while the second concentrates on developing the student’s ability to work mathematically.

The Algebra Card (see p. xxiii) is used to practise basic arithmetic and algebra skills. Corresponding terms in columns can be added, subtracted, multiplied or divided by each other or by other numbers. This is a great way to start a lesson.

Literacy in Maths sections help students to develop maths literacy skills and provide opportunities for students to communicate mathematical ideas. They present mathematics in the context of everyday experiences.

• Maths Terms relevant to the content are defi ned at the end of each chapter. These terms are also tested in a Drag and Drop Interactive activity that follows this section in each chapter. • ID Cards (see pp. xvii-xxii) review the

language of Mathematics by asking students to identify common terms, shapes and symbols. They should be used as often as possible, either at the beginning of a lesson or as part of a test or examination.

Student CD Companion Website

Technology Applications apply concepts covered in each chapter and are designed for students to work independently:

• Activities and investigations using technology such as Excel spreadsheets and The Geometer's Sketchpad.

• Drag and Drop Interactives to improve speed in basic skills.

• Animations to develop key skills by manipulating visually stimulating demonstrations of key mathematical concepts.

Foundation Worksheets provide alternative exercises for students who need to consolidate work at an earlier stage or who need additional work at an easier level. Students can access these on the Student CD by clicking on the Foundation Worksheet icons. These can also be copied from the Teacher CD or from the Teacher Resource Centre on the Companion Website.

Foundation Worksheet 3:01

Generalised arithmetic PAS5.2.1

1 Write expressions for:

a the sum of 3a and 2b b the average of m and n

2 a Find the cost of x books at

75c each.

b Find the age of Bill, who is 25 years old, in another

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Treatment of Outcomes

Each outcome relevant to the Year 9 Student Book is listed on the left-hand side. The places where these are treated are shown on the right. The syllabus strand Working Mathematically involves

questioning, applyingstrategies, communicating, reasoning and reflecting. These are given special attention in Chapter 2 and in the assignment at the end of each chapter, but are also an integral part of each chapter.

Outcome Text references

WMS5.3.1 Asks questions that could be explored using mathematics in relation to Stage 5.3 content.

Revision: Working Mathematically, Chapter 2, and throughout the text

WMS5.3.2 Solves problems using a range of strategies including deductive reasoning.

WMS5.3.3 Uses and interprets formal definitions and generalisations when explaining solutions and or conjectures

WMS5.3.4 Uses deductive reasoning in presenting arguments and formal proofs.

WMS5.3.5 Links mathematical ideas and makes connections with, and generalisations about, existing knowledge and understanding in relation to Stage 5.3 content.

NS4.2 Compares, orders and calculates with integers. 1:01, 1:02

NS4.3 Operates with fractions, decimals, percentages, ratios and rates.

1:02–1:04, 1:06, 1:07, 2:01A, B, C, D

NS5.1.1 Applies index laws to simplify and evaluate

arithmetic expressions and uses scientific notation to write large and small numbers.

6:01–6:05

NS5.1.2 Solves consumer arithmetic problems involving earning and spending money.

9:01–9:07, 9:09

NS5.1.3 Determines relative frequencies and theoretical probabilities.

4:01–4:04, Year 10

NS5.2.1 Rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form and converts rates from one set of units to another.

1:05, 1:08–1:10

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NS5.3.1 Performs operations with surds and indices. 6:06–6:11

NS5.3.2 Solves probability problems involving compound events.

Year 10

PAS4.3 Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions.

3:01–3:03

PAS4.4 Uses algebraic techniques to solve linear equations and simple inequalities.

8:01, 8:02

PAS4.5 Graphs and interprets linear relationships on the number plane.

10:04

PAS5.1.1 Applies the index laws to simplify algebraic expressions.

6:01

PAS5.1.2 Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations.

10:01–10:04

PAS5.2.1 Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices.

3:01, 6:02, 6:03

PAS5.2.2 Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods.

8:02–8:08, 13:01–13:03, Year 10

PAS5.2.3 Uses formulae to find midpoint, distance and gradient and applies the gradient–intercept form to interpret and graph straight lines.

10:01–10:03, 10:05

PAS5.2.4 Draws and interprets graphs including simple parabolas and hyperbolas.

Year 10

PAS5.2.5 Draws and interprets graphs of physical phenomena. 15:01,15:02

PAS5.3.1 Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions.

3:04–3:08, 11:01–11:08

PAS5.3.2 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations.

8:02–8:06, 8:09–8:11, Year 10

PAS5.3.3 Uses various standard forms of the equation of a straight line and graphs regions on the number plane.

10:04, 10:06–10:09

PAS5.3.4 Draws and interprets a variety of graphs including parabolas, cubics, exponentials and circles and applies coordinate geometry techniques to solve problems.

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PAS5.3.5 Analyses and describes graphs of physical phenomena.

15:01, 15:02

PAS5.3.6 Uses a variety of techniques to sketch a range of curves and describes the features of curves from the equation.

Year 10

PAS5.3.7 Recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems.

Year 10

PAS5.3.8 Describes, interprets and sketches functions and uses the definition of a logarithm to establish and apply the laws of logarithms.

Year 10

DS4.1 Constructs, reads and interprets graphs, tables, charts and statistical information.

12:01

DS4.2 Collects statistical data using either a census or a sample and analyses data using measures of location and range.

12:02, 12:03

DS5.1.1 Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs.

12:01, 12:03, 12:04

DS5.2.1 Uses the interquartile range and standard deviation to analyse data.

Year 10

MS4.1 Uses formulae and Pythagoras’ theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles.

2:01E, 7:02

MS4.2 Calculates surface area of rectangular and triangular prisms and volume of right prisms and cylinders.

2:01E, 7:03, 7:05

MS5.1.1 Uses formulae to calculate the area of quadrilaterals and finds areas and perimeters of simple composite figures.

7:01, 7:02

MS5.1.2 Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression.

14:01–14:07, Year 10

MS5.2.1 Finds areas and perimeters of composite figures. 7:01, 7:02

MS5.2.2 Applies formulae to find the surface area of right cylinders and volume of right pyramids, cones and spheres and calculates the surface area and volume of composite solids.

7:03–7:06, Year 10

MS5.2.3 Applies trigonometry to solve problems including those involving bearings.

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The above material is independently produced by Pearson Education Australia for use by teachers. Although curriculum references have been reproduced with the permission of the Board of Studies NSW, the material is in no way connected with or endorsed by them. For comprehensive course details please refer to the Board of Studies NSW Website www.boardofstudies.nsw.edu.au

MS5.3.1 Applies formulae to find the surface area of pyramids, right cones and spheres.

Year 10

MS5.3.2 Applies trigonometric relationships, sine rule, cosine rule and area rule in problem solving.

14:08, Year 10

SGS4.2 Identifies and names angles formed by the

intersection of straight lines, including those related to transversals on sets of parallel lines, and makes use of the relationships between them.

1:01, 1:11

SGS4.3 Classifies, constructs, and determines the properties of triangles and quadrilaterals.

1:01, 1:12

SGS5.2.1 Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon.

5:02

SGS5.2.2 Develops and applies results for proving that triangles are congruent or similar.

5:04–5:06, Year 10

SGS5.3.1 Constructs arguments to prove geometrical results. 5:01, 5:03–5:06, 5:09

SGS5.3.2 Determines properties of triangles and quadrilaterals using deductive reasoning.

5:07, 5:08

SGS5.3.3 Constructs geometrical arguments using similarity tests for triangles

Year 10

SGS5.3.4 Applies deductive reasoning to prove circle theorems and to solve problems.

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Metric Equivalents

Months of the year

30 days each has September, April, June and November.

All the rest have 31, except February alone, Which has 28 days clear and 29 each leap year.

Seasons

Summer: December, January, February

Autumn: March, April, May

Winter: June, July, August

Spring: September, October, November

Length 1 m = 1000 mm = 100 cm = 10 dm 1 cm = 10 mm 1 km = 1000 m Area 1 m2_{= 10 000 cm}2 1 ha = 10 000 m2 1 km2_{= 100 ha} Mass 1 kg = 1000 g 1 t = 1000 kg 1 g = 1000 mg Volume 1 m3_{= 1 000 000 cm}3 = 1000 dm3 1 L = 1000 mL 1 kL = 1000 L 1 m3_{= 1 kL} 1 cm3_{= 1 mL} 1000 cm3_{= 1 L} Time 1 min = 60 s 1 h = 60 min 1 day = 24 h 1 year = 365 days 1 leap year = 366 days

It is important that you learn these facts off by heart.

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The Language of Mathematics

You should regularly test your knowledge by identifying the items on each card.

See page 598 for answers.

ID Card 1 (Metric Units) ID Card 2 (Symbols)

1 m 2 dm 3 cm 4 mm 1 = 2 or ≈ 3 ≠ 4 < 5 km 6 m2 7 cm2 8 km2 5 6 7 > 8 9 ha 10 m3 11 cm3 12 s 9 42 10 43 11 12 13 min 14 h 15 m/s 16 km/h 13 14 || 15 16 ||| 17 g 18 mg 19 kg 20 t 17 % 18 ∴ 19 eg 20 ie 21 L 22 mL 23 kL 24 °C 21 π 22 ∑ 23 24 P(E) 2 3 2 x See ‘Maths Terms’ at the end of each chapter.

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See page 598 for answers. . ID Card 3 (Language) 1 6 minus 2 2 the sum of 6 and 2 3 divide 6 by 2 4 subtract 2 from 6 5 the quotient of 6 and 2 6 3 2

)

6 the divisor is . . . . 7 3 2

)

6 the dividend is . . . . 8 6 lots of 2 9 decrease 6 by 2 10 the product of 6 and 2 11 6 more than 2 12 2 less than 6 13 6 squared 14 the square root of 36 15 6 take away 2 16 multiply 6 by 2 17 average of 6 and 2 18 add 6 and 2 19 6 to the power of 2 20 6 less 2 21 the difference between 6 and 2 22 increase 6 by 2 23 share 6 between 2 24 the total of 6 and 2 We say ‘six squared’ but we write 62_.

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See page 598 for answers. ID Card 4 (Language) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 All sides different

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See page 598 for answers. ID Card 5 (Language) 1 A ... 2 ... 3 ... 4 ... 5 ... points 6 C is the ... 7 ... ... 8 ... 9

all angles less than 90°

10

one angle 90°

11

one angle greater than 90°

12

A, B and C are

... of the triangle.

13

Use the vertices to name the Δ. 14 BC is the ... of the right-angled Δ. 15 a° + b° + c° = ... 16 ∠BCD = ... 17 a° + b° + c° + d° = ... 18

Which (a) a° < b° is true? (b) a° = b° (c) a° > b° 19 a° = ... 20 Angle sum = ... 21 AB is a ... OC is a ... 22 Name of distance around the circle. ... 23 ... 24 AB is a ... CD is an ... EF is a... A B A B A B P Q R S A C B –4 –2 0 2 4 A B C A B C A B C A B C b° c° a° A D B C b° a° b° d° c° a° b° a° a° A B C O O O B C D F E A

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See page 598 for answers. ID Card 6 (Language) 1 ... lines 2 ... lines 3 v ... h ... 4 ... lines 5 angle ... 6 ... angle 7 ... angle 8 ... angle 9 ... angle 10 ... angle 11 ... 12 ... angles 13 ... angles 14 ... angles 15 ... angles 16 a° + b° + c° + d° = ... 17 ... 18 ... angles 19 ... angles 20 ... angles 21 b... an interval 22 b... an angle 23 ∠CAB = ... 24 CD is p... to AB. A B C (less than 90°) (90°) (between 90° and 180°) (180°) (between 180° and 360°) (360°) a° + b° = 90° a° b° a° + b° = 180° a° b° a° = b° a° b° a° b° c° d° a° = b° a° b° a° = b° a° b° a° + b° = 180° a° b° A B C D E A B C D A B C A B C

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ID Card 7 (Language) 1 a... D... 2 b... C... 3 a... M... 4 p... m... 5 area is 1 ... 6 r... shapes 7 ... of a cube 8 c...-s... 9 f... 10 v... 11 e... 12 axes of ... 13 r... 14 t... 15 r... 16 t... 17 The c... of the dot are E2.

18 t... 19 p... graph 20 c... graph 21 l... graph 22 s... graph 23 b... graph 24 s... d...

AD

BC

am

pm

100 m 100 m 4 3 2 1 0 A B C D E F Cars sold Mon Tues Wed Thurs Fri Money collected Mon Tues Wed Thurs Fri Stands for $10 70 50 30 10 M T W T F Dollars Money collected 100 80 60 40 20 John’s height 1 2 3 4 5 Age (years) Use of time Hobbies Sleep Home School People present Adults Gir ls Bo ys Smoking Cigarettes smoked Length of lif e

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Algebra Card

How to use this card

If the instruction is ‘column D + column F’, then you add corresponding terms in columns D and F. eg 1 m + (−3m) 2 (−4m) + 2m 3 10m + (−5m) 4 (−8m) + 7m 5 2m+ 10m 6 (−5m) + (−6m) 7 8m + 9m 8 20m + (−4m) 9 5m + (−10m) 10 (−9m) + (−7m) 11 (−7m) + (−8m) 12 3m + 12m A B C D E F G H I J K L M N O 1 3 2·1 m −3m 5m2 _−5x _−3x _x_{+ 2 x − 3 2x + 1 3x − 8} 2 −1 −0·4 −4m 2m −2m3 _3x _5x2 _x+ 7 x − 6 4x + 2 x − 1 3 5 0·8 10m −5m 8m5 _10x −8x _x+ 5 x + 5 6x + 2 x − 5 4 −2 1·5 −8m 7m 6m2 _−15x _−4x4 _x_{+ 1 x − 9 3x + 3 2x + 4} 5 −8 −2·5 2m 10m m2 _7x _2x3 _x_{+ 8 x + 2 3x + 8 3x + 1} 6 10 −0·7 −5m −6m −9m3 _9x _x2 _x+ 4 x − 7 3x + 1 x + 7 7 −6 −1·2 8m 9m 2m6 −6x _5x2 _x+ 6 x − 1 x + 8 2x − 5 8 12 0·5 20m −4m −3m3 _−12x _4x3 _x_{+ 10 x − 8 5x + 2 x − 10} 9 7 0·1 5m −10m m7 _5x _−3x5 _x_{+ 2 x + 5 2x + 4 2x − 4} 10 −5 −0·6 −9m −7m −8m4 −3x −7x5 _x+ 1 x − 7 5x + 4 x + 7 11 −11 −1·8 −7m −8m −4m −4x −x3 _x+ 9 x + 6 2x + 7 x − 6 12 4 −1·4 3m 12m 7m2 _−7x _x10 _x_{+ 3 x − 10 2x + 3 2x + 3} 1 4 --- 2m 3 --- x 6 --- x 2 ---– 1 8 --- m 4 ---- x 3 ---– x 4 ---1 3 --- m 4 ----– 2x 7 ---– 2x 5 ---1 20 --- 3m 2 ---– x 10 --- x 5 ---– 3 5 --- m 5 ----– 2x 3 --- x 3 ---2 7 --- 3m 7 ---– 2x 5 ---– 3x 5 ---3 8 --- m 6 ----– 5x 6 --- 2x 3 ---9 20 --- 2m 5 --- 3x 4 --- x 7 ---– 3 4 --- 3m 5 --- 3x 7 ---– 3x 7 ---– 7 10 --- 4m 5 ---– x 6 ---– 2x 9 ---1 10 --- m 5 ---- x 5 --- x 3 ---2 5 --- m 3 ---- 3x 4 ---– x 6

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---1

Basic Skills

and Number

I must remember something, surely!

Learning Outcomes

NS4·2 (reviewed) Compares, orders and calculates with integers.

NS4·3 (reviewed) Operates with fractions, decimals, percentages, ratios and rates.

NS5·2·1 Rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form and converts rates from one set of units to another.

SGS4·2 Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines, and makes use of the relationships between them.

SGS4·3 Classifies, constructs and determines the properties of triangles and quadrilaterals.

WorkingMathematically Stages 4and 5.

1 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting.

Chapter Contents

1:01 The language of mathematics NS4·2, SGS4.2,3 1:02 Diagnostic tests NS4·2, NS4.3 A Integers NS4.2 B Fractions NS4.3 C Decimals NS4.3 D Percentages NS4.3

1:03 Conversion facts you should know NS4·3

Fun Spot: What was the prime minister’s name in 1978?

1:04 Rational numbers NS4·3

1:05 Recurring decimals NS5·2·1

Challenge: Try this with repeating decimals

Fun Spot: Speedy addition

1:06 Simplifying ratios NS4·3

1:07 Rates NS4·3

Investigation: Comparing speeds

1:08 Significant figures NS5·2·1

1:09 Approximations NS5·2·1

1:10 Estimation NS5·2·1

Reading Maths: Take your medicine!

1:11 Angles review SGS4·2

1:12 Triangles and quadrilaterals SGS4·3

Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically

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1:01

The Language

Outcomes NS4·2, SGS4·2,3

of Mathematics

Much of the language met so far is reviewed in the identification cards (ID Cards) found on pages xvii to xxii. These should be referred to throughout the Student Book. Make sure that you can identify every term.

Test yourself on ID Cards 1 and 2 by identifying each symbol mentally. Look up the answer to any you can’t identify and write those symbols and their meaning in your book.

Do you know how to write each expression in ID Card 3 as symbols? Read through the card and copy expressions and answers for those that are unfamiliar. (For example, for ‘the quotient of 6 and 2’ write ‘6 ÷ 2 = 3’.)

Mentally test yourself on ID Cards 4, 5, 6 and 7. Look up the answer to any you can’t identify and record these in your exercise book.

Learn the terms you did not know. This can be done by making small cards with the figures on one side and the answers on the other. Carry these with you as an aid to learning. Have others test you.

• Which terms from ID Card 6 could be used to describe parts of this photograph?

1:02

Diagnostic Tests

Outcomes NS4·2, NS4·3

Without obtaining help, complete the diagnostic tests on the next pages to determine areas that need attention. For treatment of weaknesses refer to the sections found on the Student CD. There you will find explanations and worked examples relating to these skills. Do not use a calculator.

Exercise 1:01

1 2 3 4 –2 means ‘two below zero’ or ‘two less than zero’.

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1:02A | Integers

NS4·2

1:02B | Fractions

NS4·3 CD Appendix 1 a −7 + 14 b 2 − 15 c −2 − 8 Set A 2 a 3 − (−6) b 12 + (−5) c 6 − (3 − 8) Set B 3 a −3 × 2 b −5 ×−6 c 7 × (−9) Set C 4 a (−)15 ÷ (−3) b 63 ÷ (−9) c Set C 5 a 14 − 7 × 10 b −3 + 4 ÷ 4 c (4 − 18) ÷ (−8 + 6) Set D CD Appendix 1 Write these improper fractions as mixed numerals. Set A

a b c

2 Write these mixed numerals as improper fractions. Set B

a 2 b 5 c 3

3 Simplify these fractions. Set C

a b c

4 Complete the following to make equivalent fractions.

Set D

a = b = c =

Give the simplest answer for . . .

5 a + b + c + Set E 6 a − b − c − Set E 7 a + b + c + Set F 8 a − b − c − Set F 9 a 3 + 4 b 6 + 5 c 1 + Set G 10 a 4 − 1 b 10 − 5 c 20 − Set G 11 a 7 − b 6 − 2 c 3 − 1 Set H 12 a × b × c × Set I 13 a × b × c × Set I 14 a 3 × b 1 × 1 c 5 × 2 Set J 15 a 4 × 3 b 2 × 3 c 5 × 6 Set J 16 a ÷ b ÷ c ÷ Set K 17 a ÷ b ÷ c ÷ Set K 18 a 1 ÷ b 3 ÷ 2 c 3 ÷ 2 Set L 19 a 7 ÷ 3 b 4 ÷ 7 c 6 ÷ 5 Set L 20 a 5 ÷ b 10 ÷ c 4 ÷ Set L 156 – 3 – ---7 4 --- 13 3 --- 141 10 ---1 2 --- 3 10 --- 1 7 ---16 24 --- 100 650 --- 240 3600 ---3 4 ---28 --- 17 20 ---100 --- 3 8 ---1000 ---3 8 --- 2 8 --- 9 10 --- 3 10 --- 7 9 --- 2 9 ---9 10 --- 7 10 --- 13 14 --- 9 14 --- 37 100 --- 11 100 ---3 4 --- 4 5 --- 3 10 --- 2 5 --- 7 100 --- 3 40 ---7 8 --- 3 4 --- 9 10 --- 1 4 --- 5 6 --- 3 5 ---1 2 --- 3 5 --- 7 10 --- 3 4 --- 5 6 --- 7 8 ---1 2 --- 2 9 --- 3 4 --- 1 10 --- 3 8 --- 1 5 ---1 2 --- 7 8 --- 3 5 --- 7 10 --- 1 2 --- 5 6 ---4 5 --- 3 11 --- 3 10 --- 7 10 --- 1 10 --- 3 5 ---7 8 --- 3 7 --- 15 38 --- 19 20 --- 7 10 --- 5 6 ---1 2 --- 5 7 --- 3 10 --- 4 5 --- 1 4 --- 2 3 ---4 5 --- 1 4 --- 3 8 ---8 10 --- 2 10 --- 9 20 --- 3 20 --- 7 10 --- 7 10 ---3 5 --- 1 2 --- 8 9 --- 3 4 --- 5 8 --- 4 7 ---■ 3 of 4 equal parts 3 4 --- → Numerator → Denominator ■ Fractions should always be expressed in lowest terms. 4 6 --- 2 3 ---= ■ or or Equivalent fractions 1 8 --- 1 8 --- 1 8 --- 1 8 --- 1 8 --- 1 8 --- 1 8 --- 1 8 ---1 4 --- 1 4 --- 1 4 --- 1 4 ---1 2 --- 1 2 ---1 2 --- 2 4 --- 4 8 ---7 8 --- 3 4 --- 4 7 --- 1 2 --- 5 8 --- 9 10 ---1 2 --- 9 10 --- 7 8 ---1 1 1

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1:02C | Decimals

NS4·3

CD Appendix

Put in order, smallest to largest. Set A

1 a 0·505, 0·5, 0·55 b 8·4, 8·402, 8·41 c 1·01, 1·1, 1·011 2 a 2·6 + 3·14 b 18·6 + 3 c 0·145 + 0·12 Set B 3 a 12·83 − 1·2 b 9 − 1·824 c 4·02 − 0·005 Set B 4 a 0·7 × 6 b (0·3)2 _c 0·02 × 1·7 _{Set C} 5 a 3·142 × 100 b 0·04 × 1000 c 0·065 × 10 Set D 6 a 2·1 × 104 _b _{8·04 × 10}6 _c _{1·25 × 10}2 _{Set D} 7 a 4·08 ÷ 2 b 12·1 ÷ 5 c 0·19 ÷ 4 Set E

8 Write answers as repeating decimals.

a 2·5 ÷ 6 b 5·32 ÷ 9 c 28 ÷ 3 Set F

9 a 24·35 ÷ 10 b 6·7 ÷ 100 c 0·7 ÷ 1000 Set G

10 a 6·4 ÷ 0·2 b 0·824 ÷ 0·08 c 6·5 ÷ 0·05 Set H

11 Convert these decimals to fractions.

a 0·5 b 0·18 c 9·105 Set I

12 Convert these fractions to decimals.

a 4 b c Set J 5 --- 3 8 --- 5 6 ---■ 3 tens 7 units 4 tenths 2 hundredths 5 thousandths 37·425 10 1 · 3 7 · 4 2 5 1 10 --- 1 100 --- 1 1000 ---What does 37.425 really mean?

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1:02D | Percentages

NS4·3

CD Appendix 1 Convert to fractions. Set A

a 18% b 7% c 224%

2 Convert to fractions. Set A

a 9·5% b 6 % c 12·25%

3 Convert to percentages. Set B

a b c 1

4 Convert to decimals. Set C

a 9% b 16% c 110%

5 Convert to decimals. Set C

a 23·8% b 12 % c 4 %

6 Convert to percentages. Set D

a 0·51 b 0·085 c 1·8

7 Find: Set E

a 35% of 600 m b 162% of $8

8 Find: Set E

a 7% of 84·3 m b 6 % of 44 tonnes

9 a 7% of my spending money was spent on a watch band that cost $1.12. How much spending money did I have?

Set F

b 30% of my weight is 18 kg. How much do I weigh?

10 a 5 kg of sugar, 8 kg of salt and 7 kg of flour were mixed accidentally. What is the percentage (by weight) of sugar in the mixture?

Set G

b John scored 24 runs out of the team’s total of 60 runs. What percentage of runs did John score? 11 a Increase $60 by 15%. Set H b Decrease $8 by 35%. TAX RATE 35% For every $100 earned, $35 is paid in tax. 50% of all men play tennis. This game’s only half the

fun it used to be . . . 1 4 ---11 20 --- 5 6 --- 1 4 ---1 2 --- 2 3 ---1 4

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---1:03 Conversion Facts

Outcome NS4·3

You Should Know

To the right, I have used these facts.

Percentage Decimal Fraction

1% 0·01 5% 0·05 10% 0·1 12 % 0·125 20% 0·2 25% 0·25 33 % 0· 50% 0·5 100% 1 1 1 100 ---1 20 ---1 10 ---1 2 --- 1 8 ---1 5 ---1 4 ---1 3 --- 3˙ 1 3 ---1 2 ---a 10% = 0⋅1 = Multiply each by 6. 60% = 0⋅6 = b 5% = 0⋅05 = Multiply each by 7. 35% = 0⋅35 = c 20% = 0⋅2 = Add 1 or 100% to each. 120% = 1⋅2 = 1 d 12 % = 0⋅125 = Add 1 or 100% to each. 112 % = 1⋅125 = 1 1 10 ---6 10 ---1 20 ---7 20 ---1 5 ---1 5 ---1 2 --- 1 8 ---1 2 --- 1 8

---• How many fractions can you convert to decimals and percentages in your head?

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Work out the answer to each part and put the letter for that part in the box that is above the correct answer.

Write the basic numeral for:

A –8 + 10 A –7 − 3 A –6 × 4 A 6 − (3 − 4) A (–5)2

Y Write as a mixed numeral.

M Change 1 to an improper fraction.

Write the simplest answer for:

I I − I + T − T × T ( )2 T 4 + T 2 − N ÷ N 0·05 + 3 O 0·3 − 0·02 O 0·3 × 5 E (0·3)2 _{E 3·142}× 100 _{E 6·12}÷ 6 E 20·08 ÷ 10 C 1·8 ÷ 0·2 G of 60 kg D What fraction is 125 g of 1 kg? H 5% of 80 kg H Write as a percentage. H Write 0·75 as a fraction. H Increase 50 kg by 10%. D 40% of my weight is 26 kg. How much do I weigh?

S Write 4 ÷ 9 as a repeating (recurring) decimal.

S 10 cows, 26 horses and 4 goats are in a paddock. What is the percentage of animals that

are horses?

S Increase $5 by 20%.

S 600 kg is divided between Alan and Rhonda so that Alan gets of the amount.

How much does Alan get?

Fun Spot 1:03

What was the prime minister’s name in 1978?

15 4 ---3 4 ---44 32 --- 37 100 --- 12 100 --- 3 8 --- 1 3 ---4 5 --- 2 3 --- 7 8 --- 8 7 --- 1 3 ---3 8 --- 5 8 --- 5 8 --- 1 2 --- 1 2 --- 1 8 ---3 4 ---2 5 ---3 5 ---1·02 7 0·09 2 _$6 1 65% 1·5 25 3 55 kg 314·2 4k g − 10 360 kg 4 0·28 5 9 40% −24 3·05 ₄₅kg _2·008 ₆₅kg 1 2 1 --- 9 3 --- 4 0·4 ˙ 7 --- 4 1 --- 4 17 --- 24 2 --- 15 1 --- 8 3 --- 4 3 --- 8 1 --- 8

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1:04 Rational Numbers

Outcome NS4·3

Fractions, decimals, percentages and negative numbers are convenient ways of writing rational numbers.

Real numbers are those that are rational or irrational.

• Every point on the number line represents either a rational number or an irrational number. • Any rational number can be expressed as a terminating or recurring decimal.

Irrational numbers can only be given decimal approximations, however this does allow us to compare the sizes of real numbers.

Discussion

• How many real numbers are represented by points on the number line between 0 and 2, or between − and 0?

From the list on the right, choose two equivalent numbers for:

a 2 b 130%

c 2·8 d 1

Write each set of real numbers in order. Calculators may be used.

a 0·85, 0·805, 0·9, 1 b 87·5%, 100%, 104%, 12 %

c , , and d 1 , 150%, 1·65, 2

e 1·42, , 1·41, 140% f π, 3 , 3·1, Find the number halfway between:

a 6·8 and 6·9 b 12 % and 20% c and d 6·35 and 6·4 Real numbers Rational numbers Irrational numbers

A number is rational if it can be expressed as the quotient of

two integers, , where b ≠ 0.

eg , 8, 52%, 12 %, 0·186, , −1·5, −10

An irrational number cannot be written as a fraction, , where

a and b are integers and b ≠ 0.

eg , , , , π a b ---3 4 --- 1 2 --- 0·3˙ a b ---2 7 3 4 3 5 ■ An integer is a

whole number that may be positive, negative or zero. 1 2

---Exercise 1:04

125% 114% 2 28% 280% 2 1⋅4 2⋅5 208% 13 1⋅25 1⋅3 1 250% 25% 4 5 ---1 8 ---3 10 ---1 1 2 ---1 4 ---2 1 4 ---5 8 --- 4 7 --- 2 3 --- 64 100 --- 3 4 ---2 1 4 --- 12 3 1 2 ---1 8 --- 1 5

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---a Write as decimals: , , , , , , , , .

b Explain why 0·99999 … = 1.

c Write as decimals: , , , , , .

d Write as fractions or mixed numbers: , , , . What are the next three numbers in the sequence:

a 0·125, 0·25, 0·5, . . . ? b 1·3, 0·65, 0·325, . . . ? The average (ie mean) of five numbers is 15·8.

a What is the sum of these numbers?

b If four of the numbers are 15s, what is the other number? What is meant by an interest rate of 9·75% pa?

An advertisement reads: ‘67% leased; only one tenancy remaining for lease. Building ready October.’ How many tenants would you expect in this building?

Using a diameter growth rate of 4⋅3 mm per year, find the number of years it will take for a tree with a diameter of 20 mm to reach a diameter of 50 mm.

At the South Pole, the temperature dropped 15°C in two hours, from a temperature of – 18°C. What was the temperature after that two hours?

Julius Caesar invaded Britain in 55 BC and again one year later. What was the date of the second invasion?

Chub was playing ‘Five Hundred’.

a His score was –150 points. He gained 520 points. What is his new score?

b His score was 60 points. He lost 180 points. What is his new score?

c His score was –120 points. He lost 320 points. What is his new score?

What fraction would be displayed on a calculator as:

a 0⋅3333333? b 0⋅6666666?

c 0⋅1111111? d 0⋅5555555? To change to a decimal approximation, push on a calculator. Use this method to write the following as decimals correct to five decimal places.

a b c d e f 4 1 9 --- 2 9 --- 3 9 --- 4 9 --- 5 9 --- 6 9 --- 7 9 --- 8 9 --- 9 9 ---1 90 --- 2 90 --- 3 90 --- 1 900 --- 2 900 --- 3 900 ---0·4˙ 3·1˙ –0·5˙ –4·5˙ 5 6 7 8 9 10 11 12 13 14 7 15 ---7 ÷ 15 = 8 9 --- 2 7 --- 7 13 ---20 21 --- 4 11 --- 5 18

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---Katherine was given a 20% discount followed by a 5% discount.

a What percentage of the original price did she have to pay?

b What overall percentage discount was she given on the original price?

c For what reason might she have been given the second discount?

Since I started work, my income has increased by 200%. When I started work my income was $21 500. How much do I earn now?

Find the wholesale price of an item that sells for $650 if the retail price is 130% of the wholesale price.

What number when divided by 0·8 gives 16?

What information is needed to complete the following questions?

a If Mary scored 40 marks in a test, what was her percentage?

b In a test out of 120, Nandor made only 3 mistakes. What was his percentage?

c If 53% of cases of cancer occur after the age of 65, what is the chance per 10 000 of developing cancer after the age of 65?

In the year 2000, the distance from Australia to Indonesia was 1600 km. If Australia is moving towards Indonesia at a constant rate of 7 cm per year, when (theoretically) will they collide?

a If I earn 50% of my father’s salary, what percentage of my salary does my father earn?

b If X is 80% of Y, express Y as a percentage of X.

c My height is 160% of my child’s height. Express my child’s height as a percentage of my height.

a Two unit fractions have a difference of . What are they?

b Give two unit fractions with different denominators that subtract to give . Let and represent any two rational numbers. Do we get a rational number if we:

a add them?

b subtract them?

c multiply them?

d divide one by the other? Explain your answers.

15 16 17 18 19 20 Assume that Indonesia isn’t moving

in the meantime. A unit fraction has a numerator of 1. 21 22 3 8 ---5 11 ---23 a b -- c d

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--1:05 Recurring Decimals

Outcome NS5·2·1

To write fractions in decimal form we simply divide the numerator (top) by the denominator (bottom). This may result in either a ‘terminating’ or ‘recurring’ decimal. For example:

0· 3 7 5 0· 1 6 6 6 . . . For : 8

)

3·306040 For : 6

)

1·10404040 To rewrite a terminating decimal as a fraction the process is easy. We simply put the numbers in the decimal over the correct power of 10, ie 10, 100, 1000, etc, and then simplify.

For example: 0·375 = =

To rewrite a recurring decimal as a fraction is more difficult. Carefully examine the two examples given below and copy the method shown when doing the following exercise.

Write these fractions as decimals.

1 2 3 4

0·63974974974 . . . is written as 0·63 7

Rewrite these recurring decimals using the ‘dot’ notation.

5 0·4444 . . . 6 0·631631631 . . . 7 0·166666 . . . 8 0·72696969 . . .

Rewrite these decimals in simplest fraction form.

9 0·75 10 0·875

Worked examples

Example 1

When each number in the decimal is repeated.

Write 0·636363 . . . as a fraction Let x = 0·6363 . . .

Multiply by 100 because two digits are repeated.

Then 100x = 63·6363 . . .

Subtract the two lines.

So 100x − x = 63·6363 . . . − 0·6363 . . . ie 99x = 63

∴ x =

Simplifying this fraction.

∴ x =

Prep Quiz 1:05

1 4 --- 2 5 --- 1 3 --- 5 6 ---9. 4. 3 8 --- 1 6 --- This can be

checked using your calculator.

■ Recurring decimals

are sometimes called repeating decimals. (÷125) (÷125) 375 1000 ---3 8 ---63 99 ---continued ➜➜➜ 7 11

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---Write these fractions as terminating decimals.

a b c d

e f g h

i j

Write these fractions as recurring decimals.

a b c d e

f g h i j

Write these terminating decimals as fractions.

a 0·47 b 0·16 c 0·125 d 0·85 e 0·035

By following Example 1, rewrite these recurring decimals as fractions.

a 0·4444 . . . b 0·575757 . . . c 0·173173173 . . .

d 0· e 0· f 0· 23

Determine the value of 0· .

By following Example 2, rewrite these decimals as fractions.

a 0·83333 . . . b 0·6353535 . . . c 0·197777 . . .

d 0·6 e 0·73 f 0·82

g 0·5 2 h 0·527 i 0·64 3

Example 2

When only some digits are repeated.

Write 0·617777 . . . as a fraction Let x= 0·61777 . . .

Multiply by 100 to move the non-repeating digits to the left of the decimal point.

Then 100x= 61·777 . . .

Multiply by 1000 to move one set of the repeating digits to the left of the decimal point.

And 1000x= 617·777

Subtract the previous two lines.

So 1000x− 100x= 617·777 − 61·777 ie 900x= 556

∴x=

Simplifying this fraction using your calculator.

∴x=

This answer can be checked by performing 139 ÷ 225 using your calculator.

556 900 ---139 225

---Exercise 1:05

_Decimals_N_S4_·3 1 Write as decimals. a b 2 Write as fractions. a 0·6 b 0·95 1 5 --- 17 100 ---Foundation Worksheet 1:05 1 3 4 --- 4 5 --- 5 8 --- 7 10 ---7 100 --- 35 20 --- 4 25 --- 17 50 ---19 40 --- 117 125 ---2 2 3 --- 5 9 --- 8 9 --- 2 11 --- 1 7 ---1 6 --- 1 15 --- 7 15 --- 1 24 --- 17 30 ---3 4 7. 3.6. 1. 4. 5 3. 6 4. 6. 4.9. 1. 3. 8. 7. 4.