NSM9_51_53

Mathematics

EnhancedAlan McSeveny Rob Conway

9STAGE 5.15.3

Steve Wilkes

Sydney, Melbourne, Brisbane, Perth, Adelaide and associated companies around the world

Let the wise listen and add to their learning, and let the discerning get guidance. Proverbs 1:5Pearson 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. Copyright Pearson Australia 2009 (a division of Pearson Australia Group Pty Ltd) First published 2009 by Pearson Australia 2013 2012 2011 10 9 8 7 6 5 4 Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that that educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact Copyright Agency Limited (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 5.15.3) 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

ContentsFeatures of New Signpost Mathematics Enhanced Treatment of Outcomes Metric Equivalents The Language of MathematicsID Card 1 (Metric Units) ID Card 2 (Symbols) ID Card 3 (Language) ID Card 4 (Language) ID Card 5 (Language) ID Card 6 (Language) ID Card 7 (Language)

viii xii xvi xviixvii xvii xviii xix xx xxi xxii

What kind of breakfast takes an hour to finish? The Syracuse Algorithm Maths terms Revision assignment Working mathematically

64 64 65

Chapter 3Algebraic ExpressionsGeneralised arithmetic Lets play with blocks 3:02 Substitution The history of algebra 3:03 Simplifying algebraic expressions 3:04 Algebraic fractions A Addition and subtraction B Multiplication and division Try this maths-word puzzle 3:05 Simplifying expressions with grouping symbols What is taken off last before you get into bed? 3:06 Binomial products 3:07 Special products A Perfect squares The square of a binomial B Difference of two squares 3:08 Miscellaneous examples Patterns in products Using special products in arithmetic Maths terms Diagnostic test Revision assignment Working mathematically 3:01

6869 72 73 74 74 76 76 78 79 80 82 83 85 85 85 86 87 88 89 90

Algebra Card

xxiii

Chapter 1Basic Skills and Number1:01 1:02

1

The language of mathematics 2 Diagnostic tests 2 A Integers 3 B Fractions 3 C Decimals 4 D Percentages 5 1:03 Conversion facts you should know 6 What was the prime ministers name in 1978? 7 1:04 Rational numbers 8 1:05 Recurring decimals 11 Try this with repeating decimals 13 Speedy addition 13 1:06 Simplifying ratios 14 1:07 Rates 17 Comparing speeds 19 1:08 Significant figures 19 1:09 Approximations 22 1:10 Estimation 25 Take your medicine! 28 1:11 Angles review 29 1:12 Triangles and quadrilaterals 33 Maths terms Diagnostic test Revision assignment Working mathematically 37

Chapter 4Probability4:01

95

Chapter 2Working Mathematically2:01 Solving routine problems A Rates B Ratio C Dividing a quantity in a given ratio Mixing drinks D Percentages E Measurement Solving non-routine problems What nationality is Santa Claus? Line marking Using Venn diagrams (extension) Venn diagrams

4344 44 47 48 50 51 53 56 60 60 61 62

Describing your chances 96 Throwing dice 100 4:02 Experimental probability 100 Tossing a coin 104 Chance experiments 105 4:03 Theoretical probability 105 Computer dice 110 Chance happenings 111 4:04 The addition principle for mutually exclusive events 111 Probability: An unusual case 115 What are Dewey decimals? 116 Chance in the community 117 Maths terms Diagnostic test Revision assignment Working mathematically 117

2:02 2:03

Chapter 5Deductive Geometry5:01

122

Deductive reasoning in numerical exercises 123 A Exercises using parallel lines 123

iii

B Exercises using triangles C Exercises using quadrilaterals 5:02 Polygons The angle sum of a polygon The exterior angle sum of a convex polygon Regular polygons and tessellations Spreadsheet The game of Hex 5:03 Deductive reasoning in non-numerical exercises 5:04 Congruent triangles 5:05 Proving two triangles congruent 5:06 Using congruent triangles to find unknown sides and angles 5:07 Deductive geometry and triangles 5:08 Deductive geometry and quadrilaterals Theorems and their converses What do you call a man with a shovel? 5:09 Pythagoras theorem and its converse Proving Pythagoras theorem Maths terms Diagnostic test Revision assignment Working mathematically

125 127 129 130 131 133 134 135 136 139 143 147 149 153 158 158 159 159 162

Covering floors Surface area of prisms and cylinders How did the boy know that he had an affinity with the sea? 7:04 Surface area of composite solids Truncated cubes 7:05 Volume of prisms, cylinders and composite solids Perimeter, area and volume 7:06 Practical applications of measurement Wallpapering rooms Maths terms Diagnostic test Revision assignment Working mathematically 7:03

223 224 229 230 232 233 237 238 242 243

Chapter 8Equations, Inequations and Formulae 248Equivalent equations Equations with grouping symbols If I have 7 apples in one hand and 4 in the other, what have I got? Solving equations using a spreadsheet 8:03 Equations with fractions (1) Who holds up submarines? 8:04 Equations with fractions (2) Equations with pronumerals in the denominator 8:05 Solving problems using equations Who dunnit? 8:06 Inequations Operating on inequations Read carefully (and think!) 8:07 Formulae: Evaluating the subject Spreadsheet formulae 8:08 Formulae: Equations arising from substitution 8:09 Solving literal equations (1) 8:10 Solving literal equations (2) 8:11 Solving problems with formulae Why are cooks cruel? Maths terms Diagnostic test Revision assignment Working mathematically 8:01 8:02 249 252 254 254 255 257 257 259 260 265 265 266 269 270 273 274 277 279 282 285 286

Chapter 6Indices and Surds6:01

167

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 2 is irrational 189 f-stops and 2 190 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 assignment Working mathematically 203

Chapter 9Consumer Arithmetic9:01 9:02 9:03 9:04 Working for others Extra payments Jobs in the papers Wage deductions Taxation Income tax returns What is brought to the table, cut, but never eaten? Budgeting Best buy, shopping lists and change Goods and services tax (GST) Shopper dockets Ways of paying and discounts The puzzle of the missing dollar Working for a profit

291292 296 299 300 304 306 307 308 310 314 316 317 321 322

Chapter 7Measurement7:01 7:02 Perimeter Staggered starts Skirting board and perimeter Review of area Why is it so noisy at tennis?

208209 214 215 216 222

9:05 9:06 9:07 9:08 9:09

iv

New Signpost Mathematics Enhanced 9 5.15.3

Lets plan a disco Maths terms Diagnostic test Revision assignment Working mathematically

325 325

12:04 Grouped data The aging population Maths terms Diagnostic test Revision assignment Working mathematically

423 428 429

Chapter 10Coordinate Geometry10.01 The distance between two points 10.02 The midpoint of an interval 10.03 The gradient of a line Gradients in building 10.04 Graphing straight lines What is the easiest job in a watch factory? 10.05 The gradientintercept form of a straight line: y = mx + b What does y = mx + b tell us? 10.06 The equation of a straight line, given point and gradient 10.07 The equation of a straight line, given two points 10.08 Parallel and perpendicular lines 10.09 Graphing inequalities on the number plane Why did the banana go out with a fig? Maths terms Diagnostic test Revision assignment Working mathematically

330331 336 340 345 346 351 352 352 358 360 363 367 371 372

Chapter 13Simultaneous EquationsSolving problems by guess and check 13:01 The graphical method of solution Solving simultaneous equations using a graphics calculator What did the book say to the librarian 13:02 The algebraic method of solution A Substitution method B Elimination method 13:03 Using simultaneous equations to solve problems Breakfast time Maths terms Diagnostic test Revision assignment Working mathematically

436437 438 442 442 443 443 445 448 451 452

Chapter 14TrigonometryRight-angled triangles Right-angled triangles: the ratio of sides The trigonometric ratios Trig. ratios and the calculator The exact values for the trig. ratio of 30, 60 and 45 14:05 Finding an unknown side 14:06 Finding an unknown angle 14:07 Miscellaneous exercises 14:08 Problems involving two right triangles What small rivers flow into the Nile? Maths terms Diagnostic test Revision assignment Working mathematically 14:01 14:02 14:03 14:04

455456 458 460 466 469 470 476 479 484 487 488

Chapter 11Factorising Algebraic ExpressionsFactorising using common factors Factorising by grouping in pairs Factorising using the difference of two squares The difference of two cubes 11:04 Factorising quadratic trinomials How much logic do you have? 11:05 Factorising further quadratic trinomials Another factorising method for harder trinomials 11:06 Factorising: Miscellaneous types What did the caterpillar say when it saw the butterfly? 11:07 Simplifying algebraic fractions: Multiplication and division 11:08 Addition and subtraction of algebraic fractions Maths terms Diagnostic test Revision assignment Working mathematically 11:01 11:02 11:03

377378 380 382 383 384 385 386 389 390 391 392 395 398

Chapter 15Graphs of Physical PhenomenaDistance/time graphs A Linear graphs Graphing coins Can you count around corners? B Non-linear graphs Rolling down an inclined plane 15:02 Relating graphs to physical phenomena Spreadsheet graphs Make words with your calculator Curves and stopping distances Maths terms Diagnostic test Revision assignment Working mathematically 15:01

492493 493 502 502 503 509 510 519 520 521 522

Chapter 12Statistics12:01 Frequency and cumulative frequency 12:02 Analysing data (1) Codebreaking and statistics 12:03 Analysing data (2) Which hand should you use to stir tea? Adding and averaging

402403 410 413 414 421 422

Answers Answers to ID Cards Index Acknowledgements

528 598 599 604

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Interactive Student CDStudent Book Appendixes1:02A IntegersSet A Set B Set C Set D Set A Set B Set C Set D Set E Set F Set G Set H Set I Set J Set K Set L Set A Set B Set C Set D Set E Set F Set G Set H Set I Set J Set A Set B Set C Set D Set E Set F Set G Set H Addition and subtraction of integers Integers: Signs occurring side by side Multiplication and division of integers Order of operations

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.

22 2 3 4

1:02B Fractions

5

Improper fractions to mixed numerals 5 Mixed numerals to improper fractions 5 Simplifying fractions 6 Equivalent fractions 7 Addition and subtraction of fractions (1) 7 Addition and subtraction of fractions (2) 8 Addition and subtraction of mixed numerals 9 Harder subtractions of mixed numerals 10 Multiplication of fractions 11 Multiplication of mixed numerals 11 Division of fractions 12 Division of mixed numerals 13

1:02C DecimalsArranging decimals in order of size Addition and subtraction of decimals Multiplication of decimals Multiplying by powers of ten Division of a decimal by a whole number Division involving repeating decimals Dividing by powers of ten Division of a decimal by a decimal Converting decimals to fractions Converting fractions to decimals Converting percentages to fractions Converting fractions to percentages Converting percentages to decimals Converting decimals to percentages Finding a percentage of a quantity Finding a quantity when a part of it is known Percentage composition Increasing or decreasing by a percentage

1515 15 16 16 17 17 18 18 19 20

1:02D Percentages

2222 23 24 24 25 26 28 29

Appendix Answers

Foundation Worksheets1:05 1:09 1:10 Decimals Approximation Estimation 1 2 3

1:11 1:12 3:01 3:02 3:04A 3:04B 3:05 4:02 4:03 5:02 5:03 5:05 5:09 6:01 6:02 6:03 6:04 6:07 6:08 6:09 6:10 7:01 7:02 7:03 7:04 7:05 8:01 8:02 8:03 8:04 8:05 8:06 8:07 8:09 9:02 9:04 9:06 9:07 10:01 10:02 10:03 10:04 10:05 10:06 10:08 10:09 11:01 11:02 11:04 11:08

Angles review Triangles and quadrilaterals Generalised arithmetic Substitution Simplifying algebraic fractions Simplifying algebraic fractions Grouping symbols Experimental probability Theoretical probability Formulae Non-numerical proofs Congruent triangles Pythagoras theorem The index laws Negative indices Fractional indices Scientific notation Surds Addition and subtraction of surds Multiplication and division of surds Binomial productssurds Perimeter Area Surface area of prisms Surface area of composite solids Volume Equivalent equations Equations with grouping symbols Equations with fractions (1) Equations with fractions (2) Solving problems using equations Solving inequations Formulae Solving literal equations Extra payments Taxation Best buy, shopping lists, change Goods and services tax Distance between points Midpoint Gradients Graphing lines Gradientintercept form Pointgradient form Parallel and perpendicular lines Graphing inequalities Common factors Grouping in pairs Factorising trinomials Addition and subtraction of algebraic fractions

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

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12:01 12:02 12:03 13:01 13:02A 13:03

Frequency and cumulative frequency Mean, median and mode Mean and median Graphical method of solution The substitution method Using simultaneous equations to solve problems 14:05 Using trigonometry to find side lengths 14:07 Angles of elevation and depression, and bearings 14:08 Problems with more than one triangle

54 55 56 57 58 59 60 61 62

Chapter 3:

Chapter 4: Chapter 5:

Chapter 6:

Worksheet AnswersChapter 7:

Challenge Worksheets3:05 5:02 6:03 12:04 13:03 14:03 14:06 14:08 Fractions and grouping symbols Regular polygons and tessellations Algebraic expressions and indices Australias population Solving three simultaneous equations The range of values of the trig. ratios Trigonometry and the limit of an area Solving three-dimensional problems 1 2 3 4 5 6 7 8 Chapter 8:


Worksheet Answers

Technology ApplicationsThe 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.

Chapter 11:

Activities and Investigations2:01C 3:02 3:02 Chapter 4 5:02 5:08 6:01 6:06 7:05 8:03 8:088:10 9:03 10:05 Chapter 12 13:01 14:06 15:01 15:02 Chapter 1: Sharing the prize Substitution Magic squares Probability Spreadsheet Quadrilaterals Who wants to be a millionaire? Golden ratio investigations Greatest volume Flowcharts Substituting and transposing formulae Wages Equation grapher Sunburnt country Break-even analysis Shooting for a goal World record times Filling tanks Maths terms 1A, Maths terms 1B, Significant figures, Triangles and quadrilaterals, Angles


Maths Terms 3, Addition and subtraction of algebraic fractions, Multiplication and division of algebraic fractions, Grouping symbols, Binomial products, Special products Maths terms 4, Two dice, Pack of cards Maths terms 5, Angles and parallel lines, Triangles, Quadrilaterals, Angle sum of polygons, Pythagoras theorem Maths terms 6, Index laws, Negative indices, Fractional indices, Simplifying surds, Operations with surds Maths terms 7, Perimeter, Area of sectors and composite figures, Surface area, Volume Maths terms 8, Equations with fractions, Solving inequations, Formulae, Equations from formulae, Solving literal equations Maths terms 9, Find the weekly wage, Going shopping, GST. 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 Maths terms 11, Factorising using common factors, Grouping in pairs, Factorising trinomials 1, Factorising trinomials 2, Mixed factorisations Maths terms 12 Maths terms 14, The trigonometric ratios, Finding sides, Finding angles, Bearings 1, Bearings 2

AnimationsChapter 10: Linear graphs and equations Chapter 14: Trigonometry ratios

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

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

Drag and Drops

vii

Mathem Mathematics

Enhanced

9STAGE STAGE 5 T

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.

What does the package consist of? Full-colour Student Book with free Student CD Homework Book Pearson Places Website Teacher Edition LiveText DVD

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

Student Book Improved full-colour design and layout makes the text more appealing for students s 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 difculty. Working Mathematically is fully integrated and also features as a separate section at the end of each chapter.

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

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For more information on the New Signpost Mathematics Enhanced series, visit www.pearsonplaces.com.au

Pearson Places WebsiteThe Pearson Places Website contains a wealth of support material for students and teachers: pter R sti ev i ew Q u eon

Teacher EditionA Teacher Edition is available for each Student Book. These innovative resources allow any teacher to condently 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 wraparound 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.

Chapter Review Questions for use as a diagnostic tool or for revision. These are autocorrecting and include multiplechoice, pattern-matching and 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 Geometers Sketchpad.

s

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.

Ch

a

Techn ogy ol

D ra

g - a n d - d ro p

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

Animati on

LiveText DVD 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.

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How to use this bookThe 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:

Student BookChapter-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 chapteropening pages, in each main section within each chapter and in the Foundation Worksheet references. For example, Outcome NS51. Well-graded exercises where levels of difculty are indicated by the colour of the question number. 1 4 9 1 green blue red foundation Stage 5.3 level extension

Important rules and concepts are clearly highlighted at regular intervals throughout the text. Cartoons are used to give students friendly advice or tips.

The table of values looks like this!

Find the simple interest charged for a loan of: a 2 3 b 5 7 c 3 11 a A straight line has a gradient of 2 and passes through the point (3, 2). Find the equation of the line.

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 difcult 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.

3

4

Solve each literal equation for x: a a+x=bx b ax = px + q c x + a = ax + b

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

Worked example1 Express the following in scientic notation a 243 b 60 000 c 98 800 000

x


Assignments are provided at the end of each chapter. Where there are two assignments, the rst revises the content of the chapter, while the second concentrates on developing the students 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 dened 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.

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

Student CD Companion WebsiteTechnology 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.

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 y years.

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Treatment of OutcomesEach 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, applying strategies, 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 WMS5.3.1 Asks questions that could be explored using mathematics in relation to Stage 5.3 content. Solves problems using a range of strategies including deductive reasoning. Uses and interprets formal definitions and generalisations when explaining solutions and or conjectures Uses deductive reasoning in presenting arguments and formal proofs. Links mathematical ideas and makes connections with, and generalisations about, existing knowledge and understanding in relation to Stage 5.3 content. Compares, orders and calculates with integers. Operates with fractions, decimals, percentages, ratios and rates. Applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers. Solves consumer arithmetic problems involving earning and spending money. Determines relative frequencies and theoretical probabilities. 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. Solves consumer arithmetic problems involving compound interest, depreciation and successive discounts. Text references Revision: Working Mathematically, Chapter 2, and throughout the text Revision: Working Mathematically, Chapter 2, and throughout the text Revision: Working Mathematically, Chapter 2, and throughout the text Revision: Working Mathematically, Chapter 2, and throughout the text Revision: Working Mathematically, Chapter 2, and throughout the text 1:01, 1:02 1:021:04, 1:06, 1:07, 2:01A, B, C, D 6:016:05

WMS5.3.2

WMS5.3.3

WMS5.3.4

WMS5.3.5

NS4.2 NS4.3 NS5.1.1

NS5.1.2 NS5.1.3 NS5.2.1

9:019:07, 9:09 4:014:04, Year 10 1:05, 1:081:10

NS5.2.2

9:08, Year 10

xii


NS5.3.1 NS5.3.2 PAS4.3 PAS4.4 PAS4.5 PAS5.1.1 PAS5.1.2

Performs operations with surds and indices. Solves probability problems involving compound events. Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions. Uses algebraic techniques to solve linear equations and simple inequalities. Graphs and interprets linear relationships on the number plane. Applies the index laws to simplify algebraic expressions. 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. Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices. Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods. Uses formulae to find midpoint, distance and gradient and applies the gradientintercept form to interpret and graph straight lines. Draws and interprets graphs including simple parabolas and hyperbolas. Draws and interprets graphs of physical phenomena. Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions. Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations. Uses various standard forms of the equation of a straight line and graphs regions on the number plane. Draws and interprets a variety of graphs including parabolas, cubics, exponentials and circles and applies coordinate geometry techniques to solve problems.

6:066:11 Year 10 3:013:03 8:01, 8:02 10:04 6:01 10:0110:04

PAS5.2.1

3:01, 6:02, 6:03

PAS5.2.2

8:028:08, 13:0113:03, Year 10 10:0110:03, 10:05

PAS5.2.3

PAS5.2.4 PAS5.2.5 PAS5.3.1

Year 10 15:01,15:02 3:043:08, 11:0111:08

PAS5.3.2

8:028:06, 8:098:11, Year 10 10:04, 10:0610:09 Year 10

PAS5.3.3 PAS5.3.4

xiii

PAS5.3.5 PAS5.3.6

Analyses and describes graphs of physical phenomena. Uses a variety of techniques to sketch a range of curves and describes the features of curves from the equation. Recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems. Describes, interprets and sketches functions and uses the definition of a logarithm to establish and apply the laws of logarithms. Constructs, reads and interprets graphs, tables, charts and statistical information. Collects statistical data using either a census or a sample and analyses data using measures of location and range. Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs. Uses the interquartile range and standard deviation to analyse data. Uses formulae and Pythagoras theorem in calculating perimeter and area of circles and figures composed of rectangles and triangles. Calculates surface area of rectangular and triangular prisms and volume of right prisms and cylinders. Uses formulae to calculate the area of quadrilaterals and finds areas and perimeters of simple composite figures. Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression. Finds areas and perimeters of composite figures. 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. Applies trigonometry to solve problems including those involving bearings.

15:01, 15:02 Year 10

PAS5.3.7

Year 10

PAS5.3.8

Year 10

DS4.1 DS4.2

12:01 12:02, 12:03

DS5.1.1 DS5.2.1 MS4.1

12:01, 12:03, 12:04 Year 10 2:01E, 7:02

MS4.2 MS5.1.1

2:01E, 7:03, 7:05 7:01, 7:02

MS5.1.2

14:0114:07, Year 10

MS5.2.1 MS5.2.2

7:01, 7:02 7:037:06, Year 10

MS5.2.3

14:0414:07, Year 10

xiv


MS5.3.1 MS5.3.2 SGS4.2

Applies formulae to find the surface area of pyramids, right cones and spheres. Applies trigonometric relationships, sine rule, cosine rule and area rule in problem solving. 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. Classifies, constructs, and determines the properties of triangles and quadrilaterals. Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon. Develops and applies results for proving that triangles are congruent or similar. Constructs arguments to prove geometrical results. Determines properties of triangles and quadrilaterals using deductive reasoning. Constructs geometrical arguments using similarity tests for triangles Applies deductive reasoning to prove circle theorems and to solve problems.

Year 10 14:08, Year 10 1:01, 1:11

SGS4.3 SGS5.2.1

1:01, 1:12 5:02

SGS5.2.2 SGS5.3.1 SGS5.3.2 SGS5.3.3 SGS5.3.4

5:045:06, Year 10 5:01, 5:035:06, 5:09 5:07, 5:08 Year 10 Year 10

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

xv

Metric EquivalentsLength 1 m = 1000 mm = 100 cm = 10 dm 1 cm = 10 mm 1 km = 1000 m Area 1 m2 = 10 000 cm2 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 cm3 = 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 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: Autumn: Winter: Spring: December, January, February March, April, May June, July, August September, October, November

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

xvi


The Language of MathematicsYou should regularly test your knowledge by identifying the items on each card. ID Card 1 (Metric Units) 2 3 4 dm cm mm 6 km 9 ha 13 min 17 g 21 L 22 mL 18 mg 23 kL 14 h 19 kg 24 C 10 m3 15 m/s m2 11 cm3 7 cm2 12 s 16 km/h 20 t 21 13 8 km2 9 42 14 || 17 % 22 18 23 x 19 eg 20 ie 24 P(E) 10 43 15 11 2 16 ||| ID Card 2 (Symbols) 2 3 4 or 6 7 > 123

1 m 5

1 = 5

b 22O

a

a = ............. 23

Angle sum = ............ C 24B A E DF

AB is a ............... OC is a ............... See page 598 for answers.

Name of distance around the circle. .............................

O

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

AB is a ............... CD is an ............... EF is a...............

xx


1

2

ID Card 6 (Language) 3

4

..................... lines 5A

..................... lines 6(less than 90)

v ..................... h ..................... 7 8

..................... lines(between 90 and 180)

(90)

B

C

angle ..................... 9(180)

..................... angle 10(between 180 and 360)

..................... angle 11(360)

..................... angle 12

..................... angle 13a + b = 90

..................... angle 14a + b = 180

..................... 15a = b a b

..................... angles 16a b c d

a

b

a b

..................... angles 17

..................... angles 18a = b b a

..................... angles 19a = b a b

a + b + c + d = ..... 20a + b = 180 a b

..................... 21C A D E B

..................... angles 22A D

..................... angles 23C

..................... angles 24 CA B

B

C

A

B

b............ an interval See page 598 for answers.

b............ an angle

CAB = ............

CD is p.......... to AB.

xxi

1

2

ID Card 7 (Language) 3

4

ADa............ D............ 5100 m 100 m

BCb............ C............ 6 7

ama............ M............ 8

pmp............ m............

area is 1 ............ 9 10

r............ shapes 11

............ of a cube

c............-s............ 12

f............ 13 14

v............ 15

e............ 16

axes of ............

r............ 17 43 2 1 0 A B C D E F

t............ 18Cars sold Mon Tues Wed Thurs Fri

r............ 19 Money collectedMon Tues Wed Thurs Fri Stands for $10

t............ 2070 Dollars 50 30 10 M T W T F Money collected

The c............ of the dot are E2. 21100 80 60 40 20 Johns height

t............ 22Use of time Hobbies

p............ graph 23People present Adults

c............ graph 24Length of life Smoking

Sleep

1 2 3 4 5 Age (years)

School Home

Boys

Girls

Cigarettes smoked

l............ graph See page 598 for answers.

s............ graph

b............ graph

s............ d............

xxii


Algebra CardA 1 3 B 21 C 1 -4 1 -8 1 -3 1 ----20 3 -5 2 -7 3 -8 9 ----20 3 -4 7 ----10 1 ----10 2 -5 D m 4m 10m E 2m ------3 m --4 F G H 5x 3x I x -6 J 3x K x -2 x -4 2x ----5 x -5 x -3 3x ----5 2x ----3 x -7 3x ----7 2x ----9 x -3 x -6 L M N O

3m 5m2 2m 2m3 5m 8m5 7m 10m 6m2 m2

x + 2 x 3 2x + 1 3x 8 x + 7 x 6 4x + 2 x 1 x + 5 x + 5 6x + 2 x 5 x + 1 x 9 3x + 3 2x + 4 x + 8 x + 2 3x + 8 3x + 1 x + 4 x 7 3x + 1 x + 7 x + 6 x 1 x + 8 2x 5 x + 10 x 8 5x + 2 x 10 x + 2 x + 5 2x + 4 2x 4 x + 1 x 7 5x + 4 x + 7 x + 9 x + 6 2x + 7 x 6 x + 3 x 10 2x + 3 2x + 3

2 1 04 3 5 08 15

4 2

5 8 25 6 10 07 7 6 12 8 12 9 7 05 01

m --4 8m 3m ------2 2m m --5 5m 3m ------7 8m m --6 2m 20m ------5 3m 5m ------5

x 5x2 -3 10x 2x 8x ----7 x 15x ----- 4x4 10 2x ----7x 2x3 3 9x x2 5x2 4x3

2x ----5 5x 9m 2m6 6x ----6 3x 4m 3m3 12x ----4 6m 9m3 10m m7 5x

10 5 06 11 11 18 12 4 14

9m 4m 7m 8m4 3x ------5 m - 8m 4m 4x 7m --5 m --- 12m 7m2 7x 3m 3

3x ----- 3x5 7 x -- 7x5 6 x -x3 5 3x ----- x10 4

How to use this cardIf 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

xxiii

Basic Skills and Number

1I must remember something, surely!

Chapter Contents1:01 The language of NS42, SGS4.2,3 mathematics 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? NS43 1:04 Rational numbers 1:05 Recurring decimals NS521 Challenge: Try this with repeating decimals Fun Spot: Speedy addition NS43 1:06 Simplifying ratios 1:07 Rates NS43 Investigation: Comparing speeds NS521 1:08 Signicant gures 1:09 Approximations NS521 1:10 Estimation NS521 Reading Maths: Take your medicine! SGS42 1:11 Angles review 1:12 Triangles and quadrilaterals SGS43 Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically

Learning OutcomesNS42 NS43 NS521 SGS42 SGS43 (reviewed) Compares, orders and calculates with integers. (reviewed) Operates with fractions, decimals, percentages, ratios and rates. Rounds decimals to a specied number of signicant gures, expresses recurring decimals in fraction form and converts rates from one set of units to another. Identies 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. Classies, constructs and determines the properties of triangles and quadrilaterals.

Working Mathematically Stages 4 and 5. 1 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reecting.

1

1:01 The Language of Mathematics

Outcomes NS42, SGS42,3

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.

Exercise 1:011 2

Test yourself on ID Cards 1 and 2 by identifying each symbol mentally. Look up the answer to any 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 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 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 on one side and the answers on the other. Carry these with you as an aid to learning. Have others test you.

3 4

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

1:02 Diagnostic Tests

Outcomes NS42, NS43 2 means two below zero or two less than zero.

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.

2


1:02A | IntegersCD Appendix

NS42

1 a 7 + 14 2 a 3 ( 6) 3 a 3 2 4 a ( )15 ( 3) 5 a 14 7 10

b 2 15 b 12 + ( 5) b 5 6 b 63 ( 9) b 3 + 4 4

c 2 8 c 6 (3 8) c 7 ( 9) c 156 ----------3

Set A Set B Set C Set C Set DNS43CD Appendix

c (4 18) (8 + 6)

1:02B | Fractions1 Write these improper fractions as mixed numerals. ------------a 7 b 13 c 1414 3 10

Set A Set B Set C

2 Write these mixed numerals as improper fractions. 3 --a 21 b 5 ----c 312 10 7

3 Simplify these fractions. -----------a 16 b 10024 650

c

240 ----------3600

4 Complete the following to make equivalent fractions. ------a 3 = ----b 17 = -------c 3 = -----------4 28 20 100 8 1000

Set D

Give the simplest answer for . . . 5 a 6 a 7 a 8 a 9 a 10 a 11 12 13 a a a3 2 -- + -8 8 9 7 ----- ----10 10 3 4 -- + -4 5 7 3 -- -8 4 1 3 3 -- + 4 -5 2 --41 12 2 9 --71 7 2 8 4 3 -- ----5 11 7 3 -- -8 7 1 -3 -- 5 2 7 -4 34 5 8 2 ----- ----10 10 3 1 -- -5 2 --17 3 8 4 -71 3 2 -5 1 4

b b b b b b b b b b b b b b b b

14 a 15 a

16 a 17 18 a a

19 a 20 a

9 3 ----- + ----10 10 13 9 ----- ----14 14 3 2 ----- + -10 5 9 1 ----- -10 4 7 -6 ----- + 5 3 10 4 1 -10 3 5 ----4 10 7 -6 3 2 ----5 10 3 7 ----- ----10 10 15 19 ----- ----38 20 3 -1 ----- 1 4 10 5 -21 3 4 9 3 ----- ----20 20 8 3 -- -9 4 --34 21 7 2 9 4 ----- 7 10 -10 1 5

c c c c c c c c c c c c c c c c

7 2 -- + -9 9 37 11 -------- -------100 100 7 3 -------- + ----100 40 5 3 -- -6 5 5 -1 -- + 7 6 8 --20 3 1 8 5 --31 15 2 6 1 3 ----- -10 5 7 5 ----- -10 6 1 -5 -- 2 2 4 3 -5 63 8 7 7 ----- ----10 10 5 4 -- -8 7 9 -3 5 2 ----8 10 -67 5 8 1 4 ----10

Set E Set E Set F Set F Set G Set G Set H Set I Set I Set J Set J Set K Set K Set L Set L Set L

3 of 4 equal parts 3 Numerator -4 Denominator Fractions shouldalways be expressed in lowest terms.

4 2 -- = -6 3

1 1 1 1 -- -- -- -- - - 8 8 8 8 1 1 --4 4 1 -2 1 2 -- or -- or 2 4

1 1 1 1 -- -- -- -- - - 8 8 8 8 1 1 --4 4 1 -2 4 -8

Equivalent fractions

Chapter 1 Basic Skills and Number

3

1:02C | DecimalsPut in order, smallest to largest. 1 a 0505, 05, 055 2 a 26 + 314 3 a 1283 12 4 a 07 6 5 a 3142 100 6 a 21 104 7 a 408 2 b 84, 8402, 841 b 186 + 3 b 9 1824 b (03)2 b 004 1000 b 804 106 b 121 5 c 101, 11, 1011 c 0145 + 012 c 402 0005 c 002 17 c 0065 10 c 125 102 c 019 4

NS43CD Appendix

Set A

Set B Set B Set C Set D Set D Set E

8 Write answers as repeating decimals. a 25 6 9 a 2435 10 10 a 64 02 11 b 532 9 b 67 100 b 0824 008 c 28 3 c 07 1000 c 65 005 Set F Set G Set H

Convert these decimals to fractions. a 05 b 018 c 9105 Set I

12

Convert these fractions to decimals. a4 -5

b

3 -8

c

5 -6

Set J

3 tens7 units 4 tenths 2 hundredths 5 thousandths 37425 10 3 1 7 1 ----10 1 1 -------- ----------100 1000

What does 37.425 really mean?

4

2

5

4


1:02D | PercentagesCD Appendix

NS43

1 Convert to fractions. a 18% b 7% c 224%

Set A

2 Convert to fractions. a 95% b-61 % 4

Set A c 1225% Set B c-11 4

3 Convert to percentages. a11 ----20

TAX RATE 35% For every $100 earned, $35 is paid in tax.

b

5 -6

4 Convert to decimals. a 9% b 16% c 110%

Set C

5 Convert to decimals. a 238% b-12 1 % 2

Set C c-42 % 3

6 Convert to percentages. a 051 7 Find: a 35% of 600 m b 162% of $8 8 Find: a 7% of 843 m b-61 % 4

Set D c 18 Set E

b 0085

Set E of 44 tonnes Set F50% of all men play tennis.

9 a 7% of my spending money was spent on a watch band that cost $1.12. How much spending money did I have? 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? b John scored 24 runs out of the teams total of 60 runs. What percentage of runs did John score? 11 a Increase $60 by 15%. b Decrease $8 by 35%.

Set G

This games only half the fun it used to be . . .

Set H


5

1:03 Conversion Facts You Should KnowPercentage 1% 5% 10%-12 1 % 2

Outcome NS43

Decimal 001 005 01 0125 02 025 0 3 05 1

Fraction1 -------100 1 ----20 1 ----10 1 -8 1 -5 1 -4 1 -3 1 -2

To the right, I have used these facts.

1 a 10% = 01 = ----10 Multiply each by 6. 6 60% = 06 = ----10 1 b 5% = 005 = ----20 Multiply each by 7. 7 35% = 035 = ----20 -c 20% = 02 = 1 5 Add 1 or 100% to each. -120% = 12 = 1 1 5 --d 12 1 % = 0125 = 1 2 8 Add 1 or 100% to each. --112 1 % = 1125 = 1 1 2 8

20% 25%-33 1 % 3

50% 100%

1

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

6


Fun Spot 1:03

What was the prime ministers name in 1978?

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 15 Y Write ----- as a mixed numeral. 4 -M Change 1 3 to an improper fraction.4

Write the simplest answer for: 37 12 ----I 44 I -------- -------T T N E E G H H D S S32 4 2 -- -5 3 --43 + 5 8 8

T T

100 100 7 8 -- -8 7 --25 1 8 2

I T N

3 1 -- + -8 3 1 2 ( -- ) 3 1 1 -- -2 8

005 + 3 O 03 002 O 03 5 2 E 3142 100 E 612 6 (03) 2008 10 C 18 02 3 -- of 60 kg D What fraction is 125 g of 1 kg? 4 -5% of 80 kg H Write 2 as a percentage. 5 Write 075 as a fraction. H Increase 50 kg by 10%. 40% of my weight is 26 kg. How much do I weigh? Write 4 9 as a repeating (recurring) decimal. 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 3 of the amount. 5 How much does Alan get?

7

2

$6

1

102

009

65%

04

15

25

1 -9 3 -4

7 -4

1 -4

17 ----24

2 ----15

1 -8

4

5

10

9

360 kg

24

45 kg

2008

3142

65 kg

4 kg

55 kg

028

40%

305

-13

8 -21 8

-33

4


7

1:04 Rational Numbers

Outcome NS43

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

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

a b --eg 3 , 8, 52%, 12 1 %, 0186, 03 , 15, 10 two integers, -- , where b 0.4 2

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

a An irrational number cannot be written as a fraction, -- , where b a and b are integers and b 0. eg 2 , 7 , 3 4 , 3 5 , Real numbers are those that are rational or irrational. Real numbers Rational numbers Irrational numbers

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 1 and 0?2

Exercise 1:041

From the list on the right, choose two equivalent numbers for: -a 21 b 130% 2 -c 28 d 114

125% -21 8 125

114% 14 13

-24 5 25 3 1 ----10

28% 208% 250%

280% 13 25%

2

Write each set of real numbers in order. Calculators may be used. -a 085, 0805, 09, 1 b 875%, 100%, 104%, 12 1 % 4 5 4 2 64 3 - - c -- , -- , -- and -------d 1 -- , 150%, 165, 28 7 3 100

e 142,3

2 , 141, 140%

f

-, 3 1 , 31, 4

4

12

Find the number halfway between: -a 68 and 69 b 12 1 % and 20% 2 1 1 c -- and -d 635 and 648 5

8


4

a Write as decimals:

1 2 3 4 5 6 7 8 9 -- , -- , -- , -- , -- , -- , -- , -- , -- . - - - - - - - - 9 9 9 9 9 9 9 9 9 1 2 3 1 2 3 ----- , ----- , ----- , -------- , -------- , -------- . 90 90 90 900 900 900

b Explain why 099999 = 1. c Write as decimals: d Write as fractions or mixed numbers: 04 , 31 , 05 , 45 .5

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 ready October. How many tenants would you expect in this building? Using a diameter growth rate of 43 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 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 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 03333333? b 06666666? c 01111111? d 05555555? To change push 77 ----15

6

7 8 9 10 11

12

13

14

to a decimal approximation, 15 = on a calculator.

Use this method to write the following as decimals correct to five decimal places. 7 --a 8 b 2 c ----d9 20 ----21

e

7 4 ----11

f

13 5 ----18


9

15

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 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 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 fathers 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 childs height. Express my childs 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 5 denominators that subtract to give ----- .11 3 -- . 8

16 17 18 19

20 21

Assume that Indonesia isnt moving in the meantime.

22

a c 23 Let -- and -- represent any two rational b d 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.

A unit fraction has a numerator of 1.

10


1:05 Recurring DecimalsPrep Quiz 1:05Write these fractions as decimals. ---1 1 2 2 3 1 4 5 3 . . 063974974974 . . . is written as 063 974 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 4

Outcome NS521

5 -6

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: For3 -- : 8

0 3 7 5 8)3306040

For

1 -- : 6

0 1 6 6 6 . . . 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 power of 10, ie 10, 100, 1000, etc, and then simplify. For example: 375 ( 125) 0375 = ----------1000 ( 125) =3 -8

This can be checked using your calculator.

Recurring decimalsare sometimes called repeating decimals.

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.

Worked examplesExample 1When each number in the decimal is repeated. Write 0636363 . . . as a fraction Let x = 06363 . . . Multiply by 100 because two digits are repeated. Then 100x = 636363 . . . Subtract the two lines. So 100x x = 636363 . . . 06363 . . . ie 99x = 63 63 x = ----99 Simplifying this fraction. 7 x = ----11

continued


11

Example 2When only some digits are repeated. Write 0617777 . . . as a fraction Let x = 061777 . . . Multiply by 100 to move the non-repeating digits to the left of the decimal point. Then 100x = 61777 . . . Multiply by 1000 to move one set of the repeating digits to the left of the decimal point. And 1000x = 617777 Subtract the previous two lines. So 1000x 100x = 617777 61777 ie 900x = 556 556 x = -------900 Simplifying this fraction using your calculator. 139 x = -------225 This answer can be checked by performing 139 225 using your calculator.

Exercise 1:051

Foundation Worksheet 1:05 Decimals NS43 1 Write as decimals. 17 -a 1 b --------5 100

Write these fractions as terminating decimals. ---a 3 b 4 c 54

e i2

7 -------100 19 ----40

f j

5 35 ----20 117 -------125

g

8 4 ----25

d h

7 ----10 17 ----50

2 Write as fractions. a 06 b 095

Write these fractions as recurring decimals. ---a 2 b 5 c 8 f3 1 -6

g

9 1 ----15

h

9 7 ----15

d i d

2 ----11 1 ----24

e j

1 -7 17 ----30

3

Write these terminating decimals as fractions. a 047 b 016 c 0125

085

e 0035

4

5 6

By following Example 1, rewrite these recurring decimals as fractions. a 04444 . . . b 0575757 . . . c 0173173173 . . . . . . .. d 07 e 036 f 01234 . Determine the value of 0 3. By following Example 2, rewrite these decimals as fractions. a 083333 . . . b 06353535 . . . . . d 064 . e 0736 . . g 05123 h 05278 c 0197777 . . . .. f 08249 . . i 064734

12


Challenge 1:05

Try this with repeating decimals

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

Example 1 26 0 1 02 6 = -------------99 26 = ----99 Step 1 (Numerator) Subtract the digits before the repeating digits from all the digits. Step 2 (Denominator) Write down a 9 for each repeating digit and then a zero for each non-repeating digit in the decimal. Step 3 Simplify the fraction if possible.

Example 2 327 32 2 0327 = -------------------900one 9o tw 0s

295 = -------900That's pretty nifty.

59 = -------180

Try converting these repeating decimals to fractions using this method. . .. . . . .. 1 07 2 067 3 0312 4 016 5 03217

Fun Spot 1:05

Speedy additionMy writing is in colour.

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 adding these 5 numbers. 7 Rachel wrote down the answer immediately and surprised her 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?

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

Put a 2 at the front. These three digits are the same as in the last number. Make this 2-digit number two less than the one above it.


13

1:06 Simplifying RatiosPrep Quiz 1:06Simplify the fractions: 150 ----60

Outcome NS43

2

16 ----20

3 6 40c of 160c? 9 1 m of 150 cm?

72 ----84

4

125 -------625

What fraction is: 5 50c of $1? 8 100 cm of 150 cm?

7 8 kg of 10 kg? 10 $2 of $2.50?

A ratio is a comparison of like quantities eg Comparing 3 km to 5 km we write: -3 to 5 or 3 : 5 or 3 .5

Ratios are just like fractions!

Worked examples1 Jans height is 1 metre while Dianes is 150 cm. Find the ratio of their heights. 2 of the class walk to school while to those who ride bicycles.3 -5 1 -4

ride bicycles. Find the ratio of those who walk a X : 1 b 1 : Y.

3 Express the ratio 11 to 4 in the form

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 23

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.2 -3

From this ratio we can see that Jan is 2 Those walking to those cycling -- -- = 3 :15 4

as tall as Diane.

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

Multiply both terms by 20. 3 20 1 20 - = ---- ------- : ---- ------1 14 1 5 1 = 12 : 54 5

14


3 a 11 to 4 = 11 : 4 Divide both terms by 4. = = :1 This is in the form X : 1.11 ----4 -23 4

b 11 to 4 = 11 : 4 Divide both terms by 11.4 = 1 : ----11

:1

This is in the form 1 : Y.

Exercise 1:061

Express the first quantity as a fraction of the second each time. a 7 men, 15 men b 10 kg, 21 kg c 3 cm, 4 cm d $5, $50 e 8 m, 10 m f 10 bags, 100 bags g 75 g, 80 g h 6 runs, 30 runs i 25 goals, 120 goals Simplify the ratios. a 6:4 c 65 : 15 e 20 : 45 g 60 : 15 i 1000 : 5 k 55 : 20 m 10 : 105 o 4 : 104 q s u w1 -2 -: 21 2 -23 4 -63 4 11 ----16

Simplify thefractions.

2

b d f h j l n p r t v x

10 : 5 14 : 35 42 : 60 45 : 50 1100 : 800 16 : 28 72 : 2 -10 : 12 -21 : 2 2 -2 : 31 4 3 -4 2 -3 -:1 2 -:1 2

You may use x or

:1 :3-:1 2

3

In each, find the ratio of the first quantity to the second, giving your answers in simplest form. a 7 men, 9 men b 13 kg, 15 kg c 7 cm, 8 cm d $8, $12 e 16 m, 20 m f 15 bags, 85 bags g 90 g, 100 g h 9 runs, 18 runs i 50 goals, 400 goals j 64 ha, 50 ha k 25 m, 15 m l 100 m2, 40 m2 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, $6 d 1 m, 60 cm e 25 cm, 2 m f 100 m, 1 km Are units the same? g 600 mL, 1 L h 1 L, 600 mL i 5 L, 1 L 250 mL j 2 h, 40 min k 50 min, 1 h l 2 h 30 min, 5 h

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15

5

Write these ratios in the form X : 1. a 13 : 8 b 7:4 c 5:2 d 110 : 100 e 700 : 500 f 20 : 30 g 2:7 h 10 : 9 i 4:6 j 15 : 8 -- -- k 1:3 l 21 : 12 4

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

6

Write these ratios in the form 1 : Y. a 4:5 b 2:9 d 14 : 6 e 8 : 10 g 100 : 875 h 4 : 22

c 8 : 15 f 1000 : 150 i 4:6

ie 13 : 8= =13 ----8 5 1 -8

:1 :1

7

a Anne bought a painting for $600 (cost) and sold the painting for $800 (selling price). Find the ratio of: i cost to selling price ii profit to cost iii profit to selling price b John, who is 160 cm tall, jumped 180 cm to win the high jump competition. What is 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 breadth ii 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/neck ii 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 m her pulse rate rises to 120 beats per minute. Find the ratio of: i Joans normal pulse rate to Erics normal pulse rate ii 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 m braking distance), while a car travelling at the same speed takes 38 metres to stop (16 m reaction 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 distance iii 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 custard powder. 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 a half 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 is 40 cm long. If the real plane is 16 m long, what is the scale factor of my model? d My father is 180 cm tall. If a photograph of him has a height of 9 cm, what is the scale of the photograph?New Signpost Mathematics Enhanced 9 5.15.3

8

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1:07 RatesPrep Quiz 1:07If Wendy earns $16 per hour, how much would she earn in: 1 2 hours? 2 3 hours? 3 5 hours? Complete: 5 1 kg = . . . g 6 1 tonne = . . . kg 8 1 cm = . . . mm 9 1 m2 = . . . cm2

Outcome NS43

4 half an hour? 7 1 hour = . . . min 10 15 litres = . . . millilitres

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

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.

Worked examples1 84 km in 2 hours Divide each term by 2. = 42 km in 1 hour = 42 km/h or 84 km in 2 hours 84 km = --------------2h 84 km - = ----- ------2 h = 42 km/h

Units must be shown.

Example (1) is an average rate because,when you travel, your speed may vary from moment to moment. Example (2) is a constant rate, because each kg will cost the same. cents is the same kg as c/kg.

2 16 kg of tomatoes are sold for $10. What is the cost per kilogram? $10 Cost = ------------16 kg 1000 cents = ----------- -----------16 kg 125 cents = -------- -----------2 kg = 625 cents/kg

continued


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3 A plumber charges a householder $64 per hour to fix the plumbing in a house. Find the cost if it takes -him 4 1 hours.2

4 Change 72 litres per hour into cm3 per second. 72 L 72 L per h = --------1h 72 000 mL = ------------------------60 min 72 000 cm 3 = -------------------------60 60 s 72 L/h = 20 cm3/s

Rate = $64 per 1 hour-Multiply both terms by 4 1 .

= $64 Cost = $288

-41 2

per

2 1 4 -2

hours

Exercise 1:071

Write each pair of quantities as a rate in its simplest form. a 6 km, 2 h b 10 kg, $5 d 100 mL, 100 cm3 e 160 L, 4 h g $315, 7 days h 70 km, 10 L j 7000 g, 100 cm k 50 t, 2 blocks m 88 runs, 8 wickets n 18 children, 6 mothers a b c d e

c f i l o

500c, 10 kg $100, 5 h 20 degrees, 5 min -60 km, 1 h 2 75 g, 10 cm3

2

I walk at 5 km/h. How far can I walk in 3 hours? Nails cost $2.45 per kg. What is the cost of 20 kg? I can buy four exercise books for $5. How many books can I buy for $20? I earn $8.45 per hour. How much am I paid for 12 hours work? 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 is 160 American dollars, find the value of ten English pounds in American currency. h The density of iron is 75 g/cm3. What is the mass of 1000 cm3 of iron? (Density is mass per unit of volume.) i If light travels at 300 000 km/s, how far would it travel in one minute? j If I am taxed 16c for every $1 on the value of my $50 000 block of land, how much must I pay? Complete the equivalent rates. a 1 km/min = . . . km/h c $50/kg = . . . c/kg e 144 L/h = . . . mL/s g 7 km/L = . . . m/mL i 30 mm/s = . . . km/h k 800 kg/h = . . . t/day m 10 jokes/min = . . . jokes/h o 105 cm3/g = . . . cm3/kg b d f h j l n 40 000 m/h = . . . km/h $50/kg = . . . c/g 60 km/h = . . . m/s 25c/h = . . . $/week 90 beats/min = . . . beats/s 3 t/h = . . . kg/min 50c/m2 = . . . $/ha

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4

The density of a person is approximately 095 g/cm3. This means that the average weight of 1 cm3 of a person is 095 g. -------Now 095 g/cm3 = 95 g/100 cm3 = 1 g/ 100 cm3 1 g/105 cm3. 95 Use this information to answer the following questions. a Find the volume in cm3 of a man weighing 70 kg. b Find the volume in cm3 of a girl weighing 46 kg. 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?

Investigation 1:07

Comparing speedsIs the womens marathon this far?

Find the Olympic record for: 1 the mens 100 m, 200 m, 400 m, 800 m and marathon running. 2 the womens 100 m, 200 m, 400 m, 800 m and marathon running. For either mens or womens records, find the average speed for each distance in m/s and km/h. (Give answers correct to 3 significant figures.) Report on your findings. What conclusions can you draw?

1:08 Signicant FiguresNo 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 sure that a measurement is useful, each digit in the number should be significant. For example, if the height of a person, expressed in significant figures, is written as 213 m it is assumed that only the last figure may be in error.

Outcome NS521

A significant figureis a number that we believe to be correct within some specific or implied limit of error.


19

Clearly any uncertainty in the first or second figure would remove all significance from the last figure. (If we are not sure of the number of metres, it is pointless to worry about the number of centimetres.) It is assumed that in the measurement 213 m we have measured the number of metres, the number of tenths of a metre and the number of hundredths of a metre. Three of the figures have been measured so there are three significant figures. To calculate the number of significant figures in a measurement we use the rules below.Rules for determining significant figures 1 2 3

2.13 m

Coming from the left, the first non-zero digit is the first significant figure. All figures following the first significant figure are also significant, unless the number is a whole number ending in zeros. Final zeros in a whole number may or may not be significant, eg 000120 has three significant figures, 8800 may have two, three or four.

Putting this more simply: 1 Starting from the left, the first significant figure is the first non-zero digit. eg 0003 250 865 000 8007 the first non-zero digit 2 Final zeros in a whole number may or may not be significant. eg 56 000 000 73 210 18 000 Unless we are told, we cannot tell whether these zeros are significant. 3 All non-zero digits are significant. 4 Zeros at the end of a decimal are significant. 213123 0 0000 010 0 eg 30 These final zeros are significant.Any figure thats been measured is significant.

Method for counting the number of significant figuresEvery figure between the first and last significant figure is significant.

Locate the first and last significant figures, then count the significant figures, including all digits between the first and last significant figures.

20


Worked examplesExample AHow many significant figures has: 1 316 000 000 (to nearest million)? 3 42 007? 5 0000 130 50? 2 316 000 000 (to nearest thousand)? 4 31005 0?

Solutions1 316 000 000 (to nearest million) last significant figure first significant figure Number of significant figures = 3 3 42 007 last first Number of significant figures = 5 5 0000 130 50 last first Number of significant figures = 5 2 316 000 000 (to nearest thousand) last sig. fig. (thousands) first significant figure Number of significant figures = 6 4 31005 0 last first Number of significant figures = 6

Example B1 The distance of the Earth from the sun is given as 152 000 000 km. In this measurement it appears that the distance has been given to the nearest million kilometres. The zeros may or may not be significant but it seems that they are being used only to locate the decimal point. Hence the measurement has three significant figures. 2 A female athlete said she ran 5000 metres. This is ambiguous. The zeros may or may not be significant. You would have to decide whether or not they were significant from the context of the statement.

Exercise 1:081

How many significant figures are there in each of the following numerals? a 21 b 176 c 905 d 062 e 7305 f 0104 g 36 h 360 i 0002 j 0056 k 004 l 040 m 000471 n 3040 o 05 p 304 q 7001 r 0001 50 s 0000 000 125 t 0000 000 100 u 0000 000 001 How many significant figures are there in each of the following numerals? a c e g 2000 (to the nearest thousand) 53 000 (to the nearest thousand) 25 000 (to the nearest ten) 26 000 (to the nearest hundred) b d f h 2000 (to the nearest hundred) 530 000 (to the nearest thousand) 26 300 (to the nearest hundred) 8 176 530 (to the nearest ten)

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A newspaper article reported that 20 000 people attended the Carols by Candlelight concert. How accurate would you expect this number to be? (That is, how many significant figures would the number have?)Chapter 1 Basic Skills and Number

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1:09 ApproximationsPrep Quiz 1:09How many significant figures do the following numbers have? 2 006 3 01050 1 3605 Write one more term for each number sequence. 5 078, 079, . . . 6 2408, 2409, . . . 4 306, 307, . . . 7 Is 37 closer to 3 or 4? 8 Is 2327 closer to 232 or 233? 9 What number is halfway between 35 and 36? 10 What number is halfway between 006 and 007?

Outcome NS521

Here, halfway is 0.55.

Discussion To round off a decimal to the nearest whole number, we write down the whole number closest to it.73 75 79 84 88

7

8

9

73 is closer to 7

79 is closer to 8

84 is closer to 8

88 is closer to 9

75 is exactly halfway between 7 and 8. In cases like this it is common to round up. We say 75 = 8 correct to the nearest whole number. To round off 72 900 to the nearest thousand we write down the thousand closest to it.72 900

To round off 0813 4 to the nearest thousandth we write down the thousandth closest to it.08134

72 000

72 500

73 000

0813

08135

0814

72 900 is closer to 73 000 than to 72 000. 72 900 = 73 000 correct to the nearest thousand.

0813 4 is closer to 0813 than to 0814. 0813 4 = 0813 correct to the nearest thousandth.

To round off (or approximate) a number correct to a given place we round up if the next figure is 5 or more, and round down if the next figure is less than 5.

Worked examplesExample ARound off: 1 56 700 000 to the nearest million 3 86149 to one decimal place 2 0085 1 to the nearest hundredth 4 0666 15 correct to four decimal places

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Solutions A1 56 700 000 has a 6 in the millions column. The number after the 6 is 5 or more (ie 7). 56 700 000 = 57 000 000 correct to the nearest million. 3 86149 has a 1 in the first decimal place. The number after the 1 is less than 5 (ie 4). 86149 = 861 correct to one decimal place. 2 0085 1 has an 8 in the hundredths column. The number after the 8 is 5 or more (ie 5). 0085 1 = 009 correct to the nearest hundredth. 4 0666 15 has a 1 in the fourth decimal place. The number after the 1 is 5 or more (ie 5). 0666 15 = 0666 2 correct to four decimal places.

To approximate correct to a certain number of significant figures, we write down the number that contains only the required number of significant figures and is closest in value to the given number.

Example BRound off: 5 507 000 000 to 2 significant figures 7 0006 25 correct to 1 sig. fig. 6 1098 to 3 significant figures 8 0080 25 correct to 3 sig. figs. 6 The 3rd significant figure is the 9. The number after the 9 is 8, so increase the 9 to 10. Put down the 0 and carry the 1. 1098 = 110 correct to 3 significant figures. 8 The 3rd significant figure is 2. The number after the 2 is 5 or more (ie 5). 0080 25 = 0080 3 correct to 3 sig. figs.

Solutions B5 The 2nd significant figure is the 0 between the 5 and 7. The number after the zero is 5 or more (ie 7). 507 000 000 = 510 000 000 correct to 2 significant figures. 7 The 1st significant figure is 6. The number after the 6 is less than 5 (ie 2). 0006 25 = 0006 correct to 1 sig. fig.

Exercise 1:091

Foundation Worksheet 1:09 Approximation NS521, NS43 1 Write 74639 to the nearest: a integer b ten c hundred 2 Write 64937 correct to: a 3 dec. pl. b 2 dec. pl. c 1 dec. pl.

Round off these numbers to the nearest hundred. a 7923 b 1099 c 67 314 d 853461 e 60999 f 350 g 74 932 h 7850 Round off these numbers to the nearest whole number. a 93 b 795 c 451 d 27 e 2314 f 1781 g 236502 h 995

2

When you round offyou are making an approximation.


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3

Round off these numbers to the nearest hundredth. a 243128 b 79664 c 91351 d 9807 e 03046 f 0085 2 g 0097 h 1991 Round off these numbers correct to one decimal place. a 670 b 845 c 2119 d 6092 e 005 f 0035 g 2988 h 999 Round off these numbers correct to 2 significant figures. a 8170 b 3504 c 655 d 849 e 14 580 f 76 399 g 49 788 h 76 500 Round off the numbers in question 5 correct to 1 sig. fig. Round these off to 3 sig. figs. a 6948 b 35085 c 3205 e 0666 66 f 93333 g 10085 d 0081 54 h 9095

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Im supposed to get $496.48, but they gave me $496.50.

5

6 7

sig. figs. is short forsignificant figures, dec. pl. is short for decimal places.

8

-To change 1 7 to a decimal, Gregory divided 16 by 9 using 9 his calculator. Give the answer correct to: a 1 dec. pl. b 2 dec. pl. c 3 dec. pl. d 1 sig. fig. e 2 sig. figs. f 3 sig. figs.

9

Diane cut 60 cm of blue ribbon into 11 equal parts to make a suit for her new baby. After dividing, she got the answer 54 5 . Give the length of one part correct to: a the nearest centimetre b the nearest millimetre c 1 dec. pl. d 2 dec. pl. e 3 dec. pl. f 1 sig. fig. g 2 sig. figs. h 3 sig. figs. The following calculator display represents an answer in cents. 14059705 Give this answer correct to: a the nearest dollar b the nearest cent c 1 dec. pl. d 2 dec. pl. e 3 dec. pl. f 1 sig. fig. g 2 sig. figs. h 3 sig. figs. What level of accuracy do you think was used in each of these measurements and what would be the greatest error possible as a result of the approximation? a The crowd size was 18 000. b The nations gross domestic product was $62 000 000 000. What approximation has been made in each of these measurements and what would be the greatest error possible? a 64 cm b 0007 mg A number is rounded to give 215. What could the number have been? What is the smallest the number could have been? Is it possible to write down the largest number that can be rounded to give 215?

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11

60c or $0.60

Is the level of accuracy to the nearest ten, hundred or thousand?

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13

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14 15 16 17

An answer is given as 3 000 000 correct to 1 significant figure. What might the exact measure have been? Seven people decide to share a bill of $187.45 equally. How much should each person pay? What could be done with the remainder? The area of a room is needed to order floor tiles. The room dimensions, 249 m by 431 m, were rounded off to 2 m by 4 m to calculate the area. What problems might arise? A 10-digit calculator was used to change fractions into decimals. The truncating of the decimal produced an error. What error is present in the display after entering: ---a 1? b 2? c 5?3 3 9

Truncate means cut off.

18

Find an approximation for off the 34545 correct to: a 1 sig. fig. b 1 dec. pl.

345452

by first rounding c 2 sig. figs.

0 . 66666666619 20

In question 18, what is the difference between the answer to a and the real answer? To find the volume of the tunnel drawn on the right, each measurement was rounded off correct to 1 significant figure before calculation. What error in volume occurred?153 m 168 m 2 km

Note: Truncating or rounding numbers before a calculation may produce unwanted errors or inaccuracy.

1:10 EstimationPrep Quiz 1:101 2 3 4 Write 216 to the nearest hundred. Write 1768 to the nearest ten. 1561 10 1561 1000716 -------------35

Outcome NS521

5 Is

less than or greater than 1?

7 True or false? 2168 0716 < 2168 1 8 Which is the best approximation for 0316 081? a 25 b 025 c 0025 9 1076 ----------025 76 ------83

6 If 3 < 4 and 53 < 78, what sign (< or >) can we put in the box? 3 53 4 78

> 76, true or false? > 1, true or false?

Like all machines, calculators only operate correctly if they are used correctly. Even when doing simple calculations it is still possible to press the wrong button. So it is essential that you learn how to estimate the size of the answer before the calculation is even started.Chapter 1 Basic Skills and Number

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An estimate is a valuable means of checking whether your calculator work gives a sensible answer. If your estimate and the actual answer are not similar, then it tells you that a mistake has been made either in your estimate or your calculation. The following examples will show you how to estimate the size of an answer.

Worked examplesEstimate the size of each of the following calculations. 1 1461 715 + 32 2 756 5173 3 00253 045 2173 0815 4 ---------------------------------73 5 86 ---------------------------28 1618 2 756 5173 85 40

Both numbersare multiplied by 100 to simplify the question.

Solutions1 1461 715 + 32 15 7 + 3 11 3 00253 045 = 253 45 3 45 1 ----- or 007 15

or

means is approximately equal to.

Where possible, reduce fractions.

2173 0815 4 ---------------------------------73 21 1 -------------7 3

5

86 ---------------------------28 1618 9 ----------------13 163 ----16 1 -- or 53

0 2

These hints may be useful. When estimating, look for numbers that are easy to work with, eg 1, 10, 100. Remember, its an estimate. When you approximate a number you dont have to take the nearest whole number. Try thinking of decimals as fractions. It often helps. - eg 76 0518 8 1 or 4 2 When dealing with estimates involving fraction bars, look for numbers that nearly cancel out.

The golden rule of estimating: does your answer make sense?

eg ----------------------------------- ----------- = 12Check that the answer makes sense.

2

17 68 5 8 8 91

26 1

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Exercise 1:101

Foundation Worksheet 1:10 Estimation NS521 1 Estimate: a 49 + 102 b 615 989 2 Find an approximate answer to: a 161 79 b (71)2 99

Estimate the answers to the following calculations. a 79 + 081 + 1356 b 4256 1581 + 92 c 56 (72 + 59) d 1431 897 e 7395 142 f 073 005 453 g 0916 0032 1834 h (156 + 682) 531 i 156 + 682 531 j (1456 + 3075) (0561 2052)

2

Estimate the answers for each of the following (giving your answer as an integer, ie a whole number). 56 78 219 426 08 05 006 053 a ---------------------b ---------------------------c ---------------------d ---------------------------129 689 037 0005 73 98 e ------------------------156 32 i 196 58 ------------------------36 172 f j 212 715 ---------------------------158 089 2053 768 ------------------------------4116 + 137 36 + 97 g ---------------------158 1289 k --------------( 52 ) 2 0916 426 o -----------------------------------0561These are a bit harder, arent they?

782 564 h ---------------------------98 + 296 l ( 37 ) 2 + ( 45 ) 2 ------------------------------------271 465

1056 m -----------------1895

( 861 ) 2 n -----------------861

416 + 395 p --------------------------------( 356 ) 2

q (36)2 + 2 97 56 r s t 978 628 ---------053 765 ( 37 + 156 ) -----------------------------------------29 158 106 ( 35 ) 2 96 076 + -------------------------------2

Note: The fraction bar acts a little like grouping symbols. You work out the numerator and denominator separately. In 416 + 395 you must work out the addition first. The square root sign also acts like grouping symbols.

3

When estimating the size of a measurement, both the number and the unit must be considered. In each case, choose the most likely answer by estimation. a The weight of the newborn baby was: i 350 g ii 78 kg iii 31 kg iv 50 pounds b The length of a mature blue whale is about: i 27 m ii 3 km iii 32 cm iv 98 m c 12% discount on a television set marked $2300 is: i $86.60 ii $866 iii $276 iv $27.60 d I just borrowed more than $80 000 from the bank. Next year the interest on the loan is: i $873 ii $9400 iii $185.60 iv $21 140


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a A pile of paper is 32 cm thick. If there are 300 sheets in the pile, estimate the thickness of one sheet of paper. b Peter estimated that there were 80 people sitting in an area of 50 m2 at the Carols by Candlelight service. He estimated that about 2000 m2 of area was similarly occupied by the crowd. To the nearest 100, what would be Peters estimate of the crowd size? c Would 86 844 be between 8 80 and 9 90? Explain why. Two measurements were rounded off correct to two significant figures and then multiplied to estimate an area. The working was: 92 m 081 m = 7452 m2. Between which two measurements would the real area lie? How many of the figures in this estimate are useful given the possible spread of the area?

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Literacy in Maths

1:10 Take you medicine!

Many deaths have occurred because people have misread or not understood directions on medicine bottles. Instructions are often difficult to read and require sophisticated measuring instruments. Before beginning the investigations listed below, use the picture to answer these questions. The information shown was on the label of a 100 mL bottle of a certain medicine. Read the information carefully and answer the questions below. 1 What is the dosage for an 8-year-old child and how often should it be taken? 2 A 5-year-old girl has a dosage of 2 mg. What is the usual number of times she should take the dose? What is the maximum number of doses she Dosage may take? Children 26 years: 3 What is the youngest age for which 2 mg three times a day. The dose is not to exceed the adult dose is recommended? 12 mg a day. A picture Children 714 years: 4 In a 100 mL bottle, how many 4 mg three times a day. might help. millilitres of alcohol are present? The dose is not to exceed 16 mg a day. 5 How many milligrams of the chemical Adults: cyproheptadine hydrochloride would 4 mg three to four times a day. The dose is not to be in a 100 mL bottle of this medicine? exceed 32 mg per day.Cyproheptadine A Investigate the labelling on medicine Hydrochloride 1.2 mg per 5 mL bottles and other medicinal preparations Alcohol 5% (eg pet worming tablets, etc). B Suggest ways in which directions could be given that would make them easier to understand. C Redesign the label in the picture so that it reflects your answer to part B. D Present your findings in the form of a wr

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