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1 Percentages mc-ty-percent In this unit we shall look at the meaning of percentages and carry out calculations involving percentages. We will also look at the use of the percentage button on calculators. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: calculate a percentage of a given quantity; increase or decrease a quantity by a given percentage; findtheoriginalvalueofaquantitywhenithasbeenincreasedordecreasedbyagiven percentage; express one quantity as a percentage of another. Contents 1. Introduction. Finding percentage amounts 3 3. Finding the original amount before a percentage change 4 4. Expressing a change as a percentage 5 5. Calculating percentages using a calculator c mathcentre 009

2 1. Introduction Theword percentage isveryfamiliartousasitisusedregularly inthemediatodescribe anything from changes in the interest rate, to the number of people taking holidays abroad, to the success rate of the latest medical procedures or exam results. Percentages are a useful way ofmakingcomparisons,apartfrombeingusedtocalculatethemanytaxesthatwepaysuchas VAT,incometax,domesticfueltaxandinsurancetax,tonamebutafew. So percentages are very much part of our lives. But what does percentage actually mean? Now per cent means out of ; and out of, in mathematical language, means divide by. Soifyouscore85%(usingthesymbol % forpercentage)onatestthen,iftherewereapossible marks altogether, you would have achieved 85 marks. So 85% = 85. Let us look at some other common percentage amounts, and their fraction and decimal equivalents. 75% = 75 = 3 4 = % = 50 = 1 = 0.5 5% = 5 = 1 4 = % = 10 = 1 10 = 0.1 5% = 5 = 1 0 = Itisworthnotingthat50%canbefoundbedividingby,andthat10%iseasilyfoundby dividing by 10. Nowletuslookatwritingfractionsaspercentages. Forexample,sayyouget18marksoutof 0inatest.Whatpercentageisthis? First,writetheinformationasafraction.Yougained18outof0marks,sothefractionis Sinceapercentagerequiresadenominatorof,wecanturn 18intoafractionoutofby 0 multiplying both numerator and denominator by 5: 18 0 = = 90 = 90%. Sincewearemultiplyingboththenumeratorandthedenominatorby5,wearenotchangingthe value of the fraction, merely finding an equivalent fraction. Inthatexampleitwaseasytoseethat,inordertomakethedenominator,weneededto multiply0by5.butifitisnoteasytoseethis,suchaswithascoreof,say,53outof68,then wesimplywritetheamountasafractionandthenmultiplyby : = % = 77.94% whichis78%tothenearestwholenumber.althoughitiseasiertouseacalculatorforthistype ofcalculation,itisadvisablenottousethe%buttonatthisstage. Weshalllookatusingthe percentagebuttononacalculatorattheendofthisunit. c mathcentre 009

3 Key Point Percentage means out of, which means divide by. To change a fraction to a percentage, divide the numerator by the denominator and multiply by %. Exercises 1 (a)7outofevery10peoplequestionedwhoexpressedapreferencelikedacertainbrandof cereal. What is this as a percentage? (b)inatestyougained4marksoutof40.whatpercentageisthis? (c)30outof37gamblingsitesontheinternetfailedtorecognisethedebitcardofachild.what isthisasapercentage?. Finding percentage amounts Formanycalculations,weneedtofindacertainpercentageofaquantity. Forexample,itis commoninsomecountriestoleaveatipof10%ofthecostofyourmealforthewaiter.saya meal costs 5.40: 10%of 5.40 = =.54. Asmentionedbefore,aneasywaytofind10%issimplytodivideby10. Howeverthewritten method shown above is useful for more complicated calculations, such as the commission a salesmanearnsifhereceives%ofthevalueofordershesecures. Inonemonthhesecures 50,000 worth of orders. How much commission does he receive? % of 50, 000 = 50, 000 = 5, 000. ManythingsthatwebuyhaveVATaddedtotheprice,andtocalculatethepurchasepricewe havetopayweneedtofind 17 1 %andadditontotheprice.thiscanbedoneintwoways. Forexample,thecostofacomputeris 634plusVAT.Findthetotalcost. VAT = 17 1 %of 634 = = sototalcost = = c mathcentre 009

4 Or,insteadofthinkingofthetotalcostas%ofthepriceplus 17 1 %oftheprice,wecan thinkofitas %oftheprice,sothat %of 634 = 634 = Although 17 1 %seemsanawkwardpercentagetocalculate,thereisaneasymethodyoucanuse sothatyoudonotneedacalculator.letuslookatthesameexampleagain % is (divide by 10) 5% is (half of 10%) 1 % is (halfof5%) so 17 1 % is (addtheabove). In a similar way to a percentage increase, there is a percentage decrease. For example, shops often offer discounts on certain goods. A pair of trainers normally costs 75, but they are offered for10%offinthesale.findtheamountyouwillpay. Now10%of 75is 7.50,sothesalepriceis = Whatyouarepayingisthe%ofthecost,minus10%ofthecost,soineffectyouarepaying 90%ofthecost.Sowecouldcalculatethisdirectlybyfinding90%ofthecost. 90%of 75 = = Finding the original amount before a percentage change LetuslookatanexamplewherethepriceincludesVAT,andweneedthepriceexcludingVAT. Thecostofacomputeris 699includingVAT.CalculatethecostbeforeVAT. Nowacommonmistakehereistotake 17 1 %ofthecostincludingvat,andthensubtract.but thisiswrong,becausethevatisnot 17 1 %ofthecostincludingthevat,whichiswhatwe havebeengiven.instead,thevatis 17 1 %ofthecostbeforethevat,andthisiswhatweare tryingtofind.sowehavetouseadifferentmethod. Nowwehavebeentoldthat 699representsthecostincludingVAT,sothatmustequalthe costbeforevat,plusthevatitself,whichis 17 1 %ofthecostbeforevat.sothetotalmust be % % = 1171 %ofthecostbeforevat.thus,tofind1%wedivideby 1171.So %ofthepriceexcludingvat = 699, 1%ofthepriceexcludingVAT = TofindthecostbeforeVATwewant%,sonowweneedtomultiplyby.Then thepriceexcludingvat = = c mathcentre 009

5 Letuslookatanothersituationwhereweneedtofindanoriginalamountbeforeapercentage increase has taken place. An insurance company charges a customer 30 for his car insurance. The price includes governmentinsurancepremiumtaxat5%.whatisthecostbeforetaxwasadded? Here,the 30represents105%ofthecost,sotocalculatetheoriginalcost,%,weneed to calculate 30 = Hereisonemoresimilarcalculation,butthistimetherehasbeenareductionincost. Ashophasreducedthecostofacoatby15%inasale,sothatthesalepriceis What wastheoriginalcostofthecoat? Inthiscase, 17.50represents85%(thatis, % 15%)oftheoriginalprice.Soifwewrite thisasafraction,wedivideby85tofind1%andthenmultiplybytofindtheoriginalprice = 150. Key Point Ifyouaregivenapercentagechangeandthefinalamount,writethefinalamountas%plus (or minus) the percentage change, multiplied by the original amount. 4. Expressing a change as a percentage We might wish to calculate the percentage by which something has increased or decreased. To dothisweusetherule actual increase or decrease %. Soyouwritetheamountofchangeasafractionoftheoriginalamount,andthenturnitintoa percentage. Fouryearsago,acouplepaid 180,000fortheirhouse.Itisnowvaluedat 350,000.Calculate the percentage increase in the value of the house. 5 c mathcentre 009

6 Percentageincrease = actualincrease % 350, , 000 = % 180, , 000 = 180, 000 % = 94%tothenearest1%. Letuslookatanexamplewherethechangehasbeenadecrease. Acarcost 1,000.After3yearsitisworth 8,000.Whatisthepercentagedecrease? Percentagedecrease = actualdecrease % 1, 000 8, 000 = % 1, 000 4, 000 = 1, 000 % = 33%tothenearest1%. Key Point Towriteanincreaseordecreaseasapercentage,usetheformula actual increase or decrease %. 5. Calculating percentages using a calculator Hereisawarningaboutusingthepercentagebuttononacalculator:theresultdependsonwhen youpressthe%buttoninyourcalculation. Sometimesithasnoeffect,sometimesitseemsto divideby,andatothertimesitmultipliesby.herearesomeexamples Pressing %givesananswerof1.Now = 0.1,sopressingthe%button hashadtheeffectofmultiplyingby.thishasfound48asapercentageof400. Pressing 1 300%givestheanswer1.5. Now = 150,sopressingthe% buttonherehasdividedby.thishasfound300%ofahalf. 6 c mathcentre 009

7 Pressing %givesananswerof00. Now = 0, 000,sopressing%here hasdividedby.thishasfound50%of400. Pressing 50% 400resultsin400onthedisplay,requiring =tobepressedtodisplayan answerof0,000.sopressingthe%buttonherehashadnoeffect. Key Point Werecommendedthatyouusethe%buttononacalculatoronlywhenyouunderstandwhat affect it is having on your calculation. Exercises (a)whatistheamountofvat(atarateof 17 1 %)whichmustbepaidonanimportedcomputer game costing 16.00? (b)avisitortothiscountrybuysasouvenircosting 7.50includingVATat 17 1 %.Howmuch VAT can be reclaimed? (c)attheendof1999youboughtsharesinacompanyfor. During000theshares increasedinvalueby10%.during001thesharesdecreasedinvalueby10%.howmuchwere thesharesworthattheendof001? (Give your answers to the nearest penny.) Answers 1.. (a) 70% (b) 60% (c) 81% (a).80 (b) 4.10 (c) c mathcentre 009

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