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CambridgeUniversityPress & AssessmentacknowledgestheAboriginalandTorresStraitIslanderpeoplesofthisnation.We acknowledgethetraditionalcustodiansofthelandsonwhichourcompanyislocatedandwhereweconductourbusiness.Wepay ourrespectstoancestorsandElders,pastandpresent.CambridgeUniversityPress & Assessmentiscommittedtohonouring AboriginalandTorresStraitIslanderpeoples’uniqueculturalandspiritualrelationshipstotheland,watersandseasandtheirrich contributiontosociety.

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. He iscurrentlytheHeadofMathematicsatahighschoolinnorth-westSydney.Heispassionateabout mentoringstaff andseeingthemgrowanddevelop.Sincereturningtotheclassroomin2014,hehasguided studentstobecomeindependentlearnerswhodrivetheirowneducation.

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Acknowledgements

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Everyefforthasbeenmadetotraceandacknowledgecopyright.Thepublisherapologisesforanyaccidental infringementandwelcomesinformationthatwouldredressthissituation.

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Tocome

1 Ratiosandrates

HowfastarewemovinghereonEarth?

WhenwearestandingstillonEarth,weare actuallymovingveryfast,inthreedifferent ways.

Ourplanetrotates 360 degreesevery 24 hours, soapersonstandingontheequatoris travellingatabout 1700 kmperhour.

AtthisspeedyoucouldtravelfromSydneyto Londoninabout 9.4 seconds.

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

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WearealsoorbitingaroundtheSunata speedofapproximately 30 kmpersecond.

Ourplanetalsomovesbecausetheuniverseis expanding,butthespeedofthismovementis veryinsignificantcomparedtothoseabove.

1A Calculationswithratios

1B Thecapture-recapturetechnique

1C Dividingaquantityinagivenratio

1D Scaledrawings

1E Buildingplans

1F Perimeter,areaandvolumefromplans

1G Ratesandbestbuys

1H Distance-timegraphs

1I Energyconsumptionrates

1J Fuelconsumptionrates

NSWSyllabus

Inthischapter,astudent:

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

•Solvespracticalproblemsinvolving ratiosandrates(MST-12-S2-05).

©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 isusedtodescribethelengthofthehorizontal sidecomparedtotheverticalside.Commonaspectratios include16:9,16:10,3:2and4:3.

Iftheaspectratiois16:9,then:

• 16and9havenocommonfactorsotherthan1,sothisratioisinsimplestform

• thescreenmightbe16cmwideand9cmhigh(16cmby9cm)

• wecouldmultiplybothnumbersby2andget32cmby18cm

• wecouldmultiplybothnumbersbyanynumber

• wecoulddividebothsidesbyanynumber

• wecoulddividebothnumbersby16andcallthisratio1: 9 16

• wecoulddividebothnumbersby9andcallthisratio 16 9 :1

• thesidesareintheproportionasshownhere,butthesquarescouldbelargerorsmaller.

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:

Simplifyingaratio

Writetheseratiosinsimplestform.

Toconvertonehalftoawholenumber,multiply by2.

Checkthatratioisinsimplestform.

Toconvertthesedecimalstowholenumbers, multiplyby2or10.

Checkthatratioisinsimplestform.

Ifthedenominatorswereequaltheycouldbe ignored,solookforLCMof7and21,whichis3. Multiplythenumeratoranddenominatorofthefirst fractionby3.

Ignorethedenominators.

Checkthatratioisinsimplestform.

Nowyoutry

Exercise1A

Note:Reduceallanswerstosimplestform.

1 Expresseachratioinsimplestform. 15:3 a 10:40 b 24:16 c 14:30 d 8:12 e 49:14 f 81:27 g 48:32 h 17:51 i 9:18:9 j 5:10:20 k 27:9:3 l

2 Expresseachratioinsimplestform.

$24:$18 a 3kgto12kg b 56t:16t c

BUILDING

3 Thefollowingratioshavedifferentunitsoneachside.Expresseachratioinsimplestformafterfirst changing tothesameunitsoneachside.(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

4 Thereare14redblocksand10whiteblocksinabag.Whatistheratioof: redtoblack a blacktored b redtoallblocks? c

5 Aclothingstorehasadiscountsale.Adressmarkedat$250issoldfor$200.Whatistheratioofthe discounttothemarkedprice?

6 MadeleineandNathaninvest$4500and$2500intoamanagedfund.Whatistheratioof Madeleine’ssharetoNathan’sshare?

7 Therewere80blocksoflandavailableforsaleinanewlandrelease.Afteronemonth32blocks havebeensold.Whatistheratioofsoldblockstothetotalnumberofblocks?

8 Writedown3ratioswhichareequivalentto2:3.

9 Writedown3ratioswhichareequivalentto16:10.

10 Asquarehasasidelengthof6cm.Ifthesidelengthisdoubled,whatistheratiooftheareaofthe originalsquaretothatofthelargersquare?

11 Example1 Expresseachratioinsimplestform.

Findthemissingnumberintheseequivalentratios.

Expresseachratioinsimplestform.

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

15 SamanthaandMathildeownarestaurant.Samanthagets 3 5 oftheprofitsandMathildereceivesthe remainder.

a Whatistheratioofprofits?

b Lastweektheprofitwas$2250.HowmuchdoesMathildereceive?

c Thisweektheprofitis$2900.HowmuchdoesSamanthareceive?

16 Expresseachofthefollowingratiosintheform x :1.

17 Expresseachofthefollowingratiosintheform1: y

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

Lollyprices

Jellybabies

Liquorice

Skittles

$5.95perkg

$6.95perkg

$11.90perkg

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

1B

Thecapture-recapturetechnique

Learningintentions

• Tounderstandhowthecapture-recapturetechniqueworks

• Tousethecapture-recapturetechniquetoestimateanunknownpopulation Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• QuestionsinvolvingthistechniqueregularlyappearinHSCExaminations

The‘capture-recapture’techniqueisanapplicationofratio.Itissometimesusedtoestimateanimal populations.Thismethodcanalsobeusedtoestimatethenumberofpeopleneedingparticularservices.The techniqueworksbyfirstcapturingarandomsampleofthepopulation.Thisinitialsampleistaggedandthen released.Atalatertimeasecondsampleiscaptured,andthenumberofrecapturedortaggedmembersare recorded.

Thecapture-recapturetechnique

Thisdiagramexplainsthetechnique.Itisalsoausefultemplateforsolvingproblems.

Inthecapture:

Total population to be estimated : Tagged animals

Intherecapture:

Total captured : Tagged animals

Thetworatiosareassumedtobeequal.

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

Example2

Usingthecapture-recapturetechnique

Adamisabiologistwhoisestimatingthepopulationofcarpinalake.Herandomlycapturesandtags 120carp.Twomonthslaterhesamples80carpandfindshehasrecaptured5carphehadtagged. Estimatethenumberofcarpinthelake.

Solution

P :120

80:5

P 80 = 120 5

Explanation

Let P bethepopulationtobeestimated.120of thesecarpweretagged.

Intherecapture,80carpwerecaught,ofwhich5 weretagged.

Formanequationfromthetworatiosusing fractions.

80 × 80 = 120 5 × 80

P = 1920

Thereareapproximately1920carpinthelake.

Nowyoutry

Multiplybothsidesby80togivetheestimate.

Usethediagramtocheckthattheanswerseems reasonable: 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 80 rabbitsandthenreleasesthem.Aweeklaterhecatches70rabbitsintheparkandfindsthat10of themaretagged.EstimatethenumberofrabbitsintheparkforPatrick.

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

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

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

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

DEVELOPING

6 Todeterminethenumberoffrogsinapond,20frogsarecaughtandtaggedandthenreleasedinto thepond.Twodayslater10frogsarecaughtofwhich4havetags.Estimatethenumberoffrogsin thepond.

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

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

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. Each persongetshalfofthemoney.

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)

Example3

Dividingaquantityinagivenratio

MikhailandIlyaweregiven$450toshareintheratio4:5.Howmuchdideachget?

Addthetwopartsoftheratio. Changetheratiointofractions.Multiplyeach fractionby$450todetermineeachshare.

Mikhailgot$200,Ilyagot$250

Nowyoutry

Divide$1000intheratio5:3.

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

20 × 6000 = 2100

Xiagot$1500,Yuigot$2400,Zigot$2100

Nowyoutry

Divide$1000intheratio1:3:4.

Exercise1C

Explanation

Addthethreepartsoftheratio. Changetheratiointofractions.Multiplyeach fractionby$6000todetermineeachshare.

Checkthattheanswerisreasonable. Dotheyaddto$6000?

BUILDING

1 Example3 Calculatehowmucheachpersonreceivesif$100issharedinthefollowingratios.

2 Divide240intothefollowingratios.

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3 Shareeachamountintheratiogiven.

a $20intheratio4:1

b $20intheratio7:3

c $15intheratio1:2

d 77drinksintheratio3:4

e 100lolliesintheratio7:13

f 45kgintheratio4:5

g 160booksintheratio5:3

h 360pencilsintheratio2:7

i 50gintheratio1:3

j 60kmintheratio8:7

4 Abagof500gramsofchocolatesisdividedintotheratio7:3.Whatisthemassofthesmaller amount?

5 Ataconcerttherewere7girlsforevery5boys.Howmanygirlswereintheaudienceof8616?

6 Example4 Hazel,PreetishandAndrewinvestinabusinessintheratio6:5:1.Thetotalamountinvestedis $240000.Howmuchwasinvestedbythefollowingpeople?

Hazel a Preetish b Andrew c

Theratioofresidentialareatoparksinalocalcommunityis17:3.Thetotalareaofthelocal communityis40km2.Whatistheareaofparks?

8 Hayleyis15yearsoldandherbrotheris5yearsyounger.If$200issharedbetweenthemintheratio oftheirages,howmuchwillHayleyreceive?

Incountrytownacensusshowedthattherewere5adultstoevery7children.Ifthepopulationofthe townwas7200,howmanychildrenlivedthere?

DEVELOPING

Apunchismadefrompineapplejuice,lemonadeandorangejuiceintheratio5:3:2.

a Howmuchlemonadeisneededifonelitreofpineapplejuiceisused?

b Howmuchpineapplejuiceisrequiredtomake15litresofpunch?

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

Inafruitcakerecipetheratioofmixedfruittoflourtosugaris5:3:2.A250gpacketofmixed fruitisusedtomakethecake.Howmuchsugarandflourarerequired?

13 Adeliveryloadof8.5tonnesistobedividedbetweentwostoresintheratio11:6.Howmuchwill eachstorereceive?

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

15 Ajamismadebyadding5partsfruitto4partsofsugar.Howmuchfruitshouldbeaddedto 2 1 2 kilogramsofsugarinmakingthejam?

CHALLENGING

16 Theperimeterofarectangleis42cm.Theratiooflengthtobreadthis5:2.

a Whatisthelength?

b Whatisthebreadth?

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

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

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

20 TheratioofthepopulationsoftownAtotownBis3:8,andtheratiooftownBtotownCis5:2. Ifthetotalpopulationofthethreetownsis92655,findthepopulationofeachtown.

Communicateyourreasoning,understandingandfluency

21 SolvethefollowingproblemandwriteasolutionforJarrahtoconvincehimyouarecorrect. Jarrahhassomefuelforhislawnmowerinacontainer.Theratioofpetroltooilis50:1.Hehas 2.5litresoffuelalreadymixedatthisratio,butheneedsadifferentratioof30:1fortheoutboard motor onhisboat.Howmanymillilitresofoildoesheneedtoaddtothisfuelifhewantstouseall ofitinhisboat?Answercorrecttothenearestmillilitre.

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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)

• nounits,usingaratiosuchas1:100.

Example5

Usingascale

Ascaledrawinghasascaleof1:50.

a Findtheactuallengthifthedrawinglengthis30mm.

b Findthedrawinglengthiftheactuallengthis4.5m.Answertothenearestmillimetere.

Solution

a Drawing length : Actual length 30 mm : × 30

50 × 30 = 1500 × 30 1 : 50 Theactuallengthis1500mm.

Explanation

Writetheratio. Writetheenlargementfactor.

Usetheenlargementfactor. Statethelength.

Continuedonnextpage

b Drawinglength:Actuallength

× 90 × 90 1 : 50 : 4500 mm

1 × 90 = 90

Thedrawinglengthis90mm.

Nowyoutry

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 youranswerinmetres.

2cm a 10mm b

3.4cm c 28mm d 8.5cm e 49mm f

2 Ascaledrawinghasascaleof1:25000.Whatisthedrawinglengthoftheseactuallengths? Expressyouranswerinmillimeters.

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

4 Thescaleonamapis1:1000.Calculatetheactualdistancesifthesearethedistancesonthemap. Expressyouranswerinmetres.

Road20cm a Shops10cm b

Pathway5cm c Parkingarea10mm d Bridge34mm e Park80mm f

5 Thescaleonamapisgivenas1cm = 15km.Whatisthe actualdistanceifthedistanceonthemapis:

a 2.5cm?

b 45cm?

SYDNEY

6 Thescaleonamapis1:5000.Calculatethemapdistancesiftheseareactualdistances.Express youranswerinmillimetres.

7 Thescaleonamapisgivenas1mm = 50m.Ifthedistancebetweentwopointsis350m,whatis themapdistancebetweenthesepoints?

8 Ascaledrawinghasascaleof1:75.Findthe:

a actuallengthifthedrawinglengthis15mm

b drawinglengthiftheactuallengthis3m.

CHALLENGING

9 TheParkesradiotelescopedishhasadiameterof64metres.Theimageshownusesaphotographof thedish.

a Determineascalefortheimage.

b Estimatetheheightofthetopofthedishabovetheground.

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

11 Ascaledrawingofthespaceshuttleisshownhere.Theactual lengthofthespaceshuttleis43metres.

a Whatisthescalefactor?

b Calculatethelengthofthewingspantothenearestmetre.

c Calculatethebreadth(b)oftheshuttletothenearestmetre.

d Whatisthelengthofthenoseoftheshuttletothe nearestmetre?

Communicateyourreasoning,understandingandfluency

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

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Usethisinformationaboveandthephotographofthebridgetoanswerthesequestions,showingall calculations.

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

b Usingthatscale,withthetableassealevel,approximatelyhowtallwillhermodelbein millimetres?

c Thedeckofthebridgeis49metreswide.HowmanymillimetreswillthisbeonSarah’smodel?

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.

Example6

Drawingaplanandelevation

Drawtheplanview,frontelevationandsideelevationof thisobject.

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

Abuildingplanisshownforthegroundfloorofa home.Thescaleis1:150.

a Howmanyinternaldoorsarethere?

b WhatisthemeaningofPWDR?

c Whatisthelengthofthehouse?

d Whataretheinternaldimensionsofthedouble garage?

Solution

Shower:showerwithoutabathtub

Toilet:toiletlocatedonwall

Bathtub:bathtubshowinglocationofdrain

Explanation

a 4internaldoors Pantry,laundry,powderroom,garage.

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b Powderroom Continuedonnextpage

GARAGE

c Drawinglengthis12.6cm

Actuallength = 12.6 × 150cm = 1890cmor18.9m

d Drawinglengthis4.1 × 3.5cm

Actuallength = 4.1 × 150cm = 615cmor6.15m

Actualbreath = 3.5 × 150cm = 525cmor5.25m

Dimensionsare6.15 × 5.25m.

Nowyoutry

UsingtheplaninExample 7:

a howwideisthehouseatitswidestpoint,inmetres?

Useyourrulertomeasurethelengthofthehouse.

Useyourrulertomeasurethedimensionsofthe garage.

b whataretheinternaldimensions(inmetres)ofthesittingroom?

Exercise1E

1 Example6 Drawtheplan,frontelevationandsideelevationfortheseobjects.

BUILDING

2 Whataretheobjectsrepresentedbytheseviews?

3 Drawtheplanview,frontelevationandsideelevationfortheseobjects.

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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.

8 Example7 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.

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.

Thefrontelevationofahouseisshown.

a Whatistheheightofthefirststorey?Answerinmetres.

b Whatistheheightofthegarage?Answerinmetres.

c Whatistheangleofthepitchoftheroof?

d Howmanywindowsareatthefrontofthehouse?

Abuildingplanisshown below.Thehouselength is20mandthebreadthis 18m.

a Whatarethedimensions ofthelivingroom? Answerinmetres.

b Whatistheareaofthe livingroom?Answer correcttothenearest squaremetre.

c Thehouseandveranda werebuiltonaconcrete slab200mmthick.How muchconcretewasused? Answercorrecttothe nearestcubicmetre.

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Abuildingplanisshownforahome.Thehouselengthis 24m(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 dimensionsofthefamilyroomare5.5metresby6.0metres.

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 Lengthofswimmingpoolis3.1cm.

Scale3.1cmto50m(or5000cm).

Scaleisapproximately1:1613.

b Boatlengthis2.0cm.

Boatlength = 2.0 × 1613 = 3226cm = 32m

Lengthoftheboatis32metres,correcttothe nearestmetre.

Explanation

Nowyoutry

Thenextboattoarriveatthejettyis3.5cmlonginthephotographinExample 8.Howlongistheactual boat,correcttothenearestmetre?

Area

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

Area

Toestimatetheareaofland,divideitintosquaregridsandcountthenumberofsquares. Tocalculatetheexactareaofland,usetheappropriateformula.

Example9

Calculatingtheareaofland

AwarehouseisshownonGoogleEarth.Theredscalebarrepresents42m attheground.

a Calculatethelengthsofthesidesofthepentagonoccupiedbythe warehousebuildingusingthescale.Answercorrecttothenearesttenth ofametre.

b Calculatetheareaoflandoccupiedbythewarehousebuilding.Answer correcttothenearestsquaremetre.

Solution

a Scalebaris1.2cmrepresents42cm Scaleis1cmto35m

AB = 4.2 × 35 = 147m

BC = 1.2 × 35 = 42m

CE = 1.6 × 35 = 56m

EF = 3.2 × 35 = 112m

AF = 2.5 × 35 = 87.5m

Explanation

Areaofwarehouseis12066m2

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:

= 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

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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 = 6,and 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 BUILDING

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?

g Calculatetheareaofthelandmarked ABC.Assumearightangleat C

Example9 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,correcttothenearestmetre orsquaremetre.

a Whatistheradiusofbuilding A?

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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?

Adiagramhasascaleof1:100.Whatistheactualareaoftheseshapes,giventhefollowingdrawing lengths?Expressyouranswercorrecttothenearestsquaremetre.

a Squarewithasidelengthof10mm

b Rectanglewithalengthof2cmandabreadthof3cm

c Right-angledtrianglewithabaselengthof25mmandperpendicularheightof40mm

d Circlewitharadiusof20mm

e Semicirclewithadiameterof50mm

CHALLENGING

Ashatooktwomeasurementsat120mintervalsacrossthe irregular-shapedreservoirinthediagram.Themeasurements were100mand80m.

a Estimatethesurfaceareaofthereservoir.Answercorrectto thenearestsquaremetre.

b Ashameasuredthedepthofwaterandfoundittobe5m everywhere.Estimatethevolumeofthereservoir.Answer correcttothenearestcubicmetre.

11 150mmofrainfellonthereservoirinQuestion 10,whichhasverticalbankssonowaterflowsinor outandthedepthdoesnotvary.

a Whatisthevolumeofwaterinthereservoiraftertherain?

b Aftertherain,thefarmthatdependsonthereservoirneedsanextra15000cubicmetresofwater forthenextmonth.Howmuchrainneedstofallinthattimesothatwhentheextrawaterhas beenpumpedout,thewaterlevelinthereservoirremainsthesame?Answercorrecttothenearest millimetere.

Communicateyourreasoning,understandingandfluency

12 Mauriceusedthreeapplicationsofthetrapezoidalruletoestimate theareaofthissectionofland.Explainwhyhisestimateislarger thantheactualareaoftheland.

1G Ratesandbestbuys

Learningintentions

• Tounderstandthedifferencebetweenratiosandrates

• Toconvertratesbetweendifferentunits

• Tosolveproblemsinvolvingratesinavarietyofpracticalcontexts

• Tosolveproblemsinvolvingdistance,speedandtime 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 $2.99peronekilogramand80km/hrepresents 80kilometresperonehour.

Weareconstantlyinterestedinratesofchangeandhowthings changeoveraperiodoftime.Therearemanyexamplesofrates suchas:

• Runningrate:Yourrunningpaceinmetrespersecond.

• Typingrate:Yourtypingspeedinwordsperminute.

• Wagerate:Theamountofmoneyyouarepaidperhour.

• Fuelconsumption:Litresper100kmorkmperlitresarebothused.

Theunitarymethod

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Usingtheunitarymethod

• Growthrate:Theaveragegrowthrateofachildfrom0to15yearsofage.

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

Usedivisiontogetdownto1,thenmultiplytofindtherequiredamount.

Theunitarymethodinvolvesfindingoneunitofaquantitybydivision.Thisresultisthenmultipliedtosolve theproblem.

Example11

Usingtheunitarymethodforfuelconsumption

Acarcantravel360kmwith30litresofpetrol.Atthisrate,howfarcanittravelwith7litresofpetrol?

Solution

30 litres for 360 km

÷ 30 × 7 ÷ 30 × 7

1 litre for 12 km

7 litres for 84 km

Thecarcantravel84km.

Nowyoutry

Explanation

Writethegiveninformation. Usedivisiontogetdownto1litre. Multiplyby7togettherequiredamount.

Answerthequestionandcheckitisreasonable (itshouldbeconsiderablylessthan360km).

a Acartravelled600kmwith50litresofpetrol.Atthisrate,howfarcoulditdrivewith37litres ofpetrol?

b Acartravelled600kmwith50litresofpetrol.Atthisrate,howmanylitresofpetrolwoulditneedfor a1000kmjourney?

Example12

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-packarecheaper.

Compare24individualdrinkstothe24-pack.

Speed

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

Speed

S = D T or T = D S or D = S × T

D = Distance

S = Speed

T = Time

Example13

‘Roadsign’ontherightisusedto remembertheformulas.Hidethe requiredquantitytodeterminethe formula.

Solvingproblemsinvolvingspeed

a Findtheaveragespeedofacarthattravels341kmin5hours.

b Howlongwillittakeavehicletotravel294kmataspeedof56km/h?

Solution

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

b T = D S = 294 56 = 5.25hor5h15min

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Nowyoutry

Explanation

Writetheformula.

Substitute341for D and5for T intotheformula.

Evaluate.

Writetheformula.

Substitute294for D and56for S intotheformula.

Evaluateandexpresstheanswercorrecttothe nearesthour.

UsainBoltran100metresin9.58seconds.Whatwashisaveragespeedinmetrespersecond,correctto onedecimalplace?

Convertingbetweenrates

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

Forexample,youcould:

1 convertthenumberofkilometrestometres,and; 2 convertthenumberofhourstoseconds,then; 3 reducetheratetosimplestform,inthesamewayyoureducearatiotosimplestform.

However,theaimisforthenumberofsecondstobe1,becausem/smeans‘metresin1second’.

Note: Some peopleuseruleslike‘dividethespeedby3.6’.Theseruleshavelimitedapplicability.

Example14

Convertingbetweenrates

Convert100km/hintometrespersecond.

Solution

÷ 3600

100kmin1hour

100000min60minutes

100 000 m in 3600 seconds

_______ m in 1 second

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

Nowyoutry

÷ 3600

Explanation

Writetherategiven. Convert100kmtometresand1hourtominutes.

Convertfromminutestoseconds.

Reduce,justlikearatio,totheform___:1 Inthiscase,dividebothnumbersby3600.

Ahippopotamuscanrunataspeedof30km/hovershortdistances.

a At thisspeed,howmanymetrescanitruninonesecond?Giveyouranswercorrecttoonedecimal place.

b Howmanysecondswouldthehippopotamustaketorun100metres?Giveyouranswercorrecttothe nearestsecond.

Exercise1G

1 Example11 Converttotherateshown.

$100in4hisarateof$ /h a 240min20sisarateof m/s b 700Lin10hisarateof L/h c $39in12hisarateof$ /h d

BUILDING

UNCORRECTEDSAMPLEPAGES

$1.20for2kgisarateof$ /kg e 630kmin60Lisarateof km/L f 1200revin4minisarateof rev/min g Ariseof20◦ in4hisarateof ◦/h h

2 Expresseachrateinsimplestformusingtheratesshowninthebrackets.

300kmon60L[kmperL] a 15min10s[mpers] b $640for5m[$perm] c 56Lin0.5min[Lpermin] d 78mgfor13g[mgperg] e 196gfor14L[gperL] f 20gfor8m2 [gperm2] g 75mLfor5min[mLpermin] h

Usetheratesprovidedtoanswerthefollowingquestions.

a Costofapplesis$2.50/kg.Whatisthecostof5kg?

b Carpet costs$28/m2 . Whatisthecostfor7m2?

c Costsavingsare$35/day.Howmuchissavedin5days?

d Cost ofachemicalis$65/100mL.Whatisthecostof300mL?

e Cost ofmushroomsis$5.80/kg.Whatisthecostof 1 2 kg?

f Distancetravelledis1.2km/min.Whatisthedistancetravelledin30minutes?

g Concentration ofachemicalis3mL/L.HowmanymLofthechemicalisneededfor4L?

h Concentration ofadrugis2mL/g.HowmanymLisneededfor10g?

4 Example12 Whichisthebestbuy:10litresfor$5or12litresfor$7.40?

5 Example13

Usetheinformationprovidedonspeedtoanswerthefollowingquestions.

a Iwalkat5km/h.HowfarcanIwalkin4hours?

b A cartravelsat80km/h.Howfarwillittravelin2.5hours?

c A planeistravellingat600km/h.Howfarwillittravelin30minutes?

d A traintook7hourstotravel665km.Whatwasitsaveragespeed?

e Ryderrunsa42.4kmmarathonin2hours30minutes.Calculatehisaveragespeed.

f Aspacecrafttravelsat1700km/hforadistanceof238000km.Howmanyhoursdidittake?

6 Example14 Converteachratetotheunitsshowninthebrackets. 39240m/min[m/s] a 2m/s[cm/

/s[m/s] e

/mL[mg/mL] g

7 Miaearns$37.50perhourworkinginacafe.

/s[km/s] f

/kL[mL/kL] h

a HowmuchdoesMiaearnforworkinga9-hourday?

b HowmanyhoursdoesMiaworktoearn$1200?

c WhatisMia’sannualincomeifsheworks40hoursaweek?Assumesheworks52weeks intheyear.

8 Adeliverydriverdelivers1parcelonaverageevery20minutes.Howmanyhoursdoesittaketo deliver18parcels?

9 Waterisdrippingfromatapatarateof5L/h.Howmuchwaterwillleakinoneday?

10 A cricketteamscoresrunsatarateof5runs/overina match. Howmanyoversdidittaketoscore90runs?

Abulldozerismovingsoilatarateof22t/h.Howlongwillittaketomove55t?

IfLeocanmarchat7km/h,howfarcanhemarchin2.5hours?

Edwardsaves$40/week,howlongshouldittaketosave$1000?

14 Alexandrajogs100metresin20seconds.Howmanysecondswouldittakehertojogonekilometre?

Acartravelsatarateof50metreseachsecond.Howmanykilometresdoesittravelin: oneminute? a onehour? b

Convertthefollowingspeedstometrespersecond.Answercorrecttothenearestwholenumber. 60km/h a 260km/h b

17 Anathleteruns100metresin10seconds.Whatishisaveragespeedinkilometresperhour?

DEVELOPING

18 Naturalgasischargedatarateof1.4570centsperMJ.

a Findthechargefor12560MJofnaturalgas.Answertothenearestdollar.

b Thechargefornaturalgaswas$160.27.Howmanymegajouleswereused?

19 Kirra’scouncilrateis$2915p.a.forlandvaluedat$265000.Lucyhasacouncilrateof$3186on landworth$295000fromanothercouncil.

a WhatisKirra’scouncilchargeasarateof$/$1000valuation?

b What isLucy’scouncilchargeasarateof$/$1000valuation?

20 Fatima’scaruses9litresofpetroltotravel100kilometres.Petrolcosts$1.50perlitre.

a Whatisthecostoftravelling100kilometres?

b Howfarcanshedriveusing$50worthofpetrol?Answertothenearestkilometre. CHALLENGING

21 Apopularsoftdrinkissoldintheseways:

• apackof12300mLbottlesfor$12

• apackoffour330mLbottlesfor$8.50

• a600mLbottlefor$4

• a1.25Lbottlefor$3.25

• a2Lbottlefor$4.20. Whichisthebestbuy?

22 Amotorbikeismovingatasteadyspeed. Whenthespeedis90km/hthebike consumes 5litresofpetrolforevery 100kilometrestravelled.

a Thepetroltankholds30litres.How manykilometrescanthebiketravel onafulltankofpetrolwhenitsspeed is90km/h?

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

23 Aplanetravellednon-stopfromLosAngelestoSydney,whichisadistanceof12057kilometres,in 13hoursand30minutes.Theplanestartedwith180kilolitresoffuel,andonlandinghadenough fueltoflyanother45minutes.

a Whatwastheplane’saveragespeedinkilometresperhour?Answercorrecttothenearestwhole number.

b Howmuchfuelwasused?Answercorrecttothenearestkilolitre.

Communicateyourreasoning,understandingandfluency

24 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.

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c Calculatethecostofoperatingthecar,indollarsper100km.

d Calculatetheaveragedistancetravelled,inkilometrespermonth.

e Basedonthese8months,howmuchwillthecarcosthimforthewholeyear?

1H 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.

Example15

Calculatingspeedfromadistance-timegraph

Thisdistance-timegraphshowsapersontravellingfromPoint A toPoint B ataconstantspeed.

Howlongdidthejourneytake?

Solution

Explanation

a 5hours 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?

Example16

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

÷ 5 ÷ 5

__ km in 1 hour

Heraveragespeedwas60 ÷ 5 = 12km/h.

Nowyoutry

Usingthedistance-timegraphinExample 16:

Setitupasarate. Simplifyitlikearatio.

a whatwastheaveragespeedforthelastpartofthejourney,correcttoonedecimalplace?

b excludingthebreaks,whatwasMira’saveragespeedforthewholejourney?

Exercise1H BUILDING

1 Writeashortdescriptionaboutwhatishappeningateachpoint(A,B,C to F)alongAman’sjourney.

2 Drawareasonablyaccuratedistance-timegraphwhichincludesbothofthefollowingjourneys.

a Enzolefthomeandwalked2kmin30minutes,thenrestedfor30minutes,thenwalkedhome againin1hour.

b Marialeftthesamehomeandwalked2kmin45minutesthenturnedandwalkedhomein1hour. DEVELOPING

3 Example15 Basedonthisgraph,whatwastheaveragespeedduringthe journey?

4 Example16 Thisdistance-timegraphshowsoneofJane’strainingrides.

a Describewhathappenedduringthefirst90minutes, includingthedistancetravelledandthetimetaken.

b HowfardidJaneridealtogether?

c Whatwasheraveragespeedforthesecondpartofthe journey,afterthefirstbreak?

d Whatwastheaveragespeedforthelastpartofthe journey?

e Includingthebreaks,whatwasheraveragespeedforthe wholejourney,correcttoonedecimalplace?

f Excludingthebreaks,whatwasheraveragespeedforthe wholejourney?

Communicateyourreasoning,understandingandfluency

5 Studythetravelgraphbelow.Itdescribesthemovementsofasurferoveran8-minutetimeframe, starting50metresfromthebeach.

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Distance from beach

Time

Writeaparagraphaboutthemovementsofthesurfer,includingestimatesofthetimespentmovingor beingstationary.

1I Energyconsumptionrates

Learningintentions

• Torecognise,understandandconvertbetweenunitsofenergy

• Toapplyunitsofenergytocalculatekilowatthours

• Tosolveproblemsinvolvingtheconsumptionofenergy Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• ThisapplicationofratesmaynothavebeentreatedinYears7to11

• QuestionsinvolvingenergyconsumptionoccasionallyappearinHSCExaminations

Energyisthecapacitytodowork.Energyexistsinnumerousforms,suchasheat.Thejoule(symbolJ)isthe unitofenergyusedbytheInternationalSystemofUnits(SI).Heatenergysuchasthatproducedbyburning naturalgasinthehomeisusuallymeasuredinmegajoules(onemillionjoules,symbolMJ).

Power

Poweristherateatwhichenergyisgeneratedorconsumed.ThewattistheInternationalSystemofUnits (SI)unitofpowerandisequaltoonejoulepersecond.ThesymbolforthewattisW.Asforthejoule,the standardSIprefixessuchasmilli,kiloandmegaarethenaddedandcommonlyusedtomeasurepower.

Electricalenergy

Thejouleisnotapracticalunitformeasuringelectricalenergyinmost settings.Itismorehelpfultothinkofelectricalenergyintermsofthe powerdrawnbyanelectricaldeviceandthelengthoftimethedevice isinuse.Forexample,whenadevicewithapowerratingof100Wis turnedonforonehour,theamountofenergyusedis100watt-hours (WhorW-h).Thisisthesameamountofenergya50Wdevicewould usein2hours.

Thekilowatt-hour(kWhorkW-h)iscommonlyusedtomeasureelectrical energyinhouseholdelectricitymeters.

Itrepresentstheamountofelectricalenergywhena1000Wpowerloadisdrawnforonehour.Itistheresult ofmultiplyingpowerinkilowattsandtimeinhours.Notetheconversion1kWh = 3.6MJ. Consumptionisarateexpressedasanamountovertime.Electricalenergyconsumptioncanbeexpressed inkilowatt-hoursormegawatt-hoursperunitoftime,ormegajoulesperunitoftime.Theaverageannual energyuseperhouseholdinAustraliais11MWh/yor3MJ/y,whichalsoequatestoabouteighttonnesof CO2 emissions.

Energyconsumption

Energyconsumptionistheamountofenergyconsumedperunitoftime.

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Energycosts

1 Findtheenergy(kWh)bymultiplyingthepowerrating(kW)bythehoursused.

2 Findthecostbymultiplyingtheenergybytheelectricitycharge.

Energyratingofappliances

InAustraliaanenergyratinglabelisprovidedforappliancessuchasfridges,TVs,washers,dryersandair conditioners.Itallowsconsumerstocomparetheenergyefficiencyofsimilarproducts.Energyratinglabels all haveasimplestarrating.Themorestarsonthelabel,themoreenergyefficienttheappliance.Theenergy consumption figureisthenumberintheredbox.Itindicatestheamountofelectricity(kWh)theappliance typicallyusestoruninayear.Thelowerthenumberthelesstheappliancewillcosttorun.Therunning costoftheapplianceiscalculatedbymultiplyingtheenergyconsumptionfigurebytheelectricityprice rate.Electricitysuppliersusuallygivepricesperkilowatt-hour.ThescreenshotbelowisfromtheAustralian Governmentenergyratingwebsite(www.energyrating.gov.au/calculator).

Example17

Calculatingthecostofrunningappliances

Findthecostofrunningtheseappliancesiftheaveragepeakrateforelectricityis$0.21/kWh.

a A 3.5kWairconditionerforeighthours.

b A200Wtelevisionfor20hours.

Solution

a Electricity = 3.5 × 8 = 28kWh

Cost = 28 × 0.21 = $5.88

b Electricity = 0.2 × 20 = 4kWh

Cost = 4 × 0.21 = $0.84

Explanation

Determinetheenergy(kWh)bymultiplyingthe powerrating(kW)bythehoursused.

Determinethecostbymultiplyingtheenergyby theelectricitycharge.

Determinetheenergy(kWh)bymultiplyingthe powerrating(kW)bythehoursused.

Determinethecostbymultiplyingtheenergyby theelectricitycharge.

Nowyoutry

Determinethecostofrunningthefollowingappliances,giventheaveragerateforelectricityis $0.17perkWh.

a Adishwasherwithanenergyconsumptionof670kWhperyear.

b A2.4kWfanheaterforfivehours.

c A20WLEDlightbulbforayear.

Energy-efficienthousing

BASIXorBuildingSustainabilityIndexisaschemetoregulatetheenergyefficiencyofresidentialbuildings. It isanonlineprogramtoassessandcompareahouseorunitdesignwithenergyandwatertargets.

BASIXusesinformationsuchassitelocation,housesize,buildingmaterials,water,thermalcomfortand energy.Sustainablehousesfeaturerainwatertanks,efficientshowerheads,solarhotwater,performance glazing andenergy-efficientlighting.

Example18

Calculatingelectricitycosts

ChristianisinstallingasolarPV(photovoltaic)system.Hisaveragedaily consumptionis21.5kWh.ItispredictedthesolarPVsystemwillmeet thisneedandexport6.3kWhofenergydailytothegrid.Christian’s energyretailercharges$0.2167perkWhandpays$0.08perkWhfor energy.WhatistheexpectedsavingfromthesolarPVsystemeachday?

Solution

Dailyconsumption = 21.5 × 0.2167 = $4.65905

Exported = 6.3 × 0.08 = $0.504

Saving = $4.65905 + $0.504 = $5.16

Nowyoutry

Explanation

Multiplytheenergybyelectricitychargefordaily consumption.

Multiplytheenergybyelectricitychargethatis exportedtothegrid.

Addthetwoamountstodeterminethesaving.

HowmuchmoneywouldChristianhavesavedthatdayifthedailyconsumptionwasreducedby20%and alltheenergysavedwasexportedtothegrid? Exercise1I BUILDING

Example17 TheNguyenfamilyhasaclothesdryerwitha3.3kWrating.

a Howmanykilowattsofelectricitydoesituseinaweekifitisswitchedonforanaverageof 2hoursperday?

b Whatisthecostofrunningthedryerforaweekifelectricityis$0.2819perkWh?

2 Adishwasherhasanenergyconsumptionfigureof 250kWhperyear.Calculatethecostofrunningthe dishwasherforayearusingthefollowingelectricity tariffs.

a $0.1985/kWh

b $0.2019/kWh

c $0.2754/kWh

3 Marcususesa25Wceilingfaneveryhouroftheday.

a Howmuchenergydoestheceilingfanuseduringtheweek?

b Whatisthecostofusingtheceilingfanforaweekifelectricityis$0.2999perkWh?

4 Aliceusesanovenwitharatingof1200Wforanaverageof7hoursperweek.Findthecostofthe electricityfortheovenatarateof24.11centsperkilowatt-hour.

5 Lukereceivedhisnaturalgasaccount.

a Findthecostofthefirst6200MJ.Answertothenearestcent.

b Findthecostofthenext12400MJ.Answertothenearestcent.

c Whatisthetotalcharge?Answertothenearestcent.

6 Hamishusesa12Wlaptopforatotalof10hourseachday.Heischargedatarateof25.07centsper kilowatt-hour.

a Howmanykilowatt-hoursareusedbythelaptopduringtheyear?

b Whatisthecostofusingthelaptopforayear?Answertothenearestcent.

7 Example18 Eve’sdailyconsumptionofelectricityis19.5kWh.ShehasasolarPVsystemtomeetthisneed andexport7.5kWhofenergytothegrid.Eve’senergyretailercharges$0.2425perkWhandpays $0.10perkWhforenergy.HowmuchisthesolarPVsystemsavingEveeachday?

8 ThepiechartshowsthesourcesofbenefittoNSWofBASIX.ThetotalbenefitofBASIXforNSW to2050isestimatedtobe$294000000.

a WhatpercentageofthebenefitsofBASIXwasenvironmental?

b WhatpercentageofthebenefitsofBASIXwashouseholdenergy?

c WhatisthehouseholdenergysavingofBASIXforNSWto2050?

d WhatisthehouseholdwatersavingofBASIXforNSWto2050?

e WhatistheenvironmentalsavingofBASIXforNSWto2050?

9 Thetableshowsthetypesofheating systemsinstalledinnewhousesin oneyear.

a Whatisthetotalnumberofnew houses?

b Copyandcompletethetableby calculatingthepercentageofeach heatingsystem,correcttothe nearestpercent.

c Whichisthemostpopularheating system?

Noactiveheatingsystem 6332

Gasfixedfluedheater 2333

Gashydronicsystem 327

1-phaseair-conditioning 3357

3-phaseair-conditioning 6867

Air-conditioningducted 872

Woodheating 1635

d Howmanyhousesinstalledair-conditioning?

e Constructapiechartshowingtheproportionsofeachheatingsystem.

10 Thepowerratingofsomeelectricalappliancesis shownbelow.

a Howmuchenergyisusedbyaclothesdryerfor6 hours?Answerinkilowatt-hours.

b Howmuchenergyisusedbyanairconditioner forhalfanhour?Answerinwatt-hours.

c Howmanyhourswasthedishwasherusedifthe costofoperatingthedishwasherwas$13.44?The costofelectricityis$0.2625perkilowatt-hour.

DEVELOPING

d Howmanyhourswastherefrigeratorusedifthe costofoperatingtherefrigeratorwas$1.38?Thecostofelectricityis23centsperkilowatt-hour.

11 Anenergycompanychargesforgasovera3-monthperiodareshownbelow.

a Alexisused4400MJofgasinthisperiod.Whatisthecostofthisgasusage?

b Matthewused5450MJofgasinthisperiod.Whatisthecostofthisgasusage?Answercorrectto thenearestcent.

c WhatpercentageofMatthew’sgasusagewaschargedatthelowerrate?Answercorrecttoone decimalplace.

d Stellareceivedagasbillfor$117.18.Howmuchgasdidsheuseinthisperiod?

e Clairehasdecidedtoreduceherenergybills.Shehasatargetof$70forgas.Whatisthe maximumnumberofMJsheisallowedinthisperiod?Answercorrecttothenearestwhole number.

f Jakeused6220MJofgasinthisperiod.Thegaschargesareincreasingby5%inthenext quarter.However,Jakehaspurchasedanewgasheaterwitha5-starratingthatwillreducehis consumptionby1000MJ.WhatisJake’sexpectedbillnextquarter?

Communicateyourreasoning,understandingandfluency

12 Answerthefollowingquestionsusingthiselectricityaccount,fromtheyear2017.

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a Whenwasthiselectricityaccountissued?

b Whatisthetotalamountdue?

c Whatistheduedate?

d HowmuchGSTwascharged?

e Whatisthechargeforthepeakuseofelectricity?

f Howmuchhasthebillincreasedfromthelastbill?

g Expressthecostincreaseasapercentageofthelastbill.Answercorrecttotwodecimalplaces.

h Thefiguresintheaccountarefromtheyear2017.Howmuchwouldthiscostinthecurrentyear? Youcouldaskfriendsandfamilytoshowyoutheirelectricityinvoices.

1J Fuelconsumptionrates

Learningintentions

• Tousefuelconsumptionratestocomparetheefficiencyofvehicles

• Tosolveproblemsinvolvingfuelconsumptionrates

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.

Example19

Calculatingfuelconsumptionrates

Acartravelled850kilometresusing78.2litresofpetrol.

Whatwasthefuelconsumptionrate,in:

a litresper100kilometres?

b kilometresperlitre,correcttoonedecimalplace?

Nowyoutry

Acartravelled500kilometresusing40litresofpetrol.

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

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

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Whatwasthefuelconsumptionrate,in: litresper100kilometres? a kilometresperlitre? b

Exercise1J

1 Example19 Findthefuelconsumptionforeachofthefollowing,inL/100kmandkm/L.

a Lincoln’scaruses45.5litresofpetroltotravel520km.

b Asmallcaruses36.9litresofpetroltotravel600km.

c Gemma’ssedanuses55.1litresofLPGtotravel950km.

d Asportscartravelled250kmusing28.5litresofpetrol.

e Anthony’smotorbikeuses167.5litresofLPGtotravel2500km.

f 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.

BUILDING

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d Atripof658kmfromMelbournetoCanberra.

e Atripof732kmfromAdelaidetoMelbourne.

f Atripof309kmfromCanberratoSydney.

g Atripof3429kmfromBrisbanetoDarwin.

3 Charlietravels45kmtoworkand45kmfromworkeachday.

a Howmanykilometresdotheytraveltoandfromworkina5-dayworkingweek?

b Charliedrivesafour-wheeldrivewithafuelconsumptionof8L/100kmtoandfromwork. HowmanylitresofpetroldoesCharlieusetravellingtoandfromwork?Answercorrecttoone decimalplace.

c WhatisCharlie’spetrolbillforworkifpetrolcosts$1.20perlitre?

4 AfamilycarusesLPGatarateof15L/100kmandthegastankholds72litres.Howfarcanittravel on atankofLPG?

5 Kaidrivesatruckwithapetrolconsumptionof16L/100kmandapetroltankthatholds90litres.

He isplanningatripfromMoorebanktoMelbourne.ThedistancefromMoorebanktoMelbourne is840km.KaifilledupthepetroltankatMoorebank.Howmanymoretimeswillheneedtofillhis tankbeforearrivingatMelbourne?Givereasonsforyouranswer.

DEVELOPING

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

a Howmuchwillthetripcostviathehighway?

b Howmuchwillthetripcostavoidingthehighway?

c Howmuchmoneyissavedbytravellingviathehighway?

7 AustinownsanSUVwithafuelconsumptionof10.9L/100kminthecityand8.4L/100kminthe country.Austintravels12000kmperyearinthecityand20000kmperyearinthecountry.The averagecostofpetrolis$1.48perlitreinthecityand12centshigherinthecountry.

a Findthecostofpetroltodriveinthecityfortheyear.

b Findthecostofpetroltodriveinthecountryfortheyear.

c WhatisthetotalcostofpetrolforAustininoneyear?

d WhatisthetotalcostofpetrolforAustininoneyeariftheaveragecostofpetrolincreasedto $2.00inthecity?

8 EdenbuysafamilycarthatusesULPwithafuelconsumptionof9.9litres/100km.Tobybuysthe LPG versionofthefamilycarwithafuelconsumption14.8litres/100km.BothEdenandToby average450kminaweekinthesameconditions.TheaveragepriceofULPis$1.50perlitreand LPGis$0.85perlitre.

a HowmanylitresoffuelareusedbyEdeninaweek?

b HowmanylitresoffuelareusedbyTobyinaweek?

c Calculateeachcar’syearlyconsumptionoffuel.

d WhatisEden’syearlyfuelbill?

e WhatisToby’syearlyfuelbill?

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

g ResearchthecurrentfuelpricesusingULPandLPG.Howlongwillittakeforthefuelsavingto exceedtheinitialcosts?

9 Conductsomeresearchaboutthefuelconsumptionofsmall,energyefficientcars,andlargercars, such asSUVs.

a Whatisthelowestfuelconsumptionyoucanfind?Whatisthehighest?

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

c HowmanykilometresdoestheaverageAustraliandriveperyear?

d Calculatethecostofpetrolperyearforthecarsyouchose.Istheremuchdifference?

e Write upacomparisonofasmallefficientcarversusalargegas-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

Energyrate

Fuelconsumption

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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.

Energyistherateofpowerperunitoftime(kilowatthour-kWh) Runningcostoftheapplianceiscalculatedbymultiplyingtheenergy consumptionbytheelectricitycharge.

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.Ithasuniformdepthof 1.5metres.

Thescaleis1:500.

Estimatetheperimeter,inmetres,byassumingthecurveis asemicircle. a

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Estimatethearea,insquaremetres,usingthetwoapplicationsofthe trapezoidalrule. b

Estimatethenumberoflitresofwaterrequiredtofillthepond. c Judgingbytheshape,isthisanunderestimateoroverestimateofthevolume? d

1G 8

Icanusetheunitarymethodtosolveproblemsinvolvingrates. Acakefor6peoplerequires4eggs.Howmanyeggswillberequiredforacakebig enoughtoserve15people?

1G 9

Icandeterminewhichproductisthebestbuy. e.g.Adrinkcomesintwosizes:

• 1.25litresfor$3.25

• 2litresfor$4.20. Whichisthebestbuy?

1G 10

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

1G 11

Icanconvertaratetootherunitsofmeasurement.

e.g.Adrippingtapwastes25millilitresofwatereveryminute.Convertthisrateto litresperhour.

1H 12 Icaninterpretandanalyseadistance-timegraph.

e.g.Considerthisdistance-timegraph.

Whatwastheaveragespeedonthefirstpartofthejourney?

Whatwastheaveragespeedonthelastpartofthejourney,correcttoone decimalplace?

Whatwastheaveragespeedfortheentirejourney,includingtherestbreak?

Whatwastheaveragespeedfortheentirejourney,withouttherestbreak, correcttoonedecimalplace?

1I 13 Icancalculatethecostofrunningelectricalappliances.

e.g.Thepowerratingforelectrical appliancesisshowninthetable.

Howmuchenergydoesalaptop computerusefor12hours?Answer inkilowatt-hours.

a Howmuchenergydoesamicrowave usefor 1 2 hour?Answerin watt-hours.

Electricalappliance Powerrating

Electricshaver 15W

Hairdryer 1000W

Iron 900W

Laptopcomputer 50W

Microwave 1200W

Vacuumcleaner 600W

b Forhowmanyhourswasthe vacuumcleanerused,ifthecostofoperatingthevacuumcleanerwas$6.36? Thepriceofelectricityis$0.2650perkilowatt-hour.

1J 14 Icancalculatethefuelconsumptionrateofamotorvehicle. e.g.Rata’scaruses11.26litresper100km.

Howmanylitresofpetrolwillhiscaruseonatripof155kmfromSydneyto Newcastle?Answercorrecttothreedecimalplaces.

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Thecostofpetrolis$1.60perlitre.Howmuchwillthepetrolcostforthe 155kmtrip?

c Thecostofpetrolis$1.35perlitre.Howmuchwillthepetrolcostforthe 294kmtrip?

Howmanylitresofpetrolwillhiscaruseonatripof294kmfromArmidale toTaree?Answercorrecttothreedecimalplaces.

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?

8 1I Howmuchenergydoesa600-watthairdryerusein5hours? 1.2kWh A 3kWh B

9 1I Flynnusesa1.2-kilowattdishwasherfor5hours.Hepays25.72centsperkilowatt-hour.Whatisthe costofthisusage?

10 1J Whatisthefuelconsumptionforavehiclethattravelled340kmusing51litresofpetrol? 7L/100km A

/100km B 15L/100km C

/100km D

Short-answerquestions

1 1A Simplifythefollowingratios.

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

3 1A TheratioofVictoria’sheighttoWillow’sis8:7. IfVictoriais176cmtall,howtallisWillow?

4 1A Expresseachofthefollowingscalesasaratiointheform1:a. 1cmto3m a 1mmto6cm b 1mto2.5km c

5 1B Ascientistcatches200fishfromapond,tagsthemandreleasesthem.Onemonthlater,theycatch50 fish,ofwhich20havetags.Estimatethenumberoffishinthepond.

6 1C DanielandEddieownabusinessandsharetheprofitsintheratio3:4.

a Theprofitlastweekwas$3437.HowmuchdoesDanielreceive?

b Theprofitthisweekis$2464.HowmuchdoesEddiereceive?

7 1D Thescaleonamapisgivenas1cm = 5km.Ifthedistancebetweentwopointsonthemapis 46mm,whatistheactualdistancebetweenthesepoints?Answerinkilometres.

8 1D Ascaledrawinghasascaleof1:50000.Whatisthedrawinglengthoftheseactuallengths? Expressyouranswerinmillimeters. 4km a 1250m b 5000cm c 6.5km d 20000mm e 2125m f

SAMPLEPAGES

9 1D

Thescaleonamapis1:400.Calculatetheactualdistancesifthefollowingarethedistancesonthe map.Expressyouranswerinmetres.

Bikepath180cm a

10 1E

Towncentre20cm b Street5cm c

Beach210mm d River62cm e

Asectionofafloorplanisshownopposite. Thelongerdimensionofthelaundryis3metres.

a Estimateasuitablescaleforthefloorplan.

b Whatsymbolisusedforthebath?

c Whatdoes‘W.C.’representontheplan?

d Whatarethedimensionsofthelaundry?

Park60mm f

11 1F

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

a Findtheapproximatevolumeofthebodyofwater.

b Findtheapproximatevolumeofwaterthatfallsonthesameareawhenthereis9mmofrain.

12 1G Converteachratetotheunitsindicated.

$15/kgto$/g a 14400m/htom/ min b

120cm/stomm/min c 4800kg/gtokg/mg d 14L/gtomL/kg e

$3600/gtoc/mg f

13 1G If20metresofcurtainmaterialcosts$580,whatwouldbethecostof35metresofthesame material?

14 1G Amotorbiketravelsataspeedof100km/h.Howfardoesittravelin3.5hours?

15 1G A5kgbagofricecosts$9.20.Whatisthecostofthefollowingamounts? 10kg a 40kg b 3kg c 7kg d 500kg e 250kg f

16 1J Acartravels960kmon75litresofpetrol.Howfardoesittravelon50litres?

Extended-responsequestions

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

2 1G PumpAcanfillatankin5minutes;pumpBcanfillthetankin10minutes.Howlongwillittaketo fillthetankifbothpumpsareworkingtogether?

LAUNDRY BATHROOM

Sliding door

W.C.

Robe

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,17oftheteamshadatleasttwo playerswiththesamebirthday.Sojustover halfoftheteamsexperiencedthis coincidence,whichseemsextraordinary.

Fouroftheteamshad two pairsofplayers withmatchingbirthdays:

•Panama

•Brazil

•Columbia

•Denmark

Twooftheteams,MoroccoandNigeria,even had three pairsofplayerswithmatching birthdays.

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

Themathematicsofthiscounter-intuitive situationisexplainedhere: https://www.bbc.com/future/article/ 20230830-the-unexpected-maths-problemat-work-during-the-womens-world-cup

2A Languageofprobability

2B Definitionofprobability

2C Tablesandtreediagrams

2D Rangeofprobabilities

2E Complementaryevents

2F Relativefrequency

2G Expectedfrequency

2H Venndiagramsandtwo-waytables

NSWSyllabus

Inthischapter,astudent:

•Modelsandsolvesproblemsinvolving theprobabilitiesofmultistageevents (MST-12-S2-09).

©NSWEducationStandardsAuthority

Onlineresources

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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.

Impossible

Evenchance Certain Nochance Somewhatunlikely

Discussion

Whichoneoftheseisclosestto0:‘slimchance’or‘fatchance’?

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

Probability

Probabilityisthechancethatsomethingwillhappen. Fractions,decimalsandpercentagesareusedforprobabilities. Allprobabilitiescanbeplacedonascalefrom0(impossible)to1,or100%(certain)

Example1

Usingthelanguageofprobability

Yinghasapear,anappleandanorange.Sherandomlyselectsone pieceoffruit.Describethechanceofthefollowingeventsusing thewords‘certain’,‘likely’,‘even’,‘unlikely’or‘impossible’.

a Yingselectsabanana.

b Yingselectsanorange.

c Yingselectsapieceoffruit.

d Yingselectsapearoranapple.

Solution

a Impossible

b Unlikely

Explanation

Nochanceofselectingabanana.

Onechanceoutofthreeofselectingtheorange.

c Certain Selectingapieceoffruitmusthappen.

d Likely

Nowyoutry

Twochancesoutofthreeofselectingthepearortheapple.

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

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

Twounbiasedcoinsaretossed.

a Whatisthesamplespace?

b Howmanypossibleoutcomesarethere?

Solution

Explanation

a Firstcoin:HorT. Eachcoincouldlandonaheadoratail.

Secondcoin:HorT. Ifthefirstcoinwasahead,thenthesecondcoincouldbea headoratail.

Samplespace = {HH,HT,TH,TT} Ifthefirstcoinwasatail,thenthesecondcoincouldbea headoratail.

b Numberofpossibleoutcomesis4. Counttheelementsinthesamplespace.

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

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i Obtainingaheadoratailwhenacoinistossed

j Nightfollowingday

BUILDING

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

a Therebeing31daysinJanuary

b Tossingacoinandgettingatail

c Randomlychoosingapersonwhohasneverwatchedtelevision

d WinningfirstdivisionLotto

e Choosingaredballfromabagcontaininganequalnumberofredandwhiteballs

f SnowonMountKosciuszkoduringacolddayinwinter

g Newbornbabybeingagirl

h Drawingaheartfromanormalpackofcards

i Tossingacoinandturningupahead

j AustraliaparticipatinginthenextcricketWorldCup

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

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.

Blue

White Yellow Red Green

a Listthesamplespace.

b Howmanyelementsinthesamplespace?

Thereare10counterseachlabelledwiththelettersfromAtoJ.

a Acounterisselectedatrandom.Listallthepossibleoutcomes.

b Describeinwordstheprobabilityofselectingavowel. 10

ARugbyUnionmatchisbeingplayedbetweenAustraliaandSouthAfrica.

a Listthesamplespaceforthepossibleoutcomesofthematch.

b Howmanyelementsinthesamplespace?

c Iseachoutcomeequallylikely?Why?

Jacobislearningtoplaychess.HehasamatchagainsttheAustralianchampion.Therearethree outcomesforthematch–win,draworlose.

a Whatisthesamplespace?

b Aretheseeventsequallylikely?Why?

DEVELOPING

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

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

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

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

a Listthesamplespaceforthefirstnote.

b Howmanydistinctelementsareinthesamplespaceofthefirstnote?

c Assumethefirstnotechosenwasa$10note.Listthesamplespaceofthesecondnote.

d Assumethefirstnotechosenwasa$20note.Listthesamplespaceofthesecondnote.

e Assumethefirstnotechosenwasa$50note.Listthesamplespaceofthesecondnote.

CHALLENGING

15 Afaircoinandastandarddiearethrown.Describeinwordstheprobabilityofthe followingoutcomes.

a Headandanumberlessthan10

b Headandanumbergreaterthan10

c Tailanda1

d Tailorheadandanevennumber

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

17 Laurahasan80%chanceofwinningatennistournament.TheothertwoplayersMayaandEmma areequallylikelytowin.

a Whatisthesamplespace?

b Aretheoutcomesequallylikely?Why?

c WhatistheprobabilitythatMayawinsthetournament?

Communicateyourreasoning,understandingandfluency

18 Tesssaystwothings.

Statement1: “Whenyoutosstwocoinstogether,therearethreepossibilities:2heads,2tailsor oneheadandonetail.”

Statement2: “Therefore,thechanceoftossingtwoheadsisone-third.”

Re-writeStatement1andStatement2sothattheyarecorrect.

19 Irolledafairstandarddie3times.Irolleda4,threetimesinarow.

Rachelsaid:“Thereisareallygoodchancethatthenextonewillbea4.”

Sophiesaid:“Thereisalmostnochancethatthenextonewillbea4.”

Aditisaid:“Ifthatisafairdie,thechancethatthenextonewillbea4isone-sixth.”

WriteaparagraphinsupportofAditi’sstatement.

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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

• Tossingahead

• Rollinganumberlessthan5

• Tossingaheadthenatail • Choosingaredball

•

• Winningatennismatch

Twosimilarprobabilityexperiments,withanimportantdifference

BagAcontainsaredball,awhiteballanda greenball.Oneischosenatrandom.

ThethreepossibleoutcomesR,WandGare ‘equallylikelyoutcomes’.

Thesamplespaceis {R,W,G}

An eventis‘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}?

Is thesamplespace {R,W}? Are theretwooutcomesorthreeoutcomes? 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

Example4

Calculatingtheprobability

Acoinischosenatrandomfrom7onedollarcoins and3twodollarcoins.Calculatetheprobabilitythat thecoinisa:

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

Adeckofplayingcards

Anormaldeckofplayingcardshas 52cards.

• Therearefoursuitscalledclubs,spades, heartsanddiamonds.

• Ineachsuitthereare13cardsfromace toking.

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• Thereare3picturecardsineachsuit (jack,queenandking).

• Halfthecardsareredandhalfareblack.

Example5

Calculatingtheprobabilityfromplayingcards

Asinglecardisrandomlychosenfromthepack.Whatistheprobabilitythatitisa:

a redfour

b diamond

c picturecard.

Solution

Explanation

a P(Red4) = 2 52 = 1 26 2ofthe52cardsareredfours. Reducetosimplestform.

b P(Diamond) = 13 52 = 1 4

c P(Picture) = 12 52 = 3 13

Nowyoutry

13ofthe52cardsarediamonds. Reducetosimplestform.

12ofthe52cardsarepicturecards. Reducetosimplestform.

Thefouracesareremovedfromthepackofcards.Onecardisthenchosenatrandom.Whatisthe probabilitythatthecardisa:

a redfour?

b diamond?

c picturecard?

Exercise2B

1 Example4 Whatistheprobabilityofthefollowingexperiments?

a Acarddealtfromanormaldeckofcardsisadiamond.

b Adayselectedatrandomfromtheweekisontheweekend.

c Aheadresultswhenacoinistossed.

d Aletterchosenrandomlyfromthealphabetisavowel.

e Atworesultswhenadieisrolled.

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

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BUILDING

2 Abagcontains5blueand3redballs.Findtheprobabilityofselectingthefollowingatrandom.

a Ablueball

b Aredball

c Notaredball.

Example5

Rangichoosesoneballatrandomfromhisgolfbag.Thetablebelowshowsthetypeandnumberof golfballsinhisbag.

Typeofgolfball Number

Titleist 3

Srixon 4

Callaway 13

FindtheprobabilityofRangichoosing: aSrixon a aCallaway b aTitleist c notaSrixon. d

Anunbiasedcoinistossedthreetimes.Onthefirsttwotossestheresultistails.Whatisthe probabilitythattheresultofthethirdtosswillbeatail?

5 InAmberAvethereare3highschoolstudents,4primaryschoolstudentsand5preschoolstudents. OnestudentfromAmberAveischosenatrandom.Whatistheprobabilitythataprimaryschool studentischosen?

6 Aboxcontains3blue,4greenand2whitecounters.Findtheprobabilityofdrawingatrandoma counterthatis: blue a green b whiteorblue c notblue. d

Acardischosenatrandomfromastandarddeckof52playingcards.Findtheprobabilityofchoosing: 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

Caseyisdealtfourcardsfromanormal deck:twoacesandtwokings.Whatis theprobabilitythatthenextcardis:

a anotherace?

b anotherking?

c notanace?

d notaking?

12 Twocardsaredrawnatrandomfromanormaldeckofcards.Whatistheprobabilitythatthesecond cardis:

a atwoifthefirstcardwasatwo?

b anaceifthefirstcardwasanace?

c thesixofclubsiffirstcardwasa10?

d atwoifthefirstcardwasaking?

e adiamondifthefirstcardwasadiamond?

f apicturecardifthefirstcardwasapicturecard?

Afour-digitnumberisformedfromthedigits2,3,4and5withoutreplacement.Whatisthe probabilitythatthenumber:

a startswiththedigit4?

b isgreaterthan3000?

c endswitha2ora3?

d is2345?

14 Ifsixstudentsenteraswimmingrace,thenthechanceofanyparticularstudentwinningmustbe

Isthisstatementtrueorfalse?Givereasonstosupportyouropinion.

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

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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

Usinganarrayforamultistageevent

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

a Constructatabletolistthesamplespace.

b WhatistheprobabilityofselectingR1R1orR2R2?

Solution Explanation

Listtheoutcomesofthefirststage(firstcard)downthe firstcolumn.Therearethreeoutcomes:R1,R2andB1. Listtheoutcomesofthesecondstage(secondcard)across thetoprow.Therearethreeoutcomes:R1,R2andB1. Writetheoutcomeineachcellusingtheintersectionof therowandcolumn. b P(R1R1orR2R2) = 2 9 Formafraction.

Nowyoutry

RepeatExample 6,butthistimethefirstcardisnotreplacedbeforethesecondcardischosen.

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

Drawbranchesforfirststage–tossingacoin. Tossingacoinhastwooutcomes(headortail)sothereare twobranches.

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

Usethebranchesofthetreetolistthesamplespace.Writethe outcomesdowntheright-handside.

b P(H1) = 1 12

Nowyoutry

Numberoffavourableoutcomes(H1)is1.Thetotalnumber ofoutcomesis12. Formafraction.

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

a Constructatreediagramtoshowthesamplespace.

b Whatistheprobabilitythatthesumofthetwocardsislessthan5?

Probabilitytreesforoutcomeswhicharenotequallylikely

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.

Example8

Drawingaprobabilitytree

Asmallpackofcardscontains5redcardsand2bluecards.Onecardischosenandthenreplaced. Asecondcardisthenchosen.

a Drawaprobabilitytree.

b Whatistheprobabilitythatbothcardsarered?

c Whatistheprobabilitythatthetwocardsaredifferentcolours?

d What istheprobabilitythatoneormoreofthecardsarered?

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.Theyareusually thesameaseachother.

Addthethreedifferentoutcomes.

Thefouroutcomesaddto1,sosubtract P(BB) from1,asthatistheonlyoutcomeyoudonotneed toinclude.Thisisaveryusefulshortcut.

RepeatExample 8,butthistimethefirstcardis not replacedbeforethesecondcardischosen.

Exercise2C

1 RileyandBaileyareplanningtohavetwochildren.

Male Female Male Female

BUILDING

a Useatabletolistthenumberofelementsinthesamplespace.Considerthesexofeachchildas anevent.

b Whatisthetotalnumberofoutcomes?

2 Twofairdicearethrownandtheirsumrecorded.

a Useatabletolistallpossibleoutcomes.

b Howmanyelementsareinthesamplespace?

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

Useatabletolistallpossibleoutcomes.

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)areplaced facedownonatable.Onecardisselectedat randomandtheresultrecorded.Thiscardis returnedtothetable.Asecondcardisthen selectedatrandom.

a Useatabletolisttheelementsinthe samplespace.

b Findtheprobabilityofthefollowing selections: i KJ ii twokings iii nokings.

8 Example7 Twocoinsaretossedandtheresultsrecorded.

a Listthesamplespacebycompletingthistreediagram.

b Findtheprobabilityofthefollowingresults: i twotails ii headthenatail iii oneheadandonetail.

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

TherearethreequestionsinaTrueorFalsetest.

a Fillintheremainderofthepartiallydrawntree diagramandlistthesamplespace.

b Howmanyelementsinthesamplespace?

c Findtheprobabilityofthefollowingrandom selections:

i alloftheselectionsarefalse

ii twotrueselectionsandonefalseselection

iii atleastoneoftheselectionsisfalse.

Atwo-digitnumberisformedusingthedigits1,2and3.Thesame numbercannotbeusedtwice.Thefirstdigitchosenisthetensdigit andtheseconddigitchosenistheunitsdigit.

a Listthesamplespacefromthetreediagram.

b Whatistheprobabilityofchoosing23?

Aspinnerhasanequalnumberofredandgreensections.Thisspinner isspuntwice.

a Useatreediagramtolistthetotalpossibleoutcomes.

b Whatistheprobabilityofthesamecolourwhenthespinneris spuntwice?

Ebonytossesacoinandspinsaspinner,whichhasared,anamberandagreensection. Eachcolouronthespinnerisequallylikely.

a Useatreediagramtolistthesamplespace.

b Whatistheprobabilityofaheadonthecoinandeitheraredorgreensectiononthespinner?

14 Fourcards(ace,king,queenandjack)areplacedfacedownonatable.Onecardisselectedat randomandtheresultrecorded.Thiscardisnotreturnedtothetable.Asecondcardisthenselected atrandom.Useatreediagramtolistallthepossibleoutcomes.

CHALLENGING

15 Therearefourcandidatesforthepositionsofleaderanddeputyleader.Thefourcandidatesare Mohammed,Bridget,ConnorandDanielle.

a Constructatreediagramwiththeleaderasthefirsteventandthedeputyleaderasthesecond event.Useatreediagramtolistthesamplespace.

b WhatistheprobabilityofBridgetbeingtheleaderordeputyleader?

16 Atwo-digitnumberisformedusingthedigits3,5and7.Thesamenumbercanbeusedtwice.The firstdigitchosenisthetensdigitandtheseconddigitchosenistheunitsdigit.

a Constructatreediagramtolistthesamplespace.

b Whatistheprobabilityofforminganumberlessthan70?

Thenumberoflaptopssoldinthepasttwoweeksisshownbelow.

Example8

a OfallbrandXlaptopssold,whatfractionwassoldinweek2?

b Whatpercentageofallsalesinweek1wasbrandY?

c Duringthetwoweeks,whatwastheprobabilitythatanylaptopsoldwouldbebrandX?

RepeatQuestion 6,assumingthefirstcardisnotreplacedbeforethesecondcardischosen.

19 RepeatQuestion 7,assumingthefirstcardisnotreplacedbeforethesecondcardischosen.

20 RepeatQuestion 8,foracoinwhichisbiasedsothatitlandsonheads60%ofthetime.

Communicateyourreasoning,understandingandfluency

21 Yourcalculatorhasabutton .Whatdoesitdo?Searchtheinternet. Writedownthevalueof:

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Thelastoneisa68digitnumber.Thatisthenumberofdifferentwaysapackofcards can beshuffled.

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.

Usingtherangeofprobability

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

Explanation

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

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

Solvetheequationbymaking P(B)thesubjectofthe equation.

Simplifythefractionifpossible.

Probabilityofabluecardis 3 10 . Writetheanswerinwords.

Nowyoutry

Foraparticularcouple,thechancetheywillhaveamalechildis10%.Theyareplanningtohavetwo children.Whatisthechancethattheywillnothavetwomalechildren?

Exercise2D

1 Example9

BUILDING

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

a Whatisthevalueof P(A)?

b Whatisthevalueof P(B)?

c Whatistheprobabilityofselectingaticketwiththeletter‘D’?

d What istheprobabilityofselectingtickets A,B or C?

e Whatistheprobabilityofselectingticket C?

2 Abagcontainsblack,yellowandwhitecards.Theprobabilityofdrawingablackcardis57%andthe probabilityofdrawingayellowcardis8%.Whatisthevalueofthefollowing,expressedasafraction insimplestform?

(Black) a

(Yellow) b

3 InaparticulareventtheprobabilityofJunwinningagold medalis 3 8 andasilvermedalis 1 4 .Thereisnobronze medal.

a WhatareJun’schancesofwinningagoldorasilver medal?

b WhatareJun’schancesofnotwinninganymedals?

(White) c

4 Somepicturecardsfromadeckofcardsareplacedfacedownonthetable.Theprobabilityofdrawing akingis0.25andaqueenis0.60.Whatisthevalueofthefollowingexpressedasadecimal?

a P(King)

b P(Jack)

c P(Jack) + P(King)

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

5 Therearefouroutcomesofanexperiment.Threeoftheoutcomeshaveprobabilitiesof20%,25%and 40%respectively.Whatistheprobabilityofthefourthoutcome?

6 Abiaseddieisrolled.Theprobabilityofobtaininganevennumberis0.4andtheprobabilityofa1or a3is0.3.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 and white.If P(B) = 1 5 , P(G) = 2 13 , P(P) = 2 9 and P(R) = 1 6 ,findtheprobabilityofthefollowing outcomes.

Blackorgreendisc a

Blackorreddisc c

Black,greenorreddisc e

Pinkorreddisc b

Black,greenorpinkdisc d

Black,green,pinkorreddisc f

8 Acardischosenatrandomfromsomeplayingcards.Theprobabilityofaspadeis0.24,the probabilityofaclubis0.27andtheprobabilityofaheartis0.23.Findtheprobabilityofthefollowing outcomes.

Blackcard a Redcard b

Cluboraheart c

Diamond e

Spadeoraheart d

Diamondoraclub f

9 JuliaandNatashaareplayingagameinwhichastandardsix-sideddieisrolled.Juliawinsifaneven numberisrolled.Natashawinsifanumbergreaterthanthreeisrolled.Whatistheprobabilitythatthe numberrolledisneitherevennorgreaterthanthree?

10 Abagcontainswhite,greenandredmarbles.Theprobabilityofselectingawhitemarbleis 2 7 andthe probabilityofselectingagreenmarbleis 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,Rhowmanyofthefollowinglettersarein theword?

c Whatistheword?(Hint:anAustralianplace.)

Communicateyourreasoning,understandingandfluency

13 Yourcalculatorhasabutton .Whatdoesitdo?Searchtheinternet. Writedownthevalueof:

ThelastonehassomethingtodowithLotto.Writedownwhatitis.

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

Example10

Usingthecomplement

Lisaselectsacardatrandomfromanormalpack. Findtheprobabilityofobtainingthefollowingoutcomes.

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?

Exercise2E

1 Whatistheeventthatisthecomplementofthefollowingevents?

a Selectingablackcardfromanormalpackofcards

b WinningfirstprizeinLotto

c Throwinganevennumberwhenadieisrolled

d Obtainingatailwhenacoinistossed

e Drawingaspadefromanormalpackofplayingcards

f Choosingagreenballfromabagcontainingablue,aredandagreenball

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2 Findthevalueof P(E),giventhefollowinginformationaboutevent E

BUILDING

ThechancesoftheSydneySwanswinningthepremiership aregivenas29%.WhatarethechancesthattheSydney Swanswillnotwinthepremiership?

a Expressyouranswerasadecimal.

b Expressyouranswerasafraction. 4

Theprobabilityofobtaininga3onabiaseddieis0.6.Whatistheprobabilityofnotobtaininga3?

5 TheprobabilityofarainydayinMarchis 11 15 .WhatistheprobabilitythataparticulardayinMarch doesnothaverain?

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

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

(2) + P(2) e

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(3) + P(3) f

(2) + P(8) g

(8) + P(8) d

(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?

• third person:Whatistheprobabilitythispersonhasadifferentbirthdaytothefirst 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 caraccidentintheyearahead,sotheycansetthepriceoftheirinsurancepolicies.

Sometimesitispossibletoconductaprobabilityexperimentanduserelativefrequencyasanestimate fortheprobability.

Forexample,whenathumbtackfallstothegrounditcouldland intwodifferentways,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.

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%

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Relativefrequencyof2headsis0.27.

Explanation

Addthefrequencycolumntodeterminethetotal numberoffrequencies.

Writetheformulaforrelativefrequency.

Substitutethefrequencyandtotalnumberof frequenciesintotheformula.

Simplifythefractionifpossibleorexpressasa decimal.

Writetheanswerinwords.

Writetheformulaforrelativefrequency. Substitutethefrequencyandtotalnumberof frequenciesintotheformula.

Simplifythefractionifpossibleorexpressasa decimal.

Writetheanswerinwords.

Writetheformulaforrelativefrequency.

Substitutethefrequencyandtotalnumberof frequenciesintotheformula.

Simplifythefractionifpossibleorexpressasa decimal.

Writetheanswerinwords.

Nowyoutry

Athumbtackwastossed200times.Theseweretheresults:

Pointup 121

Pointdown 79

a Findtherelativefrequency,asapercentage,for‘pointup’.

b Findtherelativefrequency,asapercentage,for‘pointdown’.

Exercise2F

1 Example11 Thefrequencytableshowstheoutcomesof anexperiment.Whatistherelativefrequency forthefollowingoutcomes?Expressasa fractioninsimplestform.

2 Thefrequencytableshowstheoutcomes ofanexperiment.Whatistherelative frequencyforthefollowingoutcomes? Answercorrecttothreedecimalplaces.

3 Thefrequencytableshowstheoutcomes ofanexperiment.Whatistherelative frequencyforthefollowingoutcomes? Answerasapercentagecorrecttoone decimalplace.

4 Thefrequencytableshowstheoutcomesof anexperiment.Whatistherelativefrequency forthefollowingoutcomes?Answerasa percentage.

BUILDING

Calculatetherelativefrequencyforeachofthesenumbersifthetotalfrequencyis48.Writeyour answerasafractioninsimplestterms.

Calculatetherelativefrequencyforeachofthesenumbersifthetotalfrequencyis40.Writeyour answerasapercentage.

DEVELOPING

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

8 ApistolshooterattheOlympicGameshitsthetarget24outof25attempts.Whatisthe relativefrequencyoftheshooterhittingthetarget?Givetheanswerasadecimal,correcttotwo decimalplaces.

9 Thebirthstatisticsinalocalcommunitywere142femalesand126males.Whatistherelative frequencyforafemale?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 x?

b Whatisthevalueof y?

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.

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2G Expectedfrequency

Learningintentions

• Tounderstandtheconceptofexpectation

• 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 on atripthatpassesthrough54intersections?

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 Theprobabilityofaworkerinanindustrialplanthavinganaccident is0.12.Theindustrialplantemploys175workers.Whatisthe expectednumberofaccidents?

7 Adieisrolled480timesandtheresultsrecorded.

a Whatistheprobabilityofthrowinga4?

b Howmany4sareexpected?

c Whatistheprobabilityofthrowinganoddnumber?

d Howmanyoddnumbersareexpected?

e Whatistheprobabilityofthrowinganumbergreaterthan2?

f Howmanynumbersgreaterthan2areexpected?

g Whatistheprobabilityofthrowinganumberdivisibleby3?

h Howmanynumbersdivisibleby3areexpected?

DEVELOPING

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

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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.Twodiscsaredrawninsuccessionfromthebag.The firstdiscisnotreplacedbeforetheseconddiscisdrawn.Thisprocessisrepeated352times.

a Howmanyofthefirstdiscsareexpectedtobeyellowdiscs?

b Howmanyofthefirstdiscsareexpectedtobereddiscs?

c Howmanydoubleyellowdiscsareexpected?

d Howmanydoublereddiscsareexpected?

e Howmanyareexpectedtohaveafirstdiscyellowandaseconddiscred?

f Howmanyareexpectedtohaveafirstdiscredandaseconddiscyellow?

a CellC8hasaformulathatmultipliescellsB8andB4.Enterthisformula.

b FilldownthecontentsofC8toC11usingtheformulaincellC8.

c Changethenumberoftrialsfrom230to800.ObservethechangeinC8:C11.

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

a Howmanyofthesefamiliesareexpectedtohavethreemalechildren?

b Howmanyofthesefamiliesareexpectedtohaveexactlyonemalechild?

c Howmanyofthesefamiliesareexpectedtohaveexactlytwomalechildren?

d Howmanyofthesefamiliesareexpectedtohavenomalechildren?

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CHALLENGING

13 Twocardsareselectedatrandomfromanormalplayingpackwithreplacement.Thisexperimentis repeated2704timeswiththecardsbeingreplacedeachtime.

a Whatistheexpectednumberoftimestheresultwouldbetwospades?

b Whatistheexpectednumberoftimestheresultwouldbetwoaces?

c Whatistheexpectednumberoftimestheresultwouldbetwopicturecards?

d Whatistheexpectednumberoftimestheresultwouldbetwocardswithanumberlessthan9?

14 Twodicearetossedoneaftertheotherontoatable.Thiseventisrepeated144times.

a Onhowmanyoccasionswouldyouexpecttheresulttobea6thena1?

b Onhowmanyoccasionswouldyouexpecttheresulttobetwo3s?

c Onhowmanyoccasionswouldyouexpecttheresulttobetwooddnumbers?

Communicateyourreasoning,understandingandfluency

15 Meirollstwostandarddice,thenworksoutthedifferencebetweenthetwoscores.Forexample,if sherollsa5anda2,thedifferenceis3. Shepays$1everytimesheplaysthisgame.

Ifherdifferenceis0,1or2,shelosesherdollar.

Ifherdifferenceis3,4or5,shegetsherdollarback,plusanotherdollar.

a Copyandcompletethesetables.

b Usethetablestoexplainwhythisgameisnotfair.

c IfMeiplaysthisgame100times,howmuchmoneywouldsheexpecttolose?

d Changetherulestomakethegamefair.

16 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.

2H Venndiagramsandtwo-waytables

Learningintentions

• TounderstandthatVenndiagramswithtwoattributescanbeusedtocontaintheinformationina two-waytable

• TobeabletotransferinformationbetweenaVenndiagramandatwo-waytableandviceversa

• TobeabletosolveproblemsusingVenndiagramsandtwo-waytables Past,presentandfuturelearning

• Someaspectsofthissectionmaybenewtostudents

• TheseconceptsbuilduponpriorlearningfromYears7to10

• QuestionsinvolvingprobabilityregularlyappearinHSCExaminations,butVenndiagramswerenot examinedpriorto2027.

Considerthisscenario:

Somebodyaskedyoutochooseacardrandomlyfromastandardpackof52cards.Theythenaskedyouto answerthesetwoquestions:

A:Wasyourcardred?

B:Wasyourcardapicturecard?

Thediagrambelowshowsthecontentsofthepack.Note:

• 26cardsarered

• 12cardsarepicturecards,ofwhich6arered,so6cardsareredpicturecards

• 20cardsareneitherredcardsnorpicturecards.

Therearefourdifferentresponsestothetwoquestions:

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A:Wasyourcardred? Yes Yes No No

B:Wasyourcardapicturecard? Yes No Yes No

Inthisexample,thefourresponsesarenotequallylikely.

Thenumberofwaysofachievingeachofthefouroutcomescanbeexpressedinatwo-waytableandaVenn diagram(namedafterJohnVenn(1834 1923)).

Two-waytablesandVenndiagrams

Thesearedisplaysoftheresponsestoasurveyorresultsofanexperimentinwhichtwoquestionswere askedthathavetwoanswers(YESorNO).Theycanbeusedtoestimatetheprobabilitythatother peoplewillanswerthesamequestionsincertainways. Two-waytable

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Thenumberintheyellowcellis:

• thesumofthebottomrowand

• therightcolumn;or

• thesumofthefourorangecells

Thenumberintheyellowcellofthetableisthesum ofallfournumbers(theboxes)intheVenndiagram.

Theshadedcellcontainsthenumberof peoplewhoanswered‘YES’toboth questions.

Venndiagram: Thesectioninwhichthetwocircles overlapiscalledthe‘intersection’.

Inwords: Both A and B aretrue.

Insymbols: A ∩ B

Note:Thesyllabusdoesnotmandatethatset notationsuchasthismustbeused.Thisnotationis unlikelytobeusedinHSCquestions.However,itcan beusedbystudentsintheirsolutions.

Total

Two-waytable

Venndiagram A Not A

Theshadedcellcontainsthenumberof peoplewhoanswered‘YES’tooneorboth questions. AB

Venndiagram: Allthreesectionsinsidethecircles, called‘theunion’.

Inwords: A or B aretrue.Theword‘or’isinclusive. Itmeans‘A’or‘B’or‘both’.Itiscalledthe‘union’.It does notmean‘oneortheother’.

Insymbols: A ∪ B

Theshadedcellcontainsthenumberof peoplewhoanswered‘NO’toboth questions.

Venndiagram: Thenumberinthepurplecellis foundinthesectionoutsidethecirclesbutinsidethe rectangle.

Inwords: Neither A nor B

Example13

Transferringinformationfromatwo-waytabletoaVenndiagram

a Completethetwo-waytable.

b Usethetwo-waytabletodrawaVenndiagram.

Findthesumofeachrowandeachcolumn.

Note:

• thenumber6isinsidetheoverlappingsection

• thepositionsofthenumbers6,7,8and9inthe tableandtheVenndiagram

• thesumofallfournumbersis30

• thetotalinthe A circleis14

• thetotalinthe B circleis13.

Nowyoutry

a Completethetwo-waytable.

b Usethetwo-waytabletodrawaVenndiagram.

TransferringinformationfromaVenndiagramtoatwo-waytable

UsetheVenndiagramtoconstructatwo-waytable.

Nowyoutry

UsetheVenndiagramtoconstructatwo-waytable.

Note:

• thenumber10isinsidetheoverlappingsection

• thepositionsofthenumbers5,8,10and20in theVenndiagramandthetable

• thesumofallfournumbersis43

• thetotalinthe A column15

• thetotalinthe B rowis18.

Example15

Usingatwo-waytabletocalculateprobabilities

Agroupofpeoplewereaskedthesetwoquestions:

A:HaveyoubeentotheNorthIslandofNewZealand?

B:HaveyoubeentotheSouthIslandofNewZealand? Theirresponsesarelistedinthetwo-waytable.

a Completethetwo-waytable.

b Ifonepersonischosenatrandomfromthisgroup, whatistheprobabilitythat:

i theyhavebeentobothislands?

ii theyhavebeentotheSouthIsland?

iii theyhavebeentotheSouthIslandbutnottheNorthIsland?

Solution Explanation

Findthesumofeachrowandeachcolumn.

20ofthe100peoplehavebeentobothislands.

32ofthe100peoplehavebeentotheSouthIsland.

12ofthe100peoplehavebeentotheSouthIsland butnottheNorthIsland.

Nowyoutry

UsingExample 15,ifonepersonischosenatrandomfromthisgroup,whatistheprobabilitythat:

a theyhavebeentoneitherisland?

b theyhavebeentotheNorthIsland?

c theyhavebeentotheNorthIslandbutnottheSouthIsland?

Example16

UsingaVenndiagramtocalculateprobabilities

InYear12thereare120students.Ofthese,70playamusical instrument(I),60playasport(S),and20doneither.

a Usingthisinformation,completetheVenndiagram.

b AYear12studentisselectedatrandom.Whatisthe probabilitythatthestudentplaysaninstrumentand asport?

c Twostudentsarechosenatrandom.Whatisthe probabilitythattheybothplayaninstrumentonly?

Solution

b P(instrumentandsport) = 30 120 = 1 4

c P(twostudentsplayinstrumentonly) = 40 120 × 39 119 = 13 119

Nowyoutry

UsingExample 16:

Explanation

Startwith120students,thensubtractthe20who doneither,leaving100students.

60 + 70 = 130and130 100 = 30,so30students doboth.

70 30 = 40studentswhoplayaninstrumentonly.

60 30 = 30studentswhoplaysportonly.

30ofthe120studentsplayboth.

Whenthefirststudentischosen,39ofthe remaining119playaninstrument.

a AYear12studentisselectedatrandom.Whatistheprobabilitythatthestudentplaysasportbutnot aninstrument?

b Twostudentsarechosenatrandom.Whatistheprobabilitythattheybothplaysportonly?

Exercise2H

1 MatchthediagramsA,B,C,D,Ewiththecorrectdescription.

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BUILDING

2 Customersatarestaurantweregiventhefollowingchoicestoaddtotheirdessert:

A:Wouldyoulikecream?

B:Wouldyoulikeicecream?

Theresultsfrom100customersareshowninthistable.

a Copythetableandfillinthefivemissingtotals.

b Howmanypeople:

i chosebothcreamandicecream? ii choseneithercreamnoricecream? iii chosecream? choseicecream? iv chosecreamwithouticecream? v choseicecreamwithoutcream? vi didnothaveicecream? vii didnothavecream? viii chosecreamoricecream? ix

3 Example13 ConvertthetableinQuestion 2 intoaVenndiagramby writinganumberineachofthefourboxes.

4 DrawthisVenndiagramforeachofthefollowing,then shadeintheregionwhichshows:

5 StudentsinYear12wereasked:

A:Wouldyouliketorepresenttheschoolinsoccer?

B:WouldyouliketorepresenttheschoolinAFL?

Theresultsfrom100studentswereasfollows:

Howmanypeople: chosebothsoccerandAFL? a choseneithersoccernorAFL? b chosesoccer? c choseAFL? d chosesoccerbutnotAFL? e choseAFLbutnotsoccer? f didnotchoosesoccer? g didnotchooseAFL? h chosesoccerorAFL? i

6 Example14 ConverttheVenndiagraminQuestion 5 intoa two-waytable.

7 Copyandcompletethefollowingtwo-waytables.

8 InthefollowingVenndiagrams,someofthenumbersaremissing,asindicatedbyemptyboxes. Whatarethemissingnumbers?

Overalltotalis30. A 5 67 B a

Overalltotalis50.

Totalof A is20.Totalof B is30.Grandtotalis100. A 5 B c

Grandtotalis100.Totalof A is60.Totalof B is50.

9 Foreachtwo-waytable,fillinthemissingnumbersthendrawaVenndiagram.

Agroupofpeoplewereaskedthesetwoquestions:

A:Doyoudrinkcoffeeeveryweek?

B:Doyoudrinkteaeveryweek?

• 50peopleansweredYEStobothquestions

• 10peopleansweredNOtobothquestions

• 75peopleansweredYEStoquestionA

• 65peopleansweredYEStoquestionB a Copyandcompletethistwo-waytable.

b CopyandcompletethisVenndiagram.

11 a Example15 Copyandcompletethistwo-waytable:

b Calculatethefollowingprobabilities:

CHALLENGING

14 InYear12thereare100students.Ofthese,40playsoccer(S),50playnetball(N)and15doneither.

a Usingthisinformation,completetheVenndiagram.

SN

b AYear12studentisselectedatrandom.Whatistheprobabilitythatthestudentplaysasoccer andnetball?

c Twostudentsarechosenatrandom.Whatistheprobabilitythattheybothplaysoccer?

15 Inagroupof30students,itisfoundthat10playbothcricketandsoccer,5playcricketonlyand 7playsocceronly.

a RepresentthisinformationinaVenndiagram.

b Howmanystudentsplayneithercricketnorsoccer?

c Howmanystudentsplaycricket?

d Howmanystudentsplaycricketorsoccerorboth?

e Ifastudentisselectedatrandomfromthegroup,whatistheprobabilitythatthestudentplays cricketorsoccerbutnotboth?

f Ifanotherstudentischosen,whatistheprobabilitythatbothofthemplaycricketorsoccerbut notboth?

16 ConsiderthisVenndiagram,inwhichtheletters r,s,t and u representthenumberofelementsin eachsection.

A s r tu B

Writeanalgebraicfractionforthefollowingprobabilities.

a P(A and B)

b P(A or B)

c P(neither A nor B)

d P(A butnot B)

e P(A)

f P(B)

g P(not B)

Communicateyourreasoning,understandingandfluency

17a Constructatwo-waytableandaVenndiagraminwhich P(A or B) = 3 4 .Thisquestionhasmany correctanswers.Explainhowyouknowthatyoursisoneofthem.

b Constructatwo-waytableandaVenndiagraminwhich P(A and B) = 3 4 .Thisquestionhasmany correctanswers.Explainhowyouknowthatyoursisoneofthem.

c Constructatwo-waytableandaVenndiagraminwhich P(A butnot B) = 3 4 .Thisquestionhas manycorrectanswers.Explainhowyouknowthatyoursisoneofthem.

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Keyideasandchaptersummary

Languageof probability

Definitionof probability

Arraysandtree diagrams

Probabilityisthechanceofsomethinghappening. Samplespaceisthesetofallpossibleoutcomes. Eachoutcomeisanelementofthesamplespace. Equallylikelyoutcomeshaveanequalchanceofoccurring. Multistageeventconsistsoftwoormoreevents.

Ifalltheoutcomesareequallylikely,

Probability(Event) = Numberoffavourableoutcomes Totalnumberofoutcomes or

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

Atable(alsoknownasanarray)isanarrangementofinformationinrows andcolumns.

Atreediagramisanotherwaytolistallthepossibleresultsofamultistage experiment.

Rangeof probabilities

Probabilityofaneventthatisimpossibleis0andcertainis1.

Probabilityofaneventrangesfrom0to1.

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

Complementary events

Relativefrequency

Expectedfrequency ofanevent

Complementofanevent E istheeventnotincluding E

Probabilityofaneventanditscomplementwillsumto1.

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

E –Eventoroutcome

E –ComplementofeventEortheoutcomesnotincludingevent E

Relativefrequencyisanestimatefortheprobabilityofanevent.

Relativefrequency = Numberoftimesadatavaluewasobserved

Totalnumberofdatavalues

Expectedfrequencyisthenumberoftimesthataparticulareventshouldoccur.

Expectedfrequency = n × p = np

n = numberoftimestheexperimentisrepeated

p = probabilityoftheevent

Venndiagramsand two-waytables

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?

SAMPLEPAGES

2G 8 Icandetermineanexpectedfrequencybasedontheprobabilityofanevent.

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

2H 9 Icantransferbetweentwo-waytablesandVenndiagramsandviceversa. e.g.

a Copyandcompletethistwo-waytable,thenconvertitintoaVenndiagram.

Not A Total

5 6 Not B 7 8

b UsetheVenndiagramtocompletethetwo-waytable.

2H 10

Icansolveproblemsusingatwo-waytable.

e.g.UsingQuestion 9b,findthefollowing:

a P(not B)

b P(both A and B)

c numberof A only

d numberofeither A or B orboth A and B

2H 11 IcansolveproblemsusingaVenndiagram.

Atasausagesizzle,customerscanaddsomeonion(O)oranegg(E)tothesausage intheirroll.

Fromthefirst45customers,25peoplechoseonion,30choseeggand15hadboth.

a ConstructaVenndiagramtorepresentthisinformation.

b Howmanypeoplehadneitheronionnoregg?

c Howmanypeoplechoseonionbutnotegg?

d Ifoneofthesepeopleischosenatrandom,whatistheprobabilitythatthey choseeggbutnotonion?

e Find P(didnotchooseonion).

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?

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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 probability ofthefollowingoutcomes.

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,Sarah,Max,HayleyandDavid.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?

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.

13 2F Aclassfrequencytableshowsthescoresinatest.

59 5

69 6

79 8

89 6

Whatistherelativefrequencyforthefollowingoutcomes?Answercorrecttotwodecimalplaces.

14 2F LastyearOscarboughtapacketofbiscuitseveryweekandfound30ofthesepacketscontained brokenbiscuits.Whatistherelativefrequencyofthisevent?Answerasadecimalcorrecttotwo decimalplaces.

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

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

Extended-responsequestion

1 2H Aliedetectorwasusedtoindicatetheguiltorinnocenceof200suspects.

a Findtheprobabilityapersonselectedatrandom,withanaccuratetest,madeatruestatement?

b Findtheprobabilityapersonselectedatrandom,withafalsestatement,hasanaccuratetest?