Ratios mcTYratios Aratioisawayofcomparingtwoormoresimilarquantitiesb ywritingtwoormorenumbers separatedbycolons
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Ratios mcTYratios Aratioisawayofcomparingtwoormoresimilarquantitiesb ywritingtwoormorenumbers separatedbycolons

Thenumbersshouldbewholenumbersan dshouldnotincludeunits Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthistopicyoushouldbea

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Ratios mcTYratios Aratioisawayofcomparingtwoormoresimilarquantitiesb ywritingtwoormorenumbers separatedbycolons




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Presentation on theme: "Ratios mcTYratios Aratioisawayofcomparingtwoormoresimilarquantitiesb ywritingtwoormorenumbers separatedbycolons"— Presentation transcript:


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Ratios mc-TY-ratios-2009-1 Aratioisawayofcomparingtwoormoresimilarquantities,b ywritingtwoormorenumbers separatedbycolons.Thenumbersshouldbewholenumbers,an dshouldnotincludeunits. Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature. Afterreadingthistext,and/orviewingthevideotutorialo nthistopic,youshouldbeableto: calculatetheratiooftwoormoresimilarquantities,wheth erornottheyareexpressedin thesameunits; divideaquantityintoanumberofpartsingivenratios; useratiostoscaleup,orscaledown,alistofingredients.

Contents 1. Introduction 2. Simplifyingratios 3. Usingratiostosharequantities 4 www.mathcentre.ac.uk 1 math centre2009
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1. Introduction Aratioisawayofcomparingtwoormoresimilarquantities. R atioscanbeusedtocompare costs,weights,sizesandotherquantities. Forexample,supposewehaveamodelboatwhichis1mlong,whe reastheactualboatis25m long.Thentheratioofthelengthofthemodeltothelengthof theactualboatis1to25.This iswrittenas 1 : 25 Notetherearenounitsincluded,andnotealsotheuseofthec olontorepresenttheratio. Ratiosarealsousedtodescribequantitiesofdifferentingr edients inmixtures.

Pharmacists makingupmedicines,manufacturersmakingbiscuitsandbui ldersmakingcementallneedtomake mixturesusingingredientsinthecorrectratio. Iftheydon ttheremaybedireconsequences! Soknowingaboutratiosisnotonlyveryimportant,butextre melyusefulandcrucialincertain circumstances. Forexample,mortarforbuildingabrickwallismadebyusing 2partsofcementto7partsof sand.Thentheratioofcementtosandis2to7,andiswrittena 2 : 7 2. Simplifying ratios Tomakepastryforanapplepie,youneed4ozflourand2ozfat.T heratioofflourtofatis 4 : 2 Butthisratiocanbesimplifiedinthesamewaythattwofracti

onscanbesimplified. Wejust cancelbyacommonfactor.So 4 : 2 = 2 : 1 Theratio2to1isthesimplestformoftheratio4to2.Andther atiosareequivalent,because therelationshipbetweeneachpairofnumbersisthesame. Forexample,ifwehavearatio250to150,wecansimplifyitby dividingbothnumbersby10 andthenby5toget5to3: 250 : 150 25 : 15 5 : 3 Theratio5to3isthesimplestformoftheratio250to150,and allthreeratiosareequivalent. Ratiosarenormallyexpressedusingwholenumbers,soarati oof1to1.5wouldbewrittenas 10to15,andthenas2to3initssimplestform: 1 : 1 10 : 15 2 : 3 www.mathcentre.ac.uk 2 math centre2009
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Similarly,aratio to wouldbewrittenas to ,andthenas2to5initssimplestform: 2 : 5 Nowitisveryimportantinaratiotousethesameunitsforthe numbers,asotherwisetheratio willbeincorrectandthecomparisonwillbewrong.Takethis ratio:15penceto 3.Theratio isnot15to3andthen5to1. Thecomparisoniswrong. Wemustha vethesameunitsfor eachnumber,soweconvertthemtothesameunits.Itdoesntm atterwhichunityouuse,but ofcourseitisjustusecommonsensetochoosetheunitwhichg ivesthesimplestnumbers. In thiscaseitisobviousthatweshouldusepence,so15penceto 300penceisthensimplifiedto

3to60bydividingby5.Wethensimplifyitfurtherbydividin gby3toget1to20.Thatisthe ratioinitssimplestform.So 15 p : 15 : 3 5 : 1 iswrong,whereas 15 p : 15 : 300 3 : 60 1 : 20 iscorrect. Key Point Aratioisawayofcomparingtwoormoresimilarquantities.A ratioof2cmto5cmiswritten as2: 5. Aratioisnormallywrittenusingwholenumbersonly, withnounits,initssimplest form. Thenumbersinaratiomustbewrittenusingthesameunits. If theyarenot,theyshouldbe convertedtothesameunits.Itdoesnotmatterwhichunitsar eusedfortheconversion. Exercises 1.Expresstheseratiosintheirsimplestform: (a) 2to10 (b) 80to20 (c) to1 (d) 50p: 3.50

(e) 6m:30cm (f) 1.5:1 (g) 10min:4hr (h) :3 2.Inaclassthereare15girlsand12boys.Whatistheratioof girlstoboys? 3.Annahas75pence.Rashidhas 1.20.WhatistheratioofRasidsmoneytoAnnasmoney? www.mathcentre.ac.uk 3 math centre2009
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3. Using ratios to share quantities Ratioscanbeusedtoshare,ordivide,quantitesofmoney,we ightsandsoon. Example MrsSharpandMrWestshareaninheritanceof 64,000intheratio5:3.Howmuchdothey eachget? Solution Tocalculatetheanswerswefirstlookatthenumbersinvolved andseethetotalnumberofparts intowhichtheinheritanceissplit.Theratiois5to3.Sothe

totalnumberofpartsis5plus3, whichis8. Sharp 5 Wes 3 64,000 5 + 3 = 8 Nowwecanworkoutwhatonepartisworth,andthenhowmucheac hpersongets. part 64 000 000 SoMrsSharpreceives5parts,whichis 000 = 40 000 andMrWestreceives3parts, whichis 000 = 24 000 Wecancheckourcalculationsbyaddingthetwoamountstoget her.Theyshouldadduptothe totalvalueoftheinheritance.So 40 000 + 24 000 = 64 000 whichdoesequaltheoriginalinheritance. Wecanalsocheckthiscalculationinanotherway. Wecanwork backwards,bytakingthetwo amountsandfindingtheirratio.Thetwoamountsarebothgive ninthesameunits,pounds,and so 40 000 : 24

000 40 : 24 5 : 3 Example Concreteismadebymixinggravel,sandandcementintherati o3:2:1byvolume.Howmuch gravelwillbeneededtomake 12 m ofconcrete? www.mathcentre.ac.uk 4 math centre2009
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Solution gravel 3 cement 1 12m concrete sand 2 First,weworkoutthetotalnumberofpartsintowhichthecon creteisdivided: 3 + 2 + 1 = 6 partsaltogether. Usingthenumbersintheratio,weknowthe nthatgravelmakesup3parts, sand2parts,andcement1part.Sothereare6partsaltogethe r,andwehave 12 m ofconcrete, andtherefore1partmustequal 2 m .Thenasthereare3partsofgravel,thevolumeofgravel neededmustbe 2 m whichis 6 m 3 +

2 + 1 = 6 parts parts = 12 m part 12 = 2 m sogravel(3parts) = 3 2 m = 6 m Wenowneedtochecktheanswer.Gravelrepresents3partsout ofatotalof6,inotherwords ahalf.Sohalfofthetotalvolumeofconcreteisgravel,andt hatishalfof 12 m ,whichis 6 m Sothatisindeedthecorrectanswer. Example Withthesameformulaforconcrete,supposewehave 6 m ofsandandanunlimitedamountof theotheringredients.Howmuchconcretecouldwemake? Solution Inthisexample,theratioofgraveltosandtocementisstill 3: 2: 1,sothetotalnumberof partsintowhichtheconcreteisdividedisstill 3 + 2 + 1 = 6 .Butthistimeweknowthevolume

ofsand,andwehavetoworkoutthetotalvolumeofconcreteth atispossibletomake. gravel 3 cement 1 6m 3 sand 2 Twopartsofthetotalrepresents 6 m ofsand. Soonepartis ,inotherwords 3 m ,and thusthetotalof6partsofconcreterepresents 6 m ,making 18 m . So 18 m ofconcrete www.mathcentre.ac.uk 5 math centre2009
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canbemadeifwehave 6 m ofsandandanunlimitedamountoftheotheringredients: 3 + 2 + 1 = 6 parts parts = 6 m part = 3 m so parts = 6 3 m = 18 m Alternatively,wecouldhavetackledthisquestionbyusing fractions. Sandrepresents2parts outofatotalof6,whichisathird. Soifathirdofthetotalis 6 m

thenthetotalamountof concretethatcouldbemadewouldbe3times6,giving 18 m . Thisisagoodcheckthatour answeriscorrect. Example HereisalistoftheingredientstomakeaquantityoftheGree kfoodhoumoussufficient for6people. 2clovesgarlic 4ozchickpeas 4tbsoliveoil 5floztahinipaste (houmousfor6people) Whatamountswouldbeneededsothattherewillbeenoughfor9 people? Solution Theratiooftheamountsis2: 4: 4: 5for6people. Foroneperso nwescaletheamounts down,sowedivideby6. Thenfor9peoplewemultiplyby9,andw eseeaftercancellingthat weneed3clovesofgarlic,6ozchickpeas,6tbsofoliveoil,a nd flozoftahinipaste: 2

: 4 : 4 : 5 (6people) (1person) (1person) 3 : 6 : 6 : 15 = 7 giving 3clovesgarlic 6ozchickpeas 6tbsoliveoil floztahinipaste (houmousfor9people). Wecouldhavedonethesecalculationsmorequicklybymultip lyingeachamountbythefraction 9/6,or3/2initssimplestform.Butitisoftensafertoworko utwhattheamountsareforone person,andthenscaleupordownafterwardsaccordingly. Inconversionproblems,itisoftenbettertoworkoutwhaton eoftherequiredamountsrepresents, andthenscaleupordown. Example If 1isworth1.65euros,whatisthevalueof50eurostotheneare stpenny? www.mathcentre.ac.uk 6 math centre2009
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Solution Wearegiventhat1.65eurosisworth 1or100pence,so1euroisworth100/1.65pence. Then50eurosequals100/1.65times50pence,whichis5000/1 .65pence. Puttingthisintoa calculatorgives3030.3030,whichis3030pencetotheneare stpenny,or30.30.So 65 euros = 100 pence euro 100 65 pence 50 euro 100 65 50 pence 5000 65 pence = 3030 pence 30 30 Key Point Whendividingaquanityinagivenratio,itisusefultoworko ut thetotalnumberofpartsintowhichthequantityistobedivi ded,and thevalueofonepart. Exercises 4.Amapscaleis1:20,000.Onthemap,thedistancebetweentw opoints and is8.5cm. Whatistheactualdistancebetween and 5.

Arminderdoesascaledrawingofhislivingroom. Heusesas caleof1: 100. Hemeasures thelengthofthelivingroomas13.7m.Howlongisthelivingr oomonthescaledrawing? 6. Arecipetomakelasagnafor5peopleuses300gramsofmince dbeef. Howmuchminced beefwouldbeneededtoserve9people? 7. Theratioofboystogirlsinayouthclubis4: 5. Thereare28 boys. Howmanygirlsare thereintheyouthclub? 8.Onepoundisworth1.65euros. Whatis20poundsineuros? Whatis60eurostothenearestpenny? www.mathcentre.ac.uk 7 math centre2009
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9. Bettyis12yearsoldandhersisterLizis3yearsold. Their grandfathergivesthem 150,

whichistobedividedbetweenthemintheratiooftheirages. Howmuchdoeseachofthem get? 10.Divide 360 intothreeanglesintheratio1:2:3. 11.Bluecoppersulphateismadefrom 32partsofcopper 16partsofsulphur 32partsofoxygen 48partsofwater wherealltheproportionsarebyweight. Howmuchwateristherein5kgofcoppersulphate? Howmuchcoppersulphatecouldbemadewith96kgofcopperand plentyofallother ingredients? 12.Herearetheingredientsformaking18rockcakes: 9ozflour 6ozsugar 6ozmargarine 8ozmixeddriedfruit 2largeeggs. Robertwantstomake12rockcakes.Howmuchmargarinedoeshe need?

Jennyhasonly9ozofsugarandhasplentyofalltheotheringr edients.Whatisthegreatest numberofrockcakesshecanmake? Answers 1. (a) 1:5 (b) 4:1 (c) 1:3 (d) 1:7 (e) 20:1 (f) 3:2 (g) 1:24 (h) 4:9 2.5:4 3.8:5 4.1700m 5.13.7cm 6.540gm 7.35 8. (a) 33euros (b) 36.36,or3636pence 9.Bettyreceives 120,Lizreceives 30 10. 60 120 and 180 11. (a) 1875gm(or1.875kg) (b) 384kg 12. (a) 4oz (b) 27 www.mathcentre.ac.uk 8 math centre2009