Essential Maths for the WA Curriculum Year 10 Ch1&2 – uncorrected sample pages by cambridge_aused

ESSENTIAL MATHEMATICS

FOR THE WA CURRICULUM

UNCORRECTEDSAMPLEPAGES

David Greenwood

Sara Woolley

Jenny Goodman

Jennifer Prosser

Stuart Palmer

INCLUDES INTERACTIVE TEXTBOOK POWERED BY CAMBRIDGEHOTMATHS

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Understandingnumber

Algebraictechniques

Linearandnon-linear

Modellingwithnumber

Probabilityand

Modellingwith

Numberandalgebra

Algebraictechniques

Linearandnon-linear equationsand inequalities

Linearandnon-linear patternsand relationships

Modellingwithnumber andalgebra

7 Geometry

Warm-upquiz 534

7A Parallellines CONSOLIDATING 536

7B Triangles CONSOLIDATING 541

7C Quadrilaterals 547

7D Polygons 552

7E Congruenttriangles 557 Progressquiz 564

7F Similartriangles 565

7G Applyingsimilartriangles 571

7H Applicationsofsimilarityinmeasurement 576

Maths@Work:Poolbuilder 583 Modelling 585

8 Indices,exponentialsandlogarithms

8A Indexnotationandindexlawsfor multiplicationanddivision 603

8B Moreindexlawsandthezeroindex 608

8C Negativeindices 614

8D Scientiﬁcnotation 619

8E Graphsofexponentials 624 Progressquiz 628

8F Exponentialgrowthanddecay 629

8G Introducinglogarithms OPTIONAL 635

8H Logarithmicscales OPTIONAL 639 Maths@Work:Electricaltrades 648 Modelling 650 Digitaltoolsandcomputationalthinking 652 Puzzlesandgames 654 Chaptersummaryandchecklist 655 Chapterreview 658

Measurementand geometry

Two-dimensionalspace andstructures

Modellingwith measurementand geometry

Numberandalgebra

Algebraictechniques

Linearandnon-linear patternsand relationships

Modellingwithnumber andalgebra 9

Measurementand geometry

9A ReviewingPythagoras’theorem CONSOLIDATING

9B Findingthelengthofashorterside

9C ApplicationsofPythagoras’theorem 675

9D Trigonometricratios CONSOLIDATING 681

9E Findingsidelengths 687

9F Solvingforthedenominator 692 Progressquiz 698

9G Findingangles 700

Two-dimensionalspace andstructures

AbouttheAuthors

DavidGreenwood isanexperiencedmathematicseducatorandauthorwhohastaughtatboth ScotchCollegeMelbourneandTrinityGrammarSchoolMelbourne,whereheservedasHead ofMathematicsfor23years.HeistheleadauthoroftheEssentialMathematicsseriesfor CambridgeUniversityPressandhasauthoredmorethan100mathematicstitlesacrossarange ofyearlevels.Hisprofessionalinterestsincludecurriculumplanning,high-qualitycontent creation,andtheeffectiveuseoftechnologytoenhancemathematicsteachingandlearning.

SaraWoolley wasbornandeducatedinTasmania.ShecompletedanHonoursdegreein MathematicsattheUniversityofTasmaniabeforecompletinghereducationtraining attheUniversityofMelbourne.ShehastaughtmathematicsfromYears5to12since 2006andiscurrentlyaHeadofMathematics.Shespecialisesinlessondesignand creatingresourcesthatdevelopandbuildunderstandingofmathematicsforallstudents.

JennyGoodman hastaughtinschoolsforover28yearsandiscurrentlyteachingataselective highschoolinSydney.Jennyhasaninterestintheimportanceofliteracyinmathematics education,andinteachingstudentsofdifferingabilitylevels.ShewasawardedtheJonesMedal foreducationatSydneyUniversityandtheBourkePrizeforMathematics.Shehaswritten for CambridgeMATHSNSW andwasinvolvedinthe Spectrum and SpectrumGold series.

JenniferProsser wasraisedinPerthandcompletedherBachelorofScienceandBachelorof EducationattheUniversityofWesternAustralia.Afterbeginningasascienceteacher,Jennifer transitionedtomathematicsandhasnowtaughtmathstostudentsfromYears7to12forover 15years.ShehasbeenanActingHeadofMathematicsandisanauthoroftheYear11and12 Applicationstextbookswithinthe CambridgeSeniorMathematicsforWesternAustralia series.

StuartPalmer wasbornandeducatedinNewSouthWales.Heisafullyqualiﬁedhigh schoolmathematicsteacherwithmorethan25years’experienceteachingstudents fromallwalksoflifeinavarietyofschools.HehasbeenHeadofMathematicsintwo schools.Heisverywellknownbyteachersthroughoutthestatefortheprofessional learningworkshopshedelivers.StuartalsoassiststhousandsofYear12studentsevery yearastheypreparefortheirHSCExaminations.AttheUniversityofSydney,Stuart spentmorethanadecaderunningtutorialsforpre-servicemathematicsteachers.

UNCORRECTEDSAMPLEPAGES

Acknowledgements

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Introduction

FollowingthereleaseoftheWesternAustralianCurriculum,weareproudtointroducetheﬁrsteditionof EssentialMathematics COREfortheWACurriculum.ComparedtopreviousAustralianCurriculumeditionsofthe EssentialCORE series,schoolswillﬁnd manynewandrevisedtopicsinthisWAseries,andsomesubstantialimprovementsandnewfeaturesacrosstheprint,digitaland teacherresources.

Newcontentandsomerestructuring

Matchingtheintentofthenewcurriculum,thereismoreﬁnancialmathsateachyearlevel,fromtransactionalstatementsand ﬁnancialrecordsinYear7,tocoverageofincometaxandinvestmentsandloansinYear10.In Year7,thereisnewcontenton ratiosandproportions,netsofsolids,Venndiagrams,compositeshapesandAustraliantimezones.Allgeometrytopicsarenow containedinasinglechapter(Chapter7).In Year8,thereisnewcontentonthreedimensionalsolids,Pythagoras’theorem,index laws,simplequadraticequations,two-stepexperimentsandgradient-interceptform.For Year9,thereisnewcontentondirect proportion,errorsandaccuracy,dataandsamplingandquadraticequationsandgraphs.In Year10,thereisnewcontenton errorsandaccuracy,Inequalities,statisticaltwo-waytablesandsketchingparabolas.

ThenewWACurriculumplacesincreasedemphasison investigations and modelling,andthisiscoveredwithModelling activitiesattheendofchaptersandreviseddownloadableInvestigations.

TheWACurriculumhasintroducedOptionaltopicsatYears9and10,suchasrearrangingformulaeanddirectionandbearings.A selectnumberofthesehavebeencoveredinthosetitles,alongwithsomeExtendingtopicsthatarebeyondthecurriculumbutare consideredworthcovering.SchoolswillﬁndalloftheseOptionaltopicsclearlylabelledinthebooks.

Othernewfeatures

• Digitaltoolsandcomputationalthinking activitieshavebeenaddedtotheendofeverychaptertoaddressthe curriculum’sincreasedfocusontheuseofdigitaltoolsandtheunderstandingandapplicationofalgorithms.

• Targetedskillsheets –downloadableandprintable–havebeenwrittenforeverylessonintheseries,withtheintentionof providingadditionalpracticeforstudentswhoneedsupportatthebasicskillscoveredinthelesson,withquestionslinkedto workedexamplesinthebook.

• EditablePowerPointlessonsummaries arealsoprovidedforeachlessonintheseries,withtheintentionofsavingthetime ofteacherswhowerepreviouslycreatingthesethemselves.

• Numeracysupportfeatures inprintanddigitalhelpstudentsfromYear7onward,progressingfromfoundationnumeracy tofullOLNA-stylepreparationbyYear10.

Diagnosticassessmenttool

Alsonewforthiseditionisaﬂexible,comprehensivediagnosticassessmenttool,availablethroughtheOnlineTeachingSuite.This tool,featuringaround 10, 000 newquestions,allowsteacherstosetdiagnosticteststhatarecloselyalignedwiththetextbook content,viewstudentperformanceandgrowthviaarangeofreports,setfollow-upworkwithaviewtohelpingstudents improve,andexportdataasneeded.

Guidetotheworkingprograms

EssentialMathematicsCOREfortheWACurriculum containsworkingprogramsthataresubtlyembeddedintheexercises. ThesuggestedworkingprogramsprovidetwopathwaysthroughthebooktoallowdifferentiationforBuildingand Progressingstudents.

EachexerciseisstructuredinsubsectionsthatmatchthenewWACurriculumproﬁciencystrands(withProblem-solvingand Reasoningcombinedintoonesectiontoreduceexerciselength),aswellas‘Goldstar’( ).Thequestions* suggestedforeach pathwayarelistedintwocolumnsatthetopofeachsubsection.

• Theleftcolumn(lightestshade)showsthequestionsintheBuildingworkingprogram.

• Therightcolumn(darkestshade)showsthequestionsintheProgressingworkingprogram.

Gradientswithinexercisesandproﬁciencystrands

Theworkingprogramsmakeuseoftwo gradientsthathavebeencarefullyintegrated intotheexercises.Agradientrunsthrough theoverallstructureofeachexercise–where there’sanincreasinglevelofsophistication requiredasastudentprogressesthrough theproficiencystrandsandthenontothe ‘GoldStar’question(s)–butalsowithineach proficiencystrand;thefirstfewquestions inFluencyareeasierthanthelastfew,for example,andthefirstfewProblem-solvingand reasoningquestionsareeasierthanthelastfew.

Therightmixofquestions

Questionsintheworkingprogramshavebeenselectedtogivethemostappropriatemixoftypesofquestionsforeachlearning pathway.StudentsgoingthroughtheBuildingpathwayaregivenextrapracticeattheUnderstandingandbasicFluencyquestions andonlytheeasiestProblem-solvingandreasoningquestions.TheProgressingpathway,whilenotchallenging,spendsalittleless timeonbasicUnderstandingquestionsandalittlemoreonFluencyandProblem-solvingandreasoningquestions.TheProgressing pathwayalsoincludesthe‘Goldstar’question(s).

Choosingapathway

Thereareavarietyofwaysofdeterminingtheappropriatepathwayforstudentsthroughthecourse.Schoolsandindividual teachersshouldfollowthemethodthatworksbestforthem.Ifrequired,theWarm-upquizatthestartofeachchaptercanbe usedasadiagnostictool.Thefollowingarerecommendedguidelines:

• Astudentwhogets 40% orlowershouldheavilyrevisecoreconceptsbeforedoingtheBuildingquestions,andmayrequire furtherassistance.

• Astudentwhogetsbetween 40% and 75% shoulddotheBuildingquestions.

• Astudentwhogets 75% andhighershoulddotheProgressingquestions.

Forschoolsthathaveclassesgroupedaccordingtoability,teachersmaywishtoseteithertheBuildingorProgressingpathwaysas thedefaultpathwayforanentireclassandthenmakeindividualalterationsdependingonstudentneed.Forschoolsthathave mixed-abilityclasses,teachersmaywishtosetanumberofpathwayswithintheoneclass,dependingonpreviousperformance andotherfactors.

* Thenomenclatureusedtolistquestionsisasfollows:

3,4:completeallpartsofquestions3and4

• 1–4:completeallpartsofquestions1,2,3and4

• 10(½):completehalfofthepartsfromquestion 10(a,c,e,.....orb,d,f,.....)

• 2–4(½):completehalfofthepartsofquestions2,3and4

• –:completenoneofthequestionsinthissection.

• 4(½),5:completehalfofthepartsofquestion4 andallpartsofquestion5

•

Guidetothisresource

PRINTTEXTBOOKFEATURES

1 NEW Newlessons: authoritativecoverageofnewtopicsinthenewWACurriculumintheformofnew,road-testedlessons throughouteachbook.

2 WACurriculum: contentstrands,sub-strandsandcontentdescriptionsarelistedatthebeginningofthechapter(seethe teachingprogramformoredetailedcurriculumdocuments)

3 Inthischapter: anoverviewofthechaptercontents

4 NEW Quickreference: Multiplication,primenumber,fractionwallanddivisibilityrulestablesatthebackofthebook

5 Chapterintroduction: setscontextforstudentsabouthowthetopicconnectswiththerealworldandthehistoryof mathematics

6 Warm-upquiz: aquizforstudentsonthepriorknowledgeandessentialskillsrequiredbeforebeginningeachchapter

7 Sectionslabelledtoaidplanning: Allnon-coresectionsarelabelledas‘Consolidating’(indicatingarevisionsection)or withagoldstar(indicatingatopicthatcouldbeconsideredchallenging)tohelpteachersdecideonthemostsuitablewayof approachingthecoursefortheirclassorforindividualstudents.

8 Learningintentions: setsoutwhatastudentwillbeexpectedtolearninthelesson

9 Lessonstarter: anactivity,whichcanoftenbedoneingroups,tostartthelesson

10 Keyideas: summarisestheknowledgeandskillsforthesection

11 Workedexamples: solutionsandexplanationsofeachlineofworking,alongwithadescriptionthatclearlydescribesthe mathematicscoveredbytheexample.Workedexamplesareplacedwithintheexercisesotheycanbereferencedquickly, witheachexamplefollowedbythequestionsthatdirectlyrelatetoit.

12 Nowyoutry: try-it-yourselfquestionsprovidedaftereveryworkedexampleinexactlythesamestyleastheworkedexample togivestudentsimmediatepractice

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13 Workingprograms: differentiatedquestionsetsfortwoabilitylevelsinexercises

14 Puzzlesandgames: ineachchapterprovideproblem-solvingpracticeinthecontextofpuzzlesandgamesconnectedwith thetopic

theexercisebeginsatUnderstandingandthenFluency,withtheﬁrstquestionalwayslinkedto

achecklistofthelearningintentionsforthechapter,withexamplequestions

17 Chapterreviews: withshort-answer,multiple-choiceandextended-responsequestions;questionsthatare‘GoldStar’are clearlysignposted

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18 Maths@Work: asetofextendedquestionsacrosstwopagesthatgivepracticeatapplyingthemathematicsofthechapter toreal-lifecontexts

19 NEW Digitaltoolsandcomputationalthinking activityineachchapteraddressesthecurriculum’sincreasedfocusonthe useofdifferentdigitaltools,andtheunderstandingandimplementationofalgorithms

20 Modellingactivities: anactivityineachchaptergivesstudentstheopportunitytolearnandapplythemathematicalmodelling processtosolverealisticproblems

21 NEW FocusonOLNAsuccess: foundationmathsskilldevelopmentwithclearstrategiestoimprovenumeracyandtoreinforce concepts.ThisfeatureincludeddetailedOLNA-styleworkedexamplestobuildunderstandingandassessmentreadiness.

INTERACTIVETEXTBOOKFEATURES

22 NEW TargetedSkillsheets,oneforeachlesson,focusonasmallsetofrelatedFluency-styleskillsforstudentswhoneed extrasupport,withquestionslinkedtoworkedexamples

23 Workspaces: almosteverytextbookquestion–including allworking-out–canbecompletedinsidetheInteractive Textbookbyusingeitherastylus,akeyboardandsymbol palette,oruploadinganimageofthework

24 Self-assessment: studentscanthenself-assesstheirown workandsendalertstotheteacher.SeetheIntroductionon pagexformoreinformation

25 Interactivequestiontabs canbeclickedonsothatonly questionsincludedinthatworkingprogramareshownonthe screen

26 HOTmathsresources: ahugecateredlibraryofwidgets, HOTsheetsandwalkthroughsseamlesslyblendedwiththe digitaltextbook

27 Desmosgraphingcalculator,scientiﬁccalculatorand geometrytoolarealwaysavailabletoopenwithineverylesson

28 Scorcher: thepopularcompetitivegame

29 Workedexamplevideos: everyworkedexampleislinkedto ahigh-qualityvideodemonstration,supportingbothin-class learningandtheﬂippedclassroom

30 Arevisedsetof differentiatedauto-markedpractice quizzes perlessonwithsavedscores

31 Auto-markedmaths literacyactivitiesteststudentson theirabilitytounderstandandusethekeymathematical languageusedinthechapter 29

32 Auto-markedpriorknowledgepre-test (the‘Warm-upquiz’oftheprintbook)fortestingtheknowledgethatstudents willneedbeforestartingthechapter

33 Auto-markedprogressquizzesandchapterreviewquestions inthechapterreviewscanbecompletedonline

DOWNLOADABLEPDFTEXTBOOK

34 InadditiontotheInteractiveTextbook,a PDFversionofthetextbook hasbeenretainedfortimeswhenuserscannotgo online.PDFsearchandcommentingtoolsareenabled.

ONLINETEACHINGSUITE

35 NEW DiagnosticAssessmentTool included withtheOnlineTeachingSuiteallowsforﬂexible diagnostictesting,reportingandrecommendations forfollow-upworktoassistyoutohelpyour studentstoimprove

36 NEW PowerPointlesson summariescontainthe mainelementsofeachlessoninaformthatcanbe annotatedandprojectedinfrontofclass

37 NEW Numeracysuccess toolidentiﬁesskills gapsimpactingachievementoftheminimum standardinnumeracyandrecommendsfollow updigitalteachingmaterial,whilecontext-based practicequestionsstrengthenapplicationandbuild assessmentreadiness.

38 LearningManagementSystem withclass andstudentanalytics,includingreportsand communicationtools

39 Teacherviewofstudents’workand self-assessment allowstheteachertoseetheir class’sworkout,howstudentsintheclassassessed theirownwork,andany‘redﬂags’thattheclass hassubmittedtotheteacher

40 Powerfultestgenerator withahugebankof levelledquestionsaswellasready-madetests

41 Revampedtaskmanager allowsteachersto incorporatemanyoftheactivitiesandtoolslisted aboveintoteacher-controlledlearningpathways thatcanbebuiltforindividualstudents,groupsof studentsandwholeclasses

42 Worksheets,Skillanddrill,mathsliteracy worksheets,and twodifferentiatedchapter testsineverychapter,providedineditableWord documents

43 Moreprintableresources: allPre-tests andProgressquizzesandApplicationsand problem-solvingtasksareprovidedinprintable worksheetversions

1 Consumer arithmetic

Essentialmathematics:Whyskillswithpercentagesand consumerarithmeticareimportant

Masteringmoneymanagementskillsareanessentialfoundationforyoutoachievepersonal ﬁnancialindependenceandhavesuccessinyourbusinessventures.

Essentialskillsusingpercentagesincludecalculationsofproﬁts,discounts,costprice,sellingprice andGST.Bycomparingdiscountedsellingprices,thebestdealcanbefound.

Incometaxcalculationshelpworkersandbusinessestobeawareoflegaltaxobligations,keep recordsforeligibledeductions,andnotbefacedwithunexpectedandcostlytaxdebtsattheend ofaﬁnancialyear.

Whenyoujointheworkforceandearnawage,itisimportanttoprepareapersonalbudget.This includesasavingsplantopayﬁxedandvariablecost-of-livingexpensesandmoneyputasidefor personaluse.

Calculationsusingsimpleandcompoundinterestratesenableapersontocomputeandcompare thefullcostofloans,debtrepaymentsandthepotentialfuturevalueofinvestments.

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Inthischapter

1AReviewofpercentages (Consolidating)

1BApplicationsofpercentages

1CIncome

1DIncometaxation

1EBudgeting

1FSimpleinterest

1GCompoundinterest

1HInvestmentsandloans

1IComparinginterestusing digitaltools

WACurriculum

Thischaptercoversthefollowing contentdescriptorsintheWA Curriculum:

NUMBERANDALGEBRA

WA10MNAF1,WA10MNAF2, WA10MNAM1

©SchoolCurriculumandStandards Authority

Onlineresources

Ahostofadditionalonlineresources areincludedaspartofyourInteractive Textbook,includingHOTmathscontent, videodemonstrationsofallworked examples,auto-markedquizzesand muchmore.

SAMPLEPAGES

1 Findthefollowingtotals.

$87560 ÷ 52 (tothenearestcent) e

2 Expressthefollowingfractionswithdenominatorsof 100

3 Writeeachofthefollowingfractionsasdecimals.

4 Roundthefollowingdecimalstotwodecimalplaces.

5 Givethevaluesofthepronumeralsinthefollowingtable.

6 Calculatethefollowingannualincomesforeachofthesepeople.Use 52 weeksinayear. Jai: $1256 perweek a Sushena: $15600 permonth b Anthony: $1911 perfortnight c Crystal: $17.90 perhour,for 40 hoursperweek,for 50 weeksperyear d

7 Withoutacalculator,ﬁnd:

8 Findthesimpleinterestonthefollowingamounts.

$400 at 5% p.a.for 1 year a $5000 at 6% p.a.for 1 year b $800 at 4% p.a.for 2 years c

9 Completethefollowingtable,givingthevaluesofthepronumerals.

10 Thefollowingamountsincludethe 10% GST.Bydividingeachoneby 1.1,ﬁndtheoriginal costsbeforetheGSTwasaddedtoeach.

$55 a $61.60 b $605 c

1A 1A Reviewofpercentages

Learningintentions

• Tounderstandthatapercentageisanumberoutof 100

• Tobeabletoconvertdecimalsandfractionstopercentagesandviceversa

• Tobeableto ndthepercentageofaquantity

Keyvocabulary: percentage,denominator

Itisimportantthatweareabletoworkwith percentagesinoureverydaylives.Banks,retailers andgovernmentsusepercentageseverydaytowork outfeesandprices.

Lessonstarter:Whichoptionshould Jamiechoose?

Jamiecurrentlyearns $68460 p.a.(peryear)andisgiven achoiceoftwodifferentpayrises.Whichshouldshe chooseandwhy?

ChoiceA:Increaseof $25 perweek

ChoiceB:Increaseof 2% onperannumsalary

Keyideas

Bankswillusepercentagestoworkoutaccountfees andhowmuchinteresttocharge.

A percentage means‘outof 100’.Itcanbewrittenusingthesymbol %,orasafractionor adecimal.

Forexample: 75 percent = 75%= 75 100 or 3 4 or 0.75.

Toconvertafractionoradecimaltoapercentage,multiplyby 100

Toconvertapercentagetoafraction,writeitwitha denominator of 100 andsimplify.

Toconvertapercentagetoadecimal,divideby 100 15%= 15 ÷ 100 = 0.15

Toﬁndapercentageofaquantity,writethepercentageasafractionora decimal,thenmultiplybythequantity,i.e. x% of P = x 100 × P

Exercise1A

Und er stand ing

UNCORRECTEDSAMPLEPAGES

1 Completethefollowingusingthewords multiply or divide

a Toconvertadecimaltoapercentage by 100

b Toconvertapercentagetoadecimal by 100.

c Toconvertafractiontoapercentage by 100

d Toconvertapercentagetoafraction by 100

2 Completethefollowingtoexpressasafractioninpart a andadecimalinpart b a 7%= 7 i 23%= ii

b 18%= i 5%= ii

3 Completethefollowing.

Example1Convertingtoapercentage

Writeeachofthefollowingasapercentage.

HintforQ3:Cancelanyfractions beforemultiplying.

Writeusingadenominatorof 100 bymultiplying numeratoranddenominatorby 5 Alternatively,multiplythefractionby 100

Multiplythefractionby 100

Cancelcommonfactors,thensimplify. c

Multiplythedecimalby 100 Movethedecimalpointtwoplacestotheright.

Nowyoutry

Writeeachofthefollowingasapercentage.

Converteachfractiontoapercentage.

Writethesedecimalsaspercentages.

HintforQ4:Firstwriteusinga denominatorof 100 or, alternatively,multiplyby 100

HintforQ5:Tomultiplyby 100, movethedecimalpointtwoplaces totheright.

Example2Writingpercentagesassimpliﬁedfractions

Writeeachofthefollowingpercentagesasasimpliﬁedfraction.

Solution

Explanation

Writethepercentagewithadenominatorof 100.

Writethepercentagewithadenominatorof 100 Simplify 58 100 bycancelling,usingtheHCFof 58 and 100, whichis 2

Writethepercentagewithadenominatorof 100

Doublethenumerator (6 1 2 ) andthedenominator(100) sothatthenumeratorisawholenumber.

Nowyoutry

Writeeachofthefollowingpercentagesasasimpliﬁedfraction.

Writeeachpercentageasasimpliﬁedfraction.

HintforQ6:Write withadenominator of 100,thensimplify ifpossible.

Example3Writingapercentageasadecimal

Convertthesepercentagestodecimals.

Solution

a 93%= 93 ÷ 100 = 0.93

b 7%= 7 ÷ 100 = 0.07

c 30%= 30 ÷ 100 = 0.3

Nowyoutry

Convertthesepercentagestodecimals.

Explanation

Dividethepercentageby 100.Thisisthesameasmoving thedecimalpointtwoplacestotheleft.

Dividethepercentageby 100

Dividethepercentageby 100 Write 0.30 as 0.3.

7 Convertthesepercentagestodecimals.

Example4Findingapercentageofaquantity

Find 42% of $1800

Solution

42% of $1800 = 0.42 × 1800 = $756

Explanation

Rememberthat‘of’meanstomultiply. Write 42% asadecimalorafraction: 42%= 42 100 = 0.42

Thenmultiplybytheamount.

Ifusingacalculator,enter 0.42 × 1800

Withoutacalculator: 42 ✟✟ 100 1 × 18✚✚ 00 = 42 × 18 = 756

Nowyoutry

Find 36% of $2300

8 Useacalculatortoﬁndthefollowing.

Problem-solving and reasoning

9 A 300 gpiecontains 15 gofsaturatedfat.

a Whatfractionofthepieissaturatedfat?

b Whatpercentageofthepieissaturatedfat?

HintforQ9: 15 goutof 300 g.

10 About 80% ofthemassofahumanbodyiswater.IfHugois 85 kg,howmanykilogramsofwaterare inhisbody?

11 Remaspends 12% ofthe 6.6 hourschooldayinmaths.Howmanyminutesarespentinthe mathsclassroom?

12 Inacricketmatch,Brettspent 35 minutesbowling.

Histeam’stotalﬁeldingtimewas 3 1 2 hours. Whatpercentageoftheﬁeldingtime,correcttotwo decimalplaces,didBrettspendbowling?

HintforQ12:Firstconvert hourstominutes,andthen writeafractioncomparing times.

13 Alargedoglost 8 kg,andnowweighs 64 kg.Whatpercentageofitsoriginalweightdiditlose?

14 47.9% ofalocalcouncil’sbudgetisspentongarbagecollection.Ifaratepayerpays $107.50 per quarterintotalratecharges,howmuchdotheycontributeinayeartogarbagecollection?

Australia’sstatistics —15

15 BelowisthepreliminarydataonAustralia’spopulationgrowth,asgatheredbytheAustralianBureau ofStatisticsforagivenyear.

a Calculatethepercentagechangeforeachstateandterritoryshown usingthepreviousyear’spopulation,andcompletethetable.

b WhatpercentageofAustralia’soverallpopulation,correctto onedecimalplace,islivingin: NSW? i Vic? ii WA? iii

c Useaspreadsheettodrawapiechart(i.e.sectorgraph) showingthepopulationsoftheeightstatesandterritoriesinthetable.Whatpercentageof thetotalisrepresentedbyeachstate/territory?Roundyouranswertothenearestpercent.

d Inyourpiechartforpart c,whatistheanglesizeofthesectorrepresentingVictoria?

1B 1B Applicationsofpercentages

Learningintentions

• Tounderstandwhatapercentageincreaseordecreaseofaquantityrepresents

• Tobeabletoincreaseanddecreaseanamountbyagivenpercentage

• Tobeabletousepercentageincreaseanddecreasetocalculateasellingpriceoradiscountedprice

• Tobeabletodeterminethepro tmadeonanitemandcalculatethisasapercentagepro t

Keyvocabulary: discount,pro t,sellingprice,costprice

Therearemanyapplicationsofpercentages.Pricesare oftenincreasedbyapercentagetocreateaproﬁtor decreasedbyapercentagewhenonsale.

Whengoodsarepurchasedbyastore,thecosttothe owneriscalledthecostprice.Thepriceofthegoods soldtothecustomeriscalledthesellingprice.This pricewillvaryaccordingtowhetherthestoreishaving asaleordecidestomakeacertainpercentageproﬁt.

Lessonstarter:Discounts

Discussasaclass:

• Whichisbetter: 20% offora $20 discount?

Duringasale,aretailshopwilloftenofferapercentage discount,whereapercentageofthesellingpriceis subtractedtoformanewdiscountedsellingprice.

• Ifadiscountof 20% or $20 resultedinthesameprice,whatwastheoriginalprice?

• Whyarepercentagesusedtoshowdiscounts,ratherthanadollaramount?

Keyideas

Toincreasebyagivenpercentage,multiplybythesumof 100% andthegivenpercentage.

Forexample:Toincreaseby 12%,multiplyby 112% or 1.12

Todecreasebyagivenpercentage,multiplyby 100% minusthegivenpercentage.

Forexample:Todecreaseby 20%,multiplyby 80% or 0.8

Proﬁtsanddiscounts:

• Thenormalpriceofthegoodsrecommendedbythemanufactureriscalledtheretailprice.

• Whenthereisasaleandthegoodsarepricedlessthantheretailprice,theyaresaid tobe discounted

• Proﬁt istheamountofmoneymadebysellinganitemorserviceformorethanitscost.

• Proﬁt = sellingprice - costprice,where sellingprice istheamounttheitemissold forand costprice istheoriginalcosttotheseller.

• Percentageproﬁt = proﬁt costprice × 100

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• Percentagediscount = discount costprice × 100

Exercise1B

Und er stand ing

1 Bywhatnumberdoyoumultiplytoincreaseanamountby:

2 Bywhatnumberdoyoumultiplytodecreaseanamountby:

3 Usethewords sellingprice or costprice tocompletethefollowing.

a Aproﬁtismadewhenthe ismorethanthe

b Adiscountinastorereducesthe

c Proﬁt = -

4 Decidehowmuchproﬁtorlossismadeineachofthefollowingsituations.

= $15 a

1–44

× 1.08 =

%+ 8%= 108% Write 108% asadecimal(orfraction)andmultiply bytheamount. Showtwodecimalplacestorepresentthecents.

Example6Decreasingbyagivenpercentage

Decrease $8900 by 7%

Solution

$8900 × 0.93 = $8277.00

Explanation

100%- 7%= 93%

Write 93% asadecimal(orfraction)andmultiplyby theamount.

Remembertoputtheunitsinyouranswer.

Nowyoutry

Decrease $2700 by 18%

Decrease $1500 by 5%

Decrease $470 by 20%

Decrease $550 by 25%.

Decrease $119.50 by 15%

Decrease $400 by 10%

Decrease $80 by 15%

$49.50 by 5%.

Decrease $47.10 by 24%

Example7Calculatingproﬁtandpercentageproﬁt

Thecostpriceforanewcaris $24780 anditissoldfor $27600

Calculatetheproﬁt. a Calculatethepercentageproﬁt,totwodecimalplaces.

Solution

Explanation

a Proﬁt = sellingprice - costprice Writetherule. = $27600 - $24780

Substitutethevaluesandevaluate. = $2820

b Percentageproﬁt = proﬁt costprice × 100 Writetherule. = 2820 24780 × 100

Substitutethevaluesandevaluate. = 11.38% Roundyouranswerasinstructed.

Nowyoutry

Thecostpriceforanewrefrigeratoris $888 anditissoldfor $997

Calculatetheproﬁt. a Calculatethepercentageproﬁt,totwodecimalplaces. b

7 Copyandcompletethe tableonproﬁtandpercentageproﬁt.

Costprice Sellingprice Proﬁt Percentageproﬁt

a $10 $16

b $240 $300

c $15 $18

d $250 $257.50

e $3100 $5425

f $5.50 $6.49

Uncorrected

HintforQ7: Percentageproﬁt = proﬁt costprice × 100

Example8Findingthesellingprice

Aretailerbuyssomecalicomaterialfor $43.60 aroll.Hewishestomakea 35% proﬁt.

Whatwillbethesellingpriceperroll?

a Ifhesells 13 rolls,whatproﬁtwillhemake? b

Solution

a Sellingprice = 135% of $43.60

= 1.35 × $43.60

= $58.86 perroll

b Proﬁtperroll = $58.86 - $43.60

Explanation

Fora 35% proﬁt,theunitpriceis 135%

Write 135% asadecimal (1.35) andevaluate.

Sellingprice - costprice = $15.26

Totalproﬁt = $15.26 × 13

= $198.38

Nowyoutry

Thereare 13 rollsat $15.26 proﬁtperroll.

Aretailerbuysswimsuitsfor $32 persuit.Shewishestomakea 30% proﬁt.

Whatwillbethesellingpriceofeachswimsuit?

a Ifshesells 20 swimsuits,whatproﬁtwillshemake? b

8 AretailerbuyssomeChristmassnowglobesfor $41.80 each. Shewishestomakea 25% proﬁt.

a Whatwillbethesellingpricepersnowglobe?

b Ifshesellsaboxof 25 snowglobes,whatproﬁtwill shemake?

9 Asecond-handcardealerboughtatrade-incarfor $1200 andwishestoresellitfora 28% proﬁt.What istheresaleprice?

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Example9Findingthediscountedprice

Ashirtworth $25 isdiscountedby 15%

Whatisthesellingprice? a Howmuchisthesaving? b

Solution

Explanation

a Sellingprice = 85% of $2515% discountmeanstheremustbe 85% left (100%- 15%)

= 0.85 × $25

Convert 85% to 0.85 andmultiplybytheamount. = $21.25

b Saving = 15% of $25

= 0.15 × $25

Yousave 15% oftheoriginalprice.

Convert 15% to 0.15 andmultiplybytheoriginal price. = $3.75

orsaving = $25 - $21.25 = $3.75

Nowyoutry

Saving = originalprice - discountedprice

Asuitcaseworth $220 isdiscountedby 35%

Whatisthesellingprice? a Howmuchisthesaving? b

10 Samanthabuysawetsuitfromthesportsstorewheresheworks.Itsoriginalpricewas $79.95 Employeesreceivea 15% discount. Whatisthesellingprice? a HowmuchwillSamanthasave? b

11 Atravelagentoffersa 12.5% discountonairfaresifyoutravel duringMayorJune. ThenormalfaretoLondon(return trip)is $2446 a Whatisthesellingprice? b Howmuchisthesaving?

12 Astoresellssecond-handgoodsat 40% offtherecommendedretail price.Alawnmowerisvalued at $369

Whatisthesellingprice? a Howmuchwouldyousave? b

Problem-solving and reasoning

13–1615–18

13 Skijacketsaredeliveredtoashopinpacksof 50 for $3500.Theshopownerwishestomakea 35% proﬁt.

a Whatwillbethetotalproﬁtmadeonapack?

b Whatistheproﬁtoneachjacket?

14 Apairofsportsshoesisdiscountedby 47%.Therecommendedpricewas $179. a Whatistheamountofthediscount?

b Whatwillbethediscountedprice?

15 JeansarepricedataMaysalefor $89.Ifthisisasavingof 15% offthesellingprice,whatdothejeansnormallysellfor?

16 Discountedtyresarereducedinpriceby 35%.Theynowsellfor $69 each.Determine: a thenormalpriceofonetyre b thesavingifyoubuyonetyre.

17 Thelocalshoppurchasesacartonofcontainersfor $54.Eachcontainerissoldfor $4 Ifthecartonhad 30 containers,determine: theprofitpercontainer a thepercentageprofitpercontainer,totwodecimalplaces b theoverallprofitpercarton c theoverallpercentageprofit,totwodecimalplaces.

18 Aretailerbuysabookfor $50 andwantstosellitfora 26% proﬁt.The 10% GSTmustthenbeaddedontothecostofthebook.

a Calculatetheproﬁtonthebook.

b HowmuchGSTisaddedtothecostofthebook?

c Whatistheadvertisedpriceofthebook,including theGST?

d Findtheoverallpercentageincreaseoftheﬁnalselling pricecomparedtothe $50 costprice. Buildingagazebo

19 Christopherdesignsagazeboforanewhouse.Hebuysthetimberfromaretailer,whosourcesitat wholesalepriceandthenmarksitupbeforesellingtoChristopheratretailprice.Thetablebelow showsthewholesalepricesaswellasthemark-upforeachtypeoftimber.

DetermineChristopher’soverallcostforthematerial,includingthemark-up. a Determinetheprofitmadebytheretailer. b Determinetheretailer’soverallpercentageprofit,totwodecimalplaces. c Iftheretailerpays 27% ofhisprofitsintax,howmuchtaxdoeshepayonthissale? d

1C 1C Income

Learningintentions

• Tounderstandarangeofdifferentwaysinwhichemployeescanbepaid

• Toknowhownetincomeiscalculatedfromgrossincomeanddeductions

• Tobeabletocalculatewagesforovertimeandshiftwork

• Tobeabletocalculatecommission

Keyvocabulary: wages,commission,salary,fees,grossincome,overtime,deductions,netincome,timeandahalf, doubletime,deductions

Youmayhaveearnedmoneyforbaby-sittingor deliveringnewspapersorhaveapart-timejob.As youmoveintotheworkforceitisimportantthatyou understandhowyouarepaid.

Lessonstarter:Whoearnswhat?

Asaclass,discussthedifferenttypesofjobsheldby differentmembersofeachperson’sfamily,anddiscuss howtheyarepaid.

• Whatarethedifferentwaysthatpeoplecanbe paid?

• Whatdoesitmeanwhenyouworkfewerthan full-timehours?

• Whatdoesitmeanwhenyouworklongerthan full-timehours?

Whatothertypesofincomecanpeopleintheclassthinkof?

Keyideas

Methodsofpayment

Employeescanbepaidindifferentways,according totheirtypeofwork.Forexample,employeescanbe paidanhourlyrate,asalary,acommissionorafee.

Hourly wages:Youarepaidacertainamountperhourworked.

Commission:Youarepaidapercentageofthetotalamountofsales.

Salary:Youarepaidasetamountperyear,regardlessofhowmanyhoursyouwork.

Fees:Youarepaidaccordingtothechargesyouset,e.g.doctors,lawyers,contractors.

Sometermsyoushouldbefamiliarwithinclude:

• Grossincome:thetotalamountofmoneyyouearnbeforetaxesandotherdeductions

• Deductions:moneytakenfromyourincomebeforeyouarepaid,e.g.taxation,union fees,superannuation

• Netincome:theamountofmoneyyouactuallyreceiveafterthedeductionsaretaken fromyourgrossincome.

Netincome = grossincome - deductions

Paymentsbyhourlyrate

Ifyouarepaidbythehour,youwillbepaidanamountperhourforyournormalworkingtime. Ifyouwork overtime (hoursbeyondthenormalworkinghours),theratesmaybedifferent.

UNCORRECTEDSAMPLEPAGES

Usually,normalworkingtimeis 38 hoursperweek.

Normal: 1.0 × normalrate

Timeandahalf: 1.5 × normalrate

Doubletime: 2.0 × normalrate

Ifyouworkshiftwork,thehourlyratesmaydifferfromshifttoshift. Forexample:

6 a.m.– 2 p.m.

$24.00/hour (regularrate)

2 p.m.– 10 p.m. $27.30/hour (afternoonshiftrate)

10 p.m.– 6 a.m. $36.80/hour (nightshiftrate)

Exercise1C

Und er stand ing

1 Matchthejobdescriptionontheleftwiththemethodofpayment ontheright.

Mayaispaid $85600 peryear a hourlywage A

Danielleearns 3% ofallthesalesshemakes b fee B

Jettearns $18.90 perhourworked c commission C

Stuartcharges $450 foraconsultation d salary D

2 Callumearns $1090 aweekandhasannualdeductionsof $19838

1–33

HintforQ2: Net = total - deductions WhatisCallum’snetincomefortheyear?Assume 52 weeks inayear.

3 IfTaoearns $15.20 perhour,calculatehis:

= 1.5 × hourlyrate time-and-a-halfrate a double-timerate b

Fluency

Example10Findinggrossandnetincome(includingovertime)

4–85–9

Paulineispaid $13.20 perhouratthelocalstockyardtomuckoutthestalls.Hernormalhoursof workare 38 hoursperweek.Shereceivestimeandahalfforthenext 4 hoursworkedanddouble timeafterthat.

Whatwillbehergrossincomeifsheworks 50 hours? a Ifshepays $220 perweekintaxationand $4.75 inunionfees,whatwillbeherweekly netincome? b

a Grossincome = 38 × $13.20

Normal 38 hours

Overtimeratefornext 4 hours:timeanda half = 1.5 × normal

Overtimeratefornext 8 hours:double time = 2 × normal b Netincome = $792 - ($220 + $4.75) = $567.25

Netincome = grossincome - deductions

UNCORRECTEDSAMPLEPAGES

Nowyoutry

Tobyispaid $17.50 perhourathissupermarketjob.Hisnormalhoursofworkare 38 hoursperweek. Hereceivestimeandahalfforthenext 6 hoursworkedanddoubletimeafterthat. Whatwillbehisgrossincomeifheworks 48 hoursinaweek? a Ifhepays $240 perweekintaxationand $6.50 inunionfees,whatwillbehisweeklynetincome? b

4 Jackispaid $14.70 perhour.Hisnormalhoursofworkare 38 hoursperweek.Hereceivestimeand ahalfforthenext 2 hoursworkedanddoubletimeafterthat.

a Whatwillbehisgrossincomeifheworks 43 hours?

b Ifhehas $207.20 ofdeductions,whatwillbehisweeklynetincome?

5 Copyandcompletethistable.

Example11Calculatingshiftwork

Michaelisashiftworkerandispaid $31.80 perhourforthemorningshift, $37.02 perhourforthe afternoonshiftand $50.34 perhourforthenightshift.Eachshiftis 8 hours.Inagivenfortnighthe worksfourmorning,twoafternoonandthreenightshifts.Calculatehisgrossincome.

Solution Explanation

Grossincome = 4 × 31.80 × 84 morningshiftsat $31.80 perhourfor 8 hours + 2 × 37.02 × 82 afternoonshiftsat $37.02 perhourfor 8 hours

+ 3 × 50.34 × 83 nightshiftsat $50.34 perhourfor 8 hours

= $2818.08

Nowyoutry

Grossincomebecausetaxhasnotbeenpaid.

Kateisashiftworkerandispaid $26.20 perhourforthemorningshift, $32.40 perhourforthe afternoonshiftand $54.25 perhourforthenightshift.Eachshiftis 8 hours.Inagivenfortnightshe worksﬁvemorning,threeafternoonandtwonightshifts.Calculatehergrossincome.

6 Gregworksshiftsataprocessingplant.Inagivenrostered fortnightheworks:

• 3 dayshifts($31.80 perhour)

• 4 afternoonshifts($37.02 perhour)

• 4 nightshifts($50.34 perhour).

a Ifeachshiftis 8 hourslong,determineGreg’sgross incomeforthefortnight.

b Iftheanswertopart a isGreg’saveragefortnightly income,whatwillbehisgrossincomefora year(i.e. 52 weeks)?

HintforQ6: Afortnight = 2 weeks

Manyhospitalworkersworkshiftwork.

Example12Calculatingincomeinvolvingcommission

Jeffsellsmembershipstoagymandreceives $225 perweekplus 5.5% commissiononhissales. Calculatehisgrossincomeaftera 5-dayweek.

Solution

Totalsales = $4630

Commission = 5.5% of $4630 = 0.055 × $4630 = $254.65

Grossincome = $225 + $254.65 = $479.65

Nowyoutry

Explanation

Determinethetotalsales: 680 + 450 + 925 + 1200 + 1375.

Determinethecommissiononthetotalsalesat 5.5% by multiplying 0.055 bythetotalsales.

Grossincomeis $225 pluscommission.

Jinsellsvacuumcleanersandreceives $250 perweekplus 4.3% commissiononhersales.

Calculatehergrossincomeaftera 5-dayweek.

7 Acarsalespersonearns $5000 amonthplus 3.5% commissiononallsales.InthemonthofJanuary salestotalwas $56000.Calculate: theircommissionforJanuary a theirgrossincomeforJanuary. b

8 Arealestateagentreceives 2.75% commissiononthesaleofahousevaluedat $1250000. Findthecommissionearned.

9 Sarahearnsanannualsalaryof $77000 plus 2% commissiononallsales.Find: a herweeklybasesalarybeforesales

b hercommissionforaweekwhenhersalestotalled $7500

c hergrossweeklyincomefortheweekinpart b

d herannualgrossincomeifovertheyearhersalestotalled $571250

Problem-solving and reasoning

10 IfSimonereceives $10000 onthesaleofapropertyworth $800000,calculateherrateofcommission. HintforQ10:Whatpercentage of $800000 is $10000?

11 Jonahearnsacommissiononhissalesoffashionitems.Forgoods tothevalueof $2000 hereceives 6% andforsalesover $2000 he receives 9% ontheamountinexcessof $2000.Inagivenweek hesold $4730 worthofgoods.Findthecommissionearned.

SAMPLEPAGES

12 Williamearns 1.75% commissiononallsalesattheelectrical goodsstorewhereheworks.IfWilliamearns $35 incommission onthesaleofonetelevision,howmuchdidtheTVsellfor?

1.75% is $35

13 Refertothepayslipbelowtoanswerthefollowingquestions.

KugerIncorporated

EmployeeID: 75403A

Name:ElmoRodriguez

PayMethod:EFT

Bankaccountname:E.Rodriguez

Bank:MathsvilleCreditUnion

BSB: 102-196 AccountNo: 00754031 TaxStatus:GenExempt

Page: 1

PayPeriod: 21/05/2026

a WhichcompanydoesElmoworkfor?

b WhatisthenameofElmo’sbankandwhatishisaccountnumber?

c HowmuchgrosspaydoesElmoearnin 1 year?

d HowoftendoesElmogetpaid?

e Howmuch,peryear,doesElmosalarysacriﬁce?

f HowmuchisElmo’shealthfundcontributioneachweek?

g Calculate 1 year’sunionfees.

h Usingtheinformationonthispayslip,calculateElmo’sannualtaxandalsohisannualnetincome.

i IfElmoworksMondaytoFridayfrom 9 a.m.to 5 p.m.eachdayforanentireyear,calculatehis effectivehourlyrateofpay.UseElmo’sfortnightlypaymentasastartingpoint.

1D 1D Incometaxation

Learningintentions

• TounderstandhowthekeycomponentsoftheAustraliantaxationsystemwork

• Tobeabletocalculateaperson’staxableincome

• Tobeabletocalculateaperson’staxpayableusingAustraliantaxbrackets Keyvocabulary: taxation,employer,employee,taxreturn,taxableincome,taxbracket,levy,deductions, p.a.(perannum)

Ithasbeensaidthatthereareonlytwo surethingsinlife:deathandtaxes!The AustralianTaxationOfﬁce(ATO)collects taxesonbehalfofthegovernmenttopay foreducation,hospitals,roads,railways, airportsandservices,suchasthepolice andﬁrebrigades.

InAustralia,theﬁnancialyearrunsfrom July 1 toJune 30 thefollowingyear. Peopleengagedinpaidemploymentare normallypaidweeklyorfortnightly.Most ofthempaysomeincometaxeverytime theyarepaidfortheirwork.Thisis knownasthePay-As-You-Gosystem (PAYG).

Attheendoftheﬁnancialyear (June 30),peoplewhoearnedanincome

Theamountofincometaxanemployeemustpayeachﬁnancialyear willdependonsettaxratesestablishedbytheATO.

completeanincometaxreturntodetermineiftheyhavepaidthecorrectamountofincometaxduringthe year.

Iftheypaidtoomuch,theywillreceivearefund.Iftheydidnotpayenough,theywillberequiredto paymore.

TheAustraliantaxsystemisverycomplexandthelawschangefrequently.Thissectioncoversthemain aspectsonly.

Lessonstarter:TheATOwebsite

TheAustralianTaxationOfﬁcewebsitehassomeincometaxcalculators.Useonetoﬁndouthowmuch incometaxyouwouldneedtopayifyourtaxableincomeis:

$10400 perannum(i.e. $200 perweek)

$20800 perannum(i.e. $400 perweek)

$31200 perannum(i.e. $600 perweek)

$41600 perannum(i.e. $800 perweek).

Doesapersonearning $1000 perweekpaytwiceasmuchtaxasapersonearning $500 perweek?

Doesapersonearning $2000 perweekpaytwiceasmuchtaxasapersonearning $1000 perweek?

Keyideas

The Employee You (the employee and taxpayer)

The Employer

The boss (your employer)

ThePAYGtaxsystemworksinthefollowingway.

The ATO

The Australian Taxation Ofﬁce

• Theemployeeworksforandgetspaidbytheemployereveryweek,fortnightormonth.

• Theemployercalculatesthetaxthattheemployeeshouldpayfortheamountearned bytheemployee.

• TheemployersendsthattaxtotheATOeverytimetheemployeegetspaid.

• TheATOpassestheincometaxtothefederalgovernment.

• OnJune 30,theemployergivestheemployeeapaymentsummarytoconﬁrmtheamount oftaxthathasbeenpaidtotheATOonbehalfoftheemployee.

• BetweenJuly 1 andOctober 31,theemployeecompletesa taxreturn andsendsittothe ATO.Somepeoplepayaregisteredtaxagenttodothisreturnforthem.

• Onthistaxreturn,theemployeeliststhefollowing.

– All formsofincome,includinginterestfrominvestments.

– Legitimatedeductionsshownonreceiptsandinvoices,suchaswork-relatedexpenses anddonations.

• Taxableincome iscalculatedusingtheformula: Taxableincome = grossincome - deductions

• TherearetablesandcalculatorsontheATOwebsite.Thetablebelowshowsthetaxrates introducedforthe2024/2025ﬁnancialyear.Taxratescanchange;refertotheATOwebsite forcurrenttaxratesforthe2025/2026ﬁnancialyear.

–$18200 Nil $18201–$45000 16cforeach $1 over $18200

$45001–$135000 $4288 plus 30cforeach $1 over $45000

$135001–$190000 $31288 plus 37cforeach $1 over $135000 $190001 andover $51638 plus 45cforeach $1 over $190000 Thistablecanbeusedtocalculatetheamountoftaxyou shouldhave paid(i.e.thetax payable),asopposedtothetaxyou did payduringtheyear(i.e.thetaxwithheld).Each rowinthetableiscalleda taxbracket

• YoumayalsoneedtopaytheMedicare levy.ThisisaschemeinwhichallAustralian taxpayersshareinthecostofrunningthemedicalsystem.Formanypeoplethisiscurrently 2% oftheirtaxableincome.

• Itispossiblethatyoumayhavepaidtoomuchtaxduringtheyearandwillreceivea taxrefund.

• ItisalsopossiblethatyoumayhavepaidtoolittletaxandwillreceivealetterfromtheATO askingforthetaxliabilitytobepaid.

Exercise1D

Und er stand ing

1–32,3

Note:Thequestionsinthisexerciserelatetothetaxtablegiveninthe Keyideas,unlessstatedotherwise.

1 Completethisstatement:Taxableincome = incomeminus .

2 Basedonthetableinthe Keyideas,determineifthefollowingstatementsaretrueorfalse?

a Ataxableincomeof $10400 requiresnotaxtobepaid.

b ThehighestincomeearnersinAustraliapay 45 centstaxforeverydollartheyearn.

3 Inthe2024/2025ﬁnancialyear,Ann’staxableincomewas $135000,whichputsherattheverytopof themiddletaxbracketinthetaxtable.Ben’staxableincomewas $190000,whichputshiminahigher taxbracket.IgnoringtheMedicarelevy,howmuchextrataxdidBenpaycomparedtoAnn?

Fluency

Example13Calculatingincometaxpayable

4,54–6

Duringthe2024/2025ﬁnancialyear,Richardearned $1050 perweek($54600 perannum)fromhis employerandothersources,suchasinterestoninvestments.Hehasreceiptsfor $375 for work-relatedexpensesanddonations.

CalculateRichard’staxableincome. a Usethis2024/2025taxtabletocalculateRichard’staxpayableamount.

b Taxableincome Taxonthisincome

0–$18200 Nil

$18201–$45000 16cforeach $1 over $18200

$45001–$135000 $4288 plus 30cforeach $1 over $45000

$135001–$190000 $31288 plus 37cforeach $1 over $135000

$190001 andover $51638 plus 45cforeach $1 over $190000

RichardmustalsopaytheMedicarelevyof 2% ofhistaxableincome.Howmuchisthe Medicarelevy?

c AddthetaxpayableandtheMedicarelevyamounts. d Expressthetotaltaxinpart d asapercentageofRichard’staxableincome,toonedecimalplace. e

f Solution

Duringtheﬁnancialyear,Richard’semployersentatotalof $6000 intaxtotheATO.HasRichard paidtoomuchtaxornotenough?Calculatehisrefundorliability.

Explanation

a Grossincome = $54600

Deductions = $375

Taxableincome = $54225

b Taxpayable:

$4288 + 0.3 × ($54225 - $45000) = $7055.50

Taxableincome = grossincome - deductions

Richardisinthemiddletaxbracketinthetable, inwhichitsays:

$4288 plus 30cforeach $1 over $45000

Note: 30 centsis $0.30.

Continuedonnextpage

c 2 100 × 54225 = $1084.50

d $7055.50 + $1084.50 = $8140

e 8140 54225 × 100 = 15.0% (to 1 d.p.)

f Richardpaid $6000 intaxduringtheyear. Heshouldhavepaid $8140.Richardhas notpaidenoughtax.Hemustpayanother $2140 intax.

Nowyoutry

Medicarelevyis 2% ofthetaxableincome.

ThisisthetotalamountoftaxthatRichard shouldhavepaid.

ThisimpliesthatRichardpaidapproximately 15.0% taxoneverydollar.Thisissometimes readas‘15 centsinthedollar’.

Thisisknownasashortfalloraliability.He willreceivealetterfromtheATOrequesting paymentofthedifference.

$8140 - $6000 = $2140

Duringthe2024/2025ﬁnancialyear,Francescaearned $82300 perannumfromheremployerand othersources,suchasinterestoninvestments.Shehasreceiptsfor $530 forwork-relatedexpenses anddonations.

CalculateFrancesca’staxableincome. a Usethe2024/2025taxtablefromthe Keyideas tocalculateFrancesca’staxpayableamount.

b FrancescamustalsopaytheMedicarelevyof 2% ofhertaxableincome.Howmuchisthe Medicarelevy?

AddthetaxpayableandtheMedicarelevyamounts. d Expressthetotaltaxinpart d asapercentageofFrancesca’staxableincome,toonedecimal place.

e Duringtheﬁnancialyear,Francesca’semployersentatotalof $15000 intaxtotheATO. HasFrancescapaidtoomuchtaxornotenough?Calculateherrefundorliability.

4 Duringthe2024/2025ﬁnancialyear,Liamearned $94220 perannumfromhisemployerandother sources,suchasinterestoninvestments.Hehasreceiptsfor $615 forwork-relatedexpensesanddonations. CalculateLiam’staxableincome. a

Usethe2024/2025taxtablefromthe Keyideas tocalculateLiam’staxpayableamount. b LiammustalsopaytheMedicarelevyof 2% ofhistaxableincome.HowmuchistheMedicare levy?

AddthetaxpayableandtheMedicarelevyamounts. d

e Expressthetotaltaxinpart d asapercentageof Liam’staxableincome,toonedecimalplace.

f Duringtheﬁnancialyear,Liam’semployersenta totalof $21000 intaxtotheATO.HasLiampaid toomuchtaxornotenough?Calculatehisrefund orliability.

UNCORRECTEDSAMPLEPAGES

5 Usethe2024/2025taxtableinthe Keyideas tocalculatetheincometaxpayableonthesetaxable incomes.

$30000 a $60000 b $150000 c $200000 d

6 Leehascometotheendofherfirstfinancialyearemployedasawebsitedeveloper. OnJune 30 shemadethefollowingnotesaboutthefinancialyear.

Grossincomefromemployer

Grossincomefromcasualjob

Interestoninvestments

Donations

Work-relatedexpenses

$58725

$7500

$75

$250

$425

Taxpaidduringtheﬁnancialyear $11000

HintforQ6:Taxableincome = allincomes - deductions

CalculateLee’staxableincome. a Usethe2024/2025taxtableshowninthe Keyideas tocalculateLee’staxpayableamount. b LeemustalsopaytheMedicarelevyof 2% ofhertaxableincome.HowmuchistheMedicarelevy?

c AddthetaxpayableandtheMedicarelevy. d Expressthetotaltaxinpart d asapercentageofLee’staxableincome,toonedecimalplace. e HasLeepaidtoomuchtaxornotenough?Calculateherrefundorliability. f

Problem-solving and reasoning

7,8,10,117,9,11–13

7 Alec’sMedicarelevyis $1750.Thisis 2% ofhistaxableincome.WhatisAlec’staxableincome?

Taraissavingforanoverseastrip.Hertaxableincomeis usuallyabout $20000.Sheestimatesthatshewillneed $5000 forthetrip,sosheisgoingtodosomeextrawork toraisethemoney.HowmuchextrawillTaraneedtoearn inordertosavetheextra $5000 aftertax?

HintforQ8:Usethe2024/2025tax tableinthe Keyideas toconsider howmuchextrataxshewillpay.

WhenSaledusedthetaxtabletocalculatehisincome taxpayable,itturnedouttobe $19288.Whatis histaxableincome?

10 Explainthedifferencebetweenataxrefundandataxliability.

HintforQ9:Usethe2024/2025tax tablegiveninthe Keyideas to determineinwhichtaxbracket Saledfalls.

11 Gordanalookedatthelastrowofthetaxtableandsaid,‘Itissounfairthatpeopleinthattaxbracket mustpay 45 centsineverydollarintax.’ExplainwhyGordanaisincorrect.

12 Themostsignificantrecentchangeto Australianincometaxrateswasfirst appliedinthe2024/2025financial year.Considerthetaxtablesfor thetwoconsecutivefinancialyears 2023/2024and2024/2025.Notethat theamountslistedfirstineachtable areoftencalledthetax-freethreshold (i.e.theamountthatapersoncanearn beforetheymustpaytax).

a Therearesomesigniﬁcantchanges betweentheﬁnancialyears 2023/2024and2024/2025. Describethreeofthem.

2023/2024

Taxableincome Taxonthisincome

0 - $18200 Nil

$18201 - $37000 19cforeach $1 over $18200

$37001 - $80000 $3572 plus 32.5cforeach $1 over $37000

$80001 - $180000 $17547 plus 37cforeach $1 over $80000

$180001 andover $54547 plus 45cforeach $1 over $180000

2024/2025

Taxableincome Taxonthisincome

0–$18200 Nil

$18201–$45000 16cforeach $1 over $18200

$45001–$135000 $4288 plus 30cforeach $1 over $45000

$135001–$190000 $31288 plus 37cforeach $1 over $135000

$190001 andover $51638 plus 45cforeach $1 over $190000

b Thefollowingpeoplehadthesametaxableincomeduringbothﬁnancialyears.Findthe differenceintheirtaxpayableamountsandstatewhethertheywereadvantagedor disadvantagedbythechanges,ornotaffectedatall?

Ali:Taxableincome = $5000 i

Charlotte:Taxableincome = $50000 iii

Xi:Taxableincome = $25000 ii

Diego:Taxableincome = $80000 iv

13 Belowisthe2024/2025taxtableforpeoplewhoarenotresidentsofAustraliabutareworkingin Australia.

–$190000

andover $60850 plus 45cforeach $1 over $190000

Comparethistabletotheoneinthe Keyideas forAustralianresidents. Whatdifferencewoulditmaketotheamountoftaxpaidbythesepeopleiftheywerenon-residents ratherthanresidents?

Bill:Taxableincome = $5000 a

Jen:Taxableincome = $25000

Scott:Taxableincome = $100000 c

Melinda:Taxableincome = $200000 d

14a Chooseanoccupationorcareerinwhichyouareinterested.Imaginethatyouareworkinginthat job.Duringtheyearyouwillneedtokeepreceiptsforitemsyouhaveboughtthatarelegitimate work-relatedexpenses.Dosomeresearchontheinternetandwritedownsomeofthethingsthat youwillbeabletoclaimaswork-relatedexpensesinyourchosenoccupation.

bi Imagineyourtaxableincomeis $80000.Whatisyourtaxpayableamount?

ii Imagineyouhaveareceiptfora $100 donationtoaregisteredcharity.Thisdecreasesyour taxableincomeby $100.Byhowmuchdoesitdecreaseyourtaxpayableamount?

1E 1E Budgeting

Learningintentions

• Toknowthetypesofexpensesthatareincludedinabudget

• Tounderstandhowabudgetisaffectedby xedandvariableexpenses

• Tobeabletocalculatesavingsandotherexpensesbasedontheinformationinabudget

• Tobeabletocalculatethebestbuy(cheapestdeal)fromarangeofoptions

Keyvocabulary: budget, xedexpenses,variableexpenses

Oncepeoplehavebeenpaidtheirincomefortheweek,fortnightormonth,theymustplanhowto spendit.Mostfamiliesworkonabudget,allocatingmoneyforﬁxedexpensessuchasthemortgageor rentandthevarying(i.e.changing)expensesofpetrol,foodandclothing.

Lessonstarter:Expensesforthemonth

Writedowneverythingthatyouthinkyourfamilywouldspendmoneyonfortheweekandthemonth, andestimatehowmuchthosethingsmightcostfortheentireyear.Wheredoyouthinksavingscouldbe made?Whatwouldbesomeadditionalannualexpenses?

Keyideas

A budget isanestimateofincomeandexpensesforaperiodoftime.

Managingmoneyforanindividualissimilartooperatingasmallbusiness.Expensescanbe dividedintotwoareas:

• Fixedexpenses (thesedonotchangeduringatimeperiod):paymentofloans,mortgages, regularbillsetc.

• Variableexpenses (thesecostschangeoveratimeperiod):clothing,entertainment,food etc.(theseareestimates).

Whenyourbudgetiscompletedyoushouldalwayscheckthatyourﬁguresarereasonable estimates.

Bylookingatthebudgetyoushouldbeabletoseehowmuchmoneyisremaining;thiscanbe usedassavingsortobuynon-essentialitems.

Exercise1E

1 Classifyeachexpenselistedbelowasmostlikelyaﬁxedexpenseoravariableexpense.

a monthlyrent

b monthlyphonebillpaymentplan

c take-awayfood

d stationerysuppliesforwork

2 Binhhasanincomeof $956 aweek.Hisexpenses,bothﬁxedandvariable,total $831.72 ofhis income.HowmuchmoneycanBinhsaveeachweek?

3 Roslynhasthefollowingmonthlyexpenses.Mortgage = $1458,mobilephone = $49,internet = $60, councilrates = $350,water = $55,electricity = $190.WhatisthetotalofRoslyn’smonthlyexpenses?

Fluency

Example14Budgetingusingpercentages

4–74,5,7,8

Fionahasanetannualincomeof $36000 afterdeductions.Sheallocatesherbudgetona percentagebasis.

a HowmuchshouldFionasave?

Determinetheamountofﬁxedexpenses,includingthemortgage,carloanandeducation.

b Istheamountallocatedforfoodreasonable?

Solution

Explanation

a Fixedexpenses = 55% of $36000 Themortgage,carloanandeducationare 55% intotal.

= 0.55 × $36000 Change 55% toadecimalandmultiplybythenetincome.

= $19800

b Savings = 10% of $36000

Savingsare 10% ofthebudget.

= 0.1 × $36000 Change 10% toadecimalandmultiplybythenetincome.

= $3600

c Food = 25% of $36000

Foodis 25% ofthebudget.

= 0.25 × $36000 Change 25% toadecimalandcalculate.

= $9000 peryear,or

Dividetheyearlyexpenditureby 52 tomakea $173 perweek decisiononthereasonablenessofyouranswer. Thisseemsreasonable.

Nowyoutry

Kylehasanetannualincomeof $64200 afterdeductions.Heallocateshisbudgetonapercentage basis.

Determinetheamountofﬁxedexpensesincludingtherentandbills. a HowmuchshouldKylesave? b Istheamountallocatedfortransportreasonable? c

4 Paulhasanannualincomeof $75000 afterdeductions.Heallocateshisbudgetonapercentagebasis.

a Determinetheamountofﬁxedexpenses,includingthemortgageandloans.

b HowmuchshouldPaulhaveleftoverafterpayingforhismortgage,carloanandpersonalloan?

c Istheamountallocatedforfoodreasonable?

5 Lachlanhasanincomeof $2120 permonth.Ifhe budgets 5% forclothes,howmuchwillheactually havetospendonclotheseachmonth?

Example15Budgetingusingﬁxedvalues

Runningacertaintypeofcarinvolvesyearly,monthlyandweeklyexpenditure.Considerthefollowing vehicle’scosts.

lease • $210 permonth

registration • $475 peryear

• $145 perquarter

insurance

servicing • $1800 peryear

petrol • $37 perweek

Determinetheoverallcosttorunthiscarforayear.

Whatpercentageofa $70000 salarywouldthisbe,correcttoonedecimalplace?

Solution

a Overallcost = 210 × 12

+ 475

+ 145 × 4

+ 1800

+ 37 × 52

Explanation

Leasingcost: 12 monthsinayear

Registrationcost

Insurancecost: 4 quartersinayear

Servicingcost

Petrolcost: 52 weeksinayear = $7299

Theoverallcosttorunthecaris $7299

b % ofsalary = 7299 70000 × 100 = 10.4% (to 1 d.p.)

Nowyoutry

Theoverallcostisfoundbyaddingthe individualtotals.

Percentage = carcost totalsalary × 100 Roundasrequired.

Runningaboatinvolvesyearly,monthlyandweeklyexpenditure.Considerthefollowing boat’scosts.

registration

• $342 peryear insurance

• $120 perquarter

servicing

• $360 peryear

fuel

• $300 permonth

storingboat

• $2400 peryear

Determinetheoverallcosttorunthisboatforayear.

Whatpercentageofan $82000 salarywouldthisbe,correcttoonedecimalplace?

6 Elianaisastudentandhasthefollowingexpensesinherbudget.

• rent $270 perweek

• electricity $550 perquarter

• phoneandinternet $109 permonth

• car $90 perweek

• food $170 perweek

• insurance $2000 ayear

DetermineEliana’scostsforayear. a

HintforQ6:Use 52 weeksina year, 12 monthsinayearand 4 quartersinayear.

WhatpercentageofEliana’snetannualsalaryof $45000 wouldthisbe,correctto onedecimalplace? b

7 ThecostsofsendingastudenttoModkinPrivate Collegeareasfollows.

• feesperterm(4 terms) $1270

• subjectleviesperyear $489

• buildingfundperweek $35

• uniformsandbooksperyear $367

Determinetheoverallcostperyear. a Iftheschoolsendsoutabilltwiceayear (biannual),coveringalltheitemsabove,what wouldbetheamountofeachpayment?

b Howmuchshouldbesavedperweektomake thebiannualpayments?

8 Asmallbusinessownerhasthefollowingexpensestobudgetfor.

• rent $1400 amonth

• phoneline $59 amonth

• wages $1200 aweek

• electricity $430 aquarter

• water $120 aquarter

• insurance $50 amonth

Whatistheannualbudgetforthesmallbusiness? a Howmuchdoesthebusinessownerneedtomakeeachweekjusttobreakeven? b Ifthebusinessearns $5000 aweek,whatpercentageofthisneedstobespentonwages?

Problem-solving and reasoning

9 Francine’spetrolbudgetis $47 fromherweeklyincomeof $350. Whatpercentageofherbudgetisthis?Giveyouranswertotwodecimalplaces. a Ifpetrolcosts $1.59 perlitre,howmanylitresofpetrol,correcttotwodecimalplaces,isFrancine budgetingforinaweek?

10 Grantworksa 34-hourweekat $15.50 perhour.Hisnetincomeis 65% ofhisgrossincome. Determinehisnetweeklyincome. a IfGrantspends 12% ofhisnetincomeonentertainment,determinetheamountheactuallyspends peryearonentertainment.

b Grantsaves $40 perweek.Whatpercentageofhisnetincomeisthis(totwodecimalplaces)?

11 Darioearns $432 perfortnightatatake-awaypizzashop.Hebudgets 20% forfood, 10% forrecreation, 13% fortransport, 20% forsavings, 25% fortaxationand 12% forclothing. Determinetheactualamountbudgetedforeachcategoryeveryfortnight. a Dario’swageincreasesby 30%

Determinehowmuchhewouldnowsaveeachweek. b Whatpercentageincreaseistheanswertopart b ontheoriginalamountsaved? c DeterminetheextraamountofmoneyDariosavesperyearafterhiswageincrease. d Iftransportisaﬁxedexpense,itspercentageofDario’sbudgetwillchange.Determinethenew percentage.

Example16Calculatingbestbuys

Softdrinkissoldinthreeconvenientpacksatthelocalstore.

• cartonof 36 (375 mL) cansat $22.50

• asix-packof (375 mL) cansat $5.00

• 2-litrebottlesat $2.80

Determinethecheapestwaytobuythesoftdrink.

Solution Explanation

Buyingbythecarton:

Cost = $22.50 ÷ (36 × 375) TotalmL = 36 × 375 = $0.0017 permL

DividetoworkoutthecostpermL.

Buyingbythesix-pack: Cost = $5 ÷ (6 × 375)

= 6 × 375 = $0.0022 permL

Buyingbythebottle:

Cost = $2.80 ÷ 2000

= $0.0014 permL

Â Thecheapestwaytobuythe softdrinkistobuythe 2-litrebottle.

Nowyoutry

ComparethethreecostspermL.

Abrandoftoiletrollsaresoldinthreepacktypesatthesupermarket.

• apackof 18 rollsfor $8.82

• apackof 6 rollsfor $3.30

• apackof 4 doublelengthrollsfor $3.68

Determinethecheapestwaytobuythetoiletrolls.

Teabagscanbepurchasedfromthesupermarketinthreeforms.

• 25 teabagsat $2.36

• 50 bagsat $4.80

• 200 bagsat $15.00

Whatisthecheapestwaytobuyteabags?

HintforQ12:Calculatethecost perteabagineachcase.

13 Aweeklytrainconcessionticketcosts $16.Adayticketcosts $3.60.Ifyouaregoingtoschoolonly 4 daysnextweek,isitcheapertobuyoneticketperdayoraweeklyticket?

14 Aholidaycaravanparkoffersitscabinsatthefollowingrates.

$87 pernight • (Sunday–Thursday)

$187 foraweekend • (FridayandSaturday)

$500 perweek •

a Determinethenightlyrateineachcase.

b Whichpriceisthebestvalue?

15 Tomatosauceispricedat:

•

200 mLbottle $2.35

500 mLbottle $5.24

a FindthecostpermLofthetomatosauceineachcase.

b Whichisthecheapestwaytobuytomatosauce?

c Whatwouldbethecostof 200 mLatthe 500 mLrate?

d Howmuchwouldbesavedbybuyingthe 200 mLbottleatthisrate?

e Suggestwhythe 200 mLbottleisnotsoldatthisprice. Minimumcostoftennisballs

16 Safeservehasasaleontennisballsforonemonth. Whenyoubuy:

• 1 container,itcosts $5

• 6 containers,itcosts $28

• 12 containers,itcosts $40

• 24 containers,itcosts $60

Youneed 90 containersforyourclubtohaveenoughforaseason.

a Determinetheminimumcostifyoubuyexactly 90 containers.

b Determinetheoverallminimumcost,andthenumberofextracontainersyouwillhaveinthis situation.

1F 1F Simpleinterest

Learningintentions

• Tounderstandhowsimpleinterestiscalculated

• Tobeabletocalculateinterestusingthesimpleinterestformula

• Tobeabletodeterminetherateofinterestbasedontheinterestearned

• Tobeabletocalculatetheamountowingonaloanandcalculaterepayments

Keyvocabulary: principal,rateofinterest,simpleinterest,annual,invest,borrow

Borrowedorinvestedmoneyusuallyhasanassociatedinterestrate.Theconsumerneedstoestablishthe typeofinteresttheyarepayingandtheeffectsithasontheamountborrowedorinvestedovertime. Someloansorinvestmentsdeliverthefullamountofinterestusingonlytheinitialloanorinvestment amountintheinterestcalculations.Thesetypesaresaidtousesimpleinterest.

Whenchoosingahome loan,youneedto considerthetypeand amountofinterestyou willbepaying,including whethertherateisﬁxed orvariable,howoften interestiscalculated,and howthesefactorsaffect thetotalcostoftheloan overtime.

Lessonstarter:Howlongtoinvest?

MarcusandBrittneyeachhave $200 intheirbankaccounts.Marcusearns $10 ayearininterest.Brittney earns 10% p.a.simpleinterest.

Forhowlongmusteachoftheminvesttheirmoneyforittodoubleinvalue?

Keyideas

Simpleinterest isatypeofinterestthatiscalculatedontheamount invested or borrowed

Thetermsneededtounderstandsimpleinterestare:

• Principal(P):theamountofmoneyborrowedorinvested

• Rateofinterest(r):the annual (yearly)percentagerateofinterest(e.g. 3% p.a.)

• Time (t):thenumberofyearsforwhichtheprincipalisborrowedorinvested

• Interest (I ):theamountofinterestaccruedoveragiventime.

Theformulaforcalculatingsimpleinterestis:

I = principal × rate × time

I = Prt 100 (Sincetherateisapercentage)

Totalrepaid = amountborrowed + interest

Exercise1F

Und er stand ing

1 Intheformula I = Prt 100:

I isthe a P isthe b r isthe c t isthe d

2 Calculateinterestearned (I ) if:

Fluency

Example17Usingthesimpleinterestformula

Usethesimpleinterestformula, I = Prt 100,toﬁnd:

theinterest (I ) when $600 isinvestedat 8% p.a.for 18 months a theannualinterestrate (r) when $5000 earns $150 interestin 2 years. b

Solution

a P = 600

1,22

3–64–7

Explanation

Writeouttheinformationthatyouknowandthe formula. r = 8

t = 18 months = 18 12 = 1.5 years I = Prt

× 8 × 1.5

Substituteintotheformulausingyearsfor t =

Theinterestis $72 in 18 months. b

Writetheformulaandtheinformationknown. Substitutethevaluesintotheformulaandsolve theequationtoﬁnd r.

Dividebothsidesby 100

Thesimpleinterestrateis 1.5% peryear.Writetherateasapercentage.

Nowyoutry

Usethesimpleinterestformula, I = Prt 100,toﬁnd:

theinterest (I ) when $450 isinvestedat 5% p.a.for 30 months a theannualinterestrate (r) when $3500 earns $210 interestin 3 years. b

3 Usethesimpleinterestformula, I = Prt 100,toﬁnd:

a theinterest (I ) when $500 isinvestedat 6% p.a.for 24 months

b theannualinterestrate (r) when $3000 earns $270 interestin 3 years.

4 Copyandcompletethistableofvaluesfor I , P, r and t

a $700 5% p.a. 4 years

b $2000 7% p.a. 3 years

c $3500 3% p.a. 22 months

d $750 2 1 2 % p.a. 30 months

e $22500 3 years

$2025

f $1770 5 years $354

HintforQ4:Use I = Prt 100

Example18Calculatingrepaymentswithsimpleinterest

$3000 isborrowedat 12% p.a.simpleinterestfor 2 years. Whatisthetotalamountowedoverthe 2 years? a Ifrepaymentsoftheloanaremademonthly,howmuchwouldeachpaymentneedtobe? b

Solution

a P = $3000, r = 12, t = 2 I = Prt

= 3000 × 12 × 2

Totalamount = $3000 + $720 = $3720

Explanation

Listtheinformationyouknow. Writetheformula.

Substitutethevaluesandevaluate.

Totalamountistheoriginalamount plus theinterest.

b Amountofeachpayment = $3720 ÷ 24 2 years = 24 months = $155 permonth Thereare 24 paymentstobemade. Dividethetotalby 24

Nowyoutry

$5400 isborrowedat 9% p.a.simpleinterestfor 4 years. Whatisthetotalamountowedoverthe 4 years? a

Ifrepaymentsoftheloanaremademonthly,howmuchwouldeachpaymentneedtobe? b

5 $5000 isborrowedat 11% p.a.simpleinterestfor 3 years.

a Whatisthetotalamountowedoverthe 3 years?

b Ifrepaymentsoftheloanaremademonthly,howmuchwould eachpaymentneedtobe?

HintforQ5:Calculatetheinterest ﬁrst.

6 Underhirepurchase,Johnboughtasecond-handcarfor $11500.Hepaidnodepositanddecidedto paytheloanoffin 7 years.Ifthesimpleinterestis 6.45%,determine: thetotalinterestpaid a thetotalamountoftherepayment b thepaymentspermonth.

7 $10000 isborrowedtobuyasecond-handBMW.Theinterestiscalculatedatasimpleinterestrateof 19% p.a.over 4 years.

Whatisthetotalinterestontheloan? a Howmuchistoberepaid? b Whatisthemonthlyrepaymentonthisloan? c

Problem-solving and reasoning

8 HowmuchinterestwillGiorgioreceiveifheinvests $7000 instocksat 3.6% p.a.simpleinterestfor 4 years?

9 Rebeccainvests $4000 for 3 yearsat 5.7% p.a.simpleinterestpaid yearly.

Howmuchinterestwillshereceiveintheﬁrstyear? a

WhatisthetotalamountofinterestRebeccawillreceiveoverthe 3 years? b

HowmuchmoneywillRebeccahaveafterthe 3-yearinvestment? c

Hint:Substituteintothe formula I = Prt 100 andsolve theresultingequation. Aninvestmentof $15000 receivesaninterestpaymentover 3 yearsof $7200.Whatwastherateofsimpleinterest perannum?

11 Jonathonwishestoinvest $3000 at 8% perannum.How longwillheneedtoinvestforhistotalinvestmenttodouble?

12 Ivanwishestoinvestsomemoneyfor 5 yearsat 4.5% p.a.paidyearly.Ifhewishestoreceive $3000 ininterestpaymentsperyear,howmuchshouldheinvest?Roundyouranswertothenearest dollar.

13 Gretta’sinterestpaymentonherloantotalled $1875.Iftheinterestratewas 5% p.a.andtheloanhad alifeof 5 years,whatamountdidsheborrow? Whichwayisbest?

14 Ashedmanufactureroffersﬁnancewitharateof 3.5% p.a.paidattheendof 5 yearswithadeposit of 10%,orarateof 6.4% repaidover 3 yearswithadepositof 20%. Melaniedecidestopurchaseafullyerectedfour-squareshedfor $12500

UNCORRECTEDSAMPLEPAGES

Howmuchdepositwillsheneedtopayineachcase? a Whatisthetotalinterestshewillincurineachcase? b Ifshedecidedtopaypermonth,whatwouldbethemonthlyrepaymentineachcase? c Discussthebeneﬁtsofthedifferenttypesofpurchasingmethods. d

Completethefollowing.

6 1C Findthegrossincomeforaparticularweekinthefollowingworksituations. Pippaisadoor-to-doorsalesrepresentativeforanairconditioningcompany.Sheearns $300 perweekplus 8% commissiononhersales.Inaparticularweekshemakes $8200 worthofsales.

a Ariispaid $15.70 perhourinhisjobasashopassistant.Theﬁrst 36 hoursheworksinaweek arepaidatthenormalhourlyrate,thenext 4 hoursattimeandahalfandthen doubletimeafterthat.Ariworks 42 hoursinaparticularweek.

7 1D Duringthe2024/2025ﬁnancial year,Cameronearned $76300 per annum.Hehadreceiptsfor $425 fordonationsandwork-related expenses.

CalculateCameron’staxable income.

Usethis2024/2026taxtabletocalculateCameron’staxpayableamount,tothenearestcent.

b CameronalsomustpaytheMedicarelevyof 2% ofhistaxableincome.Howmuchisthelevy, tothenearestcent?

Duringtheﬁnancialyear,Cameron’semployersentatotalof $14500 intaxtotheATOonhis behalf.Byaddingtogetheryouranswersfromparts b and c,calculatetheamountCameron mustpayorwillberefundedonhistaxreturn. d

phoneandinternet $119 permonth

electricity $72 perquarter

othercarcosts $120 permonth

clothing $260 permonth

carregistration $700 peryear

food $110 perweek

medicalandotherinsurance $160 per month

Determinetheoverallcostofrunningthehouseholdfortheyear.(Use 52 weeksinayear.) a Whatpercentageofan $82000 annualsalarydoesyouranswertopart a represent?Round youranswertoonedecimalplace. b 9 1F Usethesimpleinterestformula I = Prt 100 toﬁnd: theamountowedwhen $4000 isborrowedat 6% p.a.for 3 years

theinvestmentperiod,inyears,ifaninvestmentof $2500 at 4% p.a.earns $450 ininterest.

1G 1G Compoundinterest

Learningintentions

• Tounderstandhowcompoundinterestiscalculated

• Tobeabletoapplythecompoundinterestformulatocalculatethetotalamount

• Tobeabletousethecompoundinterestformulawithdifferenttimeperiodssuchasmonths

Keyvocabulary: compoundinterest,principal,rateofinterest

Forsimpleinterest,theinterestisalwayscalculatedontheprincipal amount.Sometimes,however,interestiscalculatedontheactual amountpresentinanaccountateachtimeperiodthatinterestis calculated.Thismeansthattheinterestisaddedtotheamount, thenthenextlotofinterestiscalculatedagainusingthisnew amount.Thisprocessiscalledcompoundinterest.

Compoundinterestcanbecalculatedusingupdatedapplicationsof thesimpleinterestformulaorbyusingthecompoundinterest formula.

Lessonstarter:Investingusingupdated simpleinterest

Compoundinterestiscalculatedby addinginteresttotheinitialprincipal, thencalculatingthenextinterest amountbasedonthenewtotal,and repeatingthisprocess.

Considerinvesting $400 at 12% perannum.Whatisthebalanceat theendof 4 yearsifinterestisaddedtotheamountattheendofeachyear? Copyandcompletethetabletoﬁndout.

3rdyear

4thyear

Asyoucansee,theamountfromwhichinterestiscalculatediscontinuallychanging.

Keyideas

Compoundinterest isatypeofinterestthatispaidonaloanorearnedonaninvestment,which iscalculatednotonlyontheinitialprincipalbutalsoontheinterestaccumulatedduring theloan/investmentperiod.

Compoundinterestcanbefoundbyusingupdatedapplicationsofthesimpleinterestformula. Forexample, $100 compoundedat 10% p.a.for 2 years.

Year 1: 100 + 10% of 100 = $110

Year 2: 110 + 10% of 110 = $121,socompoundinterest = $21.

Thetotalamountinanaccountusingcompoundinterestforagivennumberoftimeperiods isgivenby:

A = P(1 + r 100)n ,where:

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• Principal (P) = the amountofmoneyborrowedorinvested

• Rateofinterest (r) = thepercentageappliedtotheprincipalperperiodofinvestment

• Periods (n) = thenumberoftimeperiodstheprincipalisinvestedfor

• Amount (A) = thetotalamountofyourinvestment

Interest = amount (A) - principal (P)

Exercise1G

Und er stand ing

1 Consider $500 investedat 10% p.a.compoundedannually.

1–33

Howmuchinterestisearnedintheﬁrstyear?

a Whatisthebalanceoftheaccountoncetheﬁrstyear’s interestisadded?

c Whatisthebalanceoftheaccountattheendof thesecondyear?

Howmuchinterestisearnedinthesecondyear?

HintforQ1:Forthesecond year,youneedtouse $500 plustheinterestfromtheﬁrst year.

2 $1200 isinvestedat 4% p.a.compoundedannuallyfor 3 years.Completethefollowing.

a Thevalueoftheprincipal P is b 4% isthe , r

c Thenumberoftimeperiodsthemoneyisinvestedis

3 Fillinthemissingnumbers.

a $700 investedat 8% p.a.compoundedannuallyfor 2 years.

A = (1.08)

b $1000 investedat 15% p.a.compoundedannually for 6 years.

A = 1000 ( )6

c $850 investedat 6% p.a.compoundedannuallyfor 4 years.

A = 850 ( )

Fluency

Example19Usingthecompoundinterestformula

HintforQ3:Forcompound interest, A = P(1 + r 100 )n

Determinetheamountafter 5 yearswhen $4000 iscompoundedannuallyat 8%

Solution

Explanation

P = 4000, n = 5, r = 8 Listthevaluesforthetermsyouknow.

A = P(1 + r 100 )n Writetheformula. = 4000(1 + 8 100 )5 Substitutethevalues. = 4000(1.08)5 Simplifyandevaluate. = $5877.31 Writeyouranswertotwodecimalplaces, (nearestcent).

Nowyoutry

Determinetheamountafter 4 yearswhen $3000 iscompoundedannuallyat 6%

4 Determinetheamountafter 5 yearswhen:

$4000 iscompoundedannuallyat 5% a

$8000 iscompoundedannuallyat 8.35% b

$6500 iscompoundedannuallyat 16% c

$6500 iscompoundedannuallyat 8% d

HintforQ4: A = P(1 + r 100 )n

5 Determinetheamountwhen $100000 iscompoundedannuallyat 6% for: 1 year a 2 years b 3 years c 5 years d 10 years e 15 years f

Example20Convertingratesandtimeperiods

Calculatethenumberofperiods (n) andtheratesofinterest (r) offeredperperiodforeachof thefollowing.

6% p.a.over 4 yearspaidmonthly a 18% p.a.over 3 yearspaidquarterly b

Solution

a n = 4 × 12 = 48 r = 6 ÷ 12 = 0.5

Explanation

4 yearsisthesameas 48 months, as 12 months = 1 year. 6% p.a = 6% inoneyear. Divideby 12 toﬁndthemonthlyrate.

b n = 3 × 4 = 12 r = 18 ÷ 4

Therearefourquartersin 1 year. = 4.5

Nowyoutry

Calculatethenumberofperiods (n) andtheratesofinterest (r) offeredperperiodforeach ofthefollowing.

3% p.a.over 2 yearspaidmonthly a 7% p.a.over 4 yearspaidbi-annually(twiceyearly) b

6 Calculatethenumberofperiods (n) andtheratesofinterest (r) offeredperperiodforthefollowing. (Roundtheinterestratetothreedecimalplaceswherenecessary.)

6% p.a.over 3 yearspaidbiannually a 12% p.a.over 5 yearspaidmonthly b

4.5% p.a.over 2 yearspaidfortnightly c

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10.5% p.a.over 3.5 yearspaidquarterly d 15% p.a.over 8 yearspaidquarterly e

9.6% p.a.over 10 yearspaidmonthly f

HintforQ6:‘Bi-annually’ means‘twiceayear’. 26 fortnights = 1 year

Example21Findingcompoundedamountsusingmonths

Tony’sinvestmentof $4000 iscompoundedat 8.4% p.a.over 5 years.Determinetheamounthewill haveafter 5 yearsiftheinterestispaidmonthly.

Solution Explanation

P = 4000

n = 5 × 12

Listthevaluesofthetermsyouknow.

Convertthetimeinyearstothenumberofperiods(inthis question,months),i.e. 60 months = 5 years. = 60

r = 8.4 ÷ 12

Converttherateperyeartotherateperperiod(i.e.months) bydividingby 12. = 0.7

A = P(1 + r 100 )n Writetheformula.

= 4000(1 + 0.007)60

Substitutethevalues 0.7 ÷ 100 = 0.007

= 4000(1.007)60 Simplifyandevaluate.

= $6078.95

Nowyoutry

Sally’sinvestmentof $6000 iscompoundedat 4.8% p.a.over 4 years.Determinetheamountshewill haveafter 4 yearsiftheinterestispaidmonthly.

7 Aninvestmentof $8000 iscompoundedat 12% p.a.over 3 years.Determinetheamounttheinvestor willhaveafter 3 yearsiftheinterestiscompoundedmonthly.

8 Calculatethevalueofthefollowinginvestmentsifinterestiscompoundedmonthly.

a $2000 at 6% p.a.for 2 years

b $34000 at 24% p.a.for 4 years

c $350 at 18% p.a.for 8 years

d $670 at 6.6% p.a.for 2 1 2 years

e $250 at 7.2% p.a.for 12 years Problem-solving and

HintforQ8:Convertyearsto monthsandtheannualrate tothemonthlyrate.

9,1010–12

10a Calculatetheamountofcompoundinterestpaidon $8000 attheendof 3 yearsforeachratebelow.

12% compoundedannually i

12% compoundedbiannually(twiceayear) ii

12% compoundedmonthly iii

12% compoundedweekly iv

12% compoundeddaily v

HintforQ10: 1 year = 12 months 1 year = 52 weeks 1 year = 365 days

b Whatisthedifferenceintheinterestpaidbetweenannualanddailycompoundinginthiscase?

11 Thefollowingareexpressionsrelatingtocompoundinterestcalculations.Determine theprincipal (P),numberofperiods (n),rateofinterestperperiod (r),annualrateofinterest (R) andtheoveralltime (t)

300(1.07)12 ,biannually a

5000(1.025)24 ,monthly b

1000(1.00036)65 ,fortnightly c

3500(1.000053)30 ,daily d

10000(1.078)10 ,annually e

HintforQ11:For 12 time periodswithinterestpaid twiceayear,thisis 6 years.

12 Ellenneedstodecidewhethertoinvesther $13500 for 6 yearsat 4.2% p.a.compoundedmonthlyor 5.3% compoundedbiannually.DecidewhichinvestmentwouldbethebestforEllen.

13 Youhave $100000 toinvestandwishtodoublethatamount.Usetrialanderrorinthefollowing.

a Determine,tothenearestwholenumberofyears,thelengthoftimeitwilltaketodoubleyour investmentusingthecompoundinterestformulaatratesof:

b Iftheamountofinvestmentis $200000 andyouwishtodoubleit,determinethetimeitwilltake usingthesameinterestratesasabove.

c Arethelengthsoftimetodoubleyourinvestmentthesameinpart a andpart b?

1H 1H Investmentsandloans

Learningintentions

• Tounderstandthataloancanberepaidininstalmentsthatincludeinterest

• Tobeabletocalculatethetotalpaymentforapurchaseorloaninvolvingrepayments

• Tobeabletocalculatebankinterestusingtheminimummonthlybalance

Keyvocabulary: investment,loan,repayment,interest,deposit,debit

Whenyouborrowmoney,interestischarged,andwhenyou investmoney,interestisearned.Whenyouinvestmoney,the institutioninwhichyouinvest(e.g.bankorcreditunion)pays youinterest.However,whenyouborrowmoney,theinstitution fromwhichyouborrowchargesyouinterest,sothatyoumust paybackthemoneyyouinitiallyborrowed,plustheinterest.

Lessonstarter:Creditcardstatements

RefertoAllan’screditcardstatementbelow.

a Howmanydaysweretherebetweentheclosingbalanceand theduedate?

b Whatistheminimumpaymentdue?

Ifyouareapprovedforabankloan,you willneedtorepaytheborrowedamount plusanyinterestchargedbythebank.

c IfAllanpaysonlytheminimum,onwhatbalanceistheinterestcharged?

d HowmuchinterestischargedifAllanpays $475.23 on 25/5?

Keyideas

Interest ratesareassociatedwithmanyloanandsavingsaccounts.

Bankaccounts:

• accrueinteresteachmonthontheminimummonthlybalance

• mayincuraccount-keepingfeeseachmonth.

Investments areamountsputintoabankaccountorsimilarwiththeaimofearninginterest onthemoney.

Loans (moneyborrowed)haveinterestchargedtothemontheamountleftowing(i.e. thebalance).

Repayments areamountspaidtothebank,usuallyeachmonth,torepayaloanplusinterest withinanagreedtimeperiod.

Exercise1H

Und er stand ing

1 Stateifthefollowingareexamplesofinvestments,loansorrepayments.

a Karapays $160 permonthtopayoffherholidayloan.

b Samdepositsa $2000 prizeinanaccountwith 3% p.a.interest.

1–33

c Georgiaborrows $6500 fromthebanktoﬁnancesettinguphersmallbusiness.

2 Donnacanaffordtorepay $220 amonth.Howmuchcansherepayover: 1 year? a 18 months? b 5 years? c

3 Saraﬁnabuysanewbedona‘buynow,paylater’offer.Nointerestischargedifshepaysforthebed in 2 years.Saraﬁna’sbedcosts $2490 andshepaysitbackoveraperiodof 20 monthsin 20 equal instalments.Howmuchiseachinstalment?

Fluency

Example22Repayingaloan

4–84,6–9

Wendytakesoutapersonalloanof $7000 tofundhertriptoSouthAfrica.Repaymentsaremade monthlyfor 3 yearsat $275 amonth.Find: thetotalcostofWendy’strip a theinterestchargedontheloan. b

Solution

Explanation

a Totalcost = $275 × 36 = $9900 3 years = 3 × 12 = 36 months

Cost = 36 lotsof $275

b Interest = $9900 - $7000 = $2900 Interest = totalpaid - amountborrowed

Nowyoutry

Jacobtakesoutapersonalloanof $13000 tobuyacar.Hemakesrepaymentsmonthlyfor 2 yearsat $680 amonth.Find: thetotalcostofthecar a theinterestchargedontheloan. b

4 Jasonhasapersonalloanof $10000.Heisrepayingtheloanover 5 years.Themonthlyrepaymentis $310

CalculatethetotalamountJasonrepaysoverthe 5-yearloan. a Howmuchinterestishecharged? b

5 Raﬁqborrows $5500 tobuyasecond-handmotorbike.Herepays theloanin 36 equalmonthlyinstalmentsof $155

Calculatethetotalcostoftheloan. a HowmuchinterestdoesRaﬁqpay? b

HintforQ4:Howmanymonthly repaymentsin 5 years?

6 Almaborrows $250000 tobuyahouse.Therepaymentsare $1736 amonthfor 30 years.

HowmanyrepaymentsdoesAlmamake? a WhatisthetotalamountAlmapaysforthehouse? b Howmuchinterestispaidoverthe 30 years? c

Example23Payingoffapurchase

Harrybuysanew $2100 computeronthefollowingterms: 20% deposit

•

• monthlyrepaymentsof $90 for 2 years.

Find:

thedepositpaid a thetotalpaidforthecomputer b theinterestcharged. c

Solution

a Deposit = 0.2 × 2100 = $420

b Repayments = $90 × 24 = $2160

Totalpaid = $2160 + $420 = $2580

Explanation

Find 20% of 2100

2 years = 24 months

Repay 24 lotsof $90

Totalpaid = deposit + repayments

c Interest = $2580 - $2100 Interest = totalpaid - originalprice = $480

Nowyoutry

Amirapays $3180 foraholidayapartmentrentalonthefollowingterms: 30% deposit

•

• monthlyrepaymentsof $195 for 1 year.

Find: thedepositpaid a thetotalpaidfortheapartment b theinterestcharged. c

7 Georgebuysacarfor $12750 onthefollowingterms: 20% depositandmonthlyrepayments of $295 for 3 years. Calculatethedeposit. a Findthetotalofalltherepayments. b Findthecostofbuyingthecarontheseterms. c FindtheinterestGeorgepaysontheseterms. d

Example24Calculatinginterest

Anaccounthasaminimummonthlybalanceof $200 andinterestiscreditedmonthlyonthisamount at 1.5%

Determinetheamountofinteresttobecreditedattheendofthemonth. a Ifthebankchargesaﬁxedadministrationfeeof $5 permonthandotherfeestotalling $1.07 permonth,whatwillbethenetamountcreditedordebitedtotheaccountatthe endofthemonth?

b Solution

a Interest = 1.5% of $200 = 0.015 × $200 = $3

b Netamount = 3 - (5 + 1.07) =-3.07

$3.07 willbedebitedfromtheaccount.

Nowyoutry

Explanation

Interestis 1.5% permonth. Change 1.5% toadecimalandcalculate.

Subtractthedeductionsfromtheinterest.

Anegativeamountiscalleda debit

Anaccounthasaminimummonthlybalanceof $180 andinterestiscreditedmonthlyonthisamount at 2.2%

Determinetheamountofinteresttobecreditedattheendofthemonth. a Ifthebankchargesaﬁxedadministrationfeeof $4.50 permonthandotherfeestotalling $1.18 permonth,whatwillbethenetamountcreditedordebitedtotheaccountattheendof themonth?

8 Abankaccounthasaminimummonthly balanceof $300 andinterestiscredited monthlyat 1.5%

Determinetheamountofinteresttobe creditedeachmonth. a

Ifthebankchargesaﬁxedadministration feeof $3 permonthandfeesof $0.24 permonth,whatwillbethenet amountcreditedtotheaccountattheend ofthemonth? b

9 Anaccounthasnoadministrationfee.The monthlybalancesforMay–Octoberareshown inthetable.Iftheinterestpayableonthe minimummonthlybalanceis 1%,howmuch interestwillbeadded:

foreachseparatemonth? a overthe 6-monthperiod? b

Problem-solving and reasoning

10 Supersoundoffersthefollowingtwodealsonasoundsystemworth $7500

• DealB: 15% offforcash. •

DealA:nodeposit,interestfreeandnothingtopayfor 18 months.

a NickchoosesdealA.Find: thedeposithemustpay i theinterestcharged ii thetotalcostifNickpaysthesystemoffwithin 18 months. iii

b PhilchoosesdealB.WhatdoesPhilpayforthesamesound system?

c HowmuchdoesPhilsavebypayingcash?

11 CamdenFinanceCompanycharges 35% ﬂatinterestonallloans. a Meiborrows $15000 fromCamdenFinanceover 6 years. Calculatetheinterestontheloan. i Whatisthetotalrepaid (i.e.loan + interest)? ii Whatisthevalueofeachmonthlyrepayment? iii

b Lancelleborrows $24000 fromthesamecompanyover 10 years. Calculatetheinterestonherloan. i Whatisthetotalrepaid? ii Whatisthevalueofeachmonthlyinstalment? iii

HintforQ10: 15% offis 85% oftheoriginalamount.

AlistoftransactionsthatEmmamadeovera 1-monthperiodisshown.Thebankcalculatesinterest daily at 0.01% andaddsthetotaltotheaccountbalanceattheendofthisperiod.Ithasan administrativefeeof $7 permonthandotherfeesoverthistimetotal $0.35 Copyandcompletethebalancecolumnofthetable.

Determinetheamountofinterestaddedoverthismonth. b

c Suggestwhattheregulardepositsmightbefor. d

Determinetheﬁnalbalanceafterallcalculationshavebeenmade.

HintforQ12:Inpart b, interestiscalculatedonthe end-of-the-daybalance.

13 Thetableshowstheinterestandmonthlyrepaymentsonloanswhenthesimpleinterestrateis 8.5%

a Usethetabletoﬁndthemonthlyrepaymentsforaloanof:

$1500 over 2 years i $2000 over 3 years ii $1200 over 18 months. iii

b DamienandLisacanaffordmonthlyrepaymentsof $60.Whatisthemosttheycanborrow andonwhatterms?

14 Partofacreditcardstatementisshownhere.

This is the amount you owe at the end of the statement period

MINIMUM PAYMENT DUE

This is the minimum payment that must be made towards this account

TO MINIMISE FURTHER INTEREST CHARGES

This is the amount you must pay to minimise interest charges for the next statement period

Thiscardcharges 21.9% p.a.interestcalculateddailyontheunpaidbalance.Toﬁndthedailyinterest amount,thecompanymultipliesthisbalanceby 0.0006.Whatdoesitcostininterestperdayifonly theminimumpaymentismade?

15 Whenyoutakeoutaloanfromalendinginstitutionyouwillbeaskedtomakeregularpayments(usually monthly)foracertainperiodoftimetorepaytheloancompletely.Thelargertherepayment,theshorter thetermoftheloan.

Loansworkmostlyonareducingbalance,andyoucanﬁndouthowmuchbalanceisowingat theendofeachmonthfromastatement,whichisissuedonaregularbasis.

Let’slookatanexampleofhowthebalanceisreducing.

Ifyouborrow $15000 at 17% p.a.andmakerepaymentsof $260 permonth,attheendoftheﬁrst monthyourstatementwouldbecalculatedasshown.

Interestdue = 15000 × 0.17 12 = $212.50

Repayment = $260

Amountowing = $15000 + $212.50 - $260 = $14952.50

Thisprocesswouldberepeatedforthenextmonth:

Interestdue = 14952.50 × 0.17 12 = $211.83

Repayment = $260

Amountowing = $14952.50 + $211.83 - $260 = $14904.33

Asyoucansee,theamountowing isdecreasingandsoistheinterest owedeachmonth.Meanwhile,more ofyourrepaymentisactuallyreducing thebalanceoftheloan.

Astatementmightlooklikethis:

Checktoseethatallthecalculationsarecorrectonthestatementabove.

Asthisprocessisrepetitive,thecalculationsarebestdonebymeansofaspreadsheet.Tocreatea spreadsheetfortheprocess,copythefollowing,extendingyoursheettocover 5 years.

1I 1I Comparinginterestusingdigitaltools

Learningintentions

• Tounderstandhowdigitaltoolscanbeusedtoef cientlycompareinterestcalculations

• Tobeabletousedigitaltoolstocalculateinterestand nalamountsandcompareinterestplans

Keyvocabulary: simpleinterest,compoundinterest

Bothcompoundinterestandsimpleinterestcalculations involveformulas.Digitaltoolsincludingscientiﬁc andCAScalculators,spreadsheetsorevencomputer programscanbeusedtomakesimpleandcompound interestcalculations.

Theseallowforquick,repeatedcalculationswherevalues canbeadjustedandtheinterestfromdifferentaccounts compared.

Lessonstarter:Whoearnsthe most?

• Ceannainvests $500 at 8% p.a.compoundedmonthly over 3 years.

Theuseofdigitaltoolscanhelpperformrepeated compoundinterestandsimpleinterestcalculations quickly.

• Huxleyinvests $500 at 10% p.a.compoundedannuallyover 3 years.

• Loreliinvests $500 at 15% p.a.simpleinterestover 3 years.

– Howmuchdoeseachpersonhaveattheendofthe 3 years?

– Whoearnedthemost?

Keyideas

Youcancalculatethetotalamountofyourinvestmentforeitherformofinterestusingdigitaltools. Usingformulasincalculators

• Simpleinterest I = Prt 100

• Compoundinterest A = P(1 + r 100 )n

Simplecode

Tocreateprogramsforthetwotypes ofinterest,enterthedatashowninthe boxes.

Thiswillallowyoutocalculatebothtypes ofinterestforagiventimeperiod.Ifyou invest $100000 at 8% p.a.paidmonthlyfor 2 years,youwillbeaskedfor P, R = r 100, t or n andthecalculatorwilldotheworkfor you.

UNCORRECTEDSAMPLEPAGES

Note: Somemodiﬁcationsmaybeneeded fortheCASorothercalculatorsorother digitaltools.

Define simpleinterest()= Prgm

Request "Principal (p): ",p

Request "Rate (%) (r): ",r

Request "Time (t): ",t

I := p r t 100

Disp "Simple interest = ",I

EndPrgm

Define compoundinterest()= Prgm

Request "Principal (p): ",p

Request "Rate (%) (r): ",r

Request "Number of periods (n): ",n

A := p 100 1+ r n

Disp "Total Amount = ",A

EndPrgm

Spreadsheet

Thespreadsheetsshownbelowcanbecompletedtocompileasimpleinterestandcompound interestsheet.

FillintheprincipalinB3 andtherateperperiodinD3.Forexample,for $4000 invested at 5.4% monthly,B3 willbe 4000 andD3 willbe 0.054 12

Exercise1I

Und er stand ing

1 Writedownthevaluesof P, r and n foraninvestmentof $750 at 7.5% p.a.,compoundedannually for 5 years.

2 Writedownthevaluesof P, r and t foraninvestment of $300 at 3% p.a.simpleinterestover 300 months.

3 Whichisbetteronaninvestmentof $100 for 2 years: HintforQ3:Forsimpleinterest

A simpleinterestcalculatedat 20% p.a.? B compoundinterestcalculatedat 20% p.a.and paidannually? Fluency

Example25Usingdigitaltools

Findthetotalamountofthefollowinginvestments,usingdigitaltools. $5000 at 5% p.a.compoundedannuallyfor 3 years. a $5000 at 5% p.a.simpleinterestfor 3 years. b

Solution

Explanation

a $5788.13 Use A = P(1 + r 100 )n oraspreadsheet (see Keyideas).

b $5750 Use I = Prt 100 withyourchosendigitaltool.

Nowyoutry

Findthetotalamountofthefollowinginvestments,usingdigitaltools. $6000 at 4% p.a.compoundedannuallyfor 5 years. a $6000 at 4% p.a.simpleinterestfor 5 years. b Uncorrected 3rd sample

4 a Findthetotalamountofthefollowinginvestments,usingdigitaltools.

$6000 at 6% p.a.compoundedannuallyfor 3 years. i

$6000 at 3% p.a.compoundedannuallyfor 5 years. ii

$6000 at 3.4% p.a.compoundedannuallyfor 4 years. iii

$6000 at 10% p.a.compoundedannuallyfor 2 years. iv

$6000 at 5.7% p.a.compoundedannuallyfor 5 years. v

b Whichoftheaboveyieldsthemostinterest?

5 a Findthetotalamountofthefollowinginvestments,using digitaltoolswherepossible.

$6000 at 6% p.a.simpleinterestfor 3 years. i

$6000 at 3% p.a.simpleinterestfor 6 years. ii

$6000 at 3.4% p.a.simpleinterestfor 7 years. iii

$6000 at 10% p.a.simpleinterestfor 2 years. iv

$6000 at 5.7% p.a.simpleinterestfor 5 years. v

b Whichoftheaboveyieldsthemostinterest?

Problem-solving and reasoning

6 a Determinethetotalsimpleandcompoundinterestaccumulated onthefollowing.

i $4000 at 6% p.a.payableannuallyfor: 1 year

ii $4000 at 6% p.a.payablebiannuallyfor:

iii $4000 at 6% p.a.payablemonthlyfor: 1

b Wouldyoupreferthesamerateofcompoundinterestor simpleinterestifyouwereinvestingmoneyandpayingofftheloanininstalments?

c Wouldyoupreferthesamerateofcompoundinterestorsimpleinterestifyouwere borrowingmoney?

7 a Copyandcompletethefollowingtableifsimpleinterestisapplied.

b Explaintheeffectontheinterestwhenwedoublethe: rate i period ii overalltime. iii

8 Copyandcompletethefollowingtableifcompoundinterestisapplied.Youmayneedtouseacalculator andtrialanderrortoﬁndsomeofthemissingvalues.

$18000

9 Ifyouinvest $5000,determinetheinterestrateperannum(totwodecimalplaces)ifthetotalamount isapproximately $7500 after 5 yearsandifinterestis: compoundedannually a compoundedquarterly b compoundedweekly.

c Commentontheeffectofchangingtheperiodforeachpaymentontherateneededtoachieve thesametotalamountinagiventime.

10 a Determine,toonedecimalplace,theequivalentsimpleinterestrateforthefollowinginvestments over 3 years.

$8000 at 4% compoundedannually. i

$8000 at 8% compoundedannually. ii

b Ifyoudoubleortriplethecompoundinterestrate,howisthesimpleinterestrateaffected?

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Financemanager

Abookkeeperandanaccountsmanagerare bothoccupationsthatdealwithnumbersand budgets.Theyrequireemployeestohavegood communicationandmathematicalskills.

Employeesalsoneedacommitmenttodetail andtobehonest,astheydealwithother people’smoney.

Excellentnumberskillsareessentialinthese ﬁelds.Bookkeepersneedtoworkwith spreadsheets,percentages,taxsystemsand businessplans.

Completethesequestionsthataﬁnancemanager mayfaceintheirday-to-dayjob.

1 Considertheinformationshownherefroma sectionofabusinessbudgetfora 3-month period.

Roundallanswerstotwodecimalplaces. Calculatethetotalincomeforthemonth ofJuly.

Calculatethetotalincomeforthemonth ofAugust.

c Whichmonthhadthehighestincome andbyhowmuch?

Calculatethetotalincomeforthemonth ofSeptember.

Whatcontributedtothisincreasein income?

Whatpercentageofthetotalincome forthe 3 monthsshowncamefroma ﬁxedfee?

Whatwasthemonthlyﬁxedfeebefore the 25% reductionoccurred?

2 Theofﬁceexpensesforthesamecompanyandforthesame 3-monthperiodaregiveninthetable.

a Calculatethepercentage ofthetotalofﬁce expensesforJulyspent onrent.

b Whatisthecostof electricityshowninthe table,andinwhatmonth isitshown?

c Whydoestheelectricity notappearintheother twomonths?

d Whatistheprojectedcostofelectricityfortheyear?

3 TheemploymentexpensesforthethreemonthsofOctober,NovemberandDecemberareshown.

a Calculatethetotalemploymentexpensesforthemonthof December.

b Whatisthewholenumberpercentageincreaseof November’stotalemploymentexpensesfromNovemberto December?Whatwasthecauseofthisincrease?

c Thecompanyhas 11 full-timeemployees.Whatisan employee’saverage: salarypermonth? i annualsalary? ii

d ThecompanyhastotalexpensesforthemonthofNovemberof $92117.Whatpercentageofthe totalexpensesforNovembercomesfromtheemploymentexpenses?

Usingdigitaltools

4 Atruckingbusinesshasinvestedinanewprimemoverforhaulingcattlebyroadtrain.Ithasabank loanof $230000 at 9% perannumchargedmonthly.ThebusinessrequiresanExcelspreadsheetto showtheprogressofthedebtrepayment.

a DevelopthefollowingtableinanExcelspreadsheetbyenteringformulasintotheyellowshaded cellstocalculatetheirvalues.Usethenotesbelowtohelpyou.

Notes:

HintforQ4:Afterentering yourformulas,checkspeciﬁc resultswithacalculator.

• Theinterestduepermonthis 1 12 of 9% ofthestartingbalanceforthatmonth.

• Theprincipal(i.e.debt)paidwillbethescheduledpaymentminustheinterestdue.

• Theendingbalanceswillequalthestartingbalanceminustheprincipalpaid.

• Thenextmonth’sstartingbalanceequalsthepreviousmonth’sendingbalance.

b Extendthetablefor 12 paymentsandanswerthefollowingquestions. WhatistheamountofdebtremainingonJuly 1? i WhatistheinterestpaidinOctober? ii UseanExcelformulatoﬁndthedifferencebetweentheprincipalpaidinDecemberandthe principalpaidinJanuary. iii Enter‘sum’formulastodeterminethetotalinterestpaidintheyearandthetotalprincipal paidoffintheyear. iv

Investinginart

Matildaisakeenartinvestorandhastheopportunitytopurchaseanewworkfromanauctionhouse.

Theauctioneersaysthattheestimatedvalueofthepaintingis $10000.Matilda’smaininvestmentgoal isforeachofherinvestmentstoatleastdoubleinvalueevery 10 years.

Presentareportforthefollowingtasksandensurethatyoushowclearmathematicalworkings, explanationsanddiagramswhereappropriate.

1Preliminarytask

a IfMatildapurchasesthepaintingfor $10000 andassumesagrowthrateof 5% p.a.,calculate thevalueoftheinvestmentafter: 1 year i 2 years ii 3 years. iii

b Theruleconnectingthevalueofthe $10000 investment($A)growingat 5% p.a.after t years isgivenby A = 10000 1 + 5 100 t

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Checkyouranswerstopart a bysubstituting t = 1, t = 2 and t = 3 intothegivenrule andevaluatingthevalueof A usingyourcalculator,oraspreadsheet.

Constructasimilarruleforaninvestmentvalueof $12000 andagrowthrateof 3% ii

Constructasimilarruleforaninvestmentvalueof $8000 andagrowthrateof 8% iii

c Using A = 10000 1 + 5 100 t ﬁndthevalueofa $10000 investmentat 5% p.a.after 10 years.

i.e.Calculatethevalueof A if t = 10.

2Modellingtask

TheproblemistodetermineaninvestmentgrowthratethatdeliversatleastadoublingofMatilda’s initialinvestmentamountafter 10 years.

a Writedownalltherelevantinformationthatwillhelpsolvethisproblem.Whatformulaneedsto beappliedinthistask?

b Explainwhatthenumbers 10000 and 5 meanintherule A = 10000 1 + 5 100 t inrelationto Matilda’sinvestment.

c Usetherule A = 10000 1 + r 100 t todeterminethevalueofMatilda’s $10000 investmentafter 10 yearsforthefollowinggrowthrates(r%).

r = 4 i

r = 7 ii

r = 10 iii

d Chooseyourownvaluesof r usingonedecimalplaceaccuracyanddeterminethegrowthrate forwhichtheinvestmentdoublesinvalueafter 10 years.

e Byconsideringvaluesof r eithersideofyourchosenvaluefoundinpart d,demonstratethatyour answeriscorrecttoonedecimalplace.

f Reﬁneyouranswersothatitiscorrecttotwodecimalplaces.

g Summariseyourﬁndingsanddescribeanykeyﬁndings.Youmightliketoshowyourresultsina spreadsheetsimilartotheonebelow.

3Extensionquestions

a Decideifchangingtheinitialinvestmentvaluechangesthetotalpercentageincreaseinvalue afterthesamenumberofyears.Justifyyouranswer.

b IfMatildaonlypaid $8000 fortheartworkbutstillwantedittobevaluedat $20000 after 10 years,determinethegrowthratethatshewillneedtheworktohave?Roundtotwodecimal places.

Analyse and represent

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Interpret and verify

Communicate

Comparingsimpleandcompoundinterest

Keydigitaltools:Graphingandspreadsheets

Intheworldofﬁnance,itisimportanttoknowthe differencebetweensimpleandcompoundinterest. Thedifferencesinthevalueofinvestmentsandloans canbeverysigniﬁcantoverthelongterm.

Youwillrecalltheserulesfortheamount A:

• Simpleinterest: A = P

• Compoundinterest: A =

1Gettingstarted

Imagineinvesting $100000.

a Calculatethetotalvalueoftheinvestmentusingthefollowingsimpleinterestterms. 4% p.a.for 5 years i 5% p.a.for 10 years. ii

b Calculatethetotalvalueoftheinvestmentusingthefollowingcompoundinterestterms. 4% p.a.for 5 years i 5% p.a.for 10 years. ii

c Compareyouranswersfromparts a and b anddescribewhatyounotice.Canyouexplainwhythe compoundinterestreturnsarehigherthanthesimpleinterestreturns?

2Usingdigitaltools

Twopeopleinvest $100000 inthefollowingways:

• A:Simpleinterestat r1 % for t years

• B:Compoundinterestat r2 % for t years.

a UsegraphingsoftwarelikeDesmostoconstructagraphofthetotalvalueoftheinvestments A and B onthesamesetofaxes.Useslidersfor r1 and r2 asshown.

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b Noteinthepreviousexamplethat r1 iscurrently 6 and r2 iscurrently 5.Dragthesliderstochange thevalueoftheinterestratesandnotethechangesinthegraphs.

c Chooseacombinationof r1 and r2 sothatthevaluesoftheinvestmentsareroughlyequalnear thefollowingnumberofyears.

5 i 10 ii

d Setthecompoundinterestrate, r2 ,at 4%.Dragthe r1 slidertoﬁndasimpleinterestratesothat thevaluesoftheinvestmentsareapproximatelyequalafter 10 years.

3Applyinganalgorithm

Asimpleinterestratewhichisequivalenttoacompoundinterest ratecanbefoundusinganalgorithmicapproachinsidea spreadsheet.

a Considerthisﬂowchartwhichﬁndsthevalueofasimple interestinvestmentover t years.Bychoosing t = 4, runthroughthealgorithmandcompletethistablefor eachpass.

b Writeasimilarﬂowchartbutthistimeforthe compoundingcase.

c Applythesealgorithmsbysettingupaspreadsheetlike thefollowingtocomparethetotalvalueofasimpleand compoundinterestinvestmentof $100000 over t years.

d Afterﬁllingdownfromcellsinrow 6,comparethevaluesoftheinvestmentsovera 12-yearperiod. Experimentwiththenumbersinrow 2,changingtheinitialinvestmentamountandtheinterest rates.

e Usinga $100000 investmentandacompoundinterestrateof 5%,useyourspreadsheettoﬁndan equivalentsimpleinterestratethatdeliversanequalinvestmentvalueafter 10 years.

1 Findanddeﬁnethe 10 termsrelatedtoconsumerarithmeticandpercentageshiddeninthis wordﬁnd.

2 Howdoyoustopabullchargingyou?Answerthefollowingproblemsandmatchtheletterstothe answersbelowtoﬁndout.

3 Howmanyyearsdoesittake $1000 todoubleifitisinvestedat 10% p.a.compoundedannually?

4 ThechanceofJaydenwinningagameofcardsissaidtobe 5%.Howmanyconsecutivegames shouldJaydenplaytobe 95% certainhehaswonatleastoneofthegamesplayed?

Decrease 20 by 8%

= 20 × 92% = 20 × 0.92

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Simple interest

I = simple interest

P = principal ($ invested)

r = rate per year, as a percentage

t = number of years

Compound interest

A = ﬁnal balance

P = principal ($ invested)

r = rate per time period, as a percentage

n = number of time periods money is invested for

Deciding how income is spent on ﬁxed and variable expenses. Percentages

Budgets

Spreadsheets can be used to manage money or compare loans.

Income

Gross income = total of all money earned

Net income = gross income deductions

Money given from wages (income tax) to the government (Refer to the income tax table in Chapter 1D Income taxation.)

Balance owing = amount left to repay

Repayment = money given each month to repay the loan amount and the interest

Chapterchecklist

AversionofthischecklistthatyoucanprintoutandcompletecanbedownloadedfromyourInteractiveTextbook.

1A 1

Icanconverttoapercentage.

e.g.Writeeachofthefollowingasapercentage.

a 7 40 b 0.24

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1A 2

Icanwritepercentagesassimpliﬁedfractionsanddecimals.

e.g.Writeeachofthefollowingpercentagesasbothasimpliﬁedfractionandadecimal.

a 53% b 4% c 10.5%

1A 3

1B 4

1B 5

Icanﬁndthepercentageofaquantity.

e.g.Find 64% of $1400.

Icanincreaseanddecreasebyagivenpercentage.

e.g.Fortheamountof $800: a increase $800 by 6% b decrease $800 by 15%

Icancalculatepercentageproﬁt.

e.g.Jimmybuysasecond-handdeskfor $145 andrestoresittoagoodcondition.Ifhesellsiffor $210, calculatehisproﬁtandthepercentageproﬁt,correcttoonedecimalplace.

1B 6

Icanﬁndthesellingprice.

e.g.Jobuyst-shirtsfor $24 eachandwishestomakea 28% proﬁtonthepurchase.Whatshouldbeher sellingpriceandwhatwillbetheproﬁtonthesaleof 20 t-shirts?

1B 7

1C 8

Icancalculateadiscount.

e.g.A $849 televisionisdiscountedby 18%.Whatisthesellingpriceofthetelevision?

Icanﬁndgrossandnetincomeinvolvingovertime.

e.g.Anikaearns $21.40 perhourandhasnormalworkinghoursof 38 hoursperweek.Sheearnstime andahalfforthenext 4 hoursworkedanddoubletimeafterthat.Shepays $190 perweekintaxand otherdeductions.

Calculatehergrossandnetincomeforaweekinwhichsheworks 45 hours.

1C 9 Icancalculateincomeinvolvingcommission.

e.g.Tiaearns $300 perweekplusacommissionof 6% onhersalesofsolarpanels.Ifshesells $8200 worthofsolarpanelsinaweek,whatishergrossincomefortheweek?

1D 10 Icancalculateincometaxpayable.

e.g.Noahearns $78406 peryear,includinginterestoninvestments.Hehasreceiptsfordonationsand workrelatedexpensesof $445.

a CalculateNoah’staxableincome.

b Usethe2024/2025taxtableintheKeyideasinChapter1D Incometaxation tocalculateNoah’s taxpayableamount,tothenearestcent.

c IfNoahalsohastopay $1559 fortheMedicarelevy,calculatehistaxrefundifhisemployersent $16000 totheATO.

1E 11

Icanbudgetusingpercentages.

e.g.Ashhasanetannualincomeof $54800 afterdeductions.Sheallocatesherbudgetonapercentage basis.

%) 25 10 5

a Determinetheamountofﬁxedexpenses(rentandtheloan).

b Determinehowmuchshebudgetstosaveeachmonth.

1E 12

Icanbudgetfromﬁxedvalues.

e.g.Runningacertaintypeofmotorbikeinvolvesthefollowingcosts:

• registration $520 peryear

• insurance $120 perquarter

• servicing $310 peryear

• petrol $ 64 permonth

Determinetheoverallcosttorunthebikeforayearandwhatpercentageofan $80000 salarythis wouldbe,correcttoonedecimalplace.

1E 13 Icancalculateabestbuy.

e.g.Packetsofchipscanbeboughtinthefollowingwaysatthestore:

• 20 packs(20 gramseach)for $5.50

• 6 packs(20 gramseach)for $3.35

• 2 sharebags(60 gramseach)for $4

Determinethecheapestwaytobuythechips.

1F 14

Icanusethesimpleinterestformulatoﬁndinterest.

e.g.Usethesimpleinterestformulatocalculatetheinterestwhen $800 isinvestedat 5% p.a.for 3 years.

1F 15 Icancalculaterepaymentsusingsimpleinterest.

e.g.Ifasimpleinterestloanof $4000 isborrowedfor 2 yearsatasimpleinterestrateof 4% p.a.,what isthetotalamountowedoverthe 2 years,andifrepaymentsaremademonthly,howmuchwouldeach paymentneedtobe?

1F 16 Icanusethesimpleinterestformulatoﬁndtherateofinterest.

e.g.Usethesimpleinterestformulatocalculatetherateofinterestwhen $2800 earns $294 interestin 3 years.

1G 17 Icanusethecompoundinterestformula.

e.g.Determinetheamountafter 6 yearswhen $8000 iscompoundedannuallyat 3%

1G 18 Icanusecompoundinterestwithdifferenttimeperiods.

e.g.Aninvestmentof $5500 iscompoundedat 6% p.a.over 4 years.Determinetheamountafter 4 yearsifinterestispaidmonthly.

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1H 19 Icanworkwithrepaymentstocalculateapurchasecost.

e.g.Vanessapaysforan $8600 travelpackagewithatravelagentwitha 30% depositandmonthly repaymentsof $300 for 2 years.

Calculate:

a thedepositpaid

b thetotalamountpaidforthetravelpackageandhencetheinterestpaid.

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1H 20 Icancalculateinterestearnedonanaccount.

e.g.Anaccounthasaminimummonthlybalanceof $140 andinterestiscreditedmonthlyonthis amountat 1.8%.Determinetheamountofinteresttobecreditedattheendofthemonthandthetotal amountcreditedordebitedifthebankcharges $5 permonthinaccountkeepingfees.

1I 21 Icanusedigitaltoolstocalculateinterestandﬁnalamounts.

e.g.Usedigitaltoolstoﬁndthetotalamountofthefollowinginvestments.

a $7000 at 4% p.a.compoundedannuallyfor 5 years.

b $7000 at 4% p.a.simpleinterestfor 5 years.

Short-answerquestions

1 1A Convertthefollowing: 11 20 toapercentage a 0.12 toapercentage b 36% toasimpliﬁedfraction c 3.5% toadecimal. d 2 1A Find 16% of $9000 3 1B Increase $968 by 12%

4 1B Thecostpriceofanitemis $7.60.Ifthisisincreasedby 50%,determine: theretailprice a theproﬁtmade.

5 1B Anairfareof $7000 isdiscounted 40% ifyouﬂyoff-peak. Whatwouldbethediscountedprice?

6 1E Josephinebudgets 20% ofherincomefor entertainment.Ifheryearlyincomeis $37000, howmuchcouldbespentonentertainmentin: ayear? a amonth? b aweek(taking 52 weeksinayear)? c

7 1C Dinaworksa 34-hourweekat $25.43 perhour.Hernetincomeis 62% ofherwage. Workoutherweeklynetincome. a If 15% isspentonclothing,determinetheamountshecanspendeachweek. b Ifshesaves $100,whatpercentage(totwodecimalplaces)ofhergrossweeklyincome isthis? c

8 1E Frankhasthefollowingexpensestorunhiscar:

$350 permonth

$885 peryear

$315 perquarter

$1700 peryear

$90 perweek

a Findthetotalcostofrunninghisvehiclefor 1 year. b Whatpercentage(tothenearestpercentage)oftheoverallcosttorunthecaristhecostofthe petrol?

9 1D Ronanworks 36 hoursinaweekat $39.20 perhour.Hepays $310 intaxand $20.50 in superannuationintheweek.Determine: hisgrosswageinaweek a hisnetpayinaweek. b

10 1D Lilreceivesanannualtaxableincomeof $90000

$51638 plus 45cforeach $1 over $190000 a

Thetablebelowshowstaxratesinthe2024/2025ﬁnancialyear.Usingtable,calculatethe amountoftaxshepaysovertheyear.

0–$18200 Nil

$18201–$45000 16cforeach $1 over $18200

$45001–$135000

$135001–$190000

$190001 andover

$4288 plus 30cforeach $1 over $45000

$31288 plus 37cforeach $1 over $135000

IfLilpaysthe 2% Medicarelevyonhertaxableincome,ﬁndthisamount. b

11 1C Zanereceives 4.5% commissiononsalesof $790.Determinetheamountofhiscommission.

12 1F Findtheinterestpaidona $5000 loanunderthefollowingconditions.

8% p.a.simpleinterestover 4 years a 7% p.a.simpleinterestover 3 yearsand 4 months b

13 1G Findtheinterestpaidona $5000 loanunderthefollowingconditions.

4% p.a.compoundedannuallyover 3 years a 9.75

1H Avehicleworth $7000 ispurchasedonaﬁnancepackage.Thepurchaserpays 15% deposit and $250 permonthover 4 years. Howmuchdepositispaid? a Whatarethetotalrepayments? b Howmuchinterestispaidoverthetermoftheloan? c

Multiple-choicequestions

1 1A 28% of $89 isclosestto:

3 1E Ifabudgetallows 30% forcarexpenses,howmuchisallocatedfromaweeklywage of $560?

5 1C IfSimonreceives $2874 onthesaleofapropertyworth $195800,hisrateofcommission, toonedecimalplace,is:

6 1C Inagivenrosteredfortnight,Bilalworksthefollowingnumberof 8-hourshifts:

• threedayshifts($10.60 perhour)

• threeafternoonshifts($12.34 perhour)

• ﬁvenightshifts($16.78 perhour).

Histotalincomeforthefortnightis:

A $5000 loanisrepaidbymonthlyinstalmentsof $200 for 5 years.Theamountofinterest chargedis:

Extended-responsequestions

1 $5000 isinvestedat 4% p.a.compoundingannuallyfor 3 years. Whatisthevalueoftheinvestmentafterthe 3 years? a Howmuchinterestisearnedinthe 3 years?

Using r = 100I Pt ,whatsimpleinterestrateresultsinthesameamount?

Howmuchinterestisearnedontheinvestmentifitiscompoundedmonthlyat 4% p.a.forthe 3 years? d

2 YourbankaccounthasanopeningJulymonthlybalanceof $217.63.Youhavethefollowing transactionsoverthemonth.

Designastatementofyourrecordsif $0.51 istakenoutasafeeon 15 July.

a Findtheminimumbalance. b Ifinterestiscreditedmonthlyontheminimumbalanceat 0.05%,determinetheinterestforJuly, roundedtothenearestcent.

2 Measurement

Essentialmathematics:whymeasurementskillsare important

Accuratemeasurementskillsandcalculationsareessentialforyourhomeimprovementprojects andalsoforsafeindustrialoperationsandastablebuiltenvironment.

Spatialdesignersrelyonaccuratemeasurements.Forexample,interiordesignersplanfurniture layoutandlightingselections,digitalgamedesignerscreateimmersiveenvironmentswith proportionalaccuracy,farmersoptimizecroplayoutsandirrigationsystems,andeventplanners arrangeseating,staging,lightingandsoundsystems.

HVACtechniciansdesigneffectiveheatingandcoolingsystemsandcalculateperimeters,surface areasandvolumesofairinroomsandducts,includinginofﬁces,restaurants,kitchens,hospitals, schoolsandmovietheatres.

Windturbinesrequireprecisemeasurementssothateachblade’scylindricaljoining-piececonnects perfectlytothehubandthentothecylindricalshaftandcirculargearsthatdrivetheelectric generator.Windpower, P,iscalculatedusingtheformula P = 1 2 qAv3 ,where q isairdensity, A is thesweptcircularareaoftheblades,and v isthewindspeed.

SAMPLEPAGES

Inthischapter

2AConversionofunits (Consolidating)

2BPerimeter (Consolidating)

2CCircumference (Consolidating)

2DArea

2EAreaofcirclesandsectors

2FMeasurementerrorsandaccuracy

2GSurfaceareaofprisms

2HSurfaceareaofacylinder

2IVolumeofsolids

2JFurtherproblemsinvolvingprisms andcylinders

WACurriculum

Thischaptercoversthefollowingcontent descriptorsintheWACurriculum:

NUMBERANDALGEBRA

WA10MNAC1,WA10MNAA2,WA10MNAM1

MEASUREMENTANDGEOMETRY

WA10MMGTH1,WA10MMGTH2, WA10MMGM1

©SchoolCurriculumandStandardsAuthority Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractiveTextbook, includingHOTmathscontent,video demonstrationsofallworkedexamples, auto-markedquizzesandmuchmore.

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1 Nametheseshapes.Choosefromthewords trapezium, triangle, circle, rectangle, square, semicircle, parallelogram and rhombus a b c d e f g h

2 Writethemissingnumber.

3 Findtheperimeteroftheseshapes.

4 Findtheareaoftheseshapes.

5 Findtheareaofthesetrianglesusing A = 1 2 bh.

6 Use C =p d and A =p r2 toﬁndthecircumferenceandareaofthiscircle.Roundyouranswerto twodecimalplaces.

2A 2A Conversionofunits

Learningintentions

• Toreviewthemetricunitsofmeasurement

CONSOLIDATING

• Tobeabletoconvertbetweenmetricunitsforlength,areaandvolume

Keyvocabulary: unit,length,area,volume

Toworkwithlength,areaorvolumemeasurements,itis importanttobeabletoconvertbetweendifferentunits. Forexample,timberiswidelyusedinbuildingsforframes, rooftrussesandwindows,thereforeitisimportantto orderthecorrectamountsothatthecostofthehouse isminimised.Althoughplansgivemeasurementsin millimetresandcentimetres,timberisorderedinmetres (oftenreferredtoaslinealmetres),sowehavetoconvertall ourmeasurementstometres.

Buildingahousealsoinvolvesmanyareaandvolume calculationsandunitconversions.

Lessonstarter:Houseplans

Beingabletoconvertbetweenunitsof measurementisanimportantskillforabuilder.

Allhomesstartfromaplan,whichisusuallydesignedbyanarchitectandshowsmostofthebasic featuresandmeasurementsthatareneededtobuildthehouse.Measurementsaregiveninmillimetres.

• Howmanybedroomsarethere?

• Whatarethedimensionsofthemasterbedroom(i.e.BED 1)?

• Whatarethedimensionsofthemasterbedroom,inmetres?

• Willtherumpusroomﬁtapooltablethatmeasures 2.5 m × 1.2 m,andstillhaveroomtoplay?

• Howmanycarsdoyouthinkwillﬁtinthegarage?

Keyideas

Toconvert units,drawanappropriatediagramanduseittoﬁndtheconversionfactor.

Forexample:

Conversions:

Tomultiplyby 10, 100, 1000 etc.move thedecimalpointoneplacetotherightfor eachzero.

3.425 × 100 = 342.5

Todivideby 10, 100, 1000 etc.movethe decimalpointoneplacetotheleft foreachzero.

e.g. 4.10 ÷ 1000 = 0.0041

1 Writethemissingnumbersinthesesentencesinvolvinglength.

a Thereare min 1 km.

b Thereare mmin 1 cm.

c Thereare cmin 1 m.

2 Writethemissingnumbersinthesesentencesinvolvingareaunits.

a Thereare mm2 in 1 cm2

b Thereare cm2 in 1 m2

c Thereare m2 in 1 km2

3 Writethemissingnumbersinthesesentencesinvolvingvolumeunits.

a Thereare mm3 in 1 cm3

b Thereare m3 in 1 km3

c Thereare cm3 in 1 m3

Fluency

Example1Convertinglengthmeasurements

Converttheselengthmeasurementstotheunitsshowninthebrackets.

Explanation

Multiplywhenconvertingtoasmallerunit.

b 45

Dividewhenconvertingtoalargerunit.

Nowyoutry

Converttheselengthmeasurementstotheunitsshowninthebrackets. 4.6

4 Convertthefollowingmeasurementsoflengthtotheunitsgivenin thebrackets.

HintforQ4:Whenconverting toasmallerunit,multiply. Otherwise,divide.

Example2Convertingareameasurements

Converttheseareameasurementstotheunitsshowninthebrackets.

Explanation

Whendividingby 10000,movethedecimalpoint 4 placestotheleft.

Continuedonnextpage

b 0.4 cm2 = 0.4 × 102 =

Nowyoutry

Converttheseareameasurementstotheunitsshowninthebrackets.

mm2 (cm2 ) a

km2 (m2 ) b

5 Convertthefollowingareameasurementstotheunitsgiveninthe brackets.

)

m2 (cm2 ) b 5 km2 (m2 ) c

mm2 (cm2 ) d 537 cm2 (mm2 ) e

m2 (cm2 ) f

Example3Convertingvolumemeasurements

Convertthesevolumemeasurementstotheunitsshowninthebrackets.

Nowyoutry

Convertthesevolumemeasurementstotheunitsshowninthebrackets. 0.21 m3 (cm3 ) a

6 Convertthesevolumemeasurementstotheunitsgiveninthe brackets.

Problem-solving and reasoning

7 Anathletehascompleteda 5.5 kmrun.Howmanymetresdidtheathleterun?

8 Determinethemetresoftimberneededtoconstructthefollowingframes.

9 Findthetotalsumofthemeasurements,expressingyour answerintheunitsgiveninthebrackets.

10 cm, 18 mm (mm)

HintforQ9:Converttotheunitsin brackets.Adduptoﬁndthesum. a 1.2 m, 19 cm, 83 mm (cm) b 453 km, 258 m (km) c

0.3 m2 , 251 cm2 (cm2 ) e

10 Asnailismovingatarateof 43 mmeveryminute.Howmanycentimetres willthesnailmovein 5 minutes?

11 Whydoyouthinkthatbuildersmeasuremanyoftheirlengthsusingonly millimetres,eventheirlonglengths?

Specialunits

12 Manyunitsofmeasurementapartfromthoserelatingtomm,cm,mandkmareusedinoursociety. Someofthesearedescribedherealongwithconversioninformation,showinghowtoconvertthemto otherunits.

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Convertthesespecialmeasurementstotheunitsgiveninthebrackets.Usetheconversion informationgiventohelp.

2B 2B Perimeter CONSOLIDATING

Learningintentions

• Tobeabletocalculatetheperimeterofashape

• Tobeableto ndanunknownlengthgiventheperimeter

Keyvocabulary: perimeter

Perimeterisameasureoflengtharoundtheoutsideofashape.Wecalculateperimeterwhenordering materialsforfencingapaddockorwhendesigningahouse.

Farmersneedtomeasuretheperimeterofpaddockswhenbuildinga fencetoensurethattheyorderthecorrectamountofmaterials.

Lessonstarter:L-shapedperimeters

TheL-shapedﬁgureshownhereincludesonlyright (90°) angles.Onlytwo measurementsaregiven.

• Canyouﬁgureoutanyothersidelengths?

• Isitpossibletoﬁnditsperimeter?Why?

Keyideas

Perimeter isthedistancearoundtheoutsideofatwo-dimensionalshape.

• Toﬁndtheperimeter,weaddallthelengthsofthesidesinthesameunits.

• Whentwosidesofashapearethesamelengththeyarelabelledwiththesamemarkings. x z y P = 2x + y + z

Exercise2B

Und er stand ing

1 Writethemissingword:Thedistancearoundtheoutsideofashapeiscalledthe

2 Writedownthevalueof x fortheseshapes.

Fluency

Example4Findingperimetersofbasicshapes

Findtheperimeteroftheseshapes.

a Perimeter = 3 + 2 + 4 + 3.5 = 12.5 cm

b Perimeter = 5 + 5.2 + 3 × 3 = 19.2 m

Nowyoutry

Findtheperimeteroftheseshapes.

Explanation

Addallthelengthsofthesidestogether.

Threelengthshavethesamemarkingsand thereforearethesamelength.

3 Findtheperimeteroftheseshapes.

4 Findtheperimeteroftheseshapes.

HintforQ3c–f:Sideswiththe samemarkingsarethesame length.

Problem-solving and reasoning 5–7 6–10

Example5Findingamissingsidelength

Findthevalueof x forthisshapewiththegivenperimeter.

Solution

4.5 + 2.1 + 3.4 + x = 11.9 10 + x = 11.9 x = 1.9

Nowyoutry

Explanation

Allthesidesaddto 11.9 inlength. Simplify. Subtract 10 frombothsidestoﬁndthevalueof x.

Findthevalueof x forthisshapewiththegivenperimeter.

5 Findthevalueof x fortheseshapeswiththegivenperimeters.

6 Findthevalueof x fortheseshapeswiththegivenperimeters.

HintforQ5:Addupallthesides andthendeterminethevalueof x tosuitthegivenperimeters.

= 17 m

= 22.9 cm

= 0.8

Example6Workingwithperimeterwiththreedimensions

Aconcreteslabhasthemeasurementsshown.Allanglesare 90°

Drawanewdiagram,showingallthemeasurementsinmetres. a Determinethelinealmetresoftimberneededtosurroundit. b

Solution

b Perimeter = 18.5 + 16.8 + 3.5 + 2.7 + 15 + 14.1 = 70.6 m

Thelinealmetresoftimberneededis 70.6 m.

Nowyoutry

Explanation

Convertyourmeasurementsand placethemallonthediagram.

1 m = 100 × 10 = 1000 mm

Addorsubtracttoﬁndthemissing measurements.

Addallthemeasurements.

Writeyouranswerinwords.

Aconcreteslabhasthemeasurementsshown.Allanglesare 90°

Drawanewdiagramshowingallthemeasurementsinmetres. a Determinethelinealmetresoftimberneededtosurroundit. b

7 Sixconcreteslabsareshownbelow.Allanglesare 90° i Drawanewdiagramforeachwiththemeasurementsinmetres. ii Determinethelinealmetresoftimberneededforeachtosurroundit.

8 Arectangularpaddockhasaperimeterof 100 m.Findthewidthofthepaddockifits lengthis 30 m.

9 Theequilateraltriangleshownhasaperimeterof 45 cm.Finditssidelength, x x cm

10 Writeformulasfortheperimeteroftheseshapes,usingthepronumeralsgiven.

HintforQ10:Aformula forperimetercouldbe

Howmanydifferenttables?

11 Alargediningtableisadvertisedwithaperimeterof 12 m.Thelengthandwidthareawholenumberof metres(e.g. 1 m, 2 m, ).Howmanydifferent-sized tablesarepossible?

12 Howmanyrectangles(usingwholenumbermetrelengths)haveperimetersbetween 16 mand 20 m, inclusive?

2C 2C Circumference CONSOLIDATING

Learningintentions

• Toknowtheformulaforthecircumferenceofacircle

• Tobeableto ndthecircumferenceofacircle

• Tobeableto ndthecircumferenceofcircleportionsandsimplecompositeshapes

Keyvocabulary: circumference,pi,radius,diameter,circle

Toﬁndthedistancearoundtheoutsideofacircle–thecircumference–weusethespecialnumbercalled pi (p ).Piprovidesadirectlinkbetweenthediameter ofacircleandthecircumferenceofthatcircle.

Thewheelisoneofthemostusefulcomponentsin manyformsofmachinery,anditsshape,ofcourse,is acircle.Onerevolutionofavehicle’swheelmovesthe vehicleadistanceequaltothewheel’scircumference.

Thecircumferenceofawheeltellsyouhowfaravehicle movesforwardafteronefullrevolution.

Lessonstarter:Whencircumference = height

Hereisadrawingofacylinder.

• Trydrawingyourowncylindersothatitsheightisequaltothecircumferenceof thecirculartop.

• Howwouldyoucheckthatyouhavedrawnacylinderwiththecorrect dimensions?Discuss. height

Keyideas

The radius(r) isthedistancefromthecentreofa circle toapointonthecircle.

The diameter(d) isthedistanceacrossacirclethroughitscentre.

– Radius = 1 2 diameterordiameter = 2 × radius

Circumference(C ) isthedistancearoundacircle.

– C = 2p× radius = 2p r or C =p× diameter =p d

– p (pi) isaspecialnumberandcanbefoundonyourcalculator. Itcanbeapproximatedby p¥ 3.142

Exercise2C

Und er stand ing

1a Thedistancefromthecentreofacircletoitsoutsideedgeiscalledthe b Thedistanceacrossacircle,throughitscentre,iscalledthe c Thedistancearoundacircleiscalledthe .

2 Writetheformulaforthecircumferenceofacircleusing: d fordiameter a r forradius. b

3 Whatfractionofacircleisshownhere?

Fluency

Example7Findingthecircumferenceofacircle

Findthecircumferenceofthesecircles,totwodecimalplaces. 2 cm a 2.65 mm b

Solution

a C = 2p r = 2p (2) = 12.57 cm (to 2 d.p.)

b C =p d =p (2.65) = 8.33 mm (to 2 d.p )

Nowyoutry

Explanation

Writetheformulainvolvingtheradius, r Substitute r = 2

Roundyouranswertotwodecimalplaces.

Writetheformulainvolvingdiameter, d Substitute d = 2.65

Roundyouranswertotwodecimalplaces.

Findthecircumferenceofthesecircles,totwodecimalplaces. 5 m a 4.85cm b

4 Findthecircumferenceofthesecircles,totwodecimalplaces.

Example8Findingperimetersofcompositeshapes

Findtheperimeterofthiscompositeshape,totwodecimalplaces.

HintforQ4:Use C = 2p r or C =p d

Solution

P = 3 + 5 + 1 2 × 2p (2)

= 8 + 2p

= 14.28 m (to 2 d.p.)

Nowyoutry

Explanation

Simplify.

Roundyouranswerasinstructed. Addallthesides,includinghalfacircle.

Findtheperimeterofthiscompositeshape,totwodecimalplaces.

Findtheperimeterofthesecompositeshapes,totwodecimalplaces.

HintforQ5:Don’tforgettoaddthe straightsidestothefraction ( 1 4 , 1 2 or 3 4 ) ofthecircumference.

Problem-solving and reasoning

6 Davidwishestobuildacircularﬁshpond.Thediameterofthepondistobe 3 m.

a Howmanylinealmetresofbricksareneededtosurroundit?Roundyouranswertotwo decimalplaces.

b Whatisthecostifthebricksare $45 permetre?(Useyouranswerfrompart a.)

Thewheelsofabikehaveadiameterof 1 m.

a Howmanymetreswillthebiketravel(totwodecimalplaces)after: onefullturnofthewheels? i 15 fullturnsofthewheels? ii

HintforQ7:Foronerevolution, use C =p d.

b Howmanykilometreswillthebiketravelafter 1000 fullturns ofthewheels?(Roundyouranswertotwodecimalplaces.)

8 Whatistheminimumnumberoftimesawheelofdiameter 1 mneedstospintocoveradistance of 1 km?Youwillneedtoﬁndthecircumferenceofthewheelﬁrst.Giveyouranswerasa wholenumber.

Findtheperimeterofthesecompositeshapes,totwodecimalplaces.

HintforQ9:Makesureyouknowthe radiusordiameterofthecircle(s) youareworkingwith.

10a Rearrangetheformulaforthecircumferenceofacircle, C = 2p r,towrite r intermsof C

b Find,totwodecimalplaces,theradiusofacirclewiththegivencircumference.

35 cm i 1.85 m ii 0.27 km iii Targetpractice

11 Atargetismadeupofthreerings,asshown.

a Findtheradiusofthesmallestring.

b Find,totwodecimalplaces,thecircumferenceofthe: smallestring i middlering ii outsidering. iii

HintforQ10:Towrite r interms of C,dividebothsidesby 2p

c Ifthecircumferenceofadifferentringis 80 cm,whatwouldbeits radius,totwodecimalplaces?

2D 2D Area

Learningintentions

• Toknowtheformulasfortheareasofsimpleshapes

• Tobeableto ndtheareaofsimpleshapes

Keyvocabulary: area,square,rectangle,triangle,rhombus,parallelogram,trapezium,perpendicular

Inthissimplediagram,arectanglewithsidelengths 2 mand 3 mhasan areaof 6 squaremetresor 6 m2 .Thisiscalculatedbycountingthenumber ofsquares(eachmeasuringasquaremetre)thatmakeuptherectangle.

Weuseformulastohelpusquicklycountthenumberofsquareunits containedwithinashape.Forthisrectangle,forexample,theformula A = lw simplytellsustomultiplythelengthbythewidthtoﬁndthearea.

Lessonstarter:Howdoes

Lookatthistriangle,includingitsrectangularreddashedlines.

• Howdoestheshapeofthetrianglerelatetotheshapeofthe outsiderectangle?

• Howcanyouusetheformulaforarectangletohelpﬁndtheareaofthe triangle(orpartsofthetriangle)?

• Whyistherulefortheareaofatrianglegivenby A = 1 2 bh?

Keyideas

The area ofatwo-dimensionalshapeisthenumberofsquareunitscontainedwithinitsboundaries. Someofthecommonareaformulasareasfollows.

The‘height’inatriangle,parallelogramortrapeziumshouldbe perpendicular (at 90°)to thebase.

Exercise2D

Und er stand ing

1 Usingtheshapesshowninthe Keyideas,matcheachshape(a–f)withitsareaformula(A–F). square a A = 1 2 bh A rectangle

2 Theseshapesshowthebaseandaheightlength.Writedownthegivenheightofeachshape.

Example9Usingareaformulas

Findtheareaofthesebasicshapes.

Solution

a Area = lw

= 7 × 3

Explanation

Writetheformulafortheareaofarectangle.

Substitutethelengths l = 7 and w = 3 = 21 cm2

b Area = 1 2 (a + b)h

= 1 2 (3 + 5) × 2

Simplifyandincludetheunits.

Writetheformulafortheareaofatrapezium.

Substitutethelengths a = 3, b = 5 and h = 2. = 8 cm2

Simplifyandincludetheunits.

c Area = 1 2 bh

Writetheformulafortheareaofatriangle. = 1 2 × 5.8 × 3.3

Substitutethelengths b = 5.8 and h = 3.3. = 9.57 m2

Nowyoutry

Findtheareaofthesebasicshapes.

Simplifyandincludetheunits.

HintforQ3:First,choosethe correctformulaandsubstitute foreachpronumeral(letter).

4 Findtheareaofthesebasicshapes.

Problem-solving and reasoning

Arectangulartabletopis 1.2 mlongand 80 cmwide. Findtheareaofthetabletopusing: squaremetres (m2 ) a squarecentimetres (cm2 ) b

6 Twotriangularsailshavesidelengthsasshown.Findthetotalarea ofthetwosails.

Example10Applyingareaformulas

Amiradecidestousecarpettocovertheﬂoorofherrectangular bedroom.Determine:

theareaofﬂoortobecovered a thetotalcostifthecarpetcosts $32 persquaremetre. b

HintforQ5:Firstconverttothe unitsthatyouwanttoworkwith.

a Areaofﬂoor = l × w = 3.5 × 2.6 = 9.1 m2

Theroomisarectangle,souse A = l × w tocalculatethe totalﬂoorspace.

b Costofcarpet = 9.1 × 32 = $291.20 Everysquaremetreofcarpetcosts $32

Nowyoutry Richodecidestolaylawnonhistriangularbackyard. Determine: theareaoflawntobelaid a thetotalcostiflawncosts $11 persquaremetre. b

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7 Jack’sshedistohaveaﬂatrectangularroof,whichhedecidestocover withmetalsheets.

a Determinethetotalareaoftheroof.

b Ifthemetalrooﬁngcosts $11 asquaremetre,howmuchwillitcost intotal?

8 Aslidingdoorhastwoglasspanels.Eachoftheseis 2.1 mhighand 1.8 mwide.

a Howmanysquaremetresofglassareneeded?

b Whatisthetotalcostoftheglassifthepriceis $65 per squaremetre?

9 Arectangularwindowhasawholenumbermeasurementforitslengthandwidthanditsareais 24 m2 Writedownthepossiblelengthsandwidthsforthewindow.

Determinetheareaofthehousesshown(ifallanglesarerightangles),insquaremetres(totwo decimalplaces).

Findthevalueofthepronumeralintheseshapes,roundingyouranswertotwodecimalplaces eachtime.

HintforQ11:First,writethe appropriateformulaandsubstitute fortheareaandlengthpronumerals. Thensolvefortheunknown.

Fourwaystoﬁndtheareaofatrapezium

12 Findtheareaofthistrapeziumusingeachofthesuggestedmethods.

2E 2E Areaofcirclesandsectors

Learningintentions

• Toknowtheformulafortheareaofacircle

• Tobeabletocalculatewhatfractionofacircleisrepresentedbyasector

• Tobeableto ndtheareaofcirclesandsectors

Keyvocabulary: sector,circle,radius,diameter,pi

Likeitscircumference,acircle’sareaislinkedtothespecialnumberpi (p ).Theareaistheproductofpi andthesquareoftheradius,so A =p r2

Knowingtheformulafortheareaofacirclehelpsusbuildcircularobjects,planwatersprinklersystems andestimatethedamagecausedbyanoilslickfromashipincalmseas.

Lessonstarter:Whatfractionisthat?

Wecandivideacircleupintosectors.Asectorisaportionofacircle.When ﬁndingareasofsectors,weﬁrstneedtodecidewhatfractionofacircleweare dealingwith.Thissector,forexample,hasaradiusof 4 cmanda 45° angle.

• Whatfractionofafullcircleisshowninthissector?

• Howcanyouusethisfractiontohelpﬁndtheareaofthissector?

• Howwouldyousetoutyourworkingtoﬁnditsarea?

Keyideas

Theformulaforﬁndingthearea (A) ofa circle of radius r isgivenbythe equation: A =p r2

Whenthe diameter(d) ofthecircleisgiven,determinetheradiusbefore calculatingtheareaofthecircle: r = d ÷ 2

A sector isaportionofacircleincludingtworadii.

Theangleofasectorofacircledeterminesthefractionofthecircle.Afullcircleis 360°

• Thissectoris h 360 ofacircle.

• Theareaofasectorisgivenby A = h 360 ×p r2

er stand ing

1 Whichisthecorrectworkingstepforﬁndingtheareaofthiscircle?

2 Whichisthecorrectworkingstepforﬁndingtheareaofthiscircle?

=p (10)2 A A = (p 10)2 B

=p (5)2 C

= 5p E

3 Whatfractionofacircleisshownbythesesectors?Simplifyyourfraction.

Fluency

Example11Findingareasofcircles

Findtheareaofthesecircles,totwodecimalplaces. 3 m a 1.06 km b

Solution

a A =p r 2

=p (3)2

=p× 9

= 28.27 m2 (to 2 d.p.)

b Radius r = 1.06 ÷ 2 = 0.53 km

A =p r 2

=p (0.53)2

= 0.88 km2 (to 2 d.p.)

Nowyoutry

Explanation

Writetheformula.

Substitute r = 3

Evaluate 32 = 9 andthenmultiplyby p

Writeyouranswertotwodecimalplaces withunits.

Findtheradius,giventhediameterof 1.06

Writetheformula.

Substitute r = 0.53

Writeyouranswertotwodecimalplaces withunits.

Findtheareaofthesecircles,totwodecimalplaces. 5 m a 3.92cm b

Findtheareaofthesecircles,totwodecimalplaces.

Example12Findingareasofsectors

Findtheareaofthissector,totwodecimalplaces.

HintforQ4: r = d ÷ 2

Solution

Fractionofcircle = 60 360 = 1 6

Area = 1 6 ×p r 2

Explanation

Thesectoruses 60° outofthe 360° ina wholecircle.

Writetheformula,includingthefraction. = 1 6 ×p (10)2

Substitute r = 10 = 52.36 m2 (to 2 d.p.) Writeyouranswertotwodecimalplaces.

Nowyoutry

Findtheareaofthissector,totwodecimalplaces.

Findtheareaofthesesectors,totwodecimalplaces.

HintforQ5:Firstdetermine thefractionofafullcircle thatyouareworkingwith.

Problem-solving and reasoning

6–87,8,9(½)

6 Apizzawith 40 cmdiameterisdividedintoeightequalparts.Findtheareaofeachportion,toone decimalplace.

Example13Findingareasofcompositeshapes

Findtheareaofthiscompositeshape,totwodecimalplaces.

Solution Explanation

Theshapeismadeupofasemicircleandatriangle.Writethe formulasforbothshapes.

Substitute

= 3.57 cm2 (to 2 d.p.) Writeyouranswertotwodecimalplaceswithunits.

Nowyoutry

Findtheareaofthiscompositeshape,totwodecimalplaces.

Findtheareaofthesecompositeshapes,totwo decimalplaces.

HintforQ7:Findthearea ofeachshapethatmakesup thelargershape,thenaddthem. Forexample,triangle + semicircle.

8 Thelawnareainabackyardismadeupofasemicircularregionwithdiameter 6.5 manda right-angledtriangularregionoflength 8.2 m,asshown.Findthetotalareaoflawninthebackyard, totwodecimalplaces.

9 Findtheareaofthesecompositeshapes,toonedecimalplace.

HintforQ9:Useadditionor subtraction,depending ontheshapegiven.

Circularpastries

10 Arectangularpieceofpastryisusedtocreatesmallcircularpastrydiscs forthebaseofChristmastarts.Therectangularpieceofpastryis 30 cm longand 24 cmwide,andeachcircularpiecehasadiameterof 6 cm. Howmanycircularpiecesofpastrycanberemovedfromthe rectangleinthisarrangement?

Findthetotalarearemovedfromtheoriginalrectangle,totwo decimalplaces.

Findthetotalareaofpastryremaining,totwodecimalplaces.

c Iftheremainingpastrywascollectedandre-rolledtothesamethickness,howmanycircularpieces couldbecut?(Assumethatthepastrycanbere-rolledandcutmanytimes.)

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2F 2F Measurementerrorsandaccuracy

Learningintentions

• Tounderstandthedif cultyinobtainingexactmeasurements

• Toknowhowto ndtheupperandlowerboundaries(limitsofaccuracy)forthetruemeasurement

• Tounderstandthatroundingoffinintermediatecalculationsleadstoanaccumulatederror

Keyvocabulary: accuracy,precision,rounding,accumulatederror,limitsofaccuracy,absolutevalue

Humansandmachinesmeasuremanydifferentthings,suchasthetimetakentoswimarace,thelengthof timberneededforabuildingandthevolumeofcementneededtolayaconcretepatharoundaswimming pool.Thedegreeorlevelofaccuracyrequiredusuallydependsontheintendedpurposeofthemeasurement.

Allmeasurementsareapproximate.Errorscanhappenasaresultoftheequipmentbeingusedortheperson usingthemeasuringdevice.

Accuracyisameasureofhowclosearecordedmeasurementistotheexactmeasurement.Precisionisthe abilitytoobtainthesameresultoverandoveragain.

Lessonstarter:Roundingadecimal

• Apieceoftimberismeasuredtobe 86 cm,tothenearestcentimetre. Whatisthesmallestmeasurementpossiblethatroundsto 86 cmwhenroundedtothenearestcm? a Whatisthelargestmeasurementpossiblethatroundsto 86 cmwhenroundedtothenearestcm?

• Ameasurementisrecordedas 6.0 cm,tothenearestmillimetre. Whatunitswereusedwhenmeasuring? a Whatisthesmallestdecimalthatcouldberoundedtothisvalue? b Whatisthelargestdecimalthatwouldhaveresultedin 6.0 cm?

• Considerasquarewithsidelength 7.8941 cm.Whatistheperimeterofthesquareifthesidelengthis: usedwiththefourdecimalplaces? a roundedtoonedecimalplace? b truncatedatonedecimalplace(i.e. 7.8)?

Keyideas

The limitsofaccuracy tellyouwhattheupperandlowerboundariesareforthetrue measurement.

• Usually,itis ± 0.5 × thesmallestunitofmeasurement. Forexample,whenmeasuringtothenearestcentimetre, 86 cmhaslimitsfrom 85.5 cmupto (butnotincluding) 86.5 cm.

• Whenmeasuringtothenearestmillimetre,thelimitsofaccuracyfor 86.0

86.05 cm.

Accumulatederrors canalsooccurinmeasurementcalculationsthatinvolveanumberofsteps.

• Itisimportanttouseexactvaluesoralargenumberofdecimalplacesthroughoutcalculations toavoidanaccumulatederror.

Error=estimatedresult - actualresult

• The absolutevalue oftheerrorgivestheerrorasapositivenumber,ignoringanynegativesign.

• Percentageerror= error actualresult × 100 1

Exercise2F

Und er stand ing

1 Stateadecimalthatgives 3.4 whenroundedfromtwo decimalplaces.

2 Stateameasurementof 3467 mm,tothenearest: centimetre a metre b

HintforQ1: 2.67 roundedto onedecimalplaceis 2.7

3 Whatisthesmallestdecimalthatcouldresultinananswerof 6.7 whenroundedtoonedecimalplace?

4 Completethesecalculations.

a 8.7 × 3.56 roundedtoonedecimalplace i Takeyourroundedanswerfrompart ai,multiplyitby 1.8 androundtoonedecimalplace. ii

b 8.7 × 3.56 withthreedecimalplaces i Takeyourexactanswerfrompart bi,multiplyitby 1.8 androundtoonedecimalplace. ii

c Compareyouranswersfromparts aii and bii.Whatdoyounotice?Whichanswerismoreaccurate?

Example14Avoidingaccumulatederrors

Considertheshapeshown.

a Findtheareaofthesemicircle,roundingtoonedecimalplace.

b Findtheareaofthetriangleintheshape,roundingtoonedecimalplace.

c Hence,ﬁndthetotalareausingyouranswerstoparts a and b

d Nowrecalculatethetotalareabyretainingmorepreciseanswersforthe calculationstoparts a and b.Roundyourﬁnalanswertoonedecimalplace.

e Compareyouranswerstoparts c and d.Howcanyouexplainthe difference?

b Areatriangle = 1 2 × 12.49 × 7.84 = 49.0 m2 (to 1 d.p.)

Areaofasemicircle = 1 2 p r2 where r isthe diameter ÷ 2 Roundtoonedecimalplace.

Trianglearea = 1 2 bh

Continuedonnextpage

c Totalarea = 61.3 + 49.0 = 110.3 m2

d Areasemicircle = 61.2610… m2

Areatriangle = 48.9608 m2

Totalarea = 61.2610…+ 48.9608 = 110.2218 m2

Totalareais 110.2 m2 (to 1 d.p.)

e Theanswersdifferby 0.1 m2 whenrounded toonedecimalplace.

Theerrorresultsinpart c fromtherounding inintermediatestepsinparts a and b

Nowyoutry

Considertheshapeshown.

Combineroundedareasofsemicircleand triangle.

Retainanumberofdecimalplacesforboththe semicircleandtriangleareas.

Combinetheareastocalculatethetotalarea.

Roundﬁnalanswertoonedecimalplace.

Compare 110.3 m2 and 110.2 m2

Roundingerrorshaveaccumulatedtogivea differenceof 0.1 m2

Findtheareaofthesemicircle,roundingtoonedecimalplace. a

Findtheareaofthetriangleintheshape,roundingtoonedecimalplace. b Hence,ﬁndthetotalareausingyouranswerstoparts a and b c Nowrecalculatethetotalareabyretainingmorepreciseanswersfor thecalculationstoparts a and b.Roundyourﬁnalanswertoone decimalplace.

Compareyouranswerstoparts c and d.Howcanyouexplainthe difference?

5 Considertheshapeshown.

Findtheareaofthesemicircle,roundingtoonedecimalplace. a

Findtheareaofthetriangleintheshape,roundingtoonedecimalplace. b Hence,ﬁndthetotalareausingyouranswerstoparts a and b c

d Compareyouranswerstoparts c and d.Howcanyouexplainthe difference?

Nowrecalculatethetotalareabyretainingmorepreciseanswersforthe calculationstoparts a and b.Roundyourﬁnalanswertoonedecimalplace.

6 Considertheshapeshown.

Findtheareaofthequartercircle,roundingtoonedecimalplace. a Findtheareaofthetriangleintheshape,roundingtoonedecimalplace. b Hence,ﬁndthetotalareausingyouranswerstoparts a and b c Nowrecalculatethetotalareabyretainingmorepreciseanswersforthe calculationstoparts a and b.Roundyourﬁnalanswertoonedecimal place.

Compareyouranswerstoparts c and d.Howcanyouexplainthe difference?

HintforQ6:Areaofaquartercircle is 1 4 p r2 where r is 6.82 mhere.

Example15Calculatingerrors

Calculatetheabsolutepercentageerrorwhencalculatingtheareaofacirclewithradius 12 cmif 3.14 isusedasanapproximationof p.Roundyouranswertothreedecimalplaces.

Solution

Estimatedresult = 3.14 × 122 = 452.16

Actualresult =p× 122 = 452.389…

Error = 452.16 - 452.389 =-0.229

Explanation

Theestimatedresultusestheapproximatevalue of p whichis 3.14

Useacalculatorfortheactualvalueof p

Error=estimatedresults - actualvalue

Absoluteerror = 0.229… Removethenegativesign. Absolutepercentageerror = 0.229

Percentageerror = error actualresult × 100 1 Roundtothreedecimalplaces.

Nowyoutry

Calculatetheabsolutepercentageerrorwhencalculatingtheareaofasquareofsidelength √2 cmif 1.4 isusedasanapproximationof √2

7 Findtheabsolutepercentageerrorinthefollowingsituations,tothreedecimalplaceswherenecessary. Using 3.14 asanapproximationof p whenﬁndingtheareaofacirclewithradius 20 cm.

Using 1.7 asanapproximationof √3 whenﬁndingtheareaofarectanglewithwidth √3 cmand length √3 cm.

Acarpenterestimatesthecombinedlengthof 20 piecesoftimberifeachpieceis 1.25 mbutis approximatedtobe 1.2 m.

0.3 isusedtoapproximate 1 3 whencalculating 1 3 of $500 d

Example16Findinglimitsofaccuracy

Givethelimitsofaccuracyforthesemeasurements. 72 cm a 86.6 mm b

Solution

a 72 ± 0.5 × 1 cm

= 72 - 0.5 cmto 72 + 0.5 cm

= 71.5 cmto 72.5 cm

b 86.6 ± 0.5 × 0.1 mm = 86.6 ± 0.05 mm

= 86.6 - 0.05 mmto 86.6 + 0.05 mm

= 86.55 mmto 86.65 mm

Nowyoutry

Explanation

Smallestunitofmeasurementisonewholecm.

Error = 0.5 × 1 cm

Thiserrorissubtractedandaddedtothegiven measurementtoﬁndthelimitsofaccuracy.

Smallestunitofmeasurementis 0.1 mm.

Error = 0.5 × 0.1 mm = 0.05 mm

Thiserrorissubtractedandaddedtothegiven measurementtoﬁndthelimitsofaccuracy.

Givethelimitsofaccuracyforthesemeasurements.

8 Foreachofthefollowing:

Givethesmallestunitofmeasurement(e.g. 0.1 cmisthesmallestunitin 43.4 cm). i Givethelimitsofaccuracy. ii

HintforQ8:Use ± 0.5 × smallestunitofmeasurement. l

9 Whatarethelimitsofaccuracyfortheamount$4500 whenitiswritten:

totwosignificantfigures? a tothreesignificantfigures? b tofoursignificantfigures? c

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HintforQ9:Forsignificantfigures, startcountingfromthefirst non-zerodigit.

Problem-solving and reasoning

10 Writethefollowingasameasurement,giventhatthelowerandupperlimitsofthese measurementsareasfollows. 29.5 mto 30.5 m a

HintforQ10:Findthemiddleof theseintervals.

11 Marthawritesdownthelengthofherfabricas 150 cm.AsMarthadoesnotgiveherlevelofaccuracy, givethelimitsofaccuracyofherfabricifitwasmeasuredcorrecttothenearest: centimetre a 10 centimetres b millimetre. c

12 Alengthofcopperpipeisgivenas 25 cm,tothenearest centimetre.

a Whatarethelimitsofaccuracyforthismeasurement?

b If 10 piecesofcopper,eachwithagivenlengthof 25 cm,arejoinedendtoend,whatistheminimum lengththatitcouldbe?

c Whatisthemaximumlengthforthe 10 piecesofpipe inpart b?

Example17Applyingthelimitsofaccuracy

Janismeasureseachsideofasquareas 6 cm.Find: theupperandlowerlimitsforthesidesofthesquare a theupperandlowerlimitsfortheperimeterofthesquare b theupperandlowerlimitsforthesquare’sarea. c

Solution

a 6 ± 0.5 × 1 cm

= 6 - 0.5 cmto 6 + 0.5 cm

= 5.5 cmto 6.5 cm

b Lowerlimit P = 4 × 5.5

= 22 cm

Upperlimit P = 4 × 6.5

= 26 cm

c Lowerlimit A = 5.52

= 30.25 cm2

Upperlimit A = 6.52

= 42.25 cm2

Nowyoutry

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Explanation

Smallestunitofmeasurementisonewholecm.

Error = 0.5 × 1 cm

Thelowerlimitfortheperimeterusesthelower limitforthemeasurementtaken,whichis 5.5, andtheupperlimitfortheperimeterusesthe upperlimitof 6.5 cm.

Thelowerlimitfortheareais 5.52 ,whereasthe upperlimitwillbe 6.52

Janismeasureseachsideofasquareas 9 cm.Find: theupperandlowerlimitsforthesidesofthesquare a theupperandlowerlimitsfortheperimeterofthesquare b theupperandlowerlimitsforthesquare’sarea. c

13 Thesideofasquareisrecordedas 9.2 cm.

Whatistheminimumlengththatthesideofthissquarecouldbe? a Whatisthemaximumlengththatthesideofthissquarecouldbe?

b Findtheupperandlowerlimitsforthissquare’sperimeter. c Findtheupperandlowerlimitsfortheareaofthissquare.

14 Thesideofasquareisrecordedas 9.20 cm.

Whatistheminimumlengththatthesideofthissquarecouldbe? a Whatisthemaximumlengththatthesideofthissquarecouldbe? b Findtheupperandlowerlimitsforthissquare’sperimeter.

c Findtheupperandlowerlimitsfortheareaofthissquare.

HintforQ13:Use theminimumand maximumlengths forparts c and d

d Howhaschangingthelevelofaccuracyfrom 9.2 cm(seeQuestion 13)to 9.20 cmaffectedthe calculationofthesquare’sperimeterandarea?

15 Codymeasuresthemassofababytobe 6 kg.Jacintasaysthe samebabyis 5.8 kgandLukegiveshisansweras 5.85 kg.

a Explainhowallthreepeoplecouldhavedifferentanswers forthesamemeasurement.

b Writedownthelevelofaccuracybeingusedbyeach person.

c Arealltheiranswerscorrect?Discuss.

16 Tocalculatethepercentageerrorofanymeasurement,theerror(i.e. ± thesmallestunitofmeasurement) iscomparedtothegivenorrecordedmeasurementandthenconvertedtoapercentage.

Forexample: 5.6 cm

Error =± 0.5 × 0.1 =± 0.05

Percentageerror = ± 0.05 5.6 × 100 =± 0.89%(totwosigniﬁcantﬁgures)

Findthepercentageerrorforeachofthefollowing.Roundtotwosigniﬁcantﬁgures.

28 m a 9 km b

8.9 km c 8.90 km d

178 mm e $8.96 f $4.25 g 701 mL h

1 2A Convertthegivenmeasurementstotheunitsshowninbrackets.

2 2B Findtheperimeteroftheseshapes.

3 2B Aconcreteslabisshownbelow.Allanglesare 90°

a Drawanewdiagram,showingallthemeasurementsinmetres.

b Determinethelinealmetresoftimberneededtosurroundit.

4 2C/E Findthecircumference (C) andarea (A) ofthesecircles,totwodecimalplaces.

5 2D/E Findtheareaoftheseshapes.Roundyouranswertoonedecimalplaceinpart d

6 2D Arectangularkitchenflooristobereplacedwithwoodenfloorboards.Ifthefloorboards cost $46 persquaremetre,determinethecosttocoverthekitchenfloorifits dimensionsare 4.4 mby 3 m.

7 2C/E Findthearea (A) andperimeter (P) ofthecompositeshapeshown.Roundeachanswerto onedecimalplace.

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8 2F Thesidelengthofasquareismeasuredtobe 9 cm.

Givethelimitsofaccuracyforthismeasurement. a Calculatetheareaofthesquareusingthe 9 cmmeasurement. b Amoreprecisesidelengthofthesquareis 8.5 cm.Calculatethedifferenceintheareaof thesquarecalculationusingasidelengthof 8.5 cmcomparedto 9 cm.

c Usingtheinformationinpart c,calculatetheabsolutepercentageerror,tothree decimalplaces.

2G 2G Surfaceareaofprisms

Learningintentions

• Toknowthatthesurfaceareaofasolidcanberepresentedusinganet

• Tobeabletocalculatethesurfaceareaofaprism

Keyvocabulary: surfacearea,prism,net,cross-section

Thesurfaceareaofathree-dimensional objectcanbefoundbyﬁndingthe sumoftheareasofeachoftheshapes thatmakeupthesurfaceoftheobject.

Theminimum amountofwrapping paperrequiredto completelycovera giftisequaltothe surfacearea.

Lessonstarter:Whichnet?

Thesolidbelowisatriangularprismwitharight-angledtriangleasitscross-section.

• Howmanydifferenttypesofshapesmakeupitsoutsidesurface?

• Whatisapossiblenetforthesolid?Istheremorethanone?

• Howwouldyouﬁndthesurfacearea?

Keyideas

A prism isasolidwithaconstant cross-section shape.

Tocalculatethe surfacearea ofasolidorprism:

• Drawa net (i.e.atwo-dimensionaldrawingthatincludesallthesurfaces).

• Determinetheareaofeachshapeinsidethenet.

• Addtheareasofeachshapetogether. Solid Net

Exercise2G

Und er stand ing

1 Atwo-dimensionaldrawingofallthefacesofasolidiscalleda

2 Forarectangularprism,answerthefollowing.

a Howmanyfacesdoestheprismhave?

b Howmany different rectanglesformthesurfaceoftheprism?

3 Forthistriangularprism,answerthefollowing.

a Whatistheareaofthelargestsurfacerectangle?

b Whatistheareaofthesmallestsurfacerectangle?

c Whatisthecombinedareaofthetwotriangles?

d Whatisthetotalsurfacearea?

Findthesurfaceareaofthisrectangularprismbyﬁrstdrawingitsnet.

Drawthenetofthesolid, labellingthelengthsandshapes ofequalareas.

= 2 × areaofA + 2 × areaofB + 2 × areaofC = 2 × (8 × 3) + 2 × (5 × 3) + 2 × (8 × 5)

Substitutethecorrectlengths. = 158 cm2 Simplifyandincludeunits. Describeeacharea.

Nowyoutry

Findthesurfaceareaofthisrectangularprismbyﬁrstdrawingitsnet.

4 Findthesurfaceareaoftheserectangularprismsbyﬁrstdrawingtheirnets.

Example19Findingthesurfaceareaofatriangularprism

Findthesurfaceareaofthetriangularprismshown.

Solution

Explanation

Drawanetoftheobjectwithallthe measurementsandlabelthesectionsto becalculated.

Surfacearea

= 2 × area A + area B + area C + area D

= 2 × ( 1 2 × 3 × 4) + (3 × 10) + (4 × 10) + (5 × 10)

= 12 + 30 + 40 + 50

= 132 m2

Nowyoutry

Therearetwotriangleswiththesame areaandthreedifferentrectangles.

Substitutethecorrectlengths.

Calculatetheareaofeachshape.

Addtheareastogether.

Findthesurfaceareaofthetriangularprismshown.

Findthesurfaceareaofthefollowingprisms.

6 Findthesurfaceareaoftheseobjectsbyﬁrstdrawinganet.

HintforQ5:Triangularprismshave threerectanglesandtwoidentical triangles.

7 Acubewithsidelengthsof 8 cmistobepaintedalloverwithbrightredpaint.Whatisthetotal surfaceareathatistobepainted?

8 Whatistheminimumamountofpaperrequiredtowrapa boxwithdimensions 25 cmwide, 32 cmlongand 20 cmhigh?

Anopen-toppedboxistobecoveredinsideandoutwithaspecial material.Iftheboxis 40 cmlong, 20 cmwideand 8 cmhigh,ﬁnd theminimumamountofmaterialrequiredtocoverthebox.

10 Hassanwantstopainthisbedroom.Theceilingandwallsaretobe thesamecolour.Iftheroommeasures 3.3 m × 4 mandtheceilingis 2.6 mhigh,ﬁndtheamountofpaintneededif: a eachlitrecovers 10 squaremetres b eachlitrecovers 5 squaremetres.

HintforQ9:Countbothinside andoutsidebutdonotinclude thetop.

11 AskirampintheshapeofatriangularprismneedstobepaintedbeforetheMoombaClassic waterskiingcompetitioninMelbourneisheld.Thebaseandsidesoftheramprequireafully waterproofpaint,whichcovers 2.5 squaremetresperlitre.Thetopneedsspecialsmoothpaint,which coversonly 0.7 squaremetresperlitre.

a Determinetheamountofeachtypeofpaintrequired.Roundyouranswerstotwodecimalplaces wherenecessary.

b Ifthewaterproofpaintis $7 perlitreandthespecialsmoothpaintis $20 perlitre,calculatethe totalcostofpaintingtheramp,tothenearestcent.(Usetheexactanswersfrompart a tohelp.)

Findthetotalsurfaceareaoftheserightsquare-basedpyramids.

13 Ihave 6 litresofpaintandon thetinitsaysthatthecoverage is 5.5 m2 perlitre.Iwishto paintthefouroutsidewalls ofashedandtheroof,which hasfouridenticaltriangular sections.Themeasurements areshowninthediagram. WillIhaveenoughpaintto completethejob?

HintforQ12:Squarepyramidshave onesquareandfouridentical triangles.

2H 2H Surfaceareaofacylinder

Learningintentions

• Tounderstandhowthenetofacylindercanbedrawntoshowthesurfacearea

• Toknowtheformulaforthesurfaceareaofacylinder

• Tobeabletocalculatethesurfaceareaofacylinder

Keyvocabulary: cylinder,area,prism,circumference,net,cross-section

Likeaprism,acylinderhasauniform cross-sectionwithidenticalcirclesas itstwoends.Thecurvedsurfaceofa cylindercanberolledouttoforma rectanglethathasalengthequaltothe circumferenceofthecircle.

Acanisagoodexampleofacylinder. Weneedtoknowtheareaoftheends andthecurvedsurfaceareainorderto cutsectionsfromasheetofaluminium tomanufacturethecan.

Tinnedfoodmanufacturersusethesurfaceareaofacylindertoworkout howmuchmaterialisneededtomakeeachcan.

Lessonstarter:Why 2p rh?

Wecanseefromthenetofacylinder(seethediagraminthe Keyideas)thatthetotalareaofthetwo circularendsis 2 ×p r2 or 2p r2 .Forthecurvedpart,though,considerthefollowing.

• Whycanitbedrawnasarectangle?Canyouexplainthis,usingapieceofpaper?

• Whyarethedimensionsofthisrectangle h and 2p r?

• Wheredoestheformula A = 2p r2 + 2p rh comefrom?

Keyideas

A cylinder isasolidwithacircular cross-section.

• The net containstwoequalcirclesandarectangle.The rectanglehasonesidelengthequaltothecircumference ofthecircle.

• Area = 2 circles + 1 rectangle

= 2p r 2 + 2p rh

• Anotherwayofwriting 2p r2 + 2p rh is 2p r(r + h)

Exercise2H

Und er stand ing

1 Writethemissingwordorexpression.

a Theshapeofthecross-sectionofacylinderisa

b Thesurfaceareaofacylinderis A = 2p r2 +

2 Acylinderanditsnetareshownhere.

a Whatisthevalueof: r? ii h? i

b Findthevalueof 2p r,totwodecimalplaces.

c Use A = 2p r2 + 2p rh toﬁndthesurfacearea,totwo decimalplaces.

Fluency

Example20Findingthesurfaceareaofacylinder

Byﬁrstdrawinganet,ﬁndthesurfaceareaofthiscylinder,totwo decimalplaces.

Solution

Explanation

Drawthenetandlabeltheappropriatelengths.

A = 2 circles + 1 rectangle

Writewhatyouneedtocalculate.

= 2p r 2 + 2p rh Writetheformula.

= 2p (1.7)2 + 2p (1.7)(5.3)

= 74.77 m2 (to 2 d.p.)

Nowyoutry

Substitutethecorrectvalues: r = 1.7 and h = 5.3

Roundyouranswertotwodecimalplaces.

Byﬁrstdrawinganet,ﬁndthesurfaceareaofthiscylinder,totwo decimalplaces.

Byﬁrstdrawinganet,ﬁndthesurfaceareaofthesecylinders,totwodecimalplaces.

HintforQ3:Rememberthat radius = diameter ÷ 2

4 Usetheformula A = 2p r2 + 2p rh toﬁndthesurfaceareaofthesecylinders,to onedecimalplace.

Findtheareaofonlythecurvedsurfaceofthesecylinders, toonedecimalplace.

HintforQ5:Findonlytherectangular partofthenet,souse A = 2p rh Becarefulwiththeunitsinpart b!

Problem-solving and reasoning 6,7

6 Findtheoutsidesurfaceareaofapipeofradius 85 cmandlength 4.5 m,toonedecimalplace. Giveyouranswerinm2

7 Thebaseandsidesofacircularcaketinaretobelinedontheinsidewithbakingpaper.Thetinhas abasediameterof 20 cmandis 5 cmhigh.Whatistheminimumamountofbakingpaperrequired, toonedecimalplace?

Theinsideandoutsideofanopen-toppedcylindricalconcretetank istobecoatedwithaspecialwaterprooﬁngpaint.Thetankhas diameter 4 mandheight 2 m.Findthetotalareatobecoated withthepaint.Roundyouranswertoonedecimalplace.

Findthesurfaceareaofthesecylindricalportions,toonedecimalplace.

HintforQ8:Includethebase butnotthetop.

HintforQ9:Carefullyconsiderthe fractionofacirclemadeupbythe ends,andthefractionofafull cylindermadeupbythecurvedpart.

10 Asteamrollerhasalarge,heavycylindricalbarrelthatis 4 mwideandhasadiameterof 2 m.

a Findtheareaofthecurvedsurfaceofthebarrel,totwodecimalplaces.

b After 10 completeturnsofthebarrel,howmuchgroundwouldbecovered,totwodecimalplaces?

c Findthecircumferenceofoneendofthebarrel,totwodecimalplaces.

d Howmanytimeswouldthebarrelturnafter 1 kmofdistance,totwodecimalplaces?

e Whatareaofgroundwouldbecoveredifthesteamrollertravels 1 km?

2I 2I Volumeofsolids

Learningintentions

• Tounderstandhowthevolumeofsolidsrelatestoitsconstantcross-sectionandheight

• Toknowthecommonunitsforcapacity

• Toknowtheformulaforthevolumeofasolidwithauniformcross-section

• Tobeabletocalculatethevolumeofasolidwithauniformcross-section

Keyvocabulary: solid,volume,cross-section,uniform,prism,cylinder,perpendicular,capacity

Thevolumeofasolidistheamountofspaceitoccupies withinitsoutsidesurface.Itismeasuredincubicunits.

Forsolidswithauniformcross-section,theareaofthe cross-sectionmultipliedbytheperpendicularheightgives thevolume.Considertherectangularprismbelow.

3 6 4

Numberofcubicunits(base) = 4 × 6 = 24

Area(base) = 4 × 6 = 24 units2

Volume = area(base) × 3 = 24 × 3 = 72 units3

Knowinghowtocalculatethevolumeofa containerortoolboxisusefulforunderstanding howmuchitcanstore.

Lessonstarter:Volumeofatriangularprism

Thisprismhasatriangularcross-section.

• Whatistheareaofthecross-section?

• Whatisthe‘height’oftheprism?

• Howcan V = A × h beappliedtothisprism, where A istheareaofthecross-section?

Keyideas

Volume istheamountofthree-dimensionalspacewithinanobject.

Thevolumeofa solid witha uniformcross-section isgivenby V = A × h,where:

• A istheareaofthecross-section.

• h isthe perpendicular (at 90°)height.

Rectangularprism Cylinder l w h V = lwh r h

Capacity isthevolumeofagivenobjectmeasuredinlitresormillilitres.

Unitsforcapacityinclude:

Exercise2I

Und er stand ing

1 Matchthesolid(a–c)withthevolumeformula(A–C). cylinder a V = lwh A rectangularprism b V = 1 2 bh × length B triangularprism c V =p r2 h C

2 Writethemissingnumber.

a Thereare mLin 1 L.

b Thereare cm3 in 1 L.

3 Theareaofthecross-sectionofthissolidisgiven.Findthe solid’svolume,using V = A × h

Example21Findingthevolumeofarectangularprism

Findthevolumeofthisrectangularprism.

Solution

Explanation

V = A × h Writethegeneralformula.

= l × w × h l = 6, w = 5 and h = 4.

= 6 × 5 × 4 Simplifyandincludeunits.

= 120 m3

Nowyoutry

Findthevolumeofthisrectangularprism.

4 Findthevolumeoftheserectangularprisms.

HintforQ4:Use V = lwh

Example22Findingthevolumeofacylinder

Findthevolumeofthiscylinder,totwodecimalplaces.

Solution

Explanation

V = A × h Writethegeneralformula.

=p r 2 × h

=p (2)2 × 6

Thecross-sectionisacircle.

Substitute r = 2 and h = 6 = 75.40 cm3 (to 2 d.p.) Simplifyandwriteyouranswerasrequired,withunits.

Nowyoutry

Findthevolumeofthiscylinder,totwodecimalplaces. 9 m 3 m

5 Findthevolumeofthesecylinders,totwodecimalplaces.

HintforQ5:Foracylinder:

6 Atrianglewithbase 8 cmandheight 5 cmformsthebaseofaprism,as shown.Iftheprismstands 4.5 cmhigh,ﬁnd: a theareaofthetriangularbase b thevolumeoftheprism. 4.5 cm

7 Findthevolumeofthesetriangularprisms.

HintforQ7:Use V = A × h, where A isthearea ofatriangle.

Problem-solving and reasoning

8 Acylindricaldrumstandsononeendwithadiameterof 25 cmandwaterisﬁlledtoaheightof 12 cm. Findthevolumeofwaterinthedrum,incm3 ,totwodecimalplaces.

Example23Workingwithcapacity

Findthenumberoflitresofwaterthatthiscontainercanhold.

Solution

Explanation

V = 30 × 40 × 20 Firstworkoutthevolumeincm3 = 24000 cm3 Thendivideby 1000 toconverttolitres,since 1 cm3 = 1 mL = 24 L andthereare 1000 mLin 1 litre.

Nowyoutry

Findthenumberoflitresofwaterthatthiscontainercanhold.

Findthenumberoflitresofwaterthatthesecontainerscanhold.

10 Findthevolumeofthesesolids,roundingyouranswerstotwodecimalplaces wherenecessary.

HintforQ10:Find theareaofthe cross-sectionﬁrst.

11 100 cm3 ofwateristobepouredintothiscontainer.

a Findtheareaofthebaseofthecontainer.

b Findthedepthofwaterinthecontainer.

12 Inascientiﬁcexperiment,solidcylindersoficeareremovedfroma solidblockcarvedoutofaglacier.Theicecylindershavediameter 7 cm andlength 10 cm.Thedimensionsofthesolidblockareshown inthediagram.

a Findthevolumeoficeintheoriginaliceblock.

b Findthevolumeoficeinoneicecylinder,totwodecimalplaces.

c Findthenumberoficecylindersthatcanberemovedfromtheice block,usingtheconﬁgurationshown.

d Findthevolumeoficeremainingaftertheicecylindersareremoved fromtheblock,totwodecimalplaces.

13 Thevolumeofapyramidorconeisexactlyone-thirdthevolumeoftheprismwiththesamebase areaandheight,i.e. V = 1 3 × A × h

Findthevolumeofthesepyramidsandcones.Roundyouranswerstoonedecimalplacewhere necessary.

2J 2J Furtherproblemsinvolvingprisms andcylinders

Learningintentions

• Tobeabletocalculatethesurfaceareaandvolumeofacompositesolid

• Tobeabletosolveproblemsinvolvingcompositesolids

Keyvocabulary: compositesolid,prism,cylinder,net,Pythagoras’theorem,capacity

Recallthatwhenworkingwithcompositeshapeswecan ﬁndperimetersandareasbyconsideringthecombination ofthemorebasicshapesthat,together,formthecomposite shape.Similarly,wecanworkwithcompositesolidsby lookingatthecombinationofmorebasicsolids,likeprisms andcylinders.Thisleadstoﬁndingthesurfacearea,volume andcapacityofsolids.

Thewell-knownEuropeanartistsChristoandJeanne-Claude hadtheirhandsfullwithcompositeobjectswhenthey wrappedtheReichstag(Parliamentbuilding),inBerlin,in 1995.Theyusedmorethan 100000 squaremetresoffabric and 15 kmofrope.

Lessonstarter:Whichsolids?

Lookatthesecompositesolids.

UNCORRECTEDSAMPLEPAGES

• Whatarethebasicsolidsthatmakeupeachcompositesolid?

• Explainamethodforﬁndingthevolumeofeachsolid.

• Explainamethodforﬁndingthesurfaceareaofeachsolid.

• Isthereenoughinformationprovidedineachdiagramtoﬁndthevolumeandsurfacearea?Discuss.

Keyideas

Compositesolids aremadeupofmorethanonebasicsolid.

Volumesandsurfaceareasofcompositesolidscanbefoundbyconsideringthevolumesand surfaceareas(orpartsthere-of)ofthebasicsolidscontainedwithin.

TheGermanparliament,theReichstag,was wrappedinfabricin1995tocreateatemporary workofart.

Pythagoras’theorem maybeusedtohelpﬁndparticularlengths,providedthataright-angled triangleisgiven.

Recallthesecommonunitconversionsforcapacity.

Exercise2J

Und er stand ing

1 Namethetwobasicsolidsthatmakeupeachofthesecompositeshapes.

2 UsePythagoras’theoremtoﬁndthelengthofthehypotenuseintheseright-angledtriangles.

Example24UsingPythagoras’theoremtohelpﬁndthesurfaceareaofa triangularprism

Findthesurfaceareaofthistriangularprism.

Solution

c2 = a2 + b2

= 62 + 82

= 100

c = √100 = 10

Surfacearea = 2 × 1 2 × 8 × 6 + (8 × 12)

+ (6 × 12) + (10 × 12)

= 48 + 96 + 72 + 120 = 336 m2

Nowyoutry

Findthesurfaceareaofthistriangularprism.

Explanation

UsePythagoras’theoremtoﬁndthe lengthoftheslantingedge.

Thesurfaceareaismadeupoftwo congruenttriangularendsandthree differentrectangles.

3 UsePythagoras’theoremtohelpﬁndthesurfaceareaofthesetriangularprisms.

Example25Findingthesurfaceareaandvolumeofacompositesolid

Findthesurfaceareaandvolumeofthiscompositesolid,totwodecimalplaces.

Solution

Surfacearea = 6 × 22 + 2p (1)(3) = 24 + 6p = 42.85 cm2 (to 2 d p )

Explanation

Thesurfaceareaismadeupof 5 squarefacesplusone more,whichismadeupoftheremainingpartoftheright sidefaceandtheendofthecylinder.

Thecurvedsurfaceofthecylinder (2p rh) isalsoincluded. Theradiusishalfthediameter,i.e. r = 1

Continuedonnextpage

Volume = 23 +p (1)2 (3) = 8 + 3p = 17.42 cm3 (to 2 d.p.)

Nowyoutry

Thevolumeconsistsofthesumofacube(l 3 ) andacylinder(p r2 h).

Findthesurfaceareaandvolumeofthissolid,totwodecimalplaces.

4 Forthesecompositesolids,ﬁnd: thesurfacearea i thevolume. ii Giveyouranswertotwodecimalplaces.

HintforQ4:Includeonlyexposed surfacesinthesurfacearea.

5 Findthecapacityofthesecompositesolids,inlitres.Roundyouranswertotwodecimalplaces wherenecessary.

Problem-solving and reasoning

6 Hereisthedesignofaglasstennistrophy.Thebaseandthe cylindricalpartarebothmadeofglass.

Findthesurfaceareaofthetrophy,tothe nearestsquarecentimetre. a

Findthevolumeofglass,tothenearestcubiccentimetre, requiredtomakethetrophy.

6–8,106,8,9,11

7 Whensolidsarepainted,theoutersurfaceareaneedstobeconsideredtohelpﬁndtheamountof paintrequiredforthejob.Assumethat 1 Lofpaintcovers 10 m2 Completeforeachoftheseobjects.

i Findthesurfaceareainsquaremetres,roundingyouranswertotwodecimalplaces wherenecessary.

ii Findtheamountofpaintthatmustbepurchased,assumingthatyoucanbuyonlyawhole numberoflitres.

8 Whensolidsarehollow,theinsidesurfaceareasareexposedtotheair.Findthesurfacearea(i.e.inner andoutercombined)ofapipeofdiameter 0.3 mandlength 3 m.Assumethattheinnerandouter diametersarethesame.(Roundyouranswertoonedecimalplace.)

9 Thisnutisasquare-basedprismwithacylindricalholeremovedfrom thecentre.Theholehasadiameterof 1 cm.Thenutiscoatedwith anti-rustpaint.Whatareaispainted,includingtheinnercylindrical surface?(Roundyouranswertoonedecimalplace.)

10 Thesesolidshaveapproximatelythesamevolume.

Whichhasthelargersurfacearea?Dosomecalculationstoﬁndout.

11 Acompanywishestodesignacontainerforpackagingandsellinglollies.Thetwodesignsare shownhere.

a Completesomecalculationstoshowthatthetwocontainershaveapproximatelythesamevolume.

b Whichdesignhastheleastsurfacearea?Justifyyouranswer.

12 Imaginethatacompanyasksyoutomakeatrayoutofasquarepieceofcard,measuring 10 cmby 10 cm,bycuttingoutfourcornersquaresandfoldingthemtoformatray,asshown.

a Whatwillbethevolumeofthetrayifthesidelengthofthesquarecut-outsis:

The 2 cmcut-outsareshownbelow.

b Whichsquarecut-outfrompart a givesthelargesttrayvolume?

c Canyouﬁndanothersizedcut-outthatgivesalargervolumethananyofthoseinpart a?

d Whatsizedcut-outgivesthemaximumvolume?

Bricklayer

Abricklayerhasaphysicallychallengingjob thatrequiresstaminaandstrengthandalso goodcommunicationskills,astheyoftenwork aspartofateam.

Bricklayersmusthaveasolidunderstandingof howtheconstructionprocessworksandthe abilitytoreadplansandblueprints.

Mathematicalskillsareessentialinthistrade. Bricklayersmustunderstandratiosformixing mortarandcement.Goodmeasurementskills arealsoimportant,asbricklayersmustbeable toworkoutthenumberofbricksrequiredfor ajob,convertbetweendifferentunitsandtake accuratemeasurementsattheworksite,using themostappropriatetools.Anunderstanding ofgeometryandtrigonometryisalsorequired.

Completethesequestionsthatabricklayermayfaceintheirday-to-dayjob.

1 Astandardhousebrickhasdimensions l × w × h = 230 mm × 76 mm × 110 mm andthestandardthicknessofmortarwhenlayingbricksis 10 mm. Thebricksarelaidsothat l × h istheouterface.

a Whatisthelengthandheightofeachbrickincentimetres andmetres?

b Determinetheareaincm2 andm2 oftheouterfaceofonebrick.

c Determinethevolumeincubiccentimetresofeachbrick.

d Calculatethelength,inmetres(totwodecimalplaces), whenlayingthefollowingnumberofbricksinaline withmortarbetweeneachjoin. 10 bricks i 100 bricks ii

HintforQ1d:For 10 bricks,there wouldbe 9 mortarjoins.

e Calculatetheheight,inmillimetres,ofawallof 25 rowsofstandardhousebricks. Remembertoconsiderthethicknessofthemortar.

f Estimatehowmanystandardhousebricksareneededtobuildawall 4 mby 1.5 m,bydividing theareaofthewallbytheareaofabrick’sface.

SAMPLEPAGES

HintforQ1b,c:

2 Ready-mixmortarcomesin 20 kgbagsthatcost $7.95 perbag.Onebagofmortarisusedtolay 20 standardhousebricks.

Usingstandardhousebricks(seedimensiondetailsinQuestion 1),abrickwallistobebuiltthat hasaﬁnishedlengthof 8630 mm,aheightof 2750 mmandisonebrickdeep.

a Calculatetheexactnumberofstandardhousebricksneededtobuildthiswall.Rememberto considerthethicknessofthemortar.

b HowmanyReady-mixmortarbagsmustbepurchasedforthiswall?

c Ifeachhousebrickcosts 60 cents,ﬁndthetotalcost,tothenearestdollar,ofthematerials neededforthiswall.

3 Atypeoflargebrickischosenforanoutsideretainingwall.These bricksaresoldonlyinwholepacksandeachpackcovers 12.5 square metreswhenlaid.Howmanywholepacksofthesebricksmust beboughttobuildawallwithdimensions:

a 6 mby 1.5 m?

b 9 mby 2 m?

HintforQ3:Youcan’tbuyhalf apack,sorounduptothe nextwholenumber.

4 Fastwallhousebricksarelargerandlighterthanstandardbricksandcanbeusedforsingle-storey constructions.Theyaresoldinpalletsof 1000 for $1258.21,includingdeliveryandGST.Each Fastwallbrickhasdimensions l × w × h = 305 mm × 90 mm × 162 mmandthestandardthickness ofmortaris 10 mm.

a Ifapallethas 125 bricksperlayer,howmanylayersdoeseachpallethave?

b Ifthewoodenbaseofthepallethasaheightof 30 cm,whatisthetotalheightofthepallet, incm,whenloadedwithbricks?

c Findtheexactnumberofbricksneededtobuildawallthatis 20.15 mlongand 6.87 mhigh. Remembertoconsiderthethicknessofthemortar.

d Determinethecostofthebricks,tothenearestdollar,requiredtobeboughtforbuildingthe wallinpart c

UNCORRECTEDSAMPLEPAGES

Usingdigitaltools

5 Copythefollowingtableintoaspreadsheet.Thenenter formulasintotheshadedcellsand,hence,determinethe missingvalues.

HintforQ5:ForE5theformula wouldbe = G5 × A5 + (G5 - 1) × D5.ForG6theformulawould be

)

6 Copythefollowingtableintoyourspreadsheetunderneaththe tablefromQuestion 5.Entertheformulasintotheshadedcells and,hence,determinethemissingvalues.Assumethereare 1000 bricksperpalletandthatonebagofmortarisusedper 20 brickslaid.

HintforQ6:Copythetotal numberofbricksused fromtheﬁrsttable.

IncellB13 enter = ROUNDUP(B12, 0),whichwillroundthenumberfromcellB12 uptothenearest wholenumber;e.g. 1.3 willberoundedupto 2

7 Useyourspreadsheettablestoﬁndthetotalcostofmaterialsforthefollowingbrickwallsmade fromFastwallbricks.(SeeQuestion 4 forFastwallbrickdimensions.)Thespreadsheetformulaswill notneedtobechanged.

a Awallof 44 bricksperlayer(row)and 30 layers(rows)ifpalletscost $1258.36,includingGST, andmortaris $7.55 perbag.

b Awall 20.15 mlongby 8.59 mhighifpalletscost $1364.32,includingGST,andmortaris $9.25 perbag.

Buildingplaygroundequipment

Agroupofhighschoolstudentshaveraisedsomemoneyforavolunteercommunityserviceproject. Thestudentsdecidetoaskthecounciliftheycanimprovethechildren’splaygroundequipmentinthe localpark.

Presentareportforthefollowingtasksandensurethatyoushowclearmathematicalworkings, explanationsanddiagramswhereappropriate.

1Preliminarytask

Calculatethefollowingareas,roundingtheanswerstoonedecimalplace.

a Calculatetheareaofacirclewithdiameter 3.5 m.

b Determinetheareaofthefollowingtriangles.Inpart ii youwillneedtoapplyPythagoras’ theoremtocalculatethetriangle’sheight.

c Determinethetotalsurfaceareaincm2 andm2 ofacylindricalbalancingbeamwithradius 12 cmandlength 3 m.

2Modellingtask

Prepareaproposalfortwoimprovementstothepark’splaygroundequipment: constructingasetoflowconcretecylindersthatchildrencanuseassteps

•

• therepaintingofanoldplaygroundroundabout.

Theproblemistodeterminethevolumeofconcreteandpaintneededandtoﬁndoutifthecostof theprojectiswithinabudgetof $200

a Withtheaidofdiagrams,writedownalltherelevantmathematicalformulasthatareneeded tocalculate:

thevolumeandsurfaceareaofacylinder i theareaofanisoscelestriangle. ii

Concretecylinders

b Nineconcretecylindersofradius 20 cmare tobeconstructed,havingabove-ground heightsof:

15 cm, 30 cm, 45 cm, 60 cm, 75 cm, 60 cm, 45 cm, 30 cm, 15 cm.

Eachcylinderalsoextends 30 cmbelow groundforstability.

CopythefollowingtableintoanExcelspreadsheetandenterformulasintotheshadedcells. FormatcellstoNumber/onedecimalplace.Usepi()forentering p

ExtendtheExceltabletoincludeallninecylindersandthetotalsurfaceareaandvolumeusingunits ofcmandm.

Theroundabout

Theoldroundabouthasanoctagonshapedtopwith 8 isoscelestriangles;eachtrianglehasa 1 m baseandtwoequalsidesof 130 cm.Theroundabout’srectangularverticalsidesareeach 50 cmhigh and 1 minlength.

Determinetheareaofoneisoscelestriangleandonerectangleincm2 .Youwillneedtoapply Pythagoras’theoremtocalculatethetriangle’sheight.

Determinetheroundabout’ssurfacearea,includingthetopandsides,inm2 ,toonedecimalplace.

Thecouncilhasofferedtopourtheconcretecylinderssothestudentvolunteerswillonlybe responsibleforthepaintingjobs,includingthecylindersandtheroundabout.Onlythepaintingwill beincludedinthebudget.

Interpret and verify

Ifonlyonecolourisusedand 3 coatsareapplied,ﬁndthetotalareatobepainted.

Ifnon-slippaintcovers 8 m2 /litre,determinethevolumeofpaintrequired.

Ifnon-slippaintcomesin 2 Lcansat $75 eachand 4 Lcansat $110 each,calculatetheminimum costofpaint,assumingthreecoatsarerequired,anddetermineiftheprojectiswithinbudget.

g Communicate

h Summariseyourﬁndings,stating: thevolumeofconcreteneededforthecylinders i theoverallareatobepainted ii thedetailsofwhatcansofpainttobuyandthecost. iii

3Extensionquestion

a Inadifferentplaygroundthestepsarehalfcylinderswithradius 30 cmandallareofheight 50 cm.Theystandupwithoneverticalﬂatside,acurvedsideandasemicirculartop.The octagonalroundaboutismadeupof 8 isoscelestriangles,eachwithbaselength 1.4 mandtwo equalsidesof 1.83 m.Theroundabout’sverticalsidesarestill 50 cmhigh.Findthesurfaceareas andvolumesoftheplaygroundobjectsandcalculatethecostofpaintforthisplayground,using thesamecostsasinpart g.

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Maximisingandminimisingwithsolids

Keydigitaltool:Spreadsheets

Whenworkingwithsolidslikeprismsandcylinders, youmightbeinterestedineitherofthefollowing:

• minimisingthesurfaceareaforaﬁxedvolume

• maximisingthevolumeforaﬁxedsurfacearea.

Youwillrecalltheserulesforthesurface areaandvolumeofcylinders.

1Gettingstarted

Acompanyismakingdrumstoholdchemicalsandrequireseachcylindricaldrumtobe 50 litresin volumewhichis 50000 cm3

a Usethevolumeformulaforacylindertoshowthat h = 50000 p r2 .

b Findtheheightofthecylinder,totwodecimalplaces,iftheradiusis: 20 cm i 10 cm ii

c Findthesurfaceareaofthecylinder,totwodecimalplaces,iftheradiusis: 20 cm i 10 cm ii

d Findaradiusthatgivesasmallersurfaceareacomparedtotheexamplesinpart c

2Usingdigitaltools

a Constructaspreadsheettoﬁndtheheightandsurfaceareaforacylinderwithaﬁxedvolumeof 50000 cm3 .Usearadiusof 1 cmtostartandincreaseby 1 cmeachtimeasshown.

b Filldownfromcells A5, B4 and C4 toﬁndtheheightsandsurfaceareasforcylindricaldrumsof volume 50000 cm3 .Locatetheintegerradiusvaluewhichprovidestheminimumsurfacearea.

c Doyouthinkthattheintegervalueoftheradiusgivesthetrueminimumvalueofthesurface area?Givereasons.

3Applyinganalgorithm

Wewillnowsystematicallyaltertheincrementmadetotheradiusvalueinourspreadsheettoﬁnda moreaccuratesolution.

a Applythisalgorithmtoyourspreadsheetandcontinueuntilyouaresatisﬁedthatyouhavefound theradiusvaluethatminimisesthesurfaceareatotwodecimalplaces.

• Step 1:Altertheformulaincell A5 sothattheincrementissmaller.e.g. 0.1 ratherthan 1

• Step 2:Filldownuntilyouhavelocatedtheradiusvaluethatminimisesthesurfacearea.

• Step 3:Adjustcell A4 toahighervaluesoyoudon’tneedtoscrollthroughsomanycells.

• Step 4:RepeatfromStep 1 butusesmallerandsmallerincrements(0.01 and 0.001)untilyou havefoundtheradiusvaluewhichminimisesthesurfaceareatotwodecimalplaces.

b Writedownthevaluefor r, h and A totwodecimalplaceswhichgivestheminimumsurfacearea ofacylindricaldrum.

c Nowaltertheﬁxedvolumeofthecylinderandrepeattheabovealgorithm.

d Whatdoyounoticeabouttherelationshipbetween r and h atthepointwherethereisaminimum surfacearea?Experimentwithdifferentvolumestoconﬁrmyourideas.

Puzzles and games

1 ‘Iamthesameshapeallthewaythrough.WhatamI?’Findtheareaofeachshape. Matchtheletterstotheanswersbelowtosolvetheriddle.

2 Onelitreofwaterispouredintoacontainerintheshapeofarectangularprism.The dimensionsoftheprismare 8 cmby 12 cmby 11 cm.Willthewateroverﬂow?

3 Acircularpieceofpastryisremovedfromasquaresheetwithsidelength 30 cm. Whatpercentageofpastryremains?

4 Howmanydifferentnetsarethereforacube?Donotcountreﬂectionsorrotationsof thesamenet.Hereisoneexample.

5 Givetheradiusofacirclewhosevalueforthecircumferenceisequaltothevaluefor thearea.

6 Findtheareaofthisspecialshape.

7 Acube’ssurfaceareais 54 cm2 .Whatisitsvolume?

Perimeter

The distance around the outside of a shape.

Surface area

Draw a net and add the surface areas.

Triangular prism

= 2 ×× 4 × 3

Cylinder 2 endscurved part 4 m 3 m

The volume and surface area of composite solids (made up of more than one solid) can be found by combining the volume and surface area (or parts of) the individual solids.

Circumference

The distance around the outside of a circle.

Area of basic shapes

Square: A = l 2

Rectangle: A = lw

Triangle: A = bh1 2

Measurement

solids associated with the measuring instruments and how they are used.

Accuracy depends on any error

Limits of accuracy are usually

Parallelogram: A = bh

Trapezium: A = (a + b)h

Rhombus: A = xy1 2 1 2

For any prism, V = Ah where A is the area of the cross-section. ± 0.5× the smallest unit. Area of a circle

It is important to use exact values or a large number of decimal places to avoid accumulating errors in calculations.

Percentage error = error100 1 actual result

Chapterchecklist

AversionofthischecklistthatyoucanprintoutandcompletecanbedownloadedfromyourInteractiveTextbook.

2A 1

2A 2

Icanconvertbetweenmetricunitsoflength. e.g.Convertthesemeasurementstotheunitsshowninthebrackets. a 2.3 m(cm) b 270000 cm(km)

Icanconvertbetweenmetricunitsofarea. e.g.Convertthesemeasurementstotheunitsshowninthebrackets. a 32000 m2 (km2 ) b 7.12 cm2 (mm2 )

2A 3 Icanconvertbetweenmetricunitsofvolume. e.g.Convertthesemeasurementstotheunitsshowninthebrackets. a 3.7 cm3 (mm3 ) b 5900000 cm3 (m3 )

2B 4

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Icanﬁndtheperimeterofbasicshapes. e.g.Findtheperimeterofthisshape.

2B 5 Icanﬁndamissingsidelengthgiventheperimeter. e.g.Findthevalueof x forthisshapewiththegivenperimeter.

2C 6

2C 7

2D 8

Icanﬁndthecircumferenceofacircle.

e.g.Findthecircumferenceofacirclewithadiameterof 5 m,totwodecimalplaces.

Icanﬁndtheperimeterofsimplecomposite shapes.

e.g.Findtheperimeterofthiscompositeshape,totwo decimalplaces.

Icanﬁndtheareaofsquares,rectanglesand triangles.

e.g.Findtheareaofthistriangle.

2D 9 Icanﬁndtheareaofrhombuses,parallelogramsand trapeziums.

e.g.Findtheareaofthistrapezium.

2E 10 Icanﬁndtheareaofacircle.

e.g.Findtheareaofthiscircle,totwodecimalplaces.

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2E 11 Icanﬁndtheareaofasector.

e.g.Findtheareaofthissector,totwodecimalplaces.

2E 12 Icanﬁndtheareaofsimplecomposite shapesinvolvingsectors.

e.g.Findtheareaofthiscompositeshape,totwo decimalplaces.

2F 13 Icanunderstandandavoidaccumulatingerrors. e.g.Findthetotalareaofthisshapebyﬁndingtheareaofthesemicircle andtriangle,eachtoonedecimalplace,thenadding,andthenbyusing nointermediaterounding.Explainwhichanswerismoreaccurate.

2F 14 Icanﬁndtheabsolutepercentageerror.

e.g.Findtheabsolutepercentageerrorwhen 2.6 isusedtoapproximate √7 whenﬁndingthearea ofarectanglewithwidth √7 cmandlength 2√7 cm.Giveyouranswertothreedecimalplaces.

2F 15 Icanstatethesmallestunitforagivenmeasurement.

e.g.Writedownthesmallestunitofmeasurementfor 27.3 cm.

2F 16 Icanﬁndthelimitsofaccuracyforagivenmeasurement.

e.g.Givethelimitsofaccuracyforthemeasurement 65.3 m.

2G 17 Icanﬁndthesurfaceareaofarectangular prismusinganet.

e.g.Findthesurfaceareaofthisrectangularprism.

2G 18 Icanﬁndthesurfaceareaofatriangular prismusinganet.

e.g.Findthesurfaceareaofthistriangularprism.

2H 19 Icanﬁndthesurfaceareaofacylinder.

e.g.Findthesurfaceareaofthiscylinder,totwodecimal places.

2I 20 Icanﬁndthevolumeofaprism. e.g.Findthevolumeofthistriangularprism.

2I 21 Icanﬁndthevolumeofacylinder.

e.g.Findthevolumeofthiscylinder,totwodecimalplaces.

2I 22 Icanﬁndthevolumeofaprism,givingananswer inLormL.

e.g.Findthevolumeofthisrectangularprisminlitres.

2J 23 Icanﬁndthesurfaceareaandvolumeofacomposite solid.

e.g.Findthesurfaceareaandvolumeofthiscompositesolid,totwo decimalplaces.

Short-answerquestions

1 2A Convertthesemeasurementstotheunitsshowninthebrackets. 5.3 km (m) a 27000 cm2 (m2 ) b 0.04 cm3 (mm3 ) c

2 2B Findtheperimeteroftheseshapes.

3 2C/E Forthecircleshown,ﬁnd,totwodecimalplaces: thecircumference a thearea. b

4 2E Forthesecompositeshapes,ﬁnd,totwodecimalplaces: theperimeter i thearea. ii

5 2D Findtheareaoftheseshapes.

2G Findthesurfaceareaoftheseprisms.

7 2H Determinethesurfaceareaofthiscylinder,totwodecimal places.

8 2I Findthevolumeofthesesolids,totwodecimalplaceswherenecessary.

9 2F Givethelimitsofaccuracyforthesemeasurements.

10 2F Findtheabsolutepercentageerrorif 3.1 isusedtoapproximate p whenﬁndingtheareaofa circlewithradius 8 cm.Roundyouranswertothreedecimalplaces.

11 2J Findthesurfaceareaandvolumeofthiscompositesolid,to twodecimalplaces.

Multiple-choicequestions

1 2A Thenumberofcentimetresinakilometreis:

2 2B Theperimeterofasquarewithsidelength 2

3 2B Theperimeteroftheshapeshownisgivenbytheformula:

4 2C Acorrectexpressionfordeterminingthecircumferenceofacirclewithdiameter 6

5 2D Theareaofarectanglewithsidelengths 3 cmand 4 cmis:

6 2D Thecorrectexpressionforcalculatingtheareaofthistrapeziumis:

7 2E Asector’scentreanglemeasures 90°.Thisisequivalentto:

Thevolumeofacubeofsidelength 3 cmis:

10 2H Thecurvedsurfaceareaforthiscylinderisclosestto:

Extended-responsequestions

1 Anewplaygroundisbeingbuiltwiththeshapeanddimensionsas shown.

a Theplaygroundwillbesurroundedbywoodenplanks. Determinetheperimeteroftheplayground,totwo decimalplaces. i Ifthewoodtobeusedcosts $16.50/m,whatwillbethecostof surroundingtheplayarea,tothenearestdollar? ii

b Theplaygroundareaistobecoveredwithalayerofwoodchips. Findtheareaoftheplayground,toonedecimalplace.

c Ifabagofwoodchipsfromthehardwarestorecovers 7.5 m2 ,how manybagswouldberequiredtocovertheplaygroundarea?

d Arectangularsandpitistobeincludedasshown.Ifsandistobe spreadﬂatandﬁlledtoaheightof 40 cm,determinethevolume ofsandrequiredinm3

2 Acylindricaltankhasdiameter 8 mandheight 2 m.

a Findthetotalvolumeofthetank,totwodecimalplaces.

b Findthetotalvolumeofthetankinlitres,totwodecimalplaces. Note:Thereare 1000 litresin 1 m3

c Findthesurfaceareaofthecurvedpartofthetank,totwodecimalplaces.

d Findthesurfacearea,includingthetopandthebase,totwodecimalplaces.