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Enhanced

Mathematics

9Enhanced STAGE

5.15.3

Sydney, Melbourne, Brisbane, Perth, Adelaideand associated companies around the world

Alan McSevenyRob 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 3207PO Box 460, Port Melbourne, Victoria 3207www.pearson.com.au

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

Copyright Pearson Australia 2009(a division of Pearson Australia Group Pty Ltd)First published 2009 by Pearson Australia

2013 2012 201110 9 8 7 6 5 4

Reproduction and communication for educational purposesThe Australian Copyright Act 1968(the Act) allows amaximum of one chapter or 10% of the pages of this work,whichever is the greater, to be reproduced and/orcommunicated by any educational institution for itseducational purposes provided that that educationalinstitution (or the body that administers it) has given aremuneration notice to Copyright Agency Limited (CAL)under the Act. For details of the CAL licence for educationalinstitutions contact Copyright Agency Limited(www.copyright.com.au).

Reproduction and communication for other purposes

Except as permitted under the Act (for example any fairdealing for the purposes of study, research, criticism orreview), no part of this book may be reproduced, stored in aretrieval system, communicated or transmitted in any form orby any means without prior written permission. All enquiriesshould be made to the publisher at the address above.

This book is not to be treated as a blackline master; that is, anyphotocopying beyond fair dealing requires prior writtenpermission.

Publisher: Leah KellyEditor: Liz WaudDesigner: Pierluigi VidoTypesetter: Nikki M GroupCover Designers: Bob Mitchell and Ruth ComeyCopyright & Pictures Editor: Michelle JellettProject Editor: Carlie JenningsProduction Controller: Jem WolfendenCover art: Corbis Australia Pty LtdIllustrators: Michael Barter, Bruce Rankin and Wendy GortonPrinted 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 5.15.3)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

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iii

Contents

Features of New SignpostMathematics Enhanced viiiTreatment of Outcomes xiiMetric Equivalents xviThe Language of Mathematics xvii

ID Card 1 (Metric Units) xviiID Card 2 (Symbols) xviiID Card 3 (Language) xviiiID Card 4 (Language) xixID Card 5 (Language) xxID Card 6 (Language) xxiID Card 7 (Language) xxii

Algebra Card xxiii

Basic Skills and Number 1

1:01 The language of mathematics 21:02 Diagnostic tests 2

A Integers 3B Fractions 3C Decimals 4D Percentages 5

1:03 Conversion facts you should know 6

What was the prime ministers name in 1978? 7

1:04 Rational numbers 81:05 Recurring decimals 11

Try this with repeating decimals 13Speedy addition 13

1:06 Simplifying ratios 141:07 Rates 17

Comparing speeds 19

1:08 Significant figures 191:09 Approximations 221:10 Estimation 25

Take your medicine! 28

1:11 Angles review 291:12 Triangles and quadrilaterals 33

Maths terms Diagnostic test Revisionassignment Working mathematically 37

Working Mathematically 43

2:01 Solving routine problems 44A Rates 44B Ratio 47C Dividing a quantity in a given ratio 48

Mixing drinks 50

D Percentages 51E Measurement 53

2:02 Solving non-routine problems 56

What nationality is Santa Claus? 60Line marking 60

2:03 Using Venn diagrams (extension) 61

Venn diagrams 62

What kind of breakfast takes an hour tofinish? 64The Syracuse Algorithm 64

Maths terms Revision assignment Workingmathematically 65

Algebraic Expressions 68

3:01 Generalised arithmetic 69

Lets play with blocks 72

3:02 Substitution 73

The history of algebra 74

3:03 Simplifying algebraic expressions 743:04 Algebraic fractions 76

A Addition and subtraction 76B Multiplication and division 78

Try this maths-word puzzle 79

3:05 Simplifying expressions with grouping

symbols 80

What is taken off last before you get intobed? 82

3:06 Binomial products 833:07 Special products 85

A Perfect squares 85

The square of a binomial 85

B Difference of two squares 863:08 Miscellaneous examples 87

Patterns in products 88Using special products in arithmetic 89


Probability 95

4:01 Describing your chances 96

Throwing dice 100

4:02 Experimental probability 100

Tossing a coin 104Chance experiments 105

4:03 Theoretical probability 105

Computer dice 110Chance happenings 111

4:04 The addition principle for mutually exclusive

events 111

Probability: An unusual case 115What are Dewey decimals? 116Chance in the community 117


Deductive Geometry 122

5:01 Deductive reasoning in numerical exercises 123A Exercises using parallel lines 123

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

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iv

New Signpost Mathematics Enhanced 9 5.15.3

B Exercises using triangles 125C Exercises using quadrilaterals 127

5:02 Polygons 129

The angle sum of a polygon 130The exterior angle sum of a convex polygon 131Regular polygons and tessellations 133Spreadsheet 134The game of Hex 135

5:03 Deductive reasoning in non-numerical

exercises 1365:04 Congruent triangles 1395:05 Proving two triangles congruent 1435:06 Using congruent triangles to find unknown

sides and angles 1475:07 Deductive geometry and triangles 1495:08 Deductive geometry and quadrilaterals 153

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

5:09 Pythagoras theorem and its converse 159

Proving Pythagoras theorem 159Maths terms Diagnostic test Revisionassignment Working mathematically 162

Indices and Surds 167

6:01 Indices and the index laws 168

Exploring index notation 172Family trees 172

6:02 Negative indices 173

Zero and negative indices 176

6:03 Fractional indices 177

Why is a room full of married peoplealways empty? 180Reasoning 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 1916:08 Addition and subtraction of surds 1936:09 Multiplication and division of surds 195

Iteration to find square roots 197

6:10 Binomial products 1986:11 Rationalising the denominator 200

What do Eskimos sing at birthday parties? 201

Rationalising binomial denominators 202Maths terms Diagnostic test Revisionassignment Working mathematically 203

Measurement 208

7:01 Perimeter 209

Staggered starts 214Skirting board and perimeter 215

7:02 Review of area 216

Why is it so noisy at tennis? 222

Covering floors 223

7:03 Surface area of prisms and cylinders 224

How did the boy know that he had anaffinity with the sea? 229

7:04 Surface area of composite solids 230

Truncated cubes 232

7:05 Volume of prisms, cylinders and compositesolids 233

Perimeter, area and volume 237

7:06 Practical applications of measurement 238

Wallpapering rooms 242Maths terms Diagnostic test Revisionassignment Working mathematically 243

Equations, Inequations and Formulae 248

8:01 Equivalent equations 2498:02 Equations with grouping symbols 252

If I have 7 apples in one hand and 4 in theother, what have I got? 254Solving 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 thedenominator 259

8:05 Solving problems using equations 260

Who dunnit? 265

8:06 Inequations 265

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

8:07 Formulae: Evaluating the subject 270

Spreadsheet formulae 273

8:08 Formulae: Equations arising fromsubstitution 274

8:09 Solving literal equations (1) 2778:10 Solving literal equations (2) 2798:11 Solving problems with formulae 282

Why are cooks cruel? 285Maths terms Diagnostic test Revisionassignment Working mathematically 286

Consumer Arithmetic 291

9:01 Working for others 2929:02 Extra payments 296

Jobs in the papers 299

9:03 Wage deductions 3009:04 Taxation 304

Income tax returns 306What is brought to the table, cut,but never eaten? 307

9:05 Budgeting 3089:06 Best buy, shopping lists and change 3109:07 Goods and services tax (GST) 314

Shopper dockets 316

9:08 Ways of paying and discounts 317

The puzzle of the missing dollar 321

9:09 Working for a profit 322

Chapter 6

2

2

Chapter 7

Chapter 8

Chapter 9

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v

Lets plan a disco 325Maths terms Diagnostic test Revisionassignment Working mathematically 325

Coordinate Geometry 330

10.01 The distance between two points 33110.02 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 gradientintercept form of astraight line: y

= mx

+ b

352

What does y

= mx

+ b

tell us? 352

10.06 The equation of a straight line, given pointand gradient 358

10.07 The equation of a straight line, giventwo points 360

10.08 Parallel and perpendicular lines 36310.09 Graphing inequalities on the number plane 367

Why did the banana go out with a fig? 371Maths terms Diagnostic test Revisionassignment Working mathematically 372

Factorising Algebraic Expressions 377

11:01 Factorising using common factors 37811:02 Factorising by grouping in pairs 38011:03 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 hardertrinomials 389

11:06 Factorising: Miscellaneous types 390

What did the caterpillar say when it sawthe butterfly? 391

11:07 Simplifying algebraic fractions:Multiplication and division 392

11:08 Addition and subtraction of algebraicfractions 395


Statistics 402

12:01 Frequency and cumulative frequency 40312:02 Analysing data (1) 410

Codebreaking and statistics 413

12:03 Analysing data (2) 414

Which hand should you use to stir tea? 421Adding and averaging 422

12:04 Grouped data 423

The aging population 428Maths terms Diagnostic test Revisionassignment Working mathematically 429

Simultaneous Equations 436

Solving problems by guess and check 437

13:01 The graphical method of solution 438

Solving simultaneous equations using agraphics calculator 442What did the book say to the librarian 442

13:02 The algebraic method of solution 443A Substitution method 443B Elimination method 445

13:03 Using simultaneous equations to solveproblems 448

Breakfast time 451Maths terms Diagnostic test Revisionassignment Working mathematically 452

Trigonometry 455

14:01 Right-angled triangles 45614:02 Right-angled triangles: the ratio of sides 45814:03 The trigonometric ratios 46014:04 Trig. ratios and the calculator 466

The exact values for the trig. ratio of30, 60 and 45 469

14:05 Finding an unknown side 47014:06 Finding an unknown angle 47614:07 Miscellaneous exercises 47914:08 Problems involving two right triangles 484

What small rivers flow into the Nile? 487Maths terms Diagnostic test Revisionassignment Working mathematically 488

Graphs of Physical Phenomena 492

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

Graphing coins 502Can 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 519Make words with your calculator 520Curves and stopping distances 521


Answers 528Answers to ID Cards 598Index 599

Acknowledgements 604

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

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vi


Interactive Student CD

1:02A Integers 2Set A Addition and subtraction of integers 2Set B Integers: Signs occurring side by side 2Set C Multiplication and division of integers 3Set D Order of operations 4

1:02B Fractions 5Set A Improper fractions to mixed numerals 5Set B Mixed numerals to improper fractions 5Set C Simplifying fractions 6Set D Equivalent fractions 7

Set E Addition and subtraction of fractions (1) 7Set F Addition and subtraction of fractions (2) 8Set G Addition and subtraction of mixed numerals 9Set H Harder subtractions of mixed numerals 10Set I Multiplication of fractions 11Set J Multiplication of mixed numerals 11Set K Division of fractions 12Set L Division of mixed numerals 13

1:02C Decimals 15Set A Arranging decimals in order of size 15Set B Addition and subtraction of decimals 15Set C Multiplication of decimals 16

Set D Multiplying by powers of ten 16Set E Division of a decimal by a whole number 17Set F Division involving repeating decimals 17Set G Dividing by powers of ten 18Set H Division of a decimal by a decimal 18Set I Converting decimals to fractions 19Set J Converting fractions to decimals 20

1:02D Percentages 22Set A Converting percentages to fractions 22Set B Converting fractions to percentages 23Set C Converting percentages to decimals 24Set D Converting decimals to percentages 24

Set E Finding a percentage of a quantity 25Set F Finding a quantity when a part of it is known 26Set G Percentage composition 28Set H Increasing or decreasing by a percentage 29

Appendix Answers

1:05 Decimals 11:09 Approximation 21:10 Estimation 3

1:11 Angles review 41:12 Triangles and quadrilaterals 53:01 Generalised arithmetic 6

3:02 Substitution 73:04A Simplifying algebraic fractions 83:04B Simplifying algebraic fractions 93:05 Grouping symbols 104:02 Experimental probability 114:03 Theoretical probability 125:02 Formulae 135:03 Non-numerical proofs 145:05 Congruent triangles 155:09 Pythagoras theorem 166:01 The index laws 176:02 Negative indices 18

6:03 Fractional indices 196:04 Scientific notation 206:07 Surds 216:08 Addition and subtraction of surds 226:09 Multiplication and division of surds 236:10 Binomial productssurds 247:01 Perimeter 257:02 Area 267:03 Surface area of prisms 277:04 Surface area of composite solids 287:05 Volume 298:01 Equivalent equations 308:02 Equations with grouping symbols 318:03 Equations with fractions (1) 328:04 Equations with fractions (2) 338:05 Solving problems using equations 348:06 Solving inequations 358:07 Formulae 368:09 Solving literal equations 379:02 Extra payments 389:04 Taxation 399:06 Best buy, shopping lists, change 409:07 Goods and services tax 4110:01 Distance between points 4210:02 Midpoint 43

10:03 Gradients 4410:04 Graphing lines 4510:05 Gradientintercept form 4610:06 Pointgradient form 4710:08 Parallel and perpendicular lines 4810:09 Graphing inequalities 4911:01 Common factors 5011:02 Grouping in pairs 5111:04 Factorising trinomials 5211:08 Addition and subtraction of algebraic

fractions 53

Student Book

Appendixes

Foundation Worksheets

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

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vii

12:01 Frequency and cumulative frequency 5412:02 Mean, median and mode 5512:03 Mean and median 5613:01 Graphical method of solution 5713:02A The substitution method 5813:03 Using simultaneous equations to solve

problems 5914:05 Using trigonometry to find side lengths 6014:07 Angles of elevation and depression, and

bearings 6114:08 Problems with more than one triangle 62

Worksheet Answers

3:05 Fractions and grouping symbols 15:02 Regular polygons and tessellations 26:03 Algebraic expressions and indices 312:04 Australias population 413:03 Solving three simultaneous equations 514:03 The range of values of the trig. ratios 6

14:06 Trigonometry and the limit of an area 714:08 Solving three-dimensional problems 8

Worksheet Answers

The material below is found in the Companion Websitewhich is included on the Interactive Student CD as bothan archived version and a fully featured live version.

Activities and Investigations2:01C Sharing the prize3:02 Substitution

3:02 Magic squaresChapter 4 Probability5:02 Spreadsheet5:08 Quadrilaterals6:01 Who wants to be a millionaire?6:06 Golden ratio investigations7:05 Greatest volume8:03 Flowcharts8:088:10 Substituting and transposing formulae9:03 Wages10:05 Equation grapherChapter 12 Sunburnt country

13:01 Break-even analysis14:06 Shooting for a goal15:01 World record times15:02 Filling tanks

Drag and DropsChapter 1: Maths terms 1A,

Maths terms 1B,Significant figures, Triangles andquadrilaterals, Angles

Chapter 3: Maths Terms 3, Addition and subtractionof algebraic fractions, Multiplication anddivision of algebraic fractions, Groupingsymbols, Binomial products, Specialproducts

Chapter 4: Maths terms 4, Two dice, Pack of cardsChapter 5: Maths terms 5, Angles and parallel lines,

Triangles, Quadrilaterals, Angle sum ofpolygons, Pythagoras theorem

Chapter 6: Maths terms 6, Index laws, Negativeindices, Fractional indices, Simplifyingsurds, Operations with surds

Chapter 7: Maths terms 7, Perimeter, Area of sectorsand composite figures, Surface area,Volume

Chapter 8: Maths terms 8, Equations with fractions,Solving inequations, Formulae, Equationsfrom formulae, Solving literal equations

Chapter 9: Maths terms 9, Find the weekly wage,Going shopping, GST.

Chapter 10: Maths terms 10,xand yintercept and

graphs, Using y = mx + bto find thegradient, General form of a line, Paralleland perpendicular lines, Inequalities andregions

Chapter 11: Maths terms 11, Factorising usingcommon factors, Grouping in pairs,Factorising trinomials 1, Factorisingtrinomials 2, Mixed factorisations

Chapter 12: Maths terms 12Chapter 14: Maths terms 14, The trigonometric ratios,

Finding sides, Finding angles, Bearings 1,Bearings 2

AnimationsChapter 10: Linear graphs and equationsChapter 14: Trigonometry ratios

Chapter Review QuestionsThese can be used as a diagnostic tool orfor revision. They include multiple choice,pattern-matching and fill-in-the-gapsstyle questions.

DestinationsLinks to useful websites that relate directly to thechapter content.

Challenge Worksheets

Technology Applications

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viii New Signpost Mathematics Enhanced 9 5.15.3

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 meetsthe high standards set by earlier editions.

Graded exercises are colour coded to indicate

levels of difficulty.

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

difficult investigations.

Enhanced technology is used extensively

throughout, with fully integrated links to both the

Student CD and the Pearson Places Website.

TheStudent CDaccompanies 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 5Mathematics9Enhancedathem

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 and10are 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 eachStudent Book. These

innovative resources

allow any teacher to

confidently 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 Enhancedseries,

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. Printfunction 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

fill-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

Geometers Sketchpad.

Drag and Drop Interactives to

improve basic skills.

Animations to develop skills

by manipulating interactive

demonstrations of key

mathematical concepts. Quick Quizzesfor each chapter

Chap

terReviewQuestion

s

Technology

Drag-and-drop

Animation

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x New Signpost Mathematics Enhanced 9 5.15.3

Student Book

Chapter-opening pagessummarise the key content

and present the syllabus outcomes addressed in each

chapter.

Clear syllabus referencesare 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 NS51.Well-graded exerciseswhere levels of difficulty 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 passesthrough the point (3, 2). Find the equation of

the line.

4 Solve each literal equation for x:

a a+ x= bx

b ax= px+ q

c x+ a= ax+ b

Worked examplesare used extensively and are easy

for students to identify.

Worked example1 Express the following in scientific notation

a 243 b 60 000 c 98 800 000

Important rules and conceptsare clearly highlighted

at regular intervals throughout the text.

Cartoonsare used to give students friendly advice

or tips.

Prep Quizzesreview 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.

Challengeactivities and worksheetsprovide more difficult investigations and

exercises. They can be used to extend

more able students.

Fun Spotsprovide amusement and interest,

while often reinforcing coursework. They

encourage creativity and divergent thinking,

and show that mathematics is enjoyable.

Investigationsencourage students to

seek knowledge and develop research

skills. They are an essential part of any

mathematics course.

Diagnostic Testsat 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

TheNew Signpost Mathematics Enhanced 9 and 10learning 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

like this!

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xi

Assignmentsare provided at the

end of each chapter. Where there are

two assignments, the first revises the

content of the chapter, while the second

concentrates on developing the students

ability to work mathematically.

TheAlgebra Card(see p. xxiii) is usedto 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 Mathssections 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 defined 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 Applicationsapply

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:

athe sum of 3aand 2b

bthe average of mand n

2aFind the cost of xbooks at

75c each.

bFind the age of Bill, who

is 25 years old, in another

yyears.

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xii New Signpost Mathematics Enhanced 9 5.15.3

Treatment of Outcomes

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

involvesquestioning

, applying

strategies

, communicating

, reasoning

and reflecting

. These are given specialattention 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 usingmathematics in relation to Stage 5.3 content.

Revision: WorkingMathematically, Chapter 2,and throughout the text

WMS5.3.2 Solves problems using a range of strategies includingdeductive reasoning.


WMS5.3.3 Uses and interprets formal definitions andgeneralisations when explaining solutions and orconjectures


WMS5.3.4 Uses deductive reasoning in presenting argumentsand formal proofs.


WMS5.3.5 Links mathematical ideas and makes connectionswith, and generalisations about, existing knowledgeand 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, ratiosand rates.

1:021:04, 1:06, 1:07,2:01A, B, C, D

NS5.1.1 Applies index laws to simplify and evaluatearithmetic expressions and uses scientific notation towrite large and small numbers.

6:016:05

NS5.1.2 Solves consumer arithmetic problems involvingearning and spending money.

9:019:07, 9:09

NS5.1.3 Determines relative frequencies and theoreticalprobabilities.

4:014:04, Year 10

NS5.2.1 Rounds decimals to a specified number of significantfigures, expresses recurring decimals in fraction formand converts rates from one set of units to another.

1:05, 1:081:10

NS5.2.2 Solves consumer arithmetic problems involvingcompound interest, depreciation and successivediscounts.

9:08, Year 10

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xiii

NS5.3.1 Performs operations with surds and indices. 6:066:11

NS5.3.2 Solves probability problems involving compoundevents.

Year 10

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

3:013:03

PAS4.4 Uses algebraic techniques to solve linear equationsand simple inequalities.

8:01, 8:02

PAS4.5 Graphs and interprets linear relationships on thenumber plane.

10:04

PAS5.1.1 Applies the index laws to simplify algebraicexpressions.

6:01

PAS5.1.2 Determines the midpoint, length and gradient of aninterval joining two points on the number plane andgraphs linear and simple non-linear relationships

from equations.

10:0110:04

PAS5.2.1 Simplifies, expands and factorises algebraicexpressions involving fractions and negative andfractional indices.

3:01, 6:02, 6:03

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

8:028:08, 13:0113:03,Year 10

PAS5.2.3 Uses formulae to find midpoint, distance andgradient and applies the gradientintercept form tointerpret and graph straight lines.

10:0110:03, 10:05

PAS5.2.4 Draws and interprets graphs including simpleparabolas 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 quadraticexpressions.

3:043:08, 11:0111:08

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

8:028:06, 8:098:11,Year 10

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

10:04, 10:0610:09

PAS5.3.4 Draws and interprets a variety of graphs includingparabolas, cubics, exponentials and circles andapplies coordinate geometry techniques to solveproblems.

Year 10

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xiv


PAS5.3.5 Analyses and describes graphs of physicalphenomena.

15:01, 15:02

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

Year 10

PAS5.3.7 Recognises, describes and sketches polynomials, and

applies the factor and remainder theorems to solveproblems.

Year 10

PAS5.3.8 Describes, interprets and sketches functions and usesthe definition of a logarithm to establish and applythe laws of logarithms.

Year 10

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

12:01

DS4.2 Collects statistical data using either a census or a

sample and analyses data using measures of locationand range.

12:02, 12:03

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

12:01, 12:03, 12:04

DS5.2.1 Uses the interquartile range and standard deviation toanalyse data.

Year 10

MS4.1 Uses formulae and Pythagoras theorem in calculatingperimeter and area of circles and figures composed ofrectangles and triangles.

2:01E, 7:02

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

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

MS5.1.1 Uses formulae to calculate the area of quadrilateralsand finds areas and perimeters of simple compositefigures.

7:01, 7:02

MS5.1.2 Applies trigonometry to solve problems (diagramsgiven) including those involving angles of elevationand depression.

14:0114:07, Year 10

MS5.2.1 Finds areas and perimeters of composite figures. 7:01, 7:02MS5.2.2 Applies formulae to find the surface area of right

cylinders and volume of right pyramids, cones andspheres and calculates the surface area and volume ofcomposite solids.

7:037:06, Year 10

MS5.2.3 Applies trigonometry to solve problems includingthose involving bearings.

14:0414:07, Year 10

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xv

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 StudiesNSW, the material is in no way connected with or endorsed by them. For comprehensive coursedetails 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, cosinerule and area rule in problem solving.

14:08, Year 10

SGS4.2 Identifies and names angles formed by theintersection of straight lines, including those related

to transversals on sets of parallel lines, and makes useof the relationships between them.

1:01, 1:11

SGS4.3 Classifies, constructs, and determines the propertiesof triangles and quadrilaterals.

1:01, 1:12

SGS5.2.1 Develops and applies results related to the angle sumof interior and exterior angles for any convexpolygon.

5:02

SGS5.2.2 Develops and applies results for proving that triangles

are congruent or similar.

5:045:06, Year 10

SGS5.3.1 Constructs arguments to prove geometrical results. 5:01, 5:035:06, 5:09

SGS5.3.2 Determines properties of triangles and quadrilateralsusing deductive reasoning.

5:07, 5:08

SGS5.3.3 Constructs geometrical arguments using similaritytests for triangles

Year 10

SGS5.3.4 Applies deductive reasoning to prove circle theoremsand to solve problems.

Year 10

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xvi


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 dm1 cm = 10 mm1 km = 1000 m

Area

1 m

2

= 10000 cm

2

1 ha = 10 000 m

2

1 km

2

= 100 ha

Mass

1 kg = 1000 g

1 t = 1000 kg1 g = 1000 mg

Volume

1 m

3

= 1 000 000 cm

3

= 1000 dm

3

1 L = 1000 mL1 kL = 1000 L1 m

3

= 1 kL1 cm

3

= 1 mL

1000 cm

3

= 1 L

Time

1 min = 60 s1 h = 60 min

1 day = 24 h1 year = 365 days

1 leap year = 366 days

It is importantthat you learnthese factsoff by heart.

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xvii

The Language of Mathematics

You should regularly test your knowledge byidentifying 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

8

9

ha

10

m

3

11

cm

3

12

s

9

4

2

10

4

3

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

22

23 24

P(E)

2 23

x

See MathsTerms atthe end of

each chapter.

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xviii



.

ID Card 3 (Language)

1

6 minus 2

2

the sum of6 and 2

3

divide6 by 2

4

subtract2 from 6

5

the quotient of

6 and 2

63

2)6the divisor

is . . . .

73

2)6the dividend

is . . . .

8

6 lots of 2

9

decrease6 by 2

10

the productof 6 and 2

11

6 more than 2

12

2 less than 6

13

6 squared

14

the squareroot of 36

15

6 take away 2

16

multiply6 by 2

17

average of6 and 2

18

add 6 and 2

19

6 to thepower of 2

20

6 less 2

21

the differencebetween 6 and 2

22

increase6 by 2

23

share6 between 2

24

the total of6 and 2

We saysix squaredbut we write

62.

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xix



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 sidesdifferent

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xx New Signpost Mathematics Enhanced 9 5.15.3



1

A

............

2

............

3

............

4

............

5

............ points

6

Cis the ............

7

............

............

8

............

9

all angles lessthan 90

10

one angle 90

11

one angle greaterthan 90

12

A, Band Care......... of the triangle.

13

Use the verticesto name the .

14

BCis the ......... ofthe right-angled .

15

a +b +c =.........

16

BCD=.........

17

a +b +c +d =.....

18

Which (a) a b

19

a =.............

20

Angle sum =............

21

ABis a ...............OCis a ...............

22

Name of distancearound the circle..............................

23

.............................

24

ABis a ...............CDis an ...............EFis 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

ca

A D

B

C

b

a

b

d

c

a

ba

a

A B

C

O

O

O

B

C

D

FE

A

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xxi



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

CDisp.......... toAB.

A

BC

(lessthan90)

(90)

(between90and 180)

(180) (between180and360)

(360)

a+ b= 90

ab

a+ b= 180

a b

a= b

a ba

bc

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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xxii New Signpost Mathematics Enhanced 9 5.15.3



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

100m

4

3

2

1

0A B C D E F

Cars soldMonTuesWedThursFri

Money collectedMonTuesWedThursFri

Stands for $10

70

50

30

10

M T W T F

Dollars

Money collected

10080604020

Johns height

1 2 3 4 5Age (years)

Use of time

HobbiesSleep

HomeSchool

People present

Adults

Girls

Boys

Smoking

Cigarettes smokedLengthoflife

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xxii

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 21 m 3m 5m2 5x 3x x+2 x3 2x+1 3x8

2 1 04 4m 2m 2m3 3x 5x2 x+7 x6 4x+2 x1

3 5 08 10m 5m 8m5 10x 8x x+5 x +5 6x+2 x5

4 2 15 8m 7m 6m2 15x 4x4 x+1 x9 3x+3 2x+4

5 8 25 2m 10m m2 7x 2x3 x+8 x+2 3x+8 3x+1

6 10 07 5m 6m 9m3

9x x2

x+4 x7 3x+1 x+7

7 6 12 8m 9m 2m6 6x 5x2 x+6 x1 x+8 2x5

8 12 05 20m 4m 3m3 12x 4x3 x+10 x8 5x+2 x10

9 7 01 5m 10m m7 5x 3x5 x+2 x+5 2x+4 2x4

10 5 06 9m 7m 8m4 3x 7x5 x+1 x7 5x+4 x+7

11 11 18 7m 8m 4m 4x x3 x+9 x+6 2x+7 x6

12 4 14 3m 12m 7m2 7x x10 x+3 x10 2x+3 2x+3

1

4

---

2m

3

-------

x

6

---

x

2

---

1

8---

m4----

x3---

x4---

1

3---

m

4----

2x

7------

2x

5------

120------

3m2-------

x10------

x5---

3

5---

m

5----

2x

3------

x

3---

27---

3m7-------

2x5------

3x5------

3

8---

m

6----

5x

6------

2x

3------

920------

2m5-------

3x4------

x7---

3

4---

3m

5-------

3x

7------

3x

7------

710------

4m

5-------

x

6---

2x9------

110------

m5----

x5---

x3---

25---

m3----

3x4------

x6---

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1Basic Skillsand Number

1

I must remember

something, surely!

Learning Outcomes

NS42

(reviewed) Compares, orders and calculates with integers.

NS43

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

NS521

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.

SGS42

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.

SGS43

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

W

orkingM

athematically S

tages 4

and5.

1 Questioning, 2

Applying Strategies, 3

Communicating, 4

Reasoning,

5

Reflecting.

Chapter Contents

1:01

The language of

mathematics NS42, SGS4.2,3

1:02

Diagnostic tests NS42, 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 NS43

Fun Spot: What was the prime ministers

name in 1978?

1:04

Rational numbers NS43

1:05

Recurring decimals NS521

Challenge: Try this with repeating decimals

Fun Spot: Speedy addition

1:06

Simplifying ratios NS43

1:07

Rates NS43

Investigation: Comparing speeds1:08

Significant figures NS521

1:09

Approximations NS521

1:10

Estimation NS521

Reading Maths: Take your medicine!

1:11

Angles review SGS42

1:12

Triangles and quadrilaterals SGS43

Maths Terms, Diagnostic Test, Revision

Assignment, Working Mathematically

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2


1:01

The Language Outcomes NS42, SGS42,3

of Mathematics

Much of the language met so far is reviewed in the identification cards (ID Cards) found onpages 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 toany you cant 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 cardand copy expressions and answers for those that are unfamiliar. (For example, for the quotientof 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 cant 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 onone side and the answers on the other. Carry these with you as an aid to learning. Have otherstest you.

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

1:02

Diagnostic Tests

Outcomes NS42, NS43

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

.

Exercise 1:01

1

2

3

4

2 meanstwo below zero ortwo less than zero.

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3

Chapter 1

Basic Skills and Number

1:02A | Integers

NS42

1:02B | Fractions

NS43

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 equivalentfractions.

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 J15 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---------

34--- 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

34---NumeratorDenominator

Fractions shouldalways be expressedin 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---

12--- 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

4---

1

5---

1

10------

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4 New Signpost Mathematics Enhanced 9 5.15.3

1:02C | Decimals NS43

CD Appendix

Put in order, smallest to largest. Set A

1 a 0505, 05, 055 b 84, 8402, 841 c 101, 11, 1011

2 a 26 +314 b 186 +3 c 0145 +012 Set B

3 a 1283 12 b 9 1824 c 402 0005 Set B

4 a 07 6 b (03)2 c 002 17 Set C

5 a 3142 100 b 004 1000 c 0065 10 Set D

6 a 21 104 b 804 106 c 125 102 Set D

7 a 408 2 b 121 5 c 019 4 Set E

8 Write answers as repeating decimals.

a 25 6 b 532 9 c 28 3 Set F

9 a 2435 10 b 67 100 c 07 1000 Set G

10 a 64 02 b 0824 008 c 65 005 Set H

11 Convert these decimals to fractions.

a 05 b 018 c 9105 Set I

12 Convert these fractions to decimals.

a b c Set J45--- 38--- 56---

3 tens7 units

4 tenths2 hundredths5 thousandths

37425

10 1

3 7 4 2 5

1

10------

1

100---------

1

1000------------

What does

37.425

really

mean?

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5Chapter 1 Basic Skills and Number

1:02D | Percentages NS43

CD Appendix

1 Convert to fractions. Set A

a 18% b 7% c 224%

2 Convert to fractions. Set A

a 95% b 6 % c 1225%

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 238% b 12 % c 4 %

6 Convert to percentages. Set D

a 051 b 0085 c 18

7 Find: Set E

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

8 Find: Set E

a 7% of 843 m b 6 % of 44 tonnes

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

Set F

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

10 a 5 kg of sugar, 8 kg of salt and 7 kg offlour were mixed accidentally. What isthe percentage (by weight) of sugar inthe mixture?

Set G

b John scored 24 runs out of the teams totalof 60 runs. What percentage of runs didJohn 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 games only half the

fun it used to be . . .

14---

11

20------

5

6---

1

4---

1

2

---2

3

---

1

4---

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1:03 Conversion Facts Outcome NS43

You Should Know

To the right, I

have used these facts.

Percentage Decimal Fraction

1% 001

5% 005

10% 01

12 % 0125

20% 02

25% 025

33 % 0

50% 05

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% =01 =

Multiply each by 6.

60% =06 =b 5% =005 =

Multiply each by 7.

35% =035 =

c 20% =02 =

Add 1 or 100% to each.

120% =12 =1

d 12 % =0125 =

Add 1 or 100% to each.112 % =1125 =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 youconvert to decimals andpercentages 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 thecorrect 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 005 +3 O 03 002 O 03 5

E (03)2 E 3142 100 E 612 6E 2008 10 C 18 02

G of 60 kg D What fraction is 125 g of 1 kg?

H 5% of 80kg HWrite as a percentage.

H Write 075 as a fraction. H Increase 50kg 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 thatare 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 ministers 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---

102 7

009 2

$6 1

65%

15 2

53

55kg

3142

4kg

10

360kg 4

028 5 9

40%

24

305

45kg

2008

65kg

1 2

1 9--- 3 4---

04

7 4--- 1 4---17

24------ 21

5------ 1 8--- 3 4---

3 8--- 1 8---

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1:04 Rational Numbers Outcome NS43Fractions, decimals, percentages and negative numbers are convenient ways of writingrational 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 tocompare the sizes of real numbers.

Discussion

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

From the list on the right, choosetwo equivalent numbers for:

a 2 b 130%

c 28 d 1

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

a 085, 0805, 09, 1 b 875%, 100%, 104%, 12 %

c , , and d 1 , 150%, 165, 2

e 142, , 141, 140% f , 3 , 31,

Find the number halfway between:

a 68 and 69 b 12 % and 20%

c and d 635 and 64

Real numbers

Rationalnumbers

Irrationalnumbers

A number is rationalif it can be expressed as the quotient of

two integers, , where b0.

eg , 8, 52%, 12 %, 0186, , 15, 10

An irrational numbercannot be written as a fraction, , whereaand bare integers and b0.

eg , , , ,

a

b---

3

4---

1

2--- 0 3

a

b---

2 7 43

53

An integer is awhole number thatmay be positive,negative or zero.

1

2---

Exercise 1:04

125% 114% 2 28% 280%

2 14 25 208% 13

125 13 1 250% 25%

4

5---

1

8---

3

10------

1

1

2---

1

4---

2

1

4---

58--- 4

7--- 2

3--- 64

100--------- 3

4---

2 14--- 12

3

1

2---

1

8---

1

5---

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

b Explain why 099999 =1.

c Write as decimals: , , , , , .

d Write as fractions or mixed numbers: , , , .

What are the next three numbers in the sequence:

a 0125, 025, 05, . . . ? b 13, 065, 0325, . . . ?The average (ie mean) of five numbers is 158.

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 975% pa?

An advertisement reads: 67% leased; only one tenancy remaining for lease. Building readyOctober. 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 treewith a diameter of 20 mm to reach a diameter of 50mm.

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

Julius Caesar invaded Britain in 55 BC and againone year later. What was the date of the secondinvasion?

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 03333333? b 06666666?c 01111111? d 05555555?

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

41

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

147

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 thewholesale price.

What number when divided by 08 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 10000 of

developing cancer after the age of 65?

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

a If I earn 50% of my fathers salary,what percentage of my salary doesmy father earn?

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

c My height is 160% of my childs height.Express my childs height as a percentageof my height.

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

b Give two unit fractions with differentdenominators 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 isnt moving

in the meantime.

A unit fraction

has a numerator

of 1.

21

223

8---

5

11------

23a

b--

c

d--

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1:05 Recurring Decimals Outcome NS521

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)3306040 For : 6)110404040

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

For example: 0375 =

=

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

Write these fractions as decimals.

1 2 3 4

063974974974 . . . is written as 063 7

Rewrite these recurring decimals using the dot notation.

5 04444 . . . 6 0631631631 . . .

7 0166666 . . . 8 072696969 . . .

Rewrite these decimals in simplest fraction form.

9 075 10 0875

Worked examples

Example 1

When each number in the decimal is repeated.Write 0636363 . . . as a fractionLetx=06363 . . .

Multiply by 100 because two digits are repeated.

Then 100x=636363 . . .Subtract the two lines.So 100xx=636363 . . . 06363 . . .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 decimalsare sometimes calledrepeating decimals.

(125)

(

125)375

1000------------

3

8---

6399------

continued

7

11------

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12


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

047

b

016

c

0125

d

085

e

0035

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

a

04444 . . .

b

0575757 . . .

c

0173173173 . . .

d

0

e

0

f

0 23

Determine the value of 0 .

By following Example 2, rewrite these decimals as fractions.

a

083333 . . .

b

06353535 . . .

c

0197777 . . .

d

06

e

073

f

082

g

05 2

h

0527

i

064 3

Example 2

When only some digits are repeated.Write 0617777 . . . as a fractionLetx

=

061777 . . .

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

x

=

61777 . . .

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

x

=

617777

Subtract the previous two lines.So 1000

x

100

x

=

617777

61777ie 900

x

=

556

x

=

Simplifying this fraction using your calculator.

x

=

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

556

900---------

139225---------

Exercise 1:05Decimals NS43

1 Write as decimals.

a b

2Write as fractions.

a 06 b095

15---

17100---------

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.
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Here is a clever shortcut method for writing a repeating decimal as a fraction.Follow the steps carefully.

Try converting these repeating decimals to fractions usingthis method.

1 0 2 0 3 0 1 4 01 5 032

Rachel discovered an interesting trick.

1 She asked her father to write down a 5-digit number.

2 Rachel then wrote a 5-digit number below her fathers.She chose each digit of her number so that when she

added it to the digit above, she got 9.3 She then asked her father to write another 5-digit number.

4 She then repeated step 2.

5 She then asked her father to write one more 5-digit number.

6 She now challenged her father to a race in addingthese 5 numbers.

7 Rachel wrote down the answer immediately and surprisedher father. Look at the example to see how she did it.

8 She then asked her father to work out how she did it.9 What should you do if the last number chosen ends

with 00 or 01?

Challenge 1:05 Try this with repeating decimals

Example 1

1 =

=

0 2626 0

99

---------------

26

99------

Step 1(Numerator)

Subtract the digits before the repeatingdigits from all the digits.Step 2(Denominator)Write down a 9 for each repeating digitand then a zero for each non-repeatingdigit in the decimal.Step 3Simplify the fraction if possible.

Example 2

2 =

=

=

0327327 32

900

---------------------

295

900---------

59180---------

one 9tw

o0s

That's prettynifty.

7. 6. 7. 3. 2. 6. 1. 7.

Fun Spot 1:05 Speedy addition

4 1 8 4 9

5 8 1 5 0

3 8 1 4 6

6 1 8 5 3

2 1 4 1 1

2 2 1 4 0 9

Puta2atthefront.

Thesethreedigits

arethesameasinthe

lastnumber.

Makethis2-d

igit

numbertwolessthan

theoneaboveit.

My writing is

in colour.

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1:06 Simplifying Ratios Outcome NS43

Simplify the fractions:

1 2 3 4

What fraction is:

5 50c of $1? 6 40c of 160c? 7 8 kg of 10kg?

8 100cm of 150 cm? 9 1 m of 150cm? 10 $2 of $2.50?

Worked examples

1 Jans height is 1 metre while Dianes is 150 cm. Find the ratio of their heights.

2 of the class walk to school while ride bicycles. Find the ratio of those who walkto those who ride bicycles.

3 Express the ratio 11 to 4 in the form a X : 1 b 1:Y .

Solutions1 Jans height to Dianes height

=1 m to 150 cm

=100 cm to 150 cm

=100 : 150

Divide both terms by 50.

=2 : 3 or

From this ratio we can see that Jan is as tall as Diane.

2 Those walking to those cycling

= :

Multiply both terms by 20.

= :

=12: 5

Prep Quiz 1:06

50

60------

16

20------

72

84------

125

625---------

Ratios are just like fractions!A ratio is a comparison of like quantitieseg Comparing 3 km to 5km we write:

3 to 5 or 3 : 5 or .35---

3

5---

1

4---

Each term is expressed in the same units,then units are left out.

We may simplify ratios by dividing ormultiplying each term by the same number.

To remove fractions, multiply each term bythe lowest common denominator.

2

3---

2

3---

3

5---

1

4---

351-----

204

1--------

141-----

205

1--------

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Express the first quantity as a fraction of the second each time.

a 7 men, 15 men b 10kg, 21kg c 3 cm, 4cmd $5, $50 e 8 m, 10 m f 10 bags, 100 bagsg 75g, 80 g h 6 runs, 30 runs i 25 goals, 120 goals

Simplify the ratios.

a 6 : 4 b 10:5c 65:15 d 14:35e 20:45 f 42:60g 60:15 h 45:50i 1000:5 j 1100: 800k 55:20 l 16:28m 10:105 n 72:2

o 4:104 p 10 :

q : 2 r 2 : 2

s 2 : 1 t 2 : 3

u 6 : 3 v :

w : x :

In each, find the ratio of the first quantity to the second, giving your answers in simplest form.

a 7 men, 9 men b 13kg, 15 kg c 7 cm, 8cmd $8, $12 e 16 m, 20 m f 15 bags, 85 bagsg 90g, 100 g h 9 runs, 18 runs i 50 goals, 400 goals

j 64ha, 50ha k 25 m, 15 m l 100m2, 40m2

Find the ratio of the first quantity to the second. Give answers in simplest form.

a $1, 50c b $5, $2.50 c $1.20, $6d 1 m, 60 cm e 25cm, 2m f 100 m, 1 kmg 600 mL, 1L h 1 L, 600 mL i 5 L, 1 L 250mL

j 2 h, 40min k 50min, 1 h l 2 h 30 min, 5 h

3 a 11 to 4 b 11 to 4=11 : 4 =11 :4Divide both terms by 4. Divide both terms by 11.

= : 1 =1 :

=2 : 1 This is in the form 1 : Y.

This is in the form X : 1.

11

4------

4

11------

3

4---

Exercise 1:06

Simplify thefractions.

1

You may use

x or

2

1

2---

1

2---

1

2---

1

2---

3

4---

1

4---

3

4---

3

4---

1

2---

11

16------

1

2---

2

3---

1

2---

3

4

Are unitsthe same?

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Write these ratios in the form X : 1.

a 13:8 b 7 : 4c 5 : 2 d 110:100e 700:500 f 20:30g 2 : 7 h 10:9i 4 : 6 j 15:8k 1 : 3 l 2 :

Write these ratios in the form 1 : Y.

a 4 : 5 b 2 : 9 c 8:15d 14:6 e 8:10 f 1000: 150g 100:875 h 4:22 i 4 : 6

a Anne bought a painting for $600 (cost) and sold the painting for $800 (selling price).Find the ratio of:

i cost to selling priceii profit to cost

iii profit to selling priceb John, who is 160 cm tall, jumped 180 cmto win the high jump competition. Whatis the ratio of this jump to his height?Write this ratio in the form X : 1.

c A rectangle has dimensions 96 cm by 60 cm.Find the ratio of:

i its length to breadthii its breadth to length

d 36% of the bodys skin is on the legs, while 9% is on the head/neck part of the body.Find the ratio of:

i the skin on the legs to the skin on the head/neckii the skin on the legs to the skin on the rest of the body

e Joans normal pulse is 80 beats per minute, while Erics is only 70. After Joan runs 100 mher pulse rate rises to 120 beats per minute. Find the ratio of:

i Joans normal pulse rate to Erics normal pulse rateii Joans normal pulse rate to her rate after the run

f At 60 km/h a truck takes 58 metres to stop (16 m during the drivers reaction time and 42 mbraking distance), while a car travelling at the same speed takes 38 metres to stop (16 mreaction and 22 m braking). Find the ratio of:

i the trucks stopping distance to the cars stopping distance

ii the cars reaction distance to the cars braking distanceiii the trucks braking distance to the cars braking distance

a A recipe recommends the use of two parts sugar to one part flour and one part custardpowder. What does this mean?

b A mix for fixing the ridge-capping on a roof is given as 1 part cement to 5 parts sand and ahalf part of lime. What does this mean?

c The ratio of a model to the real thing is called the scale factor. My model of an aeroplane is40 cm long. If the real plane is 16 m long, what is the scale factor of my model?

d My father is 180cm tall. If a photograph of him has a height of 9 cm, what is the scale ofthe photograph?

To change 8 into 1,we need to divide by 8.

5

1

2---

1

4---

ie 13 : 8

= : 1

=1 : 1

13

8------

5

8---

6

7

8

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1:07 Rates Outcome NS43

Usually we write down how many of the first quantity correspond to one of the second quantity,eg 60 kilometres per one hour, ie 60 km/h.

If Wendy earns $16 per hour, how much would she earn in:

1 2 hours? 2 3 hours? 3 5 hours? 4 half an hour?

Complete:5 1kg =. . . g 6 1 tonne =. . . kg 7 1 hour =. . . min

8 1cm =. . . mm 9 1 m2=. . . cm2 10 15 litres =. . . millilitres

Worked examples

2 16 kg of tomatoes are sold for $10.What is the cost per kilogram?

Cost =

=

=

=625 cents/kg

Prep Quiz 1:07

A rate is a comparison of unlike quantities:eg If I travel 180 km in 3 hours my average rate of speed is or 60 km/h

or 60 km per h.

180 km

3 h-----------------

is the samecentskg

as c/kg.

continued

1 84 m in 2 ours or

Divide each term by 2.

=42km in 1 hour

=42km/h

84 m in 2 ours

=

=

=42km/h

84 km

2 h---------------

84

2------

km

h--------

Units must be shown.

Example (1) is an average rate because,when you travel, your speed may varyfrom moment to moment.

Example (2) is a constant rate, becauseeach kg will cost the same.

$10

16 kg-------------

1000

16------------

cents

kg------------

125

2

---------cents

kg

------------

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Write each pair of quantities as a rate in its simplest form.

a 6 km, 2h b 10 kg, $5 c 500c, 10 kgd 100 mL, 100 cm3 e 160L, 4h f $100, 5hg $315, 7 days h 70km, 10 L i 20 degrees, 5min

j 7000g, 100cm k 50 t, 2 blocks l 60km, h

m 88 runs, 8 wickets n 18 children, 6 mothers o 75g, 10cm3

a I walk at 5 km/h. How far can I walk in 3 hours?b Nails cost $2.45 per kg. What is the cost of 20 kg?c I can buy four exercise books for $5. How many books can I buy for $20?d I earn $8.45 per hour. How much am I paid for 12 hours work?e The run rate per wicket in a cricket match has been 375 runs per wicket. How many runs

have been scored if 6 wickets have been lost?f The fuel value of milk is measured as 670 kilojoules

per cup. What is the fuel value of 3 cups of milk?g If the rate of exchange for one English pound is160 American dollars, find the value of ten Englishpounds in American currency.

h The density of iron is 75 g/cm3. What is the mass of1000cm3of iron? (Density is mass per unit of volume.)

i If light travels at 300000 km/s, how far would it travelin one minute?

j If I am taxed 16c for every $1 on the value of my $50 000 block of land, how muchmust I pay?

Complete the equivalent rates.

a 1 km/min =. . . km/h b 40000m/h =. . . km/hc $50/kg =. . . c/kg d $50/kg =. . . c/ge 144L/h =. . . mL/s f 60 km/h =. . . m/sg 7 km/L =. . . m/mL h 25c/h =. . . $/weeki 30 mm/s =. . . km/h j 90 beats/min =. . . beats/sk 800kg/h =. . . t/day l 3t/h =. . . kg/minm 10 jokes/min =. . . jokes/h n 50c/m2=. . . $/hao 105cm3/g =. . . cm3/kg

3 A plumber charges a householder$64 per hour to fix the plumbingin a house. Find the cost if it takeshim 4 hours.

Rate =$64 per 1 hour

Multiply both terms by 4 .

=$64 4 per 4 hours

Cost =$288

4 Change 72 litres per hour into cm3per second.

72 L per h =

=

=

72 L/h =20cm3/s

1

2---

1

2---

12--- 1

2---

72 L1 h----------

72000mL

60min-------------------------

72000cm3

60 60 s

---------------------------

Exercise 1:07

1

1

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2

3

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The density of a person is approximately 095 g/cm3. This means that the average weight of1 cm3of a person is 095 g.

Now 095g/cm3=95 g/100 cm3=1 g/ cm31g/105cm3.

Use this information to answer the following questions.a Find the volume in cm3of a man weighing 70 kg.b Find the volume in cm3of a girl weighing 46kg.c Find your own volume in cm3.

d What is the least number of 70 kg men required to have a total volume of more than 1 m3?

1:08 Significant Figures Outcome NS521

No matter how accurate measuring instruments are,a quantity such as length cannot be measured exactly.Any measurement is only an approximation. A measurement is only useful when one can be confident

of its validity. To make su

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