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Contents iii Contents Abouttheauthors viii Acknowledgements x Introduction xi Guidetotheworkingprograms xii Guidetothisresource xiv 1 Integers,decimals,fractions,ratiosandrates000 Warm-upquiz 000 1A Addingandsubtractingpositiveandnegative integers (CONSOLIDATING) 000 1B Multiplyinganddividingpositiveandnegative integers (CONSOLIDATING) 000 1C Decimalplacesandsignificantfigures 000 1D Rationalnumbersandirrationalnumbers (CONSOLIDATING) 000 1E Addingandsubtractingfractions (CONSOLIDATING) 000 Progressquiz 000 1F Multiplyinganddividingfractions (CONSOLIDATING) 000 1G Ratios (CONSOLIDATING) 000 1H Rates (CONSOLIDATING) 000 Maths@Work:Cooksandchefs 000 Workingmathematically 000 Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000
Financialmathematics 000 Warm-upquiz 000 2A Percentages,fractionsanddecimals (CONSOLIDATING) 000 2B Applyingpercentages (CONSOLIDATING) 000 2C Percentageincreaseanddecrease (CONSOLIDATING) 000 2D Profitsanddiscounts (CONSOLIDATING) 000 2E Income 000 Progressquiz 000 2F Taxation 000 2G Simpleinterest 000 2H Applicationsofsimpleinterest 000 Maths@Work:Facebookcake-decoratingbusiness 000 Workingmathematically 000 UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
2
iv Contents Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 3 Expressionsandequations 000 Warm-upquiz 000 3A Algebraicexpressions (CONSOLIDATING) 000 3B Addingandsubtractingalgebraic expressions (CONSOLIDATING) 000 3C Multiplyinganddividingalgebraicexpressions (CONSOLIDATING) 000 3D Expandingalgebraicexpressions 000 3E Linearequationswithpronumeralsononeside (CONSOLIDATING) 000 3F Linearequationsinvolvingfractions 000 Progressquiz 000 3G Linearequationsinvolvingbrackets 000 3H Equationswithpronumeralsonbothsides 000 3I Usinglinearequationstosolveproblems 000 3J Usingformulas 000 Maths@Work:Plumber 000 Workingmathematically 000 Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 4 Right-angledtriangles 000 Warm-upquiz 000 4A ExploringPythagoras’theorem (CONSOLIDATING) 000 4B Findingthelengthofthehypotenuse (CONSOLIDATING) 000 4C Findingthelengthsoftheshortersides (CONSOLIDATING) 000 4D UsingPythagoras’theoremtosolvetwo-dimensional problems (CONSOLIDATING) 000 4E Introducingthetrigonometricratios (CONSOLIDATING) 000 4F Findingunknownsides (CONSOLIDATING) 000 Progressquiz 000 4G Solvingforthedenominator 000 4H Findingunknownangles 000 4I Usingtrigonometrytosolveproblems 000 Maths@Work:Carpenter 000 Workingmathematically 000 Puzzlesandchallenges 000
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Contents v Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 5 Linearrelationships 000 Warm-upquiz 000 5A PointsandlinesontheCartesianplane 000 5B The x -interceptand y -intercept 000 5C Graphingstraightlinesusingintercepts (EXTENDING) 000 5D Lineswithonlyoneintercept 000 5E Gradientfromriseandrun 000 5F Gradientanddirectvariation 000 5G Gradient–interceptform 000 Progressquiz 000 5H Findingtheequationofalineusing y = mx + c 000 5I Midpointandlengthofalinesegmentfromdiagrams000 5J Linearrelationshipsinreal-lifecontexts 000 Maths@Work:Tradinginforeigncurrencies 000 Workingmathematically 000 Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 Semesterreview1 000 6 Length,area,surfaceareaandvolume 000 Warm-upquiz 000 6A Lengthandperimeter (CONSOLIDATING) 000 6B Circumferenceofcirclesandperimeterofsectors (CONSOLIDATING) 000 6C Areaofquadrilateralsandtriangles (CONSOLIDATING) 000 6D Areaofcircles (CONSOLIDATING) 000 6E Perimeterandareaofcompositeshapes 000 6F Surfaceareaofprisms 000 Progressquiz 000 6G Surfaceareaofcylinders 000 6H Volumeofprisms 000 6I Volumeofcylinders 000 Maths@Work:Vegetableandfruitgrowers 000 Workingmathematically 000 Puzzlesandchallenges 000 UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
vi Contents Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 7 Indices 000 Warm-upquiz 000 7A Indexnotation (CONSOLIDATING) 000 7B Indexlawsformultiplyinganddividing 000 7C Thezeroindexandpowerofapower 000 7D Indexlawsextended 000 7E Negativeindices 000 Progressquiz 000 7F Scientificnotation 000 7G Scientificnotationusingsignificantfigures 000 Maths@Work:Labtechnician 000 Workingmathematically 000 Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 8 Propertiesofgeometricalfigures 000 Warm-upquiz 000 8A Anglesandtriangles (CONSOLIDATING) 000 8B Parallellines (CONSOLIDATING) 000 8C Quadrilaterals (CONSOLIDATING) 000 8D Polygons (EXTENDING) 000 8E Enlargementandsimilarfigures 000 Progressquiz 000 8F Applyingscalefactortosimilartriangles 000 Maths@Work:Animator 000 Workingmathematically 000 Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 9 Quadraticexpressionsandalgebraicfractions000 Warm-upquiz 000 9A Reviewingalgebra (CONSOLIDATING) 000 9B Expandingbinomialproducts 000 9C Expandingperfectsquares 000 UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
Contents vii 9D Differenceoftwosquares 000 9E UsingHCFtofactorisealgebraicexpressions 000 Progressquiz 000 9F Simplifyingalgebraicfractions:multiplicationand division 000 9G Simplifyingalgebraicfractions:additionandsubtraction000 Maths@Work:Automotivetechnology 000 Workingmathematically 000 Puzzlesandchallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 10 Probabilityanddataanalysis 000 Warm-upquiz 000 10A Reviewofprobability (CONSOLIDATING) 000 10B Venndiagramsandtwo-waytables (EXTENDING) 000 10C Usingarraysfortwo-stepexperiments 000 10D Usingtreediagrams 000 10E Usingrelativefrequenciestoestimateprobabilities000 10F Mean,median,modeandrange (CONSOLIDATING) 000 Progressquiz 000 10G Interpretingdatafromtablesandgraphs 000 10H Stem-and-leafplots 000 10I Groupingdataintoclasses 000 Maths@Work:Personaltrainer 000 Workingmathematically 000 PuzzlesandChallenges 000 Chaptersummary 000 Chapterchecklist 000 Chapterreview 000 Semesterreview2 000 Glossary 000 Answer 000 UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
2 Financial mathematics
Whyskillsinfinancialmathematicsareimportant
Skillsinfinancialmathematicsareessentialfor successfulbusinessmanagementandforachieving personalfinancialindependence.Percentageskillsare neededforthefinancialcalculationsperformedby individuals,accountants,bookkeepers,smallbusiness managersandemployers.
Percentagesareusedtocalculatemark-upand discountamounts,costpricesandsellingprices, profits,losses,insurancepayments,GST,businesstax, wagetax,wageincreasesandtheinterestonloans.
Personalincomecanbeearnedinvariousways:
•afixedsalaryp.a(‘perannum’.i.e.peryear),e.g. accountants,businessowners,teachers,engineers, pharmacistsandsurveyors.
•awagecalculatedfromanhourlyrate,e.g.chefs, cleaners,florists,dentalassistants,nurses, receptionists,retailsalespeople,construction workers,panelbeatersandautomechanics.
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•bythejob,e.g.personaltrainers,fruitpickers, bricklayers,fishers,tailors,hairdressers,piano tunersandcarpetlayers.
•commission,e.g.carsalespeople,realestate agents.
•royalties,e.g.songwritersandbookauthors.
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Chaptercontents
2A Percentages,fractionsanddecimals (CONSOLIDATING)
2B Applyingpercentages (CONSOLIDATING)
2C Percentageincreaseanddecrease (CONSOLIDATING)
2D Profitsanddiscounts (CONSOLIDATING)
2E Income
2F Taxation
2G Simpleinterest
2H Applicationsofsimpleinterest
NSWSyllabus
Inthischapter,astudent: developsunderstandingandfluencyin mathematicsthroughexploringandconnecting mathematicalconcepts,choosingandapplying mathematicaltechniquestosolveproblems,and communicatingtheirthinkingandreasoning coherentlyandclearly(MAO-WM-01) solvesfinancialproblemsinvolvingsimpleinterest, earningmoneyandspendingmoney (MA5-FIN-C-01)
Onlineresources
Ahostofadditionalonlineresourcesareincluded aspartofyourInteractiveTextbook,including HOTmathscontent,videodemonstrationsofall workedexamples,auto-markedquizzesand muchmore.
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Warm-upquiz 70 Chapter2Financialmathematics 1 Simplify: 12 100 a 20 100 b 35 100 c 75 100 d 60 100 e 50 100 f 2 Multiplythesedecimalsby100. 0.99 a 0.58 b 0.9 c 1.22 d 0.08 e 1.5 f 3 Whichislargerineachofthefollowingpairs? 1 2 or55% a 3 4 or70% b 089or98% c 4 Copyandcomplete. 61%= 100 a 9%= 100 b 37%= 100 c 121%= 100 d 1%= 100 e 75%= 100 = 3 f 5 Copyandcompletethefollowingtable. Fraction Decimal Percentage 1 100 0 1 0.25 50% 3 4 6 Writedown10% oftheseamounts. 100g a 70km b $450 c $8000 d $5 e 90c f 7 Find: 25% of$400 a 75% of80m b 50% of$3 c 10% of$678 d 1% of600days e 1 2 % of600days f 8 Completethefollowing. 1% of60 = 60dividedby a 10% of50 = 50dividedby b 5% of100 = 100dividedby c 331 3 % of963 = 963dividedby d 25% of88 = 88dividedby e UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
9 Find331 3 % of6300km.
10 Whichislarger:40% of50or25% of100?
Warm-upquiz Warm-upquiz 71
1week
a 1year
b 1year
c 1year
d
Howmanyhoursaretherefrom: 5a.m.to7p.m.? a 9a.m.to3p.m.? b 8∶30a.m.to9p.m.? c
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11 Copyandcomplete:
= days
= days
= weeks
= months
12
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2A Percentages,
fractionsanddecimals CONSOLIDATING
Learningintentionsforthissection:
• Toreviewtheconversionsbetweenpercentages,fractionsanddecimals
• Tomemorisethemostusefulfraction/decimal/percentageconversions
• Tobeabletoexpressonequantityasapercentageofanother
Past,presentandfuturelearning:
• ThissectionconsolidatesStage4conceptswhichareusedinStages5and6
• SomeofthesequestionsaremorechallengingthanthoseinourStage4books
• PercentagesareusedextensivelyinthischapterandalsoChapter1ofourYear10book
• ExpertisewithpercentagesisassumedknowledgeforStage6Standardandmayalsoberequiredinnon-calculator examinationssuchasNAPLANandindustryaptitudetests
Weusepercentagesformanydifferentthingsinourdailylives.Someexamplesincludehomeloans,credit cards,salesandpro ts.
Weknowfromourpreviousworkonpercentagesthattheyrepresentafractionwithadenominatorof100. ‘Percent’comesfromtheLatinword percentum,andmeans‘outof100’.
LessonStarter:Orderingwithpercentages
Tendifferentvaluesaregivenbelow.
Inpairs,decide:
whichofthenumbersisthesmallest a whichofthenumbersisthelargest b
Writethe10numbersinascendingorder.
KEYIDEAS
■ Thetablebelowcontainssomecommonlyusedfractions,decimalsandpercentages.These shouldbememorised.
72 Chapter2Financialmathematics
1 2 , 08, 005, 15%, 7 20 , 09, 9%, 3 5 , 1 3 , 03
Fraction Decimal Percentage Therefore 1 1 100% 3 = 300% 1 2 0.5 50% 1 5 = 150% 1 3 0.333 … or 0.3 33 1 3 % 2 3 = 0.6 = 66 2 3 % 1 4 0.25 25% 3 4 = 0 75 = 75% 1 5 0.2 20% 3 5 = 0 6 = 60% 1 10 0.1 10% 7 10 = 0.7 = 70% 1 100 0.01 1% 21 100 = 0 21 = 21%
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■ Therearesix conversions involvingfractions,decimalsandpercentages.
Usetenths,hundredths,etc. Simplifythefraction.
Typethedecimal. Press=. Pressthebuttonthatconverts fractionstodecimals.
Convertfractiontotenthsor hundredthsor Dividenumeratorbydenominator.
Multiplyby100. Add%symbol.
(Thedecimalpointappearstomove twoplacestotheright.)
Remove%symbol. Divideby100. (Thedecimalpointappearstomove twoplacestotheleft.)
Convertthefractiontohundredths. Add%symbol
orDividenumeratorbydenominator. Multiplyby100. Add%symbol.
Remove%symbol. Writeas‘hundredths’. Simplifyfraction.
Typethefraction. Press=. Pressthebuttonthatconverts fractionstodecimals.
Multiplyby100. Add%symbol.
Remove%symbol. Divideby100.
Multiplyby100. Add%symbol.
Remove%symbol. Divideby100. Pressthebuttonthatconverts fractionstodecimals.
■ Expressingonequantityasapercentageofanother(percentagecomposition):
• Formafraction,thenconverttoapercentage.
Forexample:4goalsfrom10attempts = 4 10 = 40 100 = 40%
Keyterminology: percentage,fraction,decimal
Exercise2A
UNDERSTANDING
1 Completethefollowingusingthewordsmultiplyordivide.
a Toconvertadecimalintoapercentage by100.
b Toconvertapercentageintoafraction by100.
c Toconvertafractionintoapercentage by100.
d Toconvertapercentageintoadecimal by100.
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2 Copyandcompletethistableofcommonpercentages. Percentage 10% 50%
2APercentages,fractionsanddecimals 73
Conversion
decimaltofraction 0 6 = 6 10 = 3 5
Withoutacalculator Withacalculator
fractiontodecimal 13 50 = 26 100 = 0.26 5 8 = 5 ÷ 8 = 0 625
0.24 × 100 = 24
0.24 = 24%
decimaltopercentage
∴
percentagetodecimal 24 ÷ 100 = 0.24 ∴ 24%= 0.24
fractiontopercentage 13 50 = 26 100 = 26% 5 8 = 5 ÷ 8 = 0 625 0.625 × 100 = 62.5 ∴ 5 8 = 62 5%
60%= 60 100 = 3 5
percentagetofraction
1–4 1,4
Fraction 1 4 1 3 2 3 Decimal 0 2 0 75
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3 Whatpercentageofeachofthefollowingdiagramshasbeenshaded?
Hint:Considercommon fractionsandpercentages fortheshadedareas.
4 Scottscored38outof50onamathsquizandSarahscored79% onthesametest.Whoscoredthehigher mark?
Nowyoutry Write1120asapercentage.
Writethepercentageasafraction,usinga denominatorof100,75100.
Multiplythisfractionby2 2 sothatwedon’t haveadecimalinthefractionanditwillbe easiertosimplify.75 100 × 2 2 = 15 200
Simplify: 15 200 = 3 40bycancellingacommon factorof5.
74 Chapter2Financialmathematics
a b c d
FLUENCY 5–9(½) 5–9(½) VIDEO DEMO
Percentagesandfractions Write12
a Write7.5% asafraction. b SOLUTION EXPLANATION a 12 25 = 12 25 × 100 = 48% Multiplythefractionby100. 12 25 × 100 = 12 ✚✚ 25 1 × ✟✟ 100 4 1 b 75%= 7.5 100 = 15 200 = 3 40
Example1
25asapercentage.
a Write12.5% asafraction. b UNCORRECTED
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Nowyoutry
2APercentages,fractionsanddecimals 75 5 Expressthefollowingfractionsaspercentages. Hint:Multiplyby 100. 1 5 a 4 5 b 8 10 c 3 10 d 1 4 e 1 8 f 3 4 g 12 20 h 14 25 i 7 20 j 9 100 k 3 40 l 6 Expressthefollowingpercentagesassimpli edfractions. Hint:Divideby 100. 19% a 23% b 99% c 5% d 22% e 45% f 74% g 75% h 25% i 1725% j 1% k 125% l VIDEO DEMO Example2 Convertingbetweenpercentagesanddecimals Write045asapercentage. a Write25% asadecimal. b SOLUTION EXPLANATION
045 =
= 45% Multiplyby100. b 25%=
÷
Divideby100.
a
045 × 100
25
100 = 025
Write0
23asapercentage. a Write48% asadecimal. b 7 Expressthefollowingdecimalsaspercentages. Hint:Movethedecimal pointtwoplacesto theright. 078 a 095 b 065 c 048 d 0.75 e 1.42 f 0.07 g 0.3 h 003 i 104 j 0.12 k 0.1225 l 8 Expressthefollowingpercentagesasdecimals. 12% a 83% b 57% c 88% d 99% e 100% f 120% g 5% h UNCORRECTED
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.
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Example3 Writingaquantityasapercentage
Write50coutof$2.50asapercentage.
50coutof$2.50 =
Converttothesameunits ($2.50 = 250c) andwriteasa fraction.
Reducethefractiontosimplestform. Multiplyby100.
Nowyoutry
Write90coutof$3.60asapercentage.
9 Ineachofthefollowingcases,expressthe rstquantityasapercentageofthesecond.
Hint:Writeasafraction andthenmultiplyby 100.
11 Tonipays31.5centsinthedollarintax.Expressthisas apercentage.
12 Badweatherstoppedacricketgamefor35minutesofascheduled 312hourmatch.Whatpercentageofthescheduledtimewaslost?
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13 Joelost4kgandnowweighs60kg.Whatpercentageofhis originalweightdidhelose?
Hint:WhatwasJoe’s originalweight?
76 Chapter2Financialmathematics VIDEO DEMO
SOLUTION EXPLANATION
1
✚✚ 50 5✟✟ 250 × 100 = 20%
5goutof200g a 40coutof$4 b 10kmoutof200km c 3soutof1minute d 200moutof1km e 100mLoutof1 2 L f 200coutof$1 g 45marksoutofapossible60marks h PROBLEM-SOLVINGANDREASONING 10–12 11–14 10 Copyandcompletethetableofthefavouritesummersportsof Year9students. Numberofstudents Fractionof Percentage Sport whochosesport thetotal ofthetotal Swimming 44 Golf 12 Volleyball 58 Cricket 36 Total
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14 Acompanyclaimsthattheapplepiesitmakesare97% fatfree.Ifthenutritionalinformationonthe sideofthepackstatesthattotalfatis7gramsofthe250grampie,istheclaimcorrect?
15
15 Completethistableby llinginestimatedtimes.Inthe rsttwocolumns,insertthetimesthatyou dideachactivity.Usethatinformationtohelpyou llintheothercolumns.Roundpercentagestoone decimalplace.
a Timeyouwenttobedlast night: Timeyouwokeup thismorning:
b Timeyoustarted breakfasttoday: Timeyoufinished breakfasttoday:
c Timeschoolstartedtoday: Timeschoolisdueto finishtoday:
Hoursandminutes spentinbed:
Minutesspenteating breakfast:
Hoursandminutes spentatschool:
d Timethismathslesson started: Timethismaths lessonwillfinish: Minutesspentinthe mathslesson:
e Timeschoolwillfinish today: Timeyouwillarrive home: Minutesspent travellinghome:
f Timeyoustartedyour homeworkyesterday: Timeyoufinished yourhomework yesterday:
g Timeyoustarted watchingTVorplaying gamesyesterday:
Timeyoufinished watchingTVor playinggames:
h Timeyouwokeuptoday: Timeyouwillgoto bedtonight:
Minutesspenton homework:
Minutesspentatthis activity:
Hoursandminutes spentawaketoday:
Percentageofadayspent inbed:
Percentageofthedayyou spentatbreakfast:
Percentageoftheday spentatschool:
Percentageoftheday spentinthemathslesson:
Percentageoftheday spenttravellinghome:
Percentageofyourday spentonhomework:
Percentageoftheday spentatthisactivity:
Percentageoftheday spentawake:
2APercentages,fractionsanddecimals 77
ENRICHMENT:Today’stimeline
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2B Applying percentages CONSOLIDATING
Learningintentionsforthissection:
• Toknowhowtocalculateacertainpercentageofagivenquantity
• Tobeabletofindtheoriginalamountfromagivenpercentage
Past,presentandfuturelearning:
• ThissectionconsolidatesStage4conceptswhichareusedinStages5and6
• SomeofthesequestionsaremorechallengingthanthoseinourStage4books
• PercentagesareusedextensivelyinthischapterandalsoChapter1ofourYear10book
• ExpertisewithpercentagesisassumedknowledgeforStage6Standardandmayalsoberequiredinnon-calculator examinationssuchasNAPLANandindustryaptitudetests
Themediaoftenquotespercentagesin newsstoriesandadvertisements.For example:
• 90% ofdentistspreferthis toothbrush.
• Ashirtisreducedby45%
• Interestratesincreasedby2%.
Theseexamplesinvolve ndinga percentageofaquantityoramountand thisisanimportantpartoftheworkwe dowithpercentages.
LessonStarter:Today’snewschallenge
Inpairs,gothroughtoday’snewspaperoronlinenewssiteand ndarticlesandadvertisementsthatuse percentages.Choosetwoandexplaintotheclasshowpercentagesareusedinthearticlesyouchoose.
KEYIDEAS
■ To ndapercentageofaquantity(withoutacalculator):
• Writethepercentageasafractionordecimal.
• Multiplybythequantity.
■ Calculatormethod: 3%of200 = 3 ÷ 100 × 200 = 6
■ Mentalstrategies:
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78 Chapter2Financialmathematics
50% 25% 10% 5% 1% 75% ÷2 ÷2 ÷ 2 ÷10 ÷10 ÷ 2 ÷100 ÷2 ÷ 2 × 3
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■ To ndtheoriginalamountwhengivenapercentage,youcanusetheunitarymethodorsolve anequation.
Forexample:3% ofanamountis 15. Whatistheoriginalamount? 3
∴ Theoriginalamountis500. ∴ Theoriginalamountis500.
Exercise2B
Nowyoutry
percentage,unitarymethod
Writethepercentageasafractionoutof100
Orwrite0.15 × 35.
Note:‘of’meansto‘multiply’.
Findthefollowingamounts.Useacalculatorifnecessary.
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2BApplyingpercentages 79
%
1 %
Unitary
5 100 % is 500 ÷ 3 ÷ 3 × 100 × 100 0 03 × A = 15 A = 300 ÷ 0.0 3 ÷ 0.03 Equation method 3% of amount is 15.
is 15
is
method
Keyterminology:
UNDERSTANDING 1–3 3 1 Findthesepercentageamountsmentally. Hint:Usethemental strategiesgivenin theKeyideas. 10% of$40 a 5% of$40 b 25% of8kg c 75% of8kg d 1% of$800 e 2% of140seconds f 2 If1% ofanamountis$4,whatis100% oftheamount? 3 Trueorfalse:34% of568 = 034 × 568? FLUENCY 4–6(½) 4–7(½) VIDEO DEMO Example4 Findingapercentageofaquantity Find15% of$35. SOLUTION EXPLANATION 15% of$35 = 3✚✚ 15 20✟✟ 100 × $35 = $525
andmultiplyby$35.
4
Hint: 10% of 20 = 10 100 × 20 10% of20 a 5% of200 b 20% of40 c 15% of50 d 8% of720 e 5% of680 f 15% of8200 g 70% of60 h 90% of500 i 75% of44 j 99% of200 k 3% of50 l UNCORRECTED
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Find18% of$60.
Tousetheunitarymethod,
Writeanequationusingthepronumeral A to representtheamount.
Solvefor
80 Chapter2Financialmathematics 5 Useacalculator,ifnecessary,to nd: 10% of$360 a 50% of$420 b 75% of64kg c 125% of240km d 375% of40apples e 875% of400m f 331 3 % of750people g 662 3 % of300cars h 83 4 % of$560 i VIDEO DEMO Example5 Findingtheoriginalamount
% oftheamountis$45. SOLUTION EXPLANATION
% oftheamount = $45 1% oftheamount
100% oftheamount =
Sotheoriginalamountis$900
Determinetheoriginalamountif5
5
= $9
$900
A = $45 005
A
A = $45 ÷
ndthevalueof1 partor1% thenmultiplyby100to nd100%. Alternatemethod:5% of
×
= $45
0.05 = $900
Determinetheoriginalamountif8% oftheamountis$40. 6 Determinetheoriginalamountif: Hint:Firstfindthe valueof 1% 10% oftheamountis$12 a 6% oftheamountis$42 b 3% oftheamountis$9 c 40% oftheamountis$2.80 d 90% oftheamountis$0.18 e 6% oftheamountis$27 f 12% oftheamountis$96 g 15% oftheamountis$54 h 7 Determinethevalueof x inthefollowingif: Hint: x istheoriginal amount. 10% of x is$54 a 15% of x is$90 b 25% of x is$127 c 18% of x is$225 d 105% of x is$126 e 110% of x is$44 f PROBLEM-SOLVINGANDREASONING 8–11 8(½),11–14 8 Withoutacalculator,evaluatethefollowing. Hint: 33 1 3 %= 1 3 10% of$58 a 5% of$84 b 1% of$46 c 21 2 % of$20 d 331 3 % of$132 e 662 3 % of$60 f 9 If1 3 of96 = 32,whatis662 3 % of96?
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A bydividingbothsidesby005. Nowyoutry
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c Write2.80asapercentage.
d Write325% asafraction.
e $2000inabankaccountincreasesto$5000overaperiodoftime.Byhowmuchhastheamount increasedasapercentage?
2BApplyingpercentages 81
a 5
b 21
c
10 If10% of$800is$80,explainhowyoucanusethisresultto nd: 1% of$800
% of$800
2 % of$800
11 About80% ofthemassofthehumanbodyiswater.IfCarlaweighs60kg,howmanykilogramsof watermakeupherbodyweight?
14 10% of1dayisthesameas x hoursand y minutes.Whatisthevalueof x and y? ENRICHMENT:Morethan 100% 15
a Find120% of60.
12 Inaclassof25students,40% havebeentoEngland.HowmanystudentshavenotbeentoEngland? 13 Explainwhy10% of24 = 24% of10.
15
% of x
.
b Determinethevalueof x if165
= 1
5.
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2C Percentage increaseanddecrease CONSOLIDATING
Learningintentionsforthissection:
• Tobeabletoincreaseordecreaseanamountbyagivenpercentage
• Tobeabletocalculatethepercentagebywhichanamounthasincreasedordecreased Past,presentandfuturelearning:
• ThissectionconsolidatesStage4conceptswhichareusedinStages5and6
• SomeofthesequestionsaremorechallengingthanthoseinourStage4books
• PercentagesareusedextensivelyinthischapterandalsoChapter1ofourYear10book
• ExpertisewithpercentagesisassumedknowledgeforStage6Standardandmayalsoberequiredinnon-calculator examinationssuchasNAPLANandindustryaptitudetests
Percentagesareoftenusedtodescribebyhow muchaquantityhasincreasedordecreased. Thepriceofacarinthenewyearmightbe increasedby5%.Ona$70000car,thisisa $3500increase.Thepriceofashirtmightbe markeddownby30%.Iftheshirtoriginally cost$60,thisprovidesan$18discount.Itis importanttonotethattheincreaseordecrease iscalculatedontheoriginalamount.
LessonStarter:Thequickermethod
Twostudents,NickyandMila,considerthequestion:$250isincreasedby15%.Whatisthe nalamount?
Nickyputshissolutionontheboardwithtwosteps.
Step1:15% of$250 = 015 × $250 = $3750
Step2:Finalamount = $250 + $37.50 = $28750
Milasaysthatthesameproblemcanbesolvedwithonlyonestepusingthenumber115.
• CanyoudescribeMila’smethod?Writeitdown.
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• Whatifthequestionwasalteredsothat$250isdecreasedby15%.HowwouldNicky’sandMila’s methodsworkinthiscase?
• Whichofthetwomethodsdoyoupreferandwhy?
82 Chapter2Financialmathematics
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KEYIDEAS
■ Toincreaseanamountbyagivenpercentage:
• Addthepercentageincreaseto100%.
• Multiplytheamountbythisnewpercentage.
Forexample,toincreaseby25%,multiplyby
100%+ 25%= 125%= 125
■ Todecreaseanamountbyagivenpercentage:
• Subtractthepercentagefrom100%.
• Multiplytheamountbythisnewpercentage.
Forexample,todecreaseby25%,multiplyby
100%− 25%= 75%= 0.75.
■ To ndapercentagechange,use:
Percentagechange = changeinprice originalprice × 100%
Forexample,ifapricechangesfrom$40to$50:
Percentagechange = 10 40 × 100%= 25%
Keyterminology: percentage
Exercise2C
1 Writethemissingnumberfortheseincreases.
a Toincreaseanumberby40%,multiplyby
b Toincreaseanumberby26%,multiplyby
c Toincreaseanumberby ,multiplyby16.
d Toincreaseanumberby ,multiplyby121.
2 Writethemissingnumberforthesedecreases.
a Todecreaseanumberby20%,multiplyby
b Todecreaseanumberby73%,multiplyby
c Todecreaseanumberby ,multiplyby094.
d Todecreaseanumberby ,multiplyby031.
3 Findthepercentagechangeif
a Originalamount = $120,change = $30
b Originalamount = $35,change = $70
decrease by 25%
Hint:Toincreaseby 40%, need 140% ofamount
Hint:Todecreaseby 20%, need 80% ofamount
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2CPercentageincreaseanddecrease 83
1.25 Original price (100%) Increased price (125%) to
increase by 25%
0.75 Original price (100%) Decreased price
to
(75%)
UNDERSTANDING 1–3 3
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VIDEO DEMO
Example6 Increasingbyapercentage
Increase$70by15%
SOLUTION
100%+ 15%= 115% = 115
$70 × 115 = $8050
Nowyoutry
Increase$120by30%
EXPLANATION
Firstadd15% to100%
Notethat15%= 0.15and100%= 1
Multiplyby115togive$70plustheincrease inonestep.
4 Completethefollowing,usingacalculatorifnecessary.
Increase56by10% a
Increase100by12% c
Increase180by15% e
Increase8by50% g
Example7 Decreasingbyapercentage
VIDEO DEMO
Decrease$5.20by40%
Increase980by20% b
Increase890by5% d
Increase450by20% f
Increase98by100% h
$5.
Nowyoutry
Decrease$64by15%
5 Completethefollowing.
Decrease80by5% a
Decrease45by50% c
Decrease8000by8% e
Decrease68by75% g
Decrease7000by100% i
EXPLANATION
Firstsubtractthe40% from100% to ndthe percentageremaining.
Multiplyby60%= 06togettheresult.
Decrease600by10% b
Decrease700by12% d
Decrease450by25% f
Decrease9000by1% h
Decrease10000by1.5% j
84 Chapter2Financialmathematics FLUENCY
4–5(½),6,7 4–5(½),6,7(½),8
SOLUTION
100
40%= 60%
06
20 × 0
6
$3
12
%−
=
.
=
.
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VIDEO DEMO
Example8 Findingapercentagechange
Thepriceofamobilephoneincreasedfrom$250to$280.Findthepercentageincrease.
b
a Thepopulationofatowndecreasesfrom3220to2985.Findthepercentagedecreaseandround toonedecimalplace.
SOLUTION EXPLANATION
a Increase = $280 $250 = $30 First ndtheactualincrease.
Percentageincrease = 30 250 × 100 1 = 12%
Dividetheincreasebytheoriginalamountand multiplyby100.
b Decrease = 3220 2985 = 235 First ndtheactualdecrease.
Percentagedecrease = 235 3220 × 100 1 = 73%(to1d.p.)
Nowyoutry
Dividethedecreasebytheoriginalpopulation andmultiplyby100.Roundasindicated.
Theheightofaplantincreasedfrom20cmto28cm.Findthepercentageincrease. a Thepriceofawashingmachinedecreasedfrom$649to$545.Findthepercentagedecreaseand roundtoonedecimalplace.
b
6 Thepriceofa ightincreasedfrom$125to$150overnight.Findthepercentageincrease.
7 Copyandcompletethetablesshowingpercentagechange.Roundtoonedecimalplacewherenecessary.
2CPercentageincreaseanddecrease 85
Original New Percentage amount amount Increase change 40 60 12 16 100 125 24 30 88 100 a Original New Percentage amount amount Decrease change 90 81 100 78 20 15 24 18 150 50 b
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8 WhentheGoodsandServicesTax(GST)wasintroduced,allpricesincreasedby10%.
Copyandcompletethetable.
Example9 Findingtheoriginalamountfromanincreaseordecrease
Afterrain,thevolumeofwaterinatankincreasedby24% to2200L.Howmuchwaterwasinthe tankbeforeitrained?Roundtothenearestlitre. SOLUTION
EXPLANATION
Writethetotalpercentage.
Theoriginalvolumeisincreasedby24% to give2200litres
100% oforiginalvalue = 1774L
Divideby124to nd1%
Multiplyby100to ndtheoriginalamount.
Writethetotalpercentage.
Writeanequationusingthepronumeral V torepresentthevolume.
Dividebothsidesby124tosolvefor V Roundtothenearestlitre.
Nowyoutry
Atthestartofanewyear,dailypublictransportticketsincreaseby4% to$920.Whatwasthecost ofadailyticketbeforetheincrease?Roundtothenearestcent.
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86 Chapter2Financialmathematics
×
÷ 1.1 Original price 100% (no GST) A B New price 110% (including GST) Hint:GST = 10% of A or GST = B A
1.1
OldPrice NewPrice GST (excludingGST) (includingGST) amount a $40 b $120
VIDEO DEMO
c $55 d $154 e $7
100%+ 24%= 124%
124% oforiginalvalue = 2200L 1% oforiginalvalue = 1774L
Alternatemethod:100%+ 24%= 124% 124% of V = 2200L 1.24 × V = 2200 V = 2200 ÷ 1.24 = 1774
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9 Findtheoriginalcostforeachofthefollowingif:
a anincreaseof10% onthecostofacanof coladrinkincreaseditto$3.30
b anincreaseof10% onthecostofameal increasedthecostto$88
c afteranincreaseof5%,thecostofapairof runningshoescameto$210
d adecreaseof30% madethecostofcar insurance$350
e adecreaseof60% broughtthepriceofa usedcardownto$5000
10 Thepriceofacomputerwasdecreasedby15% inasale.Whatisthesaleprice,iftheoriginalpricewas $2100?
11 Plumbersonasalaryof$82570weregivena21 2 % payincrease.Findtheirnewannualsalary.
12 Acarmanufacturerintendstoincreasesalesby14.7% nextyear.Ifthecompanysold21390newcars thisyear,howmanydoesitexpecttosellnextyear?
13 Thelengthofabikesprintraceisincreasedfrom800mto 1200m.Findthepercentageincrease. Hint: % increase = increase originalamount × 100
14 Thenumberofpeopleonabusdecreasedfrom25to18after onestop.Findthepercentagedecreaseinthenumberofpeopleonthebus. 15 Afterapriceincreaseof20%,thecostofentrytoamuseumroseto$2580.Findtheoriginalprice.
16 ThetotalpriceofanitemincludingGST(at10%) is$120.HowmuchGST ispaid,tothenearestcent?
17 Aninvestorstartswith$1000.
Hint: $120 represents 110% ofthepricebefore GSTisadded.
a Afterabaddaytheinitialinvestmentisreducedby10%.Findthebalanceattheendoftheday.
b Thenextdayisbetterandthebalanceisincreasedby10%.Findthebalanceattheendofthe secondday.
c Theinitialamountdecreasedby10% onthe rstdayandincreasedby10% onthesecondday. Explainwhythebalanceontheseconddaydidn’treturnto$1000.
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2CPercentageincreaseanddecrease 87
PROBLEM-SOLVINGANDREASONING 9–12 13–16
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18
Ifthecostofapairofshoeswasincreasedtwice,by10% fromanoriginalpriceof$80andthenanother 15% fromthisnewprice,the nalpricewouldbe
$80 × 110 × 115 = $10120
Useasimilartechniqueto ndthe nalpriceoftheseitems.Roundtothenearestcent.
a Skisstartingat$450andincreasingby20% and10%
b Acomputerstartingat$2750andincreasingby6% and11%
c ADVDplayerstartingat$280anddecreasingby10% and25%
d Acircularsawstartingat$119anddecreasingby18% and37%
19
Ifanamountisincreasedbythesamepercentageeachtime,powerscanbeused. Forexample,50kgincreasedby12% threetimeswouldincreaseto
50kg × 1.12 × 1.12 × 1.12
= 50kg ×(112)3
= 70.25kg (to2d.p.)
Hint:Youhavea power/indexkeyon yourcalculator.
Useasimilartechniqueto ndthe nalvalueinthesesituations.Roundtotwodecimalplaces.
a Themassofaratinitiallyat60gramsgrowsatarateof10% everymonthfor3months.
b Thecostofanewcarinitiallyat$80000increasesby5% everyyearfor4years.
c Thevalueofanapartmentinitiallyat$380000decreasesby4% peryearfor3years.
d Thelengthofapencilinitiallyat16cmdecreasesthroughbeingsharpenedby15% everyweekfor 5weeks.
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88 Chapter2Financialmathematics ENRICHMENT:Repeatedincreaseanddecrease 17,18
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2D Profits anddiscounts CONSOLIDATING
Learningintentionsforthissection:
• Toknowandunderstandtheterminologyassociatedwithprofits,lossesanddiscounts
• Tobeabletosolveproblemsinvolvingpercentageprofits,lossesanddiscounts
Past,presentandfuturelearning:
• ThissectionconsolidatesStage4conceptswhichareusedinStages5and6
• SomeofthesequestionsaremorechallengingthanthoseinourStage4books
• PercentagesareusedextensivelyinthischapterandalsoChapter1ofourYear10book
• ExpertisewithpercentagesisassumedknowledgeforStage6Standardandmayalsoberequiredinnon-calculator examinationssuchasNAPLANandindustryaptitudetests
Percentagesarewidelyusedintheworldof nance.Pro ts,losses,commissions,discountsandtaxation areoftenexpressedandcalculatedusingpercentages.
LessonStarter:Thebestdiscount
Twobookshopsaresellingthesamebookatadiscountedprice.Therecommendedretailpriceforthebook isthesameforbothshops.Eachshophasasignnearthebookwiththegivendetails:
• ShopA.Discountedby25%
• ShopB.Reducedby20% thentakeafurther10% offthat.
Whichshopoffersthebiggerdiscount?
Isthedifferenceequalto5% oftheretailprice?
Examplesofpercentagesusedinthemedia.
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2DProfitsanddiscounts 89
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KEYIDEAS
■ Pro t istheamountofmoneymadeonasale. Pro t = sellingprice costprice
■ A loss ismadewhenthesellingpriceislessthanthecostprice.
Loss = costprice sellingprice
■ Mark-up istheamountaddedtothecostpricetoproducethesellingprice.
Sellingprice = costprice + mark-up
■ Thepercentagepro torlosscanbefoundbydividingthepro t orlossbythecostpriceandmultiplyingby100.
% pro t/loss = (pro torloss costprice × 100) %
■ Discount istheamountbywhichanitemismarkeddown.
Discount =% discount × originalprice
Newprice = originalprice discountamount
Exercise2D
90 Chapter2Financialmathematics
Keyterminology: pro t,sellingprice,costprice,loss,mark-up,percentage,discount
UNDERSTANDING 1–4 1 1 Matchthede nitionintheleftcolumnwiththecorrectwordintheright-handcolumn. a Theamountofmoneymadeonasale A Loss b Theamountbywhichanitemismarkeddown B Mark-up c Theresultwhenthesellingpriceislessthanthecostprice C Pro t d Theamountaddedtothecostprice D Discount 2 Copyandcompletethetableofpro tsandlosses. Hint:Profit/lossisthe differencebetweencost priceandsellingprice. Markupincreasesprice. Discountreducesprice. Costprice($) 7 18 24 80 7 30 460 95 3250 Sellingprice($) 10 15 50 26 20 11 80 395 4430 Profit/loss($) 3 Copyandcompletethetableofmark-ups. Costprice($) 30.95 99.95 199.95 18000 Mark-up($) 10 80 395.95 700 16700 Sellingprice($) 1499 95 35499 26995 4 Copyandcompletethetableofdiscounts. Originalprice($) 100 49 95 29 95 2215 Newprice($) 72 40 90 22 70 176 299 95 Discount($) 23 45.55 178 UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
VIDEO DEMO
Example10 Calculatingsellingpricefrommark-up
Anelectricalstoremarksupallentertainmentsystemsby30%. Ifthecostpriceofoneentertainmentsystemis$8000,whatwillbeitssellingprice?
SOLUTION EXPLANATION
Mark-up = 30% of$8000 = 0.3 × 8000 = $2400
Sellingprice = $8000 + $2400 = $10400
Alternatively,multiplycostpriceby 130% or1.3.
Nowyoutry
Changepercentagetoadecimalorfraction andmultiplybythecostprice.
Sellingprice = costprice + mark-up
130% isa30% increaseonthecostprice
Acomputerstoremarksupallnotebookcomputersby24% Ifthecostpriceofonenotebookis$1200,whatwillbeitssellingprice?
5 Copyandcompletethistablebycalculatingthesellingpriceofeachitem.
VIDEO DEMO
Example11 Findingthediscountamount
Anelectricalstoreadvertisesa15% discountonallequipmentasaholidayspecial.Findthesale priceonaprojectionsystemthathasamarkedpriceof$18000.
SOLUTION EXPLANATION
Discount = 15% of$18000 = 0.15 × 18000 = $2700
Thenewprice = $18000 $2700 = $15300
Alternatively,multiplytheoriginalpriceby 85% or085.
Changethepercentagetoadecimaland evaluate.
Newpriceisoriginalpriceminusdiscount.
A15% discountleaves85%.
2DProfitsanddiscounts 91 FLUENCY 5–8 5–8
Item Costprice % mark-up Sellingprice Jeans $60 28% Toaster $40 80% Car $22000 45% Canofdrink $1 20 140% Loafofbread $1 80 85% Handbag $80 70% Electronictablet $320 35%
Continuedonnextpage
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Nowyoutry
Adepartmentstoreoffersapost-Christmasdiscountof35% onalldecorations.Findthesaleprice onawreaththathasamarkedpriceof $80.
6 Copyandcompletethetablebywritinginthemissingvalues.
Example12 Determiningprofit
Amanufacturerproducesanitemfor$400andsellsitfor$540.
Nowyoutry
Ajewellerproducesanecklacefor$36andsellsitfor$4950.
7 Findthemissingvaluesinthesetablesby ndingthepro torlossandexpressingthisasapercentage ofthecostprice.Roundtotwodecimalplaceswherenecessary.
92 Chapter2Financialmathematics
Item Costprice % discount Sellingprice Camera $900 15% Car $24000 20% Bike $600 25% Shoes $195 30% Blu-rayplayer $245 50% Electricrazor $129 20% Lawnmower $880 5% VIDEO DEMO
Determinethepro tmade. a Expressthispro
b SOLUTION EXPLANATION a Pro t = $540 $400 = $140 Pro t = sellingprice costprice b % pro t = 140 400 × 100 = 35% % pro t = pro t costprice × 100
tasapercentageofthecostprice.
Determinethepro
Expressthispro
b
tmade. a
tasapercentageofthecostprice.
a Costprice($) Sellingprice($) Profit($) Profit(%) 10 15 24 30 100 150 250 255 17 50 20
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DEMO
Example13 Calculatingtheoriginalprice
Atoystorediscountsatoyby10% inasale.Ifthesalepricewas$10.80,whatwastheoriginal price?
SOLUTION EXPLANATION 90% oftheoriginal = $10.80
1% oftheoriginal = 1080 ÷ 90
1% oftheoriginal = 0.12 100% oftheoriginal = $12
Thediscountfactor = 100%− 10%= 90%. Thus,$1080is90% oftheoriginalprice.
Usetheunitarymethodto nd1% Multiplyby100to ndtheoriginalamount.
Theoriginalpricewas$12. Writetheanswerinwords.
Alternatemethod:90% of P = $1080 0.9 × P = $10.80 P = $1080 ÷ 09 = $12
Nowyoutry
Writeanequationusingthepronumeral P to representtheoriginalprice.
Solvetheequationfor P bydividingboth sidesby09
Anoutdoorequipmentstorediscountsatentby20% inasale. Ifthesalepricewas$176,whatwastheoriginalprice?
8 Answerthefollowingquestionsrelatingto nding theoriginalprice(costprice).
a Findtheoriginalpriceifacoffeemugwas discountedby20% andsoldfor$440.
b Findthecostpriceofapairofshoesthatsold for$250afteramark-upof25%.
c Findthecostpriceafteradiscountof10% was givenonasurfboardthatsoldfor$1350.
d Findtheoriginalpriceonaconcertticketfor amajorrecordingartistifitwasmarkedupby 100% andsoldfor$250.
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2DProfitsanddiscounts 93 b Costprice($) Sellingprice($) Loss($) Loss(%) 10 8 16 12 100 80 34 19 95 80.75 VIDEO
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.80
d 10 Lennymarksupallcomputersinhisstoreby125%.Ifacomputercosthim$890,whatwillbethe sellingpriceofthecomputer?
11 Aused-cardealerpurchasesavehiclefor$13000andsellsitfor $18500.Determinethepercentagemark-uponthevehicletoone decimalplace.
12 Arefrigeratorisdiscountedby25%.IfPaulapays$460forit,whatwasthe originalprice?Roundtothenearestcent.
13 Anelectricalstorebuysacomputerfromthewholesalerfor$500.Thestore marksupthecomputerby80%.
a Whatistheamountofthemark-upindollars?
b Whatistheretailpriceofthecomputerafterthestore’smark-up?
c Thestoreoffersthecomputeronsaleforadiscountof15%.Whatisthe pricenow?
d Ifthe15% discountwascalculatedontheoriginal$500costprice,and thenthecomputerwasmarkedup80% afterthat,woulditmakea differencetothesaleprice?
Hint:Whatistheincrease asapercentageofthe originalprice?
a Howmuchwasthehouseworthin1998?
b Findthepercentageincrease,toonedecimalplace,inthevalueofthehousefrom1998to: 2002 i 2004 ii 2006 iii
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c In2018theownerswishedtosellfor125milliondollars.Whatpercentageincreasewasneededin thetwoyearsfrom2016to2018sotheycouldsellforthatprice?
94 Chapter2Financialmathematics PROBLEM-SOLVINGANDREASONING 9–11 11–13
Determinethepro
Expressthispro
ii Costprice$10,sellingprice$12 a Costprice$20,sellingprice$25
Costprice$120,sellingprice$136
c Costprice$1400,sellingprice$3850
9 Amanufacturerproducesandsellsitemsforthepricesshown.
tmade. i
tasapercentageofthecostprice.
b
ENRICHMENT:Housevalue 14 14 Thegraphshowsthechangesinthevalueofaparticularhouse.
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2E Income
Learningintentionsforthissection:
• Toknowandunderstandthedifferentwaysthatworkerscanbepaidfortheirlabour
• Touseratesandpercentagestocalculatesalaries,wages,commissions,royaltiesetc
• Tobeabletomakeconversionsbetweenpayperhour,day,week,fortnight,monthandyear Past,presentandfuturelearning:
• ThissectionintroducestheStage5CoreTopiccalled FinancialMathematicsA
• Section1DofourYear10bookrevisesandfurtherextendstheseconcepts
• FinancialMathematics isamajortopicinStage6Standard
Therearemanydifferentwaysofearningaliving.Youcanbeself-employedorworkforsomeoneelse. Yourincomeisusuallyrelatedtotheskillsyouhave.Itcanbecalculatedandpaidindifferentways.You canearn,forexample,asalary,awage,acommissionorpossiblyaroyalty.
LessonStarter:Typesofincome
Asaclass,writedownoneexampleofajobthatearnseachofthefollowingtypesofincome.
• Salary
• Wage(i.e.hourlyrateofpay)
• Overtime
• Commission
Inthenewspaperclassi edsoronline, ndajobadvertisementforeachoftheincometypesabove.
KEYIDEAS
■ Workerswhoearna wage (forexample,acasualwaiter)arepaida xedrateperhour.Hours outsidethenormalworkinghours(publicholidaysetc.)arepaidatahigherratecalled overtime Thiscanoccurinacoupleofcommonways:
• Timeandahalf: payis1.5timestheusualhourlyrate
• Doubletime: payistwicetheusualhourlyrate
■ Workerswhoearna salary (forexample,anengineer)arepaida xedamountperyear,say, $125000.Thisispaidmonthly,fortnightlyorweekly.
• 12monthsinayearandapproximately52weeksinayear = 26fortnights
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2EIncome 95
Annual salary Monthly pay ÷ 12 × 12 Annual salary Fortnightly pay ÷ 26 × 26 Annual salary Weekly pay ÷ 52 × 52
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■ Somewageandsalaryearnsarepaid leaveloading.Whentheyareonholidays,theyearntheir normalpayplusabounscalledleaveloading.Thisisusually17.5%oftheirnormalpay.
■ Commission isapercentageoftheoverallsalesamount.Salespeoplemayreceiveacommission ontheirsalesaswellasasetweeklyormonthlyfeecalleda retainer.
• Commission =% commission × totalsales
■ Somepeoplewhowritebooksormusicarepaid royalties,whichcouldbe10%ofthesalesprice ofeverybookorsongwhichissold.
■ Somepeoplewhomakeordothingsarepaid piecework.Forexample,$4foreverysheepshorn.
Keyterminology: wage,overtime,timeandahalf,doubletime,salary,commission,retainer
Exercise2E
1 Matchthepaydescription(a–d)withtheincometype(A–D). a Sallyearns$22.70perhourworkinginashop
b MathsteacherStephanieearns$72000peryear
c Rossearns5% ofallsaleshemakes
d MattearnstimeandahalfpayonSaturdays
Wage
Salary
2 Tomearns$1270anhour.Howmuchdoesheearnfor: 2hoursofwork? a 8hoursofwork? b 38hoursofwork? c
3 Sela’shourlyrateofpayis$24.Calculateherovertimerateat: timeandahalf a doubletime b
4 Williamearnsasalaryof$136875eachyear.
Approximatelyhowmuchisthis:
a eachmonth?
b eachweek(tothenearestcent)?
c eachday?
Hint:Tofindapproximatemonthly salary,divideannualsalaryby 12
Tofindapproximateweeklysalary, divideannualsalaryby 52
Tofindapproximatedailysalary, divideannualsalaryby 365
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96 Chapter2Financialmathematics
÷ 1.175 × 1.175 100% + 17.5%
117.5% A B = 1.175 Holiday pay $1762.50 Normal pay $1500
Leave loading = 17.5% of A = $262.50
=
UNDERSTANDING 1–4 1,4
A
Overtime
B
Commission
C
D
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Example14 Comparingwagesandsalaries
VIDEO DEMO
Kenearnsanannualsalaryof$90000andworksa38-hourweek.HiswifeBrookeworksparttime inretailandearns$61.80perhour.
CalculatehowmuchKenearnsperweek. a
Determinewhohasthehigherhourlyrateofpay. b
IfBrookeworksonaverage18hoursperweek,whatisheryearlyincome? c
SOLUTION EXPLANATION
a Weeklyrate = $90000 ÷ 52 = $173077
∴ Kenearns$173077perweek
b Brooke:$61.80/h
Ken:$1730.77 ÷ 38 = $4555/h
∴ Brookeispaidmoreperhour.
c Inoneweek:$61.80 × 18 = $111240
Yearlyincome = $111240 × 52 = $57844.80
Nowyoutry
$90000payinayear. Thereareapproximately52weeksinayear.
Divideby52to ndtheweeklywage.
Kenworks38hoursintheweek.
Hourlyrate = weeklyrate ÷ number ofhours Roundtothenearestcent.
Comparehourlyrates.
Weeklyincome = hourlyrate × numberofhours
Multiplyby52weekstogetyearlyincome.
Maliearnsanannualsalaryof$77200andworksa38-hourweek.HerpartnerBenworksparttime asaphotographerandearns$75perhour. CalculatehowmuchMaliearnsperweek.
b
a Determinewhohasthehigherhourlyrateofpay.
c
IfBenworksonaverage15hoursperweek,whatishisyearlyincomefromphotography?
5 Talibearns$58000peryearatafastfoodrestaurant.Hissisterworksparttimeasawaitressandearns $33.20perhour.
a HowmuchdoesTalibearneachweek?
b EachweekTalibworks38hours.Calculateifhishourlyrateofpayishigherthanhissister’s.
c Ifhissisteraverages10hoursofworkperweek,whatisheryearlyincome?
6 Paulearns$790eachweek.Howmuchdoesheearneach:
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Hint:Thereare 12 months inayear. year? a month? b
c
hour,ifheworked40hourseachweek?
2EIncome 97 FLUENCY 5–7,8(½),9 5,6,8(½),9
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Example15 Calculatingovertime
Georgioworkssomeweekendsandlatenights,andearnsovertimeforthatwork.Hishourlyrateof payis$32perhour.
CalculateGeorgio’stimeandahalfrateofpayperhour. a
CalculateGeorgio’sdoubletimerateofpayperhour. b
c
CalculateGeorgio’sweeklywageforaweekwhereheworks18hoursathisnormalrate, 2hoursattimeandahalf,and1houratdoubletime.
SOLUTION EXPLANATION
a Timeandahalf = $32 × 15 = $48
b Doubletime = $32 × 2 = $64
c $32 × 18 = $576 $48 × 2 = $96 $64 × 1 = $64
Totalwage = $736
Timeandahalfis15timesthehourlyrate.
Doubletimeis2timesthehourlyrate.
Findthesumof:
• thenormalhourlyrate ($32) multipliedbythe numberofhoursworkedatthenormalrate (18)
• thetimeandahalfhourlyrate ($48) multiplied bythenumberofhoursworkedatthatrate (2)
• thedoubletimehourlyrate ($64) multipliedbythe numberofhoursworkedatthatrate (1)
Alternatively, 18hours +(2hours × 1.5) +(1hour × 2)
= 18 + 3 + 2 = 23
Wage = 23 × 32 = $736
Nowyoutry
Calculatethenumberof‘normal’hours.
Multiplybythehourlyrate.
Kaneworkssomeweekendsandsomepublicholidaysandearnsovertimeforthatwork.Hishourly rateofpayis$24perhour.
a
CalculateKane’stimeandahalfrateofpayperhour.
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b
CalculateKane’sdoubletimerateofpayperhour.
CalculateKane’sweeklywageforaweekwhereheworks15hoursathisnormalrate,6hours attimeandahalfand3hoursatdoubletime.
98 Chapter2Financialmathematics 7 Copyandcompletethetable. Employee Hourlyrate Hoursworked Income Adam $20 40 8 Betty $15 50 8 1 2 Ceanna $19.70 15 David $24.30 38 Edward $57.85 42 Francis $30 27 George $35 20 7 25
VIDEO DEMO
c
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8 Ajobhasanormalworkinghourspayrateof$29.20perhour.Calculate thepay,includingovertime,fromthefollowinghoursworked.
a 3hoursatthenormalrateand4hoursattimeandahalf
Hint:Fortimeandahalf: × 1.5 Fordoubletime: × 2
b 4hoursatthenormalrateand6hoursattimeandahalf
c 14hoursatthenormalrateand3hoursatdoubletime
d 20hoursatthenormalrateand5hoursatdoubletime
e 10hoursatthenormalrateand8hoursattimeandahalfand3hoursatdoubletime
f 34hoursatthenormalrateand4hoursattimeandahalfand2hoursatdoubletime
Example16 Calculatingcommission
VIDEO
DEMO
Apart-timesaleswomanispaidaretainerof$1500permonth.Shealsoreceivesacommissionof 5% onthevalueofgoodsshesells.Ifshesellsgoodsworth$5600duringthemonth,calculateher earningsforthatmonth.
SOLUTION EXPLANATION
Commission = 5% of$5600
= 0.05 × $5600 = $280
Earnings = $1500 + $280 = $1780
Nowyoutry
Calculatethecommissiononsales.Change thepercentagetoadecimalandevaluate.
Earnings = retainer + commission
Arealestateagentispaidaretainerof$2200permonth.Shealsoreceivesacommissionof0.4% onthevalueofhousesshesells.Ifshesellshousesworth $1575000duringthemonth,calculate herearningsforthatmonth. 9 Copyandcompletethetable.
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2EIncome 99
Person Weeklyretainer Rateofcommission Commissionearned ($) Weeklywage ($) Adina $0 12% on $7000 Byron $160 8% on $600 Cindy $300 5% on $680 Deanne $260 5% on $40000 Elizabeth $500 8% on $5600 Faruq $900 2% on $110000 Gary $1000 1 5% on $45000 PROBLEM-SOLVINGANDREASONING 10–12 11,13–15 10
3hoursand2hoursatdoubletime
6hoursand8hoursattimeandahalf
15hoursand12hoursattimeandahalf
Calculatehowmanyhoursatthestandardhourlyratethefollowingworkinghoursarethesameas: a
b
c
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11 Jim,apart-timegardener,earned$522inaweek.Ifheworked12hours duringnormalworkinghoursand4hoursovertimeattimeandahalf, whatwashishourlyrateofpay?
12 Sallyearned$658.80inaweek.Sheworked10hoursduringtheweek, 6hoursonSaturdayattimeandahalfand4hoursonSundayatdoubletime. Whatwasherhourlyrateofpay?
Hint:Calculatethenumber ofhoursatthestandard hourlyrate.
13 AmyworksatBestBookshop.Duringoneweekshesellsbooksvaluedat$800.Ifsheearns $450perweekplus5% commission,howmuchdoessheearninthisweek?
14 Jasonworksforacampervancompany.Ifhesells$84000worthofcampervansinamonth,andhe earns$1650permonthplus4% commissiononsales,howmuchdoesheearnthatmonth?
15 Stephenearnsanhourlyrateof$3460forthe rst38hours,timeandahalfforthenext3hoursand doubletimeforeachextrahourabovethat.Calculatehisearningsifheworks44hoursinaweek.
16
16 Workersatafastfoodrestaurantarepaid$2163perhourforworkingMondaytoFridayupuntil 7p.m.andtimeandahalfafter7p.m.TheyearntimeandahalfonSaturdaysanddoubletimeon Sundays.Theyaregivenanunpaid30minutemealbreakforanyshiftover5hours.
Donna’sshiftsfortheweekaregivenbelow.
a HowmuchcanDonnaexpecttoearn,tothenearestcent,ifsheworksthehourssheisrosteredfor?
b Therestaurantdecidesonanewworkplacedeal,getsridofallovertimeandcreatesa athourly rateof$24perhour.HowmuchworseoffwillDonnabefortheweekabove?
c HowmanyextrahoursaweekdoesDonnaneedtoworkduringtheweektomakeuptheextra income?Answerinawholenumberofhours.
100 Chapter2Financialmathematics
ENRICHMENT:Payslips
Monday Tuesday Wednesday Thursday Friday Saturday Sunday ––4 30 p.m.–7 p.m. 4 30 p.m.–7 30 p.m. 5 p.m.–10 p.m. 5 p.m.–8 p.m. 10 a.m.–6 30 p.m. 10 a.m.–1 30 p.m.
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.20toproduce andshesellsthemfor$10, ndherpercentagepro tcorrecttoonedecimalplace.
adresswasdiscountedby15% andsoldfor$187
acomputergamewasmarkedupby12%
.80
c Nedworksinafastfoodrestaurantandhasanormalhourlyrateof$15.10perhour. Inaweekheworks4hoursatthenormalrate,3hoursattimeandahalfand2hoursat doubletime.
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d Harperispaidaweeklyretainerof$200andshealsoearnsa5% commissiononhersales. Inaparticularweekshemakes$8400worthofsales.
Progressquiz Progressquiz 101 1 2A Expressthefollowingintherequiredform. 32% asasimpli edfraction a 7% asadecimal b 023asapercentage c 0125asapercentage d 7 20asapercentage e 45% asasimpli edfraction f 2 2A Expressthefollowingaspercentages. 20coutof$2 a 32marksoutof40 b 5kgoutof40kg c 3 2B Findthefollowingamounts. 20% of80km a 35% of$120 b 331 3 % of180g c 42% of$500 d 4 2B Determinetheoriginalamountif: 25% oftheamountis$100 a 12% oftheamountis$36 b 8% oftheamountis$60 c 5 2C Completethefollowing. Increase82by10% a Increase220by8% b Decrease110by15% c Decrease250by6% d 6 2C a Thenumberofstudentsinaschoolincreasedfrom450to540.Findthe percentageincrease. b Thecostofpetroldroppedfrom121cperlitreto103cperlitreovernight.Findthepercentage decreasecorrecttoonedecimalplace. 7 2C Afterofferingaspecialticketdealforstudents,aconcert’sticketsalesroseby24% to961ticket sales.Howmanyticketshadbeensoldbeforetheticketdeal? 8 2D Findthesellingpriceofthefollowingitems. a A$1200tabletcomputermarkedupby16% b Apairof$126sneakersdiscountedby20% 9
10 2D
a
11 2E Findtheweeklyearningsinthefollowingworksituations.
2D Janehasastallatthemarketsellingbraceletsthatshemakes.Ifeachonecosts$6
Findtheoriginalpriceif:
b
andsoldfor$72
a Sebearnsanannualsalaryof$74100.(Assume52weeksintheyear.) b Jodieworks22hoursinaweekat$1820perhour.
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2F Taxation
Learningintentionsforthissection:
• Toknowandunderstandtheterminologygrossincome,netincomeandtaxableincome
• Tobeabletocalculatenetincomefromgrossincomeanddeductions
• TobeabletousetheATOtabletocalculatetaxpayable
Past,presentandfuturelearning:
• ThissectioncontinuestheStage5CoreTopiccalled FinancialMathematicsA
• Section1CofourYear10bookrevisesandfurtherextendstheseconcepts
• FinancialMathematics isamajortopicinStage6Standard
Allwageandsalaryearnershavesomedeductionstakenoutoftheirpay.Deductionsusuallyincludeincome tax.Taxispaidtothegovernment.Thegovernmentusesittopayforcommunitywelfare,educationanda numberofotherservices.
PAY SLIP
Payment date: 14 November 2016
Pay period: 1 November – 14 November 2016 Employee’s name:
name:
LessonStarter:Deductionsfrompay
Whattypesofdeductionscanyouandyourclassthinkofthatmightbetakenoutofsomeone’spay?
Discuss,asaclass,whatsuperannuationandtheMedicarelevyis,andwhopaysit.
KEYIDEAS
■ Grossincome = thetotalofallincomefromallsources
■ Netincome = grossincomeminus deductions suchastax,unionfeesandsuperannuation
■ Taxableincome = grossincomeminuswork-relatedexpensesanddonationstocharity
■ Incometax ispaidtothegovernment.Itisbasedonaperson’staxableincome.
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■ Incometaxpayable canbecalculatedusingthetaxtablepublishedeveryyearbytheAustralian TaxationOf ce(ATO).Everyrowinthetableiscalleda taxbracket.Thetablebelowshows thetaxratesforthe2022/2023 nancialyear.
102 Chapter2Financialmathematics
Employer’s
Job Title: Hourly rate:
$22.00 Wages: Hours worked: 18 @ $22.00 $396.00 $396.00 $320.76 Deductions Tax $75.24 Gross payment Net payment
Jason Smith BSJ Clothing Sales assistant
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The MedicareLevy isanadditionaltaxwhichthegovernmentusestosupportfundingfor medicines,doctors,hospitalsandotheraspectsofthehealthcaresystem.Answerstothe questionsinthefollowingexercisearebaseduponthistable,buttheactualratesmayvaryfrom yeartoyear.
Taxableincome Taxonthisincome
$0 $18200 Nil
$18201 $45000 19 centsforeach $1 over $18200
$45001 $120000 $5092 plus 32.5 centsforeach $1 over $45000
$120001 $180000 $29467 plus 37 centsforeach $1 over $120000
$180001 andover $51667 plus 45 centsforeach $1 over $180000
Theaboverates donot includetheMedicarelevyof 2%.
Keyterminology: grossincome,netincome,taxableincome,incometax,deductions
Exercise2F
1 Copyandcompletethetablebyinsertingthenetincomeamounts. Hint:Recallthatnet
2 FindHoang’sgrossweeklyincomeifheearns:
• $1200aweekasateacher
• $60anhourfortutoring,whenhetutorsfor3hoursaweek
• $25interestonhisbankaccountperweek
3 Johnhasagrossincomeof$45000andanetincomeof$20000.Howmuchdidhisdeductions cometo?
4 Pamelapays31.5%ofhergrossincomeintax.Howmuchtaxdoesshepayonagrossincomeof $85790?
FLUENCY
Example17 Calculatingtaxtofindnetincome
5–6(½),7,8 5–6(½),8(½) VIDEO DEMO
Liamstartsanewjobwithanannualsalaryof$52800.Hispayslipeachmonthshowsdeductions fortaxationof$968.
a CalculateLiam’snetincomeeachmonth.
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b WhatpercentageofLiam’smonthlypayisbeingpaidtothegovernmentbyhisemployer fortaxation?
c Liam’ssalaryisincreasedto$65000andthetaxationrateforLiam’ssalarychangesto24% withthe rst$18200taxfree.CalculateLiam’snetincomefortheyear.
2FTaxation 103
UNDERSTANDING 1–4 4
Grossincome Deductions = Netincome $5600 $450 $5150 $87000 $28000 $50000 $6700 $890 $76 $84650 $24790
income = grossincome minusdeductions.
Continuedonnextpage
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SOLUTION EXPLANATION
a Monthlypay = $52800 ÷ 12 = $4400
Calculategrossincomepermonth.
∴ netmonthlyincome = $4400 $968 = $3432
b % tax = 968 4400 × 100 = 22%
c Salaryfortaxpurposes = $65000 $18200 = $46800
Taxamount = 24% of$46800 = 0.24 × $46800 = $11232
∴ netincome = $65000 $11232 = $53768
Nowyoutry
Netincome = grossincome taxation
Calculatewhatfraction$968isofthemonthly income$4400.Multiplyby100toconverttoa percentage.
First$18200isnottaxed.
Calculatetaxamounton$46800.Convert percentagetoadecimalandevaluate.
Netincome = grossincome taxamount.
Poppyhasajobwithanannualsalaryof$74400.Herpayslipeachmonthshowsdeductionsfor taxationof$1488.
a CalculatePoppy’snetincomeeachmonth.
b WhatpercentageofPoppy’smonthlypayisbeingpaidtothegovernmentbyheremployer fortaxation?
c Poppy’ssalaryisincreasedto$82800andthetaxationrateforPoppy’ssalarychangesto28% withthe rst$18200taxfree.CalculatePoppy’snetincomefortheyear.
5 Foreachofthefollowing nd: theannualnetincome i
thepercentageofgrossincomepaidastax.Roundtoonedecimalplacewherenecessary. ii
a Grossannualincome = $48241,taxwithdrawn = $8206
b Grossannualincome = $67487,taxwithdrawn = $13581.20
c Grossmonthlyincome = $4041,taxwithdrawn = $606.15
d Grossmonthlyincome = $3219,taxwithdrawn = $714.62
6 Calculatetheamountoftaxtobepaidusingthefollowing annualsalariesandtaxratesifthe rst$18200istaxfree.
Hint:Netincome = grossincome deductions
Hint:Remembertosubtractthe $18200 tofindthesalaryto calculatetaxon.
salary = $30400,taxrate = 15%
b
a salary = $56500,taxrate = 21%
salary = $69700,taxrate = 24.5%
c salary = $96400,taxrate = 30.4%
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7 Edearns$1400perweekandpays27% ofhisannualincomeintax.
a CalculatetheamountofincometaxthatEdpaysinoneyear.
b FindEd’sannualnetincome.
104 Chapter2Financialmathematics
d
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Example18
Usingthetaxtabletocalculatetaxpayable
Usethetaxtablebelowto ndtheincometaxpayableforataxableincomeof$124000.
Taxableincome Taxonthisincome
$0 $18200 Nil
$18201 $45000 19centsforeach$1over$18200
$45001 $120000 $5092plus32 5centsforeach$1over$45000
$120001 $180000 $29467plus37centsforeach$1over$120000
$180001andover $51667plus45centsforeach$1over$180000
Theaboverates donot includetheMedicarelevy2%.
SOLUTION
Tax = $29467 + 0.37 ×(124000 120000) = $30947
EXPLANATION
Findthetaxbracketinwhichtheamount $124000lies.Thisis ($120000 $180000)
Writedownthevaluesinthisbracket, rememberingthat37cinthedollaris37%,or0.37.
Subtract120000from124000to nd theamountofincomethatthe37% appliesto.
Useyourcalculatorto ndtheanswer, workingfromlefttoright.
Nowyoutry
Usethetaxtableaboveto ndtheincometaxforanincomeof$75000.
8 UsethetaxtableinExample18to ndtheincometaxpayable oneachoftheseincomes.
Hint:Lookcarefullyforthe correcttaxbracket
9 Melhasanetannualincomeof$53246after21% ofherincomeis withdrawnfortaxpurposes.Whatwashergrossincome?
10 Fredearns$50000eachyearasashopclerkandan extra$1200eachyearfordoinggardeningonsome weekends.
a Calculatehisannualgrosssalary.
b Usehisgrosssalaryto ndhisincometax,using thetaxtableinExample18.
c Whatishisannualnetincome,aftertax?
d Whatishisapproximatefortnightlynetincome, tothenearestcent,aftertax?
Hint:What % ofgrossincome does $53246 represent?
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2FTaxation 105 VIDEO DEMO
$10000 a $30000
$50000
$80000 d $129000 e $156000
$200000 g $500000 h $1000000 i PROBLEM-SOLVINGANDREASONING 9,10 9–12
b
c
f
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11 Williamearns$1600perweek.UsingthetaxtableinExample18,howmuchtaxisdeductedfromhis payeachweek?Assume52weeksinayear.
12 Johnpays$10000intaxayear.
a WhichtaxbracketdoesJohnfallinto?
b WorkbackwardsfromthisamounttoworkouthowmuchJohnearnsperyear.Answertothenearest dollar.
ENRICHMENT:Othertypesofdeductionsandtaxableincome
13–15
Theloweraperson’sincome,thelesstaxtheymustpay.Peoplethereforetrytolowertheirtaxableincome byclaimingallowabletaxdeductions.
Work-relatedexpensessuchasuniforms,stationeryandjob-relatedtravelexpensesareallexamplesof allowabletaxdeductions.Theincometaxisthereforecalculatedonwhatwecallaperson’staxableincome.
Taxableincome = grossincome allowabletaxdeductions
UsethetaxtablefromExample18tocomparethefollowingjobs.
13 Anyoungaccountantearns$3120afortnight,withnoallowabletaxdeductions.
a Whatistheaccountant’staxableincome?
b Howmuchtaxdoestheaccountantoweinayear?
c Whatistheaccountant’sannualnetpay?
d Whatistheaccountant’sfortnightlynetpay?
14 Ayounglawyerearns$3120afortnightwithallowable taxdeductionstotalling$2560ayear.
a Whatisthelawyer’staxableincome?
b Howmuchtaxdoesthelawyeroweinayear?
c Whatisthelawyer’sannualnetpay?
d Whatisthelawyer’sfortnightlynetpay?
15 Howmuchlessmoneydoesthelawyerinquestion 14 haveeachweekcomparedtotheaccountantin question 13?
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106 Chapter2Financialmathematics
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2G Simple interest
Learningintentionsforthissection:
• Tounderstandtheconceptofsimpleinterestandhowitiscalculated
• Tobeabletousethesimpleinterestformulatoperformcalculations
Past,presentandfuturelearning:
• ThissectioncontinuestheStage5CoreTopiccalled FinancialMathematicsA
• Thefollowingsection,2H,furtherextendstheseconcepts
• FinancialMathematics isamajortopicinStage6Standard
• YoumaybeexpectedtorecalltheformulafrommemoryintheHSCExamination
Interestischargedwhenapersonorinstitutionborrowsmoney. Theinterestisanextraamountthatmustbepaidback,ontop oftheborrowedamount.
Interestisalsoearned,whenapersonorinstitutioninvests money.
Simpleor atrateinterestisusuallychargedorearnedeachyear. Itiscalculatedonthefullamountborrowedorinvested atthebeginningoftheloan.
LessonStarter:Developingtherule
$5000isinvestedinabankand5% simpleinterestispaidtotheinvestoreveryyear.Inthetablebelow,the amountofinterestpaidisshownforYear1,andtheamountoftotalinterestisshownforYears1and2.
• Completethetable.
• Howmuchinterestwouldtheinvestorearnin10years?
KEYIDEAS
■ Simpleinterest isinterestcalculatedeachtimeperiodontheinitialamount.
■ Tocalculatesimpleinterest,weapplytheformula:
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2GSimpleinterest 107
Year Interestpaidthatyear Totalinterest 0 $0 $0 1 5 100 × $5000 = $250 1 × $250 = $250 2 2 × $250 = $500 3 4 5
I = Prn
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VIDEO DEMO
I istheamountofsimpleinterest(in$)
P isthe principal amount;themoneyborrowedorinvested(in$),alsoknownaspresentvalue r istheinterestrateperperiod,expressedasadecimal,notapercentage t isthenumberoftimeperiods,whichcouldbedays,months,yearsetc.
■ Thefuturevalue ($A) equalstheprincipalplusinterest
A = P + I
■ p.a.means‘perannum’or‘peryear’.
Keyterminology: simpleinterest,principal,perannum
Exercise2G
UNDERSTANDING
1 $12000isinvestedat6% p.a.for42months.
a
Whatistheprincipalamount?
b
Whatistheinterestrate?
c
Whatisthetimeperiodinyears?
2 Jannearns$560p.a.insimpleinterestonaninvestment.Howmuchwouldheearnonthe investmentin: 2years? a 5years? b 10years? c
3 Usetherule I = Prn to ndthesimpleinterest (I) earnedinthese nancialsituations.
a $10000,10% p.a.,3years
b $6000,12% p.a.,5years
c $5200,4% p.a.,2years
Example19
Usingthesimpleinterestformula
Calculatethesimpleinterestearnediftheprincipalis$1000,therateis5% p.a.andthetime is3years.
SOLUTION EXPLANATION
P = 1000, r = 0.05, n = 3
I = Prn = 1000 × 005 × 3 = 150
Listtheinformationgiven,notingthat 5%= 005.
Writetheformulaandsubstitutethegiven values.
Interest = $150 Answerthequestion.
108 Chapter2Financialmathematics
1–3 1,2
FLUENCY 4–6,8 4,6–8
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Nowyoutry
Calculatethesimpleinterestearnediftheprincipalis$2000,therateis4% p.aandthetimeis 5years.
4 Findthesimpleinterestearnedon:
a
$5000at6% p.a.for1year
b
$5000at6% p.a.for3years
c
$8000at4% p.a.for5years
d
$15000at3% p.a.for7years
e
$7250at55% p.a.for3years
5 Wallyinvests$15000atarateof6% p.a.for3years.Calculate thesimpleinterestandtheamountavailableattheendof3years.
Example20 Usingothertimeperiods
Hint: Amount = principal + interest VIDEO DEMO
Calculatethesimpleintereston$7000investedat61 4 % p.a.for18months.
SOLUTION EXPLANATION
P = 7000, r = 61 4 ÷ 100 = 00625Listtheinformation.
n = 18months = 18 12 or15yearsConvert18monthsintoyearsbydividingby12.
I = Prn
I = 7000 × 0.0625 × 1.5 = 65625
Interest = $65625
Nowyoutry
Writetheformula. Substituteinthevaluesandevaluate.
Calculatethesimpleintereston$4500investedat51 2 % for30months.
6 Calculatethesimpleinterestearnedon: Hint: 365 days = 1 year
a $500at7% p.a.for18months
b $1000at5% p.a.for24months
c $2000at4% p.a.for6months
d $4700at41 2 % p.a.for15months(Roundtothenearestcent.)
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e $50000at3.75% p.a.for200days(Roundtothenearestcent.)
2GSimpleinterest 109
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Example21 Calculatingthefuturevalue
AllanandRachelplantoinvestsomemoneyfortheirchildKaylan.Theyinvest$4000for 30monthsinabankthatpays45% p.a.Calculatethesimpleinterestandthefuturevalueat theendofthe30months.
Interest = $450
Writetheformula,substituteandevaluate.
Totalamount = $4000 + $450 = $4450Totalamount = principal + interest
Nowyoutry
Joywinssomemoneywhichshedecidestoinvest.Sheinveststhe$5000for36monthsinabank accountthatpays38% p.a.Calculatethesimpleinterestandtheamountavailableattheendofthe 36months.
7 Annieinvests$22000atarateof4% p.a.for27months.Calculatethesimpleinterestandtheamount availableattheendof27months.
8 Copyandcompletethetable.
9 A nancecompanycharges14% p.a.simpleinterest.IfLynborrows$2000toberepaidover2years, calculatehertotalrepayment.
10 Markusborrows$20000tobuyacar.Heischargedsimpleinterestat18% p.a.foraperiodof5years.
a HowmuchinterestisMarkuschargedeachyear?
b CalculatethetotalinterestMarkuswillpayonthisloan.
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c WhatisthetotalamountthatMarkuswillhavepaidattheendoftheloanperiod?
110 Chapter2Financialmathematics VIDEO DEMO
SOLUTION EXPLANATION P = 4000, r = 45 100 = 0.045, n = 30 12 = 2.5 n iswritteninyearssinceinterestrateisper annum.
= Prn = 4000 × 0.045 × 2.5 = 450
I
Annualinterest, Principal rate Timeperiod Interest Futurevalue a $7000 3% 4 years b $1500 7% 8 years c $40000 2 5% 18 months d $70000 3 1 4 % 2 years e $2000 4% 30 months PROBLEM-SOLVINGANDREASONING 9,10 10,11
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11 Wendywins$5000duringachesstournament.Shewishestoinvestherwinnings,andhasthetwo choicesgivenbelow.Whichonegivesherthegreatertotalattheendofthetime?
Choice1:85% p.a.simpleinterestfor4years
Choice2:8% p.a.simpleinterestfor54months
12,13
12 Thetableshowstheamountofsimpleinterestpayable(in$,tothenearest$)onloansatacertain interestrate.
Usethetableto ndtheinterestpayableonthefollowingloans.
$5000for1year a
$500for3years b
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$100for10years c $150for1year d $85500for5years e $9550for10years f $5000for4years g $50000for9years h $100000for10years i
2GSimpleinterest 111
ENRICHMENT:Simpleinteresttablesandgraphs
Amountof loan 1year 2years 3years 5years 10years 50 6 13 19 32 65 100 13 26 39 65 129 500 65 129 194 323 645 1000 129 258 387 645 1290 5000 645 1290 1935 3225 6450 10000 1290 2580 3870 6450 12900 50000 6450 12900 19350 32250 64500 100000 12900 25800 38700 64500 129000
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13 Thegraphontherightshowstheannual interestearnedoninvestmentsforinterest ratesof4% p.a.and6% p.a.Usethegraph toanswerthefollowing.
a Findtheannualinterestearnedonan investmentof:
$300at4% p.a. i
$520at6% p.a. ii
$250at4% p.a. iii
b Whatinvestmentswouldearnannual interestof(tothenearest$5): $20at6% p.a.? i $20at4% p.a.? ii
$14at6% p.a.? iii
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112 Chapter2Financialmathematics
100200300400500 0 0 6% 4% 600 Principal ($) Interest ( $ ) Annual interest earned at 4% and 6% 5 10 15 20 25 30 35 40
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2H Applications ofsimpleinterest
Learningintentionsforthissection:
• Tobeabletousethesimpleinterestformulatosolveproblemsaboutinvestmentsandloans
• Tobeabletocalculaterepaymentamountsgiventhetermsofaloan
Past,presentandfuturelearning:
• ThissectioncontinuestheStage5CoreTopiccalled FinancialMathematicsA
• Section1FofourYear10bookrevisesandfurtherextendstheseconcepts
• FinancialMathematics isamajortopicinStage6Standard
• YoumaybeexpectedtorecalltheformulafrommemoryintheHSCExamination
Financialcalculationsareacriticalcomponentof thethinkingbehindthedecisionspeoplemakeas towheretoborroworinvestmoney.
LessonStarter:WheredoIinvest?
BankA:$4000at5% p.a.for8years
BankB:$5000at8% p.a.for4years
BankC:$8000at4% p.a.for5years
Calculatethesimpleinterestearnedoneachinvestmentoption.Whatdoyounotice?
Whichbankwouldyouchooseandwhy?
KEYIDEAS
■ Thesimpleinterestformula, I = Prn,containsfourvariables:
I isamountofinterest
P isprincipal(i.e.amountinvested)
r isinterestrate,expressedasadecimal
n isnumberoftimeperiods.
Ifvaluesaregivenforthreeofthese,thentheycanbeusedto ndthefourth.
■ Torepayaloan,youmustrepaytheamountborrowed(theprincipal)andtheinterest.
■ Repayments aretheamountofmoney,usuallythesameamounteachtimeperiod,requiredto repayaloan.
Keyterminology: simpleinterest,principal,repayment
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2HApplicationsofsimpleinterest 113
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Exercise2H
UNDERSTANDING
1 IfPhilearns$100insimpleinterestinoneyear,howlongwouldittakehimtoearn: $200 a $400 b $5000 c $250 d
2 Howmanyyearsdoesittakefor$100toearn$50 ininterestifthesimpleinterestrateis10% p.a.?
Hint:Howmuchinterestisearned on $100 at 10% p.a.for 1 year?
3 Jodierepays$200amonthfor12monthstopaybackherloanandinterest.Howmuchdoessherepay?
4 Aloanof$4000hasinterestof$500addedtoit.Calculatethesizeofeachofthe10repaymentsneeded torepaytheloan.
Example22 Determiningtheinvestmentperiod
Remyinvests$2500at8% p.a.simpleinterest,foraperiodoftime,toproduce$50interest. Forhowlongdidsheinvestthemoney?
Time = 0.25years = 025 × 12months = 3months
Nowyoutry
Solvetheremainingequationfor t bydividing bothsidesby200.
Convertdecimaltimetomonthswhere appropriate.
Jasinvests$3000at6% p.a.simpleinterest,foraperiodoftime,toproduce$450interest.Forhow longdidheinvestthemoney?
5 Alviinvests$5000at8% p.a.simpleinterestandwantstoearn$1200ininterest.Forhowmanyyears shouldAlviinvesthismoney?
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6 Samearns$288interestonhis$1600investment.Iftheinterestwascalculatedat4% p.a.,howmany yearsdidSaminvestthemoneyfor?
7 $8000earns$600interestat5% p.a.overhowmanymonths?
114 Chapter2Financialmathematics
1–4
4
FLUENCY 5,6,8,9 5,7,8,9 VIDEO DEMO
SOLUTION EXPLANATION I = 50, P = 2500, r = 8 100 = 0.08 Listtheinformation. I = Prn Writetheformula. 50 = 2500 × 008 × n 50 = 200 × n 200n = 50 n = 025
Substitutetheknowninformationand simplify.
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VIDEO DEMO
Example23 Determiningtheinterestrate
BankEastadvertises$450interestayearonaninvestmentof$7500.Calculatethesimpleinterest rateforthisinvestment.
SOLUTION
I = 450, P = 7500, n = 1, r =?
I = Prn
450 = 7500 × r × 1
450 = 7500r 7500r = 450
r = 450 7500
r = 006
EXPLANATION
Listtheinformation.
Writetheformulaandsubstitute I = 450, P = 7500and t = 1.
Simplifyandsolvefor r.
VIDEO DEMO
Interestrateis6% p.a. Write r asapercentage.
Nowyoutry
OzLoansoffers$610interestayearonaninvestmentof$12200.Calculatethesimpleinterestrate forthisinvestment.
8 Findtheannualsimpleinterestrateneededforeachofthe
Hint:Setuptheformula I = Prn where r istheunknown. followingsituations.
$4000earns$500in2years
$18000earns$3510in3years c $950earns$470.25in9years
$3000earns$945in18months
Example24 Calculatingrepayments
$2500earns$393.75in4.5years
‘Deals4You’offersaloanof$24000at16% p.a.simpleinterestiftheloanisrepaidinequal monthlyrepaymentsover5years.
Howmuchinterestischargedontheloan? a
Whatisthetotalamountoftheloanandtheinterest? b Calculatethesizeofeachrepayment. c
SOLUTION EXPLANATION
a I = Prn
P = 24000, r = 0.16, n = 5
I = 24000 × 0.16 × 5
= $19200
b Total = $24000 + $19200
= $43200
Writedownthesimpleinterestformulaand listtheinformation.
Substituteinthevaluesandevaluate.
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Total = principal + interest
2HApplicationsofsimpleinterest 115
b
a $500earns$120in12years
d
e
f
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c Repayments = 43200 60 = 720 Repaymentscometo$720 permonth.
Dividethetotalamountbythenumberofmonthsin 5years (5 × 12 = 60).
Nowyoutry
Aloanof$35000isrepaidinequalmonthlyinstalmentsover6yearsat8% p.a.simpleinterest.
Howmuchinterestischargedontheloan? a Whatisthetotalamountoftheloanandtheinterest?
9 Copyandcompletethistableand ndthemonthlyrepaymentforeachloan.
10 Calculatetheprincipalamountwhichearns$500simpleinterest over3yearsatarateof8% p.a.Roundtothenearestcent. Hint:Substituteinto I = Prn
11 Charlotteborrows$9000tobuyasecond-handcar.Theloanmustberepaidover5yearsat12% p.a. simpleinterest.Calculate:
a thetotalamounttoberepaid
b themonthlyrepaymentamountiftherepaymentsarespreadequallyoverthe5years
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116 Chapter2Financialmathematics
c
b Calculatethesizeofeachrepayment.
Amount Annualsimple Numberof Totalamount Monthly borrowed interestrate years Interest toberepaid repayment $5000 21% 5 $14000 15% 5 $10000 6% 4 $55000 8% 10 $250000 7% 30 PROBLEM-SOLVINGANDREASONING 10,11 10,12,13
andsolvefor P
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12 If$5000growsto$11000in12years, ndthesimpleinterestrate. 13 Aninvestorinvests$P andwantstodoublethis amountofmoney.
a Howmuchinterestmustbeearnedtodouble thisinitialamount?
b Whatsimpleinterestrateisrequiredtodouble theinitialamountin8years?
c Ifthesimpleinterestrateis5% p.a.: howmanyyearswillittaketodoublethe investment? i
howmanyyearswillittaketotriplethe investmentamount? ii
howdotheinvestmentperiodsinparts i and ii compare? iii
14,15
Withsimpleinterest,theprincipalandtheinterestearnedeachyearremainthesamefortheperiodof theinvestment.
However,withcompoundinterest,eachtimetheinterestiscalculateditisaddedtotheprincipaltogivea newvalue.Thismeansthatthenexttimetheinterestiscalculated,itisdonesousingalargeramount.
Inthefollowingquestionsyouwillbeaskedtodorepeatedapplicationsofsimpleinterestto ndthe nal compoundedamount.
14 $500isinvestedfor4yearsat10% p.a.interestcompoundedannually.
a Completethetableto ndthe nalvalueoftheinvestmentattheendofthistimeandthetotal.
b Howmuchinterestdidtheinvestmentearnoverthe4years?
15 a Completethefollowingtableto ndthe nalvalueofaninvestmentof$4500compoundedat5% p.a.annuallyfor5years.
b Type4500 × 1.055 intoyourcalculator.Whatdoyounoticeaboutthisanswer?
c Canyouexplainhowtheanswerstopart a andpart b relate?
2HApplicationsofsimpleinterest 117
ENRICHMENT:Introducingcompoundinterest
Time(years) Amount(A) Interest(I ) Newamount(A + I ) 1 500 500 × 0 1 = 50 500 + 50 = 550 2 550 550 × 0 1 = 3 4
Time(years) Amount(A) Interest(I ) Newamount(A + I ) 1 4500 225 4725 2 4725 3 4 5
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Facebookcake-decoratingbusiness
Moreandmoreindividualsaresettingup abusinessusingFacebook.Forexample,a successfulcake-decoratingbusinesscanberun fromhomewhilelookingafterthekidsorwhile workingnormalbusinesshoursatotherjobs.
Aswithanybusiness,anunderstandingof nancialmathematicsisimportanttothesuccess ofthebusiness.Skillssuchascalculating costsandpro ts,percentagesandtaxationare importantforanymanager.
1 Calculatethetotalcostofbuyingeachofthefollowingcaketinsets: Roundset:6,8,9and12inchesby3inchesdeepat$64. Roundset:6,8,10and12inchesby4inchesdeepat$99.
Squareset:6,8and10inchesby3inchesdeepat$45.
Squareset:6,8,10and12inchesby4inchesdeepat$86.
2 Convertthefollowingmeasurementsfrominches(USstandard)towholenumberofcentimetresby usingthefollowingconversionrate:1inch = 254cm. 3inches a 4inches b 9inches c 10inches d
3 Imaginethatyouspend2hoursoutof91
thisasapercentage,roundedtothenearestwholepercentage.
4 Fondanticingcomesindifferentcoloursandindifferent-sizedtubes.Managingyourbudgetmeans lookingforthebestbuy.
a Whichofthefollowingrepresentsthebestbuyforeachcolouroffondantlistedbelow?
b Whatistheaveragecostper100gramsforwhitefondanticing?
c Underwhatcircumstanceswouldsomeonebuyasizethatwasnotthebestbuy?
5 Acustomerhasthefollowingquotesforalarge21stbirthdaycakefromfourdifferentFacebookcake suppliers:$195,$290,$225and$215.
Foreachofthefollowing,stateanswerstothenearestwholenumber.
a Whatisthemeanoraveragecostforthistypeofcake?
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b Whatisthepercentagechangefromthelowestquotetothehighestquote?
118 Chapter2Financialmathematics Maths@Work
2workhoursontheinternetpromotingyourbusiness.Write
Whitefondant Redfondant Bluefondant 100 gat $3.25 500 gat $5 50 5 kgat $40 1 kgat $10 50 2.5 kgat $36.95 1 kgat $19 90 100 gat $3 25 500 gat $7 95 100 gat $2.95 500 gat $7 95 750 gat $11 95 1 kgat $19 90
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c Ifitcostseachsupplier$50inproducttomakethecake,calculatethepercentagepro tforeach ofthefourquotesgivenabove.
d Ifittakeseachsupplieronaverage512hourstomakeanddecoratethecake,whatiseachperson chargingperhour,excludingthe$50productcosts?
Usingtechnology
6 ImaginethatyouhavestartedaFacebookcake-making business.Toanalysepossiblepro ts,setupanExcel spreadsheetasshownbelowandenterformulasin theshadedcells.Notethatthisisasimpli edanalysis andexcludespower,gasandequipmentcosts.
Hint:Formatall $ cellsas‘currency’ withzerodecimalplaces. Formatprofit % cellsas‘percentage’ withzerodecimalplaces.
a Howmuchpro twouldbemadefromselling2Valentines’Daycakesand3chocJaffa birthdaycakes?
b List,inascendingorder,the3cakesthatbringthelowestpercentagepro t.
c List,indescendingorder,the3cakesthatpaythehighesthourlyrates.
d Suggestareasonwhythecakesin c costthecustomermoreinhourlyrates.
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Exploreand connect
Chooseand apply techniques
Savingforaholiday
Sallyisanaccountsclerkearningasalaryof$70348.Sheplanstosave20%ofherfortnightlyaftertax takehomepay,foranyextraexpensesandtoputtowardsaholiday.Infact,Sallyistryingtosavefor anoverseasholidayin12monthsandsheestimatesshewillneed$12000forherplanneditinerary.
Toreachhergoalshedecidestogetasecondjobworkingatherlocallatenightsupermarket.Theyoffer herthree2½hourshiftsaweekpaying$25anhour.Sheplanstoworkhereforatotalof20weeks, thinkingthatthiswillgiveherenoughforhertrip.
Presentareportforthefollowingtasksandensurethatyoushowclearmathematicalworkingsand explanationswhereappropriate.Giveanswerstothenearestcent.
Routineproblems
a Atherjobatthesupermarket,howmuchdoesSallyearnpershift?
b Howmuchwouldsheearnperweekatthesupermarket?
c Overthe20weeks,calculateSally’stotalgrossincomefromthesupermarket.
d Asanaccountsclerk,Sallyearnsasalary.WhatisthegrossfortnightlyincomeSallyearnsfrom thisjob?
e Whatishertotalannualtaxableincomefrombothjobs,assumingsheonlyworksforthe 20weeksatthesupermarket?
Non-routineproblems
a TheproblemistodetermineifSallyhassavedenoughtoaffordtogoonhertripaftertax. Writedownallrelevantinformationthatwillhelpyousolvethisproblem.
b The rmthatSallyworksforasanaccountsclerktakeoutincometaxfromherfortnightly payonherbehalf,theydothisbyusinghersalaryand ndingthetaxowedbyusingthe currentpersonalincometaxtablefoundontheATOwebsite.
Byusingthepersonalincometaxtablebelow,calculatetheannualtaxthatthe rmdeducts fromSally’spay.
Theaboverates donot includetheMedicarelevyof2%.
c Calculatethefortnightlynetpay(takehomepay)thatSallyreceivesfromher rm.
d Sallyaimstosave20%ofhertakehomepayfromher rmeachfortnight.Howmuchwill Sallyhavesavedattheendof12months?
e Byconsideringhertotaltaxableincomefrombothjobs,calculatehertotalincometax liabilityfortheyear.
f Hence,calculateSally’sannualnetincomefromhersecondjob.
g Calculatehertotaltaxliability,byincludingtheMedicarelevyof2%.
120 Chapter2Financialmathematics Workingmathematically
Taxableincome Taxonthisincome $0 $18200 Nil $18201 $45000 19 centsforeach $1 over $18200 $45001 $120000 $5092 plus 32.5 centsforeach $1 over $45000 $120001 $180000 $29467 plus 37 centsforeach $1 over $120000 $180001 andover $51667 plus 45 centsforeach $1
over $180000
h WhenSallytakesinaccountallher nancesandtax,willshehaveenoughleftfromhersavings totakeherproposedtrip?
i Summariseyour ndingsandmakearecommendationaboutSally’ssavingsandproposedtrip attheendoftheyear.
Extensionproblems
a Whatpercentagepayriseisneededfromher rmforSallytonolongerneedhersecondjob?
b InvestigatewhathappenstoSally’staxliabilityfromhersalaryfromher rmifshehasallowable taxdeductionsof:
Problemsolve
Workingmathematically 121 Communicate thinking andreasoning
$2000
$4000 ii
i
1 Theanswerstothecluesarehiddenintheword nd.Canyou ndall16words?
2 Findoutthefourclassicalelementsoftheworldbyansweringthefollowingsimpleinterestproblems. Matchtheletterbesideeachquestiontoitscorrespondinganswerineachgrid.
Puzzlesandchallenges 122 Chapter2Financialmathematics
S A G R O S S W H K L O M A P E R C E N T A G E C O L A R C O M M I S S I O N A S E R V I N T E R E S T R T Y H E T S L O S S T H Y E F O R T N I G H T I L W S M O T A X A T I O N Y E I O L I D I D N P Y I E K M N Q M O N T R N I S C E P Y U E D I S C O U N T M L R E P A Y M E N T A H D E D U C T I O N S I X L A xedannualincome a Apercentageofthevalueofgoodssold, whichyouearnasanincome b Workinglongerthannormalworking hours c Moneyfromyourincomegiventothe government d Twoweeks e Thetotalofallincome f Moneytakenfromtotalpay g Yearly h 12timesayear i Flat-rateinterest j Meaning‘outof100’ k Moneygiventorepayaloan l Moneyearnedonaninvestment m Anitemofferedforasalepricehashadthis happen n Theoriginalpriceofanitem o Youincurthiswhenyousellanitemforless thanyoupaidforit p
E = $600 at 6% p.a.for 1 year H = $796 at 5% p.a.for 4 months R = $12500 at 6 1 4 % p.a.for 2 years A = $7000 at 5% p.a.for 3 years I = $1000 at 1% p.a.for 100 years F = $576 50 at 19% p.a.for 18 months W = $36000 at 2% p.a.for 5 years T = $550 at 10% p.a.for 6 months $36 $1050 $1562 50 $27 50 $13 27 $1050 $1000 $1562 50 $164.30 $1000 $1562.50 $36 $3600 $1050 $27 50 $36 $1562 50 UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
Percentage composition
Percentages and decimals
Percentage Decimal
÷ 100 PercentageFraction × 100 ÷ 100 49% = 0.49
30 100 17 Finding a percentage 17% of 30
Percentage increase and decrease Increase 20 by 6% 20 × 1.06 = 21.2
Decrease 20 by 5% 20 × 0.95 = 19
× 100 100 19 19% = 3 g out of 50 g = × 100 50 3 = 6% Percentage Means ‘out of 100’ 7% = = 0.07 100 7
= 5.1 =×
=
Financial mathematics
Applications of percentages
Percentage pro t/loss pro t or loss
= cost price × 100 Mark-up and discount
Commission/tax
Simple interest
p = amount borrowed or invested
r = interest rate (as a fraction or decimal)
n = number of time periods
If $2000 is invested at 5% p.a. for 4 years:
I = Prn = $2000 × 0.06 × 4 = $400
Income and tax
Employees can be paid: wage: hourly rate with overtime at time and a half = 1.5 × hourly rate double time = 2 × hourly rate salary: annual amount commission: % of sales
net income =
gross income deductions tax: money taken from pay and given to the government
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Chaptersummary Chaptersummary 123
Percentage change = original change × 100
Percentages and fractions
0.17 × 30
I =Prn
Where I = amount of interest
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Chapterchecklistwithsuccesscriteria
AprintableversionofthischecklistcanbedownloadedfromtheInteractive Textbook.
2A 1Icanconvertbetweenpercentagesandfractions.
e.g.Write
a 7 40 asapercentage b 17 5% asafraction
2Icanconvertbetweenpercentagesanddecimals.
2A
e.g.Write
a 0 3 asapercentage b 72% asadecimal
2A 3Icanwriteaquantityasapercentage.
e.g.Write 27 cmoutof 1 8 masapercentage.
2B 4Icanfindapercentageofaquantity.
e.g.Find 35% of $75
2B 5Icanfindtheoriginalamountfromapercentage.
e.g.Determinetheoriginalamountif 12% oftheamountis $72
2C 6Icanincreaseanddecreasebyapercentage.
e.g. a Increase $80 by 12% b Decrease $40 by 8%
2C 7Icanfindapercentagechange.
e.g.Thepriceofagymmembershipincreasedfrom $320 to $370.Findthepercentageincrease correcttoonedecimalplace.
2C 8Icanfindtheoriginalamountafteranincreaseordecrease.
e.g.Adecreaseof 22% reducedthepopulationofatownto 1014.Whatwastheoriginalpopulation ofthetown?
2D 9Icancalculatethesellingpricefromamark-upordiscount.
e.g.Astoremarksupallwhitegoodsby 20%.Ifthecostpriceofafridgeis $1100,whatwillbeits sellingprice?
2D 10Icandeterminepercentageprofit.
e.g.Astallholdermakescandlesfor $8 andsellsthemfor $13.Findtheprofitandexpressthis profitasapercentageofthecostprice.
2D 11Icancalculatetheoriginalpricebeforediscount.
e.g.AdepartmentstorediscountsallChristmastreesby 15%.Ifthesalepriceofatreewas $106.25,whatwastheoriginalprice?
2E 12Icancomparewagesandsalaries.
e.g.Tonyhasanannualsalaryof $88000 andJodieearns $72 perhour.Calculate
a Tony’shourlyrateofpayifheworksa 38-hourweek
b Jodie’syearlyincomeifsheworksonaverage 22 hoursperweek.
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Chapterchecklistwithsuccesscriteria
AprintableversionofthischecklistcanbedownloadedfromtheInteractive Textbook.
2E 13Icancalculateovertime.
e.g.CalculateJulian’sweeklywageforaweekwhereheworks 12 hoursathisnormalhourlyrate of $28, 4 hoursattimeandahalfand 3 hoursatdoubletime.
2E 14Icancalculatecommission.
e.g.Asalespersonispaidaretainerof $2200 permonthandreceivesan 8% commissiononsales. Ifonemonthhemakessalesworth $10400,calculatehisearningsforthatmonth.
2F 15Icancalculatenetincome.
e.g.Daniellehasanannualsalaryof $64800.Shehasmonthlytaxdeductionsonherpayslipof $1296.Calculatehernetincomeeachmonth.
2F 16Icancalculatenetincomeusingincometaxrates.
e.g.Annahasanannualsalaryof $78000 andataxationrateof 26% withthefirst $18200 tax free.CalculateAnna’snetincomefortheyear.
2F 17Icanusethetaxtabletocalculateincometax.
e.g.UsethetaxtablefromExample 18 tofindtheincometaxforanincomeof $90000.
2G 18Icanusethesimpleinterestformulaandfindthefuturevalue.
e.g.Calculatethesimpleinterestearnediftheprincipalis $4000,therateis 3% p.a.andthetimeis 4 years.Hence,whatistheamountattheendofthe 4 years?
2G 19Icanworkwithsimpleinterestusingothertimeperiods.
e.g.Calculatethesimpleintereston $6000 investedat 4 1 2 % p.a.for42months.
2H 20Icandeterminetheinvestmentperiodorinterestrateforsimpleinterest. e.g.Joshuainvests $3500 at 6% p.a.simpleinterest,foraperiodoftime,toproduce $315 interest. Forhowlongdidheinvestthemoney?
2H 21Icancalculaterepayments.
e.g.Sydtakesoutaloantopurchaseayacht.Theloanisfor $32000 at 8% p.a.simpleinterestif theloanisrepaidinequalmonthlyinstalmentsover 4 years. Calculatetheinterestchargedontheloanandhencethetotalamountowingandthesizeofthe requiredmonthlyrepayments.
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Chapterchecklist Chapterchecklist 125
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Short-answerquestions
1 2A Copyandcompletethetableshown.
2 2B Find:
a 25% of$310
b 110% of15
3 2B Determinetheoriginalamountif:
a 20% oftheamountis30.
b 72% oftheamountis18.
4 2C
a Increase45by60%
b Decrease1.8by35%.
c Findthepercentagechangeif$150isreducedby$30.
5 2C Themassofacatincreasedby12% to14kgovera12monthperiod.Whatwasits previousmass?
6 2D Determinethediscountgivenona$15000carifitisdiscountedby12%
7 2D Acouchatacostpriceof$3500istobe markedupby25%.Findthesellingprice.
8 2D Thecostpriceofanarticleis$150.Ifitissoldfor$175:
a determinethepro tmade
b expressthepro tasapercentageofthecostprice.
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9 2E Determinethehourlyrateofpayforeachofthefollowingcases:
a apersonwithanannualsalaryof$76076workinga38-hourweek
b apersonwhoearns$429working18hoursatthehourlyrateand8hoursattime andahalf.
Chapterreview 126 Chapter2Financialmathematics
Decimal Fraction Percentage 0.6 1 3 3 1 4 % 3 4 1 2 200%
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10 2F
Jo’smonthlyincomeis$5270however20% ofthisispaidstraighttothegovernmentintaxes. WhatisJo’snetyearlyincome?
11 2G Findthesimpleinterestearnedon$1500at7% p.a.for5years.
12 2H Billinvests$6000at4% simpleinterest,foraperiodoftime,toproduce$720.Forhowlong didheinvestthemoney?
13 2F
Usethetaxtablebelowto ndtheincometaxpayableonanincomeof$78000. Taxableincome Taxonthisincome
$0 $18200 Nil
$18201 $45000 19 centsforeach $1 over $18200
$45001 $120000 $5092 plus 32.5 centsforeach $1 over $45000
$120001 $180000 $29467 plus 37 centsforeach $1 over $120000
$180001 andover $51667 plus 45 centsforeach $1 over $180000
Theaboverates donot includetheMedicarelevyof2%.
Multiple-choicequestions
Chapterreview Chapterreview 127
1 2A 2.8% asadecimalis: 2.8 A 0.28 B 0.028 C 0.0028 D 280 E 2 2A Whatpercentageof$2is50cents? 4% A 40% B 25% C 21 2 % D 400% E 3 2A 121 2 % asasimplefractionis: 12 100 A 1 8 B 3 25 C 0.125 D 12.5 E 4 2B 331 3 % of$660isthesameas: $660 ÷ 2 A $660 × 0.3 B $660 × 0.03 C $660 ÷ 3 D $660 ÷ 1 3 E 5 2B 15% of$1600isequalto: 24 A 150 B $240 C $24 D 240 E 6 2B If110% ofanumberis528,thenthenumberis: 4752 A 528 B 480 C 5808 D 475 E 7 2C 9670increasedby12% becomes: 9682 A 9658 B 108304 C 11604 D 85096 E UNCORRECTEDSAMPLEPAGES Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
8 2E Janeispaidawageof$27.80perhour.Sheworks12hoursatthisrateduringaweek,plus 4hoursonapublicholidaywhereshegetspaidattimeandhalf.Herearningsinthe weekare:
80
9 2E Simonearnsaweeklyretainerof$370and12% commissionofanysaleshemakes.Ifhe makes$2700worthofsalesinaparticularweek,hewillearn:
10
$1200isinvestedwithasimpleinterestrateof10% fortwoyears.Thefuturevalueattheend ofthetwoyearsis:
Extended-responsequestions
1 Paulinebuysadebutantedressatcost pricefromherfriendTila.Paulinepaid $420forthedresswhichisnormally markedupby55%
a Howmuchdidshesave?
b Whatisthenormalsellingpriceof thedress?
c IfTilagetsacommissionof15%: howmuchcommissiondidshe get? i
howmuchcommissiondidTilalosebysellingthedressatcostpriceratherthanthe normalsellingprice?
2 Adamstartsanewjobandworksa38-hourweekforawageof$97584.
a Calculatehishourlyrateofpay.
b Ifovertimeiscalculatedattimeandahalf,whatisAdam’sovertimerate?
c HowmuchdoesAdamearnfor4hoursofovertimework?
d HowmanyhoursofovertimedidAdamworkinaweekifhiswageforthatweekwas $1226.22?
e IfAdamusuallyworkstheamountofovertimeinpart d inthe52weeksoftheyearhe works,andhepays27% ofhispayintax,whatishisnetannualincome?
f IfAdaminvests10% ofhisnetincomeinanaccountearning8% p.a.simpleinterestfor 18months,howmuchextraincomewillhehaveearned?
UNCORRECTEDSAMPLEPAGES
Chapterreview 128 Chapter2Financialmathematics
$500.40 A $444
B $556 C $667.20 D $278 E
.
$595 A $652 B $694 C $738.40 D $649.60 E
$252 A $1452 B $1450 C $240 D $1440 E
2G/2H
ii
Uncorrected 1st sample pages • Cambridge University Press and Assessment © Palmer, et al 2023 • 978-1-009-40926-1 • Ph 03 8671 1400
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