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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
Maths terms Diagnostic test Revisionassignment Working mathematically 90
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
Maths terms Diagnostic test Revisionassignment Working mathematically 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
Maths terms Diagnostic test Revisionassignment Working mathematically 398
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
Maths terms Diagnostic test Revisionassignment Working mathematically 522
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
New Signpost Mathematics Enhanced 9 5.15.3
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.
Revision: WorkingMathematically, Chapter 2,and throughout the text
WMS5.3.3 Uses and interprets formal definitions andgeneralisations when explaining solutions and orconjectures
Revision: WorkingMathematically, Chapter 2,and throughout the text
WMS5.3.4 Uses deductive reasoning in presenting argumentsand formal proofs.
Revision: WorkingMathematically, Chapter 2,and throughout the text
WMS5.3.5 Links mathematical ideas and makes connectionswith, and generalisations about, existing knowledgeand understanding in relation to Stage 5.3 content.
Revision: WorkingMathematically, Chapter 2,and throughout the text
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
New Signpost Mathematics Enhanced 9 5.15.3
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
New Signpost Mathematics Enhanced 9 5.15.3
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
New Signpost Mathematics Enhanced 9 5.15.3
See page 598 for answers.
.
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
See page 598 for answers.
ID Card 4 (Language)
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
All sidesdifferent
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xx New Signpost Mathematics Enhanced 9 5.15.3
See page 598 for answers.
ID Card 5 (Language)
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
See page 598 for answers.
ID Card 6 (Language)
1
..................... lines
2
..................... lines
3
v.....................
h.....................
4
..................... lines
5
angle .....................
6
..................... angle
7
..................... angle
8
..................... angle
9
..................... angle
10
..................... angle
11
.....................
12
..................... angles
13
..................... angles
14
..................... angles
15
..................... angles
16
a +b +c +d =.....
17
.....................
18
..................... angles
19
..................... angles
20
..................... angles
21
b............ an interval
22
b............ an angle
23
CAB=............
24
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
See page 598 for answers.
ID Card 7 (Language)
1
a............ D............
2
b............ C............
3
a............ M............
4
p............ m............
5
area is 1 ............
6
r............ shapes
7
............ of a cube
8
c............-s............
9
f............
10
v............
11
e............
12
axes of ............
13
r............
14
t............
15
r............
16
t............
17
The c............of the dot are E2.
18
t............
19
p............ graph
20
c............ graph
21
l............ graph
22
s............ graph
23
b............ graph
24
s............ d............
AD BC am pm
100 m
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
New Signpost Mathematics Enhanced 9 5.15.3
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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6 New Signpost Mathematics Enhanced 9 5.15.3
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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7Chapter 1 Basic Skills and Number
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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8 New Signpost Mathematics Enhanced 9 5.15.3
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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9Chapter 1 Basic Skills and Number
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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10 New Signpost Mathematics Enhanced 9 5.15.3
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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11Chapter 1 Basic Skills and Number
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
New Signpost Mathematics Enhanced 9 5.15.3
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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13Chapter 1 Basic Skills and Number
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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14 New Signpost Mathematics Enhanced 9 5.15.3
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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15Chapter 1 Basic Skills and Number
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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16 New Signpost Mathematics Enhanced 9 5.15.3
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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17Chapter 1 Basic Skills and Number
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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18 New Signpost Mathematics Enhanced 9 5.15.3
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
2---
2
3
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19Chapter 1 Basic Skills and Number
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