CambridgeMATHS 12 Std 1 12

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Abouttheauthors

GregPowers iscurrentlytheHeadofMathematicsatCabramattaHighSchoolandthecoordinatorof theMathematicsHeadTeacherWesternNetwork.Heisanexperiencedclassroomteacher,havingtaught forover30yearsinarangeofdifferentschools.GreghasbeenaseniormarkerfortheHSC,educational consultantfortheMetropolitanSouthWestRegionandpresentedatnumerousMANSWinservices.He hasalsoenjoyedseveralcurriculumroleswiththeDepartmentofEducationandTraining.Gregisan experiencedauthorwhohaswrittennumeroustextsonmathematicsandtechnology.

StuartPalmer wasbornandeducatedinNSW.Heisafullyqualifiedhighschoolmathematicsteacher withmorethan25years’experienceteachingstudentsfromallwalksoflifeinavarietyofschools.Hehas beenHeadofMathematicsintwoschools.Heisverywellknownbyteachersthroughoutthestateforthe professionallearningworkshopshedelivers.StuartalsoassiststhousandsofYear12studentseveryyear astheypreparefortheirHSCExaminations.AttheUniversityofSydney,Stuartspentmorethanadecade runningtutorialsforpre-servicemathematicsteachers.

ChristinaBarbara hasbeenadedicatedhighschoolMathematicsteacherforover20yearsandiscurrently teachingataCatholichighschoolintheDioceseofParramatta.Shehastaughtawiderangeofstudents, fromthoserequiringnumeracysupportinYear7toacceleratedclassesandExtension2Mathematicsatthe HSClevel.ShealsopreparedstudentsfortheHSC,servingasanHSCStudySessionFacilitatoratWestern SydneyUniversity.Beforetransitioningtoteaching,ChristinaworkedasaBuildingEstimatorandQuantity SurveyorinSydney’sconstructionindustry.

DarrenThomas hasworkedinnumeroushighschoolsintheACTandNSWforover20years.He previouslyownedabusinessspecialisinginsupportingstudentswithlearningdifficultiesfornearly15years. HeiscurrentlytheHead ofMathematicsatahighschoolinnorth-westSydney.Heispassionateabout mentoringstaff andseeingthemgrowanddevelop.Sincereturningtotheclassroomin2014,hehasguided studentstobecomeindependentlearnerswhodrivetheirowneducation.

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Introductionandoverview

CambridgeMATHSStage6MathematicsStandard2Year12SecondEdition providescomprehensiveand alignedcoverageoftheNESAStandard2Year12syllabustobeimplementedfrom2027.Studentsusing thisresourcewillfinditanaturalextensionofworktheyhavecompletedaspartofStandardYear11.

Thesecondeditionincorporatesanumberofimportantchanges.Someofthese,suchasnewsections addedrelatingtoBuyNow,PayLaterloans(Chapter3)andGanttcharts(Chapter10),comedirectlyfrom topicsaddedtothenewStandard2syllabus.Others,suchastheadditionofmanymoreHSC-style questionsthroughout,inparticularmorechallengingones,reflectthisedition’salignmentwiththeway HSCStandard2hasbecomeamorechallengingcourseovertheyears.Atthesametime,studentsneeding tobrushuptheirStage5skillswillfindthiseditionmoreaccessiblethanever.

Thiseditionalsoincludesarangeofnewfeaturesandimprovements,manyofwhichhavecomeacrossfrom the CambridgeMATHSNSWYears7–10ThirdEdition series.Theyinclude:

• LearningIntentions atthestartofeachlesson,complementedbya checklistofSuccessCriteria at theendofeachchapter,allowingstudentstounderstandclearlywhattheyareexpectedtolearnandthen checktheirprogressastheymovethroughthechapter.

• Past,presentandfuturelearning,whichplaceseachlessoninthecontextofthesyllabusandhighlights topicsthatbuildonpreviouslearning.

• NowYouTryquestions afterworkedexamples,allowingstudentstoattemptasimilarquestion immediatelyafterseeingamodelsolutionwithstep-by-stepexplanations,beforetheybegintheexercise.

• Communicateyourreasoning,understandingandfluency questionsattheendofeveryexercise, givingstudentspracticeatcarefullyanalysingthecontextofaquestiontoextractthemathematicsand thencommunicatetheirthinking,asrequiredinexams.

• Carefulrefinementofallexercisesandchapterreviews,givingstudentsmoresupportatthebasicskills aswellas accesstoharderHSCexam-stylequestions.Questionsinchapterreviewshavenowbeen labelledwiththelessoninthechapterwheretheywereintroduced,allowingforeasierrevision.

• Workedexamples adjustedtomoredirectlyfollowthestyleusedinthe CambridgeMATHS7–10 series, withtheworkedsolutionintheleftcolumnandtheexplanationofeachstepofthesolutionprovidedon theright.

InteractiveTextbook(ITB)

TheInteractiveTextbook(ITB)isanonlineversionoftheprinttextbookpoweredbytheHOTmaths platform,andisincludedwiththeprintbookoravailableasaseparatedigital-onlypurchase.Updatedand revisedforthenewsyllabus,theInteractiveTextbookincludes:

• A StudyGuideforeachchapter –aconcisesummaryinPowerPointslidesthatcanbeusedforrevision andpreparationforassessment.

• Videodemonstrationsofallworkedexamples toencourageindependentlearning.

• Revisedquickquizzes containingauto-markedquestionsineverylesson,enablingstudentstoquickly checktheirunderstandingofasection.

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• A downloadable,printableversionoftheSuccessCriteriaChecklist foreachchapter.

• MathsLiteracyworksheets ineachchapter,providingactivitiestohelpstudentsbecomefluentwith mathematicalterminologyandbetterabletocommunicatetheirreasoning.

• Comprehensiveworkedsolutions forallquestionsareprovidedintheInteractiveTextbookasanoption thatteacherscanchoosetoenablefortheirstudents.

• RevisedInteractiveWidgets,basedonDesmostechnology,allowstudentstoexploreandvisualisekey conceptsinadynamicway.DesmostoolsarealsoembeddedintheInteractiveTextbookandcanbeused tocompletecalculations,createcustomgraphs,shapesandothervisualrepresentations.

• Spreadsheettemplates canbedownloadedtoassiststudentswithmakingastartonthemanyspreadsheet activitiesthroughoutthechapters.

TheOnlineTeachingSuite(OTS)

TheOnlineTeachingSuiteisautomaticallyenabledwithateacheraccountandisintegratedwiththe teacher’scopyoftheInteractiveTextbook.Alltheteacherresourcesareinoneplaceforeasyaccess,andall havebeeneditedandrevisedforthenewsyllabus.ThefeaturesoftheOTSinclude:

• Ateacher’sviewofastudent’sworkingandself-assessmentwhichenablesthemtomodifythestudent’s self-assessedmarks,andrespondwherestudentsflagthattheyhaddifficulty.

• Thetaskmanagerallowingteacherstodirectstudentsonacustomactivitysequencebasedontheirscores inmeasurableactivities.

• Quicklycreatecustomisedtestsfromabankofmultiple-choicequestionsusingthetestgenerator.Tests areauto-markedintheInteractiveTextbookorcanbeprintedandusedforhomeworkorassessment practice.

• Anexpandedandrevisedsuiteofchaptertestsandassessments.

• Modellingactivities thatcanbedownloadedtoallowstudentstocreateandcomparemathematical modelsandlearnaboutthemodellingprocess,andaselectionof Skillsheets fortopicslikeBuyNowPay LaterschemesandGanttchartstopromoteadeeperdiveintothesetopics.

• Editablecurriculumgridsandteachingprograms.

• Abrand-newexamgeneratortool,allowingthecreationofcustomisedprintableandonlinetrialexams.

Moreabouttheexamgeneratortool

Anewexamgeneratortool,availableatnoextrachargewithintheAssessmentareaoftheOnlineTeaching Suite,includescomprehensivebankofNESAexamquestions.AugmentedwithextraHSC-stylequestions writtenbyexperts,thetoolallowsteacherstocreatefullyformattedcustomtrialexams.Customexams canmodelend-of-yearexams,ortargetspecifictopicsortypesofquestionsthatstudentsmaybehaving difficultywith.

Featuresoftheexamgeneratorinclude:

• Filteringbyquestion-type,topicanddifficulty

• Searchablebykeywords

• Workedsolutionsforallquestions

• Multiple-choiceexamscanbeauto-markedifcompletedonline,withfilterablereports

• Allcustomexamscanbeprintedandcompletedunderexam-likeconditionsorusedasrevision.

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Acknowledgements

Theauthorandpublisherwishtothankthefollowingsourcesforpermissiontoreproducematerial:

© AustralianGovernment,DepartmentoftheEnvironmentandEnergy,1C(2); © NSWGovernment, Transport.nsw.gov.au,2C(1).

Everyefforthasbeenmadetotraceandacknowledgecopyright.Thepublisherapologisesforanyaccidental infringementandwelcomesinformationthatwouldredressthissituation.

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Tocome

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1 Ratiosandrates

HowfastarewemovinghereonEarth?

WhenwearestandingstillonEarth,weare actuallymovingveryfast,inthreedifferent ways.

AtthisspeedyoucouldtravelfromSydneyto Londoninabout 9.4 seconds.

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Ourplanetrotates 360 degreesevery 24 hours, soapersonstandingontheequatoris travellingatabout 1700 kmperhour. WearealsoorbitingaroundtheSunata speedofapproximately 30 kmpersecond.

Inaddition,theEarthandtheSunorbitthe centreofourgalaxy,theMilkyWay,ataspeed ofapproximately 200 kmpersecond.

Ourplanetalsomovesbecausetheuniverseis expanding,butthespeedofthismovementis veryinsignificantcomparedtothoseabove.

1A Calculationswithratios

1B Thecapture-recapturetechnique (EXTENDING)

1C Dividingaquantityinagivenratio

1D Scaledrawings

1E Buildingplans

1F Perimeter,areaandvolumefromplans

1G Simplifyingandconvertingrates

1H Usingtheunitarymethodtosolve problems

1I Usingratestomakecomparisons

1J Distance,timeandspeed

1K Distance–timegraphs

1L Fuelconsumptionrates

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•Interpretsratiosandratestosolve practicalproblems(MST-12-S1-05). ©NSWEducationStandardsAuthority

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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1A Calculationswithratios

Learningintentions

• Toconsolidatepriorlearningaboutthemeaning,useandapplicationsofratios

• Toreduceratiostosimplestform

• Toperformcalculationsandsolveproblemsinvolvingratios Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingratiosregularlyappearinHSCExaminations

Whatis‘aspectratio’?

Thesedays,weallspendalotoftimelookingatscreenson phones,televisionsandcomputers,whichcomeinahuge varietyofsizes.

Aspectratio isusedtodescribethelengthofthehorizontalside comparedtotheverticalside.Commonaspectratiosinclude 16:9,16:10,3:2and4:3.

Iftheaspectratiois16:9,then:

• 16and9havenocommonfactorsotherthan1,sothisratio isinsimplestform

• thescreenmightbe16cmwideand9cmhigh(16cmby9cm)

• wecouldmultiplybothnumbersby2andget32cmby18cm

• wecouldmultiplybothnumbersbyanynumber

• wecoulddividebothsidesbyanynumber

• wecoulddividebothnumbersby16andcallthisratio1: 9 16

• wecoulddividebothnumbersby9andcallthisratio 16 9 :1

• thesidesareintheproportionasshownhere,butthesquarescouldbelargerorsmaller.

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Ratio

Aratioisusedtocomparequantitieswiththesameunits,sotheunitsarenotwritten. Forexample,theratioof16cmto9cmiswrittenas16:9andpronounced“16to9”. Considerthediagrambelow.

Theratioof blacksquares to whitesquares is4:3.

Theratioof whitesquares to blacksquares is3:4.

Theratioof whitesquares to allsquares is3:7.

Ratiosarerelatedtofractionsinthefollowingway:

• theratioofblacksquarestoallsquaresis4:7

• asafraction,theblacksquaresmakeupfour-sevenths 4 7 ofthediagram.

Theratiosabovearein simplestform,becausethetwonumbershavenocommonfactorsotherthan1. Ratioswhicharenotinsimplestformcanbeexpressedinsimplestformbyusingdivisionor multiplication.Forexample:

Areyouready?Shadedversusunshaded,usingfractionsandratios

Whenashapeisdividedintoequalpartswithsomeshadedandsomenot,fractionsandratioscanbe usedtocompareshadedpartstounshadedpartsandshadedpartstoallparts.

DownloadtheactivityfromyourInteractiveTextbook.

Areyouready?Expressingratiosinsimplestform

Inthefollowingexercise,theanswersaremostlyprovidedinsimplestform.Thisactivitywill remindyouhowtoreducearatiotosimplestform,bydividingthenumbersintheratiobytheirhighest commonfactor.

DownloadtheactivityfromyourInteractiveTextbook.

Example1

Simplifyingaratio

Writetheseratiosinsimplestform.

3: 1 2 a 1.5:3.5 b 2 7 : 5 21 c

Solution

a 6 : 1 1 2 × 2 × 2 3 :

b 3 : 7 × 2 × 2 1.5 : 3.5

c 2 7 : 5 21 2 × 3 7 × 3 : 5 21 6 21 : 5 21 6:5

Nowyoutry

Writetheseratiosinsimplestform.

Note:Reduceallanswerstosimplestform.

1 Completeeachpairofequivalentratios. 1:3 = 4: a 1:7

Explanation

Toconvertonehalftoawholenumber,multiply by2.

Checkthatratioisinsimplestform.

Toconvertthesedecimalstowholenumbers, multiplyby2or10.

Checkthatratioisinsimplestform.

Ifthedenominatorswereequaltheycouldbe ignored,solookforLCMof7and21,whichis3. Multiplythenumeratoranddenominatorofthefirst fractionby3.

Ignorethedenominators.

Checkthatratioisinsimplestform.

2 Completeeachpairofequivalentratios.

3 Writethreeequivalentratiosforeachofthefollowingratios.

4 Expresseachratioinsimplestform.

5 Expresseachratioinsimplestform,withouttheunits.

$24:$18 a 3kgto12kg b 56t:16t c 40cto72c d 3hto24h e 2mmto20mm f

6 Thefollowingratioshavedifferentunitsoneachside.Expresseachratioinsimplestformafterfirst changingtothesameunits oneachside.(Forexample,for a,firstchangetheratioto‘40cto100c’ or‘0.4:1’).

40cto$1 a 3hto1day b 2mmto1cm c 1km:250m d 3m:50cm e 6km:300m f 7cm:21mm g 8months:4years h

2L:450mL i

7 Thereare14redblocksand10blackblocksinabag.Findtheratioof: redtoblack a blacktored b redtoallblocks. c

8 Aclothingstorehasadiscountsale. Adressmarkedat$250issoldfor$200. Whatistheratioofthediscounttothe markedprice?

9 MadeleineandNathaninvest$4500and $2500intoamanagedfund.Whatisthe ratioofMadeleine’ssharetoNathan’s share?

10 Therewere80blocksoflandavailable forsaleinanewlandrelease.Afterone month32blockshavebeensold.Whatis theratioofsoldblockstothetotalnumber ofblocks?

11 Writedown3ratioswhichareequivalentto2:3.

12 Writedown3ratioswhichareequivalentto16:10.

13 Asquarehasasidelengthof6cm.Ifthesidelengthisdoubled,whatistheratiooftheareaofthe originalsquaretothatofthelargersquare?

15 Findthemissingnumberintheseequivalentratios.

16 Expresseachratioinsimplestform.

17 Abagcontains10greenmarbles,14bluemarblesand6yellowmarbles.Whatistheratioofgreento yellowtobluemarblesinsimplestform?

18 Inanelection,Aidenscored250votes,Billiescored175votesandChelseascored125votes.Find theratioinsimplestformofAiden’svotes:Billie’svotes:Chelsea’svotes.

19 SamanthaandMathildeownarestaurant.Samanthagets 3 5 oftheprofitsandMathildereceivesthe remainder.

a Whatistheratioofprofits?

b Lastweektheprofitwas$2250.HowmuchdoesMathildereceive?

c Thisweektheprofitis$2900.HowmuchdoesSamanthareceive?

20 Expresseachofthefollowingratiosintheform x :1.

21 Expresseachofthefollowingratiosintheform1: y.

22 Dakarimakesablendofmixedlolliesusing5kgof jellybabies,4kgofliquoriceand1kgofskittles.Their pricesaregiveninthetable.Whatisthecostoftheblendper kilogram?

Communicateyourreasoning,understandingandfluency

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Lollyprices

Jellybabies

Liquorice

$5.95perkg

$6.95perkg

Skittles $11.90perkg

23 Arectangularscreenis160cmlongby120cmhigh.Animagewithanaspectratioof16:9is projectedontothescreen.Theimageisprojectedsothatitfillsuptheentirelengthofthetopofthe screen.Therewillbesomeemptyspaceatthebottomofthescreen. Whatarethedimensionsofthatrectangleofemptyspace?

1B Thecapture-recapturetechnique EXTENDING

Learningintentions

• Tounderstandhowthecapture-recapturetechniqueworks

• Tousethecapture-recapturetechniquetoestimateanunknownpopulation Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingthistechniqueregularlyappearinHSCExaminations

The‘capture-recapture’techniqueisanapplicationofratio.Itissometimesusedtoestimateanimal populations.Thismethodcanalsobeusedtoestimatethenumberofpeopleneedingparticularservices. Thetechniqueworksbyfirstcapturingarandomsampleofthepopulation.Thisinitialsampleistaggedand thenreleased.Atalatertimeasecondsampleiscaptured,andthenumberofrecapturedortaggedmembers arerecorded.

Thecapture-recapturetechnique

Thisdiagramexplainsthetechnique.Itisalsoausefultemplateforsolvingproblems.

Inthecapture:

Intherecapture:

Thetworatiosareassumedtobeequal.

Therefore,thethreeknownnumberscanbeusedtocalculatetheunknownnumber(i.e.thetotal populationtobeestimated).

Areyouready?Solvingequationswithfraction=fraction

Tousethecapture-recapturetechnique,youneedtobeabletosolveequationsoftheform

Thisactivitywillprepareyouforthat.

DownloadtheactivityfromyourInteractiveTextbook.

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Total population to be estimated : Tagged animals

Total captured : Tagged animals

Example2

Usingthecapture-recapturetechnique

Adamisabiologistwhoisestimatingthepopulationofcarpinalake.Herandomlycapturesandtags 120carp.Twomonthslaterhesamples80carpandfindshehasrecaptured5carphehadtagged. Estimatethenumberofcarpinthelake.

Solution

P :120

80:5

P 80 = 120 5

P 80 × 80 = 120 5 × 80

P = 1920

Thereareapproximately1920carpinthelake.

Nowyoutry

Explanation

Let P bethepopulationtobeestimated.120of thesecarpweretagged.

Intherecapture,80carpwerecaught,ofwhich5 weretagged.

Formanequationfromthetworatiosusing fractions.

Multiplybothsidesby80togivetheestimate.

Checkthattheanswerseemsreasonable:

1920:120 = 80:5

1920 ÷ 80 = 24 ✓ 120 ÷ 5 = 24 ✓

Emilyisabiologistwhoisestimatingthepopulationofcrabsinalake.Sherandomlycapturesandtags 64crabs.Threemonthslatershesamples50crabsandfindsshehasrecaptured8ofthecrabsthatshe originallytagged.WhatisEmily’sestimateforthenumberofcrabsinthelake?

Trythisexperiment

Simulatethecapture–recapturetechniqueusingdiscs.

a Mixalargenumberofdiscsorblocksinacontainer.

b Firstsample:Select10discsatrandom.Tagormarkeachdisc.

c Replacethese10discsintothecontainer.Mixthediscs.

d Secondsample:Select10discsatrandom.Determinethediscsthathavebeenselectedagainor‘recaptured’.

e Usethecapture–recapturetechniquetoestimatethenumberofdiscs.

f Repeatthesimulationbyselecting20discsforeachsample.

g Repeatthesimulationbyselecting30discsforeachsample.

Exercise1B BUILDING

1 Example2 Patrickisawildlifeofficerconcernedaboutthenumberofrabbitsinthepark.Hecatchesandtags 80rabbitsandthenreleases them.Aweeklaterhecatches70rabbitsintheparkandfindsthat10of themaretagged.EstimatethenumberofrabbitsintheparkforPatrick.

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2 Lucashasalargepondwithgoldfish.Hewouldliketoestimatethenumberofgoldfishinthepond. Herandomlycapturesandtags24goldfish.Fivedayslaterhesamples20goldfishandfindshehas recaptured4goldfish,astheyaretagged.EstimatethenumberofgoldfishinthepondforLucas.

3 Thecapture-recapturetechniquewasusedtoestimatethenumberofdollarcoinsinalargecollection. Thecollectorpulledout12dollarcoinsatrandomandtaggedthem.Twoweekslaterafterthe collectionhadbeentippedoutandputbackanumberoftimes,thecollectorpulledout42dollar coins.Threeofthesecoinsweretagged.Estimatethenumberofdollarcoinsinthiscollection.

4 Marjanawouldliketoestimatethenumberofparrotsinthenationalpark.Sherandomlycaptures andmarks36parrots.Afterthreeweeks,asecondsampleof35parrotsiscaught.Marjanafindsshe hasrecaptured6parrots.Estimatethenumberofparrotsinthenationalpark.

5 Thenumberofgrasshoppersappearstobeincreasinginaparticulartown.Jamesneededanestimate ofthegrasshopperpopulation.Hespentthefirstdaycollecting28grasshoppersandtaggingeach insect.OnthenextdayJamescollectedanother28grasshoppersandfoundthathehadrecaptured 4grasshoppers.WhatisJames’sestimateforthenumberofgrasshoppersinthetown?

DEVELOPING

6

Todeterminethenumberoffrogsinapond,20frogsarecaughtandtaggedandthenreleasedinto thepond.Twodayslater10frogsarecaught,fourofwhich4havetags.Estimatethenumberof frogsinthepond.

7 Thecapture–recapturetechniquewasusedtoestimatethenumberpencilsinalargetub.Ateacher pulledout54pencilsandtaggedthembeforeputtingthemback.Onemonthlater,aftermanyuses andshufflingaround,anotherteacherpulledout51pencilsandsawthat27ofthemweretagged. Estimatethenumberofpencilsinthelargetub.

8 Jesseisafishermanwhoisestimatingthenumberoflobstersinalake.Hecapturesandtags 28lobsters.Twomonthslaterhecaptures15lobstersandfindshehasrecaptured5ofthetagged lobsters.WhatisJesse’sestimateforthenumberoflobstersinthelake?

CHALLENGING

9 Alocalcommunityisconcernedaboutthenumberofcanetoads.Sumananeedsanestimateofthe canetoadpopulation.Shespendsthefirstdaycollecting48canetoadsandtaggingeachanimal.On thenextdaySumanacollects44canetoadsandfindsthatshehasrecaptured6canetoads.Whatis Sumana’sestimateforthenumberofcanetoadsinthecommunity?

10 Katerinawouldliketoestimatethenumberofdingoesinthenationalpark.Sherandomlycaptures andmarks22dingoes.Afterfourweeks,asecondsampleof20dingoesiscaught.Katerinafindsshe hasrecaptured4dingoes.Estimatethenumberofdingoesinthenationalpark.

11 Timisascientistwhoisconcernedaboutthenumberofbatsinthepark.Hecatchesandtags60bats andthenreleasesthem.Twoweekslaterhecatches85batsintheparkandfindsthat25ofthemare tagged.EstimatethenumberofbatsintheparkforTim.

Communicateyourreasoning,understandingandfluency

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12 Arangerisconcernedbythenumberoffoxesinawildlifepark.Sheneedstoestimatethepopulation ofthefoxes,butsheknowsthatthefoxesthathavebeentrappedandtaggedpreviouslylearntoavoid thetrapinthefuture.Ifthecapture–recapturetechniquewasused,wouldthisleadtopopulation estimatethatishigherorlowerthantheactualpopulation?Explainyouranswer.

1C

Dividingaquantityinagivenratio

Learningintentions

• Toconsolidatepriorlearningaboutratios

• Tosolveproblemsinvolvingthedivisionofanamountinagivenratio

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingratiosregularlyappearinHSCExaminations

Whentwopeopleshareanamountofmoney,theyusuallysplititevenly.Thisisa1:1ratioora50–50split. Eachpersongetshalfofthemoney.

Sometimes,onepersondeservesalargersharethantheother.Forexample,OliviaandWilliamwerepaid $100fordoingajob,butOliviaworkedfor3hoursandWilliamonlyworkedfor2hours.

Inthisinstancewewouldbedividing$100intheratio3:2.

Dividingaquantityinagivenratio

Todivide$100intheratio3:2

• 3partsand2partsmakes5parts,sotheratiois 3 5 : 2 5

• Oliviagets 3 5 of$100(whichis$60)andWilliamgets 2 5 of$100(whichis$40)

Areyouready?Dividing $30 betweentwopeopleindifferentratios

AmyandBenhavebeengiven$30tosharebetweenthem.Thisactivityinvolvescalculatinghowmuch theywilleachreceiveiftheydivideitupaccordingtoasetofgivenratios.

DownloadtheactivityfromyourInteractiveTextbook.

Example3

Dividingaquantityinagivenratio

MikhailandIlyaweregiven$450toshareintheratio4:5.Howmuchdideachget?

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Mikhailgot$200,Ilyagot$250.

Nowyoutry Divide$1000intheratio5:3.

Addthetwopartsoftheratio. Changetheratiointofractions. Multiplyeachfractionby$450todetermine eachshare.

Checkthattheanswerisreasonable: $200 + $250 = $450

Example4

Dividingaquantityinagivenratio

Amanleft$6000tobedividedamonghisthreechildren,Xia,YuiandZi,intheratio5:8:7,inthat order.Howmuchdideachchildget?

Solution

5 + 8 + 7 = 20

5 20 × 6000 = 1500

8

20 × 6000 = 2400

7 20 × 6000 = 2100

Explanation

Addthethreepartsoftheratio. Changetheratiointofractions. Multiplyeachfractionby$6000todetermine eachshare.

Xiagot$1500,Yuigot$2400,Zigot$2100. Checkthattheanswerisreasonable. Dotheyaddto$6000?

Nowyoutry

Divide$1000intheratio1:3:4.

Exercise1C

1 Findthetotalnumberofpartsinthefollowingratios. 2:9

2 Theratioofredtogreenis2:7. Whatfractionisred? a Whatfractionisgreen? b

Calculatehowmucheachpersonreceivesif$100issharedinthefollowingratios.

4 Divide240intothefollowingratios.

5 Shareeachamountintheratiogiven.

$20intheratio4:1 a

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BUILDING

$20intheratio7:3 b

$15intheratio1:2 c 77drinksintheratio3:4 d

100lolliesintheratio7:13 e

160booksintheratio5:3 g

50gintheratio1:3 i

45kgintheratio4:5 f

360pencilsintheratio2:7 h

60kmintheratio8:7 j

6 Abagof500gramsofchocolatesisdividedintotheratio7:3.Whatisthemassofthesmaller amount?

7 Ataconcerttherewere7girlsforevery5boys.Howmanygirlswereintheaudienceof8616?

Example4

8

Hazel,PreetishandAndrewinvestinabusinessintheratio6:5:1.Thetotalamountinvestedis $240000.Howmuchwasinvestedbythefollowingpeople?

Hazel a Preetish b Andrew c

9 Theratioofresidentialareatoparksinalocalcommunityis17:3.Thetotalareaofthelocal communityis40km2.Whatistheareaofparks?

10

Hayleyis15yearsoldandherbrotheris5yearsyounger. If$200issharedbetweenthemintheratiooftheirages,howmuchwillHayleyreceive?

11 Incountrytownacensusshowedthattherewere5adultstoevery7children.Ifthepopulationofthe townwas7200,howmanychildrenlivedthere?

DEVELOPING

12 Divide:

$200intheratio1:2:2 a $400intheratio1:3:4 b 12kgintheratio1:2:3 c 88kgintheratio2:1:5 d 440kgintheratio12:13:15 e $63000intheratio1:2:4. f

13 Share$600inthefollowingratios. 1:9 a 2:1:3 b 2:5:5 c 13:8:9 d

14 Apunchismadefrompineapplejuice,lemonadeandorangejuiceintheratio5:3:2.

a Howmuchlemonadeisneededifonelitreofpineapplejuiceisused?

b Howmuchpineapplejuiceisrequiredtomake15litresofpunch?

15 Angus,RubyandYuxishareaninheritanceof$500000intheratioof7:5:4.Howmuchwillbe receivedbythefollowingpeople?

Angus a Ruby b Yuxi c

16 Inafruitcakerecipetheratioofmixedfruittoflourtosugaris5:3:2.A250gpacketofmixed fruitisusedtomakethecake.Howmuchsugarandflourarerequired?

17

Adeliveryloadof8.5tonnesistobedividedbetweentwostoresintheratio11:6.Howmuchwill eachstorereceive?

18 Theloadonabridgeisappliedinthreepositions,A,BandC,intheratio5:7:5.Ifthetotalloadon thebridgeis782tonnes,whatistheloadtakenateachpoint?

19 Ajamismadebyadding5partsfruitto4partsofsugar.Howmuchfruitshouldbeaddedto 2 1 2 kilogramsofsugarinmakingthejam?

CHALLENGING

20 Theperimeterofarectangleis42cm.Theratiooflengthtobreadthis5:2.

a Whatisthelength?

b Whatisthebreadth?

21 Theratioof$5to$10notesinStephanie’spurseis3:5.Thereare24notesaltogether.Whatisthe totalvalueofStephanie’s$5notes?

22 Thethreesidesofatriangleareintheratioof1:3:5.Thelongestsideofthetriangleis16.2mm. Whatistheperimeterofthetriangle?

23 Achemicalsolutionismadebymixingacidwithwaterintheratio1:24.Thesolutionisthen takenandmixedwithwaterintheratio1:3.Howmuchacidwouldtherebein400mLofthis finalsolution?

24 TheratioofthepopulationsoftownAtotownBis3:8,andtheratiooftownBtotownCis5:2. Ifthetotalpopulationofthethreetownsis92655,findthepopulationofeachtown. Communicateyourreasoning,understandingandfluency

25 SolvethefollowingproblemandwriteasolutionforJarrahtoconvincehimyouarecorrect.

Jarrahhassomefuelforhislawnmowerinacontainer.Theratioofpetroltooilis50:1.Hehas 2.5litresoffuelalreadymixedatthisratio,butheneedsadifferentratioof30:1fortheoutboard motoronhisboat.Howmanymillilitresofoildoesheneedtoaddtothisfuelifhewantstouseall ofitinhisboat?Answercorrecttothenearestmillilitre.

1D Scaledrawings

Learningintentions

• Toconsolidatepriorlearningaboutratios

• Tofindunknownlengthsusingscaledrawingsandmaps

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingscaledrawingoccasionallyappearinHSCExaminations

Ascaledrawingisadiagramthatrepresentsanactualobject. Thescalefactorofascaledrawingistheratioofthesizeofthe drawingtotheactualsizeoftheobject.Forexample,amapisa scaledrawing.Itisnotthesamesizeastheareaitrepresents.The measurementshavebeenreducedtomakethemapaconvenient size.Thescaleofadrawingmaybeexpressedwithorwithout units.Forexample,ascaleof1cmto1mmeans1cmonthescale drawingrepresents1montheactualobject.Alternatively,ascale of1:100meanstheactualdistanceis100timesthelengthof 1unitonthescaledrawing.

Scaledrawing

Scaleofadrawing = Drawinglength:Actuallength

Scaleisexpressedintwoways:

• usingunits,suchas1cmto1m(or1cm = 1m)

• withnounits,usingaratiosuchas1:100.

Areyouready?Scale,intwoways

Scalessuchas1:100maybeclearerifconvertedtotheform1cm = 1m.Thisactivitywillshowyou howtodothat.

DownloadtheactivityfromyourInteractiveTextbook.

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Example5

Usingascale

Ascaledrawinghasascaleof1:50.

a Findtheactuallengthifthedrawinglengthis30mm.

b Findthedrawinglengthiftheactuallengthis4.5m.Answertothenearestmillimeter.

Solution

a Drawing length : Actual length 30 mm : × 30

50 × 30 = 1500 × 30 1 : 50

Theactuallengthis1500mm.

b Drawinglength:Actuallength × 90 × 90 1 : 50 : 4500 mm

1 × 90 = 90 Thedrawinglengthis90mm.

Nowyoutry

Explanation

Writetheratio.

Writetheenlargementfactor.

Usetheenlargementfactor.

Statethelength.

Writetheratio. Change4.5mto4500mm,asmmisanappropriate unitformeasuringamap.

Findandusetheenlargementfactor. Statethelength.

Theplansforanewhousehavebeendrawnusingthescale1:100.

a Adooris5mmwideontheplan.Howwideistheactualdoor?

b Akitchencounterisgoingtobe2.5metreslong.Howlongwillitbeontheplan,inmillimetres?

Exercise1D

BUILDING

1 Example5 Ascaledrawinghasascaleof1:100.Whatistheactuallengthofthesedrawinglengths?Express youranswersinmetres.

2cm a 10mm b 3.4cm c 28mm d 8.5cm e 49mm f

2 Ascaledrawinghasascaleof1:25000.Whatisthedrawinglengthoftheseactuallengths? Expressyouranswersinmillimeters.

2km a 750m b 4000cm c 3.5km d 50000mm e 1375m f

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3 Expresseachofthefollowingscalesasaratiointheform1: x 1cmto2cm a 1mmto5cm b 1cmto3km c

4 Thescaleonamapis1:1000.Calculatetheactualdistancesifthesearethedistancesonthemap. Expressyouranswersinmetres.

Road20cm a Shops10cm b

Pathway5cm c Parkingarea10mm d

Bridge34mm e Park80mm f

5 Thescaleonamapisgivenas1cm = 15km.Whatisthe actualdistanceifthedistanceonthemapis:

a 2.5cm?

b 45cm?

6 Thescaleonamapis1:5000.Calculatethemapdistancesiftheseareactualdistances.Express youranswersinmillimetres.

50m a 80m b 100m c 120m d 150m e 240m f

7 Thescaleonamapisgivenas1mm = 50m.Ifthedistancebetweentwopointsis350m,whatis themapdistancebetweenthesepoints?

8 Ascaledrawinghasascaleof1:75.Findthe:

a actuallengthifthedrawinglengthis15mm

b drawinglengthiftheactuallengthis3m.

9 TheParkesradiotelescopedishhasadiameter of64metres.Theimageshownisaphotograph ofthedish.

a Determineascalefortheimage.

b Estimatetheheightofthetopofthedish abovetheground.

CHALLENGING

10 Thescaleofamodelis2:150.Fortheactuallengthslisted,calculatethemodellengths.Express youranswersinmillimeters.

Palm Beach

Coogee

SYDNEY Mona Vale

Parramatta

Penrith Woy Woy

11

Ascaledrawingofthespaceshuttleisshownhere.Theactual lengthofthespaceshuttleis43metres.

a Whatisthescalefactor?

b Calculatethelengthofthewingspantothenearestmetre.

c Calculatethebreadth(b)oftheshuttletothenearestmetre.

d Whatisthelengthof thenoseoftheshuttletothe nearestmetre?

Communicateyourreasoning,understandingandfluency

12 SarahisgoingtomakeascalemodelofthemainarchofSydneyHarbourBridge,withoutthe pylons.Themodelwillbedisplayedonatablewhichis1.2metreslong.Thehighestpointonthe archoftheSydneyHarbourBridgeisapproximately134metresabovesealevel.Thearchspans 503metres.

Usethisinformationaboveandthephotographofthebridgetoanswerthesequestions,showingall calculations.

a WhichscalewouldgiveSarahthelargestmodelthatfitsonthetable:1:100,1:500or1:1000? Explainyouranswer,showingallcalculations.

b Usingthatscale,withthetableassealevel,approximatelyhowtallwillhermodelbein millimetres?

c Thedeckofthebridgeis49metreswide.HowmanymillimetreswillthisbeonSarah’smodel?

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d Accordingtothephotographbelow,whatistheclearancebetweensealevelandthemiddleofthe roaddeckontherealbridgeandSarah’smodel?

1E Buildingplans

Learningintentions

• Tounderstandtherelationshipbetweenbuildingplansandelevations

• Togatherinformationfromabuildingplan

• Tofindunknownlengthsandareasusingbuildingplans

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingbuildingplansoccasionallyappearinHSCExaminations

Aplanisaviewofanobjectfromthetop.Itislookingdownontheobject.Ahouseplanisahorizontal sectioncutthroughthebuildingshowingthewalls,windows,dooropenings,fittingsandappliances.An elevationisaviewofanobjectfromoneside,suchasafrontelevationorsideelevation.Houseelevations arerarelyasimplerectangularshapebutshowallthepartsofthebuildingthatareseenfromaparticular direction.Houseelevationsareaverticalsectionparalleltoonesideofthebuilding.

Plansandelevations

Aplanisaviewofanobjectfromthetop.

Anelevationisaviewofanobjectfromoneside,suchasafrontelevationorsideelevation.

Drawingaplanandelevation

Drawtheplanview,frontelevationandsideelevationof thisobject.

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Solution

Explanation

Plan Front elevationSide elevation Imaginewhatyoucanseefromabove,the frontandtheside.

Nowyoutry

Drawtheplanviewofthisobject.

Buildingplans

Afloorplanforahomeisshownbelow.Buildingplansareaverycommonexampleofscaledrawings.They aredrawnusingascalefactorsuchas1:150.Thisallowsthedimensionsofahousetobedeterminedby measurementandcalculation.

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Commonfloorplansymbols

Doorswing:indicatesdirectionthedooropens

Window:glasswindowinasolidwall

Kitchensink:two-compartmentkitchensink

Example7

Findingmeasurementsfromahouseplan

Abuildingplanisshownforthegroundfloorofahome. Thescaleis1:150.

a Howmanyinternaldoorsarethere?

b WhatisthemeaningofPWDR?

c Whatisthelengthofthehouse,includingtheportico?

d Whataretheinternaldimensionsofthedoublegarage?

Solution

Shower:showerwithoutabathtub

Toilet:toiletlocatedonwall

Bathtub:bathtubshowinglocationofdrain

Explanation

a 4internaldoors Pantry,laundry,powderroom,garage.

b Powderroom

c Drawinglengthis13.7cm

Actuallength = 13.7 × 150cm = 2055cmor20.55m

Useyourrulertomeasurethelengthofthehouse.

d Drawinglengthis4.3 × 3.6cm

Actuallength = 4.3 × 150cm = 645cmor6.45m

Actualbreath = 3.6 × 150cm = 540cmor5.40m

Dimensionsare6.45 × 5.40m.

Nowyoutry

UsingtheplaninExample 7:

a howwideisthehouseatitswidestpoint,inmetres?

Useyourrulertomeasurethedimensionsofthe garage.

b whataretheinternaldimensions(inmetres)ofthesittingroom?

Exercise1E

1 Example6 Drawtheplan,frontelevationandsideelevationfortheseobjects.

BUILDING

2 Whataretheobjectsrepresentedbytheseviews?

3 Drawtheplanview,frontelevationandsideelevationfortheseobjects.

4 Asectionofafloorplanisshownopposite.

a Whatroomisshowninthediagram?

b Whatsymbolisusedforashower?

c Whatsymbolisusedforadoor?

d Whatdoes‘WIR’representontheplan?

5 Asectionofafloorplanisshownopposite.

a Whatarethedimensionsofthekitchen?

b Whatsymbolisusedforthesink?

c Whatsymbolisusedforthecooktop?

d Whatdoes‘Ref’representontheplan?

e Whatdoes‘P’representontheplan?

6 Asectionofafloorplanisshownopposite.

a Whatarethedimensionsofthebedroom?

b Whatsymbolisusedforawindow?

c Whatdoes‘BIW’representontheplan?

d Whatisthelengthofthebedroomontheplan?

e Calculateascaleforthefloorplan.

B.I.W.

Bedroom 3 3300 × 2800 Linen

7

Asectionofafloorplanisshownopposite.

a Whatarethedimensionsofthebedroom?

b Whatsymbolisusedforthetoilet?

c Whatisthelengthofthebedroomonthe plan?

d Calculateascaleforthefloorplan.

e Whatarethedimensionsofthewalk-in robe?

f Whatistheareaofthebedroom?Answer insquaremetres,correcttotwodecimal places.

Asecond-storeybuildingplanisshownforaMastertonhome.

a Whatarethedimensionsofthethirdbedroom?Answer inmetres.

b Whatarethedimensionsofthemasterbedroom?Answer inmetres.

c Useyourrulertohelpyouestimateascaleforthisplan.

d Useyourrulerandyourscalefrompart c tofindthebreadth ofthehouse.Answerinmetres.

e Calculatetheareaofthevoid.Answercorrecttothenearest squaremetre.

f Calculatetheareaoftheensuite.Answercorrecttothe nearestsquaremetre.

9 Arumpusroomisbuiltmeasuring5.5mby4.7m.Thefloorplanusesascaleof1:100.Aconcrete slabwithadepthof100mmisusedtobuildtherumpusroom.

a Whatistheareaoftherumpusroomontheplan?Answerinsquaremillimeters.

b Whatisthevolumeofconcretefortherumpusroom?Answerincubicmillimeters.

c Whatisthevolumeofconcretefortherumpusroomiftheslabdepthis200mm?Answerincubic millimetres.

10

Thefrontelevationofahouseisshownopposite (scale1:200).

a Whatisthebreadthofthehouse?Answer inmetres.

b Whatistheheightofthechimney?Answer inmetres.

c Whatarethedimensionsofthefrontdoor? Answerinmetres.

d Whatarethedimensionsofthewindowonthe right-handside?Answercorrecttothenearest centimetre.

e Whatistheareaofthewindow?Answercorrect tothenearestsquarecentimetre.

f Whatistheareaofthesmalltriangulargable abovethefrontdoor?Answercorrecttothe nearestsquarecentimetre.

11

Thefrontelevationofahouseisshown.

a Whatistheheightofthefirststorey?Answerinmetres.

b Whatistheheightofthegarage?Answerinmetres.

c Whatistheangleofthepitchoftheroof?

d Howmanywindowsareatthefrontofthehouse?

12

13

Abuildingplanisshown below.Thehouselength is20mandthebreadth is18m.

a Whatarethedimensions ofthelivingroom? Answerinmetres.

b Whatistheareaofthe livingroom?Answer correcttothenearest squaremetre.

c Thehousewasbuilt onaconcreteslab200 mmthick.Howmuch concretewasused? Answercorrecttothe nearestcubicmetre.

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Abuildingplanisshownforahome.Thehouselengthis 25m(includesportico)andthebreadthis15m(includes garage).

a Whatisasuitablescaleforthisplan?

b Whatarethedimensionsoftheverandah?Answer correcttothenearesttenthofametre.

c Whatarethedimensionsofthedoublegarage?Answer correcttothenearesttenthofametre.

d Whatarethedimensionsofbedroom3?Answercorrect tothenearesttenthofametre.

e Calculatetheareaofthesittingroom.Answercorrect tothenearestsquaremetre.

f Whatisthecostofcarpetingthesittingroomifthecost ofthecarpetis$140persquaremetre?Answercorrect tothenearestdollar.

CHALLENGING

14 Asectionofabuildingplanisshownopposite.The dimensionsofthefamilyroomare5.0metresby6.0metres.

a Estimateasuitablescaleforthisbuildingplan.

b Whatisthecombinedlengthofthefamily,kitchen andsittingrooms?Answercorrecttothenearesttenth ofametre.

c Thefamily,dining,kitchenandsittingroomsaretobe tiled.Calculatethecombinedareaoftheserooms.Answer tonearestsquaremetre.

d Ceramictilesmeasuring300 × 300mmaretobelaidin theserooms.Howmanytilesarerequired?

e Whatassumptionhasbeenmadetotheanswerinpart d?

f Thefamily,dining,kitchenandsittingroomsarebuilton aconcreteslabwithathicknessof0.15m.Whatisthe volumeofconcreteusedfortheslab?

Communicateyourreasoning,understandingandfluency PlanningahouseonanA4sheetofpaper.

15 TakeablankA4sheetofpaper(210mmby297mm),whichwillbethesizeoftheplanofyournew rectangularblockoflandandtheone-storeyhouseyouwillbuilduponit.Thescaleis1:50. Answerthefollowingquestions,justifyingeveryanswerwithcalculations.

a Whatarethedimensionsofyourblockofland,inmetres?

b Whatistheareaofyourblockofland,insquaremetres?

c Yourhousemustnotbecloserthan900mmtoanyoftheboundariesoftheblock.Drawthe footprintofthelargesthouseyoucanbuildontoyourA4sheet.

d Whatistheareaofthelargesthouseyoucanbuild,insquaremetres?

e SearchtheinternettofindsomeplansforahouseinAustraliawithanareaof115squaremetres. AlsosearchforthetypicalbuildingcostspersquaremetreforahouseinAustralia.Howmuch mightitcosttobuythisblockoflandandbuildthishouseinthesuburbortownyoulike?

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1F Perimeter,areaandvolumefromplans

Learningintentions

• Todetermineestimatesofmeasurementsfromphotos,diagramsandplans

• Touseestimatesofmeasurementstocalculateperimeter,areaandvolume

• Toestimateareasusingthetrapezoidalrule

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• ThetrapezoidalrulewasintroducedinYear11

• QuestionsinvolvingtheseconceptsregularlyappearinHSCExaminations

Perimeter

TheperimeterofapieceoflandcanbefoundusingaerialphotographssuchasthoseobtainedfromGoogle Earth.Itismuchquickerthanactuallymeasuringtheland.

Steps: 1 Determinethebordersofthepieceofland.

2 Accessalandmeasurementwebsiteandviewtheaerialphotographoftheland.

3 Usingtheonlinetools,clickalongtheperimeteroftheland.

Perimeter

Whencalculatingtheperimeterofland,multiplysidelengthsontheaerialphotographbythescale.

Example8

Calculatingdistanceusingascale

ThisaerialviewonGoogleEarthshowsa50mOlympicswimming pooloutlinedwiththe‘path’tool.

a Whatisthescaleusedontheimage?

b Usethescaletodeterminethelengthoftheboatinthepicture correcttothenearestmetre.

Solution

a Lengthofswimmingpoolis1.7cm.

Scale1.7cmto50m(or5000cm).

Scaleisapproximately1:2941.

b Boatlengthis1.1cm.

Boatlength = 1.1 × 2941

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= 3235cm

= 32m

Lengthoftheboatis32metres,correcttothe nearestmetre.

Explanation

Nowyoutry

Thenextboattoarriveatthejettyis2.0cmlonginthephotographinExample 8.Howlongistheactual boat,correcttothenearestmetre?

Area

Theareaofablockoflandcanbeestimatedandcalculateddirectlyfromscalediagramsandaerial photographs.Toestimatetheareaoftheland,dividethediagramintosquaregridsandcountthenumberof squares.Tocalculatetheexactareaofland,usetheappropriateformula.Compositeshapesaredividedinto planeshapes.Thedimensionsoftheshapesaredeterminedusingthescale.

Area

Toestimatetheareaofland,divideitintosquaregridsandcountthenumberofsquares. Tocalculatetheexactareaofland,usetheappropriateformula.

Example9

Calculatingtheareaofland

AwarehouseisshownonGoogleEarth.Theredscalebarrepresents36m attheground.

a Calculatethelengthsofthesidesofthepentagonoccupiedbythe warehousebuildingusingthescale.Answercorrecttothenearesttenth ofametre.

b Calculatetheareaoflandoccupiedbythewarehousebuilding.Answer correcttothenearestsquaremetre.

Solution

a Scalebaris0.9cmrepresents36cm Scaleis1cmto40m

AB = 2.9 × 40 = 116m

BC = 0.8 × 40 = 32m

CE = 1.1 × 40 = 44m

EF = 2.2 × 40 = 88m

AF = 1.7 × 40 = 68m

Explanation

Areaofwarehouseis7384m

Totalarea = areaofrectangle ABDF area oftriangle CDE

Areaof ABDF

Areaof CDE

Areaofwarehouse = area ABDF area CDE Writeyouranswer,roundingtothenearest squaremetre.

Nowyoutry

ThePentagonistheheadquartersbuildingoftheUnitedStates DepartmentofDefence.

a Dothefivesidesofthebuildingappeartobeequal?

b Usethescaleinthebottomright-handcornertoestimatethe perimeterofthebuilding,inmetres.Answercorrecttothenearest 100metres.

c Bydividingthebuildingintoatrapeziumandatriangle,estimate theareacoveredbythePentagon,includingthecentralcourtyard. Answercorrecttothenearestthousandsquaremetres.

Volumeofwater

Ifwatercoversanarea(suchasthebottomofapool,tankorreservoir),toaconstantdepth,itsvolumecan becalculatedusing

V = Ah

A = areaofsurface h = depthofwater

Thetrapezoidalruleforarea A ≈ b 2 d f + dl isusedtoestimatetheareaofapieceoflandorabodyofwater withanirregularboundarysuchasareservoirordam. A isthearea, b isthebreadthbetweentheparallel sides, d f isthedistancealongthefirstparallelside, dl isthedistancealongthelastparallelside.Notethatin Year11, h wasusedforthebreadthbetweentheparallelsides,butherewewilluse b since h willbeusedfor heightorthicknessoftheshape.

Soifabodyofwatersuchasapoolhasasurfaceareaof100squaremetresandtheareaofitsbottomisalso 100squaremetres(becauseithasaconstantdepth),andthatdepthis2m,then:

V = Ah

V = 100 × 2

V = 200m3

Iftheareaofareservoiris A squaremetres,and h metresofrainfallsonit(withnowaterrunninginorout), thenitsvolumewillincreaseby Ah cubicmetres.

Volumeusing V = Ah andthetrapezoidalruleforarea

Thetrapezoidalrule: A ≈ b 2 d f + dl A:Areaofshape

b:Breadthbetweentheparallelsides

d f :Distancealongfirstparallelside

dl:Distancealonglastparallelside

V = Ah

A:Areaofthebodyofwaterorthecross sectionoftheprismobject h:Heightorthickness

Thisruleonlyappliestobodiesorprismsofuniformcross–section.

Example10

Calculatevolumewiththetrapezoidalruleforarea

Theshapeofthewaterinapoolofconstantdepthwithan irregularly-shapedbottomandtopsurfaceisshown.

a Applythetrapezoidalruleforareatoapproximatethevolumeof waterinthispool.

b Ifthepoolisemptyand100mmofrainfallsintoit,howmanylitres ofwaterarecollectedinthepool?

Solution

a A ≈ b 2 d f + dl

A ≈ 8 2 (6 + 4)

A ≈ 40m2

V = Ah

V ≈ 40 × 1.5

V ≈ 60m3

Thevolumeofwaterinthepoolis approximately60cubicmetres.

b 100mm = 100 1000 m = 0.1m.

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V ≈ 40 × 0.1 = 4m3

Thevolumeofwaterinthepoolwill beapproximately4cubicmetres.

Everycubicmetreholds1000litres 4 × 1000 = 4000litres.

Explanation

Writethetrapezoidalruleforarea.

Substitute b = 8, d f = 6and dl = 4intotheformula.

Evaluatetofindtheareaofthepoolinsquaremetres.

Writetheformulaforvolumeofarightprism.

Substitute A ≈ 40and h = 1.5intotheformula.

Evaluatetofindthevolumeofwaterinthepoolincubic metres.

Writetheanswerinwords.

Sincetheareaofthewateraddedisthesameastheareaofthe pool,wecanrepeatthestepsinpart a withtheformulaforthe volumeofarightprism,andjustchange h.Ithasbeengiven inmillimetresandneedstobechangedtometres.Thereare 1000mminametre.Divide100mmby1000.

Substitute A ≈ 40and h = 0.1intotheformula.

Writetheanswerinwords.

Nowyoutry

Thediagrambelowshowsthetopsurfaceofaswimmingpool.The wateris1.5metresdeep.

a Usethetrapezoidalruletofindtheapproximateareaofthe surface.

b Hence,findthevolumeofthepool.

c Howmanylitresofwaterwillfitinthepool?

Exercise1F

1

Example8 ThelengthoftheOlympicswimmingpoolshownopposite is50metres.

a Calculateascalefortheaerialphotograph.

b Whatarethedimensionsoftheaerialphotograph?(Answerto thenearestmetre.)

c Whatisthelengthfrom A to B?

d Whatisthelengthfrom B to C?

e Whatisthelengthfrom C to A?

f Whatistheperimeterof ABC?

Example9

2

BUILDING

g Calculatetheareaofthelandmarked ABC.Assumearightangleat C.

Asportsfieldisshownopposite.Thescaleismarkedby theredbar,whichrepresents50montheground.Answer thefollowingquestions,correcttothenearestmetreor squaremetre.

a Whatisthelengthofthegrandstandhighlightedbythe yellowbar?

b Calculatethelengthofthefield.

c Calculatethebreadthofthefield.

d Whatistheperimeterofthefield?

e Whatistheareaofthefield?

f Whatisthelengthofthediagonalofthefield?

3 Twocircularbuildings, A and B,atSydneyairportare shown.Theredscalerepresents25montheground. Answerthefollowingquestions,correcttothenearest metreorsquaremetre.

a Whatistheradiusofbuilding A?

b Whatisthediameterofbuilding B?

c Whatisthecircumferenceofbuilding A?

d Whatistheareaofbuilding A?

e Whatistheradiusofthebuilding B?

f Whatisthecircumferenceofbuilding B?

4 Calculatethedistancebetweentheseplaces,tothenearestmetre, usingonlinetools.

a GovernmentHousetotheSydneyConservatoriumofMusic

b WynyardandTownHallstations

c TumbalongParktotheArchibaldfountain

d UniversityofSydneytoMoorePark

e CentennialParktoBondibeach

f KingsCrosstoWoolloomooloo

5 Themapbelowhasascaleof1:10000. Estimatetheareaofthefollowingsections ofland.

6

Example10 Findthevolumeoftheshapeswiththefollowingdimensionsusingthetrapezoidalruleforareaand theformulaforvolumeofarightprism.

a b = 14m, d f = 28m, dl = 36mand h = 8m

b b = 9m, d f = 45m, dl = 25mand h = 6

7 Thediagramshowsthebaseofapool.Estimatethevolumeof thepoolifitsdepthiseverywhere1.8m.Answercorrectto thenearestcubicmetre.

8 Afarmisshownopposite.Answerthefollowing questionscorrecttothenearestmetreorsquaremetre.

a Whatistheperimeterofpaddock A?

b Whatistheperimeterofpaddock B?

c Whatistheperimeterofpaddock C?

d WhatistheperimeteroftheL-shapedpaddock?

e Whatistheareaofpaddock A?

f Whatistheareaofpaddock B?

g Whatistheareaofpaddock C?

h WhatistheareaoftheL-shapedpaddock?

9

10

Adiagramhasascaleof1:100.Whatistheactualareaoftheseshapes,giventhefollowingdrawing lengths?Expressyouranswercorrecttothenearestsquaremetre.

a Squarewithasidelengthof10mm

b Rectanglewithalengthof2cmandabreadthof3cm

c Right-angledtrianglewithabaselengthof25mmandperpendicularheightof40mm

d Circlewitharadiusof20mm

e Semicirclewithadiameterof50mm

Ashatooktwomeasurementsat120mintervalsacrossthe irregular-shapedreservoirinthediagram.Themeasurements were100mand80m.

a Estimatethesurfaceareaofthereservoir.Answercorrectto thenearestsquaremetre.

b Ashameasuredthedepthofwaterandfoundittobe5m everywhere.Estimatethevolumeofthereservoir.Answer correcttothenearestcubicmetre.

11 150mmofrainfellonthereservoirinQuestion 10,whichhasverticalbankssonowaterflowsinor outandthedepthdoesnotvary.

a Whatisthevolumeofwaterinthereservoir?

b Thefarmthatdependsonthereservoirneedsanextra15000cubicmetresofwaterforthenext month.Howmuchrainneedstofallinthattimesothatwhentheextrawaterhasbeenpumped out,thewaterlevelinthereservoirremainsthesame?Answercorrecttothenearestmillimetere.

12 Mauriceusedthreeapplicationsofthetrapezoidalruleto estimatetheareaofthissectionofland.Explainwhyhis estimateislargerthantheactualareaoftheland.

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1G Simplifyingandconvertingrates

Learningintentions

• Tounderstandthedifferencebetweenratiosandrates

• Toexpressratesintheirsimplestform

• Toconvertratesbetweendifferentunits

• Tosolveproblemsinvolvingratesinavarietyofpracticalcontexts Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingratesregularlyappearinHSCExaminations

Rates

Aratioisacomparisonofamountsofthesameunit.Theratio 1cmto5cmiswrittenas1:5,withouttheunits.

Arateisacomparisonofamountswithdifferentunits.For example,wemaycomparethedistancetravelledwiththetime taken.Inaratetheunitsaredifferentandmustbespecified.

Theorderofarateisimportant.Arateiswrittenasthefirst amountperoneofthesecondamount.Forexample,$2.99/kg represents$2.99peronekilogram or80km/hrepresents 80kilometresperonehour.

Weareconstantlyinterestedinratesofchangeandhowthings changeoveraperiodoftime.Therearemanyexamplesofrates suchas:

• Runningrate:Yourrunningpaceinmetrespersecond.

• Typingrate:Yourtypingspeedinwordsperminute.

• Wagerate:Theamountofmoneyyouarepaidperhour.

• Fuelconsumption:Litresper100kmorkmperlitresarebothused.

Convertingbetweenrates

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Forexample,you could:

• Growthrate:Theaveragegrowthrateofachildfrom0to15yearsofage.

1 convertthenumberofkilometrestometres,and; 2 convertthenumberofhourstoseconds,then; 3 reducetheratetosimplestform,inthesamewayyoureducearatiotosimplestform. However,theaimisforthenumberofsecondstobe1,becausem/smeans‘metresin1second’.

Note: Somepeopleuseruleslike‘dividethespeedby3.6’.Theseruleshavelimitedapplicability.

Thespeedofamovingcarcouldbeexpressedin‘kilometresperhour’(km/h)or‘metrespersecond’ (m/s).Therearemanydifferentwaystoconvertaspeedfromonepairofunitstoadifferentpair.

Example11

Convertingbetweenrates

Convert100km/hintometrespersecond.

Solution

100kmin1hour

100000min60minutes

100 000 m in 3600 seconds m in 1 second ÷ 3600 ÷3600

100km/h = 27.8m/s(toonedecimalplace)

Nowyoutry

Explanation

Writetherategiven.

Convert100kmtometresand1hourto minutes. Convertfromminutestoseconds.

Reduce,justlikearatio,totheform___:1 Inthiscase,dividebothnumbersby3600.

Ahippopotamuscanrunataspeedof30km/hovershortdistances.

a Atthisspeed,how manymetrescanitruninonesecond?Giveyouranswercorrecttoonedecimal place.

b Howmanysecondswouldthehippopotamustaketorun100metres?Giveyouranswercorrecttothe nearestsecond.

Exercise1G

1 Whichofthefollowingareexamplesofrates?

$7.50 a

2 Converttotherateshown.

$100in4hisarateof$ /h a

700Lin10hisarateof L/h c

$1.20for2kgisarateof$ /kg e

1200revin4minisarateof rev/min g

275min25sisarateof m/s i

BUILDING

/L f

240min20sisarateof m/s b

$39in12hisarateof$ /h d

630kmin60Lisarateof km/L f

Ariseof20◦ in4hisarateof ◦/h h

630Lin9hisarateof L/h j

3 Example11 Expresseachrateinsimplestformusingtheratesshowninthebrackets.

300kmon60L[kmperL] a 15min10s[mpers] b

$640for5m[$perm] c

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56Lin0.5min[Lpermin] d

78mgfor13g[mgperg] e 196gfor14L[gperL] f

20gfor8m2 [gperm2] g

$1.80for9phonecalls[c/call] i

4 Writeeachofthefollowingasasimplifiedrate.

75mLfor5min[mLpermin] h

$630for36hwork[$/h] j

18goalsin3games a 12daysin4years b

$1.50for5kilograms c $180in6hours d 49500cansin11hours e $126000topurchase9hectares f 80mmrainfallin5days g 19000revolutionsin10minutes h

15kilometresrunin60minutes i 60minutestorun15kilometres j

DEVELOPING

5 Converteachratetotheunitsshowninthebrackets.

39240m/min[m/s] a 2m/s[cm/s] b

88cm/h[mm/h] c 55200m/h[m/min] d

0.4km/s[m/s] e 57.5m/s[km/s] f

6.09g/mL[mg/mL] g 4800L/kL[mL/kL] h

300m/s[cm/s] i 36km/h[km/min] j

6 Adrippingtapfilleda9litrebucketin3hours.

Whatwasthedrippingrateofthetapinlitres/hour? a Howlongwouldittakethetaptofilla21litrebucket? b

7 If30kebabswereboughttofeed20peopleatapicnic,andthetotalcostwas$120,findthe followingrates.

kebabs/person a cost/person b

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CHALLENGING

8 ThenumberofhoursofsunshinewasrecordedeachdayforoneweekinMay.Theresultswere:

Monday:6hours

Wednesday:3hours

Friday:7hours

Sunday:7hours

Findtheaveragenumberofhoursofsunshine:

Tuesday:8hours

Thursday:5hours

Saturday:6hours

perweekday a perweekendday b perweek c perday. d

9 Sebastianfinisheda10kilometreracein37minutesand30seconds.Alexanderfinisheda 15kilometreracein53minutesand15seconds.Calculatetherunningrateofeachrunnerexpressed asminutesperkilometre.

10

Afootballclubhad12000members.Afterfivesuccessfulyearsandtwopremierships,theynow have18000members.Whathasbeentheaveragerateofmembershipgrowthperyearforthepast 5years?

Communicateyourreasoning,understandingandfluency

11 Samboughtanewcarduring2026forhisbusiness. Eightmonthslater,hecalculatedthefollowing:

• Kilometresdriven:15900km

• Servicing:$501.95

• Petrol:$2094.76

• Tolls:$961.92

• Depreciation:$3501.50

• Registration:$617.33

• Compulsorythirdpartyinsurance:$312.67

• Comprehensiveinsurance:$1019.87. CreateaspreadsheetwhichSamcanusetotrackhisspendingonhiscar.Includearowforeachof theexpenses.

Usethespreadsheettoanswerthefollowingandwritedowntheformulausedtodoyour calculations.

a Calculatethetotalamountspentonthecarduringthose8months.

b Calculatethecostofoperatingthecar,indollarspermonth.

c Calculatethecostofoperatingthecar,indollarsper100km.

d Calculatetheaveragedistancetravelled,inkilometrespermonth.

e Basedonthese8months,howmuchwillthecarcosthimforthewholeyear?

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1H

Usingtheunitarymethodtosolveproblems

Learningintentions

• Toconvertratesbetweendifferentunits

• Torecognisesituationsinwhichtheunitarymethodcanbeapplied

• Tosolveproblemsinvolvingratesinavarietyofpracticalcontexts Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingratesregularlyappearinHSCExaminations

Theunitarymethod

Manyproblemscanbesolvedusingtheunitarymethod.Theword‘unitary’comesfrom‘unit’ meaning‘one’.

Theunitarymethodinvolvesfindingoneunitofaquantitybydivision.Thisresultisthenmultipliedtosolve theproblem.

Usingtheunitarymethod

Usedivisiontogetdownto1,thenmultiplytofindtherequiredamount.

Example12

Usingoneunittofindmanyunits

Acaristravellingonafreewayataconstantspeedof100km/h.Howfarwillittravelin3.5hours?

100km/h = 100kmin1hour × 3.5 × 3.5 100 : 1 : 3.5

100 × 3.5 = 350

Thecarwilltravel350km.

Nowyoutry

Writetherateasifitisaratio.

Placethe3.5hoursunderthe1hour.

Thescalefactoris3.5.Applythescalefactor. Checkthattheanswerseemsreasonable.

Petrolcosts$2.05perlitre.Howmuchwill40litresofpetrolcost?

Example13

Usingtheunitarymethod

Johanran5kilometresin38minutes.Ifhecouldmaintainthispace,howlongwouldittakehimtorun 11kilometres?

Solution

Explanation

÷ 5 × 11 ÷ 5 × 11

5 km in 38 minutes 1 km in minutes 11 km in minutes

38 ÷ 5 × 11 = 83.6minutes

ItwouldtakeJohan83.6minutestorun11km.

Nowyoutry

Writetherateasifitisaratio.Placethe1km underthe5km(thisisthe‘unit’intheword ‘unitary’),thenthe11kmunderthe1km.

Theleftcolumnindicatesthatwedivideby5then multiplyby11.Applythoseoperationstotheright column.

Checkthattheanswerseemsreasonable.

Albertorodehisbikefor135kmin3days.Atthispace,howfarcouldheridein7days?

Exercise1H

1 Example12 Usetheratesprovidedtoanswerthefollowingquestions.

a Costofapplesis$2.50/kg.Whatisthecostof5kg?

b Taxchargeis $28/m2.Whatisthetaxfor7m2?

c Costsavingsare $35/day.Howmuchissavedin5days?

d Costofachemicalis $65/100mL.Whatisthecostof300mL?

e Costofmushroomsis$5.80/kg.Whatisthecostof 1 2 kg?

f Distancetravelledis1.2km/min.Whatisthedistancetravelledin30minutes?

BUILDING

g Concentrationofachemicalis 3mL/L.HowmanymLofthechemicalisneededfor4L?

h Concentrationofadrugis 2mL/g.HowmanymLisneededfor10g?

2 Acricketteamscores runsatarateof 5runs/overinamatch.Howmanyrunsarescored in18overs?

3 Example13 Ifonedozentennisballscost$9.60,howmuch would22tennisballscost?

4 IfLeocanmarchat7km/h,howfarcanhemarch in2.5hours?

5 If8kg ofchickenfilletscost$72,howmuchwould 3kgofchickenfilletscost?

6

7

8

9

Ifthreepairsofsockscost$12.99,howmuchwould10pairsofsockscost?

If500gofmincemeatcosts$4.50,howmuchwould4kgofmincemeatcost?

10

Acourierdelivers1parcelonaverageevery20minutes.Howmanyhoursdoesittaketodeliver 18parcels?

Waterisdrippingfromatapatarateof5L/h.Howmuchwaterwillleakinoneday?

Abulldozerismovingsoilatarateof22t/h.Howlongwillittakeatthisratetomove55tonnes?

11 Edwardsaves $40/week.Howlongshouldittaketosave$1000?

12

Aprofessionalfootballerscoresan averageof3goalsevery6games.Howmanygoalsishelikelyto scoreinafullseasonof22games?

13 Acomputerprocessorcanprocess500000kilobytesofinformationin4seconds.Howmuch informationcanitprocessin15seconds?

CHALLENGING

14

Usetheinformationprovidedonspeedtoanswerthefollowingquestions.

a Iwalkat5km/h.HowfarcanIwalkin4hours?

b Acartravels at80km/h.Howfarwillittravelin2.5hours?

c Aplaneistravellingat600km/h.Howfarwillittravelin30minutes?

d Atraintook7hours totravel665km.Whatwasitsaveragespeed?

e Ryderrunsa42.4kmmarathonin2hours30minutes.Calculatehisaveragespeed.

f Aspacecrafttravelsat1700km/hforadistanceof238000km.Howmanyhoursdidittake?

15

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Miaearns$37.50perhour workinginacafe.

a HowmuchdoesMiaearnforworkinga9-hourday?

b HowmanyhoursdoesMiaworktoearn$1200?

c WhatisMia’sannualincomeifsheworks40hoursaweek?Assumesheworks52weeks intheyear.

16 Acricketteamscoresrunsatarateof5runs/overinamatch.Howmanyoversdidittaketo score90runs?

17 Julietplanstospend$135perdayduringheroverseastrip.Atthisrate,howmanydayscanshe travelusing$3510?

18 Fatima’scaruses9litresofpetroltotravel100kilometres.Petrolcosts$1.50perlitre.

a Whatisthecostoftravelling100kilometres?

b Howfarcanshedriveusing$50worthofpetrol?Answertothenearestkilometre.

Communicateyourreasoning,understandingandfluency

19 Rogerconverted215Australiandollars(AUD)intoNewZealanddollars(NZD).Hereceived $250.82NZD.AttheendofhistriptoNewZealand,heconverted$13NZDbackintoAUD.Ifthe exchangeratedidnotchange,howmuchshouldhereceive?

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1I Usingratestomakecomparisons

Learningintentions

• Tomakedecisionsregardingbestbuys

• Tosolveproblemsinvolvingratesinavarietyofpracticalcontexts Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingratesregularlyappearinHSCExaminations

Ratesareusedtosolvepracticalproblemssuchascalculatingwages,bestbuyandcosts.

Example14

Bestbuys

Ashopsellsthesamecansofdrinkinthefollowingways:

Onecanfor$2.40,six-packofcansfor$11.95or24-packofcansfor$39.95.

a Whichisthebestbuy?

b Howmuchissavedwhenbuyingthe24-packratherthan24singlecans?

Solution

a Onesingledrinkwillcost$2.40

$11.95 ÷ 6 ≈ $1.99

$39.95 ÷ 24 ≈ $1.66

Thebestbuyisthe24-pack.

b 24 × $2.40 = $57.60

$57.60 $39.95 = $17.65 Thisisasavingof$17.65.

Nowyoutry

Abreakfastcerealcomesinthreesizes:

• 290gramsfor$7

• 450gramsfor$9.55

• 765gramsfor$10.80.

Whatisthepriceper100gramsforeachsize?

Explanation

Startwiththecostofasinglecanofdrink. Thedrinksinthesix-packarecheaper. Thedrinksinthe24-packarethecheapest.

Compare24individualdrinkstothe24-pack.

Example15

Usingratestodeterminecosts

OnMaya’stelephoneplan,everyphonecalltoaEuropeancountrycosts30centsperminute plusacallconnectionfeeof55cents.Jacqueshasadifferentplanwithnocallconnectionfee,butit costs40centsperminute tomakeacall.

Howmuchwilla10-minutecallcosteachperson?

Solution

Maya

30cents = $0.30and55cents = $0.55

Cost = 10 × $0.30 + $0.55 = $3.55

Jacques

40cents = $0.40

Cost = 10 × $0.40 = $4.00

Nowyoutry

Explanation

Multiplythe‘perminute’costbythenumberof minutes,thenaddtheconnectionfee.

Multiplythe‘perminute’costbythenumberof minutes.

UsingtheinformationinExample 15,completethistableofvalues.

Example16

Usingratestodetermineearnings

Baljitispaid$40perhour.Tarikispaid$30perhourforthefirst4hoursthentime-and-a-halfafterthat. Howmuchdoeseachpersonearnforworkinga10-hourshift?

Solution Explanation

Baljit

Earnings = $40 × 10 = $400

Tarik

First4hours:payrate = $30

Next6hours:payrate = $30 × 1.5 = $45

Earnings = ($30 × 4) + ($45 × 6) = $390

Nowyoutry

Multiplythepayratebythenumberofhours worked.

Calculatethetime-and-a-halfpayrate.

Multiplyeachpayratebythenumberofhours worked,thenaddthetwoamounts.

UsingtheinformationinExample 16,findthedurationofashiftwhichwillleadtothesamepayforboth workers.Thetablebelowmaybehelpful.

Exercise1I

1 Example14

Calculatethebestbuybetweenoption1and2.

Option1:6calculatorsfor$126 a

Option1:21gfor$8.61 b

Option1:$16.92for36L c

Option1:5batteriesfor$8.00 d

Option1:22pensfor$8.36 e

Option1:$598for23pairsofshoes f

Option1:36chocolatebarsfor$66.60 g

Option1:19kgfor$37.62 h

Option1:10Lofsoftdrinkfor$6.20 i

Option1:9gfor$4.77 j

Option1:10litresfor$5 k

BUILDING

Option2:24calculatorsfor$552

Option2:27gfor$15.39

Option2:$4.68for12L

Option2:12batteriesfor$14.76

Option2:30pensfor$10.80

Option2:$891for33pairsofshoes

Option2:20chocolatebarsfor$35

Option2:28kgfor$49.56

Option2:3Lofsoftdrinkfor$1.89

Option2:24gfor$10.08

Option2:12litresfor$7.40

2 Abreakfastcerealissoldinboxesofthreedifferentsizes: small(400g)for$5.00, medium(600g)for$7.20,large(750g)for$8.25

a

b

Findthevalueofeachboxin$/100g.

Whatisthecheapestwaytobuyaminimumof3kgofthecereal?

3 Example15 Xavierhasaninternationalmobilephonecontractthatchargesacallconnectionfeeof$0.30anda callrateof$0.43per30seconds.

a

b

c

WhatisthechargeifXaviermakesa30secondcall?

WhatisthechargeifXaviermakesa2minutecall?

WhatisthechargeifXaviermakesa5minutecall?

d WhatisthechargeifXaviermade50callswithadurationoflessthan60seconds?

WhatisthechargeifXaviermade100callswithadurationoflessthan30seconds?

4 Aninternationalmobilephoneplanhasamonthlychargeof$59ona24monthcontract.Inaddition, thecallsarechargedatarateof$0.90per60secondblockwitha$0.35connectionfee.

a Whatisthechargeforacalllasting1minute?

b Whatisthechargeforacalllasting2minutes?

c Whatisthechargeforacalllasting3minutesand30seconds?

d Whatisthechargeforacalllasting4minutesand20seconds?

e Determinethemonthlychargeformaking40calls(60seconds).

f Determinethemonthlychargeformaking90calls(60seconds).

5 Nailscost$4.80/kg.Howmanykilogramsofnailscanbeboughtfor$30?

6 Naturalgasischargedatarateof$0.014perMJ.

a Findthechargefor12500MJofnaturalgas.Answercorrecttothenearestdollar.

b Findthechargefor16654MJofnaturalgas.Answercorrecttothenearestdollar.

7

Kirrapayscouncilratesof$2915p.a.forlandvaluedat$265000.Lucypayscouncilratesof$3186 onlandworth$295000fromanothercouncil.

a WhatisKirracouncilchargeasarate$/$1000valuation?

b WhatisLucy’s councilchargeasarate$/$1000valuation? 8

Ryanworksasabuilderandcharges$45.50anhour. Howmuchdoesheearnforworkingthefollowing hours?

a 35hours

b 37hours

c 40hours

d 42hours

9

10

Zacharyisaplumberwhoearned$477foraday’swork.Heispaid$53perhour.Howmanyhours didZacharyworkonthisday?

11

Himariisahairdresserwhoearns$24.20perhour.Sheworksan8-hourday.

a HowmuchdoesHimariearnperday?

b HowmuchdoesHimariearnperweek?Assumesheworks5daysaweek.

c HowmuchdoesHimariearnperfortnight?Assumesheworks10daysafortnight.

d HowmuchdoesHimariearnperyear?Assume52weeksintheyear.

Kaiearns$32.50anhourasadriver.Heworks38hoursaweekatnormaltimeand5hoursaweekat doubletime.Findhisweeklywage.Answercorrecttothenearestcent.

12 Graceisacasualemployeewhoworked8hoursatnormaltimeand2hoursattime-and-a-half.Her normalrateofpayis$12.30perhour.Whatishertotalpayfortheabovetime?

13 Apopularsoftdrinkissoldintheseways:

• apackoftwelve300mLbottlesfor$12

• apackoffour330mLbottlesfor$8.50

• a600mLbottlefor$4

• a1.25Lbottlefor$3.25

• a2Lbottlefor$4.20.

Whichisthebestbuy?Explainhowyoureachedthisconclusion.

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1J Distance,timeandspeed

Learningintentions

• Toconvertratesbetweendifferentunits

• Tosolveproblemsinvolvingdistance,speedandtime Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingratesregularlyappearinHSCExaminations

Speed

Speedisaratethatcomparesthedistancetravelledtothetimetaken.Thespeedofacarismeasuredin kilometresperhour(km/h).Thespeedometerinacarmeasurestheinstantaneousspeedofacar.Theyare nottotallyaccuratebuthaveatoleranceof5%.GPSdevicesarecapableofshowingspeedreadingsbased onthedistancetravelledpertimeinterval.Mostcarsalsohaveanodometertoindicatethedistancetravelled byavehicle.

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‘Roadsign’ontherightisusedto remembertheformulas.Hidethe requiredquantitytodeterminethe formula.

Example17

Solvingproblemsinvolvingspeed

a Findtheaveragespeedofacarthattravels341kmin5hours.

b Howlongwillittakeavehicletotravel294kmataspeedof56km/h?

Solution

a S = D T = 341 5 = 68.2km/h

b T = D S = 294 56 = 5.25hor5h15min

Nowyoutry

Explanation

Writetheformulaforspeed.

Substitute341for D and5for T intotheformula.

Evaluateandexpressyouranswerinkm/h.

Writetheformulafortime.

Substitute294for D and56for S intotheformula.

Evaluateandexpresstheanswercorrecttothe nearesthour.

UsainBoltran100metresin9.58seconds.Whatwashisaveragespeedinmetrespersecond,correctto onedecimalplace?

Exercise1J

1 Example17 Findtheaveragespeed(inkm/h)ofavehiclewhichtravels: 160kmin2hours a

1 2 hours e 18kmin20minutes f

BUILDING

2 Findthedistancetravelledbyacaratanaveragespeedof62km/hforthefollowingjourneytimes. Giveyouranswers correcttothenearestkilometre.

3 Howlongwillittakeavehicletotravel(correcttothenearesthour):

240kmataspeedof80km/h? a 175kmataspeedof70km/h? b

160kmataspeedof48km/h? c 225kmataspeedof45km/h? d

240kmataspeedof40km/h? e 556kmataspeedof69.5km/h? f

4 TheMelbourneFormula1trackis5.303kmin length.Thetrackrecordis1minuteand24seconds.

Whatisthetrackrecordinhours? a

Whatistheaveragespeed(km/h)forthelap record?Answercorrecttotwodecimalplaces. b

5 AyannalivesinWollongongandtravelstoSydneydaily.Thecartriprequireshertotravelat differentspeeds.Mostoftenshetravels30kilometresat60km/hand40kilometresat100km/h.

Whatisthetotal distanceofthetrip? a

Howlong(inhours)doesthetriptake? b

Whatisheraveragespeed(inkm/h)whentravellingtoSydney?Answercorrecttotwodecimal places. c

6 Thomasdriveshiscartowork3daysaweek.Thedistanceofthetripis48km.Thetriptook 43minutesonMonday,50minutesonTuesdayand42minutesonWednesday.

Calculatetheaveragetimetakentotraveltowork. a

Whatistheaveragespeed(inkm/h)forthethreetrips? b

DEVELOPING

7 Alexandrajogs100metresin20seconds.Howmanysecondswouldittakehertojogonekilometre?

8 Acartravelsatarateof50metreseachsecond.Howmanykilometresdoesittravelin: oneminute? a onehour? b

9 Convertthefollowingspeedstometrespersecond.Answercorrecttothenearestwholenumber. 60km/h

260km/h

10 Anathleteruns100metresin10seconds.Ifhecouldcontinueatthisrate,whatishisspeedin kilometresperhour?

11 Asnailtravellingataconstantspeedtravels400mmin8minutes.Howfardoesittravelin 7minutes?

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12

Findtheaveragespeed(inkm/h,correcttothenearestwholenumber)ofavehiclewhichtravels:

350kmin1hour and30minutes a 600kmin2hoursand15minutes b

500kmin6hoursand10minutes c 64kmin1hourand30seconds d 36000min45minutes e 320min10seconds. f

13

Findthedistancestravelledbyacaratanaveragespeedof68km/hforthefollowingjourneytimes. Giveyouranswers correcttothenearestkilometre.

3hours15minutes a 5hoursand30minutes b 30minutes c 2minutes d 1hourand20minutes e 4hoursand10seconds f

14 Findhowlongwillittakeavehicletotravel(correcttothenearestminute):

450kmataspeedof82km/h a 50kmataspeedof60km/h b

250kmataspeedof49km/h c 580000mataspeedof62km/h d 24000mata speedof72km/h e 100kmataspeed of1km/min. f

Communicateyourreasoning,understandingandfluency

15 Themarathonisarunningracethatis42.2kmlong.Thewomen’smarathonworldrecord isapproximately2hoursand10minutes.Themen’sworldrecordisapproximately2hours. Inkilometresperhour,howmuchfasterdidthewomen’srecordholderneedtoruninordertofinish theracein2hours?Giveyouranswercorrecttoonedecimalplace.

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1K Distance–timegraphs

Learningintentions

• Tointerpretandanalysedistance–timegraphstosolveproblems

• Tosolveproblemsinvolvingdistance,speedandtime Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears9and10

• Questionsinvolvingdistance–timegraphsoccasionallyappearinHSCExaminations

Adistance–timegraphusuallycontains:

• ahorizontalaxiswhichshowsthetimetaken,inhoursor minutes

• averticalaxiswhichshowsthedistancefromsomeplace,in centimetres,metresorkilometres

• aseriesofstraightlinesorcurveswhichshowthepositionof theobjectduringthejourney.

Thedistance–timegraphaboveshowsajourneyawayfrom‘home’thenreturningtothestartingpoint.

• Thehorizontalsectionsindicate‘breaks’or‘rests’inthejourney,inwhichthepeoplewerestationary.

• Thestraight-linesectionsindicatethattheymovedataconstantrate.

• Thesteepertheline,thefastertheyweremoving.

Example18

Calculatingspeedfromadistance–timegraph

Thisdistance–timegraphshowsapersontravellingfromPoint A to Point B ataconstantspeed.

Howlongdidthejourneytake?

Howfarwasthejourney?

Solution

a 5hours

Explanation

Seetheverticalreddashedlineonthegraph.

b 25km Seethehorizontalreddashedlineonthegraph.

c 5kmin1hour = 5km/h

Alternativesolution

25 km in 5 hours

5 km in 1 hour ÷ 5 ÷ 5

Inthefirsthour,thepersonmovedfrom0kmto 5km.Thatis5kmin1hour.

Treattheinformationasyouwouldtreataratio.

Nowyoutry

Thisdistance–timegraphshowsatraintravellingfromTown A to Town B ataconstantspeed.

a Howlongdidthejourneytake?

b Howfarwasthejourney?

c Whatwastheconstantspeed?

Example19

Interpretingandanalysingadistance–timegraph

Thisdistance–timegraphshowsMira’scyclingjourney.

a Describewhathappenedduringthefirsthour,includingthe distancetravelledandthespeed.

b HowfardidMiraridealtogether?

c Whatwasheraveragespeedforthesecond(middle)partof thejourney,aftertherest?

d Includingthebreaks,whatwasheraveragespeedforthe wholejourney? 10

Solution

a Shestartedat6a.m.androde10kminone hour,soherspeedwas10km/h.

Explanation

Thefirsthourisfrom6a.m.to7a.m.Shestarted from0kmandmadeitto10km.

b 30km + 30km = 60km 30kmawayfromhomethen30kmbacktohome.

c Sherode10km.

From8:30to9:00is30minor0.5h

10 km in 0.5 hours

__ km in 1 hour × 2 × 2

Herspeedwas10 × 2 = 20km/h.

d Sherode60kmin5hours.

60 km in 5 hours

__ km in 1 hour ÷ 5 ÷ 5

Heraveragespeedwas60 ÷ 5 = 12km/h.

Nowyoutry

Usingthedistance-timegraphinExample 19:

Setitupasarate. Simplifyitlikearatio.

Setitupasarate. Simplifyitlikearatio.

a whatwastheaveragespeedforthelastpartofthejourney?

b excludingthebreaks,whatwasMira’saveragespeedforthewholejourney?

Exercise1K

1 Acarjourneyof300kmtakes5hours.The distance–timegraphforthisjourneyisshown opposite.Foreachdescriptionbelow,choosethe letteronthegraphthatmatchesit.

a Ahalf-hourrestbreakistakenaftertravelling 250km.

b Inthefirsthourthecartravels90km.

c Thecarisatrestfor1hour,90kmfromthe start.

d Thecartakes1.5hourstotravel90kmto 250km.

e Thedistancefrom250kmto350kmtakes 1hour.

BUILDING

2 Writeashortdescriptionaboutwhatishappeningateachpoint(A to G)alongAman’sjourney.

3 Abicyclejourneyisshownonthedistance–timegraph opposite.

a Whatisthetotaldistancetravelled?

b Whatisthetimetakenforthejourney?

c Howlongwasthecyclistatrest?

d Howfarhadthecyclisttravelledafter1hour?

e Howfarhadthecyclisttravelledafter4hours?

f Determinetheaveragespeedduringthelasttwohoursof thetrip.

graph of a bicycle journey

4

Drawareasonablyaccuratedistance–timegraphwhichincludesbothofthefollowingjourneys.

a Enzolefthomeandwalked2kmin30minutes,thenrestedfor30minutes,thenwalkedhome againin1hour.

b Marialeftthesamehomeandwalked2kmin45minutesthenturnedandwalkedhomein1hour.

Basedonthisgraph,whatwastheaveragespeedduringthe journey?

Thedistance–timegraphdescribesJerrell’straintrip.

a Howlongwasthereststop?

b Howfardidthetraintravelfromitsstartingpoint?

c Whatwasthetotaldistancetravelled?

d Determinetheaveragespeedduringthethirdhour ofthetrip.

e Determinetheaveragespeedduringthelasthourof thetrip.

7 Thisdistance–timegraphshowsoneofJane’strainingrides.

a Describewhathappenedduringthefirst90minutes, includingthedistancetravelledandthetimetaken.

b HowfardidJaneridealtogether?

c Whatwasheraveragespeedforthesecondpartofthe journey,afterthefirstbreak?

d Whatwastheaveragespeedforthelastpartofthe journey?

e Includingthebreaks,whatwasheraveragespeedforthe wholejourney,correcttoonedecimalplace?

f Excludingthebreaks,whatwasheraveragespeedforthe wholejourney?

8 Thedistance–timegraphdescribesElla’s cartrip.

a Howlongwasthereststop?

b Howfardidthecartravelfromits startingpoint?

c Howlongdidthetriptake?

d Determinetheaveragespeedduring thefirstfivehoursofthetrip.

e Determinetheaveragespeedduring thelastfourhoursofthetrip.

f Determinetheaveragespeedforthe entiretrip,correcttotwodecimal places.

9 Thedistance–timegraphbelowshowsashopper’swalkinashoppingmall.

Distance–time graph of a shopper’s walk

graph of a car trip

a Whatisthetotaldistancetheshoppertravelled?

b Whatwasthetotaltimetheshopperwasnotwalking?

c Whatwasthetotaldistancetheshopperhadtravelledbythefollowingtimes?

i 20seconds

ii 80seconds

iii 150seconds

Communicateyourreasoning,understandingandfluency

10 Studythetravelgraphbelow.Itdescribesthemovementsofasurferoveran8-minutetimeframe, starting50metresfromthebeach.

Distance from beach

Time

Writeaparagraphaboutthemovementsofthesurfer,includingestimatesofthetimespentmovingor beingstationary.

1L Fuelconsumptionrates

Learningintentions

• Tousefuelconsumptionratestocomparetheefficiencyofvehicles

• Tosolveproblems involvingfuelconsumptionrates

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingfuelconsumptionoccasionallyappearinHSCExaminations

Overtheyears,aspetrolhasbecomemoreexpensive,carmanufacturersaretryingtolowertheamountof fuelthatcarsuse.

Fuelconsumptionrates

Therearetwodifferentratesusedtoexpressfuelconsumptionofacar.Theyare:

• litresper100km(L/100km)

• kilometresperlitre(km/L).

Bothofthesearerates,likealltheotherratesintheprevioussections.Thefirstoneisunusual becausethesecondnumberis100km,not1km.However,itistheratemostcommonlyusedincar advertisements.

Example20

Calculatingfuelconsumptionrates

Acartravelled850kilometresusing78.2litresofpetrol.

Whatwasthefuelconsumptionrate,in:

a litresper100kilometres?

b kilometresperlitre,correcttoonedecimalplace?

Solution

Explanation

Writedownlitresfirst,thenkmsecond.Treatthe rateasifitisaratiobydividingbothnumbersby 8.5,sothatsecondnumberis100. Alternatively,usetheunitarymethod.

Writedownkmfirst,thenlitressecond.Treatthe rateasifitisaratio,bydividingbothnumbers by78.2,sothatthesecondnumberis1.

Nowyoutry

Acartravelled500kilometresusing40litresofpetrol.

Whatwasthefuelconsumptionrate,in: litresper100kilometres? a kilometresperlitre? b

Example20

1

Findthefuelconsumptionforeachofthefollowing,inL/100kmandkm/L.

a Ahmedcaruses 75litresofpetroltotravel750km.

b Grace’scaruses60litresofpetroltotravel300km.

c Anewcaruses120litresofpetroltotravel2400km.

d Matthew’scaruses100litresofpetroltotravel800km.

e Lincoln’scaruses45.5litresofpetroltotravel520km.

f Asmallcaruses36.9litresofpetroltotravel600km.

g Gemma’ssedanuses55.1litresofLPGtotravel950km.

h Asportscartravelled250kmusing28.5litresofpetrol.

i Anthony’smotorbikeuses167.5litresofLPGtotravel2500km.

j Tahlia’scaruses121.6litresofpetroltotravel3200km.

2 Maryamhasboughtausedcarwhose fuelconsumptionis7.8litrespetrolper 100kilometres.Sheisplanningtotravel aroundAustralia.Calculatethenumberof litresofpetrolMaryam’scarwilluseon thefollowingdistances.Answercorrectto thenearestwholenumber.

a Atripof4049kmfromDarwinto Perth.

b Atripof982kmfromSydneyto Brisbane.

c Atripof2716kmfromPerthto Adelaide.

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d Atripof658kmfromMelbournetoCanberra.

e Atripof732kmfromAdelaidetoMelbourne.

f Atripof309kmfromCanberratoSydney.

g Atripof3429kmfromBrisbanetoDarwin.

3 Charlietravels45kmtoworkand45kmfromworkeachday.

a Howmanykilometresdotheytraveltoandfromworkina5-dayworkingweek?

b Charliedrivesafour-wheeldrivewithafuelconsumptionof8L/100kmtoandfromwork. Howmanylitres ofpetroldoesCharlieusetravellingtoandfromwork?Answercorrecttoone decimalplace.

c WhatisCharlie’spetrolbillforworkifpetrolcosts$1.20perlitre?

4 AfamilycarusesLPGatarateof15L/100kmandthegastankholds72litres.Howfarcanittravel onatankofLPG?

5 Eviedrivesacarwithapetrolconsumptionof9litresofpetrolper100km.Petrolcosts $1.50perlitre.

a Howmanylitresoffueldoesthecarusefor300km?

b Howmanylitresoffueldoesthecarusefor50km?

c Whatisthecostoftravelling100km?

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d Whatisthecostoftravelling200km?

e Howfarcanshedriveusing$10worthofpetrol?Answertothenearestkm.

f Howfarcanshedriveusing$50worthofpetrol?Answertothenearestkm.

6 Maxdrivesatruckwithapetrolconsumptionof16L/100kmandthepetroltankholds90litres. Heisplanningatrip fromMoorebanktoMelbourne.ThedistancefromMoorebanktoMelbourneis 840km.MaxfilledupthepetroltankatMoorebank.

a Whatisthedistancetravelledononetankofpetrol?

b Howmanylitresofpetrolareneededonthistrip?

c HowmanytimeswillheneedtofillhistankbeforearrivingatMelbourne?Givereasonsforyour answer.

DEVELOPING

7 NatalieisplanningatripfromParramattatoCanberrausingacarwithafuelconsumptionof 9.6L/100km.ThedistancefromParramattatoCanberraviathehighwayis278kmandavoidingthe highwayis363km.ThecostofLPGis68.5centsperlitre.

a Howmuchwillthetripcostviathehighway?

b Howmuchwillthetripcostavoidingthehighway?

c Howmuchmoneyissavedbytravellingviathehighway?

8 AustinownsanSUVwithafuelconsumptionof10.9L/100kminthecityand8.4L/100km inthecountry.Austintravels12000kmperyearinthecityand20000kmperyearinthecountry. Theaveragecostofpetrolis$1.48perlitreinthecityand12centshigherinthecountry.

a Findthecostofpetroltodriveinthecityfortheyear.

b Findthecostofpetroltodriveinthecountryfortheyear.

c WhatisthetotalcostofpetrolforAustininoneyear?

d WhatisthetotalcostofpetrolforAustininoneyeariftheaveragecostofpetrolincreasedto $2.00inthecity?

CHALLENGING

9 EdenbuysafamilycarthatusesULPwithafuelconsumptionof9.9litres/100km.Tobybuysthe LPGversionofthe familycarwithafuelconsumption14.8litres/100km.BothEdenandToby average450km inaweekinthesameconditions.TheaveragepriceofULPis$1.50perlitreand LPGis$0.85perlitre.

a HowmanylitresoffuelareusedbyEdeninaweek?

b HowmanylitresoffuelareusedbyTobyinaweek?

c Calculateeachcar’syearlyconsumptionoffuel.

d WhatisEden’syearlyfuelbill?

e WhatisToby’syearlyfuelbill?

10

f Tobypaidanadditional$2400fortheLPGversionofthefamilycar.Howmanyyearswillit takeforthefuelsavingstoreach$2400orthebreak-evenpoint?Answercorrecttothenearest wholenumber.

g ResearchthecurrentfuelpricesusingULPandLPG.Howlongwillittakeforthefuelsavingto exceedtheinitialcosts?

Amotorbikeismovingatasteadyspeed. Whenthespeedis90km/hthebike consumes5litresofpetrol forevery 100kilometrestravelled.

a Thepetroltankholds30litres.How manykilometrescanthebiketravel onafulltankofpetrolwhenitsspeed is90km/h?

b Whenthespeedis110km/hthe bikeconsumes30%more petrolper kilometretravelled.Calculatethe numberoflitresper100kilometres consumedwhenthebiketravelsat 110km/h.

11 Aplanetravellednon-stopfromLosAngelestoSydney,adistanceof12057kilometresin13hours and30minutes.Theplanestartedwith180kilolitresoffuel,andonlandinghadenoughfueltofly another45minutes.

a Whatwastheplane’saveragespeedinkilometresperhour?Answercorrecttothenearestwhole number.

b Howmuchfuelwasused?Answercorrecttothenearestkilolitre. Communicateyourreasoning,understandingandfluency

12 Conductsomeresearchaboutthefuelconsumptionofsmall,energyefficientcars,andlargercars, suchasSUVs.

a Whatisthe lowestfuelconsumptionyoucanfind?Whatisthehighest?

b FuelforasmallcarthatcanuseE10mightbe$1.50perlitre.Foramuchlargercarthatuses dieselorpremiumunleaded98,fuelmightbe$1.80perlitre.Investigatethecurrentpricesforthe twocarsyouchose.

c HowmanykilometresdoestheaverageAustraliandriveperyear?

d Calculatethecostofpetrolperyearforthecarsyouchose.Istheremuchdifference?

e Writeupacomparisonof asmallefficientcarversusalargegas-guzzlingcar.

Keyideasandchaptersummary

Ratio

Dividingaquantity inagivenratio

Scaledrawing

Aratioisusedtocompareamountsofthesameunitsinadefiniteorder.Equivalent ratiosareobtainedbymultiplyingordividingbythesamenumber.

1 Converttheratiointotwofractions.

2 Multiplythequantitybybothfractions.

Scaleofadrawing = Drawinglength:Actuallength Scaleisexpressedintwoways:

• usingunits,suchas1cmto1m(or1cm = 1m)

• withnounits,usingaratiosuchas1:100.

Plansand elevations

Perimeter

Area

Volumeofwater andthetrapezoidal rule

Rate

Unitarymethod

Speedasarate

Aplanisaviewofanobjectfromthetop.Anelevationisaviewofanobjectfrom onesidesuchasafrontelevationorsideelevation.

Whencalculatingtheperimeteroftheland,multiplysidelengthsonanaerial photographbythescale.

Toestimatetheareaofland,divideitintosquaregridsandcountthenumberof squares.Tocalculatetheareaofland,usetheappropriateformula. Thetrapezoidalruleforareaoflandwithoneirregularboundarygivesan approximation.

Volumeofabodyofwaterofconstantdepthandatrapezoidalarea,orvolumeof rainfallfallingonatrapezoidalarea,canbecalculatedusingthetrapezoidalrulefor area A ≈ b 2 d f + dl andtheformulaforthevolumeofarightprism V = Ah

Arateisacomparisonofquantitiesexpressedindifferentunits.E.g.km/h, L/100km,$/hour.

Usedivisiontogodownto1.Multiplytofindtherequiredamount.

Speedisaratethatcomparesthedistancetravelledtothetime taken.

Distance–time graphs

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Fuelconsumption

Linegraphwithtimeonthehorizontalaxisanddistanceontheverticalaxis.

• Gradientoftheline = Verticalrise Horizontalrun = Distance Time = Speed

• Thesteepnessofaline(orgradient)indicatesthespeedoftheobject.

• Ahorizontallineindicatesthattheobjectisstationaryornotmoving.

Arate,expressedinlitres/100kmorkm/litre.

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook. ✓

Checklist

1A 1

Icanchangearatiotoasimplerform,inthreeways.

e.g.Considertheratio4:6.

Expresstheratio4:6insimplestform. a

Expresstheratio4:6intheform1: a. b

Expresstheratio4:6intheform b :1. c

1A 2 Icanusearatiotofindanunknownquantity.

e.g.If4:3isequalto a :24,whatisthevalueof a?

1B 3 Icanusethecapture-recapturetechniquetoestimateapopulation.

e.g.Abiologisttraps50carpfromadam,thentagsandreleasesthem.Onemonth laterhecaptures40carp,ofwhich16aretagged.Estimatethenumberofcarpin thedam.

1C 4 Icandivideaquantityinagivenratio.

e.g.Divide$400intheratio:

1D 5 Icanuseascaledrawingtocalculaterealdistances.

e.g.Onamapthescaleis1cm = 50m.Twopointsare3.2cmapartonthemap. Howfarapartaretheyinreallife?

1E 6

Icanuseascaleonabuildingplan.

e.g.Abuildingplanhasbeendrawnusingthescale1:100.Therefore,everymm ontheplanrepresentswhatdistance,incentimetres?

1F 7

Icancalculateperimeter,areaandvolumefromaplanandanaerialphoto,usingthe trapezoidalrulewhennecessary.

e.g.Thefollowingshaperepresentsthesurfaceofaconcrete pondusedforcultivatingfish.Ithasuniformdepthof1.5metres. Thescaleis1:500.

a

b

Estimatetheperimeter,inmetres,byassumingthecurveis asemicircle.

Estimatethearea,insquaremetres,usingthetwoapplicationsofthe trapezoidalrule.

Estimatethenumberoflitresofwaterrequiredtofillthepond. c Judgingbytheshape,isthisanunderestimateoroverestimateofthevolume?

d

1G 8 Icanconvertaratetootherunitsofmeasurement.

e.g.Adrippingtapwastes25millilitresofwatereveryminute.Convertthisrateto litresperhour.

1H 9 Icanusetheunitarymethodtosolveproblemsinvolvingrates.

Acakefor6peoplerequires4eggs.Howmanyeggswillberequiredforacakebig enoughtoserve15people?

1I 10

Icandeterminewhichproductisthebestbuy.

e.g.Adrinkcomesintwosizes:

• 1.25litresfor$3.25

• 2litresfor$4.20. Whichisthebestbuy?

1J 11

Icansolveproblemsinvolvingspeed,distanceandtime.

e.g.CathyFreeman’sbesttimeforthe400-metreracewas48.63seconds. Awombatcanrunataspeedof40km/hovershortdistances.Ifthewombatcould keepupthisspeedfor400metres,woulditbeatCathyFreeman?

1K 12 Icaninterpretandanalyseadistance–timegraph.

e.g.Considerthisdistance–timegraph.

a

Whatwastheaveragespeedonthefirstpartof thejourney?

b

Whatwastheaveragespeedonthelastpartof thejourney,correcttoonedecimalplace?

c

Whatwastheaveragespeedfortheentire journey,includingtherestbreak?

d

Whatwastheaveragespeedfortheentire journey,withouttherestbreak,correcttoone decimalplace?

1L 13

Icancalculatethefuelconsumptionrateofamotorvehicle. e.g.Rata’scaruses11.26litresper100km.

a

Howmanylitresofpetrolwillhiscaruseonatripof155kmfromSydneyto Newcastle?Answercorrecttothreedecimalplaces.

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b

Thecostofpetrolis$1.60perlitre.Howmuchwillthepetrolcostforthe 155kmtrip?

c

Howmanylitresofpetrolwillhiscaruseonatripof294kmfromArmidale toTaree?Answercorrecttothreedecimalplaces.

d

Thecostofpetrolis$1.35perlitre.Howmuchwillthepetrolcostforthe 294kmtrip?

Multiple-choicequestions

1 1A Theratioofadultstochildreninaparkis5:9.Howmanyadultsareintheparkifthereare 630children?

2 1C A360-gramlollybagisdividedintheratio7:5.Whatisthemassofthesmalleramount?

3 1D Ascaledrawinghasascaleof1:20.Whatistheactuallengthifthedrawinglengthofan objectis20mm?

4 1D Thescaleonamapisgivenas1mm = 150m.Ifthedistancebetweentwopointsis600m,whatis themapdistancebetweenthesepoints?

5 1G Youssefisacourierwhodelivers1parcelonaverageevery25minutes.Howmanyhoursdoesittake todeliver24parcels?

6 1G Ahosefillsa10Lbucketin20seconds.Whatistherateofflowinlitresperhour?

7 1G Whichofthefollowingistheslowestspeed?

Questions 8 and 9 refertothedistance–timegraphofthemovementofasnail.

8 1H Thetotalnumberofhoursthesnailwasatrestis: 2 A 3 B 5 C 10 D

9 1H Thespeedofthesnailinthelast5hourswas:

Distance–time graph of the movement of a snail

10 1L Whatisthefuelconsumptionforavehiclethattravelled340kmusing51litresofpetrol?

/100km A

Short-answerquestions

1 1A Completeeachpairofequivalentratios. 4:30 = 2: a 6:10 = :20 b 2:11 = 22: c 13:3 = :9 d 2:5 = :25 e 4:9 = :36 f

2 1A Trueorfalse?

1:3 = 5:9 a

Theratio2:3isthesameas3:2. b

Theratio3:5iswritteninsimplestform. c 30cm:1miswrittenas30:1insimplestform. d

3 1A Simplifythefollowingratios.

f 6:4 g

h

i

p

4 1C Divide:

$80intheratio7:9 a

200kgintheratio1:4 b 40mintheratio6:2 c $1445intheratio4:7:6. d

5 1A TheratioofthecostpriceofaTVtoitsretailpriceis5:12.Ifitscostpriceis$480,calculateits retailprice.

6 1C DanielandEddieownabusinessandsharetheprofitsintheratio3:4.

Theprofitlastweekwas$3437.HowmuchdoesDanielreceive? a

Theprofitthisweekis$2464.HowmuchdoesEddiereceive? b

7 1A Gabrielmixessandandcementintheratio5:2 byvolume.Ifheuses5bucketsofcement,how muchsandshouldheuse?

8 1A TheratioofVictoria’sheighttoWillow’sis8:7.IfVictoriais176cmtall,howtallisWillow?

9 1A Expresseachofthefollowingscalesasaratiointheform1: a 1cmto3m a 1mmto6cm b 1mto2.5km c

10 1D

11 1D

12 1D

Twocitiesare50kmapart.Howmanymillimetresapartaretheyonamapthathasascale of1:100000?

Thescaleonamapisgivenas1cm = 5km.Ifthedistancebetweentwopointsonthemapis 46mm,whatistheactualdistancebetweenthesepoints?Answerinkilometres.

Ascaledrawinghasascaleof1:50000.Whatisthedrawinglengthoftheseactuallengths? Expressyouranswerinmillimetres.

a

13 1D

14 1E

15 1F

16 1G

b

c 6.5km d

f

Thescaleonamapis1:400.Calculatetheactualdistancesifthefollowingarethedistancesonthe map.Expressyouranswerinmetres. Bikepath180cm a

b

Asectionofafloorplanisshownopposite. Thelongerdimensionofthelaundryis3metres.

a Estimateasuitablescaleforthefloorplan.

b Whatsymbolisusedforthebath?

c Whatdoes‘W.C.’representontheplan?

d Whatarethedimensionsofthelaundry?

Abodyofwaterhasthefollowingdimensions: b = 6m, d f = 20m, dl = 40mand h = 2.5m.Use thetrapezoidalruleforareaandtheformulaforthevolumeofarightprismtoanswerthefollowing.

a Findtheapproximatevolumeofthebodyofwater.

b Findtheapproximatevolumeofwaterthatfallsonthesameareawhenthereis9mmofrain.

Converteachratetotheunitsindicated.

$15/kgto$/g a 14400m/htom/min b 120cm/stomm/min c 4800kg/gtokg/mg d 14L/gtomL/kg e

/gtoc/mg f

17 1H If20metresofcurtainmaterialcosts$580,whatwouldbethecostof35metresofthesame material?

Robe

18 1H A5kgbagofricecosts$9.20.Whatisthecostofthefollowingamounts?

10kg a 40kg b

3kg c 7kg d

500kg e 250kg f

19 1I

Determinethebestbuybetweenoptions1and2.

Option1:5booksfor$110 a

Option1:$175for50L b

Option1:35gfor$357 c

Option1:4pictureframesfor$50 d

Option2:7booksfor$161

Option2:$54for15L

Option2:27gfor$270

Option2:6pictureframesfor$74.70

20 1J Findtheaveragespeed(inkm/h)ofavehiclewhichtravels: 784kmin8hours a 315 kmin4.5hours b 48kmin 1 4 hour c 64kmin40minutes. d

21 1J

Findthedistancetravelledbyacar(correcttothenearestkilometre)whoseaveragespeedis76km/h ifthejourneylasts:

22 1J Maddisonruns200metresin45seconds.Howmanysecondswouldittakehertorunonekilometre atthesamerate?

23 1J Aaravdrivestohismother’shouse.Ittakes45minutes.CalculateAarav’saveragespeedifhis motherlives48kmaway.Answercorrecttothenearestkm/h.

24 1J

25 1K

HowlongdoesittakeStellatodrive180kmalongthefreewaytoworkifshemanagestoaverage 100km/hforthetrip?

Thedistance–timegraphdescribesthejourneyof awalker.

a Whatisthetotaldistancetravelled?

b Howlongwasthepersonactuallywalking?

c Howfarhadthepersonwalkedafter:

i 1hour?

ii 2hours?

iii 4hours?

d Howlongdidittaketowalkadistanceof 12km?

Distance–time graph of a walker

26 1L

27 1L

Acartravels960kmon75litresofpetrol.Howfardoesittravelon50litres?

Farooqtravels51kmtoworkand51kmfromworkeachday.

a Howmanykilometresdoeshetraveltoandfromworkina5-dayworkingweek?

b Farooqdrivesacarwithafuelconsumptionof7.5L/100kmtoandfromwork.Howmanylitres ofpetroldoesFarooquse travellingtoandfromworkperweek?

c WhatisFarooq’spetrolbillforworkperweekifpetrolcostsare$1.52perlitre?

Extended-responsequestions

1 1C Thereare384passengersand144crewonacruiseship.Ifthereare22lifeboats,enoughfor everyone,howmanyareallottedtothepassengersandhowmanytothecrew?

2 1H PumpAcanfillatankin5minutes;pumpBcanfillthetankin10minutes.Howlongwillittaketo fillthetankifbothpumpsareworkingtogether?

2

Relative frequencyand probability

Birthdaycoincidencesinfootballteamsatthe2023 FIFAWomen’sWorldCup

Everyfouryears,certainnationalteamsfrom allovertheworldmanagetoqualifyforoneof thebiggestsportingeventsontheplanet:the FIFAWomen’sWorldCup.Theymustchoose 23 playerstorepresenttheircountryduring thissoccercompetition.Thirty-twocountries qualifiedforthe2023Women’sWorldCup, whichwasheldinAustralia.Wouldyouexpect thatmanyoftheseteamshadtwoormore playerswhocelebratetheirbirthdayonthe samedayeveryyear?

Forexample,thinkaboutthe 23 playersfrom Brazil.Thereare365daysthattheycould havebeenbornon,ignoringleapyears,so whatisthechancethattwoormoreofthem shareabirthday?Itseemsquitelow,butit happened.

Infact, 17 oftheteamshadatleasttwo playerswiththesamebirthday.Sojustover halfoftheteamsexperiencedthis coincidence,whichseemsextraordinary.

Fouroftheteamshad two pairsofplayers withmatchingbirthdays:

•Panama

•Brazil

•Columbia

•Denmark

Twooftheteams,MoroccoandNigeria,even hadthreepairsofplayerswithmatching birthdays.

Mathematically,thisisnotsurprisingatall.It turnsoutthatinanyrandomgroupof 23 people,thechancethattwoormorepeople willshareabirthdayisapproximately 50.73% Thischance,whichisprobablyhigherthan youguessed,comesfromthenumberofpairs thatcanbemadewith 23 peoplecompared withthenumberofdaysinayear.

Themathematicsofthiscounter-intuitive situationisexplainedhere: https://cambridge.edu.au/redirect/12148

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2A Languageofprobability

2B Definitionofprobability

2C Tablesandtreediagrams

2D Rangeofprobabilities

2E Complementaryevents

2F Relativefrequency

2G Expectedfrequency

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•Modelsandsolvesproblemsinvolving theprobabilitiesofsimpleand compoundevents(MST-12-S1-07).

©NSWEducationStandardsAuthority Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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2A Languageofprobability

Learningintentions

• Tounderstandandusethelanguageassociatedwithprobability

• Toassignwordssuchas‘likely’inreferencetothechanceaneventwilloccur

• Tobeabletolisttheelementsinasamplespace

• Tounderstandthateveryeventhasaprobabilityfrom0(impossible)to1(certain) Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Probabilityisthechancethatsomethingwillhappen.Weusefractions(from0to1),decimals(from0to1) andpercentages(from0%to100%)toassignanumericalvaluetoaprobability. TherearemanywordsandphrasesintheEnglishlanguagewhichareusedtodescribethechancethat somethingwillnothappen,mighthappenorwillhappen.

Discussion

Whichoneoftheseisclosestto0:‘slimchance’or‘fatchance’?

Wheredoes‘Buckley’schance’fitin?WhowasBuckley?Whatistheoriginofthephrase?Searchthe internet.

Probability

Probabilityisthechancethatsomethingwillhappen. Fractions,decimalsandpercentagesareusedforprobabilities.

Allprobabilitiescanbeplacedonascalefrom0(impossible)to1,or100%(certain)

TheEnglishlanguagecontainsmanywordsandphrasestodescribethechancethataneventwilloccur. Thisactivityinvitesyoutomatchthemtothescalefrom0%(impossible)to100%(certain).

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DownloadtheactivityfromyourInteractiveTextbook.

Areyouready?Alltheterminology,togetherinoneplace

Thischapterinvolvesmanywordsandphrases.Thisactivityinvitesyoutomatchwordstodefinitions, asyougothroughthechapter.

DownloadtheactivityfromyourInteractiveTextbook.

Example1

Usingthelanguageofprobability

Yinghasapear,anappleandanorange.Sherandomly selectsonepieceoffruit.Describethechanceofthe followingeventsusingthewords‘certain’,‘likely’,‘even’, ‘unlikely’or‘impossible’.

a Yingselectsabanana.

b Yingselectsanorange.

c Yingselectsapieceoffruit.

d Yingselectsapearoranapple.

Solution

Explanation

a Impossible Nochanceofselectingabanana.

b Unlikely Onechanceoutofthreeofselectingtheorange.

c Certain Selectingapieceoffruitmusthappen.

d Likely Twochancesoutofthreeofselectingthepearortheapple.

Nowyoutry

Wouldyousaythefollowingphrasesmean‘morethan50%chance’or‘lessthan50%chance’?

a Sportingchance

b Redhotchance

c Youdon’tstandthechanceofacinderinsnow

d Longodds

e Shortodds

f Youareoddsontowin

Samplespace

Samplespaceisthesetofallpossibleoutcomesorpossibleresultsofachanceexperiment.Forexample,if theexperimentistossingacoin,thenthesamplespaceisaheadandatail.Whenadieistossed,thesample spaceisthenumbers1,2,3,4,5and6.Eachoutcomeordatavalueisanelementofthesamplespace.The samplespaceisusuallylistedbetweencurlybrackets{},whicharesometimescalledbraces.

Samplespace

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Samplespaceisthesetofallpossibleoutcomesofachanceexperiment. Eachoutcomeordatavalueisanelementofthesamplespace.

Example2

Identifyingthesamplespace

a Dakuischoosingadayoftheweektostarthisholiday.Listthesamplespace.

b Howmanyelementsinthesamplespace?

Solution

a Samplespace = {Mon,Tue,Wed,Thu,Fri, Sat,Sun}

Explanation

Eachdayoftheweekisapossibleoutcome–Monday,Tuesday,Wednesday,Thursday,Friday, SaturdayandSunday.

Listeachelementbetweenthecurlybrackets.

b Thereare7elementsinthesamplespace. Counttheelementsinthesamplespace.

Nowyoutry

Howmanyelementsineachsamplespace?

Monthsoftheyear a Cardsinastandardpack b Tossingonecoin c Rollingonestandarddie d

Example3

Identifyingthesamplespace

a Twounbiasedcoinsaretossed.

i Whatisthesamplespace?

ii Howmanypossibleoutcomesarethere?

b Whatisthesamplespaceforthisspinner?

Solution

ai Firstcoin:HorT.

Secondcoin:HorT.

Samplespace = {HH,HT,TH,TT}

Explanation

Eachcoincouldlandonaheadoratail.

Ifthefirstcoinwasahead,thenthesecondcoincouldbe aheadoratail.

Ifthefirstcoinwasatail,thenthesecondcoincouldbe aheadoratail.

ii Numberofpossibleoutcomesis4. Counttheelementsinthesamplespace.

b Samplespace = {pink,yellow,green,blue}

Thesamplespaceisasetofallpossibleoutcomes.In thiscase,althoughtheprobabilitiesoflandingoneach colouraredifferent,thereareonlyfourdistinctresults possiblewhenthespinnerstops.

Thesamplespacelistsuniqueoutcomesratherthantheir frequencies.

Nowyoutry

Threecardsarenumbered1,2,and3.Acardischosenthenacoinistossed,sooneoftheoutcomesis2H.

a Listalltheoutcomesinthesamplespace

b Howmanyoutcomesarethere?

Exercise2A

1 Statewhethertheprobabilityofeacheventisimpossibleorcertain.

a ChristmasDayoccurringonthe25thDecember

b Winningarafflewithoutbuyingaticket

c Rollingthenumber7onastandardsix-sideddie

d FridaybeingthedayafterThursday

e Selectingtheletter‘A’fromlettersintheword‘SCHOOL’

f Thesumof1and2being3

g Choosinganevennumberfromthenumbers2,4,6and8

h Winningacarracewithoutacar

i Obtainingaheadoratailwhenacoinistossed

j Nightfollowingday

BUILDING

2 Example1 Describethechanceofthefollowingeventsusingthewords‘certain’,‘likely’,‘even’,‘unlikely’ or‘impossible’.

a Januaryhaving31days

b Tossingacoinandgettingatail

c Randomlychoosingapersonwhohasneverwatchedtelevision

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d WinningfirstdivisionLotto

e Choosingaredballfromabagcontaininganequalnumberofredandwhiteballs

f SnowonMountKosciuszkoduringacolddayinwinter

g Newbornbabybeingagirl

h Drawingaheartfromanormalpackofcards

i Tossingacoinandturningupahead

j Pickingavoweloraconsonantfromarandomword

3 Aliyahasabetterthanevenchanceofwinning themarathon.

a Whatwordcouldyouusetodescribethisprobability?

b Aliyaissickwithacoldandrunningmuchslower.What wordcouldbeusedtodescribeherchancesnow?

4 Twocoinsaretossed.Describeinwordstheprobabilityof thefollowingoutcomes.

a Twoheads

b Headandatail

c Twotails

5 Astandardsix-sideddieisrolled.Describeinwordsthe probabilityofthefollowingoutcomes.

a Oddnumber

b Numberbetween(andincluding)1and6

c Numberlessthan10

d Number3

6 Example2 Listthesamplespaceforthefollowingevents.

a Tossinga$2coin

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b Selectingaballfromaboxcontaining5ballscolouredred,green,white,blueandyellow

c ChoosingaletterfromthewordASCOT

d Drawingacardfromcardslabelledfrom1to5

e Thesexofanewbornkitten

f Selectingtheoutcomeofafootballmatch

7 Thearrowonthisregularpentagonisspunonce.

a Whatisthesamplespace?

b Howmanyelementsareinthesamplespace?

c Describeinwordstheprobabilityoflandingonred.

d Describeinwordstheprobabilityofnotlandingonblue.

8 Thearrowonthisregularhexagonisspunonce.

a Whatisthesamplespace?

b Howmanyelementsareinthesamplespace?

c Describeinwordstheprobabilityoflandingongreen.

Blue White

Yellow Red Green

Afairdieisrolled.

a Listthesamplespace.

b Howmanyelementsinthesamplespace?

10 Thereare10counterseachlabelledwiththelettersfromAtoJ.

a Acounterisselectedatrandom.Listallthepossibleoutcomes.

b Describeinwordstheprobabilityofselectingavowel.

11 ARugbyUnionmatchisbeingplayedbetweenAustraliaandSouthAfrica.

a Listthesamplespaceforthepossibleoutcomesofthematch.

b Howmanyelementsinthesamplespace?

c Iseachoutcomeequallylikely?Why?

12 Jacobislearningtoplaychess.HehasamatchagainsttheAustralianchampion.Therearethree outcomesforthematch–win,draworlose.

a Whatisthesamplespace?

b Aretheseeventsequallylikely?Why?

DEVELOPING

13 Fourcardsarenumbered1,2,3and4.Acardischosenthenacoinistossed.Howmanyoutcomes arethereinthesamplespace?

14 Theprobabilityofsixoutcomesis0,0.25,0.5,0.75,0.9and1.Assignoneofthesenumberstothe followingworddescriptions.

Evenchance a Likely b Certain c Verylikely d Impossible e Unlikely f

15 Abagcontainsone$10note,one$20noteandone$50note.Onenoteischosenfromthebag.Itis notreplaced.Thenanothernoteischosen.

a Listthesamplespaceforthefirstnote.

b Howmanydistinctelementsareinthesamplespaceofthefirstnote?

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c Assumethefirstnotechosenwasa$10note.Listthesamplespaceofthesecondnote.

d Assumethefirstnotechosenwasa$20note.Listthesamplespaceofthesecondnote.

e Assumethefirstnotechosenwasa$50note.Listthesamplespaceofthesecondnote.

16 Afaircoinandastandarddiearethrown.Describeinwordstheprobabilityofthe followingoutcomes.

a Headandanumberlessthan10

b Headandanumbergreaterthan10

c Tailanda1

d Tailorheadandanevennumber

17 ‘Sixstudentsenteraswimmingrace.Thechanceofaparticularstudentwinningis 1 6 .’Isthis statementtrueorfalse?Givereasonstosupportyouropinion.

18 Laurahasan80%chanceofwinningatennistournament.TheothertwoplayersMayaandEmma areequallylikelytowin.

a Whatisthesamplespace?

b Aretheoutcomesequallylikely?Why?

c WhatistheprobabilitythatMayawinsthetournament?

Communicateyourreasoning,understandingandfluency

19 Tesssaystwothings.

Statement1: “Whenyoutosstwocoinstogether,therearethreepossibilities:twoheads,twotailsor oneheadandonetail.”

Statement2: “Therefore,thechanceoftossingtwoheadsisone-third.”

Re-writeStatement1andStatement2sothattheyarecorrect.

20 Irolledafairstandarddiethreetimes.Irolleda4,threetimesinarow.

Rachelsaid:“Thereisareallygoodchancethatthenextonewillbea4.”

Sophiesaid:“Thereisalmostnochancethatthenextonewillbea4.”

Aditisaid:“Ifthatisafairdie,thechancethatthenextonewillbea4isone-sixth.”

WriteaparagraphinsupportofAditi’sstatement.

21 Designthreedifferentspinnersthathavethesamesamplespace S = {Red,Green,Blue}

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2B

Definitionofprobability

Learningintentions

• Tounderstandandapplythedefinitionofprobability

• Tounderstandthattheprobabilityofaneventcanbeexpressedasafractionoftheform‘7outof10’

• Tounderstandthatsomeoutcomesare‘equallylikely’andsomearenot

• Toexpressprobabilitiesusingfractions,decimalsandpercentages Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Thereisquitealotofprobabilityterminologytounpack.Forexample,an event canbearesultofa probabilityexperiment,suchas:

Rollinga6

• Rollinganumberlessthan5

• Tossingahead

• Tossingaheadthenatail

• Winningatennismatch

•

• Choosingaredball

Twosimilarprobabilityexperiments,withanimportantdifference

BagAcontainsaredball,awhiteballanda greenball.Oneischosenatrandom.

ThethreepossibleoutcomesR,WandGare ‘equallylikelyoutcomes’.

Thesamplespaceis {R,W,G}

Aneventis ‘randomlychooseawhiteball’ Whatistheprobabilityofchoosingawhiteball?

Theanswerisobvious.Itis1outof3andcanbe writtenasafraction,adecimalorapercentage.

P(white) = 1 3

Thereisaformulaforthissituation,usingthe letter P forprobability:

P(event) = numberoffavourableoutcomes totalnumberofoutcomes

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BagBcontainsaredballandtwowhiteballs.One ballischosenatrandom. RandWarenotequallylikelyoutcomes.

Isthesamplespace {R,W,W}?

OR

Isthesamplespace {R,W}? Aretheretwooutcomes orthreeoutcomes? Thisquestionisfrequentlydiscussedonline.

Aneventis‘randomlychooseawhiteball’ Whatistheprobabilityofchoosingawhiteball?

Thereare3balls.Twoofthemarewhite. Theprobabilityis2outof3andcanbewrittenas afraction,adecimalorapercentage.

P(white) = 2 3

Thisexperimentshowstheimportanceof beingcarefulwhencalculatingprobabilitywith outcomesthatarenotequallylikely.

Probabilitythatacertaineventwilloccur

Ifalltheoutcomesinthesamplespaceareequallylikely:

P(event) = numberoffavourableoutcomes totalnumberofoutcomes

A‘plainEnglish’,moreinclusiveversion:

P(event) = numberofwaystheeventcouldoccur numberofallthethingswhichcouldhappen

Areyouready?Expressingaprobabilityasafraction,decimalor percentage

Inthissection,youmayberequiredtostateyouranswerasasimplifiedfraction,butsometimes probabilitiesareexpressedasdecimalsorpercentages.Thisactivitywillshowyouhowthatisdone.

DownloadtheactivityfromyourInteractiveTextbook.

Example4

Calculatingtheprobability

Acoinischosenatrandomfrom7onedollar coinsand3twodollarcoins.Calculatethe probabilitythatthecoinisa: a onedollarcoin b twodollarcoin.

Solution

a P($1) = 7 10 = 0.7 = 70%

b P($2) = 3 10 = 0.3 = 30%

Nowyoutry

Explanation

7outof10coinsare$1.Formafraction.Express asafraction,decimalorpercentage.

3outof10coinsare$2.Formafraction.Express asafraction,decimalorpercentage.

Abagcontainsthreeredballs,twowhiteballsandonegreenball.Oneballischosenatrandom.Whatis theprobabilitythattheballis: white? a black? b red,whiteorgreen? c

Areyouready?Thecontentsofastandardpackof52playingcards

Forthissection,youneedtobefamiliarwiththecontentsofastandardpackof52playingcards. Whatare‘suits’?Whatare‘picturecards’?Howmanycardsarered? DownloadtheactivityfromyourInteractiveTextbook.

Adeckofplayingcards

Anormaldeckofplayingcardshas52cards.

• Therearefoursuitscalledclubs,spades,heartsanddiamonds.

• Ineachsuitthereare13cardsfromacetoking.

• Thereare3picturecardsineachsuit(jack,queenandking).

• Halfthecardsareredandhalfareblack.

Example5

Calculatingtheprobabilityfromplayingcards

Asinglecardisrandomlychosenfromthepack.Whatistheprobabilitythatitisa: redfour a diamond b picturecard. c

Solution

a P(Red4) = 2 52 = 1 26

b P(Diamond) = 13 52 = 1 4

c P(Picture) = 12 52 = 3 13

Nowyoutry

Explanation

2ofthe52cardsareredfours. Reducetosimplestform.

13ofthe52cardsarediamonds. Reducetosimplestform.

12ofthe52cardsarepicturecards. Reducetosimplestform.

Thefouracesareremovedfromthepackofcards.Onecardisthenchosenatrandom.Whatisthe probabilitythatthecardisa: redfour? a diamond? b picturecard? c

Exercise2B

1 Example4 Whatistheprobabilityofthefollowingexperiments?

a Acarddealtfromanormaldeckofcardsisadiamond.

b Adayselectedatrandomfromtheweekisontheweekend.

c Aheadresultswhenacoinistossed.

d Aletterchosenrandomlyfromthealphabetisavowel.

e Atworesultswhenadieisrolled.

f Asixischosenrandomlyfrom{2,4,6,8,10}.

BUILDING

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2 Abagcontains5blueand3redballs.Findtheprobabilityofselectingthefollowingatrandom.

a Ablueball.

b Aredball.

c Notaredball.

3 Rangichoosesoneballatrandomfromhisgolfbag.Thetablebelowshowsthetypeandnumberof golfballsinhisbag. Typeofgolfball Number

FindtheprobabilityofRangichoosing: aSrixon a aCallaway b aTitleist c notaSrixon. d

4 Anunbiasedcoinistossedthreetimes.Onthefirsttwotossestheresultistails.Whatisthe probabilitythattheresultofthethirdtosswillbeatail?

5 InAmberAvethereare3highschoolstudents,4primaryschoolstudentsand5preschoolstudents. OnestudentfromAmberAveischosenatrandom.Whatistheprobabilitythataprimaryschool studentischosen?

6 Aboxcontains3blue,4greenand2whitecounters.Findtheprobabilityofdrawingatrandoma counterthatis: blue a green b whiteorblue c notblue. d 7 Example5 Acardischosenatrandomfromastandarddeckof52playingcards.Findtheprobabilityof choosing: thesevenofclubs a aspade b aredcard c aredpicturecard d anine e thesixofhearts f anevennumber g apicturecard h ablackace. i

8 Theweatheronaparticulardayisdescribedaseitherwetordry.Thereforethereisanevenchanceof awetday.Doyouagreewiththisstatement?Giveareason.

DEVELOPING

9 Adiewith16facesmarked1to16isrolled.Findtheprobabilitythatthenumberis: anoddnumber a neithera1nora2 b amultipleof3 c greaterthan12 d lessthanorequalto15 e asquarenumber. f

10

Awheelcontainseightevenlyspacedsectorsnumbered1to8.Thewheelisspununtilitstopsat anumber.Giventhatthewheelisequallylikelytostopatanynumber,findtheprobabilitythatthe wheelstopsat:

a7 a anumbergreaterthan5 b anoddnumber c a1or2 d anumberlessthan10 e anumberdivisibleby3. f

11 Caseyisdealtfourcardsfromanormaldeck:two acesandtwokings.Whatistheprobabilitythatthe nextcardis:

a anotherace?

b anotherking?

c notanace?

d notaking?

CHALLENGING

12 Twocardsaredrawnatrandomfromanormaldeckofcards.Whatistheprobabilitythatthesecond cardis: atwoifthefirstcardwasatwo? a anaceifthefirstcardwasanace? b thesixofclubsiffirstcardwasa10? c atwoifthefirstcardwasaking? d adiamondifthefirstcardwasadiamond? e apicturecardifthefirstcardwasapicture card? f

13 Afour-digitnumberisformedfromthedigits2,3,4and5withoutreplacement.Whatisthe probabilitythatthenumber: startswiththedigit4? a isgreaterthan3000? b endswitha2ora3? c is2345? d

14 Ifsixstudentsenteraswimmingrace,thenthechanceofanyparticularstudentwinningmustbe 1 6 . Isthisstatementtrueorfalse?Givereasonstosupportyouropinion.

15 ThechanceofrainonSaturdayis70%.ThechanceofrainonSundayis50%.Therefore,thechance ofrainthisweekendis120%. Whydoesthisnotmakeanysense?

2C Tablesandtreediagrams

Learningintentions

• Tousetables(arrays),treediagramsandprobabilitytreestolistalltheoutcomesofmultistage experiments

• Todeterminetheprobabilityofmultistageevents

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Someprobabilityexperimentsare‘multistage’.Forexample:

• tossingacointwice

• tossingacointhenrollingadie

• selectingthreecardsfromapackofcards.

Tablesandtreediagramsareusedtolistallthepossibleresultsforamultistageexperiment. Ifacoinistossedthenadieisrolled,oneofthemultistageeventsisH3.

Tables,alsoknownasarrays

Atableisanarrangementofinformationinrowsandcolumns.Thetable showsallthepossibleoutcomesfortossingtwocoins.Therearetwo events:tossingthefirstcoinandtossingthesecondcoin.Theoutcomes ofthefirsteventarelisteddowninthefirstcolumn(HeadorTail).The outcomesofthesecondeventarelistedacrossthetoprow(HeadorTail). Eachcellinthetableisanoutcome.Therearefourpossibleoutcomes.

Samplespace = {HH,HT,TH,TT}

Tables(arrays)

Atableisanarrangementofinformationinrowsandcolumns. Eachcellinthetableisadatavalueoranoutcome.

Example6

Usinganarraytolistmultistageevents

Tworedcards(R1,R2)andoneblackcard(B1)areplacedinabox.Twocardsareselectedatrandom withreplacement.(i.e.thefirstcardisputbackintotheboxbeforethesecondoneischosen).

a Constructatabletolistthesamplespace.

b WhatistheprobabilityofselectingR1R1orR2R2?

Solution

a R1 R2 B1

R1 R1R1 R1R2 R1B1

R2 R2R1 R2R2 R2B1

B1 B1R1 B1R2 B1B1

b P(R1R1orR2R2) = 2 9

Nowyoutry

Explanation

Listtheoutcomesofthefirststage(firstcard)downthe firstcolumn.Therearethreeoutcomes:R1,R2andB1.

Listtheoutcomesofthesecondstage(secondcard)across thetoprow.Therearethreeoutcomes:R1,R2andB1.

Writetheoutcomeineachcellusingtheintersectionof therowandcolumn.

Formafraction.

RepeatExample 6,butthistimethefirstcardisnotreplacedbeforethesecondcardischosen.

Areyouready?Fillinginblanktreediagrams

Thisactivitywillhelpyoudrawyourowntreediagramsandfillthemincorrectly.

DownloadtheactivityfromyourInteractiveTextbook.

Treediagramsforequallylikelyoutcomes

Atreediagramisadiagramthatdetailstheoutcomesofamultistageexperimentwithequallylikelyoutcomes. Itshowseacheventasabranchofthetree.Thetreediagrambelowshowsallthepossibleoutcomesfor tossingtwocoins.Theoutcomesofthefirststagearelisted(HorT)withtwobranches.Theoutcomesofthe secondstagearelisted(HorT)withtwobranchesoneachoftheoutcomesfromthefirststage.Thesample spaceisHH,HT,THandTT.

Treediagrams

• Drawatreediagramwitheacheventasanewbranchofthetree.

• Alwaysdrawlargecleartreediagramsandlistthesamplespaceontheright-handside.

Drawingatreediagram

Acoinistossedandadieisrolled.

a Constructatreediagramofthesetwostagestoshowthesamplespace.

b Whatistheprobabilityofaheadanda1? Example7

b P(H1) = 1 12

Nowyoutry

Explanation

Drawbranchesforfirststage–tossingacoin. Tossingacoinhastwooutcomes(headortail)sothereare twobranches.

Drawbranchesforthesecondstage–rollingadie. Rollingadiehassixoutcomes(1,2,3,4,5or6)sothereare sixbranches.Drawsixbranchesforeachofthetwooutcomes fromthefirststage.

Usethebranchesofthetreetolistthesamplespace.Writethe outcomesdowntheright-handside.

Numberoffavourableoutcomes(H1)is1.Thetotalnumber ofoutcomesis12.

Formafraction.

Therearethreecardsinapack,numbered1,2,and3.Acardischosenatrandom.Itisreplaced. Asecondcardischosen.

a Constructatreediagramtoshowthesamplespace.

b Whatistheprobabilitythatthesumofthetwocardsislessthan5?

Probabilitytreesforoutcomeswhicharenotequallylikely Ext

Sometimes,apairofbranchesinatreediagrammaynotbeequallylikely.Forexample,intheyear2022in Australia,51%ofbabiesbornweremalecomparedwith49%ofbabiesbornfemale.

Iftwobabieswererandomlyselected,theprobabilitiesarewrittenonthebranches, asshown,usingdecimalsorfractions.Thetreediagramisthensometimescalleda ‘probabilitytree’.

Theprobabilitiesarethencalculatedusingmultiplication.

P(MM) = 0.51 × 0.51 = 0.2601

P(FF) = 0.49 × 0.49 = 0.2401

Ifitisnecessarytocombinetwoormoreoutcomes,theirprobabilitiesare added together.

P(samesex) = P(MM) + P(FF) = 0.2601 + 0.2401 = 0.5002

Example8

Drawingaprobabilitytree

Asmallpackofcardscontains5redcardsand2bluecards.Onecardischosenandthenreplaced. Asecondcardisthenchosen.

a Drawaprobabilitytree.

b Whatistheprobabilitythatbothcardsarered?

c Whatistheprobabilitythatthetwocardsaredifferentcolours?

d Whatistheprobabilitythat oneormoreofthecardsarered?

Solution

Explanation

Thefractionsarewrittenonthebranches. Multiplytwofractionstogettheprobabilityofeach outcome.Checkthatthesumofthelastcolumnis1.

b P(RR) = 5 7 × 5 7 = 25 49

c P(differentcolours) = 10 49 + 10 49 = 20 49

d P(oneortwored) = 10 49 + 10 49 + 25 49 = 45 49

Alternatively, 49 49 4 49 = 45 49

Nowyoutry

Multiplythetwofractions.

Addthetwodifferentoutcomes.Theyareusuallythe sameaseachother.

Addthethreedifferentoutcomes.

Thefouroutcomesaddto1,sosubtract P(BB)from1, asthatistheonlyoutcomeyoudonotneedtoinclude. Thisisaveryusefulshortcut.

RepeatExample 8,butthistimethefirstcardis not replacedbeforethesecondcardischosen.

Exercise2C

1 RileyandBaileyareplanningtohavetwochildren.

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BUILDING

a Useatabletolistthenumberofelementsinthesamplespace.Considerthesexofeachchildas anevent.

b Whatisthetotalnumberofoutcomes?

2 Twofairdicearethrownandtheirsumrecorded.

a Useatabletolistallpossibleoutcomes.

b Howmanyelementsareinthesamplespace?

3 Example6 Amenuhasthreeentrees(E1,E2andE3)andfour mains(M1,M2,M3andM4). Useatabletolistallpossibleoutcomes.

4 Threepeople(A,BandC)appliedforamanager’spositionandtwopeople(DandE)appliedforan assistantmanager’sposition.

a Useatabletolisttheallthepossibleoutcomes.

b Whatisthetotalnumberofoutcomes?

5 Onebagcontainstwodiscslabelled‘X’and‘Y’.Asecondbagcontainsfourdiscslabelled‘D’,‘E’, ‘F’and‘G’.Adiscischosenfromeachbagatrandom.Useatabletolistthesamplespace.

DEVELOPING

6 Threeyellowcards(Y1,Y2,Y3)andonegreencard(G1) areplacedinabox.Twocardsareselectedatrandom withreplacement.

a Useatabletolistthenumberofelementsinthe samplespace.

b Findtheprobabilityofthefollowingselections: i Y1Y2 ii twogreen iii twoyellow.

7 Threecards(king,queenandjack)areplacedfacedownonatable.Onecardisselectedat randomandtheresultrecorded.Thiscardisreturnedtothetable.Asecondcardisthenselected atrandom.

a Useatabletolisttheelementsinthesamplespace.

b Findtheprobabilityofthefollowingselections:

i KJ

ii twokings

iii nokings.

Example7

8

Twocoinsaretossedandtheresultsrecorded.

a Listthesamplespacebycompletingthistreediagram.

b Findtheprobabilityofthefollowingresults:

i twotails

ii headthenatail

iii oneheadandonetail.

9 Asurveyhastwoquestionswhoseanswersare‘Yes’or‘No’.Constructatreediagramtolistthe samplespace.

10 TherearethreequestionsinaTrueorFalsetest.

a Fillintheremainderofthepartiallydrawntree diagramandlistthesamplespace.

b Howmanyelementsinthesamplespace?

c Findtheprobabilityofthefollowingrandom selections:

i alloftheselectionsarefalse

ii twotrueselectionsandonefalseselection

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iii atleastoneoftheselectionsisfalse. 1st 2nd 3rd

11

12

Amaleandafemaleschoolcaptainwillbevotedonfromthreemalestudents(Dom,TomandJohn) andthreefemalestudents(Lena,SinaandGina).

a Drawatreediagramtolistallpossiblepairsforschoolcaptain.

b WhatistheprobabilitythatTomandGinawillbethevotedcombination?

c WhatistheprobabilitythatLenawillbeoneofthecaptains?

Aspinnerhasanequalnumberofredandgreensections.Thisspinnerisspuntwice.

a Useatreediagramtolistthetotalpossibleoutcomes.

b Whatistheprobabilityofthesamecolourwhenthespinnerisspuntwice?

13

14

Ebonytossesacoinandspinsaspinner,whichhasared,anamberandagreensection. Eachcolouronthespinnerisequallylikely.

a Useatreediagramtolistthesamplespace.

b Whatistheprobabilityofaheadonthecoinandeitheraredorgreensectiononthespinner?

Fourcards(ace,king,queenandjack)areplacedfacedownonatable.Onecardisselectedat randomandtheresultrecorded.Thiscardisreturnedtothetable.Asecondcardisthenselectedat random.Useatreediagramtolistallthepossibleoutcomes.

CHALLENGING

15 Therearefourcandidatesforthepositionsofleaderanddeputyleader.Thefourcandidatesare Mohammed,Bridget,ConnorandDanielle.

a Constructatreediagramwiththeleaderasthefirsteventandthedeputyleaderasthesecond event.Useatreediagramtolistthesamplespace.

b WhatistheprobabilityofBridgetbeingtheleaderordeputyleader?

16 Atwo-digitnumberisformedusingthedigits3,5and7.Thesamenumbercanbeusedtwice. Thefirstdigitchosenisthetensdigitandtheseconddigitchosenistheunitsdigit.

a Constructatreediagramtolistthesamplespace.

b Whatistheprobabilityofforminganumberlessthan70?

17 Thenumberoflaptopssoldinthepasttwoweeksis shownontheright.

a OfallbrandXlaptopssold,whatfractionwassoldin week2?

b Whatpercentageofallsalesinweek1wasbrandY?

c Duringthetwoweeks,whatwastheprobabilitythat anylaptopsoldwouldbebrandX?

18

Example8 Ext RepeatQuestion 6,assumingthefirstcardisnotreplacedbeforethesecondcardischosen.

19 Ext RepeatQuestion 7,assumingthefirstcardisnotreplacedbeforethesecondcardischosen.

20 Ext RepeatQuestion 8,foracoinwhichisbiasedsothatitlandsonheads60%ofthetime.

Communicateyourreasoning,understandingandfluency

21 Yourcalculatorhasabutton .Whatdoesitdo?Searchtheinternet. Writedownthevalueof:

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Thelastoneisa68digitnumber.Thatisthenumberofdifferentwaysapackofcards canbeshuffled.

2D Rangeofprobabilities

Learningintentions

• Tounderstandthateveryeventhasaprobabilityfrom0to1

• Tounderstandthatthesumoftheprobabilitiesofalltheeventsinanexperimentis1

Past,presentandfuturelearning

• TheseconceptsbuilduponpriorlearningfromYears7to10

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Theprobabilityofaneventthatisimpossibleis0andtheprobabilityofaneventthat iscertainis1.Probabilityisalwayswithinthisrange,orfrom0to1.Itisnotpossible tohavetheprobabilityofaneventas2.Itisalsoimportanttorealisethefollowing, whichistrueforeveryprobabilityexperiment:

Whenalltheoutcomesarelisted,thesumofalltheirprobabilitiesis1.

Rangeofprobability

Probabilityofaneventisbetween0and1.

P(A) + P(B) + ... = 1 A, B, ... areallthepossibleoutcomesorevents.

Example9

Usingtherangeofprobability

Aboxcontainsred,yellowandbluecards.Theprobabilityofselectingaredcardis 3 5 andtheprobability ofselectingayellowcardis 1 10 .Whatistheprobabilityofselectingabluecard?

Solution

P(R) + P(Y) + P(B) = 1 3 5 + 1 10 + P(B) = 1

Explanation

Writetheformulafortherangeofprobability. Substituteintotheformulatheprobabilitiesoftheother events P(R) = 3 5 and P(Y) = 1 10 P(B) =

Solvetheequationbymaking P(B)thesubjectofthe equation.

Simplifythefractionifpossible.

Writetheanswerinwords.

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Probabilityofabluecardis 3 10

Nowyoutry

Foraparticularcouple,thechancetheywillhaveamalechildis10%.Theyareplanningtohavetwo children.Whatisthechancethattheywillnothavetwomalechildren?

Exercise2D

BUILDING

1 Example9 Ahatcontainsticketslabelledas‘A’,‘B’and‘C’.Theprobabilityofselectingticket A is 3 10 andthe probabilityofselectingticket B is 7 15

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a Whatisthevalueof P(A)?

b Whatisthevalueof P(B)?

c Whatistheprobabilityofselectingaticketwiththeletter‘D’?

d Whatistheprobabilityofselectingtickets A,B or C?

e Whatistheprobabilityofselectingticket C?

2 Abagcontainsblack,yellowandwhitecards.Theprobabilityofdrawingablackcardis57%and theprobabilityofdrawingayellowcardis8%.Whatisthevalueofthefollowing,expressedasa fractioninsimplestform?

P(Black) a P(Yellow) b P(White) c

3 InaparticulareventtheprobabilityofJunwinninga goldmedalis 3 8 andasilvermedalis 1 4 .Thereisno bronzemedal.

a WhatareJun’schancesofwinningagoldorasilver medal?

b WhatareJun’schancesofnotwinninganymedals?

4 Somepicturecardsfromadeckofcardsareplacedfacedownonthetable.Theprobabilityof drawingakingis0.25andaqueenis0.60.Whatisthevalueofthefollowingexpressedasa decimal?

a P(King)

b P(Jack)

c P(Jack) + P(King)

d P(King) + P(Queen) + P(Jack)

5 Therearefouroutcomesofanexperiment.Threeoftheoutcomeshaveprobabilitiesof20%, 25%and40%respectively.Whatistheprobabilityofthefourthoutcome?

6 Abiaseddieisrolled.Theprobabilityofobtaininganevennumberis0.4andtheprobabilityofa 1ora3is0.3.Findthevalueofthefollowingprobabilities.

a P(1,2,3,4,5,6)

b P(2,4,5,6)

c P(Odd)

DEVELOPING

7 Adiscischosenatrandomfromabagcontainingfivedifferentcolours:black,green,pink,red andwhite.If P(B) = 1 5 , P(G) = 2 13 , P(P) = 2 9 and P(R) = 1 6 ,findtheprobabilityofthefollowing outcomes.

Blackorgreendisc a Pinkorreddisc b

Blackorreddisc c

Black,greenorreddisc e

Black,greenorpinkdisc d

Black,green,pinkorreddisc f

8 Acardischosenatrandomfromsomeplayingcards.Theprobabilityofaspadeis0.24,the probabilityofaclubis0.27andtheprobabilityofaheartis0.23.Findtheprobabilityofthe followingoutcomes.

Blackcard a Redcard b

Cluboraheart c

Diamond e

Spadeoraheart d

Diamondoraclub f

9 JuliaandNatashaareplayingagameinwhichastandardsix-sideddieisrolled.Juliawinsifaneven numberisrolled.Natashawinsifanumbergreaterthanthreeisrolled.Whatistheprobabilitythat thenumberrolledisneitherevennorgreaterthanthree?

10 Abagcontainswhite,greenandredmarbles.Theprobabilityofselectingawhitemarbleis 2 7 and theprobabilityofselectingagreenmarbleis 1 8 .Whatistheprobabilityofselectingaredmarble?

CHALLENGING

11 Thenumbers1to20arewrittenonseparatecards.Onecardischosenatrandom.Whatisthe probabilitythatthecardchosenisaprimenumberorisdivisibleby3?

12 OneletterisselectedatrandomfromawordcontainingthelettersT,A,M,P,R.Itisgiventhat P

a Findtheprobabilityofthefollowingoutcomes.

LettersTorA i LettersTorP ii

LettersMorP iii

LettersA,MorP iv

LetterT,A,MorP v LetterR vi

b Thewordcontains10letters.FromthelettersT,A,M,P,Rhowmanyofthefollowinglettersare intheword? T i

c Whatistheword?(Hint:anAustralianplace.)

Communicateyourreasoning,understandingandfluency

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13 Yourcalculatorhasabutton .Whatdoesitdo?Searchtheinternet. Writedownthevalueof:

2E Complementaryevents

Learningintentions

• Tobeabletodeterminethecomplementofanevent

• Tousethecomplementtodetermineprobabilities

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Sometimes,thesimplestwaytofindtheprobabilitythatsomethingwillhappenistofindtheprobabilitythat itwon’thappen,thensubtractthatfrom1.

If E issomeeventthatcouldoccurinaprobabilityexperiment,likerollinga5withadie,thenthe complementof E isalltheothereventswhichcouldoccur.

Thinkof‘complement’asmeaning‘opposite’.Thecomplementof E iswrittenas E

If E is‘rollinga5withastandarddie’, E is‘rollinga1,2,3,4or6’.

Inthiscase, P(E) = 1 6 and P(E) = 5 6

If E is‘tossingahead’, E is‘tossingatail’.

Inthiscase, P(E) = 1 2 and P(E) = 1 2

Theprobabilityofaneventandtheprobabilityofthecomplementalwayshaveasumof1.

Iftheprobabilityofaneventis0.7,theprobabilityofthecomplementis1 0.7,whichis0.3.

P(aneventdoesnotoccur) = 1 P(theeventdoesoccur)

P(E) = 1 P(E)

ComplementaryEvents

P(E) + P(E) = 1or P(E) = 1 P(E)

E –Eventoroutcome

E –Complementofevent E

Areyouready?Usethecomplement,ornot?

Sometimes,theeasiestwaytofindtheprobabilitythatsomething will happenistofindtheprobability thatitwill not happen.

DownloadtheactivityfromyourInteractiveTextbook.

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Example10

Usingthecomplement

Lisaselectsacardatrandomfromanormalpack. Findtheprobabilityofobtainingthefollowing outcomes.

a Nota10

b Notablackjack(i.e.notajackofclubsorspades)

c Notapicturecard

Solution

a P(10) = 1 P(10) = 1 4 52 = 48 52 = 12 13

b P(blackjack) = 1 P(blackjack) = 1 2 52 = 50 52 = 25 26

c P(picture) = 1 P(picture) = 1 12 52 = 40 52 = 10 13

Nowyoutry

Explanation

Writetheformulaforthecomplement. Substituteintotheformulatheprobabilityfora10 or P(10) = 4 52

Evaluate. Simplifythefraction.

Writetheformulaforthecomplement. Substituteintotheformulatheprobabilityfora blackjack or P(blackjack) = 2 52

Evaluate. Simplifythefraction.

Writetheformulaforthecomplement. Substituteintotheformulatheprobabilityfora picturecard or P(picture) = 12 52 . Evaluate. Simplifythefraction.

(Forthepurposesofthisquestion,ignoreleapyears,so1year = 365days) Remywasbornon29March.

a WhatistheprobabilitythatAbdullahwasbornonthesameday?

b WhatistheprobabilitythatAbdullahwasnotbornonthesameday?

c WhatistheprobabilitythatLanawasnotbornonthesamedayasRemyorAbdullah?

1 Whatistheeventthatisthecomplementofthefollowingevents?

a Selectingablackcardfromanormalpackofcards

b WinningfirstprizeinLotto

c Throwinganevennumberwhenadieisrolled

d Obtainingatailwhenacoinistossed

e Drawingaspadefromanormalpackofplayingcards

f Choosingagreenballfromabagcontainingablue,aredandagreenball

2 Findthevalueof P(E),giventhefollowinginformationaboutevent E

BUILDING

3 ThechancesoftheSydneySwanswinningthepremiershiparegivenas29%.Whatarethechances thattheSydneySwanswillnotwinthepremiership?

a Expressyouranswerasadecimal.

b Expressyouranswerasafraction.

Theprobabilityofobtaininga3onabiaseddieis0.6.Whatistheprobabilityofnotobtaininga3?

5 TheprobabilityofarainydayinMarchis 11 15 .WhatistheprobabilitythataparticulardayinMarch doesnothaverain?

6 Theprobabilityofdrawingaredmarblefromabagis 5 8 .Whatistheprobabilityofnotdrawingared marble?Expressyouranswerasa: fraction a decimal b percentage. c

DEVELOPING

7 Aballischosenatrandomfromabagcontainingfourdifferentcolours:brown,orange,purpleand yellow.If P(O) = 2 11 , P(P) = 2 9 and P(Y) = 1 4 ,findtheprobabilityofthefollowingoutcomes.

Notayellowball a Notanorangeball b

Notapurpleball c Orangeorapurpleball d Yelloworapurpleball e Notabrownball f Abrownball g Notanorangeorayellowball h

8 Suhaimselectsacardatrandomfromanormalpack.Findtheprobabilityofobtainingthefollowing outcomes.

Notaqueen a Notaredace b

9 Whatistheprobabilitythatapersonselectedatrandomwill: a notbebornonSaturday? b notbebornonaweekend?

CHALLENGING

10 A12-sideddiehasfacesmarked1to12.Thedieisbiased.If P(8) = 0.1, P(2) = 0.15and P(3) = 0.91,find:

P(8) a P(2) b

P(3) c

P(8) + P(8) d

P(2) + P(2) e P(3) + P(3) f

P(2) + P(8) g P(2or 8) h

11 Onecardisselectedatrandomfromanon-standardpackofplayingcards.If P(ace) = 8%, P(king) = 7%and P(queen) = 10%,findtheprobabilityofthefollowingoutcomes.

Notanace a Notaking b Notaqueen c Kingoraqueen d Aceoraqueen e Notanace,kingorqueen f

12 Theprobabilityofselectingacardlabelled‘T’from32cardsisgivenas P(T ) = 3 16

a Whatistheprobabilityofnotselectingacardlabelledwitha‘T’?

b Howmanyofthe32cardswerelabelledwitha‘T’?

Communicateyourreasoning,understandingandfluency

13 Readthefirstpageofthischapter,regardingthebirthdaycoincidencesinWorldCupteams.This questionexplainsthemathsbehindit.

a Youwanttofindtheprobabilitythattwoormorepeopleshareabirthday.Thatis,youwantto know P(twoormorepeoplehavethesamebirthday).Whatisthecomplementaryevent?

b Readcarefully:

• forgetaboutleapyears.Assumethatayearhas365days

• youwanteverybodytohavea different birthday

• thefirstpersonarrivesandsaystheirbirthdayis11March.Thatdateisnow‘taken’

• thesecondpersonarrives.Whatistheprobabilitythispersonhasadifferentbirthday?

• thirdperson:Whatisthe probabilitythispersonhasadifferentbirthdaytothefirst twopeople?

• fourthperson:Whatistheprobabilitythispersonhasauniquebirthday?

• fifthperson,etc.

Copyandcomplete:

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Probability ofunique birthday

c Multiplyalltheprobabilitiestogether,thensubtracttheanswerfrom1andconverttoapercentage correcttotwodecimalplaces.Thisis P(2ormoreofthese5peopleshareabirthday).Verylow probability.

d Extendthepatternto23people.Aspreadsheetlikethis willbeveryuseful.

e Showthattheprobabilitythattwoormorepeopleshare abirthdayis50.73%,correcttotwodecimalplaces.

f Conclusion:Bythetimeyoureachthe23rd person, thereisabetterthan50%chancethattwoormore ofthemshareabirthday.Trythisexperimentwith 23randompeople.

g WehaveassumedthatnobodyisbornonFebruary29in aleapyear.Whatisanotherassumptionwehavemade?

2F Relativefrequency

Learningintentions

• Tounderstandthattheprobabilityofaneventmaybeapproximatedbytherelativefrequency

• Toestimateprobabilitiesusingrelativefrequencies

• Torecognisethatincreasingthenumberoftrialsproducesrelativefrequenciescloserinvaluetothe theoreticalprobabilities

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Sometimesitcanbedifficultorimpossibletocalculatetheprobabilityofanevent.

Forexample,aninsurancecompanyneedstoknowtheprobabilitythatan18year-oldmaledriverwillhavea caraccident,intheyearahead,sotheycansetthepriceoftheirinsurancepolicies.

Sometimesitispossibletoconductaprobabilityexperimentanduserelativefrequencyasanestimate fortheprobability.

Forexample,whenathumbtackfallstothegroundit couldlandintwodifferentways,asshowninthephoto.

Harukitriedthis50times.Thesearetheresults:

Basedonthistrial,therelativefrequencyfortheoutcome‘pointup’canbeexpressedas 28 50 ,whichreduces to 14 25 ,or0.56asadecimal,or56%asapercentage.

Thesefigurescanbeusedasanestimateoftheprobability.

P(pointup) ≈ 56%

Thequestionswhichfacescientistsandresearchersare:

• Are25trialsenoughtodrawthatconclusion?

• Isthisinformationreliableandconclusive?

• Ifnot,howmanytrialswouldbeenough?

Generallyspeaking,ifthenumberoftrialsisincreased,therelativefrequenciesbecomecloserinvaluetothe theoreticalprobability.

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Note:Theauthorsconductedthisexperiment200timeswiththesamethumbtack.After200trials,the relativefrequencywas60.5%.

Relativefrequency

Relativefrequencyisusuallycollectedfromasurveyoraprobabilityexperiment

Relativefrequency = Numberoftimesadatavaluewasobserved Totalnumberofdatavalues

Theresultcanbeexpressedasafraction,decimalorpercentage.

Ifthenumberofdatavaluesissufficient,relativefrequencycanbeusedasanestimateofthetheoretical probability.

Example11

Findingtherelativefrequency

Anexperimentoftossingtwocoinswas completedandthenumberofheadsrecordedin thefrequencytableshown.

Findtherelativefrequencyofobtainingthe followingnumberofheads:

Solution

a 100 + 192 + 108 = 400

Rel.freq. = 100 400 = 1 4 = 0.25or25%

Relativefrequencyof0headsis0.25.

b Rel.freq. = 192 400 = 12 25 or0.48or48%

Relativefrequencyof1headis0.48.

c Rel.freq. = 108 400 = 27 100 = 0.27or27%

Relativefrequencyof2headsis0.27.

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Explanation

Addthefrequencycolumntodeterminethetotal numberoffrequencies.

Writetheformulaforrelativefrequency.

Substitutethefrequencyandtotalnumberof frequenciesintotheformula.

Simplifythefractionifpossibleorexpressasa decimal.

Writetheanswerinwords.

Writetheformulaforrelativefrequency. Substitutethefrequencyandtotalnumberof frequenciesintotheformula.

Simplifythefractionifpossibleorexpressasa decimal.

Writetheanswerinwords.

Writetheformulaforrelativefrequency. Substitutethefrequencyandtotalnumberof frequenciesintotheformula.

Simplifythefractionifpossibleorexpressasa decimal.

Writetheanswerinwords.

Nowyoutry

Athumbtackwastossed200times.Theseweretheresults:

Frequency Relativefrequency

Pointup 121

Pointdown 79

a Findtherelativefrequency,asapercentage,for‘pointup’.

b Findtherelativefrequency,asapercentage,for‘pointdown’.

Exercise2F

1

Example11 Thefrequencytableshowstheoutcomesof anexperiment.Whatistherelativefrequency forthefollowingoutcomes?Expressasa fractioninsimplestform.

2 Thefrequencytableshowstheoutcomesof anexperiment.Whatistherelativefrequency forthefollowingoutcomes?Answercorrect tothreedecimalplaces.

3 Thefrequencytableshowstheoutcomesof anexperiment.Whatistherelativefrequency forthefollowingoutcomes?Answerasa percentagecorrecttoonedecimalplace.

4 Thefrequencytableshowstheoutcomesof anexperiment.Whatistherelativefrequency forthefollowingoutcomes?Answerasa percentage.

BUILDING

5

6

Calculatetherelativefrequencyforeachofthesenumbersifthetotalfrequencyis48.Writeyour answerasafractioninsimplestterms.

Calculatetherelativefrequencyforeachofthesenumbersifthetotalfrequencyis40.Writeyour answerasapercentage.

DEVELOPING

7 Aretailstoresold512televisionslastyear,ofwhich32werefaultyandreturnedtothestore.What istherelativefrequencyofafaultytelevisionlastyear?Answerasapercentage,correcttotwo decimalplaces.

8 ApistolshooterattheOlympicGameshitsthetarget24outof25attempts.Whatisthe relativefrequencyoftheshooterhittingthetarget?Givetheanswerasadecimal,correcttotwo decimalplaces.

9 Overthecourseofamonth,alocalvetsaw142dogsand126cats.Whatistherelativefrequencyfor adog?Answerasafractioninlowestterms.

10 Inanexperimentadiewasthrown 120times.Thefrequencyofeachdata valueisshowninthefrequencytable.

a Howmanytimeswouldyouexpectto obtaineachofthedatavalues?

b Doyouthinkthatthedieisfair? Giveareasonforyouranswer.

11 Createthespreadsheetbelowusingthefrequencytableinquestion 4.

a CellB10hasaformulathataddscellsB5toB9.Enterthisformula.

b TheformulaforcellC5is‘=B5/$B$10’.Itistheformulaforrelativefrequency.Filldownthe contentsforC6toC9usingthisformula.

c CellsD5toD9havethesameformulaascellsC5toC9.Enterthisformulaandformatthecellsto apercentage.

12 Afrequencydistributiontableisshownbelow.

a Whatisthevalueof y?

b Whatisthevalueof x?

c Whatisthetotalnumberofscores?

13 Performanexperimentbyrollingadie120times.

a Useafrequencytabletorecordtheresultsoftheexperiment.

b Calculatetherelativefrequencyofeachoutcome.

c Whatresultwouldyouhavepredictedforeachoutcome?

d Compareyourresultstothoseoftheotherstudentsinyourclass.

14 Rolltwostandarddiceandworkoutthedifferencebetweenthetwoscores.Forexample,ifonedie showsa2andtheotherdieshowsa5,thedifferenceis3.Dothis36times.

a Useafrequencytabletorecordtheresultsoftheexperiment.

b Calculatetherelativefrequencyofeachoutcome.

c Whatresultwouldyouhavepredictedforeachoutcome?

d Compareyourresultstothoseoftheotherstudentsinyourclass.

15 Performanexperimentbytossingtwocoins80times.

a Useafrequencytabletorecordtheresultsoftheexperiment.

b Calculatetherelativefrequencyofeachoutcome.

c Whatresultwouldyouhavepredictedforeachoutcome?

d Compareyourresultstothoseoftheotherstudentsinyourclass.

Communicateyourreasoning,understandingandfluency

16 Ashvinsaid:“Lastyearitrainedon140daysoutof365inDubbo.Therefore,thechanceitwillrain tomorrowisabout38%.”

MaketwoormorecommentswhichcastdoubtoverAshvin’sstatement.

2G Expectedfrequency

Learningintentions

• Tounderstandtheconceptofexpectedfrequency

• Tocalculatetheexpectednumberoftimesanoutcomewilloccurbymultiplyingtheprobabilitybythe numberoftrials

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations

Theexpectedfrequencyisthenumberoftimesthataparticulareventshouldoccur.Itmaynotequalthe actualresults.Forexample,whenacoinistossedtheprobabilityofgettingaheadis 1 2 .Hence,ifacoinis tossed100times,theexpectednumberofheadsis50or 1 2 × 100.Clearly,ifacoinistossed100timesit maynotresultinexactly50heads.However,thelargerthenumberoftrialsthemoreaccuratetheexpected frequencywillbe.

ExpectedFrequency

Expectedfrequencyisthenumberoftimesthataparticulareventshouldoccur.

Expectedfrequency = n × p = np

n = numberoftimestheexperimentisrepeated

p = probabilityoftheevent

Theexpectedfrequencymaynotbeawholenumber.Itisanestimateofwhattoexpect.Forexample,when adieistossed,theprobabilityofgettingasixis 1 6 .Hence,ifadieistossed100timestheexpectednumber ofsixesis 1 6 × 100or16 2 3 .Clearly,itisnotpossibletohave 2 3 ofasix.Theexpectationisawholenumber closeto16 2 3

Findingtheexpectedfrequency

Twocoinsaretossed120timesandtheresultsrecorded.

a Whatistheexpectedfrequencyfortwoheads?

b Whatistheexpectedfrequencyforaheadandatail? Example12

Solution

a P(HH) = 1 4

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Expectedfrequency = np = 1 4 × 120 = 30

Explanation

Calculatetheprobabilityoftwoheads. Numberoffavourableoutcomesis1(HH).The totalnumberofoutcomesis4(HH,HT,TH,TT). Writetheformulaforexpectedfrequency. Substituteintotheformula. Evaluate.

b P(HTorTH) = 2 4 = 1 2

Expectedfrequency = np = 1 2 × 120 = 60

Nowyoutry

Calculatetheprobabilityforaheadandatail. Numberoffavourableoutcomesis2(HT,TH).The totalnumberofoutcomesis4(HH,HT,TH,TT). Writetheformulaforexpectedfrequency. Substituteintotheformula. Evaluate.

Whenthreecoinsaretossed,theprobabilityofHHH = 1 8 .Ifthisexperimentisconducted50times,how manytimeswouldyouexpecttotossHHH?

Exercise2G

BUILDING

1 Example12 Theprobabilityofaredtrafficlightatanintersectionis 1 3 .Howmanyredtrafficlightsareexpected onatripthatpasses through54intersections?

2 Theprobabilityofapersonlivinginacertaincommunityofdevelopingmelanomaisfouroutof nine.Thereare1404peoplelivinginthiscommunity.Whatistheexpectednumberofpeoplewho willdevelopmelanoma?

3 SorenandCaitlinareplanningtohavefivechildren.Ageneticcounsellorhascalculatedtheyhave a40%chanceofhavingachildwithgreeneyes.HowmanyofSorenandCaitlin’schildrenare expectedtohavegreeneyes?

4 Amiraisagoalshooterforhernetballteam.Sheusuallyscoreson88%ofhershots.Thisyearshe hashad225attemptsatgoal.

a HowmanygoalswouldyouexpectAmiratohavescoredthisyear?

b HowmanygoalswouldyouexpectAmiratohavemissedthisyear?

5 Akiraisaprofessionalgolferwhohasa78%chanceofbreakingpar.Heplays150gamesofgolfina year.HowmanytimeswouldyouexpectAkiratobreakparinayear?

6 Theprobabilityofaworkerinanindustrialplanthavingan accidentis0.12.Theindustrialplantemploys175workers. Whatistheexpectednumberofaccidents?

7 Adieisrolled480timesandtheresultsrecorded.

a Whatistheprobabilityofthrowinga4?

b Howmany4sareexpected?

c Whatistheprobabilityofthrowinganoddnumber?

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d Howmanyoddnumbersareexpected?

e Whatistheprobabilityofthrowinganumbergreaterthan2?

f Howmanynumbersgreaterthan2areexpected?

g Whatistheprobabilityofthrowinganumberdivisibleby3?

h Howmanynumbersdivisibleby3areexpected?

8 Fivecards(A,K,Q,Jand10)areplacedfacedownonatable.Onecardisselectedatrandomand replaced.Asecondcardisthenselectedatrandom.Thisexperimentisrepeated200times.

a Whatistheprobabilityofselectingtwoaces?

b Howmanydoubleacesareexpected?

c Whatistheprobabilityofselectinganacefollowedbyaking?

d Howmanyacesthenkingsareexpected?

e Whatistheprobabilityofexactlyoneofthecardsbeinga10?

f Howmanysingle10sareexpected?

9 Athree-digitnumberisselectedfromcardslabelled3,4and5.Thefirstcardselectedisthe hundredsdigit,thesecondcardisthetensdigitandthethirdcardistheunitsdigit.Thecardsare selectedwithoutreplacement.Thisselectionisrepeated30times.

a Whatistheprobabilitythenumberstartswiththedigit3?

b Howmanynumbersstartingwiththedigit3areexpected?

c Whatistheprobabilitythenumberis453?

d Howmanytimesisthenumber453expected?

e Whatistheprobabilitythenumberendswitha4ora5?

f Howmanynumbersendingwitha4or5areexpected?

10 Abagcontains6yellowdiscsand5reddiscs.Onediscisdrawnfromthebag,itscolournotedand thenreplacedbeforetheseconddiscisdrawn.Thisprocessisrepeated352times.

a Howmanyofthediscsareexpectedtobeyellowdiscs?

b Howmanyofthediscsareexpectedtobereddiscs?

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11 Createthespreadsheetbelow.

a CellC8hasaformulathatmultipliescellsB8andB4.Enterthisformula.

b FilldownthecontentsofC8toC11usingtheformulaincellC8.

c Changethenumberoftrialsfrom230to800.ObservethechangeinC8:C11.

12 Thereare240familieswiththreechildren.Assume P(male) = 50%.

a Howmanyofthesefamiliesareexpectedtohavethreemalechildren?

b Howmanyofthesefamiliesareexpectedtohaveexactlyonemalechild?

c Howmanyofthesefamiliesareexpectedtohaveexactlytwomalechildren?

d Howmanyofthesefamiliesareexpectedtohavenomalechildren?

CHALLENGING

13 Onecardisselectedatrandomfromanormalplayingpack.Thecardisnotedandthenreplaced. Thisexperimentisrepeated2704timeswiththecardsbeingreplacedeachtime.

a Whatistheexpectednumberoftimestheresultwouldbeaspade?

b Whatistheexpectednumberoftimestheresultwouldbeanace?

c Whatistheexpectednumberoftimestheresultwouldbeapicturecard?

d Whatistheexpectednumberoftimestheresultwouldbearedcard?

14 Twodicearetossedoneaftertheotherontoatable.Thiseventisrepeated144times.

a Onhowmanyoccasionswouldyouexpecttheresulttobea6thena1?

b Onhowmanyoccasionswouldyouexpecttheresulttobetwo3s?

c Onhowmanyoccasionswouldyouexpecttheresulttobetwooddnumbers?

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Communicateyourreasoning,understandingandfluency

15

16

Meirollstwostandarddice,thenworksoutthedifferencebetweenthetwoscores.Forexample,if sherollsa5anda2,thedifferenceis3.

Shepays$1everytimesheplaysthisgame.

Ifherdifferenceis0,1or2,shelosesherdollar.

Ifherdifferenceis3,4or5,shegetsherdollarback,plusanotherdollar.

a Copyandcompletethesetables.

b Usethetablestoexplainwhythisgameisnotfair.

c IfMeiplaysthisgame100times,howmuchmoneywouldsheexpecttolose?

d Changetherulestomakethegamefair.

Thesyllabussaysthatstudentsshould:‘Examinetheuseofstatisticsandtheassociatedprobabilities inshapingdecisionsmadebymedia,governmentsandcompanies’.Australia’smedia,governments andsomecompaniesareheavilyinvolvedinthegamblingindustry.

Usetheinternettoresearchthefollowing:

a HowmuchmoneydidpeopleinNSWloseonpokermachinesinarecentyear?

b Whataresomeoftheorganisationswhichbenefitfromthisloss?

c Theprobabilityisclear:peoplewhoplaypokermachineswilllosemoneyinthelongrun.Some peopleareaddictedtoplayingpokermachines.Ourgovernmentsareawareofthis.Whatare someofthethingsthattheNSWandAustraliangovernmentshavedonetoaddresspokermachine addiction?

Suggestion:Watchthedocumentary‘Ka-Ching!PokieNation’tofurtherexplorepokermachine addiction.

Keyideasandchaptersummary

Languageof probability

Definitionof probability

Arraysandtree diagrams

Rangeof probabilities

Complementary events

Probabilityisthechanceofsomethinghappening. Samplespaceisthesetofallpossibleoutcomes. Eachoutcomeisanelementofthesamplespace. Equallylikelyoutcomeshaveanequalchanceofoccurring. Multistageeventconsistsoftwoormoreevents.

Ifalltheoutcomesareequallylikely,

Probability(Event) = Numberoffavourableoutcomes Totalnumberofoutcomes or

P(E) = n(E) n(S )

Atable(alsoknownasanarray)isanarrangementofinformationinrows andcolumns.

Atreediagramisanotherwaytolistallpossibleresultsofamultistageexperiment.

Probabilityofaneventthatisimpossibleis0andcertainis1. Probabilityofaneventrangesfrom0to1.

P(A) + P(B) + = 1

A, B, areallthepossibleoutcomesorevents.

Complementofanevent E istheeventnotincluding E

Probabilityofaneventanditscomplementwillsumto1.

P(E) + P(E) = 1or P(E) = 1 P(E)

E –Eventoroutcome

E –ComplementofeventEortheoutcomesnotincludingevent E

Relativefrequency

Expectedfrequency ofanevent

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Relativefrequencyisanestimatefortheprobabilityofanevent.

Relativefrequency = Numberoftimesadatavaluewasobserved Totalnumberofdatavalues

Expectedfrequencyisthenumberoftimesthataparticulareventshouldoccur.

Expectedfrequency = n × p = np n = numberoftimestheexperimentisrepeated p = probabilityoftheevent

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook.

Checklist

2A 1 Icanusetheterminologyassociatedwithprobability.

e.g.Iamgoingtorandomlychooseoneofthe5vowelsweuseintheEnglish alphabet.Whatisthesamplespace?

2B 2 Icanusefractions,decimalsandpercentagestoassignprobabilitiestooutcomes.

e.g.Iamgoingtorandomlychoosealetterfromthealphabet.Whatisthe probabilitythatitisavowel?

2C 3 Icanusearraysandtreediagramstolisttheoutcomesofaprobabilityexperimentand solveproblems.

e.g.Acoinistossedanda4-sideddieisrolled.

a Listtheoutcomesinanarray.

b Drawatreediagram.

c Thecoinisbiasedandshowsupasheads60%ofthetime.Thedieis biasedandrollsa‘1’40%ofthetime.Whatistheprobabilityofthe outcomeH1?

2C 4 Icanusetreediagramstosolveproblemswithreplacementandwithoutreplacement.

e.g.Abagcontains5redballsand3greenballs.

a Oneballischosen,notedandthenreplaced.Thenanotherballischosen. Whatistheprobabilitythatthetwoballsarethesamecolour?

b Repeatpart a,giventhatthefirstballisnotreplaced.

2D 5 Icanusetherangeofprobabilities.

e.g.Inabagcontaining20colouredblocks, P(red) = 2 5 and P(black) = 1 5 .How manyblocksareredorblack?

2E 6 Icanusecomplementaryeventstosolveproblems.

e.g.InQuestion 5,whatistheprobabilityofchoosingablockwhichisneitherred norblack?

2F 7 Icanuserelativefrequenciestoestimateprobabilities.

e.g.Ichoseacardfromabag,thenreplacedit.Thiswasdone50times.30of thecardspickedwerered.Whatistherelativefrequencyofchoosingared,asa percentage?

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2G 8 Icandetermineanexpectedfrequencybasedontheprobabilityofanevent.

e.g.UsingQuestion 7,ifIconductedtheexperiment250times,howmanycards wouldyouexpecttobered?

Multiple-choicequestions

1 2A Aneventhastheprobabilityofoccurringequalto0.Whatwordcanyouusetodescribethis probability?

2 2A Howmanyelementsarethereinthesamplespacewhentwocardsareselectedwithoutreplacement fromcardslabelled1to7?

3 2B

Athree-digitnumberisformedfromthedigits5,7,8and9.Whatistheprobabilitythatthenumber willbeodd?

4 2B Onecardisselectedfromcardslabelled1,2,3,4and5.Whatistheprobabilityofanevennumber oranumberdivisibleby5?

5 2B Aletterischosenatrandomfromtheword‘NEWCASTLE’.Whatistheprobabilitythattheletter will not beavowel?

6 2C Whentwodicearerolled,howmanyoutcomesarethere?

7 2D Abagcontainsblack,whiteandgreyballs.Theprobabilityofselectingablackballis0.3andagrey ballis0.6.Whatistheprobabilityofselectingawhiteball?

8 2F Thefrequencyofaneventis6andthetotalnumberoffrequenciesis20.Whatistherelative frequency?

9 2G Twounbiasedcoinsaretossed100times.Whichcalculationillustratestheexpectednumberoftimes youwouldgetatailandahead?

10 2G Acardisselectedatrandomfromanormalpackofplayingcards.Thecardisthenreplaced.Whatis theexpectednumberoftimesyouwouldselectakingfrom260trials?

Short-answerquestions

1 2A Describeeachoftheseeventsaseitherimpossible,likely,unlikelyorcertain.

a Throwingan8whenastandarddieisthrown.

b Astudenttravellingtoschoolonthewrongbus.

c Apieceofgoldbeingfoundinyourbackyard.

2 2A Listthesamplespaceofthefollowingevents.

a Selectingacardfromcardslabelledwithanevennumberfrom1to10.

b Aspinnerwitheachsectionlabelledwithavowel.

c Choosingatrandomaballfromabagcontainingthreewhiteandfourblueballs.

3 2B Araffleticketisdrawnfromaboxcontaining50raffleticketsnumberedfrom1to50.Findthe probabilityofthefollowing outcomes.

Thenumber50 a Anevennumber b Anumberlessthan20 c Anumbergreaterthan30 d Anumberdivisibleby5 e Asquarenumber f

4 2B Whatistheprobabilityofchoosingablackcardfromastandarddeckofcards?

5 2B Fourkingsaretakenfromastandarddeckofcardsandplacedfacedownonatable.Onecardis selectedatrandom.Whatistheprobabilityofselecting: thekingofclubs? a ablackking? b apicturecard? c

6 2B Therearefourgirls,threeboysandtwoadultsinahouse.Ifonepersonischosenatrandom,whatis theprobabilitythattheperson: isagirl? a isaboy? b isagirloraboy? c

7 2B Aneight-sideddiehasthenumbers1to8onit.Whatistheprobabilityofrollingthefollowing outcomes?

Thenumber2 a Eithera3ora5 b

Thenumber9 c Anumberdivisibleby3 d

Anoddnumber e Aprimenumber f

8 2B TherearefivestudentsinagroupandtheirnamesareAdam,Shazia,Cam,AnisaandDavid.Ifone nameischosenatrandom,whatistheprobabilityofselectinganame: withfiveletters? a withtheletter‘a’? b withonevowel? c

9 2C Apaperbagcontainsthreegreen,fourbrownandfiveyellowbeads.Towinagame,Zoeneedsto drawtwogreenbeadsfromthebagintwodraws,withoutreplacement.Howmanyelementsarein thesamplespace?

10 2E Afaircoinistossedthreetimes.Theprobabilityofthrowingthreetailsis0.125,twotailsis0.375 andonetailis0.375.Whatistheprobabilityofthefollowingoutcomes?

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Notails a Threeortwotails b

Atleastonetail c Notthrowingahead d

Notthrowingtwotails e Throwingonehead f

11 2E Therearethreeoutcomesofarugbyleaguegame:win,loseordraw.If P(Win) = 5 7 and P(Lose) = 1 5 ,findtheprobabilityofthefollowing.

a Winningorlosingthematch

b Drawingthematch

c Notwinningthematch

d Notlosingthematch

12 2E Abeerselectsacardatrandomfromastandardpackofcards.Findtheprobabilityofobtainingthe followingoutcomes.

Notanace a Notaheart b Notaredsix c

13 2F Aclassfrequencytableshowsthescoresinatest.

59 5

69 6

79 8

89 6

Whatistherelativefrequencyforthefollowingoutcomes?Answercorrecttotwodecimalplaces.

a

b

c

d

14 2F LastyearOscarboughtapacketofbiscuitseveryweekandfound30ofthesepacketscontained brokenbiscuits.Whatistherelativefrequencyofthisevent?Answerasadecimalcorrecttotwo decimalplaces.

15 2G Theprobabilityofacouplehavingababywithredhairis33 1 3 %.Ifthecouplehavesixchildren,how manychildrenwithredhairareexpected?

16 2G Aprobabilityofadogcatchingheartwormis 3 8 .Ifthereare896dogsinthelocalcommunity,how manyofthemwouldyouexpecttocatchheartworm? UNCORRECTED

Extended-responsequestion

1 2C,2G Thespinnersshownareusedinagame.Eacharrowisspunonce.Thescoreisthesumofthetwo numbersthatthearrowslandon.

Atableisdrawntoshowallscoresthatcanbeobtainedinthisgame.

a Fillinthetabletoshowallpossibleoutcomes.

b Whatistheprobabilitythatthesumisodd?

c Whatistheprobabilitythatthesumisnot6?

d Ifthegameisplayed300times,howmany7swouldyouexpect?

3 Investments, depreciationand loans

Howhasinflationaffectedpricesinthelast50years?

Abasketofgroceriesworth$50dollarsin 1970wouldcostapproximately$729in2025. Thisisanincreaseof1358%over55yearsat anaverageinflationrateof5%.TheReserve BankofAustraliahasafantasticinflation calculatorathttps://www.rba.gov.au/ calculator/annualDecimal.html.

HowmanyAustraliansownshares?

Sharepricesgoupanddowndaily.Overalong periodoftimesharesontheAustralianStock Exchange(ASX)haverisensteadily.Injustthe past5years,shareshaverisenapproximately 40%.Youmightthinkthatthisdoesnotaffect manypeople.However,inAustralia7.7million peopleowninvestmentslistedontheASX.Ten millionAustraliansholdinvestmentsother thantheirsuperannuation.

Creditcards

Creditcardsareusedbymillionsofpeople aroundtheworldasasimplewaytousealine ofcredit.Aslongasyoupayoffyourmonthly spendbytheduedate,youshouldnothaveto payanyinterest.However,ifyoudonotpayoff theentireamountthecompoundinterestwill addupquicklyandyourcreditcardcan becomeadebttrap.

BuyNowPayLater(BNPL)

TheriseofBNPLschemesinAustraliahas beendrivenbypeoplewantingaccessto moneyinotherwaysthanborrowingitfroma bank.BNPLschemesworktoallowpeopleto getaccesstoitemswithouthavingtosaveup themoneybeforehand.Thechargesandfees canbecomeaseriousissueforpeoplethatdo notkeeptrackofhowmuchtheyneedtopay backeachmonthorfortnight.

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3A Simpleinterestoninvestments

3B Compoundinterestoninvestments

3C Graphsofinvestments:Simple interestandcompoundinterest

3D Appreciationandinflation

3E Straight-linedepreciation

3F Declining-balancedepreciation

3G Costsofborrowingmoney

3H Reducingbalanceloans

3I Creditcards

3J Creditcardstatements

3K Feesandchargesforcreditcard usage

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•Solvesproblemsinvolvingsimple interestandcompoundinterestto makedecisionsaboutfinancial situations(MST-12-S1-02)

•Solvesproblemsinvolvinginterestand depreciationtomakedecisionsabout loansandcreditcards(MST-12-S1-03).

©NSWEducationStandardsAuthority

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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3A Simpleinterestoninvestments

Learningintentions

• Tounderstandsimpleinterestoninvestments

• Tousethesimpleinterestformulatocalculatetheinterestandthefuturevalue

• Tosolvepracticalproblemsinvolvingsimpleinterest

Past,presentandfuturelearning

• Studentsmayalreadybefamiliarwiththecontentsofthissection

• QuestionsinvolvingsimpleinterestregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Theformulaforsimpleinterestdoesnotappearonthe2027version.

Interestistheamountearnedwheninvestingmoney.Therearedifferentwaysofcalculatinginterest. Simpleinterest(orflatinterest) isafixedpercentageoftheamountinvestedandiscalculatedontheoriginal amount.Forexample,ifweinvest$100inabankaccountthatpaysinterestattherateof5%perannum (peryear)wewouldreceive$5eachyear.Thatis,

Thisamountofinterestwouldbepaideachyear.Simpleinterestisalwayscalculatedontheinitialamount, ortheprincipal.

Aninvestmentinthecontextofinterestismoneythatyouputintoabank,orotherfinancialinstitution,in returnforinterestpayments.

Formulaforcalculatingsimpleinterest

I = Prn,where

I = Interest(simpleorflat)earnedfortheuseofmoney,paidbyborrowerstolenders

P = Principalistheinitialamountofmoneyborrowed,lentorinvested

r = Rateofsimpleinterestpertimeperiodexpressedasadecimal,e.g.5% = 0.05

n = Numberoftimeperiods(days,weeks,monthsoryears)

Areyouready?Fractions,decimalsandpercentages

Inthissectionwewillbeconvertingbetweenfractions,decimalsandpercentages.Completethetablein thisactivitytorecallhowtoconvertfromonetoanother.

DownloadtheactivityfromyourInteractiveTextbook.

Example1

Findingthesimpleinterestearnedonaninvestment

Calculatetheamountofsimpleinterestpaidonan investmentof$12000at10%simpleinterestperannum for3years.

Solution

P = 12000, r = 10% = 0.10, n = 3

I = Prn = 12000 × 0.10 × 3 = 3600

Simpleinterestis$3600.

Nowyoutry

Explanation

Writethevaluesfor P, r and n.

Writethesimpleinterestformula. Substitutethevaluesfor P, r and n.

Calculatetheamountofsimpleinterestpaidonaninvestmentof$15000at9%simpleinterestperannum for3years.

Futurevalue

Theinterestisaddedtotheprincipaltodeterminethefuturevalueofaninvestment.

Formulaforfuturevalue

A = P + I A = Futurevalue I = Interest(simpleorflat)earned P = Principalistheinitialquantityofmoneyborrowed,loanedorinvested

Calculatingvalueofaninvestment

Joelplanstomakeaninvestmentof$200000at7 1 2 %p.a.simple interestfor2years.Whatisthetotalvalueofhisinvestmentatthe endof2years? Example2

Solution

P = 200000, r = 7 1 2 % = 0.075, n = 2

I = Prn = 200000 × 0.075 × 2 = 30000

A = P + I = 200000 + 30000 = 230000

Totalvalueoftheinvestmentis$230000.

Nowyoutry

Explanation

Writethevaluesfor P, r and n

Writethesimpleinterestformula.

Substitutethevaluesfor P, r and n

Writetheformulaforfuturevalue.

Substitutethevaluesfor P and I

Josephineplanstomakeaninvestmentof$150000at9%p.a.simpleinterestfor3years.Calculatethe totalvalueofherinvestmentafter3years.

Example3

Calculatingtheinterestearnedonaperiodicinvestment

Jacksonplanstomakeaninvestmentof$150000at9%p.a.simpleinterest.Calculatethetotalinterest earnedonhisinvestmentafteraperiodof3months.

Solution

P = 150000

r = 9% = 0.09

n = 3 ÷ 12 = 0.25

I = Prn = 150000 × 0.09 × 0.25 = 3375

Totalinterestearnedis$3375.

Nowyoutry

Explanation

Writethevaluefor P

Calculate r asamonthlyinterestrate.

Calculate n asafractionofayear.

Writethesimpleinterestformula.

Substitutethevaluesfor P, r and n.

Juneplanstomakeaninvestmentof$17500at6%p.a.simpleinterest.Calculatethetotalinterestearned onherinvestmentafteraperiodof4months.

Exercise3A BUILDING

1 Example1 Calculatetheamountofsimpleinterestforeachofthefollowing.

a Principal = $1500,Interestrate = 8%p.a.,Timeperiod = 3years

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b Principal = $2000,Interestrate = 5%p.a.,Timeperiod = 5years

c Principal = $300000,Interestrate = 7%p.a.,Timeperiod = 4years

d Principal = $5500,Interestrate = 12%p.a.,Timeperiod = 8years

e Principal = $7000,Interestrate = 17%p.a.,Timeperiod = 5years

f Principal = $15000,Interestrate = 13.5%p.a.,Timeperiod = 3years

g Principal = $2000,Interestrate = 6 1 2 %p.a.,Timeperiod = 7years

h Principal = $200000,Interestrate = 9 1 4 %p.a.,Timeperiod = 2years

i Principal = $3600,Interestrate = 9%p.a.,Timeperiod = 3 1 2 years

j Principal = $40000,Interestrate = 7.25%p.a.,Timeperiod = 5 1 4 years

2 Fionainvested$30000atasimpleinterestrateof10%for3years.Howmuchinterestdidsheearn?

3 Example2

Asumof$100000wasinvestedinafixed–termaccountfor4years.

a Calculatethesimpleinterestearnediftherateofinterestis5.5%perannum.

b Findthevalueoftheinvestmentattheendof4years.

4 Larryinvested$2000at5%pa.Whatwasthevalueoftheinvestmentafter3years?

5 Thesimpleinterestrateisgivenas4.8%perannum.

a Whatistheinterestrateperquarter?

b Whatistheinterestratepermonth?

c Whatistheinterestratepersixmonths?

d Whatistheinterestrateperninemonths?

6 Example3 Calculatetheamountofsimpleinterestforeachofthefollowing.

a Principal = $800,Interestrate = 12%p.a.,Timeperiod = 1month

b Principal = $1600,Interestrate = 18%p.a.,Timeperiod = 6months

c Principal = $60000,Interestrate = 9.6%p.a.,Timeperiod = 3months

d Principal = $20000,Interestrate = 6%p.a.,Timeperiod = 9months

e Principal = $7250,Interestrate = 11%p.a.,Timeperiod = 10months

f Principal = $140000,Interestrate = 15%p.a.,Timeperiod = 15months

7 Jodiinvested$3000at7.5%p.a.Howmuchinterestdidsheearninthefirst3months?

8 Createthespreadsheetshown.

a CellD5hasaformulathatcalculatesthesimpleinterest.Enterthisformula.

b TheformulaforcellE5is‘=A5 + D5’.FilldownthecontentsofE6toE12usingthisformula.

9

10

11

12

Rileywantstoearn$4000ayearininterest.Howmuchmustheinvestifthesimpleinterestrateis 10%p.a.?

Aishainvests$7000at5%for2years.Shethenreinveststhenewamountfor4yearsat6%.How muchisAisha’sinvestmentworthafter6years?

Catherinewantstobuyacarworth$22000.Howlongwillsheneedtoinvest$15000at8.5% interesttobeabletoaffordtobuythecar?Answercorrecttoonedecimalplace.

Samirainvests$16000for2 1 2 years.Whatistheminimumrateofsimpleinterestneededforherto earn$3000?

Communicateyourreasoning,understandingandfluency

13

Jackinvests$20000at6%p.a.simpleinterestfor5years. Arloinvests$20000at0.5%permonthsimpleinterestfor5years.

a HowmuchisJack’sinvestmentworthafter5years?

b HowmuchisArlo’sinvestmentworthafter5years?

c ExplainwhyJackandArloearnthesameamountofinterest.

14 Findthevalueof A and B inthereceiptbelow.

3B Compoundinterestoninvestments

Learningintentions

• Tounderstandcompoundinterestoninvestments

• Tousethefuturevalueformulatoperformcalculationsinvolvingcompoundinterest

• Tosolvepracticalproblemsinvolvingcompoundinterest

Past,presentandfuturelearning

• Studentsmayalreadybefamiliarwiththecontentsofthissection

• QuestionsinvolvingcompoundinterestregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheet publishedbyNESA.Theformulaforthefuturevalueofacompoundinterestinvestmentappearson the2027version.

Compoundinterestiscalculatedontheprincipalatthestart,andthenateachtimeperiod,ontheprincipal plusinterest.Itcalculatesinterestontheinterest.Forexample,if$100isinvestedinasavingsaccountata compoundinterestrateof10%perannum:

Thesecalculationsshowthattheinterestearnedincreasedeachyear.Inthefirstyearitwas$10,thesecond year$11andthethirdyear$12.10.

Compoundinterestformula

A = P(1 + r)n

A = Amount(finalbalance)orfuturevalueoftheloan

P = Principalistheinitialamountofmoneyborrowedorpresentvalueoftheloan

r = Rateofinterestpercompoundingtimeperiodexpressedasadecimal

n = Numberofcompoundingtimeperiods

Totalcompoundinterestearnedorpaid

Thetotalcompoundinterestiscalculatedbysubtractingthe principalfromthefinalbalancei.e.theamountofmoneyat theend.

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Interestearnedorowed

I = A P

A = Amountofmoneyorfinalbalance

I = Interest(compound)earnedorpaid

P = Principalistheinitialamountofmoney

Example4

Findingthecompoundinterestearned

Paigeinvests$5000over5yearsatacompoundinterestrateof6.5%p.a.Calculate:

a theamountoftheinvestmentafter5years,correcttothenearestcent

b theinterestearnedafter5years,correcttothenearestcent.

Solution

a P = 5000, r = 6.5% = 0.065, n = 5

A = P(1 + r)n

= 5000(1 + 0.065)5

= 6850.433317 ...

≈ $6850.43

Amountoftheinvestmentafter5yearsis

$6850.43.

b I = A P

= 6850.43 5000

= $1850.43

Interestearnedafter5yearsis$1850.43.

Nowyoutry

Explanation

Writethevaluesfor P, r and n.

Writethecompoundinterestformula. Substitutethevaluesfor P, r and n

Writetheformulaforfuturevalue. Substitutethevaluesfor A and P.

Petainvests$7000over4yearsatacompoundinterestrateof7.25%p.a.Calculate:

a theamountoftheinvestmentafter4years,correcttothenearestcent

b theinterestearnedafter4years,correcttothenearestcent.

Futurevalue

Recallthatcompoundinterestiscalculatedontheinitialamountborrowedorinvestedplusanyinterestthat hasbeenearned.Itcalculatesinterestontheinterest.Thefuturevalueformula FV = PV (1 + r)n andthe compoundinterestformula A = P(1 + r)n arecommonlyusedtocalculatethecompoundinterest.Theyare thesameformula.Thefuturevalue (FV ) isthesameastheamount (A) andthepresentvalue (PV ) isthe sameastheprincipal (P)

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Futurevalueandpresentvalueformula

A = P(1 + r)n becomes FV = PV (1 + r)n or PV = FV (1 + r)n

FV = Futurevalueoftheinvestmentoramount(finalbalance)

PV = Presentvalueoftheinvestmentorprincipal(initialquantityofmoney)

r = Rateofinterestpercompoundingtimeperiodexpressedasadecimal(orafraction)

n = Numberofcompoundingtimeperiods

A = Amount

Example5

Calculatingfuturevalueandpresentvalueofaninvestment

a Blakeinvests$7000over5yearsatacompoundinterestrateof4.5%p.a.Calculatethefuturevalue after5years.Answercorrecttothenearestcent.

b Calculatethepresentvalueofaninvestmentthathasafuturevalueof$500000over8yearswithan interestrateof8.5%perannumcompoundedmonthly.Answercorrecttothenearestcent.

Solution

a PV = 7000, r = 4.5% = 0.045, n = 5

FV = PV (1 + r)n = 7000(1 + 0.045)5

≈ $8723.27

Thefuturevalueis$8723.27.

b FV = 500000, r = 8.5% 12 = 0.085 12 , n = 8 × 12 = 96 PV = FV (1 + r)n = 500000 1 + 0.085 12 96

≈ $253916.41

Thepresentvalueis$253916.41.

Nowyoutry

Explanation

Writethevaluesfor PV , r and n

Writethefuturevalueformula.

Substitutethevaluesfor PV , r and n.

Writethevaluesfor PV , r and n.

Writetheformulaforpresentvalue.

Substitutethevaluesfor FV , r and n

a Zohyrinvests$8500over6yearsatacompoundinterestrateof6.5%p.a.Calculatethefuturevalue after6years,correcttothenearestcent.

b Calculatethepresentvalueofaninvestmentthathasafuturevalueof$250000over10yearsat 5.5%p.a.compoundedmonthly.Answercorrecttothenearestcent.

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Areyouready?Usingyourcalculatortocheckyouranswers

Inatestorexamination,itiswisetohaveamethodforcheckingthatyouranswerforthefuturevalueis correct.Thisactivityshowsyouhowtodothat.

DownloadtheactivityfromyourInteractiveTextbook.

Calculatingcompoundinterest

Compoundinterestiscalculatedbysubtractingtheprincipalfromtheamountinvested.Alternatively,finance companiesprovideaninvestmentcalculatorasanestimateofthevalueofaninvestment.

Compoundinterestearnedorowed

I = A P becomes I = FV PV

FV = Futurevalueoftheinvestment(finalbalance)

PV = Presentvalueoftheinvestmentorprincipal(initialquantityofmoney)

I = Interest(compounded)earned

Areyouready?Compoundingannually,monthlyetc.

Doesitmakeanydifferencetothefuturevalueifyouswitchfromcompoundingannuallyto compoundingmonthly?Thisactivityanswersthatquestion.

DownloadtheactivityfromyourInteractiveTextbook.

Example6

Findingthecompoundinterestcompoundedquarterly

Tysoninvests$10000over10yearsatacompoundinterestrateof7 1 2 %p.a.compounded quarterly.Calculate:

a thevalueoftheinvestmentafter10years,correcttothenearestcent b theinterestearnedafter10years,correcttothenearestcent.

Solution

a PV = 10000, r = 7.5% 4 = 0.01875, n = 10 × 4 = 40

FV = PV (1 + r)n = 10000(1 + 0.01875)40 = 21023.492772 ... ≈ $21023.49 Valueofinvestmentafter10yearsis$21023.49.

b I = FV PV

= 21023.49 10000 = $11023.49 Interestearnedafter10yearsis$11023.49.

Explanation

Writethevaluesfor PV , r and n.

Writethefuturevalueformula. Substitutethevaluesfor PV , r and n

Writetheformulaforinterestearned. Substitutethevaluesfor FV and PV

Nowyoutry

Jordaninvests$7000over4yearsatacompoundinterestrateof7.25%p.a.compoundedquarterly. Calculate:

a theamountoftheinvestmentafter4years,correcttothenearestcent

b theinterestearnedafter4years,correcttothenearestcent.

Exercise3B

BUILDING

1 Example4 Calculatetheamountofinterestearned,correcttothenearestcent,foreachofthefollowing investments.

a Principal = $800,Compoundinterestrate = 4%p.a.,Timeperiod = 3years

b Principal = $9000,Compoundinterestrate = 6 1 2 %p.a.,Timeperiod = 4years

c Principal = $12000,Compoundinterestrate = 11%p.a.,Timeperiod = 2 1 2 years

d Principal = $22000,Compoundinterestrate = 5.5%p.a.,Timeperiod = 4 1 4 years

2 Example5a Calculatethefuturevalueofeachofthefollowing,correcttothenearestcent.

a Presentvalue = $400,Compoundinterestrate = 3%p.a.,Timeperiod = 2years

b Presentvalue = $26000,Compoundinterestrate = 8%p.a.,Timeperiod = 4years

c Presentvalue = $48000,Compoundinterestrate = 3.95%p.a.,Timeperiod = 10years

d Presentvalue = $3000,Compoundinterestrate = 5 1 2 %p.a.,Timeperiod = 5years

e Presentvalue = $18000,Compoundinterestrate = 10%p.a.,Timeperiod = 2 1 2 years

3 Usetheformula A = P(1 + r)n tocalculatethevalue ofaninvestmentof$10000overaperiodof2years withaninterestrateof0.8%compoundingmonthly. Answercorrecttothenearestcent.

UNCORRECTEDSAMPLEPAGES

4 Usetheformula FV = PV (1 + r)n tocalculatethe valueofaninvestmentof$16000,overaperiodof 2yearswithaninterestrateof5%compounding annually.Answercorrecttothenearestcent.

5

Calculatethepresentvalue,correcttothenearestdollar,foreachofthefollowing:

a futurevalueof$1243,interestrateat6%p.a.compoundedbiannuallyfor5years

b futurevalueof$8200,interestrateat4%p.a.compoundedquarterlyfor8years

c futurevalueof$1580,interestrateat4.8%p.a.compoundedmonthlyfor4years

d futurevalueof$19600,interestrateat8%p.a.compoundedweeklyfor3years

e futurevalueof$3800,interestrateat5%p.a.compoundedfortnightlyfor7years.

Kariminvested$13500over7yearsat6.2%p.a.interestcompoundingquarterly.Calculate:

a thevalueoftheinvestmentafter7years,correcttothenearestcent

b thecompoundinterestearned,correcttothenearestcent.

7 Jettsoldhiscarfor$35600.Heinvestedthisamountat7.2%p.a.withinterestcompounded annually.Whatisthevalueofhisinvestmentin15years?Answercorrecttothenearestcent.

8 Bonniewishestoinvest$5000foraperiodof8years.Thefollowinginvestmentstrategiesare suggestedtoher.Howmuchinterestisearnedoneachinvestmentstrategy?Answercorrecttothe nearestdollar.

a Simpleinterestat7%p.a.

b Compoundinterestat7%p.a.compoundedannually

c Simpleinterestat14%p.a.

d Compoundinterestat14%p.a.compoundedannually.

9 Whichofthefollowingisthebestinvestmentover10years?Justifyyouranswer.

InvestmentA:Simpleinterestat5%p.a.with$70000. InvestmentB:Compoundinterestat5%p.a.compoundedannuallywith$70000.

10

Calculatethepresentvalueforeachofthefollowing,correcttothenearestcent.

a Futurevalue = $34000,Interestrate = 4%p.a.,Timeperiod = 4years

b Futurevalue = $87000,Interestrate = 5%p.a.,Timeperiod = 12years

c Futurevalue = $190000,Interestrate = 3%p.a.,Timeperiod = 15years

d Futurevalue = $200000,Interestrate = 12 1 4 %p.a.,Timeperiod = 5years

11 Calculatethefuturevalueforeachofthefollowing,correcttothenearestcent.

a Presentvalueof$680investedfor4yearsat5%p.a.compoundedfortnightly

b Presentvalueof$5000investedfor6yearsat6%p.a.compoundedquarterly

c Presentvalueof$1400investedfor3yearsat4.2%p.a.compoundedmonthly

d Presentvalueof$780investedfor5yearsat9.8%p.a.compoundedweekly

12 Createthespreadsheetshown.

UNCORRECTEDSAMPLEPAGES

a CellD5hasaformulathat calculatesthecompoundinterest. Enterthisformula.

b TheformulaforcellE5is ‘ = D5 A5’.Filldownthecontents ofE6toE12usingthisformula.

13

Whichofthefollowingisthebestinvestmentover25years?Justifyyouranswer.

InvestmentA:Simpleinterestat4%p.a.with$100000

InvestmentB:Compoundinterestat4%p.a.compoundedannuallywith$100000

InvestmentC:Compoundinterestat4%p.a.compoundedbiannuallywith$100000

InvestmentD:Compoundinterestat4%p.a.compoundedquarterlywith$100000

InvestmentE:Compoundinterestat4%p.a.compoundedmonthlywith$100000

WhatsumofmoneywouldCalebneedtoinvesttoaccumulateatotalof$100000attheendof 7yearsat8%p.a.interestcompoundingbiannually?Answercorrecttothenearestcent.

15

Whatsumofmoneyneedstobeinvestedtoaccumulateatotalof$40000after10yearsat9.25%p.a. interestcompoundingmonthly?Answercorrecttothenearestcent.

16

Howmuchmoreinterestisearnedona$40000investmentiftheinterestat6%p.a.iscompounded annuallyover6years,comparedwiththesimpleinterestat6%p.a.earnedoverthesametime? Answercorrecttothenearestcent.

17 Howmuchmoreinterestisearnedon$60000ifinterestof8%p.a.iscompoundedquarterly over6yearsthanifsimpleinterestof8%isearnedoverthesametime?Answercorrecttothe nearestcent.

Communicateyourreasoning,understandingandfluency

18 Mikaylainvests$200000for10yearsat6%p.a.interestcompoundedquarterly.Abbyalsoinvests $200000for10years,butherinterestrateis6%p.a.compoundedmonthly.

a CalculatethevalueofMikayla’sinvestmentatmaturity.Answercorrecttothenearestcent.

b ShowthatthecompoundedvalueofAbby’sinvestmentisgreaterthanthevalueofMikayla’s investment.

c ExplainwhyAbby’sinvestmentisworthmorethanMikayla’sinvestment.

19 Explainwhyyouwouldalwayschoosetohaveaninvestmentthathascompoundinterestpaid monthlyoveraninvestmentthathascompoundinterestpaidannually.

3C

Graphsofinvestments:Simpleinterestand compoundinterest

Learningintentions

• Tounderstandthedifferencebetweensimpleinterestgraphs,whicharestraightlines,andcompound interestgraphs,whichareexponential

• Tocomparethegrowthofinvestmentswhichinvolvesimpleinterestandcompoundinterestfor differentratesandperiods

• Tocompareandcontrastdifferentinvestmentstrategies

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvinggraphsofinvestmentsregularlyappearinHSCExaminations

Simpleinterestgraphs

Simpleinterestgraphsareusedtomodelinvestments.Whengraphingsimpleinterest,putthenumberoftime periods(n)onthehorizontalaxisandtheinterestearned(I)ontheverticalaxis.Simpleinterestwillincrease byaconstantamounteachtimeperiod.Thiswillalwaysresultinastraight-linegraph.

Simpleinterestgraphs

1 Constructatableofvaluesfor I and n usingthesimpleinterestformula.

2 Drawanumberplanewith n onthehorizontalaxisand I ontheverticalaxis.Plotthepoints.

3 Jointhepointstomakeastraightline.

Example7

Constructingasimpleinterestgraph

Drawagraphshowingtheamountofsimpleinterestearnedoveraperiodof10yearsif$1000isinvested at8%p.a.Usethegraphtoestimatetheinterestearnedafter7.5years.

Solution

Explanation

Writethevaluesfor P and r.

Writethesimpleinterestformula. Substitutethevaluesfor P and r togenerate theequation I = 80n

Toconstructatableofvalues:

1 createyourownvaluesfor n 2 usetheequation I = 80n tofillinthetable foreachvalueof n

Drawanumberplanewith: verticalaxislabelledas I 1 horizontalaxislabelledas n. 2 Plotthepointsandusearulertodrawa straightline.

Usearulertodrawadottedverticallineat n = 7.5tomeetthestraightline. Fromthispointdrawadottedhorizontalline tothe I-axis.

Writetheapproximatevaluewherethe horizontallinemeetsthe I-axis.

Interestafter7.5yearsisapproximately$600.

Nowyoutry

Drawagraphshowingtheamountofsimpleinterestearnedoveraperiodof10yearsif$1500was investedat5%p.a.Usethegraphtoestimatetheinterestearnedafter8.5years.

Compoundinterestgraphs

Whengraphinginterestputthenumberoftimeperiods(n)onthehorizontalaxisandtheinterestearned(I) ontheverticalaxis, likewithsimpleinterest.Compoundinterestwillincreasebyadifferentamounteach timeperiod.Thiswillresult inanexponentialcurve.

Compoundinterestgraphs

1 Constructatableofvaluesfor I and n usingthecompoundinterestformula.

2 Drawanumberplanewith n onthehorizontalaxisand I ontheverticalaxis.Plotthepoints.

3 Jointhepointstomakeanexponentialcurve.

Example8

Constructingacompoundinterestgraph

Drawagraphshowingtheinterestearnedoveraperiodof10yearsif$1000isinvestedatacompound interestrateof8%p.a.Usethegraphtoestimatetheinterestearnedafter7.5years.

Solution Explanation

PV = 1000, r = 0.08

FV = PV (1 + r)n = 1000 × (1.08)n

Writethevaluesfor PV and r

Writethefuturevalueformula.

Substitutethevaluesfor PV and r togenerate theequation FV = 1000(1.08)n Continuedonnextpage

Compoundinterestearnedon$1000at8%p.a.

Toconstructatableofvalues: createyourownvaluesfor n 1

3

usetheequation FV = 1000(1.08)n tofill inthetableforeachvalueof n 2 subtract PV (1000)fromeachvalueof FV tofillintheInterest(I)column.

Drawanumberplanewith: verticalaxislabelledas I 1 horizontalaxislabelledas n 2

Plotthepointsusingthevaluesof n and I fromthetable.

Jointhepointstocreateacurvedline.

Usearulertodrawadottedverticallineat n = 7.5tomeetthecurvedline.

Fromthispointdrawadottedhorizontalline tothe I-axis.

Writetheapproximatevaluewherethe horizontallinemeetsthe I-axis.

Interestontheloanafter7.5yearsisabout$780.

Nowyoutry

Drawagraphshowingtheamountofcompoundinterestearnedoveraperiodof10yearsif$1500was investedat7%p.a.Usethegraphtoestimatetheinterestearnedafter8.5years.

Example9

Comparingthegrowthofsimpleinterestandcompoundinterest investmentsgraphically.

Aninvestmentof$10000earnssimpleinterestat5%perannum,whileanotherinvestmentof$10000 earnscompoundinterestat5%perannum,compoundedannually.Useaspreadsheetorgraphingsoftware toanswerthefollowingquestions.

a Createatableshowingthevalueofeachinvestmentforeachyearovera10-yearperiod.

b Usetechnologytographbothinvestmentsonthesamesetofaxes.

c Describethedifferencesinthegrowthpatternsshowninyourgraph.

Solution

a

Explanation

Useformulastocalculatetheamounttheinvestmentis worthattheendofeachyear.

b Graphbothinvestmentsonthesameaxes. SelectcellsA1:C12andinserttherecommended graph.

c Simpleinterestgrowslinearlyataconstantrate. Compoundinterestgrowsexponentially, increasingfasterovertime.

After10years,compoundinterestis$1,288.95 higherthansimpleinterest(usingthetable).

Nowyoutry

Highlighttheverticalaxisandeditittostartat $9000. Labeltheaxis.

Usethecellsinthespreadsheet. C12-B12 = $1288.95

RepeatExample9foraninvestmentof$18000thatearnssimpleinterestat4.25%perannum,and anotherinvestmentof$18000thatearnscompoundinterestat4.25%perannum,compoundedannually.

Exercise3C

1 Completethetableofvaluesforeachofthefollowing.

BUILDING

2 Example7 Harveyinvested$2000at6%p.a.simpleinterestfor5years.

Simplifythesimpleinterestformula(I = Prn)bysubstitutingvaluesfortheprincipalandthe interestrate. a Usethisformulatocompletethefollowingtableofvalues.

Drawanumberplanewith n asthehorizontalaxisand I astheverticalaxis. c

d Usethegraphtofindtheinterestafter2 1 2 years.

Plotthepointsfromthetableofvalues.Jointhepointstomakeastraightline.

e Extendthegraphtofindtheinterestafter6years.

f Estimatetheinterestearnedafter6yearsusingthegraph. g

3 Oliveinvested$800at7%perannumsimpleinterestfor6years.

a Simplifythesimpleinterestformula(I = Prn)bysubstitutingvaluesfortheprincipalandthe interestrate.

b Usethisformulatocompletethefollowingtableofvalues.

0 1 2 3 4 5 6

c Drawanumberplanewith n asthehorizontalaxisand I astheverticalaxis.

d Plotthepointsfromthetableofvalues.Jointhepointstomakeastraightline.

e Usethegraphtofindtheinterestafter2 1 2 years.

f Extendthegraphtofindtheinterestafter7years.

g Estimatetheinterestearnedafter7yearsusingthegraph.

4 Ellieiscomparingthreedifferentinterestratesforapossibleinvestment.

a Onthesamenumberplane,drawagraphtorepresenttheinterestearnedover5yearson:

i $1000investedat4%perannumsimpleinterest

ii $1000investedat6%perannumsimpleinterest

iii $1000investedat8%perannumsimpleinterest.

b Howmuchdoeseachinvestmentearnafter5years?

c Usethegraphtoestimatetheinterestearnedafter3 1 2 years.

d Findthetimeforeachinvestmenttoearn$200ininterest.

5 Completethetableofvaluesforeachofthefollowing.

6

Example7 Avainvested$2000at6%perannuminterestcompoundingannuallyfor5years.

a Substitutethepresentvalueandtheinterestrateintotheformula FV = PV (1 + r)n toobtainan expressionforthefuturevalue.

b Substitutethefuturevalueexpressionandthepresentvalueintotheformula

c Usetheseformulastocompletethefollowingtableofvalues.Answercorrecttothenearestdollar.

UNCORRECTEDSAMPLEPAGES

d Drawanumberplanewith n asthehorizontalaxisand I astheverticalaxis.

e Plotthepointsfromthetableofvalues.Jointhepointstomakeacurve.

f Extendthegraphtofindtheinterestafter6years.

g Comparetheinterestearnedafter5yearsbyAvawithHarveyinQuestion 2

7 Rajinvested$800at7%p.a.compoundinterest,paidannually,for6years.

a Substitutethepresentvalueandtheinterestrateintotheformula FV = PV (1 + r)n toobtainan expressionforthefuturevalue.

b Substitutethefuturevalueexpressionandthepresentvalueintotheformula I = FV PV

c Usetheseformulastocompletethefollowingtableofvalues.Answercorrecttonearestdollar.

0 1 2 3 4 5 6

d Drawanumberplanewith n asthehorizontalaxisand I astheverticalaxis.

e Plotthepointsfromthetableofvalues.Jointhepointstomakeacurve.

f Usethegraphtofindtheinterestearnedafter2 1 2 years.

g Comparetheinterestearnedafter6yearsbyRajwithOliveinQuestion 3

8 Frankieiscomparingthreedifferentinterestratesforapossibleinvestment.

a Drawonthesamenumberplaneagraphtorepresenttheinterestearnedover5yearson:

i $1000investedat4%perannuminterestcompoundingannually ii $1000investedat6%perannuminterestcompoundingannually iii $1000investedat8%perannuminterestcompoundingannually.

b Whatistheapproximatevalueoftheinterestoneachinvestmentafter5years?Answercorrectto thenearestdollar.

c Whatistheapproximatevalueoftheinterestoneachinvestmentafter3 1 2 years?Answercorrect tothenearestdollar.

d Findtheapproximatetimeforeachinvestmenttoearn$200ininterest.Answercorrecttoone decimalplace.

e Comparetheinterestearnedafter5yearsbyFrankiewithEllieinQuestion 4 usingthe8% interestrateforeach.

DEVELOPING

9 Drawagraphshowingtheamountofinterestpaidoveraperiodof6yearsif$1000isinvestedata compoundinterestrateof10%p.a.Usethegraphtoestimatetheinterestafter5 1 2 years.

10

Mickiscomparingthreedifferentinterestratesforapossibleinvestment.

a Onthesamenumberplane,drawagraphtorepresenttheinterestearnedfor6monthson:

i $100000investedat6%p.a.simpleinterest

ii $100000investedat9%p.a.simpleinterest

iii $100000investedat12%p.a.simpleinterest.

b Howmuchdoeseachinvestmentearnafter2months?

c Howmuchdoeseachinvestmentearnafter6months?

d Findthetimeforeachinvestmenttoearn$3000ininterest.

11 Tahneeiscomparingthreedifferentinterestratesforapossibleinvestment.

a Useaspreadsheettodrawagraphtorepresenttheinterestearnedover6monthsforeachofthese scenarios,puttingallthreegraphsonasinglenumberplane.

i $100000investedat6%p.a.interestcompoundingmonthly.

ii $100000investedat9%p.a.interestcompoundingmonthly.

iii $100000investedat12%p.a.interestcompoundingmonthly.

b Whatistheapproximatevalueofeachinvestmentafter2months,correcttothenearestdollar?

c Howmuchdoeseachinvestmentearnafter6months,correcttothenearestdollar?

d Findthetimeforeachinvestmenttoearn$3000ininterest,correcttothenearestmonth.

CHALLENGING

12 Thetablebelowgivesdetailsforaninvestmentproduct.Thecompoundinterestearnedis paidquarterly.

Investment Rateofcompoundinterest

Ethanispreparedtoinvest$50000intheaboveproduct.

a Drawagraphtorepresenttheinterestearnedineachofthese$50000investments.Plotallfour graphsonasinglenumberplane.

b WhatistheapproximatevalueofinvestmentBafter2years?

c WhatistheapproximatevalueofinvestmentCafter18months?

d FindtheapproximatetimeforinvestmentDtoearn$10000ininterest.

13 Davishasthefollowingoptionsforaninvestmentof$10000.

A 10%simpleinterest,compoundedquarterly.

B 8.5%compoundinterest,compoundedmonthly.

a Howmuchinterestwouldeachoptionearnafter3years?WhichoptionshouldDavischoose?

b Drawagraphtorepresenttheamountofinterestearnedoneachinvestmentover5years.Draw bothgraphsonasinglenumberplane.

14

15

c Usethegraphtoshowhowlongittakesforoption B toearnmoreinterest.

d Whatistheeffectofcompoundingthesimpleinterestquarterly?

e Explainwhyoption B withalowerinterestrateearnsmoreinterestovertimethanoption A.

Aninvestmentof$5000earnssimpleinterestat6%perannum,whileanotherinvestmentof$5000 earnscompoundinterestat6%perannum,compoundedannually.

a Calculatethevalueofeachinvestmentafter5years.

b Comparethetotalinterestearnedbyeachinvestment.

c Explainwhythecompoundinterestinvestmentgrowsfasterthanthesimpleinterestinvestment.

Aninvestmentof$8000earnssimpleinterestat4.5%perannum,whileanotherinvestmentof$8000 earnscompoundinterestat4.5%perannum,compoundedannually.

a Createatableshowingthevalueofeachinvestmentforeachyearovera10-yearperiod.

b Usetechnologytographbothinvestmentsonthesamesetofaxes.

c Describethedifferencesinthegrowthpatternsshowninyourgraph.

16 Aninvestmentof$8000earnssimpleinterestat4.5%perannum,whileanotherinvestmentof$8000 earnscompoundinterestat4.5%perannum,compoundedannually.

a Createatableshowingthevalueofeachinvestmentforeachyearovera12-yearperiod.

b Usetechnologytographbothinvestmentsonthesamesetofaxes.

c Describethedifferencesinthegrowthpatternsshowninyourgraph.

17 Amberneedstochoosebetweenthesethreeoptionswheninvesting$25000for5years.Usea spreadsheettomodelthegrowthofeachoptionbelow.WhichoptionshouldAmberchooseandwhy?

A 5%p.a.compoundedannually.

B 4.9%p.a.compoundedmonthly.

C 4.8%p.a.compoundedweekly.

UNCORRECTEDSAMPLEPAGES

3D Appreciationandinflation

Learningintentions

• Toapplythecompoundinterestformulatotheappreciatingvalueofassetsandtheinflatingvalueof goodsandservices

• Tosolveproblemsinvolvingappreciationandinflation

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingappreciationandinflationregularlyappearinHSCExaminations.

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublished byNESA.Theformulaforthefuturevalueofacompoundinterestinvestmentappearsonthe2027 version.Itmaybeusedforappreciationandinflation.

Appreciation

Appreciationistheincreaseinvalueofitemssuch asart,goldorland.Thisincreaseinvalueisoften expressedastherateofappreciation.Calculatingthe appreciationissimilartocalculatingthecompound interest.Forexample,apaintingworth$100000that hasanannualrateofappreciationof10%willbe worth$110000afteroneyear(anincreaseof$10000). Inthesecondyearitsvaluewillincreaseby$11000. Theamountofappreciationhasincreased.

Appreciation

FV = PV (1 + r)n or A = P(1 + r)n

FV = Futurevalueoftheitem

PV = Presentvalueoftheitem

r = Rateofappreciationpercompoundingtimeperiodexpressedasadecimal

n = Numberofcompoundingtimeperiods Findingtheappreciatedvalue Omarboughtaunitfor$690000.Iftheunitappreciatesat9%p.a.,

Solution

PV = 690000, r = 0.09, n = 7;

FV = PV (1 + r)n = 690000(1 + 0.09)7 = 1261346.993 ...

≈ $1261347

Unitisvaluedat$1261347.

Nowyoutry

Explanation

Writethevaluesfor PV , r and n.

Writetheformulaforappreciation. Substitutethevaluesfor PV , r and n.

Manavboughtaunitfor$725000.Iftheunitappreciatesat8%p.a.,whatisthevalueoftheunitafter 10years?Answercorrecttothenearestdollar.

Inflation

InflationisariseinthepriceofgoodsandservicesorConsumer PriceIndex(CPI).Itismeasuredbycomparingtheprices ofafixedbasketofgoodsandservices.Ifinflationrises thenaperson’sspendingpowerdecreases.Theinflationrate istheannualpercentagechangeintheCPI.InAustralia, theReserveBankaimstokeeptheinflationrateina2% to3%band.

Calculatinginflationissimilartocalculatingappreciationor compoundinterest.

Inflation

InflationrateistheannualpercentagechangeintheCPI.

Usetheformula FV = PV (1 + r)n tocalculatethefuturevalueofanitemfollowinginflation.

Example11

Findingthepriceofgoodsfollowinginflation

Whatisthepriceofa$400clothesdryerafterthreeyearsiftheinflation ratewas3.2%p.a.?Answercorrecttothenearestcent.

Solution

PV = 400, r = 0.032, n = 3

Explanation

Writethevaluesfor PV , r and n FV = PV (1 + r)n = 400(1 + 0.032)3

Writetheformulafor FV tocalculateinflation. Substitutethevaluesfor PV , r and n.

UNCORRECTEDSAMPLEPAGES

≈ $439.64

Clothesdryerwillcost$439.64.

Nowyoutry

Whatisthepriceofa$850fridgeafter4yearsiftheinflationratewas2.8%p.a.?Answercorrecttothe nearestcent.

Exercise3D

1 Usethefuturevalueformulatofindtheappreciatedvalueofthefollowing. Presentvalueof$680,appreciationof5%for4years. a Presentvalueof$1250,appreciationof8%for7years. b Presentvalueof$5000,appreciationof2.5%for10years. c Presentvalueof$23000,appreciationof4%for3years. d Presentvalueof$68000,appreciationof1.5%for15years.

2

BUILDING

e

Avintagecarwasboughtfor$70000andappreciatedattherateof6%p.a.Whatwillbethevalueof thecarafter4years?Answercorrecttothenearestcent.

3 Thepriceofahousehasincreasedby4.5%foreachofthelasttwoyears.Itwasboughtfor$490000 twoyearsago.Whatisthenewcurrentvalue?Answercorrecttothenearestdollar.

4 Rohanboughtthefollowingantiques.

a Tallboyvaluedat$4450.Eachyear itsvalueappreciatedby5%.Calculate thevalueofthetallboyafter3years. Answercorrecttothenearestcent.

b Tablevaluedat$6200.Eachyearits valueappreciatedby4%.Calculatethe valueofthetableafter5years.Answer correcttothenearestcent.

c Chairvaluedat$1250.Eachyearits valueappreciatedby9%.Calculatethe valueofthechairafter4years.Answer correcttothenearestcent.

5 Acollectionofdollswasvaluedat $1500fouryearsago.Ifitappreciatedat 12%p.a.,finditscurrentvalue.Answer correcttothenearestcent.

6 Thepriceofadiamondringhasincreased from$3400to$5300duringthepastfive yearsduetoinflation.Whatistherisein thepriceofthering?

7 Example11 Theaverageinflationforthenextfiveyearsispredictedtobe3%p.a.Calculatethepriceofthe followinggoodsinfiveyears’time.Answercorrecttothenearestcent.

8 Iftheinflationrateis5%p.a.,whatwouldyouexpecttopay,correcttothenearestdollar,infour years’timeforahousethatcosts: $280000? a $760000? b $324000? c $580000? d $1260000? e $956000? f

9 Thegraphontherightshowsthevalueofcricket memorabiliaforthepast6years.

a Whatwasthevalueofthememorabiliaafter5years?

b Whatwastheinitialvalue?

c Howmuchdidthememorabiliaappreciateeachyear?

d Findtheequationofthestraight-linegraphintermsof V and n

10

Thefollowingblocksoflandhaveincreasedinvaluethisyear.Whatistherateofappreciation? Answercorrecttotwodecimalplaces.

a $328000to$352000

b $256000to$278000

11 Jevainvested$700onrugbymemorabilia.Itappreciated over7yearsat8.2%p.a.interestcompoundingannually.

a Whatisthevalueoftheinvestmentafter7years? Answercorrecttothenearestcent.

b Calculatetheamountofappreciation.

12 Aninvestmentisappreciatingatarateof8%ofitsvalueeachyear.Kumardecidestoinvest$20000.

a Whatwillbetheinvestment’svalueafter5years?Answercorrecttothenearestcent.

b Howmuchdoestheinvestmentincreaseduringthefirst5years?

c Whenwilltheinvestmentatleastdoubleinvalue?Answertothenearestyear.

13 Thecostofasculpturehasincreasedduringthepasttwoyearsfrom$45200to$49833following inflation.Whatwastheannualinflationrate?Answercorrecttotwodecimalplaces.

14 Explainwhathappenstothepriceofgoodsiftherateofappreciationisnegative.

15 Biancabuysahousefor$800000.Shesellsitafter10years.Thehouseappreciatesatarateof 10%p.a.forthefirst6years.Afterthistherateofappreciationdroppedto8%p.a.Howmuchis Bianca’shouseworth,correcttothenearestthousanddollars:

a 6yearsaftershepurchasedit?

b 10yearsaftershepurchasedit?

3E Straight-linedepreciation

Learningintentions

• Tocalculatethedepreciationofanassetusingthestraight-linemethod

• Tousethestraight-linedepreciationformula

• Tosolveproblemsinvolvingstraight-linedepreciation Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingdepreciationregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Theformulaforstraight-linedepreciationappearsonthe2027version.

Straight-linedepreciationoccurswhenthevalueoftheitemdecreasesbythesameamounteachperiod.For example,ifyoubuyacarfor$20000anditdepreciatesby$2000eachyear,thevalueofthecarafterone yearis$20000 $2000,or$18000.Afterthesecondyearthevalueofthecaris$20000 $2000 $2000, or$16000.

Straight-linedepreciation

S = V0 Dn

S = Salvagevalueorcurrentvalueofanitem;alsoreferredtoasthebookvalue

V0 = Purchasepriceoftheitem;valueoftheitemwhen n = 0

D = Depreciatedamountpertimeperiod

n = Numberoftimeperiods

Example12

Calculatingthesalvagevalue

Yasminepays$14500forausedcar.Itdepreciates$1100eachyear.Calculatethesalvagevalueofthecar afterthreeyears.

Solution

V0 = 14500, D = 1100, n = 3

S = V0 Dn = 14500 1100 × 3 = $11200

Thesalvagevalueofthecaris$11200.

Nowyoutry

Bakaripays$15000forausedcar.Itdepreciatesat$1250eachyear.Calculatethesalvagevalueofthe carafter4years. UNCORRECTED

Explanation

Writethevaluesfor V0, D and n.

Writethesalvagevalueformula. Substitutethevaluesfor V0, D and n.

Areyouready?Usingyourcalculatortocheckyouranswers

Inatestorexamination,itiswisetohaveamethodforcheckingthatyouranswerforthesalvagevalue iscorrect.Thisactivityshowsyouhowtodothat.

DownloadtheactivityfromyourInteractiveTextbook.

Example13

Calculatingthestraight-linedepreciation

Anewcarispurchasedfor$25800.After4yearsitssalvagevalueis$15160.Whatistheannualamount ofdepreciation,iftheamountofdepreciationisconstant?

Solution

S = 15160, V0 = 25800, n = 4

S = V0 Dn

15160 = 25800 D × 4

D × 4 = 25800 15160

D = 25800 15160 4 = $2660

Annualdepreciationis$2660.

Nowyoutry

Explanation

Writethevaluesfor S, V0 and n.

Writethesalvagevalueformula.

Substitutethevaluesfor S, V0 and n

Sallypurchasesanewcarfor$75000.After3yearsitssalvagevalueis$54000.Whatistheannual amountofdepreciation,iftheamountofdepreciationisaconstant?

Exercise3E

1 Example12 Findthesalvagevalueofthefollowingitems.

a Purchaseprice(V0)of$680,depreciationof$150eachyearfor4years.

b Purchaseprice(V0)of$1250,depreciationof$40eachyearfor7years.

c Purchaseprice(V0)of$5000,depreciationof$250eachyearfor10years.

d Purchaseprice(V0)of$23000,depreciationof$2500eachyearfor3years.

e Purchaseprice(V0)of$68000,depreciationof$22250eachyearfor2years.

BUILDING

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2 Farrahboughtausedcarfor$15200.Sheestimatesthathercarwilldepreciateinvalueby$3040 eachyear.

a Whatisthelossinvalue(depreciation)duringthefirstyear?

b Whatisthevalueofthecarattheendofthefirstyear?

c Whatisthelossinvalue(depreciation)duringthe secondyear?

d Whatisthevalueofthecarattheendofthesecondyear?

e Whatisthelossinvalue(depreciation)duringthe thirdyear?

f Whatisthevalueofthecarattheendofthethirdyear?

3

4

Harrisonpays$9500foramotorbike.Itdepreciates$850eachyear.Whatwillbethevalueofthe bikeafter: threeyears? a fiveyears? b sevenyears? c nineyears? d

Faizbuysacarfor$55500anditisdepreciatedatarateof10%ofitspurchasepriceeachyear. Whatisthesalvagevalueofthecarafterfouryears?

5 Thegraphshowsthedepreciationofacarover fouryears.

a Whatistheinitialvalue?

b Howmuchdidthecardepreciateeachyear?

c Whatisthevalueofthecarafter3years?

d Whenwasthecarworth$8000?

e Whatisthevalueofthecarafter3 1 2 years?

f Whatisthevalueofthecarafter6months?

6

Gurvinderboughtacommercialvanthreeyearsago.Ithasasalvagevalueof$36000and depreciated$4650eachyear.HowmuchdidGurvinderpayforthevan?

7 Jamieboughtausedcarfouryearsago.Ithasasalvagevalueof$16400anddepreciated$1250each year.HowmuchdidJamiepayfortheusedcar?

8 Matteohasacarworth$9220.Itdepreciatesby$420eachyear.

a Whenwillthecarbeworth$5440?

b Whenwillthecarbeworth$3340?

9 Example13 Auteispurchasedfor$18600.Aftertwoyearsithasdepreciatedto$14800usingthestraightline methodofdepreciation.

a Whenwilltheutebeworth$9100?

b Whenwilltheutebeworth$1500?

10 Atruckispurchasednewfor$64000.After3yearsitsmarketvalueis$44800.

a Whatistheannualamountofdepreciation,iftheamountofdepreciationisconstant?

b Determinethevalueofthetruckafter7years.

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11 ParkerboughtanSUVcosting$38000.ItisexpectedthattheSUVwillhaveaneffectivelifeof 10yearsandthenbe soldfor$14000.AssumetheSUVdepreciatedbythesameamounteachyear. Whatistheannualdepreciation?

12 Autilityvanispurchasednewfor$24000.After3yearsitsbookvalueis$15000.Whatisthe annualamountofdepreciation,iftheamountofdepreciationisconstant?

13 Acaravanisboughtfor$82000.Itisexpectedtobeusedfor4yearsandthensoldfor$50000. Assumethecaravandepreciatesbythesameamounteachyear.

a Howmuchdoesthecaravandepreciateeachyear?

b Whatisthetotalamountofdepreciationfor4years?

c Copyandcompletethefollowingdepreciationtableforthefirstfouryears.

d Graphthevalueindollarsagainsttheageinyears.

14 Calvinpurchasedacomputerfor$5000.Hesolditfor$2500fiveyearslater.

a Calculatetheannualamountofdepreciation,assumingtheamountofdepreciationisaconstant.

b Useaspreadsheettocreateatablethatshowshowmanyyearsitwilltakeforthecomputerto havenovalue.

c Useaspreadsheettodrawagraphtorepresentthedepreciationover10years.

3F Declining-balancedepreciation

Learningintentions

• Tounderstandthedifferencebetweenstraight-linedepreciationanddecliningbalancedepreciation

• Tocalculatethedepreciationofanassetusingthedecliningbalancemethod

• Tousethedecliningbalancedepreciationformula

• Tosolveproblemsinvolvingdecliningbalancedepreciation Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingdepreciationregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Theformulafordecliningbalancedepreciationappearsonthe2027version.

Declining-balancedepreciationoccurswhenthevalueoftheitemdecreasesbyafixedpercentageeach timeperiod.Forexample,ifyoubuyacarfor$20000anditdepreciatesby10%eachyearthenthevalue ofthecarafteroneyearis$20000 $2000,or$18000.Afterthesecondyearthevalueofthecaris $18000 $1800,or$16200.Noticethattheamountofdepreciationhasdecreasedinthesecondyear. Depreciationcalculationshavesimilaritieswithcompoundinterest,exceptthatthedepreciationissubtracted fromthevaluenotaddedtoit.Thenotation V0 isusedtorepresenttheinitialvalueorthevaluebeforeany depreciationhasoccurred.

Declining-balancedepreciation

S = V0(1 r)n

S = Salvagevalueorcurrentvalueofanitem.Alsoreferredtoasthebookvalue.

V0 = Purchasepriceoftheitem.Valueoftheitemwhen n = 0.

r = Rateofdepreciationpertimeperiodexpressedasadecimal.

n = Numberoftimeperiods.

Example14

Calculatingthecurrentvalueusingdeclining-balance depreciation

Aryapurchasedanewcarfiveyearsagofor$32000.Duringthefirsttwoyearsithaddepreciatedby 25%p.a.andduringthenextthreeyearsithaddepreciatedat20%p.a.Whatisthecurrentvalueofthe car?

Solution

V0 = $32000, r = 0.25, n = 2

S = V0(1 r)n

S = 32000(1 0.25)2 = $18000

V0 = $18000, r = 0.20, n = 3

S = V0(1 r)n

S = 18000(1 0.20)3 = $9216

Thecurrentvalueofthecaris$9216.

Nowyoutry

Explanation

Writethevaluesfor V0, r and n.

Writethedeclining-balancedepreciationformula.

Substitutethevaluesfor V0, r and n

Writethevaluesfor V0, r and n

Writethedeclining-balancedepreciationformula.

Substitutethevaluesfor V0, r and n.

Thepurchasevalueofanewcarsixyearsagowas$72000.Duringthefirst2yearsitdepreciatedat 22%p.a.Duringthenext4yearsitdepreciatedat18%p.a.Whatisthecurrentvalueofthecarcorrectto thenearestdollar?

Areyouready?Usingyourcalculatortocheckyouranswers

Inatestorexamination,itiswisetohaveamethodforcheckingthatyouranswerforthecurrentvalue iscorrect.Thisactivityshowsyouhowtodothat.

DownloadtheactivityfromyourInteractiveTextbook.

Example15

Calculatingthepurchaseprice

Bodhibuysacarthatdepreciatesattherateof26%perannum.Afterfiveyearsthecarhasasalvagevalue of$17420.HowmuchdidBodhipayforthecar,correcttothenearestdollar?

Solution

S = 17420, r = 0.26, n = 5

S = V0(1 r)n 17420 = V0 × (1 0.26)5

V0 = 17420 (1 0.26)5 = $78503.59621 ... ≈ $78504

Bodhipaid$78504forthecar.

Nowyoutry

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Explanation

Writethevaluesfor S, r and n.

Writethedeclining-balancedepreciationformula.

Substitutethevaluesfor S, r and n

Make V0 thesubjectoftheequation.

Gregbuysacarthatdepreciatesattherateof22%p.a.Afterfouryearsthevalueofthecaris$20200. HowmuchdidGregpayforhiscar,correcttothenearestdollar?

Example16

Calculatingthepercentagerateofdepreciation

Madisonboughtadeliveryvan fouryearsagofor$27500.Using thedeclining-balancemethod fordepreciation,sheestimates itspresentvaluetobe$8107. Whatannualpercentagerateof depreciationdidsheuse?Answer correcttothenearestwhole number.

Solution

S = 8107, V0 = 27500, n = 4

Explanation

Writethevaluesfor S, V0 and n. S = V0(1 r)n 8107 = 27500 × (1 r)4 (1 r)4 = 8107 27500

1 r = 4 8107 27500 r = 1 4 8107 27500 = 0.26314528 ... ≈ 26% Rateofdepreciationis26%.

Nowyoutry

Writethedeclining-balancedepreciationformula. Substitutethevaluesfor S, V0 and n. Make(1 r)4 thesubjectoftheequation.

Takethefourthrootofbothsides.

Make r thesubjectandsolve.

Morganboughtacarfiveyearsagofor$32300.Usingthedeclining-balancemethodfordepreciation,she estimatesitspresentvaluetobe$9200.Whatannualpercentagerateofdepreciationdidsheuse?Answer correcttothenearestwholenumber.

Exercise3F

BUILDING

1 Example14 Amotorvehicleisboughtfor$22000.Itdepreciatesat16%perannumandisexpectedtobeused for5years.Whatisthesalvagevalueofthemotorvehicleafterthefollowingtimeperiods?Answer correcttothenearestcent.

Oneyear a Twoyears b Threeyears c Fiveyears d Sevenyears e

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2 Quinnpurchasedausedcarfor$6560twoyearsago.Usethedeclining-balancemethodto determinethesalvagevalueoftheusedcarifthedepreciationrateis15%perannum.Answer correcttothenearestdollar.

3

Kennedypurchasedamotorcyclefor$17500.Itdepreciatesat28%peryear.Answercorrecttothe nearestdollar.

a Whatisthebookvalueofthemotorcycleafter3years?

b Howmuchhasthemotorcycledepreciatedoverthe3years?

4 Anewcarisboughtfor$52000.Itdepreciatesat22%perannumandisexpectedtobeused for4years.Howmuchhasthecardepreciatedoverthe4years?Answercorrecttothenearestdollar.

5 Chloepurchasedacarfor$19900.Itdepreciatesat24%peryear.Answercorrecttothenearest dollar.

a Whatisthesalvagevalueofthecarafter5years?

b Howmuchhasthecardepreciatedoverthe5years?

6 Thedepreciationofausedcaroverfouryearsisshowninthegraphbelow.

7

a Whatistheinitialvalueoftheusedcar?

b Howmuchdidtheusedcardepreciateduringthefirstyear?

c Whenisthevalueoftheusedcar$2000?

d Whenisthevalueoftheusedcar$1500?

e Whatisthevalueoftheusedcarafter4years?

f Whatisthevalueoftheusedcarafter1 1 2 years?

Alibuysacarthatdepreciatesattherateof20%perannum.Aftersixyearsthecarhasasalvage valueof$6400.HowmuchdidAlipayforthecar,correcttothenearestdollar?

8 Diwabuysacomputerthatdepreciatesatarateof15%p.a.After3years,thecomputerhasasalvage valueof$1000.HowmuchdidDiwapayforthecomputer,tothenearestcent?

9 Chantellebuysatruckthatdepreciatesat21%p.a.After5years,thetruckhasasalvagevalueof $48000.Whatwastheoriginalpriceofthetruck,tothenearestdollar?

10

11

Ahatchbackvehiclewaspurchasedfor$16980three yearsago.Byusingthedeclining-balancemethodof depreciation,thecurrentvalueofthevehicleis$9614. Whatistheannualpercentagerateofdepreciation,correct totwodecimalplaces?

Anewcarisvaluedat$35000.Afteroneyearusingthe declining-balancemethod,itisvaluedat$25500.

a Determinetheannualpercentagerateofdepreciation. Answercorrecttothreedecimalplaces.

b Whatisthevalueofthenewcarafterthreeyears? Answercorrecttothenearestdollar.

12

PhilipandAmyspent$200000onaluxurycar7yearsago.Itscurrentvalueis$104350.Usingthe declining-balancemethod,findthepercentagedepreciationrateoverthisperiod.Answercorrectto onedecimalplace.

13 Jessicaspent$18820tobuyanewcarforherbusiness.Howmanyyearswouldittakeforthiscarto depreciateto$4520?Assumedeclining-balancemethodofdepreciationwitharateofdepreciationof 30%p.a.Answercorrecttothenearestyear.

14 Amotorvehicleisboughtfor$32000.Itdepreciatesat16%perannumandisexpectedtobe usedfor5years.

a Howmuchdoesthemotorvehicledepreciateinthefirstyear?

b Copyandcompletethefollowingdepreciationtableforthefirstfiveyears.Answercorrecttothe nearestdollar.

c Graphthevalueindollarsagainsttheageinyears.

CHALLENGING

Communicateyourreasoning,understandingandfluency

15 Laurinapurchasedacarfor$66000.Thecardepreciatedovertime.Laurinawantedtocomparethe straight-linedepreciationtodecliningbalancedepreciation.

a UsingExceloraspreadsheetsoftware,createandcompletethespreadsheetbelow.

b WhatwilltheformulaneedtobeincellD8?

c WhatwilltheformulaneedtobeincellE8?

d WhatwilltheformulaneedtobeincellG8?

e WhatwilltheformulaneedtobeincellH8?

f Usingthecompletedspreadsheet,findthevalueofLaurina’scarafter:

i 4yearsusingdeclining-balancedepreciation

ii 4yearsusingstraight-linedepreciation.

g Drawagraphofboththestraight-linedepreciationanddecliningbalancedepreciationonthe samenumberplanetocomparethetwomethodsofdepreciation.

h ExplainthefactorsthatLaurinawillneedtoconsiderwhenchoosingwhichmethodof depreciationtochoose.

16 Listthekeysimilaritiesanddifferencesbetweenthecompoundinterestformulaandthedeclining balanceformula.

3G Costsofborrowingmoney

Learningintentions

• Tounderstandthevariouscostsinvolvedwithborrowingmoney

• Tocalculatethecostofdifferentloans

• Tosolveproblemsinvolvingvariousformsofcreditandloans

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingloansregularlyappearinHSCExaminations

Buynowpaylater(BNPL)

InSection 8C oftheYear11textbook,welookedathowBNPLschemeswork.Inthissectionwewill lookmorecloselyatthecostsassociatedwithBNPLschemes.BNPLschemesusuallyrequirethefirst monthlypaymentupfront–someserviceslikeAfterpaydon’trequireaninitialpayment.Wewillanalysethe advantagesandrisks.BelowisatableoutliningkeyadvantagesandrisksofBNPL.

AdvantagesofBNPL

Convenienceandflexibility

Interest-freeoptions

Fasterapprovalandlowerrequirements

Budgetmanagement

Accesstohigher-valueItems

Example17

RisksofBNPL

Creditlawsdonotapply

Difficultytrackingspending

Unaffordabledebt

Feesaddupquickly

Creditratingcanbeaffected

Calculatingthecostsofborrowingusingbuynowpaylater

HughmadethefollowingpurchasesusingaBNPLscheme:

January15th :Computer$700,fourmonthlypaymentsof$175

February15th :Clothes$200,fourmonthlypaymentsof$50

March15th :Carservice$500,fourmonthlypaymentsof$125

EachBNPLagreementchargesa $25latepaymentfee foranyinstalmentthatisnotpaidontime.

a HowmuchdidHughhavetopayin:

February? i March? ii April? iii May? iv

b InApril,Hughwasonlyabletopay$300ontimeandpaidtheremainingAprilinstalmentsoneweek late.InMay,hedelayedhisentireMaypaymentschedulebyoneweek.HowmuchdidHughpayin latefees?

Solution

Explanation

ai Computer$175 + Clothes$50 = $200 InFebruaryonlythecomputerandclothesBNPLareactive.

ii Computer$175 + Clothes$50 + Carservice$125 = $350 InMarchallthreeBNPLschemesareactive.

iii Computer$175 + Clothes$50 + Carservice$125 = $350 InAprilallthreeBNPLschemesareactive.

iv Clothes$50 + Carservice$125 = $175 InMayonlytheclothesandcarserviceBNPLareactive.

b April

1 × $25 = $25

May

2 × $25 = $50

Totalpenalty = $25 + $50 = $75 Hughpaysatotalof$75inpenalties.

Nowyoutry

Paymentschedule

April:$300willpaycomputerandcarservicedebtsoonly chargedonclothes.

May:Chargedforoutstandingdebtonclothesandcarservice.

Findthesumofbothmonths.

JaydenmadethefollowingpurchasesusingaBNPLscheme:

January20th:Phone$800,fourmonthlypaymentsof$200

February20th:Clothes$150,fourmonthlypaymentsof$37.50

March20th:Birthdaypresent$200,fourmonthlypaymentsof$50

EachBNPLagreementchargesa $25latepaymentfee foranyinstalmentthatisnotpaidontime.

a HowmuchdidJaydenhavetopayin:

i February?

ii March?

iii April?

iv May?

b InApril,Jaydenwasonlyabletopay$250ontimeandpaidtheremainingAprilinstalmentsoneweek late.InMay,hedelayedhisentireMaypaymentschedulebyoneday.HowmuchdidJaydenpayin latefees?

Short-termloans

Ashort-termloancanbetakenoutwithabankorwithothershort-termlenders.Recently,morepeoplehave beenusingnon-banklendersforshort-termloans.Alistofadvantagesandrisksofshort-termfinancing usingnon-banklenderscanbeseenbelow.

Advantagesofshort-termloans

Fastapplicationprocess

Almostimmediateaccesstofunds

Easyrepaymentprocess

Moreflexibility

Example18

Risksofshort-termloans

Highfeesandinterestrates

Theamountyouborrowislimited

Borrowingcanbecomeacycle

Debtalwayshassomerisk

Calculatingthecostsofashort-termloan

Leanneborrowed$1500usingashort-termfinancecompany.Therewerenosetupfees.Thetermswere nodepositand5monthlyinstalmentsof$300.Ifshedoesnotpayontime,shewillincura3%penaltyon theoutstandingbalanceanda$2processingfee.

a Leannepaysherthirdpaymenttwodayslate.Howmuchareherpenalties?

b Leannepaysherfourthpaymentfivedayslate.Howmuchareherpenalties?

c WhatisthetotalofallthepenaltiespaidbyLeanne?

Solution

a Outstandingbalance = $900

Penalties = 900 × 0.03 + 2 = $29

b Outstandingbalance = $600 Penalties = 600 × 0.03 + 2 = $20

c Totalpenalties = 29 + 20 = $49

Nowyoutry

Explanation

Twopaymentsmadesooutstandingbalanceis 1500 300 300 = $900.

Threepaymentsmadesooutstandingbalanceis 1500 300 300 300 = $600.

Elizabethborrowed$1600usingashort-termfinancecompany.Therewerenosetupfees.Thetermswere nodepositandfourmonthlyinstalmentsof$400.Ifshedoesnotpayontime,shewillincura5%penalty ontheoutstandingbalanceanda$2.50processingfee.

a Elizabethpayshersecondpaymenttwodayslate.Howmuchareherpenalties?

b Elizabethpaysherthirdpaymentfivedayslate.Howmuchareherpenalties?

c WhatisthetotalofallthepenaltiespaidbyElizabeth?

Long-termloans

Long-termloansenabletheborrowingoflargeramountsofmoney.Therearemanyfeesthatbankschargeto setuplong-termloans.Long-termloansareusedmostfrequentlyforbuyingahouseoracar.Belowisalist ofcommonfeesassociatedwithalong-termloan.

• Establishmentfees coverthelender’scostsofprocessingtheloanapplication.Theycanbeafixed amountorapercentageoftheloan.

• Originationfees coverthecostofsettinguptheloan.Theyareapercentageoftheloan.

• Valuationfees coverthecostchargedtoassesstheproperty’svalue.Theyareafixedamount.

• Conveyancingfees arepaidtoaconveyancerorsolicitortotransferlegaltitleoftheproperty.Theyarea fixedamount.

• Mortgageregistrationfees arepaidtothegovernmenttoregisterthemortgage.Theycanbeafixed amountorapercentageoftheloan.

• Lendersmortgageinsurance mustbepaidifthedepositislessthan20%ofthepropertyvalue.Thisisa percentageoftheloan.

• Stampduty ispaidtothegovernmentandisbasedonvalueandlocationofproperty.

Example19

Calculatingthecostsofalong-termloan

Angelplanstotakeoutaloanfor$600000forahousevaluedat$700000.Hehastopaythefollowing feeswhichwillneedtobeaddedtotheloan:

• Establishmentfees:0.1%oftheloan

• Originationfees:0.05%oftheloan

• Valuationfees:$250

• Conveyancingfees:$350

• Mortgageregistrationfees:$171.70

• Stampduty:$343

Calculatetheactualamounttheloanwillneedtobetocoverthefees.

Solution Explanation

Fees = 600000 × 0.001 + 600000 × 0.0005 + 250 + 350 + 171.70 + 343 = $2014.70

Loan = 600000 + 2014.70 = $602014.70

Nowyoutry

Addallthecostsforeachfee.

Addfeestooriginalloanamount.

Burtplanstotakeoutaloanfor$700000forahousevaluedat$800000.Hehastopaythefollowingfees whichwillneedtobeaddedtotheloan:

• Establishmentfees:0.1%oftheloan

• Originationfees:0.05%oftheloan

• Valuationfees:$275

• Conveyancingfees:$375

• Mortgageregistrationfees:$171.70

• Stampduty:$348.

Calculatetheactualamounttheloanwillneedtobetocoverthefees,correcttothenearestdollar.

Feesandchargesforaloan

Banksandfinancialinstitutionschargetheircustomersforborrowingmoney.Aloanaccountiscreatedand anaccountservicefeeischargedpermonth.Inadditiontothisfeethereareanumberofotherloanfeesand charges,dependingonthefinancialinstitution.Manyofthesefeesarenegotiableandcustomersareadvised tocomparethefeesandchargeswiththeinterestratecharged.Feesandchargesforaloanmayinclude:

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• loanapplicationfee–costsinsettinguptheloan.

• loanestablishmentfee–initialcostsinprocessingtheloanapplication.

• accountservicefee–ongoingaccount-keepingfee.

• valuationfee–assessmentofthemarketvalueofaproperty.

• legalfee–legalprocessingofaproperty.

1 Lucindapurchasedawashingmachinefor$1500usingaBNPLscheme.Shepaida10%depositand thenfourequalmonthlyrepayments.Howmuchwaseachrepayment?

2 Example17

HollymadethefollowingpurchasesusingBNPLcompany.Allarrangementshadnodeposit. Item

a HowmuchdidHollyspendin:

i September?

ii October?

iii November?

b AssumingallBNPLschemesusedbyHollyrequiredfourequalmonthlyrepayments,howmuch didallthreeofthesepurchasescostoverall?

3 Sajaadborrowed$1320usingashort-termfinancecompany.A$30setupfeewasadded.Theterms werenodepositand5monthlyinstalments.Hepaid4instalmentsof$275.Howmuchwaslefttobe paidinthefinalinstalment?

4 Example18 Faithborrowed$1740usingashort-termfinancecompany.Therewerenosetupfees.Theterms werenodepositand6monthlyinstalmentsof$290.Ifshedoesnotpayontime,shewillincura 5%penaltyontheoutstandingbalanceanda$2.50processingfee.Faithforgetstomakehersecond payment.Whatarehertotalfees?

5 Brandonwantstopurchaseahousefor$900000.Hehasa$200000deposit.Lendersmortgage insuranceispayableifBrandonhaslessthana20%deposit.Show,usingworkingout,andexplain whyBrandondoesnotneedtopaymortgageinsurance.

6 Example19

Walterplanstotakeoutaloanfor$500000forahousevaluedat$610000.Thefeescheduleisset outinthetablebelow.

Calculatetheactualamounttheloanwillneedtobetocoverthefees.

7 Hoangpurchasedaloungefor$2500usingaBNPLscheme.Hepaida25%depositand5equal monthlyrepaymentsonhiscreditcard.

a Howmuchwaseachrepayment?

b TheBNPLcompanycharged1%onallcreditcardpayments.Howmuchwasaddedtoeach repaymentforusingacreditcard?

8 Ismaelborrowed$1000usingashort-termfinancecompany.Theset-upfeeswere5%whichwere addedtotheamountborrowed.Thetermswere10%depositand5equalmonthlyinstalments. Ifhedoesnotpayontime,hewillincura3.5%penaltyontheoutstandingbalanceanda $2processingfee.

a Howmuchiseachmonthlypayment?

b Ismaelpayshisthirdpaymentfourdayslate.Whatarehispenalties?

c Howmuchwillhisfourthpaymentneedtobetocoverthepenalties?

9 Breewantstopurchaseahousefor$759000.Shehasa$150000deposit.Lendersmortgage insuranceispayableifBreehaslessthana20%deposit.

a DoesBreeneedtopaymortgageinsurance?

b Mortgageinsuranceiscalculatedusingtheformula: MortgageInsurance = sizeofloan valueofproperty × 1000 HowmuchmortgageinsurancedoesBreeneedtopay? CHALLENGING

10 Niazwantstobuyaunitvaluedat$725000.Hedoesnotwanttopaymortgageinsurancewhichis payableifhehaslessthana20%deposit.Hehas$105000saved.Hecansave$5000amonth.

a Howlongwillitbeuntilhecanaffordtobuytheunitwithoutpayingmortgageinsurance?

b UsingtheformulainQuestion 9,calculatethemortgageinsuranceNiazwouldpayrightnow.

c MakearecommendationtoNiazastowhetherheshouldbuytheunitnoworwait.(Remember theunitmaysellifhewaitstoolong.)

11 Kodywantstoborrow$1000.Sometimesheearns$50aweekandsometimesheearns$100aweek.

a ExplaintoKodytheadvantagesandrisksinvolvedinusingaBNPLscheme.

b ExplaintoKodytheadvantagesandrisksinvolvedinusingashort-termloan.

Modellingactivity:Comparingfinancingoptions

UNCORRECTEDSAMPLEPAGES

Thisactivityexploresfourdifferentfinancingoptions—creditcards,personalloans,BuyNowPay Later(BNPL),andrevolvingcredit—byusingaspreadsheettoanalysecosts,interest,fees,andthe impactoflatepaymentsacrossseveralreal-worldpurchasingscenarios(purchasingalaptop,bike,car, andaholiday).TheactivitycanbedownloadedfromyourInteractiveTextbook.Itisrecommendedthat thisactivitybedoneafteryouhavecompletedsections 3H-3L.

3H Reducing-balanceloans

Learningintentions

• Tounderstandhowreducingbalanceloanswork

• Toperformcalculationsinvolvingfeesandchargesassociatedwithloans

• Tosolvepracticalproblemsinvolvingreducingbalanceloans

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingloansregularlyappearinHSCExaminations

Reducing-balanceloansarecalculatedonthebalanceowing.Aspaymentsare made,thebalanceowingisreducedandthereforetheinterestchargedisreduced. Thiscansavethousandsofdollarsonthecostofaloan.Thecalculationsfor reducing-balanceloansarecomplicatedandfinancialinstitutionspublishtables relatedtoloans.

Loanrepayments

Totaltobepaid = Monthlyrepayment × Numberofrepayments

Totaltobepaid = Principal + Interest

Example20

Usingatableforareducing-balanceloantofindinterestpaid

Thetablebelowshowsthemonthlyrepaymentsforareducing-balanceloan.Calculatetheamountof interesttobepaidonaloanof$200000over13years.

Solution Explanation

Totaltobepaid = Monthlyrepayment × Numberofrepayments

Loanis$200000andtimeperiodis13years. Findtheintersectionvalue(i.e.where13years intersectswith$200000($2856)).

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= 2856 × 13 × 12

= $445536

Multiplytheintersectionvaluebythenumber ofyearsand12(numberofmonthsinayear)to determinethetotaltobepaid.

Continuedonnextpage

Totaltobepaidfortheloanis$445536.

Totaltobepaid = Principal + Interest

445536 = 200000 + I

I = $245536

Interestpaidis$245536.

Nowyoutry

Substitutethetotaltobepaid ($445536) and principal ($200000) intotheformula.

UsethetablefromExample 20 tocalculatetheamountofinterestpaidonaloanof$250000over 14years.

Example21

Usingatableforareducing-balanceloantofindamountsaved

Thetableshowsthemonthlypaymentsforeach$1000borrowed.Zainabisplanningtoborrow $280000tobuyahouseat8%perannumoveraperiodof20years.

a WhatisZainab’smonthlypaymentonthisloan?

b HowmuchwouldZainabpayintotaltorepaythisloan?

c HowmuchwouldZainabsaveifsherepaidtheloanover15years?

Solution

a $8.36

Monthlyrepayment = $8.36 × 280 = $2340.80

b Totaltobepaid

= Monthlyrepayment × Numberofrepayments

= 2340.80 × 20 × 12 = $561792

Totaltobepaidfortheloanis$561792.

c 15years:Monthlyrepayment = $9.56 × 280 = $2676.80

Totaltobepaid:

= Loanrepayment × Numberofrepayments

Explanation

Findtheintersectionvaluefromthetablefor interestrate8%p.a.andtimeperiod20years.

Multiplytheintersectionvaluebythenumberof thousandsborrowed (280000 ÷ 1000 = 280).

Multiplythemonthlyrepaymentbythenumber ofyearsand12(numberofmonthsinayear)to determinethetotaltobepaid.

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= 2676.80 × 15 × 12

= $481824

Amountsaved = $561792 $481824 = $79968

Theamountsavedis$79968.

Repeattheabovecalculationsusing15years insteadof20years.

Subtractthetotaltobepaidfor15yearsfromthe totaltobepaidfor20years.

Nowyoutry

Loganisplanningtoborrow$460000tobuyahouseat7%p.a.over15years.

UsingthetableinExample 21:

a WhatisLogan’smonthlypaymentforthisloan?

b HowmuchwouldLoganpayintotalforthisloan?

c HowmuchwouldLogansaveifherepaidtheloanover10years?

Graphofareducing-balanceloan

Thegraphbelowshowstheamountowedaftereachmonthonareducing-balanceloan.Theamount borrowedis$50000ataninterestrateof10%p.a.Itillustratesthedifferencebetweenmakingrepayments of$500permonthand makingrepaymentsof$1000permonth.Whenpaying$500amonth,ittakes 215monthstopayoff theloan,andtheinterestchargedis$57500.However,whenpaying$1000amonth, itonlytakes65monthstopayoff theloan,andtheinterestchargedis$15000.Eachgraphisagradual curveaseachpaymentreducestheamountowedandslowlydecreasestheinterestcharged.

Reducing-balance loan

1 Example20

Tylerisconsideringaninvestmentloanfromthebankatareducibleinterestrateof9.9%p.a. (i.e.areducing-balanceloan).Thetablebelowshowsthemonthlyrepaymentforaninvestmentloan.

Investmentloan

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a Whatisthemonthlyrepaymentforaloanof$75000over15years?

b Whatisthemonthlyrepaymentforaloanof$150000over10years?

c Whatisthemonthlyrepaymentforaloanof$100000over5years?

d Whatisthemonthlyrepaymentforaloanof$50000over15years?

e Whatisthemonthlyrepaymentforaloanof$125000over5years?

2 Kevinisapplyingforaninvestmentloanfrom abankof$75000over5yearsusingthetable inQuestion 1.

a Whatisthemonthlyrepayment?

b Whatisthetotalamountpaidforthisloan?

c Whatistheinterestpaidonthisloan?

3 Thetablebelowshowsthemonthlyrepaymentsper$1000onabankloan.

Calculatethemonthlyrepaymentonthefollowingloans.

a $310000at7.50%p.a.for15years

b $120000at7.00%p.a.for10years

c $450000at7.75%p.a.for10years

d $180000at7.25%p.a.for15years

4 Binhisborrowing$35000at7%p.a.for10years.UsethetableinQuestion 3 toanswer thesequestions.

a Whatisthemonthlyrepayment?

b Howmuchinterestwillhepay?

5 Example21 Thistableshowsthemonthlypaymentsforaloanof$1000forvaryinginterestrates.Jackis planningtoborrow$340000tobuyahouseat10%p.a.overaperiodof15years.

Periodofloan

a CalculateJack’smonthlypaymentonthisloan.

b HowmuchdoesJackpayintotaltorepaythisloan?

c HowmuchinterestdoesJackpayonthisloan?

d HowmuchwouldJacksaveifherepaidtheloanover10years?

6 HannahandMitchellborrow$180000over20yearsatareducibleinterestrateof8.5%p.a. Theypay$1754permonth.

a Calculatethetotalamounttobepaidonthisloan.

b Howmuchinterestdotheypayontheloan?

7 Thegraphoppositeshowstheamountowed eachmonthonareducing-balanceloan.Use thegraphtoestimatetheanswerstothese questions.

a Howmuchwasborrowed?

b Howmuchisowedafter20months?

c Howmuchisowedafter40months?

d Howmuchisowedafter60months?

e Whenistheamountowing$20000?

f Whenistheamountowing$60000?

g Afterhowmanymonthswilltheloanbe paidoff?

8 MrandMrsSmithintendtoborrow$400000at8%p.a.reducibleinterestforatermof20years. UsethetableinQuestion 5 forthefollowingcalculations.

a Whatistheirmonthlyrepayment?

b Findthetotalamounttoberepaid.

c Whatistheinterestpaidforthisloan?

d Howmuchwouldtheysaveiftheyrepaidtheloanover15years?

e Howmuchwouldtheysaveiftheyrepaidtheloanover10years?

f Whatistheequivalentflatinterestratechargedperannumonthisloan?Answercorrectto twodecimalplaces.

CHALLENGING

9 Alintaborrows$160000over8yearsatareducibleinterestrateof21.2%p.a.Shepays $1600perfortnight.

a HowmuchwillAlintapaybackaltogether?

b Whatistheinterestpaidforthisloan?

c Whatistheequivalentflatinterestratechargedperannumonthisloan?Answercorrectto onedecimalplace.

10 Theformulabelowconvertsaflatrateofinteresttotheapproximateequivalentreducible interestrate.

E = 2NR N + 1

E = Reduciblerateofinterestperannumasadecimal

R = Flatrateofinterestperannumexpressedasadecimal

N = Numberofrepaymentsintheloan

Usetheformulatocalculatetheapproximatereducibleinterestratefortheseloans.Answer correcttotwodecimalplaces.

a Loanof$20000over3yearsat12%p.a.flatinterestwithmonthlyrepayments.

b Loanof$50000over5yearsat8.5%p.a.flatinterestwithmonthlyrepayments.

c Loanof$58000over4yearsat9%p.a.flatinterestwithquarterlyrepayments.

d Loanof$7000over12monthsat15%p.a.flatinterestwithbiannualrepayments.

e Loanof$62000over6yearsat11%p.a.flatinterestwithquarterlyrepayments.

f Loanof$25000over2yearsat19%p.a.flatinterestwithbiannualrepayments.

g Loanof$70000over24monthsat15%p.a.flatinterestwithmonthlyrepayments.

Communicateyourreasoning,understandingandfluency

11

12

Thetablebelowsimulatesthefirst6paymentsona$440000loanat7%p.a.withinterest compoundedmonthly.

a WhatwilltheformulaneedtobeincellD4?

b WhatwilltheformulaneedtobeincellE4?

c WhatwilltheformulaneedtobeincellG4?

d WhatwilltheformulaneedtobeincellC5?

e UsingExceloraspreadsheetsoftwaretocreateandcompletetheabovespreadsheet.

f Usingthecompletedspreadsheet,find: theamountofinterestpaidinmonth10 i thebalanceattheendof12months. ii

g Howmuchwaspaidtowardstheloanin12months?

h Whatisthetotalamounttheloanhasreducedbyin12months?

i Experimentwithwhatchangesoccurwhenyouincreaseordecreasethemonthlyrepayment amount.

j Experimentwithwhatchangesoccurifalumpsumispaidoff theloanafter3,6or9years.

Opentheprovidedspreadsheetmodelforreducing-balanceloans.

Thecurrentsettingsare:

Principal:$150,000 • Annualinterestrate:4%

Paymentsperyear:12 • Term:15years • Theregularrepaymentshownis$1,109.53.

a Usethesliderstoincreasetheannualinterestrateto8%whilekeepingthetermat15years.

i Whatisthenewregularrepayment?

ii Howmuchmorewillyoupayeachmonthcomparedtotheoriginalsetting?

b Nowreducethetermto10yearswhilekeepingtheinterestrateat8%.

i Whatisthenewrepaymentamount?

ii Explainwhytherepaymentchangeswhenthetermisshortened.

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c Finally,increasetheprincipalto$200000withthetermat10yearsandinterestrateat8%.

i Whatistherepaymentnow?

ii Comparethistotherepaymentwhentheprincipalwas$150000.

d Basedonyourobservations,explainhoweachfactor(interestrate,term,andprincipal)affectsthe repaymentamount.

3I Creditcards

Learningintentions

• Tounderstandhowcreditcardswork

• Tocalculatetheinterest,feesandotherchargesonpurchasesandpaymentsmadewithcreditcards

• Tosolveproblemsinvolvingcreditcards

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingcreditcardsregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublished byNESA.Theformulaforthefuturevalueofacompoundinterestinvestmentappearsonthe2027 version.Itmaybeusedforcreditcards.

Creditcardsareusedtobuygoodsandservicesandpayforthemlater.Thetimewheninterestisnotcharged onyourpurchasesiscalledtheinterest-freeperiod.Ifpaymentisnotreceivedwhenthestatementisdue theninterestischargedfromthedateofpurchase.Interestoncreditcardsisusuallycalculateddailyonthe outstandingbalanceusingcompoundinterest.

Theinterestrateisusuallymuchhigherthanforotherkindsofloansandcreditfacilities.

Creditcards

r = Rateofinterestpercompoundingtimeperiodexpressedasadecimal

n = Numberofcompoundingtimeperiods

I = Interest(compound)chargedontheoutstandingbalance

Example22

Calculatingthecostofusingacreditcard

Lunahasacreditcardwithacompoundinterestrate of18%p.a.andnointerest-freeperiod.Lunaused hercreditcardtopayforclothingcosting$280.She paidthecreditcardaccount14dayslater.Whatisthe totalamountshepaidfortheclothing,includingthe interestcharged?

Solution

P = 280, r = 0.18 ÷ 365, n = 14

FV = PV (1 + r)n = 280 1 + 0.18 365 14 = $281.94

Totalamountpaidis$281.94.

Nowyoutry

Explanation

Writethevaluesfor P, r and n

Writethefuturevalueformula. Substitutethevaluesfor P, r and n

Carterhasacreditcardwithacompoundinterestrateof22%p.a.andnointerest-freeperiod.Carterused theircreditcardtopayfortheirgrocerieswhichcost$196.50.Theypaidthecreditcardaccount21days later.Whatwasthetotaltheypaidfortheirgroceriesincludingtheinterestcharges?

Exercise3I

1

2 Example22

UNCORRECTEDSAMPLEPAGES

BUILDING

Acreditcardhasadailyinterestrateof0.05%perday.Findtheinterestchargedonthese outstandingbalances.Answercorrecttothenearestcent.

$840for12days a

$5680for30days c

$240for6days e

Emirhasacreditcardwithaninterestrateof 0.04%compoundingperdayandnointerest-free period.Heuseshiscreditcardtopayfora mobilephonecosting$980.Calculatethetotal amountpaidforthemobilephoneifEmirpaid thecreditcardaccountinthefollowingtime period.Answercorrecttothenearestcent.

10dayslater a 20dayslater b 30dayslater c 40dayslater d 50dayslater e 60dayslater f

$742.40for20days b

$128for18days d

$1450for15days f

3 Calculatethecompoundinterestchargedontheseoutstandingbalances.Answercorrecttothe nearestcent.

a Balance = $6820,Dailyinterestrateof0.08%,Timeperiod70days

b Balance = $23648,Dailyinterestrateof0.06%,Timeperiod35days

c Balance = $1550,Dailyinterestrateof0.05%,Timeperiod20days

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d Balance = $35800,Dailyinterestrateof0.09%,Timeperiod100days

e Balance = $4500,Dailyinterestrateof0.05%,Timeperiod27days

f Balance = $7680,Dailyinterestrateof0.04%,Timeperiod180days

4 Dasula’screditcardcharges0.045%compoundinterestperdayonanyoutstandingbalances. CalculatehowmuchinterestDasulaischargedonanamountof$450,whichisoutstandingonhis creditcardfor35days.Answercorrecttothenearestcent.

5 Cheyannereceivedanewcreditcardwithnointerest-freeperiodandadailycompoundinterestrate of0.05%.Sheusedhercreditcardtopurchasefoodfor$320andpetrolfor$50on18July.This amountstayedonthecreditcardfor24days.Whatisthetotalinterestcharged?Answercorrectto thenearestcent.

6 JhyeusedhiscreditcardtobuyaholidaytoNewZealand.Thecostofthepackagewas$6500.The chargeonthecreditcardis1%interestpermonthontheunpaidbalance.HowmuchdoesJhyeowe forhisholidayaftersixmonths?Answercorrecttothenearestcent.

7 Calculatetheamountowed,tothenearestcent,foreachofthefollowingcreditcardtransactions. Thecreditcardhasnointerest-freeperiod.

a Transactions = $540,Compoundinterestrate = 14%p.a.,Timeperiod = 15days

b Transactions = $270,Compoundinterestrate = 11%p.a.,Timeperiod = 9days

c Transactions = $1400,Compoundinterestrate = 18%p.a.,Timeperiod = 22days

d Transactions = $480,Compoundinterestrate = 16%p.a.,Timeperiod = 18days

e Transactions = $680,Compoundinterestrate = 10%p.a.,Timeperiod = 9days

8 Calculatetheinterestchargedforeachofthefollowingcreditcardtransactions.Thecreditcardhas nointerest-freeperiod.Answercorrecttothenearestcent.

a Transactions = $680,Compoundinterestrate = 15%p.a.,Timeperiod = 20days

b Transactions = $740,Compoundinterestrate = 12%p.a.,Timeperiod = 13days

c Transactions = $1960,Compoundinterestrate = 17%p.a.,Timeperiod = 30days

d Transactions = $820,Compoundinterestrate = 21%p.a.,Timeperiod = 35days

e Transactions = $1700,Compoundinterestrate = 19%p.a.,Timeperiod = 32days

9 Pedrohasacreditcardwithacompoundinterestrateof18.25%perannum.

a Whatisthedailypercentageinterestrate,correcttotwodecimalplaces?

b Pedrohasanoutstandingbalanceof$4890foraperiodof30days.Howmuchinterest,tothe nearestcent,willhebecharged?

10 Alyssausesacreditcardwithanointerest-freeperiodandacompoundinterestrateof15.5%p.a. fromthepurchasedate.DuringAprilshemakesthefollowingtransactions:

a Whatisthedailycompoundinterestrate,correcttothreedecimalplaces?

b Alyssa’saccountisdueon30April.Whatisthetotalamountdueifyoudisregardtheamountof interesttobepaid?

c HowmuchinteresthasAlyssapaidontheIGAtransactionduringthemonth?Answercorrectto thenearestcent.

d HowmuchinteresthasAlyssapaidontheKmarttransactionduringthemonth?Answercorrectto thenearestcent.

DEVELOPING

11 Vihaan’screditcardhasupto55daysinterestfreeandisdueonthe22ndofeachmonth.The interestrateis15.4%p.a.compoundingdaily.Vihaanbuysfurniturecosting$5160on25October. Howmuchinterestishechargedifhepaysthebalanceon22December?Interestischargedfromthe dateofpurchaseifthetotalamountowinghasnotbeenpaidbytheduedate.

12 HarrybuystwoBlu-rayDVDsfor$29.90eachandashirtfor$84.95onhiscreditcard.Thisamount staysonhiscreditcardfor75days.Thereisa45-dayinterest-freeperiodandadailyinterestof 0.05%compoundonhiscreditcard.

a HowmuchdidHarryspendonhiscreditcard?

b CalculatetheamountHarryowesonthecreditcard.

c Whatwastheinterestchargedonthesepurchases?

d WhatwouldbetheinterestchargedifHarry’scredit carddidnothavea45-dayinterest-freeperiod?

13

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AminaandJoshuaeachusetheircreditcardstobuyholidaypackagestoAdelaide.Thecostofthe packageis$1700foreachperson.

a ThechargeonAmina’screditcardis0.9%compoundinterestpermonthontheunpaidbalance.It hasnointerest-freeperiod.Aminapays$800afteronemonthandanother$500thenextmonth. Howmuchdoesshestilloweonhercreditcard?

14

b ThechargeonJoshua’screditcardisinterest-freeinthefirstmonth,and1.4%compoundinterest permonthonanyunpaidbalance.Joshuapays$800afteronemonthandanother$500thenext month.Howmuchdoeshestilloweonhiscreditcard?

Avery’sAugustcreditcardstatementshowsanopeningbalanceof$1850,apurchaseof$2450on 5August,andanotherpurchaseof$55on14August.Theminimumpaymentis3%oftheclosing balance.Theinitialcreditchargeis1.6%compoundingpermonthofanyamountoutstanding.

a WhatistheclosingbalanceonthiscreditcardforAugust?

b CalculatetheamountofinterestchargedforthemonthofAugust.

c Whatistheminimumpayment,tothenearestcent,requiredforAugust?

d WhatistheopeningbalanceforOctoberifAverypaidtheminimumpaymentinSeptemberfor interestchargedinAugustandmadenopurchasesinSeptember? Communicateyourreasoning,understandingandfluency

15 Opentheprovidedspreadsheet CreditCardInterestCalculator

Thespreadsheetshowsanoutstandingbalanceof$2400,anannualinterestrateof18%,andthe numberofcompoundingperiodsperyearas365.Thetimeindaysuntilthecarddebtispaidoff is currentlysetto45days,andtheamountowingisdisplayedas$2453.84.

a Changethetimeindaysto60.

i Whatisthenewamountofinterestowing?

ii Howmuchinteresthasbeenaddedcomparedtotheoriginal45–dayscenario?

b Changethetimeindaysto90.

i Whatistheamountofinterestowingnow?

ii Howdoestheinterestcomparetothe60 dayscenario?

c Explainwhytheamountowinggrowsfasterasthenumberofdaysincreases.Usetheconceptof compoundinterestinyourexplanation.

d Experimentbychangingtheannualinterestrateto20%.

i Whathappenstotheamountofinterestowingafter45days?

ii Whydoesincreasingtheinterestratehavesuchalargeeffect?

16 Researchcurrentcreditcardinterest rates,homeloaninterestratesandshort-terminterestratesat fourbanksofyourchoice.

a Completethetablebelow.

Interestratecomparison

Bank Creditcard Homeloan Short-termloan

b Explainwhythereissuchadifferencebetweeneachoftheinterestrates.

3J Creditcardstatements

Learningintentions

• Tounderstandandinterpretcreditcardstatements

• Toperformcalculationsbasedoncreditcardstatements

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingcreditcardsregularlyappearinHSCExaminations

Creditcardstatementsareissuedeachmonthandcontaininformationsuchasaccountnumber,opening balance,newcharges,payments,refunds,rewardpoints,paymentduedata,minimumpaymentandclosing balance.Thecreditcardstatementincludesthedateandcostofeachpurchaseandcouldberegardedasa ledger.Aledgerdocumentsyourspending.

Iftheminimumpaymentisnotmadebytheduedate,theconsequencescanbeexpensive.Youmaybe chargedalatepaymentfeeand,ofcourse,youwillbechargedinterestonit.

Readingacreditcardstatement

Solution Explanation

a 5353180100010001

1 Readthenumberafter‘MasterCard’.

b Openingbalanceis$207.72. 2 Read‘Openingbalance’.

c Paymentduedateis30Nov.

3 Readthebox‘Paymentduedate’.

d Minimumpaymentis$25.00. 4 Readthebox‘Minimumpayment’.

Nowyoutry

AnswerthefollowingquestionsusingthecreditcardstatementfromExample 23

a Whatisthetotalpointsbalance?

b Whatistheclosingbalance?

c Whatarethedatesthatthestatementbeginsandends?

d Whatwasthecostofnewcharges?

Exercise3J

1 Example23 Usethecreditcardstatementbelowtoanswerthesequestions.

BUILDING

a Whatistheduedate?

b Whatisthecostofthepurchases?

c Whatistheclosingaccountbalance?

d Whatistheminimumamountdue?

e Whatpaymentwasmadelastmonth?

f Howmuchinterestwascharged?

g Whatwastheopeningbalance?

h Whatisthecardholder’screditbalance?

2 Thetransactionsonacreditcardareshownbelow.

UNCORRECTEDSAMPLEPAGES

Whatisthecreditlimit? a Whatistheaccountbalance? b Howmanytransactionsareshown? c Whatistheavailablecredit? d Howmuchwasspenton29November? e Howmuchwasspenton28November? f Wherewas$49.13spenton28November? g Wherewas$24.38spenton30November? h Whatisthepaymentduedate? i Whatistheminimumamountdue? j

3 Createthespreadsheetbelow.

a Howmanytransactionsareshownontheledger?

b HowmuchhasbeenspentatManlyValePharmacy?

c Iftheaccountbeginson15Novemberandendson14December,howmanydaysdoesit accountfor?

d Ifthecardhasa$5000creditlimit,whatistheavailablecrediton14December?

e Iftheminimumpaymentis$10andispaidontheduedate,whatisthebalanceowing?

f Thiscreditcardcharges0.06%perdaycompoundinterestontheunpaidbalanceafterthe minimumpaymentwasmade.Whatistheinterestchargedontheclosingbalanceonthe 15December?Answercorrecttothenearestcent.

4 Considerthecreditcardstatementshownbelow.

VisaAccountnumber

Payment&othercreditsreceived

a Whatistheopeningbalance?

b Whatisthecreditlimit?

c Whatistheavailablecredit?

d Whatistheclosingbalance?

e Howmuchhasbeenspentonpurchases,cashadvancesandspecialpromodebitsthismonth?

f Howmuchinterestandotherchargeswerepaidlastmonth?

g Thiscreditcardcharges0.05%perdaycompoundinterestontheunpaidbalances.

i Whatistheinterestchargedontheclosingbalancethedayafterthepaymentisdueifno paymentismade?Answercorrecttothenearestcent.

ii Howmuchinterestwouldbepaidontheclosingbalanceforayearifnopaymentwasmade? Answercorrecttothenearestcent. CHALLENGING

5 Thetransactionsonacreditcardwithanopeningbalanceof$5246.84areshownbelow.

a Whatistheopeningbalance?

b Whatisthetotalofnewchargesandcreditsmadeonthisaccount?

c Ifthecardhasa$30000creditlimit,whatistheavailablecrediton1December?

d Howmanytransactionswerecompletedusingcard6360?

e Howmanytransactionswerecompletedusingcard6378?

f Whichtransactionwasacredit?

g WhatwasthetransactiondateforColes?

h Whatistheclosingbalance?

i Thiscreditcardcharges21.9%p.a.compoundinterestontheunpaidbalances.Whatistheinterest chargefor120daysontheclosingbalance?Answercorrecttothenearestcent.

6 Paymentdetailsforacreditcardareshownhere.

a Whatistheclosingbalanceifthe minimumpaymentwaspaidonthe 30September?

b Thiscreditcardcharges18.75%p.a.compoundinterest ontheunpaidbalance.Whatistheinterestchargedforthe next60daysiftheminimumpaymentwasmade?Answer correcttothenearestcent. Communicateyourreasoning,understandingandfluency

7 Answerparts a–e belowfromthefollowingcreditcardstatementshown.

a Whatisthenameofthebank?

b Whatistheclosingbalance?

c Explainwhytheclosingbalanceisthesameasthenewcharges.

d ExplainwhyMrABSeahaspaidnointerestthismonth.

e MrABSeadoesnotgetpaidfromhisjobuntil15th January.Hemadeapaymentof$30on 10th Januaryandthenpaidthebalanceon15th January.Howmuchinterestwasaddedtohisnext creditcardstatementasaresultofhislatepayment?

3K Feesandchargesforcreditcardusage

Learningintentions

• Tounderstandandinterpretthefeesandchargesassociatedwithcreditcardusage

• Toperformcalculationsoffeesandchargeswhenusingacreditcard

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingcreditcardsregularlyappearinHSCExaminations

Banksandfinancialinstitutionschargetheircustomersanannualcardfeeformaintainingacreditcard account.Inadditiontothisfee,customersmaybechargedfeesforlatepayment,cashadvancesandbalance transfers.Thelatepaymentfeeappliesiftheminimumpaymenthasnotbeenreceivedbytheduedate. Interestischargedforretailpurchasesandtheamountstillowingfromthepreviousmonth.

Feesandchargesforcreditcardusage

• Annualcardfee–maintainingcreditcardaccount

• Interestcharge–interestchargedforretailpurchases

• Latepaymentfee–whenminimumpaymenthasnotbeenreceivedbytheduedate

• Cashadvances–withdrawingcashfromthecreditcardaccount

• Balancetransfers–movingbalancetoanotheraccount,oftenheldatanotherinstitution

Example24

Calculatingfeesandchargeswhenusingacreditcard

Hillaryhasadebitof$6000onacreditcardwithan interestrateof14.75%p.a.thatcompoundsdaily. On1January,shedecidedtotransferthebalanceto anewcardwitha0%balancetransferfor6months. However,after6monthsthenewcardreverted toaninterestrateof19.75%p.athatcompounds daily.ShowbyhowmuchHillaryisbetteroff after 12months.

Solution

Writethevaluesfor P, r and n

Writethefuturevalueformula.

Substitutethevaluesfor P, r and n. Continuedonnextpage

Newcard

P = 6000, r = 0.1975 ÷ 365, n = 184

FV = PV (1 + r)n = 6000 1 + 0.1975 365 184

≈ $6627.94

Savings = $6953.39 $6627.94 = $325.45

Hillaryis$325.45betteroff withthenewcard.

Nowyoutry

n becomesthenumberofdaysfrom1Julyto 31December(184days).

Writethevaluesfor P, r and n. Writethefuturevalueformula.

Substitutethevaluesfor P, r and n

Savings = Oldcard newcard.

Graysonhasadebtof$7500onacreditcardwithaninterestrateof13.25%p.a.compoundeddaily. On1January,hedecidestotransferhisbalancetoanewcardwitha0%balancetransferfor6months. However,after6monthsthenewcardrevertstoaninterestrateof21%p.a.compoundeddaily.How muchisGraysonbetteroff byafter12months?

Exercise3K

BUILDING

1 Alicia’sbankchargedanannualcreditcardfeeof$350,acashadvancefeeof$2.50andalate paymentfeeof$20.CalculateAlicia’sbankingcostsfortheyearifshemade:

a 11cashadvancesand4latepayments

b 20cashadvancesand12latepayments

c 50cashadvancesand6latepayments

d 0cashadvancesand12latepayments

e 100cashadvancesand0latepayments

f 0cashadvancesand0latepayments.

2 Thetablebelowshowsthecreditcardusagechargesforfourbanks.

a WhatisthecostofthecashadvancefeeatbankB?

b WhatisthecostofthelatepaymentfeeatbankD?

c Whichbankhasthelowestannualfee?

d Whichbankhasthehighestcashadvancefee?

e CalculatethedifferencebetweenthelatepaymentfeesatbankCandbankD.

f CalculatethedifferencebetweenthecashadvancefeesatbankBandbankC.

3

g Whatistheaverageannualfeeforthesebanks?

h Whatistheaveragelatepaymentfeeforthesebanks?

i Whataretheannualbankingcostsfor30cashadvancesandonelatepaymentat: BankA? i BankB? ii BankC? iii BankD? iv

j Whataretheannualbankingcostsfor100cashadvancesandsixlatepaymentsat: BankA? i BankB? ii BankC? iii BankD? iv

Elijah’sbankchargedanannualcreditcardfeeof$320,cashadvancefeeof$2.30andlatepayment feeof$18.WhatareElijah’sbankingcostsfortheyearifhemade80cashadvancesandhadonelate paymentfee?

4 Thetablebelowshowsthebankingchargesforcreditcardusage.

Findthecostofthefollowingactivities.Answercorrecttothenearestcent. 1cashadvanceof$400. a 10cashadvancesof$50. b 100cashadvancesof$100. c 1balancetransferof$3580. d 1balancetransferof$8620. e 1balancetransferof$10000. f 2latepaymentsand3cashadvancesof$300. g 10latepaymentsand20cashadvancesof$200. h 5latepaymentsand50cashadvancesof$80. i

5 Example24 Georgepurchasedalaptoponhiscreditcardfor$2500.Interestischargedat0.04%perday.He decidedtotransferthebalancetoanewcardwith0%interestfor90days.Howmuchdoeshesavein thefirst180daysonthefollowingnewcards?Answercorrecttothenearestcent.

Interestrateof0.03%perday a

Interestrate0.04%perday b

Interestrateof0.05%perday c

Interestrate0.06%perday d

Interestrateof0.07%perday e

Interestrate0.08%perday f

6

Annabellehasadebitof$8000onacreditcardwithaninterestrateof12%p.a.Onthe1stJanuary, shedecidedtotransferthebalancetoanewcardwitha0%balancetransferfor6months.Howmuch doesshesaveinthefirst12monthsonthefollowingnewcardswhoseinterestratecompoundsdaily? Answercorrecttothenearestcent.

Interestrateof12%p.a. a

Interestrateof16%p.a. c

Interestrateof20%p.a. e

Interestrate14%p.a. b

Interestrate18%p.a. d

Interestrate22%p.a. f

7 Prisha’sAugustcreditcardstatementshowsanopeningbalanceof$1850,apurchaseof$2450on 5August,andanotherpurchaseof$1055on14August.Theminimumpaymentis3%ontheclosing balance.Thecreditchargeis0.05%perdaycompoundinterestontheoutstandingbalance.Answer thefollowingquestionscorrecttothenearestcent.

a WhatwasthetotalcostofthepurchasesinAugustonthecreditcard?

b WhatistheclosingbalanceonthiscreditcardforAugust?

c WhatistheminimumpaymenttothenearestcentrequiredforAugust?

d CalculatetheamountofinterestchargedforthemonthofSeptemberifPrishapaidtheminimum paymentinAugust.

Communicateyourreasoning,understandingandfluency

8 Trevoropensanewcreditcardaccounton10Decemberwiththeinterestandfeesshownbelow.

• Interestchargedat17.25%p.a.compoundeddaily

• Nointerest-freeperiod

• $0onlinepayments

• $3.50cashpayments

• $10establishmentfeepayablethedaytheaccountisopened. Trevorpurchasesaflightvaluedat$1767withthecreditcardonthe10December. Trevorworkedoutthebalanceowingonthecreditcardafter29daysinthefollowingtwoways. Explainwhichwayiscorrectandwhy.

b

a Trevormadeacashrepaymentof$800,29daysaftertheyboughttheflight.Explainwhythenew balanceowingis$1005.02.

Keyideasandchaptersummary

Futurevalue formula FV = PV (1 + r)n or PV = FV (1 + r)n or I = FV PV

FV = Futurevalueoftheinvestmentoramount(finalbalance)

PV = Presentvalueoftheinvestmentorprincipal(initialquantityofmoney)

r = Rateofinterestpercompoundingtimeperiodasadecimal (i.e.convertpercentageratestodecimal)

n = Numberofcompoundingtimeperiods

I = Interest(compounded)earned.

Investmentgraphs 1

Appreciationand inflation

Declining-balance depreciation

Constructatableofvaluesfor A and n usingthesimpleinterestformulaorthe futurevalueformula.

2 Drawanumberplane: n horizontalaxis, A theverticalaxis.Plotandjointhe points.

Usetheformula FV = PV (1 + r)n forappreciationandinflation.Inflationrateisthe annualpercentagechangeintheCPI.

S = V0(1 r)n

S = Salvagevalueorcurrentvalue

V0 = Purchasepriceoftheitem

r = Rateofinterestpertimeperiod(decimal)

n = Numberoftimeperiods.

Reducing-balance loans

Creditcards

Feesandcharges forcreditcard usage

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Totaltobepaid = Loanrepayment × Numberofrepayments

Totaltobepaid = Principal + Interest

Dailyinterestrate = Annualinterestrate 365

FV = PV (1 + r)n

FV = Futurevalueortheamountowingonthecreditcard

PV = Presentvalueorthepurchasesmadeonthecreditcard

r = Rateofinterestpercompoundingtimeperiodasadecimal

n = Numberofcompoundingtimeperiods.

Annualcardfee,interestcharge,latepaymentfee,cashadvances,balancetransfers.

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook.

Checklist

3A 1 Icanfindthesimpleinterestearnedonaninvestment.

e.g.Calculatetheamountofsimpleinterestpaidonaninvestmentof$22000at 5%simpleinterestperannumfor7years.

3A 2 Icancalculatethevalueofaninvestment.

e.g.Albertplanstomakeaninvestmentof$250000at7%p.a.simpleinterestfor 6years.Calculatethetotalvalueoftheinvestmentafter6years.

3A 3 Icancalculatethesimpleinterestearnedonaperiodicinvestment.

e.g.Indigoplanstomakeaninvestmentof$150000at5.5%p.a.simpleinterest. Calculatethetotalinterestearnedontheinvestmentafter5months.

3B 4 Icanfindthecompoundinterestearnedonaninvestment.

e.g.Aahilinvests$17000over3yearsatacompoundinterestrateof8.25%p.a. Calculate:

theamountoftheinvestmentafter3years,correcttothenearestcent a theinterestearnedafter3years,correcttothenearestcent.

b

3B 5 Icancalculatethefuturevalueandpresentvalueofaninvestment. e.g.

Nnekainvests$7500over7yearsatacompoundinterestrateof7.5%p.a. Calculatethefuturevalueafter7years,correcttothenearestcent.

b

a Calculatethepresentvalueofaninvestmentthathasafuturevalueof$150000 over5yearsat9.5%p.a.compoundedmonthly.

3B 6 Icanfindthecompoundinterestcompoundedquarterly.

e.g.Winstoninvests$5500over5yearsatacompoundinterestrateof6%p.a. compoundedquarterly.Calculate:

theamountoftheinvestmentafter5years,correcttothenearestcent a theinterestearnedoverthe5years,correcttothenearestcent. b

3C 7 Icanconstructasimpleinterestgraph.

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e.g.Drawagraphshowingtheamountofsimpleinterestearnedoveraperiodof 12yearsif$1800wasinvestedat5%p.a.Usethegraphtoestimatetheinterest earnedafter9.5years.

3C 8 Icanconstructacompoundinterestgraph.

e.g.Drawagraphshowingtheamountofcompoundinterestearnedoveraperiod of10yearsif$1500wasinvestedat7%p.a.Usethegraphtoestimatetheinterest earnedafter9.5years.

3D 9 Icanfindtheappreciatedvalue.

e.g.Aviboughtaunitfor$825000.Iftheunitappreciatesat9.5%p.a.,whatisits valueafter5years?Answercorrecttothenearestdollar.

3D 10 Icanfindthepriceofgoodsfollowinginflation.

e.g.Whatisthepriceofa$1050fridgeafterfiveyearsiftheinflationratewas 1.8%p.a.?

3E 11 Icancalculatethesalvagevalue.

e.g.Kobypays$11000forausedcar.Itdepreciatesat$950eachyear.Howmuch willthecarbeworthafter3years?

3E 12 Icancalculatethestraight-linedepreciation.

e.g.Naomipurchasesanewcarfor$25000.After5yearsitssalvagevalueis $10000.Whatistheannualamountofdepreciation,iftheamountofdepreciation isaconstant?

3F 13 Icancalculatethedeclining-balancedepreciation.

e.g.Thepurchasevalueofanewcarfouryearsagowas$62000.Duringthe first2yearsitdepreciatedat25%p.a..Duringthenext2yearsitdepreciatedat 15%p.a.Whatisthecurrentvalueofthecarcorrecttothenearest$100?

3F 14 Icancalculatethepurchaseprice.

e.g.Gabriellabuysacarthatdepreciatesattherateof24%p.a.After6yearsthe valueofthecaris$12150.HowmuchdidGabriellapayforhercar,correcttothe nearestdollar?

3F 15 Icancalculatethepercentagerateofdepreciation.

e.g.Morticiaboughtacarthreeyearsagofor$30000.Usingthedeclining-balance methodfordepreciation,sheestimatesitspresentvaluetobe$11100.Whatannual percentagerateofdepreciationdidsheuse?Answercorrecttothenearestwhole number.

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3G 16

Icancalculatethecostsofborrowingusingbuynowpaylater.

e.g.HaydenmadethefollowingpurchasesusingaBNPLscheme:

January10th :Clothes$200,fourmonthlypaymentsof$50

February10th :Phone$750,fourmonthlypaymentsof$187.50

March10th :Bike$400,fourmonthlypaymentsof$100

a HowmuchdidHaydenhavetopayin:

i February?

ii March?

iii April?

iv May?

b Haydenwasonlyabletoaffordtopaythe$187.50forthephoneinApril. Asaresult,hepaid penaltyfeesof17.5%ofhisoutstandingdebtanda$5 processingfee.HowmuchwerehistotalpenaltiesinApril?

3G 17 Icancalculatethecostsofashort-termloan.

e.g.YungWeiborrowed$1400usingashort-termfinancecompany.Therewereno setupfees.Thetermswerenodepositand4monthlyinstalmentsof$350.Ifshe doesnotpayontime,shewillincura4%penaltyontheoutstandingbalanceanda $2.50processingfee.

YungWeipaysherthirdpaymenttwodayslate.Howmuchareherpenalties? a YungWeipaysherfourthpaymentfivedayslate.Howmuchareherpenalties? b WhatisthetotalofallthepenaltiespaidbyYungWei? c

3G 18 Icancalculatethecostsofalong-termloan.

e.g.Bimalplanstotakeoutaloanfor$900000forahousevaluedat$955000.He isrequiredtopaythefollowingfeeswhichwillneedtobeaddedtotheloan:

• Establishmentfees:0.1%oftheloan

• Originationfees:0.05%oftheloan

• Valuationfees:$222

• Conveyancingfees:$350

• Mortgageregistrationfees:$171.70

• Stampduty:$284

Calculatetheactualamounttheloanwillneedtobetocoverthefees.

3H 19 Icanuseatableforareducing-balanceloantofindinterestpaid.

e.g.UsethetablefromExample 20 onpage157tocalculatetheamountofinterest paidonaloanof$150000over12years.

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3H 20 Icanuseatableforareducing-balanceloantofindamountsaved.

e.g.Charlotteisplanningtoborrow$560000tobuyahouseat6%p.a.over 20years.

UsingthetableinExample 21 onpage158showingthemonthlyrepaymentsfor each$1000borrowed:

WhatisCharlotte’smonthlypaymentforthisloan? a

HowmuchwouldCharlottepayintotalforthisloan? b HowmuchwouldCharlottesaveifsherepaidtheloanover15years? c

3I 21 Icancalculatethecostofusingacreditcard.

e.g.Dukehasacreditcardwithacompoundinterestrateof21%p.a.andno interestfreeperiod.Dukeusedhiscreditcardtopayforheadphoneswhichcost $386.Hepaidthecreditcardaccount10dayslater.Whatwasthetotalhepaidfor theheadphonesincludingtheinterestcharges?

3J 22 Icanreadacreditcardstatement.

e.g.UsingthecreditcardstatementfromExercise 3J Question 7 answerthe followingquestions.

Whatisthecreditcardenquiriesphonenumber? a

WhatistheMasterCardnumber? b Whenisthecreditcarddue? c Whatistheinterestpayableonoverdueamounts? d

3K 23 Icancalculatethefeesandchargeswhenusingacreditcard.

e.g.Bettyhasadebtof$9000onacreditcardwithaninterestrateof14.25%p.a. compoundeddaily.Shedecidestotransferherbalancetoanewcardwitha0% balancetransferfor90days.However,after90daysthenewcardrevertstoan interestrateof18.75%p.a.compoundeddaily.Showbyhowmuchsheisbetteroff after12months.

3H-3K 24 Icancomparecreditcardsandpersonalloanstomeetshort,mediumandlongterm financialgoals.

e.g.Downloadthe‘Comparisonmodellingtask’fromyourInteractiveTextbook.

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Multiple-choicequestions

1 3A Findthesimpleinterestearnedon$3214at12%p.afor6years.

2 3A $22000isinvestedfor5yearsat11%simpleinterest.Howmuchistheinvestmentworthin5years?

3 3B Bellainvests$8000at10%p.a.interestcompoundingannually.Whatisthefuturevalueafter 3years?Answercorrecttothenearestdollar.

4 3B Clarainvests$3000at10%p.a.interestcompoundingannually.Whatistheinterestearnedafter 2years?Answercorrecttothenearestdollar.

5 3C Whichtableiscorrect?

6 3D Apaintingwasboughtfor$460000andappreciatedattherateof7%p.a.Whatwillbethevalueof thepaintingafter4years?Answercorrecttothenearestdollar.

7 3E

Acardepreciatesinvaluefrom$39000to$12250in4yearsusingthedeclining-balancemethodfor depreciation.Whatistheannualrateofdepreciationcorrecttothenearestwholenumber?

8 3H Thetableshowsthemonthlyrepaymentof$1000onareducing-balanceloan.Whatisthemonthly repaymenton$290000at8.75%for20years?

9 3H Lachlanborrows$245000over20yearsatareducibleinterestrateof6.5%p.a.Hepays$1856per month.Whatisthetotalpaidonthisloan?

10 3I Acreditcardhasadailyinterestrateof0.05%perday(nointerestfreeperiod).Findtheinterest chargedon$1530for14days.Answercorrecttothenearestcent.

Short-answerquestions

1 3A Findthesimpleinterestearnedif$1100isinvestedfor3yearsat5%p.a.

2 3B

Calculatethefuturevalue,correcttothenearestcent,foreachofthefollowing.

a Presentvalue = $920,compoundinterestrate = 5%p.a.,timeperiod = 4years

b Presentvalueof$2100investedfor3yearsat6.1%p.a.compoundedmonthly

3 3B

Calculatethepresentvalue,correcttothenearestcent,foreachofthefollowing.

a Futurevalue = $26000,interestrate = 4.9%p.a.,timeperiod = 3years

b Futurevalueof$10400,interestrateat9%p.a.compoundedquarterlyfor5years

4 3B WhatsumofmoneywouldAmelianeedtoinvesttoaccumulateatotalof$200000attheendof 10yearsat12%p.a.interestcompoundingbiannually?Answercorrecttothenearestcent.

5 3C Drawagraphtoshowtheamountofsimpleinterestearnedin10yearsif$2500wasinvestedat4%. Usethegraphtoestimatetheinterestearnedafter6years.

6 3C Darcyisaskedtoinvest$10000fortenyears.Drawagraphtoshowthedifferenceininterestearned ifthemoneywasinvestedat6%p.a.simpleinterestand6%p.a.compoundinterest.

7 3D Aninvestmentisappreciatingatarateof4%ofitsvalueeachyear.Alkiradecidestoinvest $480000.

a Whatwillbethevalueoftheinvestmentafter10years?Answercorrecttothenearestdollar.

b Howmuchdoestheinvestmentincreaseduringthefirsttenyears?

8 3D Theaverageinflationforthenextthreeyearsispredictedtobe2.5%p.a.Calculatethepriceofthe followinggoodsinthreeyears’time.Answercorrecttothenearestcent.

2Lofsoftdrinkfor$2.80 a Applepiefor$4.60 b Hamburgerfor$6.00 c Bottleofwaterfor$1.60 d Punnetofstrawberriesfor$4.50 e 500gofchickenbreastfor$8.90 f

9 3E ZhangWeipurchasedacarfor$19900.Itdepreciatesat 24%peryear.

a Whatisthesalvagevalueofthecarafterfiveyears? Answercorrecttothenearestdollar.

b Howmuchhasthecardepreciatedoverthefiveyears?

10 3F

LiNatakesoutaloanof$21000over36months.Therepaymentrateis$753.42permonth.

a HowmuchwillLiNapaybackaltogether?Answercorrecttothenearestdollar.

b Whatistheequivalentsimpleinterestrateperannumfortheloan,correcttoonedecimalplace?

11 3F

12 3G

13 3G

Jabariborrows$280000andrepaystheloaninequalfortnightlyrepaymentsof$1250over20years. WhatistheflatrateofinterestperannumonJabari’sloan,correcttotwodecimalplaces?

Derekpurchasedanairfryerfor$275usingaBNPLscheme.Hepaida15%depositandthen fiveequalmonthlyrepayments.Howmuchwaseachrepayment?

Jasonborrowed$470usingashort-termfinancecompany.A$30setupfeewasthenadded.The termswerenodepositandfivemonthlyinstalments.Hepaidfourinstalmentsof$100.Howmuch waslefttobepaidinthefinalinstalment?

14 3H Thetablebelowshowsthemonthlyrepaymentsforaloanof$1000forvaryinginterestrates. Jillborrows$527000at9%over15years.

15 3I

a WhatwereJill’smonthlyrepayments?

b HowmuchdoesJillpayintotalfortheloan?

c HowmuchinterestdoesJillpayonthisloan?

d Howmuchwouldshesaveifshepaidoff theloanin10years?

Carahasacreditcardwithaninterestrateof17%p.a.compoundingdailyandnointerest-free period.Carausedhercreditcardtopayforshoescosting$170.Shepaidthecreditcardaccount 26dayslater.Whatisthetotalamountshepaidfortheshoesincludingtheinterestcharged?Answer correcttothenearestcent.

16 3I Zemira’sbankchargedanannualcreditcardfeeof$300,acashadvancefeeof$4.00andalate paymentfeeof$20.CalculateZemira’sbankingcostsfortheyearifshemade: 9cashadvancesand5latepayments a 15cashadvancesand7latepayments. b

17 3J UsingthecreditcardstatementinExercise 3J Question 7,answerthefollowingquestions.

a Whatisthenameofthepersonwhoownsthecreditcard?

b WhatistheABNoftheStandardBank?

c Whatistheopeningbalance?

d Whatistheminimumpayment?

Extended-responsequestion

1 3K Benjaminusesacreditcardwithanointerest-freeperiodandaninterestrateof18.5%p.a. compoundingdailyfromandincludingthepurchasedateandduedate.Benjamin’saccountisdueon February28.DuringFebruaryhemakesthefollowingtransactions.

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TransactionDetails

06February Coles $278.00

07February Myer $87.00

18February BigW $259.00 18February JAG $120.00

20February Bunnings $460.00 21February Woolworths $300.00

HowmuchinterestwillBenjaminpayduringthemonthonthefollowingtransactions?Answer correcttothenearestcent.

Colestransaction a Myertransaction b BigWtransaction c JAGtransaction d Bunningstransaction e Woolworthstransaction f

PracticePaper1

SectionI

AttemptQuestions1–15(15marks). Allowabout20minutesforthissection.

1 A4Ltinofpaintismadeusingamixtureofblue,whiteandgreenpaintintheratio3:5:2.How muchbluepaintisneededpertin?

2 Acaristravellingataconstantspeed.Ittravels60kmin3hours.Thissituationisdescribedbythe linearequation d = mt

Whatisthevalueof m?

3 Abankcharges0.05753%compoundinterestperdayontheamountowingonacreditcard.Whatis theinterestchargedinfourweeksonabalanceof$1200?

4 Theratioof1.5mto10cmis:

5 Whatistheamountofinterestpaidona$150000investmentover25yearsiftheinterestrateis 7.2%p.a.compoundingannually?Answercorrecttothenearestdollar.

6 Acarusesonaverage7Lper100kminfuel.Howmuchfuelwouldbeusedonatripof382km?

7 Whentwocoinsaretossedtogether,whatistheprobabilitythatonlyoneofthemshowsheads?

8 Afaircoinistossed10timesandtheresultis10headsinarow.Whichstatementaboutthenexttossis themostcorrect?

Thechanceoftossingaheadonthenexttossisbasically0. A Thereisaverygoodchanceoftossingaheadonthenexttoss. B Thatexperimentwasrigged. C Itisunusualtotossahead10timesinarow,butthechanceoftossingaheadonthenexttoss isstill50%. D

9 Whentwocoinsaretossed,whatistheprobabilityoftossingoneortwoheads?

10 Twostandardsix-sideddicearerolled.Thetwoscoresareaddedtogethertogiveatotalscore.Whatis themostlikelytotalscore?

11 Sadiehasplantedredandwhiterosebushesintheratio2:3.Howmanywhiterosebushesarethereif sheplantedatotalof30rosebushes?

12 Whichsymbolrepresentsatoiletinafloorplan?

13 Whichofthefollowingamountswouldearnapproximately$9ininterestin29daysifcompounded dailyatarateof0.037%perday?

A

B

14 Fiveyearsago,thepriceofashirtwas$50.Duetoinflation,thepriceincreasedannuallyby5%.What isthepriceoftheshirtnow?

15 Suneilbreathesabout15timeseachminute.Howmanytimeswouldhebreathein9hours?

SectionII

AttemptQuestions16–18(45marks). Allowabout70minutesforthissection. Allnecessaryworkingshouldbeshownineveryquestion.

Question16(15marks)

a Agroupofthreepeopleshareasumof$1200intheratio5:3:8.Howmuchdoeseach personreceive? (2marks)

b Thediagramshowsasiteplan.

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i Whatisthetotalareaoflandoccupiedbythehouseandgarage? (1mark)

ii Whatpercentageofthesiteistakenupbythehouseandgarage?Answercorrecttoone decimalplace. (2marks)

iii Thereisabrickwall,800mmhigh,alongthenorthernboundaryofthesite.Whatisthe areaofthebrickwallinsquaremetres? (1mark)

iv 600bricksareneededtobuild10squaremetresofwall.Howmanybrickswereusedto buildthewallinpart iii? (1mark)

v Thegutteringaroundtheperimeterofthehouseistobereplaced.Gutteringcosts $60permetre.Howmuchwillthenewgutteringcost? (2marks)

c RanveerandIvyeachinvest$50000for10years.

• Ranveer’sinvestmentearnssimpleinterestatarateof7%perannum.

• Ivy’sinvestmentearnsinterestatarateof6%perannum,compoundingmonthly. Whowillhavethegreateramountattheendof10yearsandbyhowmuch? (3marks)

d Thetablebelowshowsthebankingchargesforcreditcardusage.

Fee Charge

Cashadvance 5%

Latepayment $30

Balancetransfer 1.25%

i Findthecostof40cashadvancesof$100. (1mark)

ii Findthecostofabalancetransferof$7000. (1mark)

iii Findthecostof6latepaymentsand5cashadvancesof$200. (1mark)

Question17(15marks)

a Arectangularposteristobemadetoadvertiseenergy-efficienthousing.Itistobe1.6m longand1mwide.

i Drawascaledrawingoftherectangleusingthescale1:20. (1mark)

ii Outlinetwofeaturesofanenergy-efficienthousethatcouldbepartoftheposter.(2marks)

b Apaintingwasboughtfor$695atthebeginningof2017.Itisexpectedthatthepainting willappreciateinvalueby8%eachyear.Whatwillbethevalueofthepaintingattheend of2027?Answercorrecttothenearestdollar. (2marks)

c Lighttravelsataspeedof2.9979 × 108 metrespersecond.Howmanykilometresdoes lighttravelinonehour?Answerinscientificnotationcorrecttothreesignificantfigures.(3marks)

d Acardischosenfromastandardpackof52playingcards. Whatistheprobabilitythatthecardis:

i anace (1mark)

ii aredace (1mark)

iii aredcard (1mark)

iv aredcardoranace (1mark)

v redorblack (1mark)

vi aredpicturecard (1mark)

vii neitherrednorapicturecard. (1mark)

Question18(15marks)

a Ezraplayscardswithanormaldeckofplayingcardsanddrawsacardfromthedeck.

i Whatarethechancesofdrawinga3orablackcard? (1mark)

ii Whatarethechancesofnotdrawinganace? (1mark)

b Harriethasadebtof$15890withaninterestrateof17%p.a.compoundedannually.

i Howmuchinterestwillbechargedifshemakesnopaymentsfor2years? (2marks)

ii Harrietcouldhavetakentheloanat15%p.a.,compoundedmonthlyfortwoyears. Wouldshehavepaidmoreorlessinterestthisway? (2marks)

c Acoinisbiasedsothattheprobabilityoftossingaheadis75%.

i Ifthecoinistossedonce,whatistheprobabilityoftossingtails? (1mark)

ii Ifthecoinistossedtwice,whatistheprobabilityoftossingtailstwice? (1mark)

d Toniboughtanewcarfor$45000.Whenwillthedepreciatedvalueofthecar fallbelow$10000usingthedeclining-balancemethodofdepreciationiftherateof depreciationis15%p.a.? (2marks)

e Asumof$290000wasinvestedinabankaccountfor6years.Answerthefollowing questionstothenearestdollar.

i Findthesimpleinterestearnediftherateofinterestis3.6%p.a. (1mark)

ii Findthevalueofthisinvestmentattheendof6yearsatthissimplerateofinterest.(2marks)

iii Findthevalueofthisinvestmentattheendof6yearsiftheinterestrateis compoundingannuallyat3.6%p.a. (2marks)

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4

Trigonometry

Someusesoftrigonometry

• TrigonometrymakesGPSandGoogleMaps work: Satellites,TV,Wi-FiandGPSuse trigonometrytopinpointyourexactlocation onEarthbytriangulatingsignals.Every‘You arehere’dotispoweredbytrigonometry.

• Thesineandcosinegraphsareinfiniteand repeating: Theseare periodicfunctions, meaningtheirpatternsrepeatforever. Greatformodellingcycles—liketides, seasons,evenyoursleeprhythm.

• Musichastrigonometryhiddeninsideit: Allmusic/soundwavescanbemodelled usingtrigonometricfunctionsandtheir graphs.Anindividualnotecanbemodelled

withasinewavewhileachordcanbe modelledwithmultiplesinewaves.

• Trigonometrywasasecretweaponin WorldWarII: Navigation,artillerytargeting andradarsystemsreliedheavilyon trigonometrytocalculatedistances,angles andbearings.

• Trigonometrymakesrollercoasters possible: Engineersusesineandcosine functionstodesignandanalyseroller coastertracks,especiallyincalculating angles,distancesandtheshapeofcurves andloops,ensuringsafeandthrillingrides.

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Chaptercontents

4A Pythagoras’theorem

4B UsingPythagoras’theoremtosolve problems

4C Trigonometricratios

4D Trigonometriccalculations

4E Usingtrigonometrytofindan unknownside

4F Usingtrigonometrytofindan unknownangle

4G Solvingpracticalproblems

4H Anglesofelevationanddepression

4I Truebearingsandcompass bearings

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•AppliesPythagoras’theoremand trigonometrytosolveproblems involvingright-angledtriangles (MST-12-S1-04).

©NSWEducationStandardsAuthority

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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4A

Pythagoras’theorem

Learningintentions

• TobeabletousePythagoras’theoremtowriteanequationrelatingthethreesidesofaright-angled triangle

• TousePythagoras’theoremtofindunknownsidesinright-angledtriangles Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.TheformulaforPythagoras’theoremdoesnotappearonthe2027version.

Pythagoras’theorem

Pythagoras’theoremlinksthesidesofaright-angledtriangle.Inaright-angled trianglethesideoppositetherightangleiscalledthehypotenuse.Thehypotenuseis alwaysthelongestside.

Pythagoras’theorem

Pythagoras’theoremstatesthatthesquareofthehypotenuseisequaltothe sumofthesquaresoftheothertwosides.

Hypotenuse (longest side)

Pythagoras’theoremisusedtofindamissingsideofaright-angledtriangleiftwoofthesidesaregiven. Itcanalsobeusedtoprovethatatriangleisrightangled.

Areyouready?Whichsideisthehypotenuse?

Itiscrucialthatyouareabletocorrectlyidentifythehypotenuseofatriangle.

DownloadtheactivityfromyourInteractiveTextbook.

Areyouready?Writetheequation

Havingidentifiedthehypotenuse,itistimetowritetheequationfortherelationshipbetweenthethree sidesofthetriangle.

DownloadtheactivityfromyourInteractiveTextbook.

UNCORRECTEDSAMPLEPAGES

Example1

UsePythagoras’theoremtofindthelengthofasideina right-angledtriangle

UsePythagoras’theoremtofindthevalueof x,correcttotwodecimalplaces.

Solution

a h2 = a2 + b2

Explanation

WritePythagoras’theorem. Substitutethelengthofthesides. Takethesquareroottofind x Expresstheanswercorrecttotwodecimalplaces.

WritePythagoras’theorem. Substitutethelengthofthesides. Make x2 thesubject. Takethesquareroottofind x Expresstheanswercorrecttotwodecimalplaces.

Nowyoutry

UsePythagoras’theoremtofindthevalueof x.Answercorrecttotwodecimalplaces.

Pythagoreantriple

APythagoreantriple(orPythagoreantriad)isasetofthreepositiveintegerswhichsatisfyPythagoras’ theorem.Forexample,theset {3,4,5} isaPythagoreantriple.

Example2

CheckingPythagoreantriples

WhichofthefollowingsetsarePythagoreantriples? 5,12,13 a 6,8,12 b

Solution

a a 2 + b2 = h2

52 + 122 = 132

25 + 144 = 169

169 = 169

∴ 5,12,13isaPythagoreantriple

b a 2 + b2 = h2

62 + 82 = 122

36 + 64 = 144

100 144

∴ 6,8,12isnotaPythagoreantriple

Nowyoutry

WhichofthefollowingsetsarePythagoreantriples?

9,40,41 a

Exercise4A

Explanation

Let a = 5, b = 12and h = 13andcheckthat a2 + b2 = h2 .

Pythagoras’theoremissatisfied,so {5,12,13} isa Pythagoreantriple.

Let a = 6, b = 8and h = 12andcheckthat a2 + b2 = h2 .

Pythagoras’theoremis not satisfied,so {6,8,12} is notaPythagoreantriple.

10,45,51 b

BUILDING

1

Example1a Findthelengthofthehypotenuse,correcttoonedecimalplace.

2

Example1b Findthevalueof x,correcttotwodecimalplaces.

DEVELOPING

3 Calculatethelengthofthesidemarkedwiththepronumeral.(Answercorrecttothenearestmm.)

4 Example2 Show,bysubstitutingintoPythagoras’theorem,whichofthefollowingsetsarePythagoreantriples. 8,9,10 a 7,24,25 b 8,15,16 c 11,60,61 d

CHALLENGING

5 Find,correcttoonedecimalplace,thelengthofthediagonalofarectanglewithdimensions 7.5metresby5.0metres.

UNCORRECTEDSAMPLEPAGES

6a Findthevalueof y,correcttotwodecimalplaces.

b Findthevalueof x usingthevalueof y frompart a,correcttoonedecimalplace.

7a Arectangularblockoflandmeasures25mby50m.Johnwantstoputafence alongthediagonal.Howlongwillthefencebe?Answercorrecttothree decimalplaces.

b Thebaseofaladderleaningagainstawallis1.5mfromthebaseofthewall. Iftheladderis4.5mlong,findhowhighthetopoftheladderisfromthe ground,correcttoonedecimalplace.

c Thelengthofthediagonalofasquaretableis150cm.Whatistheside lengthofthetable?Answercorrecttoonedecimalplace.

Communicateyourreasoning,understandingandfluency

8 {3,4,5} isthesimplestPythagoreantriple.YoucanfindmanyotherPythagoreantriplesby multiplyingeachnumberinthetriplebythesamewholenumber.Forexample, {30,40,50} isalsoa Pythagoreantriplebecause302 + 402 = 502 .

Find10morePythagoreantriplesusingwholenumberslessthan100.Provethattheseare PythagoreantriplesbysubstitutingthevaluesintoPythagoras’theorem.

9 Provethat ∆ABC isaright-angledtriangle.

4B

UsingPythagoras’theoremtosolveproblems

Learningintentions

• TousePythagoras’theoremtofindunknownsidesinright-angledtriangles

• Tobeabletosolvepracticalproblemsinvolvingthesidesofaright-angledtriangle

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.TheformulaforPythagoras’theoremdoesnotappearonthe2027version.

Pythagoras’theoremisusedtosolvemanypracticalproblems.TheseproblemsarerepresentedbyarightangledtriangleandrequiretheuseofPythagoras’theoremtodeterminethehypotenuseorthelengthofa shorterside.

SolvingaproblemusingPythagoras’theorem

1 Readthequestionandunderlinethekeyterms.

2 Drawadiagramandlabeltheinformationfromthequestion.

3 Decidewhethertofindthehypotenuseorthelengthofashorterside.

4 UsePythagoras’theoremtocalculateasolution.

5 Checkthattheanswerisreasonableandunitsarecorrect.

8 Explaintheanswerinwordsandensurethequestionhasbeenanswered.

Example3

SolvingproblemsusingPythagoras’theorem

Ahelicopterisataheightof200mabovethegroundandisa horizontaldistanceof320mfromalandingpad.Findthedirect distanceofthehelicopterfromthelandingpad,correcttotwo decimalplaces.

Solution

200 m h m

320 m

Explanation

Drawadiagramandlabeltheinformationfromthe question.

Labelthehypotenusewith h.Thisrepresentsthe distanceofthehelicopterfromthelandingpad.

h2 = a 2 + b2 = 3202 + 2002

h = √3202 + 2002

≈ 377.36m

∴ Thehelicopteris377.36metresfromthe landingpad.

Nowyoutry

WritePythagoras’theorem. Substitutethelengthofthesides. Takethesquareroottofind x. Expresstheanswercorrecttotwodecimalplaces. Writeyouranswerinwords.

Adroneisataheightof85mabovethegroundandisahorizontaldistanceof120mfromitscontroller. Findthedirectdistanceofthedronefromitscontroller,correcttotwodecimalplaces.

Exercise4B

1 Example3 Afarmgatethatis1.6mhighissupportedbyadiagonalbarof length3.0m.Findthewidthofthegate,correcttoonedecimal place.

2 Aladderrestsagainstabrickwallasshowninthediagramontheright. Thebaseoftheladderis2.0mfromthewall,andreaches3.4mupthe wall.Findthelengthoftheladder,correcttoonedecimalplace.

BUILDING

3

Thebaseofaladderleaningagainstawallis2.8mfromthebaseofthe wall.Iftheladderis5.8mlong,findhowhighthetopoftheladder isfromtheground,correcttoonedecimalplace.

4 Find,correcttoonedecimalplace,thelengthofthediagonalofa rectanglewithdimensions15metresby10metres.

5 Inatriangle ABC,angle ABC isarightangle. AB is12cmand BC is16cm. Findthelengthof AC.

6 Afarmerownsarectangularblockoflandmeasuring30mby60m.Thefarmer wantstoputafencealongthediagonal.Howlongwillthefencebe?Answer correcttothreedecimalplaces.

7 Usethemeasurementsonthediagramtodeterminethedistancetheboatis outtosea.Answercorrecttothenearestmetre.

8 Thehypotenuseofaright-angledtriangleis40cmlongandoneoftheshortersidesmeasures20cm. Whatisthelengthoftheremainingsideinthetriangle?Answercorrecttotwodecimalplaces.

9 Twobuildings,25mapart,areconnectedbyawire. Ifthebuildingsare20mand32mtall,findthelengthof thewireconnectingthetwobuildings,correcttothenearest metre.

10 Ineachofthefollowingdiagrams,usetwoapplicationsofPythagoras’theoremtofindthevalueof x. Giveallanswerscorrecttoonedecimalplace.

11 Findtheareaofthefollowingtriangles.Giveyouranswerscorrecttoonedecimalplace.

12a Apersonwalks9kmeastandthen7kmnorth.Howfararetheyfromtheirstartingpoint? Giveyouranswercorrecttoonedecimalplace.

b AshipleavesPort A andsails22kmsouthandthen18kmwest.Findthestraight-linedistanceof theshipfromPort A.Giveyouranswercorrecttoonedecimalplace.

c Akiteisflyingwithastringlengthof25m.Thekiteis7mhorizontallyawayfromtheperson holdingthestring.Whatistheverticalheightofthekiteabovetheground?

13 Findtheperimeterofthefollowingshapes,givingyouranswerscorrecttotwodecimalplaces.

Communicateyourreasoning,understandingandfluency

14 Explaintheerrormadeineachsetofworking.

4C Trigonometricratios

Learningintentions

• Tounderstandthateachtrigonometricratioisarelationshipbetweenoneangleandtwosides

• Tobeabletosetupanequationusingatrigonometricratio

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thethreetrigonometricratiosareprovidedonthe2027version.

Trigonometricratiosaredefinedusingthesidesofaright-angledtriangle.Thehypotenuseisoppositethe rightangle,theoppositesideisoppositetheangle θ andtheadjacentsideistheremainingside.

Namingsidesofaright-angledtriangle

Hypotenuse (H) q

Adjacent (A)

Opposite (O)

Theoppositeandadjacentsidesarelocatedinrelationtothepositionofangle θ.If θ wastheotherangle inthetoprightofthetriangle,thesideswouldswaptheirlabels.Theletter θ istheGreeklettertheta.Itis commonlyusedtolabelanangle.

Areyouready?Namethethreesides:Hypotenuse,Opposite,Adjacent

Successwithtrigonometrydependsuponnamingthethreesidesofaright-angledtrianglecorrectly. DownloadtheactivityfromyourInteractiveTextbook.

Example4

Namingthesidesofaright-angledtriangle

Whatarethevaluesofthehypotenuse,theoppositesideandtheadjacentside ofthetriangleshown? 4 5 3 q

Solution

Hypotenuseis5. Oppositesideis4. Adjacentsideis3.

Explanation

Hypotenuseisoppositetherightangle. Oppositesideisoppositetheangle θ Adjacentsideisbetweentherightangleandthe angle θ

Nowyoutry

Whatarethevaluesofthehypotenuse,theoppositesideandtheadjacentsideofthetriangleshown?

Thetrigonometricratios

Thetrigonometricratiossin θ,cos θ andtan θ aredefinedusingthesidesofaright-angledtriangle.

= opposite hypotenuse

= adjacent hypotenuse

Trigonometricratios

Themnemonic‘SOHCAHTOA’ispronouncedasasingleword.

SOH: Sine-Opposite-Hypotenuse

CAH: Cosine-Adjacent-Hypotenuse

TOA: Tangent-Opposite-Adjacent

Theorderofthelettersmatchestheratioofthesides.

Themeaningofthetrigonometricratios

Considerthethreetrianglesdrawnbelow.

Thethreetrianglesdrawnaboveshowtheratiooftheoppositesidetothehypotenuseas0.5 1 2 , 2 4 or 3 6 . Thisiscalledthesineratio.Allright-angledtriangleswithanangleof30◦ haveasineratioof0.5.Ifthe angleisnot30◦ theratiowillbedifferent,butanytworight-angledtriangleswiththesameanglewillhave thesamevaluefortheirsineratio.

Similarly,thethreetrianglesdrawnbelowshowtheratiooftheoppositesidetotheadjacentsideas 1 1 1 , 2 2 or 3 3 .Thisiscalledthetangentratio.Allright-angledtriangleswithanangleof45◦ haveatangent ratioof1.

Theratiooftheoppositesidetothehypotenuse(sineratio),theratiooftheadjacentsidetothehypotenuse (cosineratio)andtheratiooftheoppositesidetotheadjacentside(tangentratio)willalwaysbeconstant irrespectiveofthesizeoftheright-angledtriangle.

Example5

Findingthetrigonometricratio

Findthesine,cosineandtangentratiosfortheangle θ inthetriangleshown. 15 17 8 q

Namethesidesoftheright-angledtriangle.

Writethesineratio(SOH).

Substitutethevaluesfortheoppositesideandthe hypotenuse.

Writethecosineratio(CAH).

Substitutethevaluesfortheadjacentsideandthe hypotenuse.

Writethetangentratio(TOA).

Substitutethevaluesfortheoppositesideandthe adjacentside.

Nowyoutry

Findthesine,cosineandtangentratiosfortheangle θ inthetriangleshown.

Findingatrigonometricratio

Findsin θ insimplestformgiventan θ = 6 8

Explanation

Drawatriangleandlabeltheoppositeandadjacent sides.

FindthehypotenuseusingPythagoras’theorem. SubstitutethelengthofthesidesintoPythagoras’ theorem.

Takethesquareroottofindthehypotenuse(H). Evaluate.

Writethesineratio(SOH).

Substitutethevaluesfortheoppositesideandthe hypotenuse.

Simplifytheratio.

Nowyoutry

Findsin

1 Example4 Statethevaluesofthehypotenuse,oppositesideandadjacentsideineachtriangle.

2 Statethevaluesofthehypotenuse,oppositesideandadjacentsideineachtriangle.

3

Writetheratiosforsin θ,cos θ andtan θ foreachtriangleinquestion 1

4 Writetheratiosforsin θ,cos θ andtan θ foreachtriangleinquestion 2

5 Namethetrigonometricratiorepresentedbythefollowingfractions.

6 Findthesine,cosineandtangentratiosinsimplestformforeachangle.

7 Findthesineandcosineratiosinsimplestformforangles ABC and BAC foreachtriangle.

8 Example6

DEVELOPING

Drawaright-angledtriangleforeachofthefollowingtrigonometricratiosanduseittofind: i thelengthoftheunknownside ii theothertwotrigonometricratiosinsimplestform.

9 In △ABC, ∠ABC = 90◦ , AB = 27cmand BC = 36cm.

a Draw △ABC toincludethedimensionsabove.

b Findthelength AC.

c Findthefollowingassimplifiedfractions.

10

Drawaright-angledtriangleforeachofthefollowingtrigonometricratios.UsePythagoras’theorem tofindthevalueoftheunknownside,thenfindthevalueofthefollowing,insimplestform.

a Ifsin θ = 12 37 ,findthevalueofcos θ andtan θ

b Ifsin θ = 21 29 ,findthevalueofcos θ andtan θ

c Ifcos θ = 30 50 ,findthevalueofsin θ andtan θ

d Ifcos θ = 9 41 ,findthevalueofsin θ andtan θ

e Iftan θ = 20 21 ,findthevalueofsin θ andcos θ.

f Iftan θ = 28 21 ,findthevalueofsin θ andcos θ

Communicateyourreasoning,understandingandfluency

11 Drawthefollowingtwotrianglesusingaprotractorandaruler.

a Measurethelengthofthehypotenuse,adjacentandoppositesidesineachtrianglecorrecttothe nearestmm.

b Whatisthevalueofthecosineratiofor60◦ inbothtriangles?

c Whatisthevalueofthesineratiofor60◦ inbothtriangles?

4D Trigonometriccalculations

Learningintentions

• Tobeabletocorrectlycalculatethevalueofatrigonometricexpression

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thethreetrigonometricratiosareprovidedonthe2027version.

Findingatrigonometricratio

Acalculatorisusedtofindatrigonometricratioofagivenangle.Itrequiresthe sin , cos and tan keys. Thetrigonometricratiokeyispressedfollowedbytheangle.Thedegrees,minutesandseconds ◦ ′′′ or DMS isthenselectedtoenterminutesandseconds.Somecalculatorsmayrequireyoutochoosedegrees, minutesandsecondsfromanoptionsmenu.

Findingatrigonometricratio

Findthevalueofthefollowing,correcttotwodecimalplaces.

Nowyoutry Findthevalueofthefollowing,correcttotwodecimalplaces.

Findingananglefromatrigonometricratio

Acalculatorisusedtofindagivenanglefromatrigonometricratio.Checkthat thedegreemodeisselected.Tofindanangle,usethesin 1 ,cos 1 andtan 1 keys.Toselectthesekeys,presstheSHIFTor2ndfunctionkey.

Roundinganglestothenearestminute

Intrigonometry,anangleisusuallymeasuredindegrees,minutesandseconds.Makesurethecalculatoris setuptoacceptanglesindegrees. D or DEG shouldbedisplayedonyourcalculator.

Degree Minutes 1degree = 60minute 1minute = 60seconds

1◦ = 60′ 1′ = 60′′

Convertdecimaldegreestodegreesandminutesusing and DMS onyourcalculator. Writetheanswercorrecttothenearestminute.

Notethatwhenroundinganglestothenearestminute,sincethereare60secondsinaminute,30secondsor morewillroundup.Forexample,34◦12′32′′ isroundedto34◦13′ .

Example8

Findingananglefromatrigonometricratio

a Givensin θ = 0.6123,findthevalueof θ correcttothenearestdegree.

b Giventan θ = 1.45,findthevalueof θ correcttothenearestminute.

Solution

a sin θ = 0.6123

= 37.75599438

38

Explanation

PressSHIFT sin 1 0.6123 or exe . b tan θ = 1.45

= 55◦24′27.76′′

55◦24′

Nowyoutry

PressSHIFT tan 1 1.45 or exe

Converttheanswertominutesbyusing ◦ ′′′ or DMS .

a Givensin θ = 0.5736,findthevalueof θ correcttothenearestdegree.

b Giventan θ = 0.7855,findthevalueof θ correcttothenearestminute.

Example9

Findingananglefromatrigonometricratio

a Givensin θ = 4 5 ,findthevalueof θ correcttothenearestdegree.

b Giventan θ = 2 √5 ,findthevalueof θ correcttothenearestminute.

Solution

a sin θ = 4 5 θ = 53.130102

53◦

b tan θ = 2 √5

Explanation

PressSHIFT sin 1 4 5 or exe

PressSHIFT tan 1 2 √5 or exe

Convertyouranswertominutesbyusing ◦ ′′′ or DMS . Numberofsecondsisgreaterthan30,round 48′37.13′′ upto49′ .

Nowyoutry

a Givensin θ = 13 18 ,findthevalueof θ correcttothenearestdegree.

b Givencos θ = 2 √5 ,findthevalueof θ correcttothenearestminute.

Exercise4D

1 Whatisthevalueofthefollowinganglesinminutes?

BUILDING

2 Whatisthevalueofthefollowinganglesindegrees? 120minutes a 480minutes b 60minutes c 600minutes d 360minutes e 240minutes f 900minutes g 720minutes h 30minutes i 15minutes j 45minutes k 20minutes l

3 Example7a Findthevalueofthefollowingtrigonometricratios,correcttotwodecimalplaces.

4 Example7b Findthevalueofthefollowingtrigonometricratios,correcttotwodecimalplaces.

5

Example7c Findthevalueofthefollowingtrigonometricratios,correcttoonedecimalplace.

6 Findthevalueofthefollowingtrigonometricratios,correcttoonedecimalplace.

Example7d Findthevalueofthefollowingtrigonometricratios,correcttotwodecimalplaces.

7

8 Example8a Giventhefollowingtrigonometricratios,findthevalueof θ correcttothenearestdegree.

9

Example8b Giventhefollowingtrigonometricratios,findthevalueof

10

Example9a Giventhefollowingtrigonometricratios,findthevalueof θ correcttothenearestdegree.

Giventhefollowingtrigonometricratios,findthevalueof θ correcttothenearestdegree.

12 Giventhefollowingtrigonometricratios,findthevalueof θ correcttothenearestminute.

13 Giventhatsin θ = 0.4andangle θ islessthan90◦,findthevalueof:

a θ correcttothenearestdegree

b cos θ correcttoonedecimalplace

c tan θ correcttotwodecimalplaces.

14

Giventhatcos θ = 0.8andangle θ islessthan90◦,findthevalueof:

a θ correcttothenearestdegree

b sin θ correcttoonedecimalplace

c tan θ correcttotwodecimalplaces.

15 Giventhattan θ = 2.1andangle θ islessthan90◦,findthevalueof:

a θ correcttothenearestminute

b sin θ correcttothreedecimalplaces

c cos θ correcttofourdecimalplaces.

CHALLENGING

16 A isanacuteangle.Finditssizeindegreesandminutesforeachoftheseequations.

A = 4 5

Communicateyourreasoning,understandingandfluency

17 Inthetriangleshown,findthefollowingintermsof a, b and c.Whatdoyounotice?

andsin β

18 Complementaryanglesaddto90◦ .

a Findthecomplementaryangleto: 10◦ i 35◦ ii 58◦ iii 75◦ iv

b Evaluatethefollowingcorrecttothreedecimalplaces. cos10◦ andsin80◦ i sin35◦ andcos55◦ ii cos58◦ andsin32◦ iii sin75◦ andcos15◦ iv

c Whatdoyounoticeinpart b?

d Completethefollowinganduseacalculatortocheckyouranswers.

i sin25◦ = cos

ii sin63◦ = cos

iii cos50◦ = sin

iv cos54◦ = sin

4E Usingtrigonometrytofindanunknownside

Learningintentions

• Tochoosethecorrecttrigonometricratioforeverysituation

• Tousetrigonometricratiostofindunknownsidesinright-angledtriangles

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thethreetrigonometricratiosareprovidedonthe2027version.

Trigonometricratiosareusedtofindanunknownsideinaright-angledtriangle,givenatleastoneangleand oneside.ThemethodinvolveslabellingthesidesofthetriangleandusingthemnemonicSOHCAHTOA. Theresultingequationisrearrangedtomaketheunknownside(x)thesubjectandacalculatorisusedtofind theunknownside.

Findinganunknownsideinaright-angledtriangle

1 Namethesidesofthetriangle: H forhypotenuse, O foroppositeand A foradjacent.

2 Usethegivensideandunknownside x todeterminethetrigonometricratio.Themnemonic SOHCAHTOAhelpswiththisstep.

3 Rearrangetheequationtomaketheunknownside x thesubject.

4 Useacalculatortofind x.Remembertocheckthecalculatorissetupfordegrees.

5 Writetheanswertothespecifiedlevelofaccuracy.

Areyouready?Howtochoosethecorrecttrigonometricratio

Havingnamedthethreesidesofatrianglecorrectly,itiscrucialtochoosethecorrecttrigonometricratio andsetuptheequationproperly.

DownloadtheactivityfromyourInteractiveTextbook.

Areyouready?Howtosolvetheequation

Whenatrigonometricratioisused,anequationisformedwithafractionononeside.Theunknown mightbeinthenumeratororthedenominator.Thisactivitywillhelpyousolveequationslikethat.

DownloadtheactivityfromyourInteractiveTextbook.

Example10

Findinganunknownsideinaright-angledtriangle(numerator)

Findthelengthoftheunknownside x inthetriangleshown.Answercorrecttothree decimalplaces.

Solution

Theoppositesideistheunknownandtheadjacent sideis20.

θ = O A tan35◦ = x 20

20 × tan35◦ = x x = 20 × tan35◦

Explanation

Identifythesidesoftheright-angledtrianglethat arerelevanttothequestion. Determinetheratiothatusesthesesides(TOA).

Substitutetheknownvaluesandsubstitute x for O

Tomake x thesubject,multiplybothsidesofthe equationby20.

Press20tan 35exe or = Writetheanswercorrecttothreedecimalplaces.

Nowyoutry

Findthelengthoftheunknownside x inthetrianglesshown.Answercorrecttoonedecimalplace.

Findinganunknownsideinthedenominator

Itispossiblethattheunknownside(x)isthedenominatorofthetrigonometricratio.Forexample,inthe trianglebelow,theunknown x isthehypotenuseofthetriangle.Thisresultsinthetrigonometricratio 2 x

Tosolvethesetypesofequations,multiplybothsidesby x.Thendividebothsidesbythetrigonometric expression (sin60◦) tomake x thesubject. Findinganunknownsideinthedenominator

Example11

Findthelengthoftheunknownside x inthetriangleshown.Answercorrecttotwo decimalplaces.

Solution

Adjacent = 12,hypotenuse = x

cos θ = A H

cos40◦ = 12 x x × cos40◦ = 12 x = 12 cos40◦ x = 15.66488747 ≈ 15.66

Nowyoutry

Usingtrigonometrytofindanunknownside

Explanation

Identifytherelevantsidesoftheright-angled triangle.

Determinetheratiothatusesthesesides(CAH).

Substitutetheknownvalues.

Multiplybothsidesoftheequationby x.

Dividebothsidesbycos40◦ .

Press12 ÷ cos 40 = orexe .

Writetheanswercorrecttotwodecimalplaces.

Findthelengthoftheunknownside x inthetrianglesshown.Answercorrecttoonedecimalplace.

1 Example10

Findthelengthoftheunknownside x ineachtriangle,correcttotwodecimalplaces.

BUILDING

2 Findthelengthoftheunknownside x ineachtriangle,correcttotwodecimalplaces.

3 Example11 Findthelengthofthehypotenuse x ineachtriangle,correcttotwodecimalplaces.

4 Findthelengthoftheunknownside x ineachtriangle,correcttoonedecimalplace.

5 Findthelengthoftheunknownside x ineachtriangle,correcttothreedecimalplaces.

6 Findthelengthoftheunknownside x ineachtriangle,correcttotwodecimalplaces.

7a Afirefighteris12mfromaburningbuilding.Theangletothetopwindowis70◦.Whatminimum ladderlengthisneededtoreachthewindow?

b Adronerisesatanangleof50◦ fromtheground.Calculatetheverticalheightabovegroundafter thedronehastravelled25m.

c Ahillhasaslopeangleof18◦.Howfarmustyouwalkupthehilltogainaverticalheightof 25m?

d Arampmakesanangleof15◦ withthegroundandis5mlong.Howfaralongthegrounddoes therampextend?

e Aflagpolestandsonlevelground.Fromapointontheground10maway,theangletothetopof theflagpoleis55◦.Findtheheightoftheflagpole.

f Aroadrisesatanangleof6◦.Howmuchdoestheroadriseverticallyoverahorizontaldistance of80m?

Communicateyourreasoning,understandingandfluency

8a Thediagonallengthofasquareis35mlong.Findthelengthofonesideofthesquarecorrectto onedecimalplace.

b Thelengthofthediagonalsofarhombusare6cmand8cm.Findthelengthofonesideofthe rhombus.

4F

Usingtrigonometrytofindanunknownangle

Learningintentions

• Tochoosethecorrecttrigonometricratioforeverysituation

• Tousetrigonometricratiostofindunknownanglesinright-angledtriangles

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thethreetrigonometricratiosareprovidedonthe2027version.

Trigonometricratiosareusedtofindanunknownangleinaright-angledtriangle,givenatleasttwosides. ThemethodinvolveslabellingthesidesofthetriangleandusingthemnemonicSOHCAHTOA.The resultingequationisrearrangedtomake(θ)thesubjectandacalculatorisusedtofindtheunknownangle.

Findinganunknownangleinaright-angledtriangle

1 Namethesidesofthetriangle: H forhypotenuse, O foroppositeand A foradjacent.

2 Usethegivensidesandunknownangle θ todeterminethetrigonometricratio.Themnemonic SOHCAHTOAhelpswiththisstep.

3 Rearrangetheequationtomaketheunknownangle θ thesubject.

4 Useacalculatortofind θ.Remembertocheckthecalculatorissetupfordegrees.

5 Writetheanswertothespecifiedlevelofaccuracy.

Example12

Findinganunknownangleinaright-angledtriangle

Findtheangle θ inthetriangleshown.Answercorrecttothenearestminute.

Solution

Opposite = 13,hypotenuse = 22

sin θ = O H

sin θ = 13 22

θ = 36.22154662

θ ≈ 36◦13′

Explanation

Identifytherelevantsidesoftheright-angledtriangle. Determinetheratiothatusesthesesides(SOH).

Substitutetheknownvalues.

Lookforthesin 1 buttononyourcalculator.

Evaluatesin 1 13 22 onyourcalculator.

Convertdecimaldegreestodegreesandminutesusing or DMS onyourcalculator.

Writetheanswercorrecttothenearestminute.

Nowyoutry

1 Example12

Findtheunknownangle θ ineachtriangle.Answercorrecttothenearestdegree.

3 Findtheunknownangle θ ineachtriangle.Answercorrecttothenearestminute.

4 Findtheunknownangle θ ineachtriangle.Answercorrecttothenearestdegree.

5 Findtheunknownangle θ ineachtriangle.Answercorrecttothenearestdegree.

6 Findtheangle θ and ϕ ineachtriangle.Answercorrecttothenearestminute.

7 A12mladderstandingonlevelgroundreaches8.5mupaverticalwall.Findtheangletheladder makeswiththeground.Giveyouranswercorrecttothenearestdegree.

8 ABCD isarectanglewith AC = 85cmand AD = 72cm.Find ∠ACD correcttothenearestminute.

9 Atreeis26mtallandcastsashadow19.5mlong.Whatistheangle,correcttothenearestdegree, thattheraysofthesunmakewiththeground?

10 Findthevalueof θ correcttothenearestminute.

Communicateyourreasoning,understandingandfluency

11a Provethat ∆ABC isaright-angledtriangle.

b Calculatethesizeoftheangle ABC correcttothenearestminute.

12 Find θ correcttothenearestminute.

13 Explaintheerrorintheworkingoutbelowandwritethecorrectsolution.

4G Solvingpracticalproblems

Learningintentions

• Tochoosethecorrecttrigonometricratioforeverysituation

• Tousetrigonometricratiostofindunknownsidesandanglesinright-angledtriangles

• Tosolvepracticalproblemsinvolvingright-angledtrianglesusingtrigonometricratios Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• Questionsinvolvingright-angledtrianglesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thethreetrigonometricratiosareprovidedonthe2027version

Trigonometryisusedtosolvemanypracticalproblems,suchasdeterminingtheheightofatreeor calculatingthewidthofariver.Whensolvingatrigonometricproblem,makesureyoureadthequestion carefullyanddrawadiagram.Labelalltheinformationgiveninthequestiononyourdiagram.

Solvingatrigonometricwordedproblem

1 Readthequestionandunderlinethekeyterms.

2 Drawadiagramandlabeltheinformationfromthequestion.

3 Usetrigonometrytocalculateasolution.

4 Checkthattheanswerisreasonableandunitsarecorrect.

5 Writetheanswerinwordsandensurethequestionhasbeenanswered.

Applicationrequiringthelengthofaside

Averticaltentpoleissupportedbyaropetiedtothetopofthepoleandto apegontheground.Theropeis3minlengthandmakesanangleof29◦ tothehorizontal.Whatistheheightofthetentpole?Answercorrectto twodecimalplaces.

UNCORRECTEDSAMPLEPAGES

Explanation

Drawadiagramandlabeltherequiredheightas x. Hypotenuse = 3m,opposite = x Namethesidesoftheright-angledtriangle.

sin θ = O H

sin29◦ = x 3

x = 3 × sin29◦ = 1.454428 ... ≈ 1.45

∴ Heightofthetentpoleis1.45m.

Nowyoutry

Determinetheratio(SOH).

Substitutetheknownvalues. Multiplybothsidesoftheequationby3.

Press3sin 29 = . Writetheanswercorrecttotwodecimalplaces.

Writetheanswerinwords.

Averticaltentpoleissupportedbyaropetiedtothetopofthepoleand toapegontheground.Theropeis4.5minlengthandmakesanangle of35◦ tothehorizontal.Whatistheheightofthetentpole?Giveyour answercorrecttotwodecimalplaces.

Example14

Applicationrequiringthelengthofaside

Averticaltentpoleissupportedbyaropetiedtothetopofthepoleandtoapegon thelevelground.Thepegis2.5mfromthebaseofthepoleandtheropemakesan angleof63◦ tothehorizontal.Whatisthelengthoftheropebetweenthepegand thetopofthetentpole?Answercorrecttotwodecimalplaces.

Solution

Adjacentis2.5m,hypotenuseistheunknown x

Lengthoftheropeis5.51m.

Nowyoutry

UNCORRECTEDSAMPLEPAGES

Explanation

Identifytheunknownsideandcallit x,andidentify therelevantsidesoftheright-angledtriangle. Determinetheratiothatusesthesesides(CAH). Substitutetheknownvalues,and x for H

Multiplybothsidesoftheequationby x Dividebothsidesoftheequationbycos63◦,so x is nowthesubjectoftheequation.

Press2.5 ÷ cos 63exe or = Writetheanswercorrecttotwodecimalplaces. Writetheanswerinwords.

Apoleissupportedbyawirethatrunsfromthetopofthepoletoapointonthelevel ground5mfromthebaseofthepole.Thewiremakesangleof42◦ withtheground. Findtheheightofthepole,correcttotwodecimalplaces.

42° 5 m

Applicationsrequiringanangle

Trigonometryhasmanyapplications,suchasinbuildingandconstruction.Anyverticalpartsofastructure makearightanglewithhorizontalparts.Slopinglinesinthestructurecompletearight-angledtriangle,and trigonometrycanbeusedtocalculateitsotheranglesandsidelengths.

Example15

Applicationrequiringanangle

TheslopingroofofashedusessheetsofColorbondsteelthatare4.5m longonashedthatis4mwide.Thereisnooverlapoftheroofpastthe sidesofthewalls.Findtheangletheroofmakeswiththehorizontal. Answercorrecttothenearestdegree.

Solution

Adjacent = 4m,hypotenuse = 4.5m

Explanation

Drawadiagramandlabeltherequiredangleas θ.

Theroofmakesanangleof27◦

Namethesidesoftheright-angledtriangle.

Determinetheratio(CAH).

Substitutetheknownvalues. Make θ thesubjectoftheequation.

PressSHIFTcos 1 (4 ÷ 4.5) = orexe

Writetheanswercorrecttothenearestdegree.

Writetheanswerinwords.

Nowyoutry

AslopingroofofashedusesColorbondsteelsheetsthatare5.5mlongonashedthatis4.3mwide. Thereisnooverlapoftheroofpastthesidesofthewalls.Findtheanglethattheroofmakeswiththe horizontal.Giveyouranswercorrecttothenearestdegree.

Exercise4G

1

Example13 Aballoonistiedtoastring25mlong.Theotherendofthestringissecured byapegtothesurfaceofalevelsportsfield.Thewindblowssothatthestring formsastraightlinemakinganangleof37◦ withtheground.Findtheheight oftheballoonabovetheground.Answercorrecttoonedecimalplace.

2 Apoleissupportedbyawirethatrunsfromthetopofthepoletoapointon thelevelground7mfromthebaseofthepole.Thewiremakesanangleof51◦ withtheground.Findtheheightofthepole,correcttotwodecimalplaces.

BUILDING

3 Annnoticedatreewasdirectlyoppositeheronthefarbankofthe river.Aftershewalked50malongthesideoftheriver,shefound herlineofsighttothetreemadeanangleof39◦ withtheriverbank. Findthewidthoftheriver,correcttothenearestmetre.

4 Ashipatanchorrequires70mofanchorchain.Ifthechainisinclinedat 35◦ tothehorizontal,findthedepthofthewater,correcttoonedecimal place.

5 Example15 Averticaltentpoleissupportedbyaropeoflength3.6mtiedtothe topofthepoleandtoapegontheground.Thepoleis2minheight. Findtheangletheropemakestothehorizontal.Answercorrectto thenearestdegree.

6 A3.5mladderhasitsfoot2.5moutfromthebaseofawall.Whatangledoes theladdermakewiththeground?Answercorrecttothenearestdegree.

7 Aplanemaintainsaflightpathof19◦ withthehorizontalafterit takesoff.Ittravelsfor4kmalongtheflightpath.Find,correcttoone decimalplace:

a thehorizontaldistanceoftheplanefromitstake-off point b theheightoftheplaneabovegroundlevel.

8 Awheelchairrampisbeingprovidedtoallowaccesstothefirst floorshops.Thefirstflooris3mabovethegroundfloor.The ramprequiresanangleof20◦ withthehorizontal.Howlongwill therampbe,measuredalongitsslope?Answercorrecttotwo decimalplaces.

9 Anarcher80mfromatarget andlevelwithit,aims2mabove thebullseyeandhitsit.Whatis theangle,correcttothenearest minute,thatthearrowisinclined tothelineofsightfromthe archer’seyetothetarget?

10 Example14 Aropeneedstobefixedwithoneendattachedtothetopof a6mverticalpoleandtheotherendpeggedatanangleof 65◦ withthelevelground.Findtherequiredlengthofrope. Answercorrecttoonedecimalplace.

DEVELOPING

11 Twoladdersarethesamedistanceupawall.Theshorterladderis5m longandmakesanangleof50◦ withtheground.Thelongerladderis 7mlong.Find:

a thedistancecorrecttotwodecimalplaces,fromthegroundto wheretheladdersareonthewall

b theanglethelongerladdermakeswiththeground,correcttothe nearestdegree.

12 Apoleissupportedbyawirethatrunsfromthetopofthepoletoapointonthelevelground7.2m fromthebaseofthepole.Theheightofthepoleis5.6m.Findtheangle,correcttothenearest degree,thatthewiremakeswiththeground.

13 Find θ ineachofthefollowingdiagrams,correcttothenearestminute.

14 △ABC and △ABD areright-angledtriangles.

a Use △ABC tofindthelengthof AB correcttoonedecimalplace.

b Use △ABD tocalculatethelengthof CD correcttoonedecimalplace.(Hint:firstfindthelength of BD.)

c Find ∠CAD

15 Inthediagramshown, △RST and △QRT arebothright-angled triangles.

a For △QRT ,find θ correcttothenearestminute.

b Find y (thelengthof QT )correcttoonedecimalplace.

c For △RST ,find x (thelengthof RS )correcttoonedecimal place.

Communicateyourreasoning,understandingandfluency

16 Inthediagramshown, △EFH and △EGH arebothright-angledtriangles.

a Showthat GH = 120 × sin40◦ tan70◦ .

b Showthat FG = 63.85m.

4H Anglesofelevationanddepression

Learningintentions

• Tounderstandthedifferencebetweenanglesofelevationanddepression

• Tosolveproblemsinvolvinganglesofelevationanddepression

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• QuestionsinvolvinganglesofelevationanddepressionregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thethreetrigonometricratiosareprovidedonthe2027version.

Theangleofelevationistheanglemeasuredupwardsfromthehorizontal.Theangleofdepressionisthe anglemeasureddownwardsfromthehorizontal.

Angleofelevation

Angleofdepression

Angleofelevation = angleofdepression

Theyformalternateanglesbetweentwoparallellines.Thisinformationisusefultosolvesomeproblems.

Example16

Angleofelevation

Aparkrangermeasuredthetopofaplumeofvolcanicashtobeatan angleofelevationof41◦.Fromhermapshenotedthatthevolcanowas 7kmaway.Calculatetheheightoftheplumeofvolcanicash.Answer correcttotwodecimalplaces.

Solution

Explanation

Drawadiagramandlabeltherequiredheightas x tan θ = O A

7 km 41° x

tan41◦ = x 7

7 × tan41◦ = x x = 7 × tan41◦ x = 6.085007165 ≈ 6.09

Theheightofthevolcanicplumewas6.09km.

Nowyoutry

Determinetheratio(TOA).

Substitutetheknownvalues.

Multiplybothsidesoftheequationby7.

Writetheanswercorrecttotwodecimalplaces.

Writetheanswerinwords.

a Tistands25mawayfromthebaseofatallbuildingandmeasuretheangle ofelevationtothetopofthebuildingas72◦ . Findtheheightofthebuilding,correcttothenearestmetre.

b A22mhightreecastsalongshadow. Findthelengthoftheshadowiftheangleofelevationfromtheendof theshadowtothetopofthetreeis15◦ Roundyouranswertothenearestmetre.

Findingadistanceusingangleofdepression

Thetopofacliff is85mabovesealevel.Minhsawatallship.Heestimatedthe angleofdepressiontobe17◦ .

a Howfarwastheshipfromthebaseofthecliff?Answercorrecttothenearest metre.

b Howfaris theshipinastraightlinefromthetopofthecliff?Answercorrectto thenearestmetre. Example17

Solution

a

85 m 17° 17° x y

tan θ = O A

tan17◦ = 85 x

x × tan17◦ = 85 x = 85 tan17◦ x = 85 tan17◦ = 278.022 ≈ 278m

∴ Theshipis278metresfromthebase ofthecliff.

b sin θ = O H

sin17◦ = 85 y

y × sin17◦ = 85 y = 85 sin17◦ = 290.7258 ≈ 291m

∴ Theshipis291metresfromthetop ofthecliff.

Nowyoutry

Explanation

Drawadiagramandlabelthedistancetothebase ofthecliff as x andthedistancetothetopofthe cliff as y.

Determinetheratio(TOA).

Substitutetheknownvalues.

Multiplybothsidesoftheequationby x.

Dividebothsidesbytan17◦

Writetheanswercorrecttonearestmetre. Writetheanswerinwords.

Determinetheratio(SOH).

Substitutetheknownvalues.

Multiplybothsidesoftheequationby y.

Dividebothsidesbysin17◦

Writetheanswercorrecttonearestmetre. Writetheanswerinwords.

Aironoreseamof120mslopesdownatanangleofdepressionfromthe horizontalof38◦.Themineengineerwishestosinkaverticalshaft, x,asshown. Whatisthedepthoftherequiredverticalshaft?Answercorrecttothenearest metre.

38°

m x

Example18

Findinganglesofelevationanddepression

ShobnastandsontheobservationdeckoftheSydneyTowerEye,whichis250metresabovetheground. Fromthisvantagepoint,shespotstheSydneyOperaHouse,located1.6kmaway.

WhatistheangleofdepressionfromtheobservationdecktotheOperaHouse,correcttothenearest minute?

Solution

θ

Explanation

Angleofdepression = angleofelevation(alternate anglesinparallellines). Therefore,findtheangleofelevationinstead. Convert1.6kmto1600m.

Determinetheratio(TOA).

Substitutetheknownvalues,usingangleof elevationastheunknownangle.

Lookforthetan 1 buttononyourcalculator.

Evaluatetan 1 250 1600 onyourcalculator.

Convertdecimaldegreestodegreesandminutes using orDMS onyourcalculator. Writetheanswercorrecttothenearestminute.

Nowyoutry

Ahelicopterisflying850mabovesealevel.Itisalso1162minastraight linetoashipatsealevel.Whatistheangleofdepressionfromthehelicopter totheship?Answercorrecttothenearestdegree.

1

Example16 Arjunwalked400mawayfromthebaseofatallbuilding,onlevelground.He measuredtheangleofelevationtothetopofthebuildingtobe62◦.Findtheheight ofthebuilding.Answercorrecttothenearestmetre.

2

4

TheangleofdepressionfromthetopofaTVtowertoasatellitedishnearits baseis59◦.Thedishis70mfromthecentreofthetower’sbaseonflatland. Findtheheightofthetower.Answercorrecttoonedecimalplace.

WhenTasneemlookedfromthetopofacliff 50mhigh,shenoticedaboat atanangleofdepressionof25◦.Howfarwastheboatfromthebaseofthe cliff?Answercorrecttotwodecimalplaces.

5

Thepilotofanaeroplanesawanairportatsealevelatanangleof depressionof13◦.Heraltimetershowedthattheaeroplanewasat aheightof4000m.Findthehorizontaldistanceoftheaeroplane fromtheairport.Answercorrecttothenearestmetre.

Theangleofelevationtothetopofatreeis51◦ atadistanceof45mfromthe pointonlevelgrounddirectlybelowthetopofthetree.Whatistheheightofthe tree?Answercorrecttoonedecimalplace.

6

7 Example18

Jackmeasurestheangleofelevationtothetopofatreefromapointonlevel groundas35◦.WhatistheheightofthetreeifJackis50mfromthebaseof thetree?Answercorrecttothenearestmetre.

AtouristviewingSydneyHarbourfromabuilding130mabovesea levelobservesaferry800mfromthebaseofthebuilding.Findthe angleofdepression.Answercorrecttothenearestdegree.

8 Whatwouldbetheangleofelevationtothetopofaradiotransmitting tower130mtalland300mfromtheobserver?Answercorrecttothe nearestdegree.

9

Jarliobservesthetopofatreeatadistanceof60mfromthebase ofthetree.Thetreeis40mhigh.Whatistheangleofelevationto thetopofthetree?Answercorrecttothenearestdegree.

10

Atownis12kmfromthebaseofamountain.Thetownisalsoadistanceof 12.011kminastraightlinetothetopofthemountain.Whatistheangleof depressionfromthetopofamountaintothetown?Answercorrecttothe nearesttenthofadegree.

11 Find,correcttothenearestdegree,theangleofelevationofarailwayline thatrises7mforevery150malongthetrack.

12 Thedistancefromthebaseofthetreeis42m.Thetreeis28min height.Whatistheangleofelevationmeasuredfromgroundleveltothe topofatree?Answercorrecttothenearestdegree.

13 Theangleofelevationtothetopofatreefromapoint A onthegroundis25◦.Thepoint A is22m fromthebaseofthetree.Findtheheightofthetree.Answercorrecttothenearestmetre.

14 Aplaneis460mdirectlyaboveoneendofa1200mrunway.Findtheangleofdepressiontothefar endoftherunway.Answercorrecttothenearestminute.

DEVELOPING

15 Arocketlaunchingpadcastsashadow35mlongwhentheangleofelevationofthesunis52◦.How highisthetopofthelaunchingpad?Answercorrecttothenearestmetre.

16

Ishaanisonthetopofa65mhighcliff.Heobservesaman swimmingouttoseaatanangleofdepressionof51◦.Ishaan alsoseesaboatouttoseaatanangleofdepressionof30◦ Find,correcttothenearestmetre,thedistance:

a x ofthemanfromthebaseofthecliff

b y oftheboatfromthebaseofthecliff

c fromthemantotheboat.

17 Alighthousestandsonthetopofacliff.Asailingboatislocated75mfromthe baseofthecliff.Theangleofelevationtothetopofthecliff andthetopofthe lighthouseis42◦ and50◦ respectively.Findtheheightofthelighthouse,correct tothenearestmetre.

18 Aplaneisflyingatanaltitudeof880m.Kaylaisstandingontheground. Sheobservestheangleofelevationtotheplaneas67◦40′,and25seconds latertheangleofelevationhadchangedto24◦30′ .

a Howfarhadtheplaneflowninthattime?Answercorrecttothenearest metre.

b Whatisthespeed(tothenearestkm/h)oftheplane?

Communicateyourreasoning,understandingandfluency

19 Acommunicationtowerislocatedonthetopofahill.Theangleofelevationtothetopofthehill fromanobserver2kmawayfromthebaseofthehillis6◦.Theangleofelevationtothetopofthe towerfromtheobserveris8◦.Find,correcttothenearestmetre,theheightofthe:

a hill

b hillandthetower

c tower.

20 Theangleofelevationfromaboatouttoseatothetopofa350mcliff is13◦.Aftertheboattravels directlytowardsthecliff,theangleofelevationfromtheboatis19◦.Howfardidtheboattravel towardsthecliff?Answercorrecttothenearestmetre.

UNCORRECTEDSAMPLEPAGES

4I Truebearingsandcompassbearings

Learningintentions

• Tounderstandtruebearingsandcompassbearings

• Tosolveproblemsinvolvingtruebearingsandcompassbearings

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingbearingsregularlyappearinHSCExaminations.

Abearingisthedirectiononeobjectisfromanotherobjectoranobserverorafixedpoint.Therearetwo typesofbearings:compassbearingsandtruebearings.

Compassbearings

Compassbearingsusethefourdirectionsofthecompass:north,east, southandwest(N,E,SandW).TheNSlineisverticalandtheEWlineis horizontal.Inbetweenthesedirectionsareanotherfourdirections:north-east, south-east,south-westandnorth-west(NE,SE,SWandNW).Eachofthese directionsmakesanangleof45◦ withtheNSandEWlines.

Adirectionisgivenusingacompassbearingbystatingtheangleeitherside ofnorthorsouth.Forexample,acompassbearingofS50◦Wisfoundby measuringanangleof50◦ fromthesouthdirectiontowardsthewestside.

Areyouready?Whatcompassbearingisthat?

Applyingacompassbearingtoadiagramisacriticalskillinthissection.Thisactivitywillimprove yourconfidenceinapplyingcompassbearings.

DownloadtheactivityfromyourInteractiveTextbook.

Understandingacompassbearing

a Findthecompassbearingof:

b Drawthecompassbearing:

Solution

ai Thecompassbearingof A from O isN30◦E.

Explanation

Acompassbearingiswrittenasstartingfromnorth orsouth,anglelessthan90◦,goingtowardstheeast orwest. A from O isintheNEquadrantsoitis north30◦ goingeast.

ii Thecompassbearingof B from O isS60◦E.

B from O isintheSEquadrantsoitis south180◦ 120◦ = 60◦ goingeast.

i S40◦WisintheSWquadrant,south40◦ towardsthewest.

ii N40◦WisintheNWquadrant,north40◦ towardsthewest.

iii S10◦EisintheSEquadrant,south10◦ towards theeast.

iv N52◦EisintheNEquadrant,north52◦ towards theeast.

Nowyoutry

a Findthecompassbearingof:

A fromO i B from O ii

b Drawthecompassbearing:

S65◦W i

N65◦W ii

S35◦E iii

◦E. iv

Truebearings

Atruebearingistheanglemeasuredclockwisefromnortharoundtotherequired direction,anditiswrittenwiththeletterTafterthedegreesorminutesorseconds symbol.Truebearingsaresometimescalledthree-figurebearingsbecausetheyare writtenusingthreenumbersorfigures.Forexample,120◦Tisthedirectionmeasured 120◦ clockwisefromnorth.ItisthesamebearingasS60◦E.

Thesmallesttruebearingis000◦Tandthelargesttruebearingis360◦T. Theeightdirectionsofthecompasshavethefollowingtruebearings:

°

Clockwise from north

Thebearingsinthefollowingdiagramsaregivenusingbothmethods.

Compassbearing

Adirectiongivenbystatingtheangleeither sideofnorthorsouth,suchasS60◦E.

Truebearing

Adirectiongivenbymeasuringtheangleclockwise fromnorthtotherequireddirection,suchas120◦T.

Areyouready?Whattruebearingisthat?

Applyingatruebearingtoadiagramisacriticalskillinthissection.Thisactivitywillimproveyour confidenceinapplyingtruebearings.

DownloadtheactivityfromyourInteractiveTextbook.

Example20

Understandingatruebearing

a Findthetruebearingof:

i B from O

ii C from O

iii D from O.

b Draweachtruebearingonaseparatediagram.

i

Solution

ai Thebearingof B from O is050◦T.

ii Thebearingof C from O is210◦T.

iii Thebearingof D from O is330◦T.

Explanation

Measuretheangleclockwisefromnorth.

Measuretheangleclockwisefromnorth.

Measuretheangleclockwisefromnorth.

270◦ isduewest + 60◦ further = 330◦

Continuedonnextpage

Nowyoutry

a Findthetruebearingof:

B from O i

C from O ii

D from O iii

b Draweachtruebearingonaseparatediagram.

Drawanglesclockwisefromnorth.

Solvingproblemsusingbearings

Whensolvingbearingproblemsinvolvingtwoormorepoints,wecandrawimaginaryverticallines extendingnorthfromeachpoint.Thesenorthrunninglinesarealwaysparallel.Usingtherulesofangles formedbyparallellines,wecandetermineotherangleswithinthediagram.

Cointerioranglesinparallellines addto180◦ .

Alternateanglesinparallellines areequal. Correspondinganglesin parallellinesareequal.

Solvingproblemsusingbearings

a Ifthebearingof P from O is110◦,whatisthebearingof O from P?

b Ifthebearingof Q from O is230◦,whatisthebearingof O from Q?

Thebearingof O from P is290◦T.

Drawthenorth/southlineatpoint P.

Usecointeriorangles:180◦ 110◦ = 70◦

Tofindthebearingof O from P : 360◦ 70◦ = 290◦

Continuedonnextpage

b 230° N N Q O 50° 50°

Thebearingof O from Q is050◦T.

Nowyoutry

Drawadiagramtoanswerthefollowing.

Drawthenorth/southlineatPoint Q

Usealternateangles: 230◦ 180◦ southbearing = 50◦ Alternateanglesareequal.

a Ifthebearingof P from O is120◦,whatisthebearingof O from P?

b Ifthebearingof Q from O is250◦,whatisthebearingof O from Q?

Example22

Usingbearingsandtrigonometrytocalculatedistance

Cassieruns12kmonabearingof210◦.Howfarsouthhassherun?Answercorrecttoonedecimalplace.

Solution

N

Explanation

Drawadiagramandformaright-angledtriangle. s isforhowfarsouthsheran.

Theanglefortheright-angledtriangleisgivenby 210◦ 180◦ = 30◦

cos θ = A H

cos30◦ = s 12

s = 12 × cos30◦

s = 10.3923 ...

s ≈ 10.4

Sheran10.4kmsouth.

Nowyoutry

Determinetheratiothatusesthesesides.

Substitutetheknownvalues.

Multiplybothsidesby12.

Roundyouranswertoonedecimalplace.

Finnruns15kmonabearingof310◦.Howfarnorthhasherun?Answercorrecttoonedecimalplace.

Exercise4I BUILDING

1 Statethecompassbearingandthetruebearingofeachofthefollowingdirections. NE a NW b SE c SW d

2 Example19,20 Statethecompassbearingandthetruebearingofeachofthefollowingdirections.

3 Sketcheachofthesebearingsonaseparatediagram.

4 Example21 Foreachofthesequestions,drawadiagramandhencefindthebearing.

a Ifthebearingof P from O is065◦,whatisthebearingof O from P?

b Ifthebearingof Q from O is190◦,whatisthebearingof O from Q?

c Ifthebearingof R from O is155◦,whatisthebearingof O from R?

d Ifthebearingof S from O is310◦,whatisthebearingof O from S ?

5

Example22 Considerthefollowingscenariosandgiveallanswerscorrecttoonedecimalplace.

a Ashipsails102kmfromPort P inaNEdirection.Howfar northhastheshiptravelled?

b AbikeriderstartstheirjourneytravellingN35°Wfor28km. Howfarnorthhastheridertravelled?

c Ahikerwalks6kmonabearingof120◦ .

i Howfarsouthhasthehikerwalkedfromtheir startingpoint?

ii Howfareasthasthehikerwalkedfromtheir startingpoint?

d Acartravelsfromhome(H)onacountryroadfor 65kmonabearingof250◦

i Howfarwesthasthecartravelled?

ii Howfarsouthhasthecartravelled?

e AshipsailsfromPort S for85kmonabearingof158◦ HowfareasthastheshiptravelledfromPort S ?

f JuliawalksS60◦Wfor8.4km.

i HowfarwesthasJuliawalked?

ii HowfarsouthhasJuliawalked?

DEVELOPING

6 Inthefollowingscenarios,firstfind θ,thenfindthetruebearing.Giveallanswerscorrecttothe nearestdegree.

a Town A is45kmeastand32kmnorthofTown B.Findthetrue bearingofTown A fromTown B.

b Town S is45kmeastand32kmsouthofTown B.Findthetrue bearingofTown S fromTown B

c Town C is230kmwestand110kmsouthofTown D.Findthetrue bearingofTown C fromTown D.

d Jelenarunssouthfor6km,thensheturnsandrunswestfora further8km.Whatisthetruebearingfromherstartingpoint?

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e Toddrunssouthfor6km,thenheturnsandrunseastfora further8km.Whatisthetruebearingfromhisstartingpoint?

f Town A is80kmnorthand120kmwestofTown B.Findthe truebearingofTown A fromTown B

7 Aplaneistravellingonatruebearingof030◦ from A to B.

a Whatisthecompassbearingof A to B?

b Whatisthetruebearingof B to A?

c Whatisthecompassbearingof B to A?

8 Thediagramshowsthepositionof P, Q and R relativeto S .Inthediagram, R isNEof S , Q isNWof S and ∠PSR is155◦

a Whatisthetruebearingof R from S?

b Whatisthetruebearingof Q from S?

c Whatisthetruebearingof P from S?

d Whatisthetruebearingof S from R?

e Whatisthetruebearingof S from P?

f Whatisthetruebearingof S from Q?

9

Thebearingof E from D isN38◦E, F iseastof D and ∠DEF is87◦ .

a Findthevaluesof x and y

b Whatisthecompassbearingof E from F?

c Whatisthetruebearingof E from F?

10 Huntertravelsfrom X to Y for125kmonabearingofN32◦W.

a HowfardidHuntertravelduenorth,tothenearestkilometre?

b HowfardidHuntertravelduewest,tothenearestkilometre?

11 Diyacycledfor15kmwestandthen24kmsouth.

a Whatisthevalueof θ correcttothenearestdegree?

b WhatisDiya’struebearingfromherstartingpoint?

c WhatisDiya’scompassbearingfromherstartingpoint?

12 Aboatsails137kmfromPortStephensonabearingof065◦T.

a Howfareasthastheboatsailed?Answercorrecttoonedecimalplace.

b Howfarnorthhastheboatsailed?Answercorrecttoonedecimalplace.

13 Ashipsails5kilometreswest,then5kilometressouth.

a Whatisthecompassbearingoftheshipfromitsoriginalposition?

b Whatisthetruebearingoftheshipfromitsoriginalposition?

14 Lewistravelledfor8.5kmonabearingofS30◦Wfromhishome.

a HowfarwestisLewisfromhome?Answercorrecttotwodecimalplaces.

b HowfarsouthisLewisfromhome?Answercorrecttotwodecimalplaces.

c Whatisthecompassbearingofhishomefromhiscurrentposition?

15 km

15 Acyclistrides15kmonabearingof290◦

a Howfarnorthisthecyclist?Answercorrecttoonedecimalplace.

b Howfarwestisthecyclist?Answercorrecttoonedecimalplace.

c Whatisthetruebearingofthecyclist’sstartingpositionfromthecyclist’scurrentposition?

16 Isladrove6kmeastandthendroveafurther13kmonabearingof070◦T. HowfareastisIslafromherstartingposition?

17 Siyawalkedfor3hoursand45minutesat6km/honabearingof145◦ togettoherfriend Radha’shouse.

a Howfarsouthhasshewalked?

b WhatisthetruebearingofSiya’sstartingpointfromRadha’shouse?

Communicateyourreasoning,understandingandfluency

18 Bradleedrivesataspeedof80km/honabearingof125◦Tfor2.5hours.

a How fareastisBradleefromhisstartingposition?Answercorrecttothenearestkilometre.

b WhatisthetruebearingofBradlee’sstartingpositionfromhiscurrentposition?

19 Aplaneleft O andtravelled350kminthedirection225◦Ttoreach P.Itthenchangeddirectionand travelled315◦Ttoreachpoint Q whichisdueeastof O.

b Howfardidtheplanetravelfrom P to Q? UNCORRECTED

a Explainwhy ∠QPO is90◦

Keyideasandchaptersummary

Pythagoras’ theorem

Applying Pythagoras’theorem

Pythagoras’theoremstatesthatthesquareofthe hypotenuseisequaltothesumofthesquaresoftheother twosides h2 = a2 + b2 a

1 Readthequestionandunderlinethekeyterms.

2 Drawadiagramandlabeltheinformationfromthequestion.

3 Decidewhethertodeterminethehypotenuseorthelengthofashorterside.

4 UsePythagoras’theoremtocalculateasolution.

5 Checkthattheanswerisreasonableandunitsarecorrect.

Trigonometric ratios sin θ = O H (SOH) cos θ = A H (CAH) tan θ = O A (TOA)

(H) q

(

(A)

Usingacalculatorin trigonometry 1degree = 60minutes 1minute = 60seconds 1◦ = 60′ 1′ = 60′′

Findinganunknown side

1 Namethesidesofthetriangle.

2 Usethegivensideandunknownside x todeterminethetrigonometricratio. ThemnemonicSOHCAHTOAhelps.

3 Rearrangetheequationtomaketheunknownside x thesubjectthenusethe calculatortofind x

Findinganunknown angle 1 Namethesidesofthetriangle.

2 Usethegivensidesandunknownangle θ todeterminethetrigonometricratio. ThemnemonicSOHCAHTOAhelps.

3 Rearrangetheequationtomaketheunknownangle θ thesubjectthenusethe calculatortofind θ.

Solvingpractical problems

Anglesofelevation anddepression

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1 Readthequestionandunderlinethekeyterms.

2 Drawadiagramandlabeltheinformationfromthequestion.

3 Usetrigonometrytocalculateasolution.

Angle of elevation

Horizontal q

Horizontal q

Angle of depression

Bearings Compassbearing Adirectiongivenbystatingtheangleeithersideofnorthor south,suchasS60◦E.

Truebearing Adirectiongivenbymeasuringtheangleclockwisefromnorthtothe requireddirection,suchas120◦

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook.

Checklist

4A 1 IcanusePythagoras’theoremtofindthemissingsideinaright-angledtriangle.

e.g.Findthevalueof x,correcttoonedecimalplacewhereappropriate..

4B 2 IcanapplyPythagoras’theoremtosolvepracticalproblemsintwodimensions.

e.g.Thebaseofaladderleaningagainstawallis1.5mfrom thebaseofthewall.Iftheladderis4.5mlong,findhow highthetopoftheladderisfromtheground,correcttoone decimalplace.

4C 3 Icanfindthesine,cosineandtangentratiosinaright-angledtriangle.

e.g.Findsin θ,cos θ andtan θ intheright-angledtriangle.

4C 4 Icanlabelthehypotenuse,oppositesideandadjacentsideinaright-angledtriangle.

e.g.Labelthehypotenuse,adjacentandoppositesideinthe right-angledtriangle.

4D 5 Icanuseacalculatorintrigonometry.

e.g.Findcorrecttotwodecimalplaces:

4D 6 Icanfindananglefromatrigonometricratio.

e.g. a Givensin θ = 0.6123,find θ correcttothenearestdegree.

b Giventan θ = 1.45,findthevalueof θ correcttothenearestminute.

4E 7 Icanapplytrigonometricratiostoright-angledtrianglestofindthelengthsof unknownsides.

e.g.Findthelengthsoftheunknownsides,correcttoonedecimalplace.

4F 8 Icanapplytrigonometricratiostoright-angledtrianglestofindthesizeofunknown anglesinbothdegrees,anddegreesandminutes.

e.g.Findtheangle θ inthetriangleshown.Answercorrect tothenearestminute.

4G 9 Icansolvepracticalproblemsinvolvingtrigonometry.

e.g.A3.5mladderhasitsfoot2.5moutfromthebaseofthewall.Whatangle doestheladdermakewiththeground,correcttothenearestdegree?

4G 10 IcansolvepracticalproblemsinvolvingPythagoras’theorem,thetrigonometryof right-angledtriangles,anglesofelevationanddepression,andtruebearingsand compassbearings.

e.g.Atriangle DEG hasarightangleat E and DG is8minlength.Alineisdrawn from D to F with ∠DFG = 144◦ and ∠GDF = 7◦

a Findthelengthof DE.Answercorrectto twodecimalplaces.

b Findthesizeof ∠DFE.

c Findthelengthof FE.Answercorrectto twodecimalplaces.

4H 11 Icansolvepracticalproblemsinvolvinganglesofelevationanddepression.

e.g.Anironoreseamof120mslopesdownatanangleof depressionfromthehorizontalof38◦.Themineengineer wishestosinkaverticalshaft, x,asshown.Whatisthe depthoftherequiredverticalshaft?

Answercorrecttothenearestmetre.

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4I 12 Icanidentifyandinterprettruebearingsandcompassbearingsandconvert betweenthem.

e.g.Statethecompassbearingandtruebearing of B from O

4I 13 Icansolvepracticalproblemsinvolvingtruebearingsandcompassbearings.

e.g.Simranruns12kmonabearingof210◦.Howfarsouthhassherun?Answer correcttoonedecimalplace.

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Multiple-choicequestions

1 4A Whatisthelengthofthehypotenuseifthetwoothersidesare12cmand16cm?

Whatistheangleofelevationtothetopofatower80mtalland100mfromtheobserver?Answer indegreescorrecttoonedecimalplace.

Short-answerquestions

1 4A Findthevalueof x,correcttotwodecimalplaces.

2 4B Arectangularblockoflandmeasures20mby27m.Afenceisrequiredalongitsdiagonal.How longwillthefencebe?Answercorrecttoonedecimalplace.

3 4B Calculatethelengthof

4 4C Inthetriangleshown,statethevalueofthe:

6 4C Inthetriangleshown,whatarethefollowingtrigonometricratiosinsimplest form?

10 4F Findtheunknownangle θ ineachtriangle.Answercorrecttothenearestminute.

11 4H Susanlookedfromthetopofacliff,62mhigh,andnoticedashipatanangle ofdepressionof31◦.Howfarwastheshipfromthebaseofthecliff?Answer correcttoonedecimalplace.

12 4I Brookerodefor8.5kmonabearingofN43◦Wfromherhome.

a HowfarnorthisBrookefromhome?Answercorrecttoonedecimalplace.

b HowfarwestisBrookefromhome?Answercorrecttoonedecimalplace.

13 4D Findthevalueofthefollowingtrigonometricratios,correcttotwodecimalplaces.

Extended-responsequestions

1 4G Atriangle BCE hasarightangleat C, ∠BEC = 24◦ and BE is 20minlength.Alineisdrawnfrom B to D with ∠DBE = 29◦

a Findthelengthof BC.Answercorrecttotwodecimalplaces.

b Findthesizeofangle BDC.Answercorrecttothenearest minute.

c Findthelengthof CD.Answercorrecttotwodecimalplaces.

2 4I Ahikerwalksfor12kmonabearingof165◦.Howforsouthhasthehikerwalkedfromtheirstarting point?

5

Simultaneous linearequations

Usingalgebratomakebusinessdecisions

OliviarunsamarketstalleverySaturday. Shesellsbraceletswhichshemakesherself.

Relevantinformation:

•howmuchdoesthestallcost?

•howmuchdoesitcosttomakethe bracelets?

•howmuchmightpeoplebewillingtopay forthem?

Inthischapterwewillusealgebratounpack situationslikethese.

Hercosts(alsoknownasexpenditure)forma straightlineontheCartesianplane.

Herincome(alsoknownasrevenue)formsa differentstraightline.

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•howpriceshouldsheaskfor?

•howmanymightsheselleverySaturday?

•howmanywillsheneedtoselljustto breakeven?

•ifshesells50braceletseverySaturday, howmuchprofitorlosswillshemake?

Thepointatwhichthelinesmeetisvery importanttothesuccessorfailureofthis businessventure.

5A Straightlines (CONSOLIDATING)

5B Linearrelationships (CONSOLIDATING)

5C Solvingsimultaneousequations graphically

5D Solvingpracticalproblemsusing simultaneousequations

5E Break-evenanalysis

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•Representsandinterpretsthe relationshipsbetweenquantitiesin algebraicandgraphicalterms (MST-12-S1-01).

©NSWEducationStandardsAuthority

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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5A Straightlines CONSOLIDATING

Learningintentions

• Toconstructstraight-linegraphsontheCartesianplane

• Tounderstandthegradientandtheverticalintercept

• Tousethegradient–interceptformulatodrawstraight-linegraphs

• Todeterminetheequationofalinefromagraph Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvingstraightlinesregularlyappearinHSCExaminations

• StudentsneedtobefamiliarwiththeformulaswhichappearontheHSCReferenceSheetpublishedby NESA.Thegradient–interceptformulaisnotprovidedonthe2027version.

Linearrelationships:Thebigpicture

Thistopicshouldbefamiliartoyoufrompreviousyearsofschooling.

Inthischapter,thefollowingconceptswillbeconsolidatedandextended,withemphasisonreal-life scenarioswhicharelinearrelationships.

Alinearrelationshipcanbeviewedandrepresentedinseveraldifferentways,asfollows.

Areal-lifescenario

Asrahas $4inhermoneyboxatpresent.

Thisisasingle,one-off,once-onlyamount,atthebeginningofatimeperiod.Thetimeis0.

Shedecidestoadd$3everyweek.Thisisarepetitiveincrease.

Thisisa linearrelationship,becausesheisalwaysaddingthesameamounteveryweek.

Anumberpattern

4,7,10,13, ...

Thenumbersinthepatternaresometimescalled terms

Thenumber4isthe presentvalue

Thenumbers7,10,13, arethe futurevalues ofthisinvestment.

Atableofvalues

Thenumbers4,7,10,13, ... areplacedinthebottomrowofatable.

Thetoprowtrackstheweeks,startingfrom0.

Weeks(W) 0 1 2 3

Dollars(D) 4 7 10 13

Variables

‘Weeks’and‘Dollars’arecalled variables. Thetoprowisthe independentvariable.

Thebottomrowisthe dependentvariable.

Anequation,sometimescalleda‘rule’ora‘function’

Youmayrecallanactivitycalled‘guesstherule’,butguessingisreallynotrequired.

Youjustneedtolookinthecorrectplaces:

• Asraisadding$3,repeatedly.Thisisknownasthe gradient, slope or rateofchange

• Shebeganwith$4.Thisisknownasthe constant,orthe verticalintercept,orthe y-intercept

Chooseanynumberinthetoprow.Multiplyby3thenadd4.Thatwillgiveyouthenumberinthe bottomrow.

W × 3 + 4 = D

3W + 4 = D

Theequationisusuallywritten‘backwards’.

D = 3W + 4

Note:

• thegradient(3)hasaletterattachedtoit

• theverticalintercept(4)istheconstantterm.

Everystraightlinecanbewrittenintheform:

y = mx + c,where:

m isthe gradient,sometimescalledthe slope

c isthe y-intercept,alsoknownasthe verticalintercept

AseriesofpointsontheCartesianplane

Everycolumninthetableisapoint:(0,4)then(1,7),(2,10),(3,13), ...

AgraphontheCartesianplane

Thepointscanbejoinedwithone straightline.

Asra’ssavingsare increasingataconstantrate

Therelationshipbetweenweeksanddollarsislinear.

Thisisanexampleofa linearrelationship

Note:

• thetoprowofthetableisthehorizontalaxis

• thebottomrowofthetableistheverticalaxis

• thescaleonthehorizontalaxiscanbedifferenttothescaleonthe verticalaxis.

Summary

Everylinearrelationshipcanbeviewedandrepresentedasa:

• wordedreal-lifescenario

• numberpattern

• tableofvaluescontainingaseriesofpoints

• rulecalledanequation

• straight-linegraphontheCartesianplane.

Inthischapter,youwill:

• becomemorefamiliarwitheachofthese

• learnhowtomoveefficientlyfromonerepresentationtotheothers

• solveproblemsinvolvingreal-lifescenarios.

Areyouready?PlottingpointsontheCartesianplane

UsethegivencoordinatestoaccuratelyplotasetofpointsontheCartesianplane.Thisactivitywill consolidateyourpriorlearning.

DownloadtheactivityfromyourInteractiveTextbook.

Areyouready?Usingequationstodrawlines

InYear11,youusedequationslike y = 2x 3toconstructatableofvaluesanddrawastraightline. Thisactivitywillconsolidatethatlearning.

DownloadtheactivityfromyourInteractiveTextbook.

Graphingastraightlineusingatableofvalues

1 Constructatableofvalueswiththeindependentvariableasthefirstrowandthedependentvariable asthesecondrow.

2 Drawanumberplanewiththeindependentvariableonthehorizontalaxisandthedependentvariable astheverticalaxis.Plotthepoints.

3 Jointhepointstomakeastraightline.

Drawthegraphof y = 2x 1. Example1

Drawingastraightlinegraph

Drawatableofvaluesfor x and y. Let x = 2, 1,0,1and2.Find y using thelinearfunction y = 2x 1.

Drawanumberplanewith x asthe horizontalaxisand y astheverticalaxis. Plotthepoints( 2,3),( 1,1),(0, 1) and(1, 3).Thepoint(2, 5)hasnot beenplottedasitdoesnotfitthescaleof thenumberplane.

Jointhepointstomakeastraightline.

Nowyoutry

Drawthegraphof y = 2x 1.

Gradient

Thegradientofalineistheslopeorsteepnessofthe line.Itiscalculatedbydividingtheverticalrisebythe horizontalrun.Thelargerthegradient,thesteeperthe slope.Theletter m isusedtoindicategradient.

Gradientofastraightline m = verticalrise horizontalrun

Likereadingabook,itisbesttoviewalinefromlefttoright,consideringtherunfirst,thentherise (orfall).Afractioncanbeformedwithriseoverrun.

Positivegradient

Negativegradient

Runispositive. Riseispositive. Thelinegoes‘uphill’.

The y-intercept

Runispositive. Riseisnegative. Thelinegoes‘downhill’.

Linescancrossthe x-axisandthe y-axisat‘intercepts’.Theinterceptontheverticalaxisiscalledthe y-interceptandisdenotedbytheletter c

Gradient

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Intercept

Gradientofalineistheslopeof theline. Theinterceptofalineiswherethelinecuts theaxis.

m = rise run

Verticalinterceptisoftendenotedby c Inthetableofvalues,theverticalinterceptis alwaysthenumberinthebottomrow,underthe0.

Whentheequationofalineiswrittenas y = mx + c,itissaidtobein‘gradient–interceptform’.Thegradient is m,thecoefficientof x,andthe y-interceptis c,theconstantterm.Theindependentvariableintheformula is x andthedependentvariableintheformulais y

Gradient–interceptformisusefulinmodellingrelationshipsinmanypracticalsituations.However,the variablesareoftenchangedtoreflectthesituation.Forexample,theequation c = 25n + 100has c asthecost oftheevent($)and n asthenumberofguests.Theselettersarethedependentandindependentvariables.

Usingthegradient–interceptformula

y = mx + c

m = gradientoftheline(verticalriseoverhorizontalrun)

c = theinterceptwherethelinecutsthe y-axisorverticalaxis

Example2

Drawagraphfromatableofvalues,findgradientand y-intercept

Drawthegraphof y = 3x 2fromatableofvalues.Findthegradientand y-interceptofthisline.

Solution

x 2 1 0 1 2 y 8 5 2 1 4

Explanation

Constructatableofvaluesfor x and y. Let x = 2, 1,0,1and2.Find y usingthelinearfunction y = 3x 2.

When x = 2, y = 3x 2becomes y = 3( 2) 2 = 6 2 = 8

Repeatthisfor x = 1,0,1and2.

Drawanumberplanewith x asthehorizontalaxisand y asthe verticalaxis.

Plotthepoints( 1, 5),(0, 2),(1,1)and(2,4).Thepoint ( 2, 8)hasnotbeenplottedasitdoesnotfitthescaleofthe numberplane.

Jointhepointstomakeastraightline.Calculatethegradient ofthelineusingthe‘riseoverrunformula’andreadthevalue ofthe y-inteceptwherethelinecrossestheverticalaxis.

Writethevaluesofthegradientand y-intercept.

Thegradientisthecoefficientof x (thenumberbefore x).The y-interceptis 2.

Nowyoutry

Drawthegraphof y = 2 3x.Findthegradientand y-interceptofthisline.

Sketchingastraightlinerequiresatleasttwopoints.Whenanequationiswritteningradient–interceptform, onepointonthegraphisimmediatelyavailable:the y-intercept.Asecondpointcanbequicklycalculated usingthegradient.

Example3

Sketchingalinearfunctionusingthegradientand y-intercept

Sketchthegraphof3y + 6x = 9.

Solution

3y + 6x = 9

3y = 9 6x

y = 9 6x 3

y = 3 2x

y = 2x + 3

Gradientis 2and y-interceptis3.

x y y-intercept 1 Horizontal run

2 Vertical rise (1.5, 0)

Alternatively,

Let x = 0:

3y + 6x = 9

3y + 0 = 9 y = 3

Explanation

Rearrangetheequationintogradientform

y = mx + c.

Subtract6x frombothsides.

Dividebothsidesby3andsimplify.

Theequationisnowwrittenintheform y = mx + c.

Therefore m = 2and c = 3.

m = 2canberepresentedas

m = rise run = 2 1 (down) (across)

Agradientof 2meansthatforeveryunitacrossin thepositive x-axisdirection,yougodown2inthe negative y-axisdirection.

Plotthe y-intercept(0,3),thenmoveacross 1(horizontalrun)anddown2(verticalrise)toplot thepoint(1,1).

Jointhepoints(0,3)and(1,1)tomakeastraight line.

Let x = 0tofindthe y-intercept.

Solvefor y. Plotthepoint(0,3). y-interceptis3.

Let y = 0:

3y + 6x = 9

0 + 6x = 9 x = 1.5 x-interceptis1.5.

Let y = 0tofindthe x-intercept.

Solvefor x Plotthepoint(1.5,0).

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Plotthesetwopoints.Usearulertojointhetwo pointstograph3y + 6x = 9.

Nowyoutry

Sketchthegraphof2x + 3y = 6.

Exercise5A

1 Plotthefollowingpointsonanumberplaneandjointhemtoformastraightline.

BUILDING

2 Completethefollowingtableofvaluesandgrapheachstraightline.

3

Example1 Drawthegraphsoftheselinesbyfirstcompletingatableofvalues.

4

Example2 Findthegradientofthefollowingstraightlines.

5 Findthe y-interceptofthefollowingstraightlines.

6 Findtheequationofthefollowingstraightlinesdefinedby:

a gradient2,passingthrough(0,1)

b gradient 3,passingthrough(0,4)

c gradient0.5,passingthrough(0, 2)

d gradient0,passingthrough(0,6)

e gradient 2 5 ,passingthrough(0,4)

f gradient 1 3 ,passingthrough(0,0).

7 Whatistheequationofeachofthefollowinglines?

8 Writeanequationforeachline.

m = 5, c = 1 a m = 5, c = 1 b m = 5, c = 1 c m = 5, c = 1 d

9 PlotthepairofpointsonaCartesianplane,thenfindtheequationoftheline.

(0,3),(3,0) a ( 2,0),(0,4) b (2,0),(0,2) c (1,0),(0, 1) d ( 1,0),(0,3) e (0,4),(4,2) f

10 Expressthefollowinglinearequationsingradient–interceptform(y = mx + c). y + 2 = 2x a x + y 4 = 0 b y 1 2 x = 1 c 4x y + 2 = 0 d 1 3 x y = 1 e 4 y = 3x f

11 Drawthegraphofeachlineinquestion 10.

DEVELOPING

12 Example3

13

Drawthegraphsoftheselinesusingthegradientand y-intercept. y = x + 2 a y = 2x + 1 b y = 1 3 x + 2 c

= 1 2 x 1 d

+ 2 = x e 3x y = 0 f

Considertheequation2x y + 5 = 0.

a Whatisthevalueof y when x = 3?

b Whatisthevalueof x when y = 21?

c Doesthepoint(6,17)lieontheline?

d Whatisthegradientandthe y-intercept?

e Whatistheequationofalineparallelto2x y + 5 = 0passingthrough(0, 2)?

f Drawagraphof2x y + 5 = 0.

14 Answertrueorfalsetothefollowingquestions.

a Thepoint(3,0)liesontheline y = 3x.

b Theline y = x 7passesthroughthepoint(1, 6).

c Thepoint(1,4)liesontheline x + 2y = 9.

d Theline2x 3y = 0passesthroughthepoint(0,0).

e Thepoint(1, 1)liesontheline4x y + 1 = 0.

f Theline3x 4y + 1 = 0passesthroughthepoint(2,0).

15 Itisknownthat y variesdirectlywith x.When x = 3, y = 12.

a Writealinearequationintheform y = mx todescribethissituation.

b Drawthegraphof y against x.

16 Itisknownthat y variesdirectlywith x.When x = 6, y = 2.

a Writealinearequationintheform y = mx todescribethissituation.

b Drawthegraphof y against x.

17 Amotorcyclististravellingatconstantspeed.Hetravels350kmin7hours.

a Writealinearequationintheform d = mt todescribethissituation.

b Drawthegraphof d against t.

Communicateyourreasoning,understandingandfluency

18 DrawthesethreestraightlinesonthesameCartesianplane: y = 3xy = 3x + 1 y = 3x 2

CHALLENGING

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Whatdoyounoticeaboutthesethreelines?

Completethissentence:Iftwoormorelineshavethesamegradient,________. Whatistheequationofthelineparallelto y = 3x whichpassesthroughthepoint(1, 4)?

Linearrelationships CONSOLIDATING

Learningintentions

• Toapplylinearrelationshipstopracticalsituations,includingdirectvariation

• Tounderstand,useandinterpretlinearmodels

Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvinglinearrelationshipsregularlyappearinHSCExaminations

Linearmodellingoccurswhenapracticalsituationismodelledbyalinearrelationship.Forexample,the gradient–interceptformofastraight-linegraphcansometimesbeusedtomodelcateringcosts.Acatering companychargesabaseamountof$100plusarateof$20perguest.Usingthisinformation,wecanwrite downalinearequationtomodelthecostoftheevent.

Let c bethecostoftheevent($)and n bethenumberofguests.Wecanthenwrite

Note:Thenumberofguests(n)mustbegreaterthanzeroandawholenumber.

Thegraphofthislinearmodelhasbeendrawn.Thereare twoimportantfeaturesofthislinearmodel:

1 gradientistherateperguest($20)

2 thevertical-interceptisthebaseamount($100).

Somelinearmodelshavelimitationsandrestrictions

Inthecateringcostscenario:

• thenumberofguests(n)mustbeapositivenumber,notnegative

• thenumberofguestsmustbeawholenumber

• thevenuefortheeventmayhavealimitof100people

• thecateringcompanymightonlybeabletocaterforamaximumof100people.

Otherexamplesofrestrictions:

• ifawatertankisbeingfilled,thereisamaximumamountwhichcanfitinthetank,sothemodelonly appliesuptothatmoment

• ifawatertankisbeingemptied,themodelstopswhenthetankisempty

• ateenagermaybegrowingtallerby12cmperyear,buteventuallytheywillstopgrowingtaller,sothe modelonlyappliesuptothattime

• ifyouarespendingthemoneyyousavedupforyourholiday,themodelstopswhenthemoneyrunsout.

Linearmodels

Asectionofastraightlineisusedtomodelalinearrelationship.

Example4

Usinglinearmodelsfordirectvariation

ThegraphoppositeisusedtoconvertAustraliandollars(AUD)to NewZealanddollars(NZD).Usethegraphtoconvert:

a 40AUDtoNZD

b 25NZDtoAUD.

Solution

a 50NZD

b 20AUD

Nowyoutry

Explanation

Readfromthegraph (whenAUD = 40,NZD = 50).

Readfromthegraph (whenNZD = 25,AUD = 20).

Bradcanmake4surfboardsevery5days.Drawagraphtorepresentthissituation.

Example5

Interpretinglinearmodels

Talasellsinsurance.Sheearnsabasesalaryanda commissiononeachnewinsurancepolicyshesells. ThegraphshowsTala’sweeklyincome(I)plotted againstthenumberofnewpolicies(n)shesellsinthat week.Therelationshipbetween I and n islinear

a WhatisTala’sbasesalary?

b WhatisTala’ssalaryinaweekinwhichshesells 8newpolicies?

c HowmanypoliciesdoesTalaneedtoselltoearn $700fortheweek?

d Findtheequationofthestraightlineintermsof I and n.

e Usetheequationtocalculatetheweeklyincomewhen Talasells3newpolicies.

f HowmuchdoesTalaearnforeachnewpolicy?

Solution

a $400

b $800

c 6newpolicies

d c = 400

m = rise run

m = 800 400 8 0

m = 400 8

m = 50

I = mn + c

I = 50n + 400

e I = 50 × 3 + 400 = $550

Explanation

Readfromthegraph(when = 0, I = 400).

Readfromthegraph(when = 8, I = 800).

Readfromthegraph(when = 700, n = 6).

Determinetheverticalintercept(400).

Findthegradientbychoosingtwosuitablepoints. (0,400)and(8,800).

Calculatethegradient(m)betweenthesepoints usingthegradientformula. Substitutethegradientand y-interceptintothe gradient–interceptform y = mx + c. Usetheappropriatevariables(for y, n for x).

Substitute n = 3intotheequation. Evaluate. Checktheanswerusingthegraph.

f $50 Thegradientofthegraphisthecommissionfor eachnewpolicy.

Nowyoutry

Qinghas$10inhermoneybox.Sheisgoingtoadd$6everyweek.

a Whatistheequationofthelinearrelationship,using D fordollarsand w forweeks?

b Howmuchwillshehave2yearsfromnow?

c Howlongwillittakehertoreach$1000?

Exercise5B

1 Completethefollowingtablesofvaluesandgrapheachline.

BUILDING

2 Drawthegraphsoftheselinesusingatableofvaluesfrom1to10.

a C = 2n + 10

b V = 5t + 30

c M = 1 2 n 10

3 Example4 a TheconversiongraphoppositeisusedtoconvertAustralian dollarstoBritishpounds.Usethegraphtocalculatethese exchanges.

i 30Australiandollarstopounds

ii 50Australiandollarstopounds

iii 20poundstoAustraliandollars

iv 10poundstoAustraliandollars

b Whatisthegradientoftheconversiongraph?

c Whatistheequationofthisgraph?

4 Therelationshipoftheageofmachinery(a)in yearstoitsvalue (v)in$1000is v = 4a + 20.

a Constructatableofvaluesforageagainst value.Usevaluesof a from0to4.

b Drawthegraphofage(a)againstvalue(v).

c Whatisthe initialcostofthemachinery?

d Whatistheageofthemachineryifits currentvalueis$15000?

e Whatisthevalueofthemachineryafter 3 1 2 years?

f Whatwillbethevalueofthemachineryafter 2years?

g Whenwillthemachinerybeworthhalfits initialcost?

5 Example5 Anultrasoundmachinewaspurchasedby amedicalcentrefor$150000.Itsvalueis depreciatedeachmonthasshowninthegraph.

a Whatwasthevalueofthemachineafter 40months?

b Whatwasthevalueofthemachineafterfive years?

c Whendoesthelinepredictthemachinewill havenovalue?

d Findtheequationofthestraightlineinterms of v and t.

e Usetheequationtopredictthevalueofthe machineafter6months.

6 Acaristravellingatconstantspeed.Ittravels360kmin9hours.

a Writealinearequationintheform d = mt todescribethissituation.

b Drawthegraphof d against t

Value of ultrasound machine

7

8

Thecost($C)ofhiringataxiconsistsoftwoelements:afixedflagfallandafigurethatvarieswith thenumber(n)ofkilometrestravelled.Iftheflagfallis$4andthecostperkilometreis$2,determine arulethatgives C intermsof n

Theweeklywage,$w,ofavacuumcleanersalespersonconsistsofafixedsumof$350plus$20for eachcleanersold.If n cleanersare soldperweek,constructarulethatdescribestheweeklywageof thesalesperson.

Agardeningshopwilldelivermulchfor$55perscoopplus$45delivery.

9 If c isthecostand s isthenumberofscoops,whatistheequationofthislinearrelationship?

10

Weiconverted100Australiandollars(AUD)to60euros(EUR).

a DrawaconversiongraphwithAustraliandollarsonthehorizontalaxisandeurosonthe verticalaxis.

b Howmanyeurosis25Australiandollars?Usetheconversiongraph.

c HowmanyAustraliandollarsis45euros?Usetheconversiongraph.

d Findthegradientandverticalinterceptfortheconversiongraph.

e WriteanequationthatrelatesAustraliandollars(AUD)toeuros(EUR).

11 Acarrentalcompanycharges$85,plusanadditional24centsperkilometre.

DEVELOPING

a Writearuletodeterminethetotalcharge,$C,forhiringacarandtravelling x kilometres.

b Whatwouldbethecosttotravel250kilometres?

12 Anelectriciancharges$50tocalland$80perhour.Assumethattherelationshipbetweenchargeand timeislinear.

a Findaformulafor$C,thecharge,intermsofthetimespent(t hours).

b Drawagraphof C against t.

c Calculatethecostofajobthattakes2hours45minutestocomplete.

13 Thecostofhiringataxiis$5flagfalland$1.50perkilometretravelled.

a Constructatableofvaluesusing0,10,20,30and40asvaluesforthekilometrestravelled(d). Calculatethecostofthetaxihire(C)foreachvalueof d

b Drawthegraphofthekilometrestravelled(d)againstcostofthetaxihire(C).

c Usethegraphtofindthecostofthetaxihireaftertravelling15kilometres.

d Usethegraphtofindthedistancetravelledifthecostofthetaxihirewas$35.

e Findthegradientandverticalinterceptforthisgraph.

f Findtheequationofthestraightlineintermsof d and C.

CHALLENGING

14 Anelectronicbanktellerregistered$775afterithadcounted120notesand$975afterithad counted160notes.

a Findaformulaforthesumregistered($C)intermsofthenumberofnotes(n)counted.

b Wasthereasumalreadyontheregisterwhencountingbegan?Ifso,howmuch?

15 Aprintingfirmcharges$35forprinting600sheetsofheadednotepaperand$47forprinting 800sheets.

a Findaformula,assumingtherelationshipislinear,forthecharge,$C,intermsofnumberof sheetsprinted, n.

b Howmuchwouldtheychargeforprinting1000sheets?

16 Thetyresonaracingcarhadlost3mmoftreadaftercompleting250kmofaraceand4mmof treadaftercompleting1000km.Assumingthatalinearrelationshipexistsbetweenlossoftreadand distancecovered,findthetotallossoftread, d mm,after s kmfromthestartoftherace.Whatwould bethetreadlossbytheendofa2000kmrace?Giveyouranswercorrecttoonedecimalplace.

Communicateyourreasoning,understandingandfluency

17 TalafromExample 5 hasbeenofferedanewpaydealbyheremployer:

• basesalaryof$800

• commissionof$25oneachnewinsurancepolicyshesells. Usingthisnewmodel,comparedtothecurrentmodelinExample 5:

a underwhatcircumstanceswillTalaearnthesameincome?

b underwhatcircumstanceswillTalaearnmoreincome?

c underwhatcircumstanceswillTalaearnlessincome?

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5C Solvingsimultaneousequationsgraphically

Learningintentions

• Tofindthepointofintersectionoftwolinesbydrawingtheirgraphs Past,presentandfuturelearning

• Studentsshouldalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvingsimultaneousequationsregularlyappearinHSCExaminations

Areyouready?Gradientand y-interceptforequationofline

Whenpresentedwiththeequationofalinewhichyouneedtograph,itishelpfultodeterminethe gradientandthe y-intercept.Thesearehelpfulwhencheckingthatyourgraphiscorrect.

DownloadtheactivityfromyourInteractiveTextbook.

Twostraightlineswillintersect(unlesstheyareparallel).Thepointatwhichtwostraightlinesintersectcan befoundbysketchingthetwographsonaCartesianplaneandreadingoff thecoordinatesofthepointof intersection.Findingthepointofintersectionissaidtobe‘solvingtheequationssimultaneously’.Inaddition tographingthestraightlines,thepointofintersectioncouldbedeterminedbylookingatthetableofvalues. Ifthesamevaluefor x and y occursinbothtablesitisthepointofintersection.

Solvingapairofsimultaneousequationsgraphically

1 Drawanumberplane.

2 Graphbothlinearequationsonthenumberplane.

3 Readthepointofintersectionofthetwostraightlines.

4 Interpretthepointofintersectionforpracticalapplications.

Example6

Graphicalsolutionofsimultaneouslinearequations

Findthesimultaneoussolutionof y = x + 3and y = 2x

Solution

y = x + 3

Gradientis +1, y-interceptis3.

y = 2x

Gradientis 2, y-interceptis0. 2

Explanation

Usethegradient–interceptformtodeterminethe gradient(coefficientof x)and y-intercept(constant term)foreachline.

Drawanumberplane.

Sketch y = x + 3usingthe y-interceptof3anda gradientof1(blueline).

Sketch y = 2x usingthe y-interceptof0anda gradientof 2(redline).

Continuedonnextpage

Thelinesintersectatthepoint( 1,2).

Simultaneoussolutionis x = 1and y = 2.

Simultaneoussolutionis x = 1and y = 2.

Nowyoutry

Findthepointofintersectionofthetwolines. Thesimultaneoussolutionisthepointof intersection.

Alternatively,constructatableofvaluesfor x and y.Let x = 2, 1,0,1and2.Find y usingthe linearfunction y = x + 3.

Repeattofind y usingthelinearfunction y = 2x

Thesamevalueof x and y occursinbothtables when x = 1and y = 2.

Findthepointofintersectionof y = 2x + 1and y = 3 2x

Exercise5C

1 Whatisthepointofintersectionforeachofthesepairsofstraightlines?

BUILDING

2 Example6

3

Plotthefollowingpointsonanumberplaneandjointhemtoformtwo straightlines.Whatisthepointofintersectionofthesestraightlines?

4

Plotthefollowingpointsonanumberplaneandjointhemto formtwostraightlines.Whatisthepointofintersectionof thesestraightlines?

Plotthefollowingpointsonanumberplaneandjointhemtoform twostraightlines.Whatisthepointofintersectionofthesestraight lines?

5 Completethefollowingtableofvaluesandplotthepointsonanumberplane.Findthesimultaneous solutionofthesepairsofequations.Allsolutionsarewholenumbers.

= 2x + 3

DEVELOPING

6 Completethefollowingtableofvaluesandplotthepointsonanumberplane.Whatisthesolutionto eachpairofsimultaneousequations?Someofthesolutionsarenotwholenumbers.

3

7 Drawthegraphsofthefollowingpairsofequationstofindtheirsimultaneoussolution.Someofthe solutionsarenotwholenumbers. y = x and y =

CHALLENGING

8 Estimatethesolutiontoeachofthefollowingpairsofsimultaneousequationsbygraphingthemon thesamenumberplane.Answercorrecttoonedecimalplace. 4y = x and x + y = 1 a

Communicateyourreasoning,understandingandfluency

9 Hereisaveryinterestingpartytrick:

Step1: Startwith4thenadd3,thenadd3againtomakethispattern:4,7,10…

Step2: Nowmakethatintoastraightline:4x + 7y = 10

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Step3: Nowyoutryitwithyourownnumbers:thinkofanynumber,thenaddanumber,thenadd itagain.Examplesinclude:

• 2,5,8

• 1,3,5

• 3,4,5

• 7,6,5

Step4: Chooseoneoftheaboveexamplesormakeyourownpattern,thenchooseanother.Then makethemintoastraightline.Forexample: x + 3y = 5and7x + 6y = 5.

Drawyourchosenstraightlinesandfindthepointofintersectionofthetwolines.

Compareyourresultwithothers.Youareinforabigsurprise!

Modellingactivity:Findingthebreak-evenpointforasmallbusiness

Thistaskguidesyouthroughbuildingaspreadsheetamodeltoconductabreak-evenanalysisfor asmallbusiness.Thismodellingtaskshouldbedoneafteryouhavecoveredsections 5C-5F. TheactivitycanbedownloadedfromyourInteractiveTextbook.

5D Solvingpracticalproblemsusingsimultaneousequations

Learningintentions

• Tosolvepracticalproblemsbyfindingthepointofintersectionoftwolines Past,presentandfuturelearning

• Studentsmayalreadybefamiliarwiththecontentsofthissection

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvingsimultaneousequationsregularlyappearinHSCExaminations

Areyouready?Fromwordstoequations

Sometimes,youwillberequiredtoconvertedwordedscenariosintoequations.Thisactivitywill improveyourskillsatdoingthat.

DownloadtheactivityfromyourInteractiveTextbook.

Whentwopracticalsituationsaredescribedusingsimultaneousequations,thepointofintersectionhasan importantandoftendifferentmeaningdependingonthesituation.Forexample,whenincomeisgraphed againstcoststhepointofintersectionrepresentsthepointwhereabusinesschangesfromalosstoaprofit.

Simultaneousequationsasmodelsforreallifescenarios

Whensimultaneousequationsofreallifescenariosaresolved,thepointofintersectionisoftenthe solutiontoaproblem.

Example7

Usingsimultaneousequationsasmodels

Zainabuysandsellsbooks.Incomereceivedbysellingabookiscalculated usingtheformula I = 16n.Costsassociatedinsellingabookarecalculated usingtheformula C = 8n + 24.

a Whatistheincomewhen6booksaresold?

b Whatarethecostswhen6booksaresold?

c Drawthegraphof I = 16n and C = 8n + 24onthesamenumberplane.

d Usethegraphtodeterminethenumberofbooksneededtobesoldfor thecoststoequaltheincome.

Solution

a I = 16n = 16 × 6 = 96

∴ Incomeforsixbooksis$96.

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b C = 8n + 24 = 8 × 6 + 24 = $72

∴ Costsforsixbooksare$72.

Explanation

Substitute6for n intotheformulaforincome I = 16n

Substitute6for n intotheformulaforcosts C = 8n + 24.

d Incomeisequaltocostswhen n = 3 ∴ 3books

Nowyoutry

Drawanumberplane.

Usethegradient–interceptformtodetermine thegradientandverticalinterceptforeachline. Gradientisthecoefficientof n.Verticalinterceptis theconstantterm.

Sketch I = 16n usingtheverticalinterceptof0and gradientof16.

Sketch C = 8n + 24usingtheverticalinterceptof 24andgradientof8.

Findthepointofintersectionofthetwolines (3,48).

Thecostofrunningadinnerisgivenby C = 200 + 50n,where n isthenumberofpeoplewhoattend. Ticketsaresoldfor$75each,sotherevenueisgivenby R = 75n.

a Drawthetwolinesonthesamenumberplane.

b Usethegraphtodeterminethenumberofpeoplerequiredfortherevenuetoequalthecosts.

Isabella’sMathematicsmarkexceededherEnglish markby15.Shescoredatotalof145forbothtests. FindIsabella’smarksineachsubjectbyplotting intersectinggraphs. Example8

Solvingproblemsusingintersectinggraphs

Solution

LettheMathematicsmarkbe m.

LettheEnglishmarkbe e.

m = e + 15

Gradientis1. Verticalinterceptis15.

m + e = 145

Explanation

ExpresstherelationshipbetweentheMathematics andtheEnglishmarkasalinearequation.

Usethegradient–interceptformtodeterminethe gradientandverticalinterceptfortheline.Gradient isthecoefficientof e.Verticalinterceptisthe constantterm.

Expressthetotalofthetwomarksasalinear equation.

UNCORRECTEDSAMPLEPAGES

m = e + 145

Gradientis 1. Verticalinterceptis145.

Usethegradient–interceptformtodeterminethe gradientandverticalinterceptfortheline.

Continuedonnextpage

Intersectionis(65,80)so e = 65and m = 80.

Isabellascored65inEnglishand80in Mathematics.

Nowyoutry

Drawanumberplane.

Sketch m = e + 15usingtheverticalinterceptof15 andgradientof1(blueline).

Sketch m = e + 145usingtheverticalintercept of145andagradientof 1(redline). Thesimultaneoussolutionisthepointof intersection.

Markthepointofintersectionusingtwodotted lines.

Writethesolutioninwordsusingthecontextofthe question.

Annahas$1000inhermoneybox.Shebeginsspending$50perweek.

a Drawagraphofthismodel.

b Whatistheequationofthisline?

Jasminehas$100inhermoneybox.Shebeginstoadd$50perweektohermoneybox.

c Drawagraphofthismodel.

d Writeanequationforthisline.

e Atwhattime,inweeks,doAnnaandJasminehavethesameamountofmoneyintheirmoneyboxes?

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1 MatildaandNatehavewages m and n respectively.

a Matildaearns$100morethanNate.Writeanequationtodescribethisinformation.

b ThetotalofMatilda’swagesandNate’swagesis$1200.Writeanequationtodescribethis information.

c Drawagraphofthetwoequationsonthesamenumberplane.Use n asthehorizontalaxisand m astheverticalaxis.

d UsetheintersectionofthetwographstofindMatilda’sandNate’swages.

2 Letonenumberberepresentedby a andtheothernumberby b

a Thesumofthetwonumbersis42.Writeanequationtodescribethisinformation.

b Thedifferenceofthetwonumbersis6.Writeanequationtodescribethisinformation.

c Drawagraphofthetwoequationsonthesamenumberplane.Use a asthehorizontalaxisand b astheverticalaxis.

d Usetheintersectionofthetwographstofindthetwonumbers.

3

Letonenumberberepresentedby p andanothernumberby q

a Thesumofthetwonumbersis15.Writeanequationtodescribethisinformation.

b Oneofthenumbersistwicetheothernumber.Writeanequationtodescribethisinformation.

c Drawagraphofthetwoequationsonthesamenumberplane.Use p asthehorizontalaxisand q astheverticalaxis.

d Usetheintersectionofthetwographstofindthetwonumbers.

4

SanaandNghiworkforthesamecompanyandtheirwagesare a and b respectively.

a Sanaearns$100morethanNghi.Writeanequationtodescribethisinformation.

b ThetotalofSana’sandNghi’swagesis$1500.Writeanequationtodescribethisinformation.

c Drawagraphoftheabovetwoequationsonthesamenumberplane.Use a asthehorizontalaxis and b astheverticalaxis.

d UsetheintersectionofthetwographstofindSana’sandNghi’swages.

5 Example7 Afactoryproducesitemswhosecostsare$1000plus$10foreveryitem.Thefactoryreceives$60 foreveryitemsold.

a Writeanequationtodescribetherelationshipbetweenthe: i costs(C)andnumberofitems(n) ii income (I) andnumberofitems (n).

b Drawagraphandfindthenumberofitemswhenincomeequalscosts.

DEVELOPING

6

Letthecostofanapplebe a andthecostofabananabe b

a Oneappleandonebananacost$1.25.Writeanequation(usingcents,notdollars)todescribethis information.

b Twoapplesandthreebananascost$3.25.Writeanequation(usingcents,notdollars)todescribe thisinformation.

c Drawagraphofthetwoequationsonthesamenumberplane.Use a asthehorizontalaxisand b astheverticalaxis.

d Usetheintersectionofthetwographstofindthecost(incents)ofeachfruit.

7 Example8

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LetZara’shourlywageratebe a andBayani’s hourlywageratebe b.

a ThesumofZara’shourlywagerateand Bayani’shourlywagerateequals$70.Write anequationtodescribethisinformation.

b TwiceZara’swagerateisthesameasthree timesBayani’swagerate.Writeanequation todescribethisinformation.

c Drawagraphofthetwoequationsonthe samenumberplane.Use a asthehorizontal axisand b astheverticalaxis.

d Usetheintersectionofthetwographstofind thewagerates.

CHALLENGING

8 Abusinesshasfivetimesasmanyjuniorworkersasitdoesseniorworkers.Iftheweeklywagefora senioris$800andforajunioris$600,findhowmanyofeachareemployedifthetotalweeklywage billis$57000.

9 Acinemahas2340seats.Alloftherowsofseatsinthecinemahaveeither50seatsor45seats.If therearethreetimesasmanyrowswith50seatsasthosewith45seats,findthetotalnumberofrows inthecinema.

10 Atabirthdaypartyfor20people,eachpersoncouldorderchocolateorstrawberryice-cream.Ifthere werefourtimesasmanyordersforchocolateasforstrawberry,howmanyordersweretakenfor chocolateice-cream?Howmanypeopleorderedstrawberry?

Communicateyourreasoning,understandingandfluency

11 Afactorymakeswatchesandclocks.Inanyweek:

• thetotalnumberofwatchesandclocksmadeis100

• themaximumnumberofclocksmadeis70

• themaximumnumberofwatchesmadeis80.

Thegraphbelowshowsthenumberofclocks(x)andwatches(y)thatcanbemade.

a Whatistheequationoftheline AD?

b Whyisthislineonlyapplicablebetweenthepoints B and C forthisfactory?

c Theprofitperweek,$P,canbefoundusingtheequation P = 40x + 50y.Comparetheprofitsat B and C

5E

Break-evenanalysis

Learningintentions

• Tousestraightlinestomodelincome(revenue)andcosts(expenses)

• Tofindthebreak-evenpointatwhichincomeandcostsareequal Past,presentandfuturelearning

• Studentsmayalreadybefamiliarwiththecontentsofthissection

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYear11

• Questionsinvolvingbreak-evenanalysisregularlyappearinHSCExaminations

Thebreak-evenpointisreachedwhencosts(orexpenses)andincomeareequal.Thereisnoprofitorlossat thebreak-evenpoint.Forexample,ifthebreak-evenpointforabusinessis100itemspermonth,thebusiness willmakealossifitsellsfewerthan100itemseachmonth;ifitsellsmorethan100itemspermonth,itwill makeaprofit.Aprofit(orloss)iscalculatedbysubtractingthecostsfromtheincome(Profit = Income–Costs).Incomeisalinearfunctionoftheform I = mx,where x isthenumberofitemssoldand m isthe sellingpriceofeachitem.Costisalinearfunctionoftheform C = mx + c,where x isthenumberofitems sold, m isthecostpriceperitemmanufacturedand c isthefixedcostsofproduction.

Break-evenanalysis

Break-evenpointoccurswhencostsequalincome.

Profit = Income Costs

Income: I = mx

Costs: C = mx

Example9

Interpretingthepointofintersectionoftwographs

Noorbuysandsellswallets.Incomereceivedbyselling walletsiscalculatedusingtheformula I = 30x.Costs associatedwithsellingwalletsarecalculatedusingthe formula C = 20x + 30.

a Usethegraphtodeterminethenumberofwallets thatNoorneedstoselltobreakeven.

b Howmuchprofitorlossdoesshemakewhenfour walletsaresold?

Solution

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a Breakevenpointoccurswhentheincome equalsthecosts. Intersectionisat(3,90).So x = 3. Numberofwallets = 3

Explanation

Considerwhenthebreak-evenpointoccurs. Readthepointofintersectionofthetwolinear graphs.

b Profit = Income Cost

I = 120and C = 110

Profit = 120 110 = $10

Profitforselling4walletsis$10.

Nowyoutry

UsingExample 9:

a whatisthelossifNoorsellsnowallets?

b whatistheprofitifshesells10wallets?

Example10

Break-evenanalysis

Profitisdeterminedbysubtractingthecostsfrom theincome.

Readfromthegraphthevaluesof I and C when x = 4.

Evaluate.

Writetheanswerinwords.

Afirmsellsitsproductat$20perunit.Thecostofproduction $C isgivenbytherule C = 4x + 48, where x isthenumberofunitsproduced.

a Findthevalueof x forwhichthecostoftheproductionof x unitsisequaltotheincomeorrevenue receivedbythefirmforselling x units.

b Checkyouransweralgebraically.

Solution

a Lettheincomeorrevenueforproducing x units be$I.Formulais:

I = 20x

Gradientis20,verticalinterceptis0.

Costofproduction $C isgivenby: C = 4x + 48

Gradientis4,verticalinterceptis48

Explanation

Setuptheincomeequationanddeterminethe gradientandverticalintercept.

Setupthecostofproductionequationand determinethegradientandverticalintercept.

Drawanumberplane.

Use x asthehorizontalaxis.

Use I and C astheverticalaxis.

Sketch I = 20x usingtheverticalinterceptof0and gradientof20.

Checkthislineusingsomevalidpointssuchas (1,20)

Sketch C = 4x + 48usingtheverticalinterceptof 48andgradientof4.

Checkthislineusingsomevalidpointssuchas (1,52)

Thepointofintersectionofthetwolineargraphs occurswhen x = 3.Thisisthebreak-evenpoint, thevalueof x forwhichcostofproductionisequal toincome.

b Checkalgebraically.

IncomeCosts

Readthevalueof x atthepointofintersectionof thetwolineargraphs.

Substitute x = 3intotheformula I = 20x

Substitute x = 3intotheformula C = 4x + 48.

Incomeequalscosts,soanswerto a iscorrect. Checkthat I isequalto C

Nowyoutry

UsingExample 10,howmanyunitsneedtobesoldtomakeaprofit$112?

Exercise5E

1 Example9 Whatisthebreak-evenpointforthefollowinggraphs?

BUILDING

2

Thegraphontherightshowsthecostofproducing boxesofchocolatesandtheincomereceivedfrom theirsale.

a Usethegraphtodeterminethenumberofboxes thatneedtobesoldtobreakeven.

b Howmuchprofitorlossismadewhen3boxes aresold?

c Howmuchprofitorlossismadewhen1boxis sold?

d Whataretheinitialcosts?

3 Thegivengraphshowsthecostofproducingcartonsofsoftdrinksandtheincomereceivedfrom theirsale.

a Usethegraphtodeterminethenumberofcartonsthatneedtobesoldtobreakeven.

b Howmuchprofitorlossismadewhenfivecartonsaresold?

c Howmuchprofitorlossismadewhentwocartonsaresold?

d Whatistheinitialcost?

e Whatisthegradientofthestraightlinethatrepresentsincome?

f Whatistheverticalinterceptofthestraightlinethatrepresentsincome?

g Writeanequationtodescribetherelationshipbetweenincomeandthenumberofcartons.

h Whatisthegradientofthestraightlinethatrepresentscosts?

i Whatistheverticalinterceptofthestraightlinethatrepresentscosts?

j Writeanequationtodescribetherelationshipbetweencostsandthenumberofcartons.

4 Thisgraphshowsthecostofproducingbottlesofwineandtheincomereceivedfromtheirsale.

a Usethegraphtodeterminethenumberofbottlesthatneedtobesoldtobreakeven.

b Writeanequationtodescribetherelationshipbetweenincomeandthenumberofbottles.

c Writeanequationtodescribetherelationshipbetweencostsandthenumberofbottles.

5 LeviandAtamaearnasalaryof L and A respectively.

a Leviearns$10000morethanAtama.Writeanequationtodescribethisinformation.

b ThetotalofLevi’ssalaryandAtama’ssalaryis$150000.Writeanequationtodescribethis information.

c Drawagraphoftheabovetwoequationsonthesamenumberplane.Use A($1000)asthe horizontalaxisand L($1000)astheverticalaxis.

d UsetheintersectionofthetwographstofindLevi’sandAtama’ssalaries.

6 FarmerArjitdecidestobuy200sheepeachmonthfor5months.Shehas1200pigsandchoosesto sell200pigseachmonthfor5months.

a Useaspreadsheettocreateatableofvaluesforthissituation.

b Useaspreadsheettocreateagraphofthissituation.

ci HowmanymonthswillitbeuntilArjithasthesamenumberofsheepandpigs?

ii Whatisthisnumberofsheepandpigs?

7 Sharonthecleanerhas$1000anddecidestospend$100eachweekfor6weeks.Herson,Charlie, has$400andwillearn$50aweekfor6weeks.

a Useaspreadsheettocreateatableofvaluesforthissituation.

b Useaspreadsheettocreateagraphofthissituation.

ci HowmanyweekswillitbeuntilCharliehasthesameamountofmoneyashismum?

ii HowmuchmoneydoSharonandCharlieeachhavewhentheybothhavethesameamount?

8 Freyastartswith$125andspends$25adayuntilsherunsoutofmoney.Tobystartswithnothing andthenearns$37.50eachdayfor5days.

a Useaspreadsheettocreateatableofvaluesforthissituation.

b Useaspreadsheettocreateagraphofthissituation.

ci HowmanyweekswillitbeuntilFreyahasthesameamountofmoneyasToby?

ii HowmuchmoneydoFreyaandTobyeachhavewhentheybothhavethesameamount?

9 Example10 Anindustrialplantproducesitemswhosecostsare$250plus$50foreveryitem.Theplantreceives $75foreveryitemsold.

a Writeanequationtodescribetherelationshipbetweenthe:

i costs (C) andnumberofitems (x)

ii income (I) andnumberofitems (x).

b Drawagraphtorepresentthecostsandincomeforproducingtheitem.

c Howmanyitemsneedtobesoldtobreakeven?

10

Junownsanurserythatgrowsnativeplants.ItcostsJun$4000permonthtocovercostssuchasrent, water,electricityandwages.Eachplantcosts$5andissoldfor$17.50.

a Writeanequationtodescribetherelationshipbetween:

i thecosts (C) andthenumberofplants (x)

ii income (I) andthenumberofplants(x).

b Drawagraphtorepresentthecostsandincomeforproducingtheplants.

c Howmanyplantsneedtobesoldeachmonthtobreakeven?

Communicateyourreasoning,understandingandfluency

11

Sakuramakesandsellsbirdhousesfor$40each.Thecostofdoingthisis$500plus$15perbird house.Shewantstomake$1900profit.Howmanybirdhouseswillsheneedtomake?

Keyideasandchaptersummary

Graphinglines 1 Constructatableofvalueswiththeindependentvariableasthefirstrowand thedependentvariableasthesecondrow.

2 Drawanumberplanewiththeindependentvariableonthehorizontalaxisand thedependentvariableastheverticalaxis.Plotthepoints.

3 Jointhepointstomakeastraightline.

Gradient–intercept formula

Linearmodels

Simultaneous equations–graphically

Simultaneous equationasmodels

Break-even analysis

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Linearequation: y = mx + c.

m = Slopeorgradientoftheline (verticalriseoverhorizontalrun).

c = y-intercept Wherethelinecutsthe y-axisorverticalaxis.

Linearmodelsdescribeapracticalsituationmathematicallyusingastraightline.

1 Drawanumberplane.

2 Graphbothlinearequationsonthenumberplane.

3 Readthepointofintersectionofthetwostraightlines.

4 Interpretthepointofintersectionforpracticalapplications(break-evenpoint).

Simultaneousequationmodelsusetwostraightlinestodescribeapracticalsituation andthepointofintersectionisoftenthesolutiontoaproblem.

Break-evenpointoccurswhencostsequalsincome.

Profit = Income–Costs

Income: I = mx

Costs: C = mx + c

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook.

Checklist

5A 1 IcanusetheequationofastraightlinetographitontheCartesianplane.

e.g.Useatableofvaluestodrawthestraightline y = 2x 4.

5A 2 Icanusethe y-interceptandgradienttographastraightlineontheCartesianplane. e.g.

a Make y thesubjectofthesubjectoftheequation2x + y 6 = 0.

b Usethegradientand y-intercepttodrawtheline.

5A 3 Icandeterminetheequationofastraightlinefromagraph. e.g.Whatistheequationofthisstraightline?

5B 4 Icanusestraightlinesasmodelsforlinearrelationships.

e.g.Ataxidrivercharges$4basefeeand$2perkilometre.Whatistheequation forthecostofataxifare(C)for n kilometres?

5B 5 Icansolveproblemsinvolvinglinearrelationships.

e.g.UsingtheequationinQuestion 4,howmanykilometrescanbetravelled for$50?

5C 6 Icansolvesimultaneousequationsgraphically.

e.g.Drawthelines y = x + 1and y = 5 x.Whatistheirpointofintersection?

5D 7 Icansolvepracticalproblemsusingsimultaneousequations. e.g.Thesumoftwonumbersis114.Onenumberistwiceasbigastheother.What arethenumbers?

5E 8 Icansolveproblemsinvolvingprofit,lossandthebreak-evenpoint. e.g.Thecostofmakinggiftpacksis$200plus$40perpack.Theysellfor $60each.

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a Howmanyneedtobesoldtobreakeven?

b Howmanyneedtobesoldtomakeaprofitof$1000?

Multiple-choicequestions

1 5A

Whatisthegradientofthelineshownatright?

2 5A Whatisthe y-interceptofthelineshownatright?

3 5A

4 5B

Astraightlinehastheequationof y = x 3.Whatisthe y-intercept?

Thecostofmanufacturingbags(C)isgivenbytheformula C = 40x + 150,where x isthenumberof bagssold.Whatisthecostofmanufacturingtwobags?

5 5B Acaristravellingataconstantspeed.Ittravels80kmin4hours.Thissituationisdescribedbythe linearequation d = mt.Whatisthevalueof m?

6 5C

Whatisthepointofintersectionofthelines y = x + 2and y = x + 2?

A (2,0)

B (0,2)

C (0, 2)

D (1,1)

Usethegraphbelowtoanswerquestions 7–9.

7 5E Whatistheprofitforselling12items?

8 5E Whatisthebreak-evenpoint?

9 5E

Whatisthelossforselling5items?

Short-answerquestions

1 5A Drawthegraphoftheselinearfunctions.

a y = x + 2

b y = 3x + 1

c y = 2x 2

2 5A Findtheequationofthefollowingstraight-linegraphs.

3 5B

Thetablebelowshowsthespeed v (inkm/s)ofaplaneattime t seconds.

a Drawanumberplanewith t asthehorizontalaxisand v astheverticalaxis.Plotthepointsand jointhemtomakeastraightline.

b Determinealinearmodelintheform y = mx + c todescribethissituation.

c Whatdoesthemodelpredictwillbetheplane’sspeedwhen t = 2.5seconds?

d Whatdoesthemodelpredictwillbetheplane’sspeedwhen t = 6seconds?

e Whatdoesthemodelpredictwillbetheplane’sspeedwhen t = 7seconds?

f Whatdoesthemodelpredictwillbetheplane’sspeedwhen t = 10seconds?

4 5B Aninternetaccessplanchargesanexcessfeeof$12perGB.

a Drawagraphofdataagainstcost.

b Usethegraphtofind d if c is30.

c Usethegraphtofind c if d is3.5.

d Estimatethecostof7GBofdata.

e Estimatethecostof10GBofdata.

f Estimatethecostof8.5GBofdata.

5 5C Whatisthepointofintersectionofthelines y = 3x + 3and y = x 2?

6 5D

Thisgraphshowsthecostofmakingpictureframesandtheincomereceivedfromtheirsale.

a Usethegraphtodeterminethenumberofpictureframesthatneedtobesoldtobreakeven.

b Howmuchprofitorlossismadewhenonepictureframeissold?

c Howmuchprofitorlossismadewhenfourpictureframesaresold?

d Whatistheinitialcost?

7 5E

Thisgraphshowsthecostofmanufacturingtablesandtheincomereceivedfromtheirsale.

a Usethegraphtodeterminethenumberoftableswhichneedtobesoldtobreakeven.

b Writeanequationtodescribetherelationshipbetweenincomeandthenumberoftables.

c Writeanequationtodescribetherelationshipbetweencostsandthenumberoftables.

d Howmuchprofitorlossismadewhen10tablesaresold?

e Howmuchprofitorlossismadewhen4tablesaresold?

f Howmuchprofitorlossismadewhen16tablesaresold?

Extended-responsequestions

1 5D

LetthecostofitemAbe x andthecostofitemBbe y.

a Thetotalcostofbothitemsis$1.80.Writeanequation(usingcents,notdollars)todescribethis information.

b OneitemAandtwoitemBcost$2.75.Writeanequation(usingcents,notdollars)todescribe thisinformation.

c Drawagraphoftheabovetwoequationsonthesamenumberplane.Use x asthehorizontalaxis and y astheverticalaxis.

d Usetheintersectionofthetwographstofindthecost(incents)ofitemA.

6

Bivariatedata analysis

Bivariatedataanalysisinsports

Coachesusebivariatedataanalysistoinform criticaldecisionsthatimpactplayerselection, restschedulesandhowtomaximisepotential onthefield.Byexaminingrelationships betweendifferentvariables,theycanrefine teamstrategies,optimiseindividual contributionsandultimatelyimprovetheir chancesofwinninggames.

•fieldingerrorsvsgamepressure

•possessiontimevspointsscored

•yearsofexperiencevsgameperformance

•playerheightvsshootingaccuracy (basketball,netball)

•battingaveragevshomeruns(baseball)

•distancerunvsballpossessiontime (soccer)

•playerweightvsposition(NRL).

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Variablesanalysedinclude:

•traininghoursvsinjuryrate

•weatherconditionsvswinrate

6A Constructingabivariatescatterplot

6B Usingabivariatescatterplot

6C Lineofbestfit

6D Interpolationandextrapolation

6E Statisticalinvestigations

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•Displaysandinterpretsbivariate datasetstosolveproblemsandmake predictions(MST-12-S1-06).

©NSWEducationStandardsAuthority

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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6A Constructingabivariatescatterplot

Learningintentions

• Tounderstandandreadbivariatescatterplotstodrawconclusions

• Toconstructbivariatescatterplotstoidentifypatternsindata Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvingbivariatescatterplotsregularlyappearinHSCExaminations

Bivariatedataisdatathathastwovariables.

Ascatterplotisusedtodetermineifthereisa relationshipbetweentwonumericalvariables.Datais collectedonthetwovariablesandoftendisplayedina tableoforderedpairs.Ascatterplotisagraphofthe orderedpairsofnumbers.Eachorderedpairisadoton thegraph.

Toillustratethisprocess,ascatterplothasbeen constructedtodeterminetherelationshipbetween theheightandarmspan.Thedatacollectedonthese variablesisshownbelowinthetableoforderedpairs.

Eachpersonhastwonumericalvariables,heightandarm span.Toconstructascatterplot,drawanumberplanewith theheightonthehorizontalaxisandarmspanonthevertical axis.Ploteachorderedpairasadot.Thescatterplotshows thereisarelationshipbetweenthesevariables.

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Constructingascatterplot

1 Drawanumberplane.

2 Determineascaleandatitleforthehorizontalor x-axis.

3 Determineascaleandatitlefortheverticalor y-axis.

4 Ploteachorderedpairofnumberswithadot.

Example1

Readingascatterplot

Theaveragenumeracyscoreforyear6studentsandtheirgeneralrateofinternetuse(%)for10countries aredisplayedinthisscatterplot.

a Whatisthescalefortheverticalaxis?

b Whatistheaveragenumeracyscoreforthecountrywhichhasaninternetuserateof24%?

c Whatistheinternetuse(%)forthecountrywhichhasanaveragenumeracyscore160?

d Howmanycountrieshaveinternetuseoflessthan50%?

e Howmanycountrieshaveanumeracyscoregreaterthan100?

f Istherearelationshipbetweenthesetwovariables?

Solution

a 1unit = 10

b 120

c 50%

d 6countries

e 6countries

f Yes,thereisarelationship.Whentheinternet useisgreaterthan20%,boththevariablesare increasing.

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Explanation

Countthenumberofdivisionsbetween0and 50 (5).Therefore1unitis50dividedby5(10).

Readfromthescatterplot(wheninternetuseis 24%thenumeracyscoreis120).

Readfromthescatterplot(whenthenumeracy scoreis160theinternetuseis50%).

Countthenumberofdotslessthan50%(left-hand side).

Countthenumberofdotsgreaterthan100 (top-half).

Lookforanypatterninthedots.Inthisscatterplot whentheinternetuseisgreaterthan20%,thereisa clearincreaseinthenumeracyscore.However,this relationshipdoesnotexistwhentheinternetuseis lessthan20%.

Nowyoutry

Thescatterplotbelowshowsthecostoffuel(incostperlitre)andtheaveragenumberofpeopleontrains perday(per1000people).

a Whatisthescalefortheverticalaxis?

b Howmanytraincommuterswilltherebeifthecostoffuelis$1.70/L?

c Ifthereare190000 commuters,whatwouldbethepriceperlitreoffuel?

d Istherearelationshipbetweenthesetwovariables?

Exercise6A

1

Example1 Thescatterplotshowstheresults for15studentsintwotests.

a Whatisthehighestmarkin test2?

b Whatisthelowestmarkin test1?

c Whatistherangefortest1?

d Whatisthemodefortest2?

e Howmanystudentsscored greaterthan6intest1?

2 Thescatterplotshowstheheadheighttoagefor 10people.

a Whatistheheadheightforthepersonwho is21yearsold?

b Whatistheageofthepersonwhohasahead heightof22cm?

c Whatisthelargestheadheight?

d Whatistheageoftheyoungestperson?

e Howmanypeoplehaveaheadheightgreaterthan 23cm?

f Isthereaclearrelationshipbetweenthese twovariables?

BUILDING

3 Thetablebelowshowstheheight(incm)ofaeucalypttreeseedlingasitgrows.

a Copythenumberplaneoppositetoconstructascatterplot usingtheabovetable.

b Whatistheincreaseintheheightoftheseedlingduringthe firstweek?

c Whatistheincreaseintheheightoftheseedlingduringthe lastweek?

d Howmanyweeksdoesittakefortheseedlingtoincrease inheightfrom9cmto12cm?

e Estimatetheheightoftheseedlingafter4.5weeks.

f Estimatethetimetakenfortheseedlingtogrowtoaheight of14cm.

4

Harutoisapoliticalcommentatorwhohasbeenstudyingtheeffectsoftelevisionexposuretimeon theapprovalratingsofninepoliticians.Thedataisshownbelow.

a Constructascatterplotofthedatagiveninthetable.

b Arethereanyconclusionstobedrawnfromthescatterplot?

5 Thetableshowsthenumberofrunsscoredandthenumberofballsfacedbybatsmeninaone-day cricketmatch.

a Constructascatterplotofthedatagiveninthetable.

b Arethereanyconclusionstobedrawnfromthescatterplot?

6 Themaximumwindspeedandmaximumtemperatureinalocationwererecordedfor2weeks. Thedataisdisplayedinthetablebelow.

a Constructascatterplotofthedatagiveninthetable.

b Arethereanyconclusionstobedrawnfromthescatterplot?

CHALLENGING

7 Thedatabelowshowsthelitresofpetrolusedoveracertaindistancetravelledinkilometres.

a Constructascatterplotusingthedataabove.

b Approximatelyhowfarwouldyoutravelon14litresofpetrol?

c Ifyoutravel200km,howmanylitresofpetrolwillyouneed?

d Describetherelationshipbetweenthetwovariables.

Communicateyourreasoning,understandingandfluency

8 Thetablebelowshowstheage(inmonths)ofsixcars,andtheminimumstoppingdistance(in metres)whenthecaristravellingat60km/h.

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a Constructascatterplotusingtheabovedata.

b Calculatethemeanageofthecarandthemeanstoppingdistance.Answercorrecttoonedecimal place.

c Doyouagreewiththefollowingstatement?Justifyyouranswer.‘Asacargetsolder,itsstopping distanceincreases.’

d Arethereanylimitationswhenmakingpredictionsfromthisdata?

Investigation:Investigatinghowthepresentationofdatacaninfluence decision-makinginagovernmentfundingmeeting

Inthisassignment,youwillexaminehowstatisticsandvisualcommunicationinfluencedecisionmakingbyreviewingandcritiquingareal-worldstyleinfographicusedinagovernmentfunding meeting.You’llthenworkwithbivariatedatatocomeupwithyourowninfographic.Thisinvestigation shouldbedoneafteryouhavecoveredsections 6A–6E.Theactivitycanbedownloadedfromyour InteractiveTextbook.

6B Usingabivariatescatterplot

Learningintentions

• Tousebivariatescatterplotstodescribethepatterns,featuresandassociationsofbivariatedatasets

• Toidentifytheindependentvariableandthedependentvariable Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvingbivariatescatterplotsregularlyappearinHSCExaminations

Whatarethefeaturesinascatterplotthatwillidentifyanddescribeany relationship?Firstlookforaclearpattern.Inthescatterplotopposite,thereis noclearpatterninthepoints:theyarejustrandomlyspreadonthescatterplot. Thereisnorelationshiporassociationbetweenthevariables.

Forthethreeexamplesbelow,thereisaclear(butdifferent)patternineachset ofpoints,soweconclude thatthereisanassociationineachcase.

Havingfoundaclearpattern,weneedtobeabletodescribetheseassociationsclearly,astheyareobviously quitedifferent.Therearethreethingswelookforinthepatternofpoints:form,directionandstrength.

Formofanassociation

Ifanassociationexistsbetweenthevariablesthenthepointsinascatterplottendtofollowalinearpatternor acurvedpattern.Thisiscalledtheformofanassociation.

Linearform

Ifthepointsseemtoapproximateastraightline, theassociationisalinearform.

y x

Non-linearform

Ifthepointsseemtoapproximateacurve,the associationisanon-linearform. y x

Formofanassociation

Linearform:whenthepointstendtofollowastraightline. Non-linearform:whenthepointstendtofollowacurvedline.

Directionofanassociation

Therearetwotypesofdirectioniftheassociationisinlinearform.

Positive

Positiveassociationexistsbetweenthevariables ifthegradientofthelineispositive.Thatis,the dotsonthescatterplottendtogoupaswego fromlefttoright.

y x

Directionofanassociation

Positive:gradientofthelineispositive. Negative:gradientofthelineisnegative.

Strengthofanassociation

Negative

Negativeassociationexistsbetweenthevariables ifthegradientofthelineisnegative.Thatis,the dotsonthescatterplottendtogodownaswego fromlefttoright.

y x

Thestrengthofanassociationisameasureofhowmuchscatterthereisinthescatterplot.

Strong

Inastrongassociation,thedots willtendtoresemblealine orsmoothcurve.Apatternis clearlyseen.Thereisonlyasmall amountofscatterintheplot.

y x

Moderate

Inamoderateassociation,the amountofscatterintheplot increasesandthepatternbecomes lessclear.Thisindicatesthatthe associationislessstrong.

y x

Weak

Inaweakassociation,theamount ofscatterincreasesfurtherand thepatternbecomesevenless clear.Linearformislessevident.

y x

Strengthofanassociation

Strong:smallamountofscatterintheplot.

Moderate:modestamountofscatterintheplot.

Weak:largeamountofscatterintheplot.

Example2

Describingbivariatedatasets

Thetablebelowshowstheheight(cm)andarmlength(cm)ofninepeople.

a Constructascatterplotusingtheabovetable.

b Describetheformoftheassociation.

c Describethedirectionoftheassociation.

d Describethestrengthoftheassociation.

e Predictthearmlengthofapersonwhois173cmtallusingthescatterplot.

f Predicttheheightofapersonwhohasanarmlengthof75cmusingthescatterplot.

Drawanumberplanewithheightasthe horizontalaxisandlengthastheverticalaxis.

Determineascaleforthehorizontalaxis. Leteachunitrepresent1cm.

Determineascalefortheverticalaxis. Leteachunitrepresent1cm.

Writetitlesforthehorizontalandvertical axes.

Plotthepointsfromthetable.

b Linearform Lookforapattern.Thepointsapproximate astraightline.

c Positive Gradientofthelineispositive.Thedotstend togoupasyoumovefromlefttoright.

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d Strong Thereisasmallamountofscatterinthe scatterplot. Drawanimaginaryverticallinefrom173cm.

Continuedonnextpage

e Thepersonhasanarmlengthofapproximately 65cm.

f Theperson’sheightisapproximately187cm.

Nowyoutry

Trytomaintainthelinearrelationshipandguess thearmlength.

Drawanimaginaryhorizontallinefrom75cm.

Trytomaintainthelinearrelationshipandguess theheight.

Thistableshowstheheights(incm)andmasses(inkg)ofsixkangaroos.

a Drawascatterplotusingthetable.

b Istheformlinearornon-linear?

c Describethedirectionoftheassociation.

d Describethestrengthoftheassociation.

e Predictthemassofakangaroothatis170cmtallusingthescatterplot.

f Predicttheheightofakangaroothathasamassof60kgusingthescatterplot.

Independentanddependentvariables

Bivariatedatahastwovariablesthatareoftenidentifiedastheindependentanddependentvariables.The independentvariableistheinput.Itisnotaffectedbytheothervariableandisrepresentedonthehorizontal axisofthescatterplot.Thedependentvariableistheoutputandis‘dependent’ontheindependentvariable. Itisrepresentedontheverticalaxisofascatterplot.

Example3

Identifyingindependentanddependentvariables

Thetablebelowshowsthetimetaken(hours)relativetothedistancetravelled(km).

a Drawascatterplotusingtheabovetable.

b Whicharetheindependentanddependentvariables?

c Describethestrengthanddirectionoftheassociation.

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b Independentvariableisdistance(d).

Dependentvariableistime(t).

Explanation

Drawanumberplanewith d asthehorizontalaxis and t astheverticalaxis.

Determineascaleforthehorizontalaxis.Leteach unitrepresent10km.

Determineascalefortheverticalaxis.Leteach unitrepresent0.2hours.

Writetitlesforthehorizontalandverticalaxes. Plotthepointsfromthetable.

Theindependentvariableistheinputand representedonthehorizontalaxisofthe scatterplot.

Thedependentvariableistheoutputand representedontheverticalaxisofthescatterplot.

c Strongandpositive Pointslieinalinewithapositivegradient.Clear pattern,strongassociation.

Nowyoutry

Thistableshowssatisfactionlevelsincomparisontoovertimehoursworked.

a Drawascatterplotusingthetable.

b Identifythedependentandindependentvariables.

c Describethestrengthanddirectionofthescatterplot.

Exercise6B

1 Example2 Describetheassociationinthefollowingscatterplotsas:

i linearornon-linear ii positiveornegative iii strong,moderateorweak.

BUILDING

2 Foreachofthefollowingpairsofvariables,indicatewhetheryouexpectanassociationtoexistand, ifso,whetheryouwouldexpecttheassociationtobepositiveornegative.

a Independentvariable:Distancetravelled Dependentvariable:Timetaken

b Independentvariable:AmountofdailyexerciseDependentvariable:Fitnesslevel

c Independentvariable:FootlengthofanadultDependentvariable:Intelligence

d Independentvariable:Numberofpages Dependentvariable:Bookprice

e Independentvariable:Temperatureabove30◦CDependentvariable:Comfortlevel

3 Example3 Chocolatesaresoldfor$12perkg.Thefollowingtableshowsweightagainstcost.

a Whichistheindependentvariable?

b Whichisthedependentvariable?

c Drawascatterplotofweightagainstcost.

d Istheformlinearornon-linear?

e Isthedirectionpositiveornegative?

f Isthestrengthstrong,moderateorweak?

4 Thefollowingtableshowsthedrugdosageagainsttheresponsetimeforthedrugtostart takingeffect.

a Whichistheindependentvariable?

b Whichisthedependentvariable?

c Drawascatterplotoftimeagainstcost.

d Istheformlinearornon-linear?

e Isthedirectionpositiveornegative?

f Isthestrengthstrong,moderateorweak?

5 Salahconductedascienceexperimentandpresentedtheresultsinatable.

a Whichistheindependentvariable?

b Whichisthedependentvariable?

c Drawascatterplotofmassagainsttime.

d Istheformlinearornon-linear?

e Isthedirectionpositiveornegative?

f Isthestrengthstrong,moderateorweak?

6 Thefollowingtableshowsleglengthcomparedtoheight.

a Whichistheindependentvariable?

b Whichisthedependentvariable?

c Drawascatterplotofleglengthagainstheight.

d Istheformlinearornon-linear?

e Isthedirectionoftheassociationpositiveornegative?

f Isthestrengthoftheassociationstrong,positiveorweak?

7 Thefollowingtableshowsforearmlengthandhandlength.

a Drawascatterplotusingtheabovetable.

b Describetheassociationbetweentheforearmlengthandhandlength.

Communicateyourreasoning,understandingandfluency

8 Thefollowingtabledemonstratestheeffectsofsleepagainsttestscoresforagroupof10students.

e Discussthelimitationsofthisscatterplot. UNCORRECTED

a Drawascatterplotusingthetable.

b Isthereanassociationbetweenthetwovariables?Ifso,whattypeofformdoestheassociation have?

c Whicharethedependentandindependentvariables?

d Describethestrengthanddirectionoftheassociation.

Lineofbestfit

Learningintentions

• Tounderstandthepurposeofalineofbestfit

• Todrawlinesofbestfitbyeye

• Tofindtheequationoflinesofbestfit

• Tousethelinetomakepredictionsandexpectations

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvinglinesofbestfitregularlyappearinHSCExaminations

Ifthepointsonascatterplottendtolieinastraightline,thenwecandrawastraightlineonthescatterplot. Thislineisknownasthelineofbestfit,whichisalsocalledtheleast-squaresregressionlineofbestfitora trendline.

Thesimplestmethodistodrawalineofbestfitbyeyethatseemstobeabalanceofpointsaboveandbelow theline,asintheexamplebelowofascatterplotshowingthepetrolusedbyavehiclecomparedtothe distancetravelled.Thelineofbestfitdoesnotneedtopassthroughanyofthedatapoints,althoughitmay passthroughsomeofthepoints.

Theaimofthelineofbestfitistomodeltheassociationbetweentwonumericalvariablesbyusingthe equationofastraightline.Theequationofastraightlineisfoundusingthegradient–interceptformula: y = mx + c,where m isthegradientand c isthe y-intercept.

Lineofbestfit

Alineofbestfitisastraightlinethatapproximatesalinearassociationbetweenpoints.Theequationof thelineofbestfitisfoundusingthegradient–interceptformula: y = mx + c

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Thelineofbestfitisusedtomakeapredictionaboutoneofthevariables.Whenitisusedtomakea predictionwithinthedatarangeitiscalledinterpolation.

Extrapolationisapredictionoutsidethedatarangeandmustbeusedcarefully,asthelineofbestfitmaynot apply(forexample,predictinganadult’sheightbasedontheirincreasingheightasachild).Interpolationand extrapolationwillbeexaminedindetailinSection 6D

Drawingalineofbestfitbyeye

Drawalineofbestfitonthescatterplot.

Nowyoutry

Drawalineofbestfitonthescatterplot.

Thelinedoesnotneedtopassthroughanyofthepointsbut shouldbalancethepointsaboveandbelowit.

Findingtheequationofalineofbestfitusing y = mx + c

FromtheYear11topicofLinearrelationships,weknowthattheequationofastraightlineis y = mx + c, where m isthegradientand c isthe y-intercept.

Whengiventhelineofbestfitortheleast-squaresregressionline,wecancalculatethegradientandthe y-interceptandhencefindtheequationofthetrendline.

Example5

Findingtheequationofthelineofbestfitusing y = mx + c

Thescatterplotshowsamother’sheight(incm)andherdaughter’sheight(incm).

Usethetwopointsgivenonthelineofbestfitto:

a determinethegradientoftheline

b determinethe y-intercept

c formanequationofthelineofbestfit.

Solution

Explanation

Gradient = m = rise run = y2 y1 x2 x1 b 160 = 8 11 ×

Substituteoneofthepointsontheline(152,160) andgradient m = 8 11 into y = mx + c tosolvefor c

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c y = 8 11 x + 544 11 orto2decimalplaces:

y = 0.73x + 49.45

Substitutethevalueof m foundinpart a andthe valueof c foundinpart b into y = mx + c togetthe equationofastraightline.

Nowyoutry

Theheightsandshoesizeof10peoplewererecordedandplottedinthescatterplotbelow.Alineofbest fitisalreadydrawn.

Usethetwopointsgivenonthelineofbestfitto:

a determinethegradientoftheline

b determinethe y-intercept

c formanequationofthelineofbestfit.

Exercise6C BUILDING

1 Example4 Usearulertodrawalineofbestfitoneachofthescatterplotsbelow.

2 Drawascatterplotandalineofbestfitbyeyeforthefollowingpoints.

a (0,0)(10,30)(20,67)(30,93)(40,126)(50,158)(60,178)

b (5,20)(10,42)(15,73)(20,94)(25,122)(30,150)(35,165)

c (0,6)(2,24)(3,39)(4,44)(5,59)(6,64)(7,79)(8,84)

d (10,55)(12,45)(14,20)(16,40)(18,30)(20,28)(22,25)

3 Thescatterplotshowsthemother’sheight (incm)andherdaughter’sheight(incm).

a Copythescatterplotanddrawalineofbest fitbyeye.

b Describethestrengthoftherelationshipas strong,moderateorweak.

c Estimatethedaughter’sheightifthe mother’sheightis170cm.

d Estimatethemother’sheightifthe daughter’sheightis162cm.

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4 Thetablebelowshowstheamountofenergy(inmegajoules,MJ)usedperdayfor12greysealsof variousmass(inkg).

a Drawascatterplotusingenergyforthehorizontalaxisandmassfortheverticalaxis.

b Drawalineofbestfitbyeye.

c Usethelineofbestfittopredictthemasswhen1.7MJofenergyisused.

d Usethelineofbestfittopredictthemasswhen2MJofenergyisused.

e Usethelineofbestfittopredicttheenergyusedwhenthemassis100kg.

f Usethelineofbestfittopredicttheenergyusedwhenthemassis80kg.

Daughter’s

5 Theweights(inkg)andpulserates (inbeatsperminute)of15chimpanzees housedatSydneyZoowererecordedand plottedinthescatterplotshowntothe right.

a Usethetrendlinetooltoinsertthe least-squareslineofbestfitontothe scatterplot.

b Displaytheequationoftheleast-squares lineofbestfitontothescatterplot.

c Usetheequationtopredictthepulse ratewhentheweightis65kg.

d Usetheequationtopredicttheweight whenthepulserateis60beatsper minute.

e Commentonthereliabilityofthe predictionsusingthelineofbestfit.

DEVELOPING

6 Theheightandmassesofsmalldogbreedsaremeasuredandrecordedbelow.

a Completethescatterplotoppositeanddrawa lineofbestfitbyeye.

b Whatistheexpectedmassofadoggiventheir heightis73cm?

c Whatistheexpectedheightofadoggiventheir massis4.8kg?

d Whatistheexpectedmassofadoggiventheir heightis48cm?

e Whatistheexpectedheightofadoggiventheir massis9.0kg?

f Createaspreadsheetandscatterplotforthe data.Usetheequationofthetrendlinetocheck youranswersinparts b–e.

7 Thetablecomparesamother’sheightincentimetreswithherson’sheightincentimetres.

a Createaspreadsheetandscatterplotusingmother’s(m)heightasthehorizontalaxisandson’s height(s)astheverticalaxis.

b Insertthetrendlineanddisplaytheequationofthetrendline.

c Usetheequationtopredicttheson’sheightifthemother’sheightis163cm.

d Usetheequationtopredictthemother’sheightiftheson’sheightis178cm.

e Commentonthereliabilityofthepredictionsusingthelineofbestfit.

8 Thetablebelowshowsthenumberofreboundspergamefor10peoplewhoplayadifferentnumber ofminutesinabasketballgame.

a Createaspreadsheetandascatterplotusing‘Minutesplayed’asthehorizontalaxisand ‘Reboundsgrabbed’astheverticalaxis.

b Insertthetrendlineanddisplayitsequation.

c Describethelinearrelationshipintermsofstrengthanddirection.

d Usethisequationtopredictthenumberofreboundsgrabbedwhenaplayerplaysfor38minutes.

e Usetheequationtopredicttheminutesplayedifaplayergrabs11rebounds.

9 Thetablebelowdepictsthetestscoresoutof40forastudentandthehoursofstudytheyhave completed.

a Whichvariableistheindependentvariable?

b Createaspreadsheetandascatterplotusingthe‘Hoursofstudy’forthehorizontalaxisand‘Test scores’fortheverticalaxis.

c Describethelinearrelationshipintermsofstrengthanddirection.

d Insertthetrendlineanddisplayitsequation.

e Usethisequationtopredictthetestscoreofastudentwhostudiedfor7hours.

10 Example5 Giventwopointsthatlieonthelineofbestfit,determine:

i thegradientoftheline

ii the y-intercept

iii theequationofthelineofbestfitdrawnthroughthegivenpoints.

a (10,40)and(60,190)

b (3,8)and(1,12)

c (14,20)and(7, 1)

d (10,12)and(4,9)

11

12

Theequationrelatingaperson’sshoesizetotheweightofthebackpacktheycarryonahiking expedition(inkg)isgivenbelow:

weight = 2.2 × shoesize + 48.1(onlyforshoesizesbetween6and12).

a Whatistheweightofabackpackcarriedbyapersonwithashoesizeof7?

b Whatistheweightofabackpackcarriedbyapersonwithashoesizeof5?

c Whatistheweightofabackpackcarriedbyapersonwithashoesizeof11?

d Whatistheshoesizeofsomeonewhocarriesabackpackthatweighs70kg? CHALLENGING

Theexaminationscore(%)andhoursofstudyof12studentswhosatthesametestwererecorded andplottedinthescatterplotbelow.Thelineofbestfitisalsoshown.

a Determinethegradientoftheline.

b Determinethe y-intercept.

c Hence,findtheequationofthelineofbestfitdrawnthroughthegivenpoints.

d Usingtheequationorotherwise,howmanyhoursofstudydidapersondoiftheyscored70%on thetest?

e Describethestrengthanddirectionoftherelationship.

f Interpretthegradientofthelineofbestfitinthecontextofthedata.

g Interpretthe y-interceptofthelineofbestfitinthecontextofthedata.

13 Theheightsandweightsofsmalldogbreedsaremeasuredandrecordedbelow.

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a Drawascatterplotusingthetable.

b Drawalineofbestfitbyeye.

c Determinethegradientofthelineofbestfit.Interpretthegradientinthecontextofthedata.

d Determinethe y-interceptandexplainwhyinterpretingthe y-interceptdoesnotmakesenseinthe contextofthedata.

e Findtheequationofthelineofbestfitdrawnthroughthegivenpoints.

Theartassessmentresultsforagroupofstudents andtheircreativityscorehavebeenrecordedinthe scatterplotbelow.Alineofbestfithasbeendrawn.

a Determinethegradientofthelineofbestfit.

b Determinethe y-intercept.

c Findtheequationofthelineofbestfitdrawn throughthegivenpoints.

d Describethestrengthanddirectionofthe relationship.

Communicateyourreasoning,understandingandfluency

15

Therainfall(inmm)andthetemperature(in ◦C)duringthelastmonthwasrecordedandplottedin thescatterplotbelow.Theleast-squaresregressionlineisalsoshown.

a Byfindingthegradientand y-intercept,determinetheequationofthelineofbestfitdrawn throughthegivenpoints.

b Usingtheequationorotherwise,howmuchrainwouldtherebeifthetemperaturewas21◦C? Answercorrecttoonedecimalplace.

c If5mmofrainfell,whatwouldthemaximumdailytemperaturebe?

d Describethestrengthanddirectionoftherelationship.

e Interpretthegradientofthelineofbestfitinthecontextofthedata.

f Interpretthe y-intercept.Doesthisvaluehavemeaninginthiscontext?

6D Interpolationandextrapolation

Learningintentions

• Tounderstandthemeaningandpurposeofinterpolationandextrapolation

• Touselinesofbestfittomakepredictionsandexpectations

• Tounderstandthelimitationsofinterpolationandextrapolation

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsinvolvinglinesofbestfitregularlyappearinHSCExaminations

Theequationofthelineofbestfitcanprovideimportantinformationandbeusedtomakepredictions. Whenthelineofbestfitiswritteninthegradient–interceptform y = mx + c,gradient(m)indicatesthe changeindependentvariableastheindependentvariableincreasesby1unit.Theverticalintercept(c) indicatesthevalueof thedependentvariablewhentheindependentvariableiszero.Inadditiontothis information,theequationofbestfitisusedforinterpolationandextrapolation.

Interpolation

Interpolationistheuseofthelinearregressionlinetopredictvalueswithintherangeofthedataset.Ifthe datahasastronglinearassociationthenwecanbeconfidentourpredictionsareaccurate.However,ifthe datahasaweaklinearassociation,wearelessconfidentwithourpredictions.

Example6

Makingpredictionsusinginterpolation

Lifeexpectancyatbirthforfemalesandmalesisshowninthistableandscatterplot.

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a Whatwasthelifeexpectancyin1935forfemales?

b Whatwasthelifeexpectancyin1995formales?

Solution

a Lifeexpectancyforfemalesin1935was approximately67years.

b Lifeexpectancyformalesin1995was approximately74years.

Nowyoutry

Explanation

Drawaverticallinefrom1935untilitintersectsthe blueline.Atthispointdrawahorizontallineuntil itreachestheverticalaxis.Readthevalue.

Drawaverticallinefrom1995untilitintersectsthe redline.Atthispointdrawahorizontallineuntilit reachestheverticalaxis.Readthevalue.

Thistableandscatterplotdepictthefindingsofastudywhichinvestigatedtheeffectsofwages (inthousands$)onsatisfactionlevels.

a Whatisthesatisfactionlevelforsomeonewhosewageis$45000?

b Ifyoursatisfactionlevelis70%,whatwagecouldyoubeearning?

Extrapolation

Extrapolationistheuseofthelinearregressionlinetopredictvaluesoutsidetherangeofthedataset. Predictedvaluesareeithersmallerorlargerthanthedataset.Theaccuracyofpredictionsusingextrapolation dependsonthestrengthofthelinearassociationsimilartointerpolation.Itmaynotbereasonableto extrapolatetoofarasthisexampleshows.

Example7

Makingpredicationsusingextrapolation

Thetablebelowshowstheageofastudentandtheirheightincentimetres.

a Constructascatterplotfromthetableusingagefrom0to30andheightfrom100to300.

b Drawthelineofbestfitanddescribetheassociationbetweenageandheight.

c Predicttheheightofthestudentwhentheyareaged19years.

d Whatarethelimitationsofthislinearmodel?

b Strongpositivelinearassociation.

c Heightofthestudentis200cmwhentheyare 19yearsold.

d Adultheightdoesnotgrowatthesamerateasachild. Usingthemodeltoextrapolateisflawed,e.g.,the predictionistheheightwillbe260cmatage30.

Nowyoutry

Explanation

Drawanumberplanewithageasthe horizontalaxisandheightasthevertical axis.

Determineascaleforthehorizontal axis.Leteachunitrepresent1year. Determineascalefortheverticalaxis. Leteachunitrepresent10cm. Writeatitleforthehorizontaland verticalaxes.

Plotthepointsfromthetable.

Thereisasmallamountofscatterinthe scatterplot.

Readtheheightfromthescatterplot whenageis19.

Extrapolationtoofarfromthedataset maynotbereliable.

Thetablebelowdepictsthetestscoreoutof50forninestudentsandthehoursofstudytheyhave completed.

a Drawascatterplotusingthehoursofstudyforthehorizontalaxisandtestscoresfortheverticalaxis.

b Drawthelineofbestfitanddescribetheassociationbetweenhoursofstudyandtestscore.

c Useyourlineofbestfittopredictthetestscoreofastudentwhostudiedfor7hours.

d Whatwouldbethescoreofastudentthatstudiedfor40hours?Whatarethelimitationsofthislinear model?

Interpolation

Extrapolation

PredictingvalueswithinthedatasetrangePredictingvaluesoutsidethedatasetrange

Example6

Exercise6D

1

2

Thescatterplotshowndisplaystherainfall(inmm)andthe percentageofcleardaysforeachmonthin2018.

a Howmanymonthshad10%ofcleardays?

b Whatwasthepercentageofcleardayswhentherainfall was70mm?

c Predicttherainfallinthemonthgiventhefollowing percentageofcleardays: 4% i 20% ii 26%. iii

d Predictthepercentageofcleardaysinthemonthgiven thefollowingrainfall:

BUILDING

3

Thescatterplotshowstheaverageannual femaleincomeplottedagainstaverageannual maleincomefor15countries.

a Whatwastheaverageannualfemale incomeforacountrywhoseaverage annualmaleincomewas$45000?

b Howmanycountrieshadanaverage annualfemaleincomeof$25000?

c Predictthefemaleincomegiventhe followingmaleincomes:

i $20000 ii $40000 iii $60000.

d Predictthemaleincomegiventhe followingfemaleincomes: i $25000 ii $30000 iii $35000.

Inastudyoferrorsmadeinatest,fortimesrangingfrom5to25seconds,theequationrelatingthe numberoferrorstotimeis:

errors = 0.53 × time + 15

Usethisequationtopredictthenumberoferrors(correcttothenearestwholenumber)withthe followingtimes.Areyouinterpolatingorextrapolating?

4

5

Forminimumtemperaturesfrom5◦ Cto20◦ C,theequationrelatingthemaximumandminimum temperature(in◦C)ataweatherstationisshownbelow:

maximum = 0.67 × minimum + 13

Usethisequationtopredictthemaximumtemperaturegiventhefollowingminimumtemperatures. Areyouinterpolatingorextrapolating?

Theequationrelatinglifeexpectancyatbirthfrom 1900tothecurrentyearforaparticularcountryis givenbelow:

lifeexpectancy = 0.21 × year 353.78

Usethisequationtopredictalifeexpectancy inthefollowingyears.Areyouinterpolatingor extrapolating?

1900 a 1950 b 1870 c

2000 d 2040 e 1970 f

6 Astudent-runbusinesssellshomemadecookies.Forproductionlevelsbetween160and190cookies, theequationrelatingprofit(in$)andthenumberofcookies(n)isshownbelow:

profit = 0.75 × n 65.63

Usethisequationtopredicttheprofitforthefollowingproductionlevels.Areyouinterpolatingor extrapolating?

150 a 175 b 200 c

7 Whenaworker’saveragepayrateisbetween$5and$25,theequationrelatingacountry’s developmentindex(%)totheaveragepayrate(indollarsperhour)isshownbelow:

developmentindex = 0.272 × payrate + 81.3

Usethisequationtopredictacountry’sdevelopmentindexwiththefollowingaveragepayrates.Are youinterpolatingorextrapolating?

a $40perhour

b $20perhour

c $10perhour

8 Whentheareainalargecityisbetween1km2 and8km2,theequationrelatingapopulationtoarea (insquarekilometres)isshownbelow:

population = 2680 × area + 5330

Usethisequationtopredictthepopulationwiththefollowingareas.Areyouinterpolatingor extrapolating?

a 2.5km2

b 5.0km2

c 7.5km2

9

Afactoryproducesvideogamecartridges.Theleast-squaresregressionlineforcostofproduction (C)asafunctionofthenumberofcartridges(n)producedisgivenby:

C = 7.2n + 500 Furthermore,thisfunctionisdeemedaccuratewhenproducingbetween100and1000cartridges.

a Whatisthecosttoproduce400cartridges?

b Howmanycartridgescanbeproducedfor$5900?

c Whatisthecosttoproduce10000cartridges?

d Whatarethe‘fixed’costsforthisproduction?

e Whichoftheparts a to c isanextrapolation? 10

Thetablebelowcomparesthelength(incentimetres)for10fishandtheirweight(ingrams).

Length(cm) 30 31 32 33 34 35 36 37 38 39

a Drawascatterplotusinglengthasthehorizontalaxisandweightastheverticalaxis.Drawaline ofbestfit.

b Usingthecoordinates(32,340)and(38,550),determinetheequationofthelineofbestfit.

c Whatistherateatwhichtheweightischangingrelativetothelength?

d Useinterpolationtopredictafish’sweightifitslengthis35.5cm.

e Useextrapolationtopredictafish’sweightifitslengthis45.5cm.

f Whatisthelengthofafishgivenitweighs300g?

g Explainthemeaningofthe y-interceptintheequationofthelineofbestfit.Whatarethe limitationsofthislinearmodel? CHALLENGING

11

Thetablebelowcomparestheliteracyscorefor10countriesandtheirinternetuse(%).

a Drawascatterplotusinginternetuseasthehorizontalaxisandliteracyastheverticalaxis.

b Drawthelineofbestfitonthescatterplot.

c Describetheassociationbetweeninternetuseandliteracyscore.

d Usingthecoordinates(9,90)and(50,295),determinetheequationofthelineofbestfit.

e Usingtheequationofthelineofbestfit,ifacountryhasaninternetuseof30%predictthe country’sliteracyscore.Answercorrecttothenearestwholenumber.

f Ifacountryhasaliteracyscoreof300,predictthecountry’sinternetuse,correcttothenearest wholenumber.

g Commentontheinternetuseforacountrywithaliteracyscoreof600.

Communicateyourreasoning,understandingandfluency

12 Thepopulationofatownovera10-yearperiodwasrecordedstartingfromtheyear2000andis showninthetable.

a Drawascatterplotwith‘Yearafter2000’onthehorizontalaxisand‘Population’onthe verticalaxis.

b Drawalineofbestfitonthescatterplot.

c Describetheassociationbetweenpopulationeachyearafter2000.

d Usingthecoordinates(2,9300)and(10,4900),findtheequationofthelineofbestfit.

e Interpretthegradientand y-interceptofthelineofbestfitinthecontextofthedata.

f Usingthisequation,whatyearafter2000didthepopulationreach8000?

g Commentonthepopulationofthetownintheyear2020.

6E Statisticalinvestigations

Learningintentions

• Toknowandunderstandthestatisticalinvestigationprocess

• Toviewandundertakestatisticalinvestigationsinvolvingbivariatedatasets

• Toexplainthedifferencebetweenvariablesthatshowcorrelationandthosethathaveacausal relationship

Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYear11

• QuestionsaboutstatisticalinvestigationsregularlyappearinHSCExaminations

Statisticalinvestigationistheprocessofgatheringstatistics.Theinformationgainedfromastatistical investigationisavitalpartofoursociety.Astatisticalinvestigationinvolvesfoursteps.

Fourstepsinastatisticalinvestigation

1 Collectthedata

Collectingdatainvolvesdecidingwhattocollect,locatingitandcollectingit.Thegatheringofstatistical datamaytaketheformofa:

• census–dataiscollectedfromthewholepopulation

• survey–dataiscollectedfromasmallergroupofthepopulation. Itisimportantthatproceduresareinplacetoensurethecollectionofdataisaccurate,up-to-date,relevant andsecure.Ifthedatacollectedcomesfromunreliablesourcesorisinaccurate,theinformationgained fromitwillbeincorrect.Whentakingasample,thedatagatheredmustberepresentativeoftheentire populationotherwisetheinformationcollectedmaybebiasedtowardsaparticularoutcome.

2 Organisethedata

Organisingdataistheprocessofarranging,representingandformattingdata.Itiscarriedoutafterthe dataiscollected.Theorganisationofthedatadependsonthepurposeofthestatisticalinvestigation.For example,tostoreandsearchalargeamountofdata,thedataneedstobecategorised.Organisinggives structuretothedata.

3 Summariseanddisplaythedata

Displayingdataisthepresentationofthedataand information.Informationmustbewellorganised, readable,attractivelypresentedandeasyto understand.Informationisoftendisplayedusing graphssuchasscatterplots,dotplots,histograms, linegraphs,stem-and-leafplotsandboxplots.Data issummarisedusingstatisticssuchasthemean, median,modeandstandarddeviation.

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4 Analysethedata

Analysingdataistheprocessofinterpretingdataandtransformingitintoinformation.Itinvolves examiningthedataandgivingmeaningtoit.Whenanalysingbivariatedata,theform,directionand strengthoftheassociationisdetermined.Scatterplotsandlinesofbestfitarecommonlyusedtoanalyse thedata.Theymakeiteasytointerpretdatabymakinginstantcomparisonsandrevealingtrends. Predictionsandconclusionsarecompletedbyinterpolatingandextrapolatingthedata.

Statisticalinvestigation

Astatisticalinvestigationinvolvesfoursteps:collectingdata,organisingdata,summarisingand displayingdataandanalysingdata.

Casestudyofastatisticalinvestigation

Coenhasbeenaskedtocompleteastatistical investigationonwhetherthebloodglucoselevel (inmg/100mL)ofanadultcanbepredictedfrom theirweight(inkg).

Coenperformedthefollowingsteps.

1 Collectingthedata–Coenaccessedthemedical dataon20adults.

2 Organisingthedata–Coencategorisedthedata intobloodglucoselevelsandweight.

3 Summarisinganddisplayingthedata–Coen presentedthebivariatedataintothetableshown oppositeandthescatterplotshownbelow.

4 Analysingthedata:

a Coendescribedthestrengthofalinearassociation. Thereisastrongpositivelinear associationbetweenweightandblood glucoselevels.

b Coencalculatedtheequationandgraphedtheline ofbestfitonthescatterplot. y = mx + c Pointsontheline:(72.0,95.7)and (78.9,99.9)

c Coenappliedtheresultsofhisstatistical investigationtopredicttheglucoselevelofa personwhoweighs75kg. Ifweight = 75kg

= 0.6 × 75 + 52.5 = 97.5

∴ Apersonweighing75kghasablood glucoselevelof97.5mg/100mL.

Nowyoutry

Shreyahasbeenaskedtocompleteastatisticalinvestigationonwhetherheightabovesealevelcanbe predictedbytemperature.

a WhatassociationwouldyouexpectShreyatofind?

b Whatotherfactorsmightaffectthesefindings?

Issuesinstatisticalinvestigations

Astatisticalinvestigationraisesanumberofethicalissuessuchasbias,accuracy,copyrightandprivacy.

• Dataneedstobefreefrombias.Biasmeansthatthedataisunfairlyskewedorgivestoomuchweighttoa particularresult.Forexample,ifasurveyaboutfavouritemusicwasonlycompletedbyteenagers,andthe resultsweregeneralisedtotheentirepopulation,itwouldhaveabias.Severalchecksshouldbemadeto limittheimpactofbias.

• Theaccuracyofthecollecteddataisavitalingredientofastatisticalinvestigation.Itdependsonthe sourceofthedataandwhetherthedatahasbeenrecordedcorrectly.Theaccuracyofthedataisoften difficulttocheckinareasonabletime.Itisoftennecessarytocomparedatafromanumberofdifferent sourcesanddeterminewhichdataisaccurate.

• Copyrightistherighttouse,copyorcontroltheworkofauthorsandartists.Itisagainstthelaw toinfringecopyright.Youarenotallowedtouseorcopytheworkofanotherpersonwithouttheir permission.Ifdataiscollectedfromtheinternet,itshouldbeassumedtobeprotectedbycopyright.

• Privacyistheabilityofanindividualtocontrolpersonaldata.Datacollectedonindividualsisnotalways accurate.Inaccuraciescanbecausedbymistakesingatheringorenteringthedata,bymismatchofthe dataandthepersonorbyinformationbeingout-of-date.Mostpeoplegiveinformationaboutthemselves toselectedpartsoftheoutsideworld.OftenpeoplearequitewillingtotellAsomethingbutwouldbe shockedifBknew.ButwhatpreventsAtellingB?

Issuesinastatisticalinvestigation

Astatisticalinvestigationraisesanumberofethicalissuessuchasbias,accuracy,copyrightand privacy.

Causation

Causationindicatesthatoneeventistheresultofthe occurrenceofanotherevent(orvariable).Thisisoften referredtoasthecauseandeffect.Thatis,oneevent isthecauseofanothereventhappening.Forexample, thebellattheendoftheperiodisaneventthatcauses studentstoleaveforthenextperiod.Whencompleting astatisticalinvestigationitisimportanttobeawarethat twoevents(orvariables)mayhaveahighcorrelationbut beunrelated.Thatis,highcorrelationdoesnotimply causation.Forexample,theincreaseintheuseofmobile phoneshasastrongcorrelationtotheincreaseinlife expectancy.However,theuseofmobilephonesdoesnot causetheincreaseinlifeexpectancy.

Causation

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Causationindicatesthatoneeventistheresultoftheoccurrenceofanotherevent(orvariable).

BUILDING

1 Copyandcompletethefollowingsentences.

a Astatistical________________involvesfoursteps:collectingdata,organisingdata, summarisinganddisplayingdata,andanalysingdata.

b Censusdataiscollectedfromthewhole________________.

c Whentakinga________________thedatagatheredmustberepresentativeoftheentire population.

d Displayingdataisthe________________ofthedataandinformation.

e Whenanalysing________________datatheform,directionandstrengthoftheassociationis determined.

f Biasmeansthatthedataisunfairly________________orgivestoomuchweighttoaparticular result.

Exercise6E

2 Trueorfalse?

a Asurveyiswhendataiscollectedfromasmallergroupofthepopulation.

b Datacollectedfromunreliablesourcescanresultinincorrectinformation.

c Dataisoftendisplayedusinggraphssuchasscatterplots,dotplots,histograms,linegraphs, stem-and-leafplotsandboxplots.

d Analysingdataistheprocessthatinterpretsdataandtransformsitintoinformation.

e Youareallowedtouseorcopytheworkofanotherpersonwithouttheirpermission.

f Datacollectedonindividualsisalwaysaccurate.

3 TheAustralianBureauofStatisticscollectsdataforoursociety.Collectingdataisonestepina statisticalinvestigation.Listallthefourstepsinvolvedinastatisticalinvestigation.

4 Explainthedifferencebetweenacensusandasurvey.

5 Howcanyoulimittheimpactofbiaseddata?

6 Thereisastrongpositivecorrelationbetweennumberofcaraccidentsandthenumberofteachersin citiesaroundtheworld.Canweconcludefromthisthatteachersarecausingcaraccidents?Givea possibleexplanation.

7 Thereisastrongpositivecorrelationbetweenthenumberofchurchesinatownandtheamountof alcoholconsumedbyitsinhabitants.Doesthismeanthatreligionisencouragingpeopletodrink? Whatcommoncausemightcounterthisconclusion?

DEVELOPING

8 Implementastatisticalinvestigationtoexplorethelinkbetweenaperson’sheartrateafteraset exercisesessionandthenumberofkilometrestheyruneachweek.Thedatabelowwascollected from15youngadults.

a Drawascatterplottodisplaythedata.Drawalineofbestfit.

b Describetheassociationbetweendistancerunandheartrate.

c Findtheequationofthelineofbestfit.

d Thelinegraphbelowwasconstructedbyomittingthevaluefor6km.Whatvalueforheartrate wouldyoureadfromthisgraphforapersonrunning6km?

e Thegraphabovewasconstructedtoencouragepeopletorunmorekilometrestoreducetheirheart rate.Commentonthemisrepresentationofthedata.

9 Theissueofprivacyhasbeenaconcernfordecades.Discussthisstatement.Whatisyourviewon privacy?Usetheinternettofindinformationaboutprivacy.CompareAustralianandoverseasdata.

CHALLENGING

10

Example8 Implementastatisticalinvestigationtosolvetheseproblems.

a StudentsinYear12havedecidedtohaveapartyaftertheHSCatalocalvenue.Youhavebeen giventheresponsibilityofchoosingthevenue.Investigateanumberofdifferentvenuesand collectdatasuchas

• thename andaddressofthevenue

• thecostperhead

• theinitialdepositrequired

• detailsofmeals

• availabledates.

DesignasurveyforYear12studentsthatwilldecideonthevenue.Studentswillvoteonthe venueandiftheyarecommittedtoattend.Thesurveyshouldalsodecideonpossibledatesand mealarrangements.

b Theprincipalwouldliketopromotethepositivefeaturesofyourschooloutsidethelocal community.Heispresentingthesefeaturesataconferencecentre.Developasolutionforthe principal.DesignasurveyforYear12studentsthatwilldecideonthebestfivefeaturesofyour schoolandusethemostappropriatemethodofpresentation.

Communicateyourreasoning,understandingandfluency

11 Discusswhetheracause-and-effectrelationshipexistsbetweenthesevariableswhenastrongpositive correlationisfound.

a Theheightsof childrenandtheirreadingspeeds.

b Acompanyexpenditureonadvertisingandsales.

c Dailytemperatureandtheconsumptionoficecream.

d Icecreamsalesanddrowningaccidents.

e CheeseconsumptionandthenumberofmoviesNicoleKidmanappearsin.

f Thedistancetravelledandthelitresoffuelacarconsumes.

g Hoursofstudyandexamresults.

h Revenuegeneratedbygamblingandelectriccarsales.

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Keyideasandchaptersummary

Scatterplot

Constructingascatterplot:

1 drawanumberplane

2 determineascaleandatitleforthehorizontalor x-axis

3 determineascaleandatitlefortheverticalor y-axis

4 ploteachorderedpairofnumberswithadot.

Usingabivariate scatterplot

Lineofbestfit

Interpolation

Extrapolation

Statistical investigations

FormofanassociationLinearform:astraightline

Non-linearform:acurvedline

DirectionofanassociationPositive:gradientofthelineispositive

Negative:gradientofthelineisnegative

StrengthofanassociationStrong:smallamountofscatter

Moderate:modestamountofscatter

Weak:largeamountofscatter

Lineofbestfitisastraightlinethatapproximatesalinearassociationbetween points.Theequationisgivenby y = mx + c,wherethegradient(orslope)isgiven by m andthe y-interceptis c.Alsoknownastheleast-squareslineofbestfitora trendline.

Predictingvalueswithintherangeofthedataset

Predictingvaluesoutsidetherangeofthedataset

Foursteps:collectingdata,organisingdata,summarisinganddisplayingdata,and analysingdata.

Astatisticalinvestigationraisesanumberofethicalissuessuchasbias,accuracy, copyrightandprivacy.

Causation

Indicatesthatoneeventistheresultoftheoccurrenceofanotherevent(orvariable).

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook.

Checklist

6A 1 Icanrepresentabivariatedatasetusingascatterplot.

e.g.Thetablebelowshowstheheight(incm)ofaeucalypttreeseedlingasit grows.

a Copythenumberplaneshowntoconstructascatterplot usingthegiventable.

b Whatistheincreaseintheheightoftheseedlingduringthe firstweek?

c Whatistheincreaseintheheightoftheseedlingduringthe lastweek?

d Howmanyweeksdoesittakefortheseedlingtoincreasein heightfrom9cmto12cm?

e Estimatetheheightoftheseedlingafter4.5weeks.

f Estimatethetimetakenfortheseedlingtogrowtoaheight of14cm.

6A 2 Icanreadbivariatedatainascatterplot.

e.g.Theaveragenumeracyscorefor year6studentsandtheirgeneralrate ofinternetuse(%)for10countries aredisplayedinthegivenscatterplot.

a Whatisthescalefortheverticalaxis?

b Whatistheaveragenumeracyscoreforthecountrywhichhasaninternetuse rateof24%?

c Whatistheinternetuse(%)forthecountrywhichhasanaveragenumeracyscore160?

d Howmanycountrieshaveinternetuseoflessthan50%?

e Howmanycountrieshaveanumeracyscoregreaterthan100?

f Istherearelationshipbetweenthesetwovariables?

6B 3 Icandescribethestrengthofalinearrelationshipbetweentwovariablesasstrong, moderateorweak,anditsdirectionaspositiveornegative.

e.g.

He ad circum f ere n ce (cm)

a Statewhethertheassociationispositiveornegative.

b Describethestrengthoftheassociationasstrong,moderateorweak.

6B 4 Icanidentifytheindependentanddependentvariablewithinabivariatedatasetwhere appropriate.

e.g.Determinetheindependentvariableinthisscatterplot.

(years) He ad circum f ere n ce (cm)

6B 5 Icandescribetheformofadatasetaslinearornon-linearbasedontheassociation betweentwovariables.

e.g.Isthedatasetlinearornon-linear?

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6C 6 Icancreatealineofbestfitonascatterplotforabivariatedataset,byeye.

e.g.Drawthelineofbestfitbyeyeonthescatterplot.

6C 7 Icandetermineandinterprettheinterceptandgradientofthelineofbestfitfroma givengraphtoformanequationoftheline.

e.g.Theheightsandshoesizesof10 peoplewererecordedandplottedinthe scatterplotshown.Alineofbestfitis alreadydrawn.

Usethetwopointsgivenonthelineof bestfitto:

a determinethegradientoftheline

b determinethe y-intercept

c formanequationofthelineof bestfit.

6D 8 Icanexaminelinesofbestfittomakepredictionsandrecogniselimitationsof interpolationandextrapolationforbivariatedatasetswithinavarietyofcontexts.

e.g.Forminimumtemperaturesfrom5◦ Cto20◦ C,theequationrelatingthe maximumandminimumtemperature(in ◦ C)ataweatherstationisgivenas: maximum = 0.67 × minimum + 13

Usethisequationtopredictthemaximumtemperaturegiventhefollowing minimumtemperatures.Areyouinterpolatingorextrapolating?

6E 9 Icanexplainthedifferencebetweenvariablesthatshowcorrelationandthosethat haveacausalrelationship.

e.g.Thereisastrongpositivecorrelationbetweennumberofcaraccidentsand thenumberofteachersincitiesaroundtheworld.Canweconcludefromthisthat teachersarecausingcaraccidents?Giveapossibleexplanation.

Multiple-choicequestions

1 6B Bloodpressurelevelsforwomenincreaseastheygetolder.Whatisthebestdescriptionforthe associationbetweenbloodpressurelevelsandawoman’sage?

2 6B

Whatisthecorrelationbetweenthevariablesinthescatterplot?

3 6B Whichofthefollowingscatterplotsshowsweaknegativecorrelation?

4 6E

Whichofthefollowingwouldmostlikelybeduetocausation?

A Thenumberofpeopleunemployedandthecostofbread.

B Familyincomeandthenumberoffamilypets.

C Dailytemperatureandice-creamconsumption.

D WineconsumptionandthenumberofmoviesmadeinAustralia.

5 6C Theequationofalineofbestfitthatenablesweeklyamountspentonentertainment(indollars)tobe predictedfromweeklyincomeisgivenby:amount = 0.10 × income + 40. Whatisthepredictedamountspentonentertainmentwithaweeklyincomeof$600?

Short-answerquestions

1 6A

Thescatterplotshowsthenavelheightandthebodyheightforninestudents.

a 6B Whichhasbeenplottedastheindependentvariable?

b Whichhasbeenplottedasthedependentvariable?

c Istheassociationbetweenthesetwovariableslinearor non-linear?

d Describetheassociationasstrong,moderateorweak.

e Whatisthebodyheightforthestudentwithanavelheightof 112cm?

f Whatisthenavelheightforthestudentwithabodyheightof 166cm?

g Usethescatterplottopredictthebodyheightofastudentwith anavelheightof110cm.

2 6A

6B

Thescatterplotshowsastudent’sheight (incm)andtheirage(inyears).

a Whatistheageforthestudentwhen theirheightis120cm?

b Whatistheheightforthestudentat 11yearsofage?

c Statewhethertheassociationis positiveornegative.

d Describethestrengthofthe associationasstrong,moderateor weak.

3 6C Thetablebelowshowsthelengthoftherightfoot(incm)andbodyheight(incm).

a Drawascatterplotusingtheabovetable.

b Drawalineofbestfitbyeye.

c Describethestrengthanddirectionoftherelationship.

4 6D Astrongpositivelinearassociationexistsbetweenthehoursspentstudyingforanexamandthemark achieved.Theequationforthisassociationismark = 4.5 × studyhours + 2.

a Predicttheexammarkifthestudentstudiedfor20hoursaweek.

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b Predicttheamountofstudyhoursrequiredtoachieveamarkof56.

5 6C

6D

Thetablebelowshowstheage(inmonths)andtheheight(incm)ofayoungplant.

6 6A

a Usingthepoints(1,48)and(15,131),findtheequationofthelineofbestfit.

b Whatisthepredictedheightoftheplantafter19months?Answercorrecttothenearest centimetre.

c Whatisthepredictedheightoftheplantafter3years?Answercorrecttothenearestcentimetre.

d Whichoftheabovequestionsinvolvesinterpolation?

Thetableshowsthebirthrate(livebirthsper1000)andthelifeexpectancy(inyears).

a Drawascatterplotusingtheabovetable.

b Statewhethertheassociationispositiveornegative.

c Describethestrengthoftheassociationasstrong,moderateorweak.

d Determinetheequationofthelineofbestfit,usingthecoordinates(40,48)and(42,45).

e Usetheequationtopredictthelifeexpectancywhenthebirthrateis35,correcttothenearest wholenumber.

f Usetheequationtopredictthebirthratewhenthelifeexpectancyis60,correcttothenearest wholenumber.

Extended-responsequestion

Amarketingfirmwantedtoinvestigatetherelationshipbetweenthenumberoftimesasongwas playedontheradio(played)andthenumberofdownloadssoldthefollowingweek(weeklysales). Thefollowingdatawascollectedfromarandomsampleoftensongs. 6B

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a Constructascatterplotusingtheabovetable.

b Drawalineofbestfit.

c Describetherelationshipbetweenweeklysalesandplayedintermsofdirection,strengthand form.

d Determinetheequationofthelineofbestfit,usingthecoordinates(34,2800)and(25,2200). Giveyouranswercorrecttothenearestwholenumbers.

e Interprettheslopeandinterceptoftheregressionlineinthecontextoftheproblem.

f Usetheequationtopredictthenumberofdownloadsofasongwhenitwasplayedontheradio 100timesinthepreviousweek.Areyouinterpolatingorextrapolating?

7 Graphsof practical situations

TheRabbit-ProofFence

InAustralia,therewasanattemptmadeinthe early1900stobuildarabbit-prooffencein WesternAustralia.Whileitsoriginalpurpose wastostopthewestwardspreadofrabbits, ithasnowevolvedintoamulti-speciesbarrier knownastheStateBarrierFence.

breachedthesurveyedlinesbeforeorduring construction.

RenamedtheStateBarrierFence,itisnow usedtomanagethemovementofemus,wild dogsandkangaroosintoagriculturalzones. Largesectionsarestillmaintainedbythe WesternAustraliangovernmentandlocal landholders.

Thesystemoriginallyconsistedofthree distinctfencescoveringthousandsof kilometres.Thefirstfencewascompletedin 1907.Itwasthelongestunbrokenfenceinthe worldatthetime,stretching1833kmfrom StarvationBoatHarbourinthesouthtoCape Keraudreninthenorthwest.Thesecondfence wasbuiltwhenrabbitswerediscoveredwest ofthefirstfence,running1166kminlength andjoiningthefirstfenceatGumCreek.The thirdfenceisthesmallestsection,running approximately257kmfromthewestcoastto meetthesecondfence.

TheRabbit-ProofFencewaslargelyineffective atstoppingrabbits,asmanyhadalready

TheRabbit-ProofFenceisoftenconfusedwith theDingoFence,oneoftheworld’slongest man-madestructures.Itstretchesover 5600kmacrossAustralia,runningthrough Queensland,NewSouthWalesandSouth Australia.Itprotectssheepfromdingoes,with itsnorthernendnearJandowaein Queensland,anditssouthernterminusonthe NullarborPlains.Youcanfindaccessible sections,suchastheonenearJandowaeor alongtheEyreHighwayinSouthAustralia,or visitmemorialsliketheoneinYelarbon.

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7A Usinglinearrelationshipstomake predictions

7B Graphsofexponentialrelationships

7C Exponentialmodelsinpractical contexts

7D Problemsinvolvinggraphsof practicalsituations

NSWSyllabus

Inthischapter,astudent:

•Developsunderstandingandfluencyin mathematicsthroughexploringand connectingmathematicalconcepts, choosingandapplyingmathematical techniquestosolveproblems,and communicatingtheirthinkingand reasoningcoherentlyandclearly (MAO-WM-01)

•Representstherelationshipsbetween quantitiesinalgebraicandgraphical forms(MST-12-S1-01).

©NSWEducationStandardsAuthority

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractive Textbook,includingvideodemonstrations ofallworkedexamples,auto-marked quizzes,downloadablespreadsheets, worksheetsandcalculatorsupport,and muchmore.

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7A Usinglinearrelationshipstomakepredictions

Learningintentions

• Touseequationsandwordedscenariostodrawlineargraphs

• Tobeabletoanalyseastraight-linegraphanduseittomakepredictions

• Tobeabletomatchagraphtoascenarioofavesselbeingfilled

Past,presentandfuturelearning

• Theconceptsinthissectionwillbenewtomanystudents

• QuestionsinvolvinglinearrelationshipsregularlyappearinHSCExaminations

Linearequationsareusedinmanyreal-worldscenarios.Inthissection,wewilllearntouselinear relationshipsinavarietyofreal-worldscenariostohelpmakepredictions.Thiscanbedonebycreatinga linearequationtomodelareal-worldproblem.Wecanobservewhatishappeningandcreateagraphtohelp displayourobservations.Intheseways,wecanmakepredictionsaboutwhatwillhappeninthefuture.

Fillingacontainerwithliquid

Howfastwillacontainerfillwithwater?Whenacontainerisauniformshape,therateinwhichthe containerfillsupwillbeconstant.Ifthecontainerisnotauniformshape,therateinwhichthecontainerfills upwillnotbeconstant.

Considerthefollowingcontainers:

Ifallthecontainersarefilledatthesamerate,thecontainerswillfillupinorderofincreasingcapacity (socontainerCwillfillupfirst,followedbycontainerAandthencontainerB).Youcanseehowtherates differinthegraphshown.Notethatbecausethecontainersareeachfilledataconstantratetherelationship betweentimeandcapacityislinear.

Example1

Recognisingthefillrateofauniformcontainer

Faradaypourswaterintothecontainershown.Whichgraphbestrepresentstheheightofthe waterinthecontainerasitfillsup?

Solution

GraphB

Explanation

Thecontainerisnotauniformshape,sothe ratetheheightofthewaterincreaseswillnot beconstant.Thenarrowpartofthecontainer willfillmorequicklyatthestart.Then,itwill fillmoreslowly.

Nowyoutry

Louispourssandintothecontainershown.Whichgraphbestrepresentstheheightofthe sandinthecontainerasitfillsup?

Areyouready?Substitutingintoaformula

Inthissection,wewillneedtorecallhowtoevaluateanequationwhensubstitutingnumbersintoa formula.Thisactivitywillremindyouhowtodothat.

DownloadtheactivityfromyourInteractiveTextbook.

Example2

Creatingalinearequationfromareal-worldsituation

Awheatsiloholds4000tonnesofwheatwhenfull.Thefullsiloisemptiedintotraincarriagesat 5tonnesperminute.

a Writeanexpressionfortheremainingwheatinthesilo(W tonnes)after t minutes.

b Howlongwillittakeuntilthesiloisempty?

c Howlongwillittakeuntilthesilocontains1000tonnesofwheat?

Solution

a W = 4000 5t

b 0 = 4000 5t

5t = 4000

t = 4000 5 t = 800

Ittakes800minutestoemptythesilo.

c 1000 = 4000 5t 5t = 3000

t = 3000 5

t = 600

Ittakes600minutesuntilthesilohas 1000tonnesofwheat.

Nowyoutry

Explanation

Theamountofwheatleftinthesilo(W)isequal totheamountofwheatatthestart(4000tonnes) minus5tonneseachminute.

Thesiloisemptywhen W = 0sosubstitute W = 0 intotheequation.

Rearrangetomake t thesubjectandsolvethe equation.

Writeyouranswerinwordsusingthecorrectunits.

Substitute W = 1000intotheequation. Rearrangetomake t thesubjectandsolvethe equation. Writeyouranswerinwordsusingthecorrectunits.

Aswimmingpoolholds30000litresofwater.Thepoolisdrainedcompletelysoitcanberepaired.The waterisdrainedat60litresperhour.

a Writeanexpressionforthevolume(V litres)leftinthepoolafter h hours.

b Howlonguntilthepoolisempty?

c Howlonguntilthepoolhas12000litresofwaterleft?

Exercise7A

BUILDING

1 Eachofthefollowingcontainersarethesamevolumewhenfull.Theyareallfilledwithwateratthe sameconstantrate.Matchthecontainerstothefollowinggraphs. i ii iii iv

2 Foreachofthefollowingsituations,writeanexpressionforthevolume(V litres)ofwaterleftina watertankafter t minutes,giventhatthetankholds:

a 4500litreswhenfullandisemptiedat50litresperminute

b 3230litreswhenfullandisemptiedat300litresperminute

c 800litreswhenfullandisemptiedat6litresperminute

d 50000litreswhenfullandisemptiedat2500litresperhour

e 1111litreswhenfullandisemptiedat64litresperhour.

3 Camilaneeds400appleseachdayforherfruitshop.

a WritealinearequationtohelpCamilafindouthowmanyapplesshewillneedperday.

b UseyourequationtodeterminehowmanyapplesCamilawillneedinJuly.

4 Jasperhas40000nailsanduses500nailseachdayinhisbuildingbusiness.

a WritealinearequationthatshowshowmanynailsJasperwilluseovertime.

b UseyourequationtofindouthowmanynailsJasperwillhaveleftafter22days.

c HowlongwillitbeuntilJasperrunsoutofnails?

DEVELOPING

5 Robertinvestsinsharesthatareworth$5.25persharetoday.Theyarepredictedtoincreasebyan averageof$0.02pershareeachday.

a WritealinearequationthattellsRoberthowmuchhisshareswillbeworthovertime.

b UseyourequationtofindouthowmuchRobert’sshareswillbeworthin14days.

c Robertplanstosellhissharesoneyearfromtoday.Howmuchwillhissharesbeworththen?

6 Ascienceclassdoesanexperiment.Thereare29studentsintheclass.Eachstudentisrequiredtofill acontainerwithwater.

a Writealinearequationtoshowhowmuchwatertheclasswilluse.

b Useyourequationtofindouthowmuchwatertheclasswilluseifthecontainershold400mLof water.

c Thewatertobeusedcomesfromawatertankwhichhas15000mLofwaterinit.Howmuch waterisleftinthetankaftertheexperiment?

7 Lukeworksinafactoryfillingboxesofmatches.Heputs50matchesineachbox.

a WritealinearequationthatshowshowmanymatchesLukeuses.

b UseyourequationtofindouthowmanymatchesLukeneedstofill400boxes.

c Lukeisinstructedtonowonlyput40matchesineachbox.HowmanyboxeswillLukenowfillif heusesthesamenumberofmatchesfrompart b?

8 Maxhasalawnmowingbusiness.Onhisown,hecanmow6lawnseachday.Maxworksoutthat eachofhisemployeeswillmow4lawnseachday.Hisbusinessneedstomow140lawnsin10days. Maxwritesthefollowingequationtohelphimworkouthowmanyemployeeshewillneedtomow thelawns.

140 = 60 + 40N

a ExplainhowMaxcameupwiththisformula.

b WhatisthenewformulaMaxshouldusetofindouthowmanyemployeeshewillneedtomow 220lawnsin10days?

c ShowalltheworkingoutneededtoexplaintoMaxthathewillneedfouremployeestomow 220lawnsin10days.

9 Observethetwocontainersshown.Thegraphsshowinghoweachcontainerwillbefilledovertimeis alsoprovided.

a ExplainwhyGraphAisastraightline.

b ExplainwhyGraphBisnotastraightline.

CHALLENGING

7B Graphsofexponentialrelationships

Learningintentions

• Torecogniseexponentialmodels

• Touseequationsandwordedscenariostodrawexponentialgraphs

Past,presentandfuturelearning

• Theconceptsinthissectionwillbenewtomanystudents

• QuestionsinvolvingexponentialrelationshipsregularlyappearinHSCExaminations

InSection7A,weencounteredlinearrelationships.Weusedstraightlinestomakepredictionsinreal-world scenarios.Inthissection,weexaminenon-linearrelationships,withemphasisonreal-lifescenarios.

Everynon-linearrelationshipcanbeviewedandrepresentedasa:

• wordedreal-lifescenario

• numberpattern

• tableofvaluescontainingaseriesofpoints

• anequation,alsoknownasarule

• curvedgraphontheCartesianplane.

Inthischapter,youwill:

• becomemorefamiliarwitheachofthese

• learnhowtomoveefficientlyfromonerepresentationtotheothers

• solveproblemsinvolvingreal-lifescenarios.

Exponentialrelationships:thebigpicture

Anexponentialrelationshipcanbeviewedandrepresentedinseveraldifferentways,asfollows.

Areal-lifescenario

Inpreviouschaptersweencounteredinvestmentswhichearncompoundinterest.Forexample,$1isinvested at5%perannum,compoundingannually.

Everyyear,theamountintheaccount:

• increasesby5%,so

• itgrowsfrom100%to105%,whichis1.05,so

• togofromoneyeartothenext,theamountismultipliedby1.05.

Thisisan exponentialrelationship,becausewearealwaysmultiplyingbythesameamounteveryyear.

Anumberpattern

Wecanalsoseeexponentialrelationshipsincertainnumberpatterns.Trythisonyourcalculator:

Typethenumber1,thenpress =

Multiplyby1.05,thenpress = = = = = 1,1.05,1.1025,1.157625,

Atfirst,thenumbersdon’tseemtobechangingbymuch.Ittakesalongtimefor$1tobecome$2.Butif youkeeponpressing = ,thenumbersstarttoincreasedramatically.Theyare‘growingexponentially’, whichisacommonlyusedphrase.

Tryitagainwith12%ratherthan5%.Itonlytakes6or7yearstodoubleyourmoney.

Belowaresomecommonexamplesofexponentialnumberpatterns.

Exponentialgrowthbyafactorof2,startingfrom1 1,2,4,8,16, ...

Theequationis y = 2x

Exponentialdecaybyafactorof 1 2 ,startingfrom1 1, 1 2 , 1 4 , 1 8 , 1 16 , ...

Theequationis y = 2 x

Exponentialgrowthbyafactorof3,startingfrom1 1,3,9,27,

Theequationis y = 3x

Exponentialgrowthbyafactorof4,startingfrom2 2,8,32,128, ...

Theequationis y = 2 × 4x

Exponentialdecaybyafactorof 1 2 ,whichis2 1,startingfrom256 256,128,64,32,16,8, ...

Theequationis y = 256 × 2 x

Atableofvalues

Thenumbers1,1.05,1.1025,1.157625, thatwelookedatearlierareplacedinthebottomrowofatable. Thetoprowtrackstheyears,startingfrom0.

‘Time’and‘Money’arecalled variables.Thetoprowinthetableaboveisthe independentvariable. Timeisusuallyinthetoprow.Thebottomrowisthe dependentvariable.

Anequation,sometimescalleda‘rule’ora‘function’ Thenumbersinthebottomrowarecalculatedasfollows:

Thiscanbegeneralisedto: y = 1 × 1.05x where: y istheterminthepattern

x isnumberofyears.

Moregenerally:

Equationsofexponentialrelationships

Exponentialrelationshipscanbemodelledby y = k × ax or y = kax,where:

• y istheterminthepattern,whichisthe y-value

• k isaconstantandthevaluewhen x = 0(k × a0 = k)

• a isthebaseornumberbywhichwemultiplytogetfromonetermtothenextforeveryonestepthat x takes.Notethat a > 0.

• x isthenumberinthetoprowofthetable.(Notethatwhenwearetalkingabouttimeasthe independentvariable,weoftenuse n or t insteadof x.)

AgraphontheCartesianplane

EverycolumninthetablesabovecontaintwonumberswhichmakeapointontheCartesianplane.The pointscanbejoinedwithacurve,notastraightline.Itisnon-linear.If y representsaninvestment,wesay thattheinvestmentis increasingatanincreasingrate.

Notes:

• thetoprowofthetableisthehorizontalaxis

• thebottomrowofthetableistheverticalaxis

• the y-interceptis1,becauseweinvested$1

• thescaleonthe x-axiscanbedifferenttothescaleonthe y-axis.

Graphsofexponentialfunctions

Keyfeaturesofexponentialgraphs

• Thegraphlieswhollyabovethe x-axisbecause ax isalwayspositive,regardlessofthevalueof x.When theconstant k ispositive,itisimpossibleforthe y-valuestobezeroornegative.

• Thegraphalwayspassesthroughthepoint(0, k)becausewhen x = 0then y = a0 = 1,regardlessofthe valueof a

• The x-axisisanasymptote.Thatis,itisalinethatthecurveapproachesbygettingcloserandclosertoit butneverreachingit.

• Thegraph y = a x isthereflectionofthegraph y = ax aboutthe y-axis.

• Increasingthevalueof a suchaschanging y = 2x to y = 3x affectsthesteepnessofthegraph.The y-values increaseatagreaterratewhenthe x-valuesincrease.

• Theexponentialfunction y = ax when a > 1isoftenreferredtoasexponentialgrowthbecause,asthe x-valuesincrease,the y-valuesincrease.

• Theexponentialfunction y = a x when a > 1isoftenreferredtoasexponentialdecaybecause,asthe x-valuesincrease,the y-valuesdecrease.

Tographanexponentialfunction:

1 constructatableofvalues

2 drawanumberplane

3 plotthepoints

4 jointhepointstomakeacurve.

Example3

Exponentialgrowth

Drawthegraphof

Constructatableofvaluesfor x and y. Let x = 2, 1,0,1,and2.

Find y usingtheexponentialfunction y = 3x .

Drawanumberplanewith x asthehorizontal axisand y astheverticalaxis.

Plotthepointsfromthetable.

Jointhepointstomakeacurve.

Nowyoutry

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Drawthegraphof

Example4

Exponentialdecay

Drawthegraphof y = 3 x .

Solution

Explanation

Constructatableofvaluesfor x and y Let x = 2, 1,0,1,and2.Find y usingthe exponentialfunction y = 3 x

Drawanumberplanewith x asthehorizontal axisand y astheverticalaxis. Plotthepointsfromthetable. Jointhepointstomakeacurve.

Note:Theexponentialcurve y = 3 x isthe reflectionof y = 3x aboutthe y-axis.Both curvespassthrough(0,1)andhavethe x-axis asanasymptote.

Nowyoutry

Drawthegraphof y = 2 x

Desmoswidget7B

Graphingexponentialfunctionswithtechnology

Spreadsheetactivity:Graphingexponentialfunctions

Exercise7B

1 Whatarethenextthreeterms?

3,6,12, a 1, 1 2 , 1 4 , b 2,6,18, c 5,10,20, d

2 Useatableandagraphof y = 2x toanswerthesequestions.

a Isitpossiblefor y tohavenegativevalues?

b Isitpossiblefor x tohavenegativevalues?

c Isitpossibletocalculate y when x = 0?Ifso,whatisit?

d Isitpossibletocalculate x when y = 0?Ifso,whatisit?

e Whatistheapproximatevalueof y when x = 0.5?

f Whatistheapproximatevalueof y when x = 1.5?

3 Example3 Completethefollowingtableofvaluesbyexpressingthe y values,correcttotwodecimalplaces. Grapheachexponentialcurve. Example4

y 0.02 b y = 4 x

4 Usethegraphof y = 4x inQuestion 3a toanswerthesequestions.

a Isitpossiblefor y tohavenegativevalues?

b Isitpossiblefor x tohavenegativevalues?

c Isitpossibletocalculate y when x = 0?Ifso,whatisit?

d Isitpossibletocalculate x when y = 0?Ifso,whatisit?

e Whatisthevalueof y when x = 1.5?

f Whatisthevalueof y when x = 2.5?

5 Sketchthesegraphsonthesamesetofaxes.

= 2x a

6 Sketchthesegraphsonthesamesetofaxes.

= 2 x a

7 Whatistheeffectonthegraphofchangingthevalueof a in y = ax? Hint:Useyourgraphsin Questions 5 and 6.

8 Completethefollowingtableofvaluesandgrapheachexponentialcurve. a y = 0.5x

DEVELOPING

9 UsethegraphsinQuestions 5a, 6a and 8 toanswerthesequestions.

a Whatarethecoordinatesonthegraphof y = 0.5x where x = 1?

b Whatarethecoordinatesonthegraphof y = 2 x where x = 1?

c Whatarethecoordinatesonthegraphof y = 0.5 x where x = 2?

d Whatarethecoordinatesonthegraphof y = 2x where x = 2?

e Isthegraphof y = 0.5x thesameas y = 2 x?Explainyouranswer.

f Isthegraphof y = 0.5 x thesameas y = 2x?Explainyouranswer.

CHALLENGING

10a Sketchthesegraphsonthesamesetofaxes.Labelthe y-intercept.

i y = 2x

ii y = 2 × 2x

iii y = 1 2 × 2x

b Howdoesmultiplyingthegraph y = ax byanumberaffectthegraph?

11 Whatdoesthegraphof y = 1x looklike?

Communicateyourreasoning,understandingandfluency

12 Considertheequation y = 100 × 3x.Answerthefollowinginwords,withoutdrawingatableora graph.

a Whatdoesthenumber100tellyouaboutthiscurve?

b Whatdoesthenumber3tellyouaboutthenumbersinthebottomrowofthetableofvalues?

c Willthisgraphmodelexponentialgrowthordecay?

13 Considertheequation y = 4000 × 4 x.Answerthefollowinginwords,withoutdrawingatableora graph.

a Whatdoesthenumber4000tellyouaboutthiscurve?

b Whatdoesthenumber4tellyouaboutthenumbersinthebottomrowofthetableofvalues?

c Willthisgraphmodelexponentialgrowthordecay?

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7C

Exponentialmodelsinpracticalcontexts

Learningintentions

• Toconstructandanalyseexponentialmodelstosolveproblemsinpracticalcontexts

Past,presentandfuturelearning

• Theconceptsinthissectionwillbenewtomanystudents

• QuestionsinvolvingexponentialrelationshipsregularlyappearinHSCExaminations

Exponentialmodellingoccurswhenapracticalsituationisdescribedmathematicallyusinganexponential equation,tableandgraph.Thequantityusuallyexperiencesfastgrowthordecay.

Exponentialmodel

Exponentialgrowth:Quantityincreasesrapidlyaccordingtothefunction

Exponentialdecay:Quantitydecreasesrapidlyaccordingtothefunction

Limitationsofexponentialmodelsinpracticalcontexts

Whenweuse t fortime:

• thetoprowofthetableofvaluesonlycontains0andpositivenumbers

• thegraphcanbedrawninthefirstquadrant.

Example5

Usinganexponentialmodel

Afishpopulationismodelledusingtheformula, N = 500 × 1.5t where N isthenumberoffishand t is thetimeinyears.

a Constructatableofvaluesfor t and N.Usevaluesfor t from0 to4.Approximatethenumber offishtothenearestwhole number.

b Drawthegraphof N = 500 × 1.5t .

c Howmanyfishwerepresent after2years?

d Howmanyextrafishwillbe presentafter4yearscompared to2years?

e Estimatethenumberoffish after18months.

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Solution

a Time ( t years) 0 1 2 3 4

Fish (N) 500 750 1125 1688 2531

b 1000

N

Explanation

Constructatableofvaluesfor t and N

Let t = 0,1,2,3and4.Find N using N = 500 × 1.5t.Expressthevaluesfor N asawholenumber.

Drawanumberplanewith t asthe horizontalaxisand N asthevertical axis.

Plotthepointsfromthetable. Jointhepointstomakethecurve.

c 1125fish Lookup t = 2inthetableandfind N

d Extrafish = 2531 1125 = 1406

e Approximately900fish.

Nowyoutry

a InExample 3,whatarethevaluesof N when t = 10and t = 20?

Subtractthenumberoffishfor2years from4years.

Readtheapproximatevalueof N from thegraphwhen t = 1.5.

b Doyouthinkthenumberoffishcangrowinthiswayinarealpopulation?Whatfactorsmightlimit thismodel?

1 Theequation N = 6t isusedtomodelthegrowthinthenumberofinsects(N)after t days.

a Copyandcompletethetableofvaluesfor t and N t 0 1 2 3 4 N

b Copythegivennumberplaneanddrawthegraphof N = 6t

c Whatwastheinitialnumberofinsects?

d Howmanyinsectswerepresentafter3days?

e Howmanyextrainsectswillbepresentafter4days comparedto2days?

f Estimatehowmanydaysittookforthenumberofinsects toexceed1000.

2 Example5 Thepopulationofanendangeredreptileisdecreasingexponentiallyaccordingtotheformula, P = 100 × 1.5 t where P isthepopulationofreptilesafter t years.

a Copyandcompletethetableofvaluesfor t and P.Expressthepopulationofreptilestothenearest wholenumber.

t 0 2 4 6 8

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b Graph P = 100 × 1.5 t usingthetableoforderedpairsinpart a

c Whatistheinitialpopulationofreptiles?

d Estimatethepopulationofreptilesafter3years.

e Estimatethepopulationofreptilesafter7years.

f Whatisthedifferenceinthepopulationofreptilesafter2yearscomparedto6years?

g Estimatethetimetaken(tothenearestyear)forthepopulationofreptilestobebelow1.

Note:Thishighlightsanotherlimitationofexponentialmodelsinpracticalconcepts.If P isa populationofanimalsexperiencingexponentialdecay,eventuallytheywillbecomeextinct, butthemodelkeepsonproducingdecimalswhichareclosetozero.

DEVELOPING

3 Thesizeofaflockofbirds, F,after t yearsisdecayingexponentiallyaccordingto F = 200 × 0.5t .

a Makeatableofvaluesfor t and F.Usevaluesfor t from0to5.Express F correcttothenearest wholenumber.

b Drawthegraphof F = 200 × 0.5t .

c Whatwasthesizeofinitialflockofbirds?

d Howmanybirdswerepresentafter3years?

e Howmanyfewerbirdswillbepresentafter4yearscomparedto2years?

f Estimatehowmanyyearsitwilltakeforthenumberofbirdstofalltolessthanonebird.

g Explainhowmanyyearsthismodelisusefulforandwhy.

4 Thenumberofbacteriagrowsexponentiallyaccordingtotheformula b = 30 × 1.2t where b isthe numberofbacteriaafter t hours.

a Constructatableoforderedpairsusing0,5,10,15and20asvaluesfor t.Expressthenumberof bacteriatothenearestwholenumber.

b Graph b = 30 × 1.2t usingthetableoforderedpairsinpart a.

c Whatistheinitialnumberofbacteria?

d Whatisthenumberofbacteriaafter4hours?

e Estimatethetimetakenforthenumberofbacteriatoreach120.

5 Oweninvested$1000intoamanagedfundthatappreciatedinvaluefor5years.Theamountof money(A)inthefundforeachyear(t)isshownbelow. t 0 1 2 3 4 5

a Drawanumberplanewith t asthehorizontalaxisand A astheverticalaxis.

b Plotthepointsfromthetableofvalues.Jointhepointstomakeacurve.

Anexponentialgrowthmodelintheform y = 1000 × ax describesthissituation.

c Whatisthemodelif x and y arereplacedwith t and A?

d Determinethevalueof a inthemodelbyusing t = 1.

e Whatistheexponentialmodelusing t and A?

f Usethemodeltofindthevalue(tothenearestdollar)ofthefundafter2 1 2 years.

g Usethemodeltofindthevalue(tothenearestdollar)ofthefundafter7years.

h Estimatethetimewhenthevalueofthefundisapproximately$1480.

Communicateyourreasoning,understandingandfluency

6 ApopulationofAnimalAlivesonasmallisland.AnotherpopulationofAnimalBwasintroducedto thesameisland.ThiscausedthepopulationofAnimalAtodecline. Thepopulationshavebeenmodelledusingtheseequations,inwhich:

• N isthenumberofanimals

• t isthenumberofyears

AnimalA: N = 5000 × 1.2 t

AnimalB: N = 200 × 1.2t

Note:Thisactivitymaybecompletedusingaspreadsheet.

IsthepopulationofAnimalAincreasingordecreasing?Howcanyoutellfromtheequation? a IsthepopulationofAnimalBincreasingordecreasing?Howcanyoutellfromtheequation? b Completethefollowingtable,usingwholenumbersforAandB.

WhatwastheinitialpopulationofAnimalBthatwasintroducedontotheisland? d ApproximatelywhenwilltheAnimalBpopulationoutnumbertheAnimalApopulation? e WhatwillhappentoAnimalAifthistrendcontinues? f DescribethelimitationstothismodelforAnimalA. g

7D Problemsinvolvinggraphsofpracticalsituations

Learningintentions

• Tobeabletomatchagraphtoascenarioofavesselbeingfilled

• Tosolveproblemsinvolvingnon-linearrelationships

Past,presentandfuturelearning

• Theconceptsinthissectionwillbenewtomanystudents

• Questionsinvolvingnon-linearrelationshipsregularlyappearinHSCExaminations

Algebraicmodellingoccurswhenapracticalsituationisdescribedmathematicallyusinganequationanda graph.Thisinvolvesgatheringdataandanalysingthedatatodeterminethebestequation.Determiningthe equationismadeeasierusingtechnology.

Algebraicmodels

• Algebraicmodelsareusedtodescribepracticalsituations.

• Algebraicmodelsmayhavelimitationsthatrestricttheiruse.

Example6

Modellingphysicalphenomena

Thistableshowsthemass(M kg)ofababyorangutananditsageafter x months.

a Plotthepointsfromthetableontoanumberplane.

b Theformula M = 1.5(1.2)x modelsthedatainthetable.Graph M = 1.5(1.2)x onthe samenumberplane.

c Usethemodeltodeterminethemassoftheorangutanafter2.5months,correctto1decimalplace.

d Thismodelonlyapplieswhen x islessthanorequalto6.Why?

Drawanumberplanewith x asthehorizontal axisand M astheverticalaxis.

Plotthepointsfromthetable.

Theformula M = 1.5(1.2)x hasthesametable ofvalues.Jointhepointstomakeacurve.

c M = 1.5(1.2)2.5 = 2.4kg

Substitute2.5for x intotheformula. Evaluate,correcttoonedecimalplace.

d M = 1.5(1.2)48 = 9479.6kg

Orangutansarelessthan100kgin generalsotheanswerof9479.6kgis unreasonable.

Nowyoutry

Usethemodelfor x greaterthan6.Let x be48monthsor 4years.Substitute48for x intotheformula.Evaluate. Writetheanswerinwords.

Thedistance(d metres)thatanobjectfallsin t secondsisshowninthistable.

a Plotthepointsfromthetableofvaluesonthenumberplane. Jointhepointstomakeacurve.

b Aquadraticmodelintheform y = ax2 describesthissituation.

i Whatisthemodelif x and y arereplacedwith t and d?

ii Determinethevalueof a inthemodelbyusing t = 1.

iii Whatisthequadraticmodelusing t and d?

c Usethemodeltofindthedistancefallenafter3 1 2 seconds.

d Usethemodeltofindthedistancefallenafter10seconds.

e Whatisthetimetakenforanobjecttofall320metres?

f Earth’satmosphereisapproximately1000kmthick.What limitationswouldyouplaceonthismodel?

Example7

Recognisingthefillrateofanon-uniformcontainer

Donnapourswaterintothecontainershown.Whichgraphbestrepresentstheheightofthewaterinthe containerasitfillsup?

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Continuedonnextpage

Solution

GraphD

Nowyoutry

Explanation

Theheightofthewaterinthecontainerincreasesquicklyto startwithandthenmoreslowlyasthecontainerwidens.

Louispourssandintothecontainershown.Whichgraphbestrepresentstheheightofthe sandinthecontainerasitfillsup?

Exercise7D

BUILDING

1 Example6 Anewpieceofequipmentispurchasedbyabusinessfor$150000.Thevalueoftheequipment (v inthousandsofdollars)depreciateseachyear(t)asshowninthetable.

a Drawanumberplaneasshown.

b Plotthepointsfromthetableofvalues.Jointhe points.

c Anexponentialmodelintheform y = 2 x × a describesthissituation.

i Whatisthemodelif x and y arereplacedwith t and v?

ii Determinethevalueof a inthemodelbyusing t = 0.

iii Whatistheexponentialmodelusing t and v?

d

Usethemodeltopredictthevalueoftheequipmentafter2.5years.Answercorrecttothenearest thousand.

e

f

g

Usethemodeltopredictthevalueoftheequipmentafter6months.Answercorrecttothenearest thousand.

Whenwillthevalueoftheequipmentbe$75000?

Usethemodeltopredictthevalueoftheequipmentafter20years.Explainyouranswer.

2 Thisgraphshowsthenumberofbirds recordedbyvolunteersinJanuaryeach yearfrom2020to2026.

a Howmanybirdswereobservedin: 2020? i 2022? ii 2025? iii

b Whatwasthelowestnumberandhighestnumberofbirdsobservedfrom2020to2026?

c Inwhichyearwasthegreatestdeclineinbirdsobservedfromthepreviousyear?

DEVELOPING

3 Example7 Jaydenfilledthreecontainerswithwater.Thefollowinggraphsshowthefillratesofeachofthe containers.Matcheachcontainertothecorrectgraph.

4 Apopulationof300000bacteriadoublesevery20minutesasseeninthetablebelow.

a Drawanumberplanewith t asthehorizontalaxisand B astheverticalaxis.

b Plotthepointsfromthetable.Jointhemtomakeacurve.

c Anexponentialmodeloftheform y = 2x × a describesthissituation.

Whatisthemodelif x and y arereplacedby t and B? i

Determinethevalueof a. ii

Whatistheexponentialmodelusing t and B? iii

5 Marcushasdrawnthefollowinggraphstoshowtherateinwhichfourcontainerswerefilledwith water.Drawapossiblecontainertomatcheachgraph.

CHALLENGING

6 Filomenainvested$100intoamanagedfundthatappreciatedinvaluefor5years.Theamountof money(A)inthefundforeachyear(t)isshowninthistable.

a Drawanumberplanewith t asthehorizontalaxisand A astheverticalaxis.

b Plotthepointsfromthetableofvalues.Jointhepointstomakeacurve.

c Anexponentialmodelintheform y = 100 × (a)x describesthissituation.

i Whatisthemodelif x and y arereplacedwith t and A?

ii Determinethevalueof a inthemodelbyusing t = 1.

iii Whatistheexponentialmodelusing t and A?

d Usethemodeltofindthevalue(correcttothenearestdollar)ofthefundafter2 1 2 years.

e Usethemodeltofindthevalue(correcttothenearestdollar)ofthefundafter7years.

f Whenisthevalueofthefundapproximately$5767?

7 Linghas500booksinhisstore.Shesells5bookseachday.

a Constructatableofvaluesforthefirst5daysthatLingsellsbooks.

b Usingbooks(B)anddays(d),createalinearequationtomodelthissituation.

c Lingwillbuymorebookswhenherstockreaches100books.Howmanydayswillitbebefore Lingbuysmorebooks?

d After20daysLingstartstosell12bookseachday.WriteanewequationtohelpLingmodelher changingcircumstances.

e GiventhatLingwillstillbuymorebookswhenherstockreaches100books,howmanydayswillit nowbebeforeLingbuysmorebooks?

Communicateyourreasoning,understandingandfluency

8 Thecostofusinganewmotorwaywasinitiallyfreeand4500vehiclesusediteachday.Atollwas introducedthatgraduallyincreased.

Let c bethetollcharge(indollars)andlet n bethenumberofvehiclesusingthemotorwayeachday.

a Anexponentialmodelintheform y = b x × a describesthis situation.

i Whatisthemodelif x and y arereplacedwith c and n?

ii Determinethevalueof a inthemodelbyusing c = 0.

iii Determinethevalueof b inthemodeliftherewere3000 vehiclesusingthemotorwayforatollchargeof$1.

iv Determineanexponentialmodelusing c and n todescribe thissituation.

b Constructatableofvaluesbyusing c and n.

c Plotthepointsfromthetableofvaluesonanumberplane.

d Howmanyvehiclesusethemotorwayifthetollchargeis$3?

e Whatistheincomereceivedwhenthetollchargeis$3?

f ‘Ahighertollalwaysmeansahighertotaldailyincome’. Doyouagreewiththisstatement?Explainyouranswer.

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Keyideasandchaptersummary

Linearmodels

Limitationson models

Linearmodelscanbeusedtomakepredictions. Createatableofvaluestomodelareal-worldsituation. Byusinganequation,wecanpredictwhatwillhappenovertime. Substituteinthegivenvalueandsolvetheequation. Linearmodelscanbeusedtoshowthefillrateofacontainerovertime.

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Exponentialgrowth

Exponentialdecay

Quantityincreasesrapidlyusing y = ax Quantitydecreasesrapidlyusing y = a x

Whenusinganymathematicalmodel,itimportanttorecognisethatthemathematics maygiveananswerbuttheanswercouldmakelittleornosenseintherealworld. Forexample,thenumberofanimalscalculatedmathematicallycouldbe 3,which wouldmakenosenseintherealworld.

Chapterchecklistwithsuccesscriteria

AprintableversionofthischecklistisavailableintheInteractiveTextbook.

Checklist

7A 1 Icanrecognisethefillrateofauniformcontainer.

e.g.Dantepourswaterintothecontainershown.Whichgraph bestrepresentstheheightofthewaterinthecontainerasitfillsup?

7A 2 Icancreatealinearequationfromareal-worldsituation.

e.g.Afishtankholds55litresofwaterwhenfull.Thefulltankisemptiedat 2litresperminute.

a Writeanexpressionfortheremainingwaterinthefishtank(W litres)after t minutes.

b Howlongwillittakeuntilthetankisempty?

c Howlongwillittakeuntilthereare33litresofwaterremaininginthefishtank?

7B 3 Icanrecognisetheequationandshapeofanexponentialrelationshipanddraw thegraph.

e.g.OnthesameCartesianplanedrawthefollowingtwographsfrom x = 3to x = 3.

y = 2x a

y = 3 × 2x b

7C 4

Icanuseexponentialmodelsinpracticalcontexts.

e.g.Atownof1000peopleisgrowingby10%perannum.Howmanyyearswillit takeforthepopulationtodouble?

7D 5 Icanrecognisethefillrateofanon-uniformcontainer.

e.g.Debrapourswaterintothecontainershown.Whichgraph bestrepresentstheheightofthewaterinthecontainerasit fillsup?

7D 6 Icanchooseanappropriatemodelrelevanttoaproblemanduseittosolvethe problem.

e.g.Answerthefollowingquestions.

Whatarethemissingnumbersinthistable?

Multiple-choicequestions

1 7A Whichcontainerbestrepresentsthegraphoftheheightofwater inthecontainerasitisfilledup?

2 7A Sapphirehasabagofdogfoodthatweighs2kg.Sapphire’sdogeats150goffoodeachday.Which equationdescribestheamountoffoodleftinthebag(F)asitchangeseachday(d)?

F = 20 150d A

F = 2000 150d B

F = 2000 + 150d C F = 20 + 150d D

3 7B Whatisthe y-interceptoftheexponentialfunction y = 2 x?

4 7C Whichgraphshowsexponentialdecay?

5 7D

Thisgraphshowstheinsectpopulation(N)plotted againstthetime(t)indays.

Whattypeoffunctionwouldmodelthisdata?

Exponential

Quadratic

Cubic

Short-answerquestions

1 7A

2 7A

Thefollowinggraphrepresentstheheightofwaterinacontainerovertimeasitisfilledup.Drawa possiblecontainerthatfitsthegraph.

3 7B

Donkeyhas500jokesthathecantelltoannoyShrek.Donkeywillnottellthesamejokeuntilhehas toldalltheotherjokesheknows.Donkeytells20jokeseveryhour.

a WriteanequationtohelpShrekworkouthowlonghehastolistentoDonkey’sjokes.

b HowmanyhourswillShrekhavetoputupwithDonkey’sjokes?

Completethefollowingtablesofvaluesbyexpressingthe y-valuescorrecttoonedecimalplace. Grapheachexponentialfunction.

4 7C Theheight, h cm,ofaplantanditsageafter x monthsisgiveninthistable.

a Plotthepointsfromtheabovetableontoanumberplane.

b Theformula h = 1.1 × 2.2x modelsthedatainthetable.Drawthisfunction.

c Usethemodeltodeterminetheheightofaplantafter1.5months.Answercorrecttoone decimalplace.

d Usethemodeltodeterminetheheightofaplantafter3.5months.Answercorrecttoone decimalplace.

5 7C Thepopulationofearthwormsgrowsexponentiallyaccordingtotheformula w = 25 × 1.1t,where w isthenumberofearthwormsafter t days.

a Constructatableoforderedpairsusing0,5,10,15and20asvaluesfor t.Expressthenumberof earthwormscorrecttothenearestwholenumber.

b Graph w = 25 × 1.1t usingthetableoforderedpairsinpart a

c Whatistheinitialnumberofearthworms?

d Whatisthenumberofearthwormsafter3days?

e Estimatethetimetakenfortheearthwormstoreachapopulationof75.

6 7D ThisgraphmodelsChristine’spay.

a HowmuchisChristine’sretainer?

b WhatvalueofsalesdoesChristineneedtoachievetobepaidmorethan$200?

c FindChristine’spayifshereachessalesof: $4000 i $6000 ii $7000. iii

d FindChristine’ssalesifsheispaid: $200 i $250 ii $400. iii

Extended-responsequestions

1 7D Agroupofdoctorsmonitorthespreadofavirusandrecordtheirfindingsinthetablebelow.

Thespreadofthevirusisgivenbytheequation

a Findthevalueof a

b Writetheequationthatmodelsthespreadofthevirus.

c Howmanypeoplewereinitiallysickfromthevirus?

d Howmanypeopleweresickafter4days?

e Howmanypeoplewillbesickafter20days?

f Thedoctorsfoundthattheyhadmadeanerror.Theinitialnumberofsickpeoplewas200.Write theequationtheyshouldhaveused.

g Constructatableofvaluesthatshowsthenewinformationfromday0today4.

h Whatwasthedifferenceinthetwomodelsbyday4?

i Whatdoesthenew(moreaccurate)modeltellusabouthowmanypeoplewillbesickafter 20days?

PracticePaper2

SectionI

AttemptQuestions1–15(15marks).

Allowabout20minutesforthissection.

1 Whichequationisexponential?

2 ThescatterplotshowstheweightsofGreatDanesat 1montholdandat1yearold.

Howisthisassociationbestdescribed?

Weakpositivenon-linear A

Weaknegativenon-linear B

Strongpositivelinear C

Strongnegativelinear D

3 Whichfractionisequivalenttosin θ inthetrianglebelow?

4 Whatisthevalueof y

5 Whichofthefollowingexpressionswouldgivetheheight(h),ofthe treeinthediagram?

Usethefollowinggraphtoanswerquestions 6 to 8

6 WhatisthegradientoflineB?

7 WhatistheequationoflineA?

8 WhatisthepointofintersectionoflineAandlineB?

9 Whichofthefollowingequationsisclosesttothislineof bestfit?

y = 0.9x + 8.7 A

y = 8.7x + 0.9 B

y = 0.9x 8.7 C

y = 8.7x 0.9 D

10 Whatisthe

11 Thecompassbearingof Y from X isS49◦W.Whatisthecompass bearingof X from Y ?

S49◦W A

S41◦W B

N49◦E C

N41◦E D

12 Forthegraphopposite, R istherevenue,indollars,earnedbya mushroomfarmerselling n kilogramsofmushrooms,and C isthecost, indollars,ofproducing n kilogramsofmushrooms.Tomakeaprofit, howmanykilogramsofmushroomsshouldthefarmer’sproduction exceed?

25 A

B

C

D

13 Supposethattheequationofthelineofbestfitwhichwouldenabletheamountofmoneyspentatthe shoppingcentre(indollars)tobepredictedfromthenumberofhoursspentattheshoppingcentreis givenby:

Shoppingexpenditure = 20 + 50 × Hoursspentshopping

Theslopeofthelinecanbeinterpretedasfollows.

Onaverage,foreachextra50minutesspentshoppinganextra$1isspent. A

Onaverage,foreachextrahourspentshoppinganextra$50isspent. B

Onaverage,customersspend$20perhourattheshoppingcentre. C

Onaverage,customersspend$50attheshoppingcentre. D

14 Asailingboattravelsdueeastfrom A to B. Itthenturnsandsailsonabearingof210◦ . Whatistheboat’sbearingfrom A whentheboatisclosestto A?

030◦ A

15

IfSuebuysthreepacketsofchipsandtwobottlesofcola,itwillcosther$13.20.However,ifshebuys twopacketsofchipsandthreebottlesofcolaitwillcostheronly$11.80.Whatisthepriceofeach item?

cola = $3.20,chips = $1.80 A

cola = $2.10,chips = $3 B

cola = $1.80,chips = $3.20 C

cola = $2,chips = $2.90 D

SectionII

AttemptQuestions16–18(45marks).

Allowabout70minutesforthissection.

Allnecessaryworkingshouldbeshownineveryquestion.

Question16(15marks)

a Thecost($C)ofhiringacarisgivenby C = 2.5x + 65,where x isthenumberof kilometrestravelled.Apersonischarged$750forthehireofthecar.Whatwasthe numberofkilometrestravelled? (2marks)

bi Completethetableofvaluesfortheequation y = 4 × 2x . (2marks) x 2 1 0 1 2 y

ii Drawthegraphof y = 4 × 2x . (2marks)

c Whatisthepointofintersectionofthelines x + 3y = 4and x + y = 2? (3marks)

d Thenumberofbacteriagrowsexponentiallyaccordingtotheformula b = 30(1.2)t , where b isthenumberofbacteriaafter t hours.

i Whatistheinitialnumberofbacteria? (1mark)

ii Copyandcompletethetableoforderedpairs.Expressthenumberofbacteriatothe nearestwholenumber.

t 0 5 10 15 b (2marks)

iii Drawthegraphof b = 30(1.2)t . (2marks)

iv Estimatethetimetakenforthenumberofbacteriatoreach120. (1mark)

Question17(15marks)

a Thegraphsshowthecostofgrowingroseplantsandtheincomereceivedfromsellingthem.

i Usethegraphtodeterminethenumberofroseplantsthatneedtobesoldtobreakeven.(1mark)

ii Howmuchprofitorlossismadewhen1roseplantissold? (1mark)

iii Howmuchprofitorlossismadewhen4roseplantsaresold? (1mark)

iv Whatistheinitialcost? (1mark)

b Thecostofpreparingmealsinaschoolcanteenvariesdirectlywiththenumberofmealsprepared.To helpthecatererspredictthecosts,datawerecollectedonthecostofpreparingmealsfordifferentlevels ofdemand.Thedataare shownbelow.

i Usingthepoints(35,154)and(55,198),findtheequationofthelineofbestfit.Answercorrectto onedecimalplace. (2marks)

ii Whatisthepredictedcostofproducing48meals? (1mark)

iii Whatisthepredictednumberofmealsifthecostis$253? (1mark)

c Astraightlinehastheequationof2x + y = 2.

i Whatisthegradientofthisline? (1mark)

ii Whatisthe y-intercept? (1mark)

iii Whatisthepointofintersectionbetween2x + y = 2and y = 2? (1mark)

di Drawthegraphsof y = 3x 3and y = 3 3x onthesameCartesianplane. (2marks)

ii Writedownthecoordinatesofthepointofintersection. (2marks)

Question18(15marks)

a Thescatterplotshowstheaverageannualfemaleincomeplottedagainstaverageannualmaleincomefor 16countries.Alineofbestfitisfittedtothedata.

i Whatwastheaverageannualfemaleincomeforacountrywhoseaverageannual maleincomewas$60000? (1mark)

ii Howmanycountrieshadanaverageannualfemaleincomegreaterthan$30000?(1mark)

iii Estimatethegradientofthelineofbestfit. (1mark)

b Inthetriangle ABC, AB = 8cm, BC = 5cmand CD = 4cm.

i Findthesizeofangle BCD,correcttothenearestdegree. (1mark)

ii UsePythagoras’theoremtofindthelengthof BD. (1mark)

iii Calculatethelengthof AD,correcttotwodecimalplaces. (1mark)

iv Findtheareaofthetriangle ABC,correcttotwodecimalplaces. (1mark)

c Liasellspillowsfor$65each.

i Writeanequationfortherevenue, R dollars,thatLiareceivesfromthesaleof x pillows.(1mark)

ii Thecost, C dollars,ofmakingpillowsisgivenby C = 500 + 40x.Findthecostof making30pillows. (1mark)

iii Drawthegraphof C = 500 + 40x andyourequationinpart i onthesamenumberplane.(2marks)

iv UsethegraphtoestimatethenumberofpillowsLiawillneedtoselltobreakeven.(1mark)

d Fortimerangingfrom30to60seconds,theequationrelatingthenumberofcorrectanswers totimeis:

Correctanswers = 0.72 × time + 30

Usethisequationtopredictthenumberofcorrectanswerswiththefollowingtimes. i 25seconds ii 50seconds iii 75seconds(3marks)

Inparts i, ii and iii,didyouinterpolateorextrapolate?

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