CONTENTSPreface

Introduction

Chapter1:Multiplication:GettingStartedWhatisMultiplication?TheSpeedMathematicsMethod

Chapter2:UsingaReferenceNumberReferenceNumbersDoubleMultiplication

Chapter3:NumbersAbovetheReferenceNumberMultiplyingNumbersinTheTeensMultiplyingNumbersAbove100SolvingProblemsinYourHeadDoubleMultiplication

Chapter4:MultiplyingAbove&BelowtheReferenceNumberNumbersAboveandBelow

Chapter5:CheckingYourAnswersSubstituteNumbers

Chapter6:MultiplicationUsingAnyReferenceNumberMultiplicationbyfactorsMultiplyingnumbersbelow20Multiplyingnumbersaboveandbelow20Using50asareferencenumberMultiplyinghighernumbersDoublingandhalvingnumbers

Chapter7:MultiplyingLowerNumbersExperimentingwithreferencenumbers

Chapter8:Multiplicationby11Multiplyingatwo-digitnumberby11Multiplyinglargernumbersby11Multiplyingbymultiplesof11

Chapter9:MultiplyingDecimals

Multiplicationofdecimals

Chapter10:MultiplicationUsingTwoReferenceNumbersEasymultiplicationby9UsingfractionsasmultiplesUsingfactorsexpressedasdivisionPlayingwithtworeferencenumbersUsingdecimalfractionsasreferencenumbers

Chapter11:AdditionAddingfromlefttorightBreakdownofnumbersCheckingadditionbycastingoutnines

Chapter12:SubtractionNumbersaround100EasywrittensubtractionSubtractionfromapowerof10Checkingsubtractionbycastingnines

Chapter13:SimpleDivisionSimpledivisionBonus:Shortcutfordivisionby9

Chapter14:LongDivisionbyFactorsWhatAreFactors?Workingwithdecimals

Chapter15:StandardLongDivisionMadeEasy

Chapter16:DirectLongDivisionEstimatinganswersReversetechnique—roundingoffupwards

Chapter17:CheckingAnswers(Division)ChangingtomultiplicationBonus:Castingtwos,tensandfivesCastingoutnineswithminussubstitutenumbers

Chapter18:FractionsMadeEasyWorkingwithfractionsAddingfractions

SubtractingfractionsMultiplyingfractionsDividingfractionsChangingvulgarfractionstodecimals

Chapter19:DirectMultiplicationMultiplicationwithadifferenceDirectmultiplicationusingnegativenumbers

Chapter20:PuttingitAllintoPracticeHowDoIRememberAllofThis?AdviceForGeniuses

Afterword

AppendixA:UsingtheMethodsintheClassroom

AppendixB:WorkingThroughaProblem

AppendixC:Learnthe13,14and15TimesTables

AppendixD:TestsforDivisibility

AppendixE:KeepingCount

AppendixF:PlusandMinusNumbers

AppendixG:Percentages

AppendixH:HintsforLearning

AppendixI:Estimating

AppendixJ:SquaringNumbersEndingin5

AppendixK:PracticeSheets

Index

Firstpublished2005byWrightbooksanimprintofJohnWiley&SonsAustralia,Ltd

42McDougallStreet,Milton,Qld4064OfficealsoinMelbourne©BillHandley2005

Themoralrightsoftheauthorhavebeenasserted.NationalLibraryofAustraliaCataloguing-in-Publicationdata:

Handley,Bill.Speedmathsforkids:Helpingchildrenachievetheirfullpotential.

Includesindex.Forprimaryschoolstudents.

ISBN0731402278.1.Mentalarithmetic.2.Mentalarithmetic–Studyandteaching(Primary).I.Title.

513.9

Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,

withoutthepriorpermissionofthepublisher.CoverdesignbyRobCowpe

PREFACEIcouldhavecalledthisbookFunWithSpeedMathematics.Itcontainssomeofthesamematerialasmyotherbooksand teachingmaterials. Italso includesadditionalmethodsandapplicationsbasedon thestrategiestaughtinSpeedMathematicsthat,Ihope,givemoreinsightintothemathematicalprinciplesandencouragecreativethought.Ihavewrittenthisbookforyoungerpeople,butIsuspectthatpeopleofanyagewillenjoyit.Ihaveincludedsectionsthroughoutthebookforparentsandteachers.AcommonresponseIhearfrompeoplewhohavereadmybooksorattendedaclassofmineis,‘Why

wasn’tItaughtthisatschool?’Peoplefeelthat,withthesemethods,mathematicswouldhavebeensomucheasier, and theycouldhaveachievedbetter results than theydid,or they feel theywouldhaveenjoyedmathematicsalotmore.Iwouldliketothinkthisbookwillhelponbothcounts.IhavedefinitelynotintendedSpeedMathsforKidstobeaserioustextbookbutratherabooktobe

playedwithandenjoyed.IhavewrittenthisbookinthesamewaythatIspeaktoyoungstudents.SomeofthelanguageandtermsIhaveusedaredefinitelynon-mathematical.Ihavetriedtowrite thebookprimarilysoreaderswillunderstand.AlotofmyteachingintheclassroomhasjustbeenexplainingoutloudwhatgoesoninmyheadwhenIamworkingwithnumbersorsolvingaproblem.Ihavebeengratified to learn thatmanyschoolsaround theworldareusingmymethods. I receive

emails every day from students and teachers who are becoming excited about mathematics. I haveproduced a handbook for teacherswith instructions for teaching thesemethods in the classroomandwithhandoutsheetsforphotocopying.Pleaseemailmeorvisitmywebsitefordetails.BillHandley,Melbourne,2005bhandley@speedmathematics.comwww.speedmathematics.com

INTRODUCTION

I have heard many people say they hate mathematics. I don’t believe them. They think they hatemathematics. It’snot reallymaths theyhate; theyhate failure. Ifyoucontinually failatmathematics,youwillhateit.No-onelikestofail.Butifyousucceedandperformlikeageniusyouwilllovemathematics.Often,whenIvisitaschool,

studentswill ask their teacher, canwedomaths for the restof theday?The teachercan’tbelieve it.Thesearekidswhohavealwayssaidtheyhatemaths.Ifyouaregoodatmaths,peoplethinkyouaresmart.Peoplewilltreatyoulikeyouareagenius.Your

teachersandyourfriendswilltreatyoudifferently.Youwilleventhinkdifferentlyaboutyourself.Andthereisgoodreasonforit—ifyouaredoingthingsthatonlysmartpeoplecando,whatdoesthatmakeyou?Smart!Ihavehadparentsandteacherstellmesomethingveryinteresting.Someparentshavetoldmetheir

childjustwon’ttrywhenitcomestomathematics.Sometimestheytellmetheirchildislazy.Thenthechildhasattendedoneofmyclassesorreadmybooks.Thechildnotonlydoesmuchbetterinmaths,butalsoworksmuchharder.Whyisthis?Itissimplybecausethechildseesresultsforhisorherefforts.Oftenparentsandteacherswilltellthechild,‘Justtry.Youarenottrying.’Ortheytellthechildtotry

harder.Thisjustcausesfrustration.Thechildwouldliketotryharderbutdoesn’tknowhow.Usuallychildrenjustdon’tknowwheretostart.Sometimestheywillscrewuptheirfaceandhitthesideoftheirhead with their fist to show they are trying, but that is all they are doing. The only thing theyaccomplishisaheadache.Bothchildandparentbecomefrustratedandangry.I am going to teach you, with this book, not only what to do but how to do it. You can be a

mathematical genius. You have the ability to perform lightning calculations in your head that willastonishyourfriends,yourfamilyandyourteachers.Thisbookisgoingtoteachyouhowtoperformlikeagenius—todothingsyourteacher,orevenyourprincipal,can’tdo.Howwouldyouliketobeable tomultiplybignumbers or do longdivision in your head?While theother kids arewriting theproblemsdownintheirbooks,youarealreadycallingouttheanswer.Thekids(andadults)whoaregeniusesatmathematicsdon’thavebetterbrainsthanyou—theyhave

better methods. This book is going to teach you those methods. I haven’t written this book like aschoolbook or textbook. This is a book to play with. You are going to learn easy ways of doingcalculations,andthenwearegoingtoplayandexperimentwiththem.Wewillevenshowofftofriendsandfamily.WhenIwasinyearnineIhadamathematics teacherwhoinspiredme.Hewouldtellusstoriesof

SherlockHolmesorofthrillermoviestoillustratehispoints.Hewouldoftensay,‘Iamnotsupposedtobeteachingyouthis,’or,‘Youarenotsupposedtolearnthisforanotheryearortwo.’OftenIcouldn’twait togethomefromschool to trymoreexamples formyself.Hedidn’t teachmathematics like theotherteachers.Hetoldstoriesandtaughtusshortcutsthatwouldhelpusbeattheotherclasses.Hemademathsexciting.Heinspiredmyloveofmathematics.WhenIvisitaschoolIsometimesaskstudents,‘Whodoyouthinkisthesmartestkidinthisschool?’

ItellthemIdon’twanttoknowtheperson’sname.Ijustwantthemtothinkaboutwhothepersonis.Then I ask, ‘Who thinks that the person you are thinking of has been told they are stupid?’No-oneseemstothinkso.Everyonehasbeentoldatonetimethat theyarestupid—but thatdoesn’tmakeit true.Wealldo

stupid things. Even Einstein did stupid things, but he wasn’t a stupid person. But people make themistake of thinking that thismeans they are not smart.This is not true; highly intelligent people do

stupidthingsandmakestupidmistakes.Iamgoingtoprovetoyouasyoureadthisbookthatyouareveryintelligent.Iamgoingtoshowyouhowtobecomeamathematicalgenius.

HowToReadThisBookReadeachchapterandthenplayandexperimentwithwhatyoulearnbeforegoingtothenextchapter.Dotheexercises—don’tleavethemforlater.Theproblemsarenotdifficult.Itisonlybysolvingtheexercisesthatyouwillseehoweasythemethodsreallyare.Trytosolveeachprobleminyourhead.Youcanwritedowntheanswerinanotebook.Findyourselfanotebooktowriteyouranswersandto

useasareference.Thiswillsaveyouwritinginthebookitself.Thatwayyoucanrepeattheexercisesseveraltimesifnecessary.Iwouldalsousethenotebooktotryyourownproblems.Remember,theemphasisinthisbookisonplayingwithmathematics.Enjoyit.Showoffwhatyou

learn.Usethemethodsasoftenasyoucan.Usethemethodsforcheckinganswerseverytimeyoumakeacalculation.Makethemethodspartofthewayyouthinkandpartofyourlife.Now,goaheadandreadthebookandmakemathematicsyourfavouritesubject.

Chapter1

MULTIPLICATION:GETTINGSTARTED

Howwelldoyouknowyourmultiplicationtables?Doyouknowthemuptothe15or20timestables?Doyouknowhow tosolveproblems like14×16,oreven94×97,withoutacalculator?Using thespeedmathematicsmethod,youwillbeabletosolvethesetypesofproblemsinyourhead.Iamgoingtoshowyouafun,fastandeasywaytomasteryourtablesandbasicmathematicsinminutes.I’mnotgoingtoshowyouhowtodoyourtablestheusualway.Theotherkidscandothat.Using the speed mathematics method, it doesn’t matter if you forget one of your tables. Why?

Because if you don’t know an answer, you can simply do a lightning calculation to get an instantsolution.Forexample,aftershowingherthespeedmathematicsmethods,Iaskedeight-year-oldTrudy,‘Whatis14times14?’Immediatelyshereplied,‘196.’Iasked,‘Youknewthat?’Shesaid,‘No,IworkeditoutwhileIwassayingit.’Wouldyou like tobe able todo this? Itmay take fiveor tenminutespracticebeforeyou are fast

enoughtobeatyourfriendsevenwhentheyareusingacalculator.

WhatisMultiplication?Howwouldyouaddthefollowingnumbers?

6+6+6+6+6+6+6+6=?Youcouldkeepaddingsixesuntilyougettheanswer.Thistakestimeand,becausetherearesomany

numberstoadd,itiseasytomakeamistake.Theeasymethodis tocounthowmanysixes thereare toaddtogether,andthenusemultiplication

tablestogettheanswer.Howmanysixesarethere?Countthem.Thereareeight.You have to find out what eight sixes added together would make. People often memorise the

answersoruseachart,butyouaregoingtolearnaveryeasymethodtocalculatetheanswer.Asamultiplication,theproblemiswrittenlikethis:8×6=

Thismeansthereareeightsixestobeadded.Thisiseasiertowritethan6+6+6+6+6+6+6+6=.Thesolutiontothisproblemis:8×6=48

TheSpeedMathematicsMethodIamnowgoingtoshowyouthespeedmathematicswayofworkingthisout.Thefirststepistodrawcirclesundereachofthenumbers.Theproblemnowlookslikethis:

Wenowlookateachnumberandask,howmanymoredoweneedtomake10?

Westartwiththe8.Ifwehave8,howmanymoredoweneedtomake10?Theansweris2.Eightplus2equals10.Wewrite2inthecirclebelowthe8.Ourequationnowlooks

likethis:

Wenowgotothe6.Howmanymoretomake10?Theansweris4.Wewrite4inthecirclebelowthe6.Thisishowtheproblemlooksnow:

Wenowtakeawaycrossways,ordiagonally.Weeithertake2from6or4from8.Itdoesn’tmatterwhichwaywesubtract, theanswerwillbethesame,sochoosethecalculationthat lookseasier.Twofrom6is4,or4from8is4.Eitherwaytheansweris4.Youonlytakeawayonetime.Write4aftertheequalssign.

Forthelastpartoftheanswer,you‘times’thenumbersinthecircles.Whatis2times4?Twotimes4meanstwofoursaddedtogether.Twofoursare8.Writethe8asthelastpartoftheanswer.Theansweris48.

Easy,wasn’t it?This ismuch easier than repeating yourmultiplication tables every day until yourememberthem.Andthisway,itdoesn’tmatterifyouforgettheanswer,becauseyoucansimplyworkitoutagain.Doyouwanttotryanotherone?Let’stry7times8.Wewritetheproblemanddrawcirclesbelowthe

numberslikebefore:

Howmanymoredoweneedtomake10?Withthefirstnumber,7,weneed3,sowewrite3inthecirclebelowthe7.Nowgotothe8.Howmanymoretomake10?Theansweris2,sowewrite2inthecirclebelowthe8.Ourproblemnowlookslikethis:

Nowtakeawaycrossways.Either take3from8or2 from7.Whicheverwaywedo it,weget thesameanswer.Sevenminus2is5or8minus3is5.Fiveisouranswereitherway.Fiveisthefirstdigitoftheanswer.Youonlydothiscalculationoncesochoosethewaythatlookseasier.Thecalculationnowlookslikethis:

Forthefinaldigitoftheanswerwemultiplythenumbersinthecircles:3times2(or2times3)is6.Writethe6astheseconddigitoftheanswer.

Hereisthefinishedcalculation:

Seveneightsare56.Howwouldyousolvethisprobleminyourhead?Takebothnumbersfrom10toget3and2inthe

circles.Takeawaycrossways.Sevenminus2is5.Wedon’tsayfive,wesay,‘Fifty...’.Thenmultiplythenumbersinthecircles.Threetimes2is6.Wewouldsay,‘Fifty...six.’Witha littlepracticeyouwillbeable togivean instantanswer.And,aftercalculating7 times8a

dozenorsotimes,youwillfindyouremembertheanswer,soyouarelearningyourtablesasyougo.

TestyourselfHerearesomeproblemstotrybyyourself.Doalloftheproblems,evenifyouknowyourtableswell.Thisisthebasicstrategywewilluseforalmostallofourmultiplication.a)9×9=b)8×8=c)7×7=d)7×9=e)8×9=f)9×6=g)5×9=h)8×7=Howdidyougo?Theanswersare:a)81b)64c)49d)63e)72f)54g)45h)56

Isn’tthistheeasiestwaytolearnyourtables?Now, cover your answers anddo themagain in your head.Let’s look at 9×9 as an example.To

calculate9×9,youhave1below10eachtime.Nineminus1is8.Youwouldsay,‘Eighty...’.Thenyoumultiply1times1togetthesecondhalfoftheanswer,1.Youwouldsay,‘Eighty...one.’Ifyoudon’tknowyourtableswellitdoesn’tmatter.Youcancalculatetheanswersuntilyoudoknow

them,andno-onewilleverknow.

Multiplyingnumbersjustbelow100Doesthismethodworkformultiplyinglargernumbers?Itcertainlydoes.Let’stryitfor96×97.

96×97=Whatdowetakethesenumbersupto?Howmanymoretomakewhat?Howmanytomake100,so

wewrite4below96and3below97.

Whatdowedonow?Wetakeawaycrossways:96minus3or97minus4equals93.Writethatdownas the first part of the answer.What dowe do next?Multiply the numbers in the circles: 4 times 3equals12.Writethisdownforthelastpartoftheanswer.Thefullansweris9,312.

Whichmethoddoyouthinkiseasier,thismethodortheoneyoulearntinschool?Idefinitelythinkthismethod;don’tyouagree?Let’stryanother.Let’sdo98×95.98×95=

Firstwedrawthecircles.

Howmanymoredoweneedtomake100?With98weneed2moreandwith95weneed5.Write2and5inthecircles.

Nowtakeawaycrossways.Youcandoeither98minus5or95minus2.98−5=93or95−2=93

Thefirstpartoftheansweris93.Wewrite93aftertheequalssign.

Nowmultiplythenumbersinthecircles.2×5=10

Write10afterthe93togetananswerof9,310.

Easy.Withacoupleofminutespracticeyoushouldbeable todotheseinyourhead.Let’s tryonenow.

96×96=Inyourhead,drawcirclesbelowthenumbers.What goes in these imaginary circles?Howmany tomake 100? Four and 4. Picture the equation

insideyourhead.Mentallywrite4and4inthecircles.Nowtakeawaycrossways.Eitherwayyouare taking4 from96.Theresult is92.Youwouldsay,

‘Ninethousand,twohundred...’.Thisisthefirstpartoftheanswer.Nowmultiply thenumbers in thecircles:4 times4equals16.Nowyoucancomplete theanswer:

9,216.Youwouldsay,‘Ninethousand,twohundred...andsixteen.’Thiswillbecomeveryeasywithpractice.Tryitoutonyourfriends.Offer toracethemandlet themuseacalculator.Evenifyouaren’tfast

enoughtobeatthemyouwillstillearnareputationforbeingabrain.

BeatingthecalculatorTobeatyourfriendswhentheyareusingacalculator,youonlyhavetostartcallingtheanswerbeforethey finish pushing the buttons. For instance, if you were calculating 96 times 96, you would ask

yourself howmany tomake 100,which is 4, and then take 4 from96 to get 92.You can then startsaying, ‘Nine thousand, twohundred . . .’.Whileyouaresaying the firstpartof theansweryoucanmultiply4times4inyourhead,soyoucancontinuewithoutapause,‘...andsixteen.’Youhavesuddenlybecomeamathsgenius!

TestyourselfHerearesomemoreproblemsforyoutodobyyourself.a)96×96=b)97×95=c)95×95=d)98×95=e)98×94=f)97×94=g)98×92=h)97×93=Theanswersare:a)9,216b)9,215c)9,025d)9,310e)9,212f)9,118g)9,016h)9,021

Didyougetthemallright?Ifyoumadeamistake,gobackandfindwhereyouwentwrongandtryagain.Becausethemethodissodifferent,itisnotuncommontomakemistakesatfirst.Areyouimpressed?Now,dothelastexerciseagain,butthistime,doallofthecalculationsinyourhead.Youwillfindit

mucheasierthanyouimagine.Youneedtodoatleastthreeorfourcalculationsinyourheadbeforeitreallybecomeseasy.So,tryitafewtimesbeforeyougiveupandsayitistoodifficult.Ishowedthismethodtoaboyinfirstgradeandhewenthomeandshowedhisdadwhathecoulddo.

Hemultiplied96times98inhishead.Hisdadhadtogethiscalculatorouttocheckifhewasright!Keepreading,andinthenextchaptersyouwilllearnhowtousethespeedmathsmethodtomultiply

anynumbers.

Chapter2

USINGAREFERENCENUMBER

Inthischapterwearegoingtolookatasmallchangetothemethodthatwillmakeiteasytomultiplyanynumbers.

ReferenceNumbersLet’sgobackto7times8:

The10attheleftoftheproblemisourreferencenumber.Itisthenumberwesubtractthenumberswearemultiplyingfrom.Thereferencenumberiswrittentotheleftoftheproblem.Wethenaskourselves,isthenumberwe

aremultiplyingaboveorbelowthereferencenumber?Inthiscase,bothnumbersarebelow,soweputthecirclesbelow thenumbers.Howmanybelow10are they?Threeand2.Wewrite3and2 in thecircles.Sevenis10minus3,soweputaminussigninfrontofthe3.Eightis10minus2,soweputaminussigninfrontofthe2.

Wenowtakeawaycrossways:7minus2or8minus3is5.Wewrite5aftertheequalssign.

Now,hereisthepartthatisdifferent.Wemultiplythe5bythereferencenumber,10.Fivetimes10is50,sowritea0afterthe5.(Howdowemultiplyby10?Simplyputa0attheendofthenumber.)Fiftyisoursubtotal.Hereishowourcalculationlooksnow:

Nowmultiply thenumbers in thecircles. three times2 is6.Addthis to thesubtotalof50for thefinalanswerof56.thefullworkingoutlookslikethis:

Whyuseareferencenumber?WhynotusethemethodweusedinChapter1?Wasn’tthateasier?Thatmethodused10and100asreferencenumbersaswell,wejustdidn’twritethemdown.Usingareferencenumberallowsustocalculateproblemssuchas6×7,6×6,4×7and4×8.Let’sseewhathappenswhenwetry6×7usingthemethodfromChapter1.Wedraw thecirclesbelow thenumbersandsubtract thenumberswearemultiplying from10.We

write4and3inthecircles.Ourproblemlookslikethis:

Nowwesubtractcrossways:3from6or4from7is3.Wewrite3aftertheequalssign.

Fourtimes3is12,sowewrite12afterthe3forananswerof312.

Isthisthecorrectanswer?No,obviouslyitisn’t.Let’sdothecalculationagain,thistimeusingthereferencenumber.

That’smorelikeit.Youshouldsetoutthecalculationsasshownaboveuntilthemethodisfamiliartoyou,thenyoucan

simplyusethereferencenumberinyourhead.

TestyourselfTrytheseproblemsusingareferencenumberof10:a)6×7=b)7×5=c)8×5=d)8×4=e)3×8=f)6×5=Theanswersare:a)42b)35c)40d)32e)24f)30

Using100asareferencenumberWhatwasourreferencenumberfor96×97inChapter1?Onehundred,becauseweaskedhowmanymoredoweneedtomake100.Theproblemworkedoutinfullwouldlooklikethis:

The technique I explained for doing the calculations in your head actually makes you use thismethod.Let’smultiply98by98andyouwillseewhatImean.

Ifyou take98and98from100yougetanswersof2and2.Then take2from98,whichgivesananswerof96. Ifyouwere saying theansweraloud,youwouldnot say, ‘Ninety-six’,youwould say,‘Nine thousand, six hundred and . . .’.Nine thousand, six hundred is the answer you getwhen youmultiply96bythereferencenumber,100.Nowmultiply the numbers in the circles: 2 times 2 is 4.You can now say the full answer: ‘Nine

thousand,sixhundredandfour.’Withoutusingthereferencenumberwemighthavejustwrittenthe4after96.Hereishowthecalculationlookswritteninfull:

TestyourselfDotheseproblemsinyourhead:a)96×96=b)97×97=c)99×99=d)95×95=e)98×97=Youranswersshouldbe:a)9,216b)9,409c)9,801d)9,025e)9,506

DoubleMultiplicationWhathappensifyoudon’tknowyourtablesverywell?Howwouldyoumultiply92times94?Aswehaveseen,youwoulddrawthecirclesbelowthenumbersandwrite8and6inthecircles.Butifyoudon’tknowtheanswerto8times6youstillhaveaproblem.Youcangetaroundthisbycombiningthemethods.Let’stryit.Wewritetheproblemanddrawthecircles:

Wewrite8and6inthecircles.

Wesubtract(takeaway)crossways:either92minus6or94minus8.Iwouldchoose94minus8becauseitiseasytosubtract8.Theeasywaytotake8fromanumberis

totake10andthenadd2.Ninety-fourminus10is84,plus2is86.Wewrite86aftertheequalssign.

Nowmultiply86bythereferencenumber,100,toget8,600.Thenwemustmultiplythenumbersinthecircles:8times6.If we don’t know the answer, we can draw two more circles below 8 and 6 and make another

calculation.Wesubtractthe8and6from10,givingus2and4.Wewrite2inthecirclebelowthe8,and4inthecirclebelowthe6.Thecalculationnowlookslikethis:

Wenowneedtocalculate8times6,usingourusualmethodofsubtractingdiagonally.Twofrom6is4,whichbecomesthefirstdigitofthispartofouranswer.Wethenmultiplythenumbersinthecircles.Thisis2times4,whichis8,thefinaldigit.Thisgives

us48.Itiseasytoadd8,600and48.8,600+48=8,648

Hereisthecalculationinfull.

Youcanalsousethenumbersinthebottomcirclestohelpyoursubtraction.Theeasywaytotake8from94istotake10from94,whichis84,andaddthe2inthecircletoget86.Oryoucouldtake6from92.Todothis,take10from92,whichis82,andaddthe4inthecircletoget86.Withalittlepractice,youcandothesecalculationsentirelyinyourhead.

NotetoparentsandteachersPeopleoftenaskme,‘Don’tyoubelieveinteachingmultiplicationtablestochildren?’Myansweris,‘Yes,certainlyIdo.Thismethodistheeasiestwaytoteachthetables.Itisthefastestway,themostpainlesswayandthemostpleasantwaytolearntables.’Andwhiletheyarelearningtheirtables,theyarealsolearningbasicnumberfacts,practisingadditionandsubtraction,memorisingcombinations of numbers that add to 10,workingwith positive and negative numbers, and learning awhole approach to basicmathematics.

Chapter3

NUMBERSABOVETHEREFERENCENUMBER

Whatifyouwanttomultiplynumbersabovethereferencenumber;above10or100?Doesthemethodstillwork?Let’sfindout.

MultiplyingNumbersinTheTeensHereishowwemultiplynumbersintheteens.Wewilluse13×15asanexampleanduse10asourreferencenumber.

Both 13 and 15 are above the reference number, 10, so we draw the circles above the numbers,insteadofbelowaswehavebeendoing.Howmuchabove10arethey?Threeand5,sowewrite3and5inthecirclesabove13and15.Thirteenis10plus3,sowewriteaplussigninfrontofthe3;15is10plus5,sowewriteaplussigninfrontofthe5.

Asbefore,wenowgocrossways.Thirteenplus5or15plus3is18.Wewrite18aftertheequalssign.

Wethenmultiplythe18bythereferencenumber,10,andget180.(Tomultiplyanumberby10weadda0 to theendof thenumber.)Onehundredandeighty isoursubtotal, sowewrite180after theequalssign.

Forthelaststep,wemultiplythenumbersinthecircles.Threetimes5equals15.Add15to180andwegetouranswerof195.Thisishowwewritetheprobleminfull:

If the numberwe aremultiplying isabove the reference numberwe put the circleabove. If thenumberisbelowthereferencenumberweputthecirclebelow.If the circled number is above we add diagonally. If the circled number is below we subtractdiagonally.Thenumbersinthecirclesaboveareplusnumbersandthenumbersinthecirclesbelowareminusnumbers.

Let’stryanotherone.Howabout12×17?Thenumbersareabove10sowedrawthecirclesabove.Howmuchabove10?Twoand7,sowe

write2and7inthecircles.

What do we do now? Because the circles are above, the numbers are plus numbers so we addcrossways.Wecaneitherdo12plus7or17plus2.Let’sdo17plus2.

17+2=19Wenowmultiply19by10(ourreferencenumber)toget190(wejustputa0afterthe19).Ourwork

nowlookslikethis:

Nowwemultiplythenumbersinthecircles.2×7=14

Add14to190andwehaveouranswer.Fourteenis10plus4.Wecanaddthe10first(190+10=200),thenthe4,toget204.Hereisthefinishedproblem:

TestyourselfNowtrytheseproblemsbyyourself.a)12×15=b)13×14=c)12×12=d)13×13=e)12×14=f)12×16=g)14×14=h)15×15=i)12×18=j)16×14=Theanswersare:a)180b)182c)144d)169e)168f)192g)196h)225i)216

j)224

Ifanyofyouranswerswerewrong,readthroughthissectionagain,findyourmistake,thentryagain.Howwouldyousolve13×21?Let’stryit:

We still use a reference number of 10. Both numbers are above 10 so we put the circles above.Thirteenis3above10,21is11above,sowewrite3and11inthecircles.Twenty-oneplus3is24,times10is240.Threetimes11is33,addedto240makes273.Thisishow

thecompletedproblemlooks:

MultiplyingNumbersAbove100Wecanuseourspeedmathsmethodtomultiplynumbersabove100aswell.Let’stry113times102.Weuse100asourreferencenumber.

Addcrossways:113+2=115

Multiplybythereferencenumber:115×100=11,500

Nowmultiplythenumbersinthecircles:2×13=26

Thisishowthecompletedproblemlooks:

SolvingProblemsinYourHeadWhenyouusethesestrategies,whatyousayinsideyourheadisveryimportant,andcanhelpyousolveproblemsmorequicklyandeasily.Let’strymultiplying16by16.ThisishowIwouldsolvethisprobleminmyhead:16plus6(fromthesecond16)equals22,times10equals2206times6is36220plus30is250,plus6is256

Tryit.Seehowyougo.

Insideyourheadyouwouldsay:16plus6...22...220...36...256

Withpractice,youcanleaveoutalotofthat.Youdon’thavetogothroughitstepbystep.Youwouldonlysaytoyourself:

220...256Practisedoingthis.Sayingtherightthinginyourheadasyoudothecalculationcanbetterthanhalve

thetimeittakes.Howwouldyoucalculate7×8inyourhead?Youwould‘see’3and2belowthe7and8.Youwould

take2fromthe7(or3fromthe8)andsay,‘Fifty’,multiplyingby10inthesamestep.Threetimes2is6.Allyouwouldsayis,‘Fifty...six.’Whatabout6×7?Youwould‘see’4and3belowthe6and7.Sixminus3is3;yousay,‘Thirty’.Fourtimes3is12,

plus30is42.Youwouldjustsay,‘Thirty...forty-two.’It’snotashardasitsounds,isit?Anditwillbecomeeasierthemoreyoudo.

DoubleMultiplicationLet’smultiply88by84.Weuse100asourreferencenumber.Bothnumbersarebelow100sowedrawthecirclesbelow.Howmanybelowarethey?Twelveand16.Wewrite12and16inthecircles.Nowsubtractcrossways:84minus12is72.(Subtract10,then2,tosubtract12.)Multiplytheanswerof72bythereferencenumber,100,toget7,200.Thecalculationsofarlookslikethis:

Wenowmultiply12times16tofinishthecalculation.

Thiscalculationcanbedonementally.Nowaddthisanswertooursubtotalof7,200.Ifyouweredoing thecalculation inyourheadyouwould simplyadd100 first, then92, like this:

7,200plus100is7,300,plus92is7,392.Simple.Youshouldeasilydothisinyourheadwithjustalittlepractice.

TestyourselfTrytheseproblems:a)87×86=b)88×88=c)88×87=d)88×85=Theanswersare:a)7,482b)7,744c)7,656

d)7,480

Combiningthemethodstaughtinthisbookcreatesendlesspossibilities.Experimentforyourself.

NotetoparentsandteachersThischapter introduces theconceptofpositiveandnegativenumbers.Wewill simply refer to themasplusandminusnumbersthroughoutthebook.Thesemethodsmakepositiveandnegativenumberstangible.Childrencaneasilyrelatetotheconceptbecauseitismadevisual.Calculatingnumbersintheeightiesusingdoublemultiplicationdevelopsconcentration.Ifindmostchildrencandothecalculationsmuchmoreeasilythanmostadultsthinktheyshouldbeableto.Kidsloveshowingoff.Givethemtheopportunity.

Chapter4

MULTIPLYINGABOVE&BELOWTHEREFERENCENUMBER

Untilnow,wehavemultipliednumbersthatwerebothbelowthereferencenumberorbothabovethereferencenumber.Howdowemultiplynumberswhenonenumberisabovethereferencenumberandtheotherisbelowthereferencenumber?

NumbersAboveandBelowWewillseehowthisworksbymultiplying97×125.Wewilluse100asourreferencenumber:

Ninety-seven is below the reference number, 100, sowe put the circle below.Howmuch below?Three,sowewrite3 in thecircle.Onehundredand twenty-five isabovesoweput thecircleabove.Howmuchabove?Twenty-five,sowewrite25inthecircleabove.

Onehundredandtwenty-fiveis100plus25soweputaplussigninfrontofthe25.Ninety-sevenis100minus3soweputaminussigninfrontofthe3.Wenowcalculatecrossways.Either97plus25or125minus3.Onehundredandtwenty-fiveminus3

is122.Wewrite122after theequals sign.Wenowmultiply122by the referencenumber,100.Onehundredandtwenty-twotimes100is12,200.(Tomultiplyanynumberby100,wesimplyputtwozerosafterthenumber.)Thisissimilartowhatwehavedoneinearlierchapters.Thisishowtheproblemlookssofar:

Nowwemultiplythenumbersinthecircles.Threetimes25is75,butthatisnotreallytheproblem.Wehavetomultiply25byminus3.Theansweris−75.Nowourproblemlookslikethis:

AshortcutforsubtractionLet’stakeabreakfromthisproblemforamomenttohavealookatashortcutforthesubtractionswearedoing.Whatistheeasiestwaytosubtract75?Letmeaskanotherquestion.Whatistheeasiestwaytotake9

from63inyourhead?63−9=

Iamsureyougottherightanswer,buthowdidyougetit?Somewouldtake3from63toget60,thentakeanother6tomakeupthe9theyhavetotakeaway,andget54.Somewouldtakeaway10from63andget53.Thentheywouldadd1backbecausetheytookaway1

toomany.Thiswouldalsogive54.Somewoulddotheproblemthesamewaytheywouldwhenusingpencilandpaper.Thiswaythey

havetocarryandborrowintheirheads.Thisisprobablythemostdifficultwaytosolvetheproblem.Remember,theeasiestwaytosolveaproblemisalsothefastest,withtheleastchanceofmakingamistake.

Most people find the easiestway to subtract 9 is to take away 10, then add 1 to the answer. Theeasiestwaytosubtract8istotakeaway10,thenadd2totheanswer.Theeasiestwaytosubtract7istotakeaway10,thenadd3totheanswer.Whatistheeasiestwaytotake90fromanumber?Take100andgiveback10.Whatistheeasiestwaytotake80fromanumber?Take100andgiveback20.Whatistheeasiestwaytotake70fromanumber?Take100andgiveback30.Ifwegoback to theproblemwewereworkingon,howdowe take75from12,200?Wecan take

away100andgiveback25.Isthiseasy?Let’stryit.Twelvethousand,twohundredminus100?Twelvethousand,onehundred.Plus25?Twelvethousand,onehundredandtwenty-five.Sobacktoourexample.Thisishowthecompletedproblemlooks:

Withalittlepracticeyoushouldbeabletosolvetheseproblemsentirelyinyourhead.Practisewiththeproblemsbelow.

TestyourselfTrythese:a)98×145=b)98×125=c)95×120=d)96×125=e)98×146=f)9×15=g)8×12=h)7×12=Howdidyougo?Theanswersare:a)14,210b)12,250c)11,400d)12,000e)14,308f)135g)96h)84

MultiplyingnumbersinthecirclesTheruleformultiplyingthenumbersinthecirclesis:

Whenboth circles are above the numbers or both circles are below the numbers, we add theanswer.Whenonecircleisaboveandonecircleisbelowwesubtract.

Mathematically, we would say: when wemultiply two positive (plus) numbers we get a positive(plus)answer.Whenwemultiplytwonegative(minus)numberswegetapositive(plus)answer.Whenwemultiplyapositive(plus)byanegative(minus)wegetaminusanswer.Let’stryanotherproblem.Wouldourmethodworkformultiplying8×42?Let’stryit.Wechooseareferencenumberof10.Eightis2below10and42is32above10.

Weeithertake2from42oradd32to8.Twofrom42is40,timesthereferencenumber,10,is400.Minus2times32is−64.Totake64from400wetake100,whichequals300,thengiveback36forafinal answer of 336. (Wewill look at an easyway to subtract numbers from 100 in the chapter onsubtraction.)Ourcompletedproblemlookslikethis:

Wehaven’tfinishedwithmultiplicationyet,butwecantakearesthereandpractisewhatwehavealreadylearnt.Ifsomeproblemsdon’tseemtoworkouteasily,don’tworry;westillhavemoretocover.Inthenextchapterwewillhavealookatasimplemethodforcheckinganswers.

Chapter5

CHECKINGYOURANSWERS

Whatwoulditbelikeifyoualwaysfoundtherightanswertoeverymathsproblem?Imaginescoring100%foreverymathstest.Howwouldyouliketogetareputationfornevermakingamistake?Ifyoudomakeamistake,Icanteachyouhowtofindandcorrectitbeforeanyone(includingyourteacher)knowsanythingaboutit.WhenIwasyoung,Ioftenmademistakesinmycalculations.Iknewhowtodotheproblems,butI

stillgot thewronganswer.Iwouldforget tocarryanumber,orfindtherightanswerbutwritedownsomethingdifferent,andwhoknowswhatothermistakesIwouldmake.IhadsomesimplemethodsforcheckinganswersIhaddevisedmyselfbuttheyweren’tverygood.

TheywouldconfirmmaybethelastdigitoftheanswerortheywouldshowmetheanswerIgotwasatleast close to the real answer. I wish I had known then the method I am going to show you now.EveryonewouldhavethoughtIwasageniusifIhadknownthis.Mathematicianshaveknownthismethodofcheckinganswersforabout1,000years,althoughIhave

madeasmallchangeIhaven’tseenanywhereelse.Itiscalledthedigitsummethod.Ihavetaughtthismethodofcheckinganswers inmyotherbooks,but this time Iamgoing to teach itdifferently.Thismethodofcheckingyouranswerswillworkforalmostanycalculation.BecauseIstillmakemistakesoccasionally,Ialwayscheckmyanswers.HereisthemethodIuse.

SubstituteNumbersTo check the answer to a calculation,we use substitute numbers instead of the original numberswewereworkingwith.Asubstituteonafootballteamorabasketballteamissomebodywhotakesanotherperson’splaceontheteam.Ifsomebodygetsinjured,ortired,theytakethatpersonoffandbringonasubstituteplayer.Asubstituteteacherfillsinwhenyourregularteacherisunabletoteachyou.Wecanusesubstitutenumbersinplaceoftheoriginalnumberstocheckourwork.Thesubstitutenumbersarealwayslowandeasytoworkwith.Letmeshowyouhowitworks.Letussaywehavejustcalculated12×14andcometoananswerof

168.Wewanttocheckthisanswer.12×14=168

Thefirstnumberinourproblemis12.Weadditsdigitstogethertogetthesubstitute:1+2=3

Three is our substitute for 12. Iwrite 3 in pencil either above or below the 12,wherever there isroom.Thenextnumberweareworkingwithis14.Weadditsdigits:1+4=5

Fiveisoursubstitutefor14.Wenowdothesamecalculation(multiplication)usingthesubstitutenumbersinsteadoftheoriginal

numbers:3×5=15

Fifteenisatwo-digitnumbersoweadditsdigitstogethertogetourcheckanswer:

1+5=6Sixisourcheckanswer.Weaddthedigitsoftheoriginalanswer,168:1+6+8=15

Fifteenisatwo-digitnumbersoweadditsdigitstogethertogetaone-digitanswer:1+5=6

Sixisoursubstituteanswer.Thisisthesameasourcheckanswer,soouroriginalansweriscorrect.Had we got an answer that added to, say, 2 or 5, we would knowwe hadmade a mistake. The

substitute answermust be the same as the check answer if the substitute is correct. If our substituteanswerisdifferentweknowwehavetogobackandcheckourworktofindthemistake.IwritethesubstitutenumbersinpencilsoIcanerasethemwhenIhavemadethecheck.Iwritethe

substitutenumberseitheraboveorbelowtheoriginalnumbers,whereverIhaveroom.Theexamplewehavejustdonewouldlooklikethis:

Ifwehavetherightanswerinourcalculation,thedigitsintheoriginalanswershouldadduptothesameasthedigitsinourcheckanswer.

Let’stryitagain,thistimeusing14×14:14×14=1961+4=5(substitutefor14)1+4=5(substitutefor14again)

Sooursubstitutenumbersare5and5.Ournextstepistomultiplythese:5×5=25

Twenty-fiveisatwo-digitnumbersoweadditsdigits:2+5=7

Sevenisourcheckanswer.Now,tofindoutifwehavethecorrectanswer,weaddthedigitsinouroriginalanswer,196:1+9+6=16

Tobring16toaone-digitnumber:1+6=7

Seveniswhatwegotforourcheckanswersowecanbeconfidentwedidn’tmakeamistake.

Ashortcut

Thereisanothershortcuttothisprocedure.Ifwefinda9anywhereinthecalculation,wecrossitout.This is called casting out nines. You can see with this example how this removes a step from ourcalculationswithoutaffectingtheresult.Withthelastanswer,196,insteadofadding1+9+6,whichequals16,andthenadding1+6,whichequals7,wecouldcrossoutthe9andjustadd1and6,whichalsoequals7.Thismakesnodifference to theanswer,but itsavessometimeandeffort,andIaminfavourofanythingthatsavestimeandeffort.Whatabouttheanswertothefirstproblemwesolved,168?Canweusethisshortcut?Thereisn’ta9

in168.Weadded1+6+8toget15,thenadded1+5togetourfinalcheckanswerof6.In168,wehave

twodigitsthataddupto9,the1andthe8.Crossthemoutandyoujusthavethe6left.Nomoreworktodoatall,soourshortcutworks.

CheckanysizenumberWhatmakesthismethodsoeasytouseis thatitchangesanysizenumberintoasingle-digitnumber.Youcancheckcalculationsthataretoobigtogointoyourcalculatorbycastingoutnines.For instance, ifwewanted to check12,345,678×89,045=1, 099,320,897,510,wewouldhave a

problembecausemostcalculatorscan’thandlethenumberofdigitsintheanswer,somostwouldshowthefirstdigitsoftheanswerwithanerrorsign.Theeasywaytochecktheansweristocastoutthenines.Let’stryit.

Allofthedigitsintheanswercancel.Theninesautomaticallycancel,thenwehave1+8,2+7,then3+5+1=9,whichcancelsagain.And0×8=0,soouranswerseemstobecorrect.Let’stryitagain.137×456=62,472

Tofindoursubstitutefor137:1+3+7=111+1=2

Therewerenoshortcutswiththefirstnumber.Twoisoursubstitutefor137.Tofindoursubstitutefor456:4+5+6=

Weimmediatelyseethat4+5=9,sowecrossoutthe4andthe5.Thatjustleavesuswith6,oursubstitutefor456.Canwefindanynines,ordigitsaddingupto9,intheanswer?Yes,7+2=9,sowecrossoutthe7

andthe2.Weaddtheotherdigits:6+2+4=121+2=3

Threeisoursubstituteanswer.Iwritethesubstitutenumbersinpencilaboveorbelowtheactualnumbersintheproblem.Itmight

looklikethis:

Is62,472therightanswer?Wemultiplythesubstitutenumbers:2times6equals12.Thedigitsin12addupto3(1+2=3).This

isthesameasoursubstituteanswersowewererightagain.Let’stryonemoreexample.Let’scheckifthisansweriscorrect:456×831=368,936

Wewriteinoursubstitutenumbers:

Thatwaseasybecausewecastout(orcrossedout)4and5fromthefirstnumber,leaving6.Wecastout8and1fromthesecondnumber, leaving3.Andalmosteverydigitwascastoutoftheanswer,3plus6twice,anda9,leavingasubstituteanswerof8.Wenowseeifthesubstitutesworkoutcorrectly:6times3is18,whichaddsupto9,whichalsogets

castout,leaving0.Butoursubstituteansweris8,sowehavemadeamistakesomewhere.Whenwecalculateitagain,weget378,936.Didwegetitrightthistime?The936cancelsout,soweadd3+7+8,whichequals18,and1+8

addsupto9,whichcancels,leaving0.Thisisthesameasourcheckanswer,sothistimewehaveitright.Doesthismethodprovewehavetherightanswer?No,butwecanbealmostcertain.Thismethodwon’t findallmistakes. For instance, saywe had 3,789,360 for our last answer; by

mistakewe put a 0 on the end. The final 0 wouldn’t affect our check by casting out nines andwewouldn’tknowwehadmadeamistake.When it showedwehadmadeamistake, though, the checkdefinitelyprovedwehadthewronganswer.Itisasimple,fastcheckthatwillfindmostmistakes,andshouldgetyou100%scoresinmostofyourmathstests.Doyouget the idea?Ifyouareunsureaboutusingthismethodtocheckyouranswers,wewillbe

using the method throughout the book so you will soon become familiar with it. Try it on yourcalculationsatschoolandathome.

Whydoesthemethodwork?Youwillbemuchmoresuccessfulusinganewmethodwhenyouknownotonlythatitdoeswork,butyouunderstandwhyitworksaswell.Firstly,10is1times9with1remainder.Twentyis2nineswith2remainder.Twenty-twowouldbe2

nineswith2remainderforthe20plus2morefortheunitsdigit.Ifyouhave35¢inyourpocketandyouwanttobuyasmanylolliesasyoucanfor9¢each,each10¢

willbuyyouonelollywith1¢change.So,30¢willbuyyouthreelollieswith3¢change,plustheextra5¢ in your pocket gives you 8¢. So, the number of tens plus the units digit gives you the ninesremainder.Secondly,thinkofanumberandmultiplyitby9.Whatis4×9?Theansweris36.Addthedigitsin

theanswertogether,3+6,andyouget9.Let’stryanothernumber.Threeninesare27.Addthedigitsoftheanswertogether,2+7,andyouget

9again.Eleven nines are 99. Nine plus 9 equals 18.Wrong answer?No, not yet. Eighteen is a two-digit

numbersoweadditsdigitstogether:1+8.Again,theansweris9.Ifyoumultiplyanynumberby9,thesumofthedigitsintheanswerwillalwaysaddupto9ifyou

keepaddingthedigitsuntilyougetaone-digitnumber.Thisisaneasywaytotellifanumberisevenlydivisible by9. If the digits of anynumber addup to 9, or amultiple of 9, then the number itself isevenlydivisibleby9.Ifthedigitsofanumberadduptoanynumberotherthan9,thisothernumberistheremainderyou

wouldgetafterdividingthenumberby9.

Let’stry13:1+3=4

Four is the digit sum of 13. It should be the remainder youwould get if you divided by 9.Ninedividesinto13once,with4remainder.Ifyouadd3 to thenumber,youadd3 to theremainder. Ifyoudouble thenumber,youdouble the

remainder.Ifyouhalvethenumberyouhalvetheremainder.Don’tbelieveme?Halfof13is6.5.Sixplus5equals11.Oneplus1equals2.Twoishalfof4,the

ninesremainderfor13.Whatever you do to the number, you do to the remainder, so we can use the remainders assubstitutes.

Whydoweuse9remainders?Couldn’tweusetheremaindersafterdividingby,say,17?Certainly,but there is somuchwork involved in dividing by 17, the checkwould be harder than the originalproblem.Wechoose9becauseoftheeasyshortcutmethodforfindingtheremainder.

Chapter6

MULTIPLICATIONUSINGANYREFERENCENUMBER

InChapters1to4youlearnthowtomultiplynumbersusinganeasymethodthatmakesmultiplicationfun.Itiseasytousewhenthenumbersarenear10or100.Butwhataboutmultiplyingnumbersthatarearound30or60?Canwestillusethismethod?Wecertainlycan.Wechosereferencenumbersof10and100becauseitiseasytomultiplyby10and100.Themethod

will work just as well with other reference numbers, but wemust choose numbers that are easy tomultiplyby.

MultiplicationbyfactorsIt is easy tomultiply by 20, because 20 is 2 times 10. It is easy tomultiply by 10 and it is easy tomultiplyby2.Thisiscalledmultiplicationbyfactors,because10and2arefactorsof20(20=10×2).So, tomultiplyanynumberby20,youmultiply itby2and thenmultiply theanswerby10,or,youcouldsay,youdoublethenumberandadda0.Forinstance,tomultiply7by20youwoulddoubleit(2×7=14)thenmultiplyyouranswerbyl0

(14×10=140).Tomultiply32by20youwoulddouble32(64)thenmultiplyby10(640).Multiplyingby20iseasybecauseitiseasytomultiplyby2anditiseasytomultiplyby10.So,itiseasytouse20asareferencenumber.Letustryanexample:23×21=

Twenty-three and 21 are just above 20, sowe use 20 as our reference number.Both numbers areabove20soweputthecirclesabove.Howmuchabovearethey?Threeand1.Wewritethosenumbersaboveinthecircles.Becausethecirclesareabovetheyareplusnumbers.

Weadddiagonally:23+1=24

Wemultiplytheanswer,24,bythereferencenumber,20.Todothiswemultiplyby2,thenby10:24×2=4848×10=480

Wecouldnowdrawalinethroughthe24toshowwehavefinishedusingit.Therestisthesameasbefore.Wemultiplythenumbersinthecircles:3×1=3480+3=483

Theproblemnowlookslikethis:

CheckinganswersLetusapplywhatwelearntinthelastchapterandcheckouranswer:

Thesubstitutenumbersfor23and21are5and3.5×3=151+5=6

Sixisourcheckanswer.Thedigitsinouroriginalanswer,483,addupto6:4+8+3=151+5=6

Thisisthesameasourcheckanswer,sowewereright.Let’stryanother:23×31=

Weput3and11above23and31:

Theyare3and11abovethereferencenumber,20.Addingdiagonally,weget34:31+3=34

or23+11=34

Wemultiplythisanswerbythereferencenumber,20.Todothis,wemultiply34by2,thenmultiplyby10.

34×2=6868×10=680

Thisisoursubtotal.Wenowmultiplythenumbersinthecircles(3and11):3×11=33

Addthisto680:680+33=713

Thecalculationwilllooklikethis:

Wecheckbycastingoutthenines:

Multiplyoursubstitutenumbersandthenaddthedigitsintheanswer:5×4=202+0=2

Thischeckswithoursubstituteanswersowecanacceptthatascorrect.

TestyourselfHerearesomeproblemstotry.Whenyouhavefinishedthem,youcancheckyouranswersyourselfbycastingoutthenines.a)21×24=b)24×24=c)23×23=d)23×27=e)21×35=f)26×24=

Youshouldbeabletodoallofthoseproblemsinyourhead.It’snotdifficultwithalittlepractice.

Multiplyingnumbersbelow20Howaboutmultiplyingnumbersbelow20?Ifthenumbers(oroneofthenumberstobemultiplied)areinthehighteens,wecanuse20asareferencenumber.Let’stryanexample:18×17=

Using20asareferencenumberweget:

Subtractdiagonally:17−2=15

Multiplyby20:2×15=3030×10=300

Threehundredisoursubtotal.Nowwemultiplythenumbersinthecirclesandthenaddtheresulttooursubtotal:2×3=6300+6=306

Ourcompletedworkshouldlooklikethis:

Nowlet’strythesameexampleusing10asareferencenumber:

Addcrossways,thenmultiplyby10togetasubtotal:18+7=2510×25=250

Multiplythenumbersinthecirclesandaddthistothesubtotal:8×7=56250+56=306

Ourcompletedworkshouldlooklikethis:

This confirms our first answer. There isn’t much to choose between using the different referencenumbers.Itisamatterofpersonalpreference.Simplychoosethereferencenumberyoufindeasiertoworkwith.

Multiplyingnumbersaboveandbelow20Thethirdpossibilityisifonenumberisaboveandtheotherbelow20.

We can either add 18 to 14 or subtract 2 from 34, and thenmultiply the result by our referencenumber:

34−2=3232×20=640

Wenowmultiplythenumbersinthecircles:2×14=28

Itisactually−2times14soouransweris−28.640−28=612

Tosubtract28,wesubtract30andadd2.640−30=610610+2=612

Ourproblemnowlookslikethis:

Let’schecktheanswerbycastingoutthenines:

Zerotimes7is0,sotheansweriscorrect.

Using50asareferencenumberThattakescareofthenumbersuptoaround30times30.Whatifthenumbersarehigher?Thenwecanuse50asareferencenumber.Itiseasytomultiplyby50because50ishalfof100.Youcouldsay50is100dividedby2.So,tomultiplyby50,wemultiplythenumberby100andthendividetheanswerby2.Let’stryit:

Subtractdiagonally:45−3=42

Multiply42by100,thendivideby2:42×100=4,2004,200÷2=2,100

Nowmultiplythenumbersinthecircles,andaddthisresultto2,100:3×5=152,100+15=2,115

Fantastic.Thatwassoeasy.Let’stryanother:

Adddiagonally,thenmultiplytheresultbythereferencenumber(multiplyby100andthendivideby2):

57+3=6060×100=6,000

Wedivideby2toget3,000.Wethenmultiplythenumbersinthecirclesandaddtheresultto3,000:3×7=213,000+21=3,021

Let’stryonemore:

Adddiagonallyandmultiplytheresultbythereferencenumber(multiplyby100andthendivideby2):

64+1=6565×100=6,500

Thenwehalvetheanswer.Halfof6,000is3,000.Halfof500is250.Oursubtotalis3,250.Nowmultiplythenumbersinthecircles:1×14=14

Add14tothesubtotaltoget3,264.Ourproblemnowlookslikethis:

Wecouldcheckthatbycastingoutthenines:

Sixand4in64addupto10,whichaddedagaingivesus1.Sixtimes1doesgiveus6,sotheansweriscorrect.

TestyourselfHerearesomeproblemsforyoutodo.Seehowmanyyoucandoinyourhead.a)45×45=b)49×49=c)47×43=d)44×44=e)51×57=f)54×56=g)51×68=h)51×72=

Howdidyougowiththose?Youshouldhavehadnotroubledoingalloftheminyourhead.Theanswersare:a)2,025b)2,401c)2,021d)1,936e)2,907f)3,024g)3,468h)3,672

MultiplyinghighernumbersThereisnoreasonwhywecan’tuse200,500and1,000asreferencenumbers.Tomultiplyby200wemultiplyby2and100.Tomultiplyby500wemultiplyby1,000andhalvethe

answer.Tomultiplyby1,000wesimplywritethreezerosafterthenumber.Let’strysomeexamples.212×212=

Weuse200asareferencenumber.

Bothnumbersareabovethereferencenumbersowedrawthecirclesabovethenumbers.Howmuchabove?Twelveand12,sowewrite12ineachcircle.Theyareplusnumberssoweaddcrossways.212+12=224

Wemultiply224by2andby100.224×2=448448×100=44,800

Nowmultiplythenumbersinthecircles.12×12=144

(Youmustknow12times12.Ifyoudon’t,yousimplycalculatetheanswerusing10asareferencenumber.)

Let’stryanother.Howaboutmultiplying511by503?Weuse500asareferencenumber.

Addcrossways.511+3=514

Multiplyby500.(Multiplyby1,000anddivideby2.)514×1,000=514,000

Halfof514,000is257,000.(Howdidweworkthisout?Halfof500is250andhalfof14is7.)Multiplythenumbersinthecircles,andaddtheresulttooursubtotal.3×11=33257,000+33=257,033

Let’stryonemore.989×994=

Weuse1,000asthereferencenumber.

Subtractcrossways.989−6=983

or994−11=983

Multiplyouranswerby1,000.983×1,000=983,000

Multiplythenumbersinthecircles.

DoublingandhalvingnumbersTouse20and50asreferencenumbers,weneedtobeabletodoubleandhalvenumberseasily.Sometimes,likewhenyouhalveatwo-digitnumberandthetensdigitisodd,thecalculationisnotso

easy.Forexample:78÷2=

Tohalve78,youmighthalve70toget35,thenhalve8toget4,andaddtheanswers,butthereisaneasiermethod.Seventy-eightis80minus2.Halfof80is40andhalfof−2is−1.So,halfof78is40minus1,or39.

TestyourselfTrytheseforyourself:a)58÷2=b)76÷2=

c)38÷2=d)94÷2=e)54÷2=f)36÷2=g)78÷2=h)56÷2=Theanswersare:a)29b)38c)19d)47e)27f)18g)39h)28

Todouble38,thinkof40−2.Double40wouldbe80anddouble−2is−4,soweget80−4,whichis76.Again,thisisusefulwhentheunitsdigitishigh.Withdoubling,itdoesn’tmatterifthetensdigitisoddoreven.

TestyourselfNowtrythese:a)18×2=b)39×2=c)49×2=d)67×2=e)77×2=f)48×2=Theanswersare:a)36b)78c)98d)134e)154f)96

Thisstrategycaneasilybeusedtomultiplyordividelargernumbersby3or4.Forinstance:19×3=(20−1)×3=60−3=5738×4=(40−2)×4=160−8=152

NotetoparentsandteachersThisstrategyencouragesthestudenttolookatnumbersdifferently.Traditionally,wehavelookedatanumberlike38asthirty,orthreetens,pluseightones.Nowweareteachingstudentstoalsoseethenumber38as40minus2.Bothconceptsarecorrect.

Chapter7

MULTIPLYINGLOWERNUMBER

Youmayhavenoticedthatourmethodofmultiplicationdoesn’tseemtoworkwithsomenumbers.Forinstance,let’stry6×4.

Weuseareferencenumberof10.Thecirclesgobelowbecausethenumbers6and4arebelow10.Wesubtractcrossways,ordiagonally.

6−6=0or

4−4=0Wethenmultiplythenumbersinthecircles:4×6=

Thatwasouroriginalproblem.Themethoddoesn’tseemtohelp.Isthereawaytomakethemethodworkinthiscase?Thereis,butwemustuseadifferentreference

number.Theproblemisnotwiththemethodbutwiththechoiceofreferencenumber.Let’s try a reference number of 5. Five equals 10 divided by 2, or half of 10. The easy way to

multiplyby5istomultiplyby10andhalvetheanswer.

Sixishigherthan5soweputthecircleabove.Fourislowerthan5soweputthecirclebelow.Sixis1higherthan5and4is1lower,soweput1ineachcircle.Weaddorsubtractcrossways:6−1=5

or4+1=5

Wemultiply5bythereferencenumber,whichisalso5.Todothis,wemultiplyby10,whichgivesus50,andthendivideby2,whichgivesus25.Nowwe

multiplythenumbersinthecircles:1×−1=−1

Becausetheresultisanegativenumber,wesubtractitfromoursubtotalratherthanaddit:

Thisisalongandcomplicatedmethodformultiplyinglownumbers,butitshowswecanmakethe

methodworkwithalittleingenuity.Actually,thesestrategieswilldevelopyourabilitytothinklaterally,whichisveryimportantformathematicians,andalsoforsucceedinginlife.Let’strysomemore,eventhoughyouprobablyknowyourlowertimestablesquitewell:

Subtractcrossways:4−1=3

Multiplyyouranswerbythereferencenumber:3×10=30

Thirtydividedby2equals15.Nowmultiplythenumbersinthecircles:1×1=1

Addthattooursubtotal:15+1=16

TestyourselfNowtrythefollowing:a)3×4=b)3×3=c)6×6=d)3×6=e)3×7=f)4×7=Theanswersare:a)12b)9c)36d)18e)21f)28

I’msureyouhadnotroubledoingthose.Idon’treallythinkthisistheeasiestwaytolearnthosetables.Ithinkitiseasiertosimplyremember

them.Somepeoplewanttolearnhowtomultiplylownumbersjusttocheckthatthemethodwillwork.Others like to know that if they can’t remember some of their tables, there is an easy method tocalculatetheanswer.Evenifyouknowyourtablesforthesenumbers,itisstillfuntoplaywithnumbersandexperiment.

Multiplicationby5Aswehaveseen,tomultiplyby5wecanmultiplyby10andthenhalvetheanswer.Fiveishalfof10.Tomultiply6by5,wecanmultiply6by10,whichis60,andthenhalvetheanswertoget30.

Testyourself

Trythese:a)8×5=b)4×5=c)2×5=d)6×5=Theanswersare:a)40b)20c)10d)30

Thisiswhatwedowhenthetensdigitisodd.Let’stry7×5.Firstmultiplyby10:7×10=70

Ifyoufinditdifficulttohalvethe70,splititinto60+10.Halfof60is30andhalfof10is5,whichgivesus35.Let’stryanother:9×5=

Tenninesare90.Ninetysplitsto80+10.Halfof80+10is40+5,soouransweris45.

TestyourselfTrytheseforyourself:a)3×5=b)5×5=c)9×5=d)7×5=Theanswersare:a)15b)25c)45d)35

Thisisaneasywaytolearnthe5timesmultiplicationtable.Thisworksfornumbersofanysizemultipliedby5.Let’stry14×5:14×10=140140÷2=70

Let’stry23×5:23×10=230230=220+10Halfof220+10is110+5110+5=115

Thesewillbecomelightningmentalcalculationsafterjustafewminutespractice.

ExperimentingwithreferencenumbersWeusethereferencenumbers10and100becausetheyaresoeasytoapply.Itiseasytomultiplyby10(addazerototheendofthenumber)andby100(addtwozerostotheendofthenumber.)Italsomakessensebecauseyourfirststepgivesyouthebeginningoftheanswer.We have also used 20 and 50 as reference numbers. Then we used 5 as a reference number to

multiplyvery lownumbers. Infact,wecanuseanynumberasa reference.Weused5asa reference

numbertomultiply6times4becausewesawthat10wasn’tsuitable.Let’stryusing9asourreferencenumber.

Subtractcrossways:6−5=1

or4−3=1

Multiply1byourreferencenumber,9:1×9=9

Nowmultiplythenumbersinthecircles,andaddtheresultto9:3×5=1515+9=24

Thefullworkinglookslikethis:

It certainly isnotapracticalway tomultiply6 times4butyousee thatwecanmake the formulaworkifwewantto.Let’strysomemore.Let’stry8asareferencenumber.

Let’stry7asareferencenumber.

Let’stry3asareferencenumber.

Let’stry2asareferencenumber.

Tryusingthereferencenumbers9and11tomultiply7by8.Let’stry9first.

Thenumbersarebelow9sowedrawthecirclesbelow.Howmuchbelow9?Twoand1.Write2and1inthecircles.Subtractcrossways.

7−1=6Multiply6bythereferencenumber,9.6×9=54

Thisisoursubtotal.Nowmultiplythenumbersinthecirclesandaddtheresultto54.2×1=254+2=56Answer

Thisisobviouslyimpracticalbutitprovesitcanbedone.Let’stry11asareferencenumber.

Sevenis4below11and8is3below.Wewrite4and3inthecircles.Subtractcrossways.7−3=4

Multiplybythereferencenumber.11×4=44

Multiplythenumbersinthecirclesandaddtheresultto44.4×3=1244+12=56Answer

Thismethodwillalsoworkifweuse8or17asreferencenumbers.Itobviouslymakesmoresensetouse10asareferencenumberforthecalculationaswecanimmediatelycallouttheanswer,56.Still,it

isalwaysfuntoplayandexperimentwiththemethods.Trymultiplying8times8and9times9using11asareferencenumber.Thisisonlyforfun.Itstill

makessensetouse10asareferencenumber.Playandexperimentwiththemethodbyyourself.

Chapter8

MULTIPLICATIONBY11

Everyoneknowsitiseasytomultiplyby11.Ifyouwanttomultiply4by11,yousimplyrepeatthedigitfortheanswer,44.Ifyoumultiply6by11youget66.Whatanswerdoyougetifyoumultiply9by11?Easy,99.Didyouknowitisjustaseasytomultiplytwo-digitnumbersby11?

Multiplyingatwo-digitnumberby11Tomultiply 14 by 11 you simply add the digits of the number, 1 + 4 = 5, and place the answer inbetweenthedigitsfortheanswer,inthiscasegivingyou154.Let’stryanother:23×11.2+3=5Answer:253

Thesecalculationsareeasytodoinyourhead.Ifsomebodyasksyoutomultiply72by11youcouldimmediatelysay,‘Sevenhundredandninety-two.’

TestyourselfHerearesometotryforyourself.Multiplythefollowingnumbersby11inyourhead:42,53,21,25,36,62Theanswersare:462,583,231,275,396and682.

Theywereeasy,weren’tthey?Whathappensifthedigitsaddto10ormore?Itisnotdifficult—yousimplycarrythe1(tensdigit)

tothefirstdigit.Forexample,ifyouweremultiplying84by11youwouldadd8+4=12,putthe2betweenthedigitsandaddthe1tothe8toget924.

Ifyouwereaskedtomultiply84by11inyourhead,youwouldseethat8plus4ismorethan10,soyoucouldaddthe1youarecarryingfirstandsay,‘Ninehundred...’,beforeyouevenadd8plus4.Whenyouadd8and4toget12,youtakethe2forthemiddledigit,followedbytheremaining4,soyouwouldsay‘...andtwenty-...four.’Let’stryanother.Tomultiply28by11youwouldadd2and8toget10.Youaddthecarried1tothe

2andsay,‘Threehundredand...’.Twoplus8is10,sothemiddledigitis0.Thefinaldigitremains8,soyousay,‘Threehundredand...eight.’Howwouldwemultiply95by11?Nineplus5is14.Weadd1tothe9toget10.Weworkwith10thesameaswewouldwithasingle-

digitnumber.Tenisthefirstpartofouranswer,4isthemiddledigitand5remainsthefinaldigit.Wesay,‘Onethousandand...forty-five.’Wecouldalsojustsay‘Ten...forty-five.’Practisesomeproblemsforyourselfandseehowquicklyyoucancallout theanswer.Youwillbe

surprisedhowfastyoucancallthemout.Thisapproachalsoappliestonumberssuchas22and33.Twenty-twois2times11.Tomultiply17

by22,youwouldmultiply17by2toget34,thenby11usingtheshortcut,toget374.Easy.

Multiplyinglargernumbersby11Thereisaverysimplewaytomultiplyanynumberby11.Let’staketheexampleof123×11.Wewouldwritetheproblemlikethis:

Wewriteazeroinfrontofthenumberwearemultiplying.Youwillseewhyinamoment.Beginningwiththeunitsdigit,addeachdigittothedigitonitsright.Inthiscase,add3tothedigitonitsright.Thereisnodigitonitsright,soaddnothing:

3+0=3Write3asthelastdigitofyouranswer.Yourcalculationshouldlooklikethis:

Nowgotothe2.Threeisthedigitontherightofthe2:2+3=5

Write5asthenextdigitofyouranswer.Yourcalculationshouldnowlooklikethis:

Continuethesameway:1+2=30+1=1

Hereisthefinishedcalculation:

Ifwehadn’twrittenthe0infrontofthenumbertobemultiplied,wemighthaveforgottenthefinalstep.Thisisaneasymethodformultiplyingby11.Thestrategydevelopsyouradditionskillswhileyou

areusingthemethodasashortcut.Let’stryanotherproblem.Thistimewewillhavetocarrydigits.Theonlydigityoucancarryusing

thismethodis1.Let’strythisexample:471×11

Wewritetheproblemlikethis:

Weaddtheunitsdigittothedigitonitsright.Thereisnodigittotheright,so1plusnothingis1.Writedown1belowthe1.Nowaddthe7tothe1onitsright:

7+1=8Writethe8asthenextdigitoftheanswer.Yourworkingsofarshouldlooklikethis:

Thenextstepsare:4+7=11

Write1andcarry1.Addthenextdigits:0+4+1(carried)=5

Writethe5,andthecalculationiscomplete.Hereisthefinishedcalculation:

Thatwaseasy.Thehighestdigitwecancarryusingthismethodis1.Usingthestandardmethodto

multiplyby11youcancarryanynumberupto9.Thismethodiseasyandfun.

AsimplecheckHereisasimplecheckformultiplicationby11problems.Theproblemisn’tcompleteduntilwehavecheckedit.Let’scheckourfirstproblem:

WriteanXundereveryseconddigitoftheanswer,beginningfromtheright-handendofthenumber.Thecalculationwillnowlooklikethis:

AddthedigitswiththeXunderthem:1+5=6

AddthedigitswithouttheX:3+3=6

If theanswer iscorrect, theanswerswillbe thesame,haveadifferenceof11,oradifferenceofamultipleof11,suchas22,33,44or55.Bothaddedto6,soouransweriscorrect.Thisisalsoatesttoseeifanumbercanbeevenlydividedby11.Let’scheckthesecondproblem:

AddthedigitswiththeXunderthem:5+8=13

AddthedigitswithouttheX:1+1=2

Tofindthedifferencebetween13and2,wetakethesmallernumberfromthelargernumber:13−2=11

Ifthedifferenceis0,11,22,33,44,55,66,etc.,thentheansweriscorrect.Wehaveadifferenceof11,soouransweriscorrect.

NotetoparentsandteachersIoftenaskstudentsinmyclassestomakeuptheirownnumberstomultiplyby11andseehowbigadifferencetheycanget.Thelargerthenumbertheyaremultiplying,thegreaterthedifferenceislikelytobe.Letyourstudentstryforanewrecord.Childrenwillmultiplya700-digitnumberby11intheirattempttosetanewrecord.Whiletheyaretryingforanewworldrecord,theyareimprovingtheirbasicadditionskillsandcheckingtheirworkastheygo.

Multiplyingbymultiplesof11Whatarefactors?WereadaboutfactorsinChapter6.Wefoundthatitiseasytomultiplyby20because2and10arefactorsof20,anditiseasytomultiplybyboth2and10.Itiseasytomultiplyby22because22is2times11,anditiseasytomultiplybyboth2and11.Itiseasytomultiplyby33because33is3times11,anditiseasytomultiplybyboth3and11.Wheneveryouhavetomultiplyanynumberbyamultipleof11—suchas22(2×11),33(3×11)or

44 (4 × 11)— you can combine the use of factors and the shortcut for multiplication by 11. For

instance,ifyouwanttomultiply16by22,itiseasyifyousee22as2×11.Thenyoumultiply16by2toget32,andusetheshortcuttomultiply32by11.Let’strymultiplying7by33.7×3=2121×11=231Answer

Howaboutmultiplying14by33?Thirty-threeis3times11.Howdoyoumultiply3by14?Fourteenhasfactorsof2and7.Tomultiply3×14youcanmultiply3×2×7.

3×2=66×7=4242×11=462Answer

Youcanthinkoftheproblem14×33as7×2×3×11.Seventimes3×2isthesameas7×6,whichis42.Youarethenleftwith42×11.Teachingyourselftolookforfactorscanmakelifeeasierandcanturnyouintoarealmathematician.One last question for this chapter: howwould youmultiply by 55? Tomultiply by 55 you could

multiplyby11and5(5×11=55)oryoucouldmultiplyby11and10andhalvetheanswer.Alwaysremembertolookforthemethodthatiseasiestforyou.Afteryouhavepractisedthesestrategies,youwillbegintorecognisetheopportunities tousethem

foryourself.

Chapter9

MULTIPLYINGDECIMALS

Whataredecimals?Allnumbersaremadeupofdigits:0,1,2,3,4,5,6,7,8and9.Digitsarelikelettersinaword.A

word ismadeupof letters.Numbersaremadeupofdigits:23 isa two-digitnumber,madefromthedigits2and3;627isathree-digitnumbermadefromthedigits6,2and7.Thepositionofthedigitinthenumbertellsusitsvalue.Forinstance,the2inthenumber23hasavalueof2tens,andthe3hasavalue of 3 ones. Numbers in the hundreds are three-digit numbers: 435, for example. The 4 is thehundredsdigitandtellsusthereare4hundreds(400).Thetensdigitis3andsignifies3tens(30).Theunitsdigitis5andsignifies5ones,orsimply5.Whenwewriteanumber, thepositionofeachdigit is important.Thepositionofadigitgives that

digititsplacevalue.Whenwewriteprices,ornumbersrepresentingmoney,weuseadecimalpointtoseparatethedollars

from thecents.Forexample,$2.50 represents2dollarsand50hundredthsofadollar.The firstdigitafterthedecimalrepresentstenthsofadollar.(Ten10¢coinsmakeadollar.)Theseconddigitafterthedecimalrepresentshundredthsofadollar.(Onehundredcentsmakeadollar.)So$2.50,or twoandahalfdollars, is the sameas250¢. Ifwewanted tomultiply$2.50by4wecould simplymultiply the250¢by4toget1,000¢.Onethousandcentsisthesameas$10.00.Digitsafteradecimalpointhaveplacevaluesaswell.Thenumber3.14567signifiesthreeones,then

afterthedecimalpointwehaveonetenth,fourhundredths,fivethousandths,sixten-thousandths,andsoon.So$2.75equalstwodollars,sevententhsofadollarandfivehundredthsofadollar.Tomultiplyadecimalnumberby10wesimplymove thedecimalpointoneplace to the right.To

multiply1.2by10wemovethedecimaloneplacetotheright,givingananswerof12.Tomultiplyby100,wemovethedecimaltwoplacestotheright.Iftherearen’ttwodigits,wesupplythemasneededbyaddingzeros.So,tomultiply1.2by100,wemovethedecimaltwoplaces,givingananswerof120.Todivideby10,wemovethedecimaloneplacetotheleft.Todivideby100,wemovethedecimal

twoplacestotheleft.Todivide14by100weplacethedecimalafterthe14andmoveittwoplacestotheleft.Theansweris0.14.(Wenormallywritea0beforethedecimaliftherearenootherdigits.)Now,let’slookatgeneralmultiplicationofdecimals.

MultiplicationofdecimalsMultiplyingdecimalnumbersisnomorecomplicatedthanmultiplyinganyothernumbers.Letustakeanexampleof1.2×1.4.Wewritedowntheproblemasitis,butwhenweareworkingitoutweignorethedecimalpoints.

Althoughwewrite1.2×1.4,wetreattheproblemas:12×14=

Weignore thedecimalpoint in thecalculation;wecalculate12plus4 is16, times10is160.Fourtimes2is8,plus160is168.Theproblemwilllooklikethis:

Ourproblemwas1.2×1.4,butwehavecalculated12×14,soourworkisn’tfinishedyet.Wehavetoplaceadecimalpointintheanswer.Tofindwhereweputthedecimal,welookattheproblemandcount the number of digits after the decimals in the multiplication. There are two digits after thedecimals:the2in1.2andthe4in1.4.Becausetherearetwodigitsafterthedecimalsintheproblem,theremustbetwodigitsafterthedecimalintheanswer.Wecounttwoplacesfromtherightandputthedecimalbetweenthe1andthe6,leavingtwodigitsafterit.

1.68AnswerAneasywaytodouble-checkthisanswerwouldbetoapproximate.Thatmeans,insteadofusingthe

numbersweweregiven,1.2×1.4,weroundoffto1.0and1.5.Thisgivesus1.0times1.5,whichis1.5,soweknowtheanswershouldbesomewherebetween1and2.Thistellsusourdecimalisintherightplace.This isagooddouble-check.Youshouldalwaysmakethischeckwhenyouaremultiplyingordividingusingdecimals.Thecheckissimply:doestheanswermakesense?Let’stryanother.9.6×97=

Wewritetheproblemdownasitis,butworkitoutasifthenumbersare96and97.

96−3=9393×100(referencenumber)=9,3004×3=129,300+12=9,312

Wheredoweputthedecimal?Howmanydigitsfollowthedecimalintheproblem?One.That’showmanydigitsshouldfollowthedecimalintheanswer.

931.2AnswerToplacethedecimal,wecountthetotalnumberofdigitsfollowingthedecimalsinbothnumberswe

aremultiplying.Wewillhavethesamenumberofdigitsfollowingthedecimalintheanswer.Wecandouble-checktheanswerbyestimating10times90;fromthisweknowtheanswerisgoingto

besomewherenear900,not9,000or90.Iftheproblemhadbeen9.6×9.7,thentheanswerwouldhavebeen93.12.Knowingthiscanenable

us to take some shortcuts that might not be apparent otherwise. We will look at some of thesepossibilitiesshortly.Inthemeantime,trytheseproblems.

TestyourselfTrythese:a)1.2×1.2=b)1.4×1.4=c)12×0.14=d)96×0.97=e)0.96×9.6=f)5×1.5=

Howdidyougo?Theanswersare:a)1.44b)1.96c)1.68d)93.12e)9.216f)7.5

Whatifwehadtomultiply0.13×0.14?13×14=182

Wheredoweputthedecimal?Howmanydigitscomeafterthedecimalpointintheproblem?Four,the1and3inthefirstnumberandthe1and4inthesecond.Sowecountbackfourdigitsintheanswer.But,waitaminute; thereareonly threedigits in theanswer.Whatdowedo?Wehave tosupply thefourth digit. So,we count back three digits, then supply a fourth digit by putting a 0 in front of thenumber.Thenweputthedecimalpointbeforethe0,sothatwehavefourdigitsafterthedecimal.Theanswerlookslikethis:.0182

Wecan alsowrite another 0before thedecimal, because there should alwaysbe at least onedigitbeforethedecimal.Ouranswerwouldnowlooklikethis:

0.0182Let’strysomemore:0.014×1.4=14×14=196

Wheredoweputthedecimal?Therearefourdigitsafterthedecimalintheproblem;0,1and4inthefirstnumberand4inthesecond,sowemusthavefourdigitsafterthedecimalintheanswer.Becausethereareonlythreedigitsinouranswer,wesupplya0tomakethefourthdigit.Ouransweris0.0196.

TestyourselfTrytheseforyourself:a)22×2.4=b)0.48×4.8=c)0.048×0.48=d)0.0023×0.23=Easy,wasn’tit?Herearetheanswers:a)52.8b)2.304c)0.02304d)0.000529

BeatingthesystemUnderstanding this simple principle can help us solve some problems that appear difficult using ourmethodbutcanbeadaptedtomakethemeasy.Hereisanexample.

8×79=Whatreferencenumberwouldweuseforthisone?Wecoulduse10forthereferencenumberfor8,

but79 iscloser to100.Maybewecoulduse50.Thespeedsmathsmethod iseasier tousewhen thenumbersareclosetogether.So,howdowesolvetheproblem?Whynotcallthe8,8.0?

Thereisnodifferencebetween8and8.0.Thefirstnumberequals8;thesecondnumberequals8too,butitisaccuratetoonedecimalplace.Thevaluedoesn’tchange.We can use 8.0 andwork out the problem as if it were 80, aswe did above.We can now use a

referencenumberof100.Let’sseewhathappens:

Nowtheproblemiseasy.Subtractdiagonally.79−20=59

Multiply59bythereferencenumber(100)toget5,900.Multiplythenumbersinthecircles.20×21=420

(Tomultiplyby20wecanmultiplyby2andthenby10.)Addtheresulttothesubtotal.5,900+420=6,320

Thecompletedproblemwouldlooklikethis:

Now,wehavetoplacethedecimal.Howmanydigitsarethereafterthedecimalintheproblem?One,the0weprovided.Sowecountonedigitbackintheanswer.

632.0AnswerWewouldnormallywritetheansweras632.Let’scheckthisanswerusingestimation.Eightiscloseto10sowecanroundupwards.10×79=790

The answer should be less than, but close to, 790. It certainlywon’t be around 7,900 or 79.Ouranswerof632fitssowecanassumeitiscorrect.Wecandouble-checkbycastingoutnines.

Eighttimes7equals56,whichreducesto11,then2.Ouransweriscorrect.Let’stryanother.98×968=

Wewrite98as98.0andtreatitas980duringthecalculation.Ourproblemnowbecomes980×968.

Ournextstepis:968−20=948

Multiplybythereferencenumber:948×1,000=948,000

Nowmultiply32by20.Tomultiplyby20wemultiplyby2andby10.32×2=6464×10=640

Itiseasytoadd948,000and640.948,000+640=948,640

Wenowhavetoadjustouranswerbecauseweweremultiplyingby98,not980.Placingonedigitafterthedecimal(ordividingouranswerby10),weget94,864.0,whichwesimply

writeas94,864.Ourfullworkingwouldlooklikethis:

Let’scastoutninestochecktheanswer.

Fivetimes8is40,whichreducesto4.Ouranswer,94,864,alsoreducesto4,soouranswerseemstobecorrect.Wehavecheckedouranswer,butcastingoutninesdoesnot tellus if thedecimal is in thecorrect

position,solet’sdouble-checkbyestimatingtheanswer.Ninety-eight is almost 100, so we can see if we have placed the decimal point correctly by

multiplying 968by 100 to get 96,800.This has the samenumber of digits as our answer sowe canassumetheansweriscorrect.

TestyourselfTrytheseproblemsforyourself:a)9×82=b)9×79=c)9×77=d)8×75=e)7×89=Theanswersare:a)738b)711c)693d)600e)623

That was easy, wasn’t it? If you use your imagination you can use these strategies to solve anymultiplicationproblem.Trysomeyourself.

Chapter10

MULTIPLICATIONUSINGTWOREFERENCENUMBERS

Thegeneralruleforusingareferencenumberformultiplicationisthatyouchooseareferencenumberthatisclosetobothnumbersbeingmultiplied.Ifpossible,youtrytokeepbothnumberseitheraboveorbelowthereferencenumbersoyouendupwithanaddition.Whatdoyoudoifthenumbersaren’tclosetogether?Whatdoyoudoifitisimpossibletochoosea

referencenumberthatisanywhereclosetobothnumbers?Hereisanexampleofhowourmethodworksusingtworeferencenumbers:8×37=

Firstly,wechoosetworeferencenumbers.Thefirstreferencenumbershouldbeaneasynumbertouseasamultiplier,suchas10or100.Inthiscasewechoose10asourreferencenumberfor8.Thesecondreferencenumbershouldbeamultipleofthefirstreferencenumber.Thatis,itshouldbe

double the first referencenumber, or three times, four timesor even sixteen times the first referencenumber.InthiscaseIwouldchoose40asthesecondreferencenumberasitequals4times10.Wethenwritethereferencenumbersinbracketstothesideoftheproblem,withtheeasymultiplier

writtenfirstandthesecondnumberwrittenasamultipleofthefirst.So,wewouldwrite10and40as(10×4),the10beingthemainreferencenumberandthe4beingamultipleofthemainreference.Theproblemiswrittenlikethis:(10×4)8×37=

Bothnumbersarebelowtheirreferencenumberssowedrawthecirclesbelow.

Whatdowewrite in thecircles?Howmuchbelow the referencenumbersare thenumberswearemultiplying?Twoand3,sowewrite2and3inthecircles.

Here is the difference.We nowmultiply the 2 below the 8 by themultiplication factor, 4, in thebrackets.Twotimes4is8.Drawanothercirclebelowthe8(underthe2)andwrite−8init.Thecalculationnowlookslikethis:

Nowsubtract8from37.37−8=29

Write29aftertheequalssign.

Nowwemultiply29bythemainreferencenumber,10,toget290.Thenwemultiplythenumbersinthetoptwocircles(2×3)andaddtheresultto290.

Let’stryanotherone:96×289=

Wechoose100and300asourreferencenumbers.Wesetouttheproblemlikethis:(100×3)96×289=

Whatdowewriteinthecircles?Fourand11.Ninety-sixis4below100and289is11below300.

Nowmultiplythe4below96bythemultiplicationfactor,3.Thisgivesus:4×3=12.Drawanothercirclebelow96andwrite12inthiscircle.Thecalculationnowlookslikethis:

Subtract12from289.289−12=277

Multiply277bythemainreferencenumber,100.277×100=27,700

Nowmultiplythenumbersintheoriginalcircles(4and11).4×11=44

Add44to27,700toget27,744.Thiscaneasilybecalculatedinyourheadwithjustalittlepractice.Thefullcalculationlookslikethis:

Thecalculationpartoftheproblemiseasy.Theonlydifficultyyoumayhaveisrememberingwhatyouhavetodonext.Ifthenumbersareabovethereferencenumbers,wedothecalculationasfollows.Wewilltake12×

124asanexample:(10×12)12×124=

Wechoose10and120asreferencenumbers.Becausethenumberswearemultiplyingarebothabovethereferencenumberswedrawthecirclesabove.Howmuchabove?Twoand4,sowewrite2and4in

thecircles.

Nowwemultiplythenumberabovethe12(2)bythemultiplicationfactor,12.2×12=24

Drawanothercircleabovethe12andwrite24init.

Nowweaddcrossways.124×24=148

Write148aftertheequalssignandmultiplyitbyourmainreferencenumber,10.148×10=1,480

Nowmultiplythenumbersintheoriginalcircles,2×4,andaddtheanswertothesubtotal.2×4=81,480+8=1,488Answer

Thecompletedproblemlookslikethis:

I try to choose reference numbers that keep both numbers either above or below the referencenumbers,sothatIhaveanadditionratherthanasubtractionattheend.Forinstance,ifIhadtomultiply9times83,Iwouldchoose10and90asreferencenumbersratherthan10and80.Althoughbothwouldwork,Iwanttokeepthecalculationsassimpleaspossible.Let’scalculatetheanswerusingbothcombinations,soyoucanseewhathappens.Firstly,wewilluse10and90.

Thatwasstraightforward.Wemultiplied9by1andwrote9inthebottomcircle.Thenweperformedthefollowingcalculations:

83−9=74

74×10=7401×7=7740+7=747

Easy.Nowlet’suse10and80asreferencenumbers.

Therewasnotmuchdifferenceinthiscase,butifyouhavealargenumbertosubtractyoumayfinditeasiertochooseyourreferencenumberssothatyouendyourcalculationwithanaddition.Youcouldalsosolvethisproblemusing10and100asreferencenumbers.Let’stryit.

There are many ways you can use these methods to solve problems. It is not a matter of whichmethod is right but whichmethod is easiest, depending on the particular problem you are trying tosolve.

TestyourselfTrytheseproblemsbyyourself:a)9×68=b)8×79=c)94×192=d)98×168=Writethemdownandcalculatetheanswers,andthenseeifyoucandothemallinyourhead.Withpracticeitbecomeseasy.Herearetheanswers:a)612b)632c)18,048d)16,464

Easymultiplicationby9Itiseasytousethismethodtomultiplyanynumberfrom1to1,000by9.Let’stryanexample:76×9=

Weuse10asourreferencefor9and80asourreferencefor76.Wecanrewritetheproblemlikethis:

Wesubtract8 from76 (76−10=66,plus2 is68).Multiply68by thebase reference,10,whichequals680.Nowmultiplythenumbersintheoriginalcircles,andaddtheresultto680.

1×4=4680+4=684Answer

Thebasenumberisalways10whenyoumultiplyby9.Let’stryanother.9×486=

The referencenumbers are10 and490.Youalwaysgoup to thenext10 for the second referencenumber.Wewriteitlikethis:

Wesubtract49from486.Isthisdifficult?Notatall.Wesubtract50andadd1:486−50=436,plus1is437

Multiplythenumbersinthecircles.1×4=4

Thisresultisalwaysthefinaldigitintheanswerwhenyoumultiplyby9.Theansweris4,374.Thecompletedproblemlookslikethis:

Theproblemsareeasytocalculateinyourhead.

UsingfractionsasmultiplesThereisyetanotherpossibility.Themultiplecanbeexpressedasafraction.WhatdoImean?Let’stry94×345.Wecanuse100and350asreferencenumbers,whichwewould

writeas(100×3½).Ourproblemwouldlooklikethis:

Herewehavetomultiply-6by3½.Todothis,simplymultiply6by3,whichis18,thenaddhalfof6,whichis3,toget21.

Subtract21from345.Multiplytheresultby100,thenmultiplythenumbersinthecircles,andthenaddthesetworesultsforouranswer.

345−21=324324×100=32,4006×5=3032,400+30=32,430Answer

Thecompletedproblemlookslikethis:

UsingfactorsexpressedasdivisionTomultiply48×96,wecouldusereferencenumbersof50and100,expressedas(50×2)or(100÷2).Itwouldbeeasiertouse(100÷2)because100thenbecomesthemainreferencenumber.Itiseasiertomultiplyby100thanitisby50.Whenwriting themultiplication,write the number firstwhich has themain reference number, so

insteadofwriting48×96wewouldwrite96×48.Thecompletedproblemwouldlooklikethis:

Thatworkedoutwell,butwhatwouldhappenifwemultiplied97by48?Thenwehavetohalveanoddnumber.Let’ssee:

Inthiscasewehavetodividethe3belowthe97bythe2inthebrackets.Threedividedby2is or1½.Subtracting1½from48givesananswerof46½.Thenwehavetomultiply46½by100.Forty-sixby100is4,600,plushalfof100givesusanother50.So,46½times100is4,650.Thenwemultiply3by2forananswerof6;addthisto4,650forouranswerof4,656.Letuscheckthisanswerbycastingoutthenines:

Wecheckbymultiplying7by3toget21,whichreducesto3,thesameasouranswer.Ouransweriscorrect.Whatifwemultiply97by23?Wecoulduse100and25(100÷4)asreferencenumbers.

Wedivide3by4 for an answerof¾.Subtract¾ from23 (subtract1 andgiveback¼).We thenmultiplyby100.

23−¾=22¼22¼×100=2,225(25isaquarterof100)

Thecompletedproblemlookslikethis:

Youcanuseanycombinationofreferencenumbers.Thegeneralrulesare:1Makethemainreferencenumberaneasynumbertomultiplyby;forexample,10,20or50.2The second referencenumbermust be amultiple of themain referencenumber; for example,doublethemainreferencenumber,orthreetimes,tentimesorfourteentimesthemainreferencenumber.

Thereisnoendtothepossibilities.Playandexperimentwiththemethodsandyouwillfindyouareperforminglikeagenius.Eachtimeyouusethesestrategiesyoudevelopyourmathematicalskills.Wouldweusetworeferencenumberstocalculateproblemssuchas8×17or7×26?No,Ithinkthe

easywaytocalculate8×17istomultiply8by10andthen8by7.8×10=808×7=5680+56=136

Howabout7×26?Iwouldsay20times7is140.Tomultiplyby20wemultiplyby2andby10.Twotimes7is14,times10is140.Then7times6is42.Theansweris140plus42,whichis182.Experiment foryourself toseewhichmethodyoufindeasiest.Youwill findyoucaneasilydo the

calculationsentirelyinyourhead.

PlayingwithtworeferencenumbersLet’smultiply13times76.Wewouldusereferencenumbersof10and70.Wewouldn’tnormallyuse80asareferencenumberbecausewewouldhaveasubtractionattheend.Let’stryafewmethodstoseehowtheywork.Firstly,using10and70.

Wemultiplythe3abovethe13bythemultiplicationfactorof7inthebrackets.Threetimes7is21;write21inacircleabovethe3.Nowweaddcrossways.76+21=97

Multiply97bythebasereferencenumberof10toget970.Nowmultiply3times6toget18.Addthisto970toget988.Hereisthefullcalculation:

Nowlet’stryusingreferencenumbersof10and80.

Multiply3times8toget24.Write24inacircleabovethe3.Add24to76toget100.Multiplyyouranswerbythebasereferencenumberof10toget1,000.Theproblemnowlookslikethis:

Multiply 3 timesminus 4 to get minus 12. Subtract 12 from 1,000 to get your answer, 988. (Tosubtract12,subtract10,then2.)

Let’stryanotheroption;let’suse10and75asreferencenumbers.

Thirteenis3above10,and76is1above75.Wemultiplythe3abovethe13by7½.Isthatdifficult?No,wemultiply3by7toget21,plushalfof

3togetanother1½.21+1½=22½

Nowweaddcrossways.76+22½=98½

Nowwemultiply98½bythebasereferencenumberof10toget985.Thenwemultiplythenumbersinthecircles.

3×1=3Thenweadd985and3togetouranswer.985+3=988Answer

Thecompleteworkinglookslikethis:

This issomethingtoplaywithandexperimentwith.Wejustusedthesameformula threedifferentwaystogetthesameanswer.Thelastmethod(usingafractionasamultiplicationfactor)canbeusedtomakemanymultiplicationproblemseasier.Let’stry96times321.Wecoulduse100and325asreferencenumbers.

Wemultiply4by3¼toget13(4×3=12,plusaquarterof4givesanother1,making13).Write13inacirclebelowthe4,under96.

Subtractcrossways.321−13=308

Multiply308bythebasereferencenumberof100toget30,800.Thenmultiplythenumbersinthecircles.

4×4=16Then:30,800+16=30,816Answer

Thatcaneasilybedoneinyourhead,andismostimpressive.

UsingdecimalfractionsasreferencenumbersHere is another variation for using two reference numbers. The second reference number can beexpressedasadecimalfractionofthefirst.Let’stryitwith58times98.Wewillusereferencenumbersof100and60,expressedas0.6of100.

(100×0.6)98×58=Thecirclesgobelow ineachcase.Write2 insidebothcircles.Multiply2 times themultiplication

factorof0.6.Theansweris1.2.Allwedoismultiply6times2toget12,anddivideby10.Ourworklookslikethis:

Subtract1.2from58toget56.8.Multiplyby100toget5,680.Multiplythenumbersinthecircles:2times2is4.

5,680+4=5,684Answer

Hereistheproblemfullyworkedout.

Theaboveproblemcanbesolvedmoreeasilybysimplyusingasinglereferencenumberof100(tryit),butitisusefultoknowthereisanotheroption.Thenextexampleisdefinitelyeasiertosolveusingtworeferencenumbers.96×67=

Weusereferencenumbersof100and70.

Wemultiply4times0.7toget2.8(4×7=28).Wesubtract2.8from67bysubtracting3andadding0.2 to get 64.2. Then wemultiply our answer of 64.2 by 100 to get 6,420.Multiplying the circlednumbers,4times3,givesus12.

6,420+12=6,432AnswerThatwasmucheasierthanusing100asthereferencenumber.Hereisanotherproblemusing200asourbasereferencenumber.189×77=

Weuse200and80asourreferencenumbers.Eightyis0.4of200.

Wemultiply11by0.4toget4.4.(Wesimplymultiply11times4anddividetheanswerby10.)Wewrite4.4belowthe11.Wenowsubtract4.4from77.Todothiswesubtract5andadd0.6.Seventy-sevenminus5is72,plus0.6is72.6.

72.6×100=7,260Wemultiply7,260by2toget14,520.(Wecoulddouble7toget14and2times26is52,makinga

mentalcalculationeasy.)Ourworksofarlookslikethis:

Nowwemultiplythenumbersinthecircles.11×3=33

Add33to14,520toget14,553.Hereisthecalculationworkedoutinfull.

Itisinterestingandfuntotrydifferentstrategiestofindtheeasiestmethod.

TestyourselfTrytheseforyourself:a)92×147=b)88×172=c)94×68=d)96×372=Theanswersare:a)13,524b)15,136c)6,392d)35,712

Howdidyougo?Iused100×1½fora),100×1¾forb),100×0.7forc)and100×3¾ford).By now youmust find these calculations very easy. Experiment for yourself.Make up your own

problems.Try to solve themwithoutwriting anythingdown.Checkyour answers by castingout thenines.Whatyouaredoingistrulyamazing.Youarecertainlydevelopingexceptionalskills.Keepplaying

with the methods and you will find it easier with practice to do everything in your head. You aredevelopinggreatmentalandmathematicalskills.

Chapter11

ADDITION

Mostofusfindadditioneasierthansubtraction.Wewilllearninthischapterhowtomakeadditioneveneasier.Somenumbersareeasytoadd.Itiseasytoaddnumberslike1and2,10and20,100and200,1,000

and2,000.Twenty-fiveplus1is26.Twenty-fiveplus10is35.Twenty-fiveplus100is125.Twenty-fiveplus200is225.These are obviously easy, but how about adding 90? The easy way to add 90 is to add 100 and

subtract10.Forty-sixplus90equals146minus10.Whatis146minus10?Theansweris136.46+90=146−10=136

Whatistheeasywaytoadd9?Add10andsubtract1.Whatistheeasywaytoadd8?Add10andsubtract2.Whatistheeasywaytoadd7?Add10andsubtract3.Whatistheeasywaytoadd95?Add100andsubtract5.Whatistheeasywaytoadd85?Add100andsubtract15.Whatistheeasywaytoadd80?Add100andsubtract20.Whatistheeasywaytoadd48?Add50andsubtract2.

TestyourselfTrythefollowingtoseehoweasythisis.Callouttheanswersasfastasyoucan.24+10=38+10=83+10=67+10=Itiseasytoadd10toanynumber.I’msureIdon’tneedtogiveyoutheanswerstothese.

Now,trythefollowingproblemsinyourhead.Callouttheanswersasquicklyasyoucan.Trytosaytheanswerinonestep.Forexample,for34+9,youwouldn’tcallout,‘Forty-four...forty-three,’youwouldmaketheadjustmentwhileyoucallouttheanswer.Youwouldjustsay,‘Forty-three.’Trythem.Aslongasyouremembertoadd9byadding10andtaking1,add8byadding10andtaking2,and

add7byadding10andtaking3,youwillfindthemeasy.

TestyourselfTrythese:a)25+9=b)46+9=c)72+9=d)56+8=e)37+8=

f)65+7=Theanswersare:a)34b)55c)81d)64e)45f)72

Youdon’thavetoaddandsubtractthewayyouweretaughtatschool.Infact,highmathsachieversusuallyusedifferentmethods toeveryoneelse.That iswhatmakes themhighachievers—not theirsuperiorbrain.Howwouldyouadd38?Add40andsubtract2.So,howwouldyouaddthefollowing?a)23+48=b)126+39=c)47+34=d)424+28=

Fora),youwouldsay23plus50is73,minus2is71.Forb),youwouldsay126plus40is166,minus1is165.Forc),youwouldsay50plus34is84,minus3is81.Andford),youwouldsay424plus30is454,minus2is452.Didyoufindtheseeasytodoinyourhead?Whatifyouhavetoadd31toanumber?Yousimplyadd30,andthenaddthe1.Toadd42youadd

40,andthenadd2.

TestyourselfTrythese:a)26+21=b)43+32=c)64+12=d)56+41=Theanswersare:a)47b)75c)76d)97

Youmayhavethoughtthatallofthoseanswerswereobvious,butmanypeoplenevertrytocalculatethesetypesofproblemsmentally.Oftenyoucanroundnumbersofftothenexthundred.Howwouldyoudothefollowingproblemin

yourhead?

Youcouldsay2,351plus500is2,851(300+500=800),minus11is2,840.Normallyyouwoulduseapenandpapertomakethecalculation,butwhenyousee489as11less

than500,thecalculationbecomeseasytodoinyourhead.

Testyourself

Trytheseproblemsinyourhead:

a)

b)

c)Fora)yousimplyadd300andsubtract3:531+300is831,minus3is828Forb)youadd200,thenadd50,andsubtract1:333+200is533,plus50is583,minus1is582Forc)youadd400andsubtract12:4,537plus400is4,937,minus12is4,925

AddingfromlefttorightFormostadditions,ifyouareaddingmentally,youshouldaddfromlefttorightinsteadoffromrighttoleftasyouaretaughtinschool.Howwouldyouaddthesenumbersinyourhead?

Iwouldadd3,000,thensubtract100toadd2,900.ThenIwouldadd40andsubtract2toadd38.Toadd2,900youwouldsay,‘Eightthousandandsixty-four,’thenadd40andsubtract2toget‘Eight

thousand,onehundredandtwo.’Thisstrategymakesiteasytokeeptrackofyourcalculationandholdthenumbersinyourhead.

OrderofadditionLet’ssaywehavetoaddthefollowingnumbers:

Theeasywaytoaddthenumberswouldbetoadd:6+4=10,plus8=18

Mostpeoplewouldfindthateasierthan6+8+4=18.So,aneasyruleis,whenaddingacolumnofnumbers,addpairsofdigitstomaketensfirst ifyou

can,andthenaddtheotherdigits.Youcanalsoaddadigittomakeupthenextmultipleof10.Thatis,ifyouhavereached,say,27in

youraddition,andthenexttwonumberstoaddare8and3,addthe3beforethe8tomake30,andthenadd8tomake38.Usingourmethodsofmultiplicationwillhelpyoutorememberthecombinationsofnumbersthataddto10,andthisshouldbecomeautomatic.Thisalsoappliesifyouhadtosolveaproblemsuchas26+32+14.Wecanseethattheunitsdigits

from26and14(6and4)addto10,soitiseasytoadd26and14.Iwouldadd26plus10tomake36,plus4makes40,andthenaddthe32forananswerof72.Thisisaneasycalculationcomparedwithaddingthenumbersintheordertheyarewritten.

BreakdownofnumbersYoualsoneedtoknowhownumbersaremadeup.Twelveisnotonly10plus2butalso8plus4,6plus6,7plus5,9plus3and11plus1.Itisimportantthatyoucanbreakupallnumbersfrom1to10into

theirbasicparts.

Howdoyouadd8plus5?Youcouldadd5plus10andsubtract2.Oryoucouldadd8plus2tomake10,thenaddanother3(tomakeupthe5)togivethesameanswerof13.Howdoyouadd7+6?Sixis3+3.Sevenplus3is10,plusthesecond3is13.

NotetoparentsandteachersOftenchildrenare just told, ‘Yousimplyhave tomemorise theanswers to8plus5or8plus4.’Andwithpractice theywillbememorised,butgivethemastrategytocalculatetheanswerinthemeantime.

CheckingadditionbycastingoutninesJust as we cast out nines to check our answers for multiplication, so we can use the strategy forcheckingouradditionandsubtraction.Hereisanexample:

Weaddthenumberstogetourtotalof143,835.Isouranswercorrect?Let’scheckbycastingoutnines,orworkingwithoursubstitutenumbers.

Oursubstitutesare6,3,2and4.Thefirst6and3canceltoleaveuswithjust2and4toadd.2+4=6

Sixisourcheckorsubstituteanswer.Therealanswershouldaddto6.Let’ssee.Aftercastingnines,wehave:

Ouransweriscorrect.Butthereisafurthershortcutwhenyouarecastingninestocheckacalculationinaddition.Youcan

castout (crossout)anydigitsanywhere in theaddition thatadd to9.Youcan’t,ofcourse,crossoutdigitsintheanswerwithdigitsyouareadding.So,youcouldhavecastoutthe6inthefourthnumberwiththe3inthethird,the7inthethirdnumberwiththe2inthefourth.Youcouldhavecastouteverydigitinthetoptwonumbersaseachdigitinthetopnumbercombineswithadigitinthesecondtoequal

9.Yourcheckcouldlooklikethis:

Notethatthisonlyworkswithaddition.Withallothermathematicalchecksyoumustfindsubstitutesforeachnumber.Withadditionyoucancrossout(castout)anydigitsofthenumbersyouareadding.Butremember,youcan’tmixdigitsinthenumbersyouareaddingwithdigitsintheanswer.Let’slookatanotherexample.

Herethe2in234canbecrossedwiththe7in671.The3in234combineswiththe6in671.The4in234combineswitha5in855.The1in671combineswiththe8in855.Alldigitsarecastoutexcepta5in855.Thatmeanstheanswermustaddto5.Let’snowchecktheanswer:1+7+6+0=141+4=5Ourcalculationiscorrect.Thefinalchecklookslikethis:

Iftheabovefigureswereamountsofmoneywithadecimal, itwouldmakenodifference.Youcanusethismethodtocheckalmostallofyouradditions,subtractions,multiplicationsanddivisions.Andyoucanhavefundoingit.

NotetoparentsandteachersThebasicnumberfactsarequicklylearntwhenchildrenmasterthesetechniquesandlearnthemultiplicationtables.Castingoutnines tocheckanswerswillgiveplentyofpracticewith thecombinationsofnumbers thatadd to9.Theseexercisesdon’tneedtobedrilled.Theyarelearnednaturallywhenstudentsusethesestrategies.Once a student knows the basic make-up of numbers, he or she can easily add small numbers without mistake and withoutborrowing.

Chapter12

SUBTRACTION

Mostofusfindadditioneasierthansubtraction.Subtraction,thewaymostpeoplearetaughtinschool,is more difficult. It need not be so. You will learn some strategies in this chapter that will makesubtractioneasy.Firstly,youneed toknow thecombinationsofnumbers thatadd to10.You learnt thosewhenyou

learntthespeedmathsmethodofmultiplication.Youdon’thavetothinktoohard;whenyoumultiplyby8,whatnumbergoesinthecirclebelow?Youdon’thavetocalculate,youdon’thavetosubtract8from10.Youknowa2goesinthecirclefromsomuchpractice.Itisautomatic.If a class is asked to subtract 9 from 56, some students will use an easy method and give an

immediateanswer.Becausetheirmethodiseasytheywillbefastandunlikelytomakeamistake.Thestudentswhouseadifficultmethodwilltakelongertosolvetheproblemand,becausetheirmethodisdifficult,theyaremorelikelytomakeamistake.Remembermyrule:

Theeasiestwaytosolveaproblemisalsothefastest,withtheleastchanceofmakingamistake.So,whatistheeasywaytosubtract9?Subtract10andadd(giveback)1.Whatistheeasywaytosubtract8?Subtract10andadd(giveback)2.Whatistheeasywaytosubtract7?Subtract10andadd(giveback)3.Whatistheeasywaytosubtract6?Subtract10andadd(giveback)4.Whatistheeasywaytosubtract90?Subtract100andadd(giveback)10.Whatistheeasywaytosubtract80?Subtract100andadd(giveback)20.Whatistheeasywaytosubtract70?Subtract100andadd(giveback)30.Whatistheeasywaytosubtract95?Subtract100andadd(giveback)5.Whatistheeasywaytosubtract85?Subtract100andadd(giveback)15.Whatistheeasywaytosubtract75?Subtract100andadd(giveback)25.Whatistheeasywaytosubtract68?Subtract70andadd(giveback)2.Doyougettheidea?Thesimplewaytosubtractistoroundoffandthenadjust.Whatis284minus68?Let’ssubtract70from284andadd2.284−70=214214+2=216

This is easily done in your head. Calculating with a pencil and paper involves carrying andborrowing.Thiswayismucheasier.Howwouldyoucalculate537minus298?Again,mostpeoplewouldusepenandpaper.Theeasy

wayistosubtract300andgiveback2.537−300=237237+2=239

Usingawrittencalculationthewaypeoplearetaughtinschoolwouldmeancarryingandborrowingtwice.To subtract 87 from a number, take 100 and add 13 (because 100 is 13more than youwanted to

subtract).

Subtract100toget332.Add13(add10andthen3)toget345.Easy.

TestyourselfTrytheseproblemsinyourhead.Youcanwritedowntheanswers.a)86−38=b)42−9=c)184−57=d)423−70=e)651–185=f)3,424−1,895=Theanswersare:a)48b)33c)127d)353e)466f)1,529

Fora)youwouldsubtract40andadd2.Forb)youwouldsubtract10andadd1.Forc)youwouldsubtract60andadd3.Ford)youwouldsubtract100andadd30.Fore)youwouldsubtract200andadd15.Forf)youwouldsubtract2,000andadd100,then5.Ineachcasethereisaneasysubtractionandtherestisaddition.

Numbersaround100Whenwesubtractanumberjustbelow100fromanumberjustabove100,thereisaneasymethod.Youcandrawacirclebelowthenumberyouaresubtractingandwriteintheamountyouneedtomake100.Thenaddthenumberinthecircletotheamountthefirstnumberisabove100.Thisturnssubtractionintoaddition.Let’stryone.

Seventy-fiveis25below100.Add25to23foraneasyanswerof48.Let’stryanother.

Thistechniqueworksforanynumbersaboveandbelowanyhundredsvalue.

Italsoworksforsubtractingnumbersnearthesametensvalue.

Here’sanother:

TestyourselfTrytheseforyourself:a)13−8=b)17−8=c)12−9=d)12−8=e)124−88=f)161−75=g)222−170=h)111−80=i)132−85=j)145−58=Howdidyougo?Theyareeasyifyouknowhow.Herearetheanswers:a)5b)9c)3d)4e)36f)86g)52h)31i)47j)87

Ifyoumadeanymistakes,gobackandreadtheexplanationandtrythemagain.

Easywrittensubtraction

Easysubtractionuseseitheroftwocarryingandborrowingmethods.Youshouldrecogniseoneorevenbothmethods.The difference between standard subtraction and easy subtraction is minor, but important. I will

explaineasysubtractionwithtwomethodsofcarryingandborrowing.Usethemethodyouarefamiliarwithorthatyoufindeasier.

SubtractionmethodoneHereisatypicalsubtraction:

Thisishowtheworkingmightlook:

Let’sseehoweasysubtractionworks.Subtract7from5.Youcan’t,soyou‘borrow’1fromthetenscolumn.Crossoutthe6andwrite5.Now,hereisthedifference.Youdon’tsay7from15,yousay7from10equals3,thenaddthenumberabove(5)toget8,thefirstdigitoftheanswer.Withthismethod,youneversubtractfromanynumberhigherthan10.Therestisaddition.Ninefrom5won’tgo,soborrowagain.Ninefrom10is1,plus5is6,thenextdigitoftheanswer.Eightfrom1won’tgo,soborrowagain.Eightfrom10is2,plus1is3,thenextdigitoftheanswer.Threefrom7is4,thefinaldigitoftheanswer.

Subtractionmethodtwo

Subtract7from5.Youcan’t,soyouborrow1fromthetenscolumn.Puta1infrontofthe5tomake15andwriteasmall1alongsidethe9inthetenscolumn.Usingoureasymethod,youdon’tsay7from15,but7from10is3,plus5ontopgives8,thefirstdigitoftheanswer.Ten(9plus1carried)from6won’tgo,sowehavetoborrow;10from10is0,plus6is6.Ninefrom2won’tgo,soweborrowagain.Ninefrom10is1,plus2is3.Fourfrom8is4.Wehaveouranswer.Youdon’thavetolearnorknowthecombinationsofsingle-digitnumbersthataddtomorethan10.

Youneversubtractfromanynumberhigherthan10.Mostofthecalculationisaddition.Thismakesthecalculationseasierandreducesmistakes.

TestyourselfTrytheseforyourself:

a)

b)Theanswersare:a)2,757b)2,238

NotetoparentsandteachersThis strategy is very important. If a student has mastered multiplication using the simple formula in this book, he or she hasmasteredthecombinationsofnumbersthataddto10.Thereareonlyfivesuchcombinations.If a student has to learn the combinations of single-digit numbers that add to more than 10, there are another twenty suchcombinationstolearn.Usingthisstrategy,childrendon’tneedtolearnanyofthem.Tosubtract8from15,theycansubtractfrom10(whichgives2),andthenaddthe5,forananswerof7.

Thereisafargreaterchanceofmakingamistakewhensubtractingfromnumbersintheteensthanwhensubtractingfrom10.Thereisverylittlechanceofmakingamistakewhensubtractingfrom 10; when children have been using the methods in this book the answers will be almostautomatic.

Subtractionfromapowerof10Thereisaneasymethodforsubtractionfromanumberendinginseveralzeros.Thiscanbeusefulwhenusing100or1,000asreferencenumbers.Theruleis:

Subtracttheunitsdigitfrom10,theneachsuccessivedigitfrom9,thensubtract1fromthedigittotheleftofthezeros.

Forexample:

Wecanbeginfromtheleftorright.Let’stryitfromtherightfirst.Subtracttheunitsdigitfrom10.10−8=2

Thisistheright-handdigitoftheanswer.Thentaketheotherdigitsfrom9.Sixfrom9is3.Threefrom9is6.Onefrom1is0.Sowehaveouranswer:632.Nowlet’stryitfromlefttoright.Onefrom1is0.Threefrom9is6.Sixfrom9is3.Eightfrom10is2.Againwehave632.Hereiswhatwereallydid.Thesetproblemwas:

Wesubtracted1fromthenumberweweresubtractingfrom,1,000, toget999.Wethensubtracted368from999withnonumberstocarryandborrow(becausenodigitsinthenumberwearesubtractingcanbehigherthan9).Wecompensatedbyaddingthe1backtotheanswerbysubtractingthefinaldigitfrom10insteadof9.Sowhatwereallycalculatedwasthis:

Thissimplemethodmakesalotofsubtractionproblemsmucheasier.Ifyouhadtocalculate40,000minus3,594,thisishowyouwoulddoit:

Take1fromtheleft-handdigit(4) toget3, thefirstdigitof theanswer.(Wearereallysubtracting3,594from39,999andthenaddingtheextra1.)

Threefrom9is6.Fivefrom9is4.Ninefrom9is0.Fourfrom10is6.Theansweris36,406.You could do the calculation off the top of your head.Youwould call out the answer, ‘Thirty-six

thousand,fourhundredandsix.’Tryit.Withalittlepracticeyoucancalloutthedigitswithoutapause.Youmayfinditeasierjusttosay,‘Three,six,four,oh,six.’Eitherwayitisveryimpressive.

TestyourselfTrytheseforyourself:a)10,000−2,345b)60,000−41,726Youwouldhavefoundtheanswersare:a)7,655b)18,274

SubtractingsmallernumbersIf thenumberyouaresubtracting isshort, thenaddzeroesbefore thenumberyouaresubtracting (atleastmentally)tomakethecalculation.Let’stry45,000−23:

Youextendthezeroesinfrontofthesubtrahend(thenumberbeingsubtracted)asfarasthefirstdigitthatisnota0inthetopnumber.Youthensubtract1fromthisdigit.Fiveminus1equals4.Subtracteachsuccessivedigitfrom9untilyoureachthefinaldigit,whichyousubtractfrom10.Thisisusefulforcalculatingthenumberstowriteinthecircleswhenyouareusing100or1,000as

referencenumbersformultiplication.Itisalsousefulforcalculatingchange.The method taught in Australian and North American schools has you doing exactly the same

calculation,butyouhavetoworkoutwhatyouarecarryingandborrowingwitheachstep.Thebenefitof this method is that it becomes mechanical and can be carried out with less chance of making amistake.

CheckingsubtractionbycastingninesForsubtraction,themethodusedtocheckanswersissimilartothatusedforaddition,butwithasmalldifference.Letustryanexample.

Istheanswercorrect?Let’scastouttheninesandsee.

Fiveminus 8 equals 6?Can that be right?Obviously not.Although in the actual problemwe aresubtractingasmallernumberfromalargernumber,withthesubstitutes,thenumberwearesubtractingislarger.Wehavetwooptions.Oneistoadd9tothenumberwearesubtractingfrom.Fiveplus9equals14.

Thentheproblemreads:14−8=6

Thisansweriscorrect,soourcalculationwascorrect.HereistheoptionIprefer,however.Callouttheproblembackwardsasanaddition.Thisisprobably

howyouweretaughttochecksubtractionsinschool.Youaddtheanswertothenumberyousubtractedtogetyouroriginalnumberasyourcheckanswer.Addingthesubstitutesupwards,weget6+8=14.6+8=14

Addingthedigitsweget1+4=5,soourcalculationchecks.1+4=5

Ouransweriscorrect.Isetitoutlikethis:

TestyourselfCheckthesecalculationsforyourself toseeif thereareanymistakes.Castout thenines tofindanyerrors.If there isamistake,correctitandthencheckyouranswer.

a)

b)

c)

d)Theywereallrightexceptforb).Didyoucorrectb)andthencheckyouranswerbycastingthenines?Thecorrectansweris7,047.

Thismethodwillfindmostmistakesinadditionandsubtraction.Makeitpartofyourcalculations.Itonlytakesamomentandyouwillearnanenviablereputationforaccuracy.

NotetoparentsandteachersThe difference is not somuchwith students’ brains but with themethods they use. Teach the strugglers themethods the highachieversuseandtheycanbecomehighachieverstoo.

Chapter13

SIMPLEDIVISION

Whendoyouneedtobeabletodivide?Youneedtobeabletodoshortdivisioninyourheadwhenyouare shopping at the supermarket, when you are watching a sporting event, when you are handlingmoneyorsplittingupfood.Ifyouhave$30todivideamong6people,youneedtobeabletodivide6into30tofindouthow

muchmoneyeachpersonshouldreceive.Youwouldfindtheyreceive$5each.30÷6=55×6=30

Ifyouarewatchingacricketmatchanda teamhasscored32runs in8overs,howmanyrunsperoverhavetheyscored?Whatistheiraveragerunrate?Youneedtodivide32by8tofindtheanswer.Theansweris4,or4runsperover.

32÷8=44×8=32

Youneedtobeabletodividesoyouknowwhenyouaregettingthemostforyourmoney.Whichisbettervalue,a2-litrebottleofdrinkfor$2.75ora1.25-litrebottlefor$2.00?Youneedtobeabletodividetoworkoutwhichisthebetterbuy.

SimpledivisionManypeoplethinkdivisionisdifficultandwouldrathernothavetodoit,butitisreallynotthathard.Iwillshowyouhowtomakedivisioneasy.Evenifyouareconfidentwithsimpledivision,itmightbeworthwhilereadingthischapter.

DividingsmallernumbersIfyouhadtodivide10lolliesamong5people,theywouldreceive2lollieseach.Tencanbedividedevenlyby5.Ifyoudivided33mathsbooksamong4people,theywouldeachreceive8books,andtherewouldbe

1bookleftover.Thirty-threecannotbeevenlydividedby4.Wecallthe1bookleftovertheremainder.Wewouldwritethecalculationlikethis:

Orlikethis:

Wecouldask,whatdowemultiplyby4togetananswerof33,orascloseto33aswecanwithoutgoingabove?Fourtimes8is32,sotheansweris8.Wesubtract32fromthenumberwearedividingtofindtheremainder(whatisleftover).

DividinglargernumbersHereishowwewoulddividealargernumber.Todivide3,721by4,wewouldsetouttheproblemlikethis:

Orlikethis:

Webeginfromtheleft-handsideofthenumberwearedividing.Threeisthefirstdigitontheleft.Webeginbyasking,whatdoyoumultiplyby4togetananswerof3?Threeislessthan4sowecan’tevenlydivide3by4.Sowejointhe3tothenextdigit,7,tomake37.

Whatdowemultiplyby4togetananswerof37?Thereisnowholenumberthatgivesyou37whenyoumultiplyitby4.Wenowask,whatwillgiveananswerjustbelow37?Theansweris9,because9×4=36.Thatisascloseto37aswecangetwithoutgoingabove.So,theansweris9(9×4=36),with1leftovertomake37.Oneisourremainder.Wewouldwrite‘9’abovethe7in37(orbelow,dependingonhowyousetouttheproblem).The1leftoveriscarriedtothenextdigitandputinfrontofit.The1carriedchangesthenextnumberfrom2to12.Thecalculationnowlookslikethis:

Orlikethis:

Wenowdivide4into12.Whatnumbermultipliedby4givesananswerof12?Theansweris3(3×4=12).Write3above(orbelow)the2.Thereisnoremainderas3times4isexactly12.Thelastdigitislessthan4soitcan’tbedivided.Fourdividesinto1zerotimeswith1remainder.Thefinishedproblemshouldlooklikethis:

Or:

The 1 remainder can be expressed as a fraction,¼. The¼ comes from the 1 remainder over thedivisor,4.Theanswerwouldbe930¼,or930.25.

This is a simple method and should be carried out on one line. It is easy to calculate theseproblemsmentallythisway.

DividingnumberswithdecimalsHowwouldwedivide567.8by3?Wesetouttheproblemintheusualway.

Thecalculationbeginsasusual.Threedividesonceinto5with2remainder.Wecarrythe2remaindertothenextdigit,making26.

Wewritetheanswer,1,below(orabove)the5wedivided.Thecalculationlookslikethis:

Wenowdivide26by3.Eighttimes3is24,sothenextdigitoftheansweris8,with2remainder.Wecarrythe2tothe7.

Threedividesinto27exactly9times(9×3=27),sothenextdigitoftheansweris9.

Becausethedecimalpointfollowsthe7inthenumberwearedividing,itwillfollowthedigitintheanswerbelowthe7.Wecontinueasbefore.Threedividesinto8twotimeswith2remainder.Twoisthenextdigitofthe

answer.Wecarrytheremainder,2,tothenextdigit.

Becausethereisnonextdigit,wemustsupplyadigitourselves.Wecanwriteawholestringofzerosafterthelastdigitfollowingadecimalpointwithoutchangingthenumber.Wewillcalculateouranswertotwodecimalplaces,sowemustmakeanotherdivisiontoseehowwe

roundoffouranswer.Wewritetwomorezerostomakethreedigitsafterthedecimal.

Threedividesinto20sixtimes(6×3=18),with2remainder.Theremainderiscarriedtothenextdigit,making20again.Becausewewillkeependingupwith2remainder,youcanseethatthiswillgoonforever.Threedividesinto20sixtimeswith2remainder,andthiswillcontinueinfinitely.

Becausewearecalculatingtotwodecimalplaces,wehavetodecidewhethertoroundoffupwardsordownwards. If thenextdigitafterour requirednumberofdecimalplaces is5oraboveweroundoffupwards;ifthenextdigitisbelow5werounddown.Thethirddigitis6soweroundtheseconddigitoffto7.Ouransweris189.27.

SimpledivisionusingcirclesJustasourmethodwithcirclescanbeusedtomultiplynumberseasily,itcanalsobeusedinreversefordivision.Themethodworksbestfordivisionby7,8and9.Ithinkitiseasiertousetheshortdivisionmethodwehavejustlookedat,butyoucantryusingthecirclesifyouarestillnotsureofyourtables.Let’stryasimpleexample,56÷8:

Hereishowitworks.Wearedividing56by8.Wesettheproblemoutasabove,or,ifyouprefer,youcansettheproblemoutlikebelow.Sticktothewayyouhavebeentaught.

Iwillexplainusingthefirstlayout.Wedrawacirclebelowthe8(thenumberwearedividingby—

thedivisor)andthenask,howmanydoweneedtomake10?Theansweris2,sowewrite2inacirclebelowthe8.Weaddthe2tothetensdigitofthenumberwearedividing(5isthetensdigitof56)andgetananswerof7.Write7belowthe6in56.Drawacirclebelowouranswer(7).Again,howmanymoredoweneedtomake10?Theansweris3,sowrite3inthecirclebelowthe7.Nowmultiplythenumbersinthecircles.

2×3=6Subtract6fromtheunitsdigitof56togettheremainder.6−6=0

Thereis0remainder.Theansweris7with0remainder.Hereisanotherexample:75÷9.

Nineis1below10,sowewrite1inthecirclebelowthe9.Addthe1tothetensdigit(7)togetananswerof8.Write8astheanswerbelowthe5.Drawacirclebelowthe8.Howmanymoretomake10?Theansweris2.Write2inthecirclebelowthe8.Multiplythenumbersinthecircles,1×2,toget2.Take2fromtheunitsdigit(5)togettheremainder,3.Theansweris8r3.Hereisanotherexamplethatwillexplainwhatwedowhentheresultistoohigh.

Eightis2below10,sowewrite2inthecircle.Twoplus5equals7.Wewrite7abovetheunitsdigit.Wenowdrawanothercircleabovethe7.Howmanytomake10?Theansweris3,sowewrite3inthecircle.Togettheremainder,wemultiplythetwonumbersinthecirclesandtaketheanswerfromtheunitsdigit.Ourworkshouldlooklikethis:

Wefind,though,thatwecan’ttake6fromtheunitsdigit,2.Ouransweristoohigh.Torectifythis,wedroptheanswerby1to6,andwriteasmall1infrontoftheunitsdigit,2,makingit12.Sixis4below10,sowewrite4inthecircle.

Wemultiplythetwocirclednumbers,2×4=8.Wetake8fromtheunitsdigit,now12;12−8=4.Fouristheremainder.Theansweris6r4.

TestyourselfTrytheseproblemsforyourself:a)76÷9=b)76÷8=c)71÷8=d)62÷8=e)45÷7=f)57÷9=Theanswersare:a)8r4b)9r4c)8r7d)7r6e)6r3f)6r3

Thismethod is useful if you are still learning yourmultiplication tables and have difficulty withdivision,orifyouarenotcertainandjustwanttocheckyouranswer.Asyougettoknowyourtablesbetteryouwillfindstandardshortdivisiontobeeasy.Nexttimeyouwatchasportingevent,usethesemethodstoseehowyourteamisgoing.

RemaindersLet’sgoback toourproblemat thebeginningof thechapter.Howwouldwedivide33mathsbooksamong4students?Youcouldn’treallysaythateachstudentisgiven8.25or8¼bookseach,unlessyouwanttodestroyoneofthebooks!Eachstudentreceives8booksandthereis1bookleftover.Youcanthendecidewhattodowiththeextrabook.Wewouldwritetheansweras8r1,not8.25or8¼.Ifweweredividingupmoney,wecouldwrite theansweras8.25,becausethis is8dollarsand25

cents.Some problems in division require a whole remainder to make sense, others need the remainder

expressedasadecimal.

Bonus:Shortcutfordivisionby9Thereisaneasyshortcutfordivisionby9.Whenyoudivideatwo-digitnumberby9,thefirstdigitofthenumberistheanswerandaddingthedigitsgivesyoutheremainder.Forinstance,dividing42by9,thefirstdigit,4,istheanswerandthesumofthedigits,4+2,istheremainder.

42÷9=4r661÷9=6r723÷9=2r5

Theseareeasy.Whatdowedoif thesumof thedigits is9orhigher?For instance, ifwedivide65by9, thefirst

digit,6,isouranswerandtheremainderisthesumofthedigits,6+5=11.Ouransweris6r11.Butthatdoesn’tmakesensebecauseyoucan’thavearemainderlargerthanyourdivisor.Ninedividesinto11onemore time, soweadd1 to theanswer (6+1), and the2 leftover from11becomes thenewremainder.Sotheanswerbecomes7r2.Wecouldalsohaveappliedourshortcuttothe11remainder.Thefirstdigitis1,whichweaddtoour

answer,andthesum(1+1)givesaremainderof2.

Testyourself

Trytheshortcutwiththefollowing:a)25÷9=b)61÷9=c)34÷9=d)75÷9=e)82÷9=Theanswersare:a)2r7b)6r7c)3r7d)8r3e)9r1

Thesemethodsarefuntoplaywithandgiveinsightsintothewaynumberswork.

Chapter14

LONGDIVISIONBYFACTORS

Mostpeopledon’tmindshortdivision(orsimpledivision)buttheyfeeluneasywhenitcomestolongdivision.Todivideanumberby6youneedtoknowyour6timestable.Todivideanumberby7,youneedtoknowyour7timestable.Butwhatdoyoudoifyouwanttodivideanumberby36?Doyouneedtoknowyour36timestable?No,notifyoudivideusingfactors.

WhatAreFactors?Whatarefactors?Wehavealreadymadeuseoffactorswhenweused20asareferencenumberwithourmultiplication.Tomultiplyby20,wemultiplyby2andthenby10.Twotimes10equals20.Weareusingfactors,because2and10arefactorsof20.Fourand5arealsofactorsof20because4times5equals20.Justaswemadeuseoffactorsinmultiplication,wecanusefactorstomakelongdivisioneasy.Let’s

trylongdivisionby36.Whatcanweuseasfactors?Fourtimes9is36,andsois6times6.Wecouldalsouse3times12.

Let’stryourcalculationusing6times6.Wewillusethefollowingdivisionasanexample:2,340÷36=

Wecansetouttheproblemlikethis:

Orlikethis:

Usethelayoutthatyouarecomfortablewith.Now,togetstartedwedivide2,340by6.Weusethemethodwelearntinthepreviouschapter.Webeginbydividingthedigitontheleft.Thedigitontheleftis2,sowedivide6into2.Twoisless

than6sowecan’tdivide2by6,sowejoin2tothenextdigit,3,tomake23.Whatnumberdowemultiplyby6togetananswerof23?Thereisnowholenumberthatgivesyou

23whenyoumultiplyitby6.Wenowask,whatwillgiveananswerjustbelow23?Threetimes6is18.Fourtimessixis24,whichistoohigh,soouransweris3.Write3belowthe3of23.Subtract18(3×6) from 23 to get an answer of 5 for our remainder.We carry the 5 remainder to the next digit, 4,makingit54.Ourworksofarlookslikethis:

Wenowdivide6into54.Whatdowemultiplyby6togetananswerof54?Theansweris9.Ninetimes6isexactly54,sowewrite9andcarrynoremainder.Nowwehaveonedigitlefttodivide.Zerodividedby6is0,sowewrite0asthefinaldigitofthe

firstanswer.Ourcalculationlookslikethis:

Nowwedivideouranswer,390,bythesecond6.Sixdivides into39 six timeswith3 remainder (6×6=36).Write6below the9 and carry the3

remaindertomake30.

Sixdivides into30exactly5 times,so thenextdigitof theanswer is5.Ouranswer is65withnoremainder.

IhavewrittentheproblemthewayIwastaught.Ifyouprefertheotherlayoutyourcalculationwouldlooklikethis:

Whataresomeotherfactorswecoulduse?Todivideby48,wecoulddivideby6,thenby8(6×8=48).Todivideby25wewoulddivideby5twice(5×5=25).Todivideby24wecoulddivideby4,thenby6(4×6=24).Orwecoulddivideby3,thenby8,or

wecouldhalvethenumberandthendivideby12.Agoodgeneralrulefordividingbyfactorsistodividebythesmallernumberfirstandthenbythe

largernumber.Theideaisthatyouwillhaveasmallernumbertodividewhenitistimetodividebythelargernumber.Dividingbynumberssuchas14and16shouldbeeasytodomentally.Itiseasytohalveanumber

beforedividingbyafactor.Ifyouhadtodivide368by16mentally,youwouldsay,‘Halfof36is18,halfof8is4.’Youhaveasubtotalof184.Itiseasytokeeptrackofthisasyoudivideby8.Eighteendividedby8is2with2remainder.The2carriestothefinaldigitofthenumber,4,giving

24.Twenty-fourdividedby8isexactly3.Theansweris23withnoremainder.Thiscanbeeasilydoneinyourhead.Ifyouhadtodivide2,247by21,youwoulddivideby3first,thenby7.Bythetimeyoudivideby7

youhaveasmallernumbertoworkwith.2,247÷3=749749÷7=107

Itiseasiertodivide749by7thantodivide2,247by7.

Divisionbynumbersendingin5Todivide by a two-digit number ending in 5, double both numbers and use factors.As long as youdoublebothnumbers,theanswerdoesn’tchange.Thinkof4dividedby2.Theansweris2.Nowdoublebothnumbers.Itbecomes8dividedby4.Theanswerremainsthesame.(Thisiswhyyoucancancelfractionswithoutchangingtheanswer.)Let’shaveatry:

1,120÷35=Doublebothnumbers.Twice11is22,andtwotimes20is40;so1,120doubledis2,240.Thirty-five

doubledis70.Theproblemisnow:2,240÷70=

Todivideby70,wedivideby10,thenby7.Weareusingfactors.2,240÷10=224224÷7=32

This is an easy calculation.Sevendivides into 22 three times (3×7=21)with 1 remainder, anddividesinto14(1carried)twice.Thisisausefulshortcutfordivisionby15,25,35and45.Youcanalsouseitfor55.Thismethod

alsoappliestodivisionby1.5,2.5,3.5,4.5and5.5.Let’stryanother:512÷35=

Fivehundreddoubledis1,000.Twelvedoubledis24.So,512doubledis1,024.Thirty-fivedoubledis70.Theproblemisnow:1,024÷70

Divide1,024by10,thenby7.1,024÷10=102.4102.4÷7=

Sevendividesinto10once;1isthefirstdigitoftheanswer.Carrythe3remaindertothe2,giving32.

32÷7=4r4Wenowhaveananswerof14witharemainder.Wecarrythe4tothenextdigit,4,toget44.44÷7=6r2

OuranswerisWehavetobecarefulwiththeremainder.The2remainderweobtainedisnottheremainderforthe

originalproblem.Wewillnowlookatobtainingavalidremainderwhenwedivideusingfactors.

FindingaremainderSometimes when we divide, we would like a remainder instead of a decimal. How do we get aremainderwhenwedivideusingfactors?Weactuallyhavetworemaindersduringthecalculation.Theruleis:Multiplythefirstdivisorbythesecondremainderandthenaddthefirstremainder.

Forexample:

Webeginbymultiplyingthecorners6×1=6Thenweaddthefirstremainder,1.Thefinalremainderis7,or7/36.

TestyourselfTrytheseforyourself,calculatingtheremainder:

a)2,345÷36=b)2,713÷25=Theanswersare:a)65r5b)108r13

WorkingwithdecimalsWecandividenumbersusingfactorstoasmanydecimalplacesaswelike.Putasmanyzerosafterthedecimalasthenumberofdecimalplacesyourequire,andthenaddone

more.Thisensuresthatyourfinaldecimalplaceisaccurate.Ifyouweredividing1,486by28andyouneedaccuracytotwodecimalplaces,putthreezerosafter

thenumber.Youwoulddivide1,486.000by28.

RoundingoffdecimalsToroundofftotwodecimalplaces,lookatthethirddigitafterthedecimal.Ifitisbelow5,youleavetheseconddigitasitis.Ifthethirddigitis5ormore,add1totheseconddigit.Inthiscase,thethirddigitafterthedecimalis1.Oneislowerthan5soweroundofftheanswerby

leavingtheseconddigitas7.Theanswer,totwodecimalplaces,is53.07.Ifwewere roundingoff toonedecimalplace theanswerwouldbe53.1, as the seconddigit, 7, is

higherthan5,soweroundoffupwards.Theanswertoeightdecimalplacesis53.07142857.Toroundofftosevendecimalplaces,welookattheeighthdigit,whichis7.Thisisabove5,sothe5

isroundedoffupwardsto6.Theanswertosevendecimalplacesis53.0714286Toroundofftosixdecimalplaces,welookattheseventhdigit,whichisa5.Ifthenextdigitis5or

greaterweroundoffupwards,sothe8isroundedoffupwardsto9.Theanswertosixdecimalplacesis53.071429.Toroundofftofivedecimalplaces,welookatthesixthdigit,whichisa9,sothe2isroundedoff

upwardsto3.Theanswertofivedecimalplaces53.07143.

TestyourselfTrytheseforyourself.Calculatethesetotwodecimalplaces:a)4,166÷42=(Use6×7asfactors)b)2,545÷35=(Use5×7asfactors)c)4,213÷27=(Use3×9asfactors)d)7,817÷36=(Use6×6asfactors)Theanswersare:a)99.19b)72.71c)156.04d)217.14

Long division by factors allows you to domanymental calculations that most people would notattempt.Iconstantlycalculatesportingstatisticsmentallywhileagameisinprogresstocheckhowmy

teamisgoing.Itisafunwaytopractisethestrategies.IfIwanttocalculatetherunsperoverduringacricketmatchIsimplydividethescorebythenumber

ofoversbowled.Whenmorethan12overshavebeenbowledIwillusefactorsformycalculations.Whynottrythesecalculationswithyourfavouritesportsandhobbies?

Chapter15

STANDARDLONGDIVISIONMADEEASY

Inthepreviouschapterwesawhowtodividebylargenumbersusingfactors.Thisprincipleiscentraltoalllongdivision,includingstandardlongdivisioncommonlytaughtinschools.Divisionbyfactorsworkedwellfordivisionbynumberssuchas36(6×6),27(3×9),andanyother

numberthatcanbeeasilyreducedtofactors.Butwhataboutdivisionbynumberssuchas29,31or37,thatcan’tbereducedtofactors?Thesenumbersarecalledprimenumbers;theonlyfactorsofaprimenumberare1andthenumberitself.Letmeexplainhowourmethodworksinthesecases.Ifwewanttodivideanumberlike12,345by29,thisishowwedoit.Wecan’tuseourlongdivision

byfactorsmethodbecause29isaprimenumber.Itcan’tbebrokenupintofactors,soweusestandardlongdivision.Wesettheproblemoutlikethis:

Wethenproceedaswedidforshortdivision.Wetrytodivide29into1,whichisthefirstdigitof12,345,andofcoursewecan’tdoit.Sowejointhenextdigitanddivide29into12.Wefindthat12isalsotoosmall—itislessthanthenumberwearedividingby—sowejointhenextdigittoget123.Nowwedivide29into123.Thisiswherewehaveaproblem;mostpeopledon’tknowthe29times

table,sohowcantheyknowhowmanytimes29willdivideinto123?Themethod iseasy.This ishoweveryonedoes longdivisionbut theydon’t alwaysexplain it this

way.Firstly,weroundoffthenumberwearedividingby.Wewouldroundoff29to30.Wedivideby30aswego,toestimatetheanswer,andthenwecalculatefor29.Howdowedivideby30?Thirtyis10times3,sowedivideby10andby3toestimateeachdigitof

theanswer.So,wedivide123by30togetourestimateforthefirstdigitoftheanswer.Wedivide123by10andthenby3.Toroughlydivide123by10wecansimplydropthefinaldigitofthenumber,sowedropthe3from123toget12.Nowdivide12by3togetananswerof4.Write4abovethe3of123.Ourworklookslikethis:

Nowwemultiply4times29tofindwhattheremainderwillbe.Fourtimes29is116.(Aneasywaytomultiply29by4istomultiply30by4andthensubtract4.)

4×(30−1)=120−4=116Write116below123andsubtracttofindtheremainder.123−116=7

WebringdownthenextdigitofthenumberwearedividingandwriteanXbeneathtoremindusthedigithasbeenused.

Wenowdivide74by29.Wedivide74by10toget7(wejustdropthelastdigittogetanapproximateanswer),andthendivide

by3.Threedividesinto7twice,so2isthenextdigitofouranswer.Multiply2by29toget58(twice30minus2),andthensubtractfrom74.Theansweris16.Thenwe

bringdownthe5,writingtheXbelow.

Wenowdivide165by29.Roughlydividing169by10weget16.Sixteendividedby3is5.Multiply29by5toget145.Subtract145from165foraremainderof20.

Ouransweris425with20remainder.Thecalculationisdone.Ouronlyrealdivisionwasby3.Long division is easy if you regard the problem as an exercisewith factors. Even thoughwe are

dividingbyaprimenumber,wemakeourestimatesbyroundingoffandusingfactors.Thegeneralruleforstandardlongdivisionisthis:Roundoffthedivisortothenearestten,hundredorthousandtomakeaneasyestimate.

Ifyouaredividingby31,roundoffto30anddivideby3and10.Ifyouaredividingby87,roundoffto90anddivideby9and10.Ifyouaredividingby321,roundoffto300anddivideby3and100.Ifyouaredividingby487,roundoffto500anddivideby5and100.Ifyouaredividingby6,142,roundoffto6,000anddivideby6and1,000.Thisway,youareabletomakeaneasyestimateandthenproceedintheusualway.Rememberthatthesecalculationsarereallyonlyestimates.Ifyouaredividingby31andyoumake

yourestimatebydividingby30,theanswerisnotexact—itisonlyanapproximation.Forinstance,whatifyouweredividing241by31?Youroundoffto30anddivideby10andby3.Twohundredandforty-onedividedby10is24.Wesimplydropthe1.Twenty-fourdividedby3is8.Wemultiply8times31andsubtracttheanswerfrom241togetourremainder.Eighttimes31is248.Thisisgreaterthanthenumberwearedividingsowecannotsubtractit.Our

answerwastoohigh.Wedropto7.Seventimes31is217.

Wesubtract217from241forourremainder.241−217=24

So241dividedby31is7with24remainder.We could have seen this by observing that 8 times 30 is 240 and 241 is only 1more. Sowe can

anticipatethat8times31willbetoohigh.Howaboutdividing239by29?Weroundoffto30anddivideby10andby3forourestimate.Twohundredandthirty-ninedividedby10is23.Twenty-threedividedby3is7.Butlikethepreviouscase,wecanseetherealanswerisprobablydifferent.Twenty-nine is less than 30 and 30 divides into 239 almost 8 times, as it is only 1 less than 240,

whichisexactly8times30.Sowemultiply8times29toget232,whichisavalidanswer.Hadwenotseenthisandchosen7asouranswer,ourremainderwouldhavebeentoolarge.Seventimes29is203.Subtract203tofindtheremainder.239−203=36

Thisishigherthanourdivisorsoweknowwehavetoraisethelastdigitofouranswer.Whatifwearedividingby252?Whatdowerounditoffto?Twohundredistoolowand300istoo

high.Theeasywaywouldbetodoubleboththenumberwearedividingandthedivisor,whichwon’tchangetheanswerbutwillgiveusaneasiercalculation.Let’stryit.2,233÷252=

Doublingbothnumbersweget4,466÷504.Wenowdivideby500(5×100)forourestimatesandcorrectaswego.

Wedivide4,466by500forourestimate.Wedivideusingfactorsof100×5:4,466÷100gives4444÷5gives8(8×5=40)

Now,8×504=4,032,subtractedfrom4,466givesaremainderof434.Iftheremainderisimportant(wearenotcalculatingtheanswerasadecimal),becausewedoubledthenumbersweareworkingwith,theremainderwillbedoublethetrueremainder.Thatisbecauseourremainderisafractionof504andnot252.So,forafinalsteptofindtheremainder,wedivide434by2togetananswerof8r217.Ifyouarecalculatingtheanswertoanynumberofdecimalplacesthennosuchchangeisnecessary.Thatisbecause217/252isthesameas434/504.

TestyourselfTrytheseproblemsforyourself.Calculatetheremainderfora)andtheanswertoonedecimalplaceforb).a)2,456÷389=b)3,456÷151=Theanswersare:a)6r122b)22.9

The answer to b) is 22.8874 to four decimal places, 22.887 to three decimal places, 22.89 to two

decimalplacesand22.9toonedecimalplace,whichwastherequiredanswer.Forb)youwouldhavedoubledbothnumberstomakethecalculation6,912÷302.Thenyouestimateeachdigitoftheanswerbydividingby100×3.Longdivisionisnotdifficult.Ithasabadreputation.Itdoesn’tdeserveit.Manypeoplehavelearnt

longdivision,buttheyhaven’tbeentaughtproperly.Anythingisdifficultifyoudon’tunderstanditandyoudon’tknowhowtodoitverywell.Breakbigproblemsdowntolittleproblemsandyoucandothem.Inthenextchapteryouwilllearnaneasymethodtodolongdivisioninyourhead.

NotetoparentsandteachersIwasspeakingtoteachersataspecialgovernmentprogrammeintheUnitedStatesandIsaidIalwaysusefactorsforlongdivision,evenwhenIamdividingbyaprimenumber.Thatwastoomuchforoneteacher,whowasoneoftheorganisersoftheprogramme,andhechallengedmetoexplainmyself.Igaveaquicksummaryofthischapter,andhesaid,‘Youknow,thatishowIhavealwaysdonelongdivision,butIhaveneverthoughttoexplainitthatwaybefore.’

Chapter16

DIRECTLONGDIVISION

Here isaveryeasymethod fordoing longdivision inyourhead. It isverysimilar toourmethodofshort division.Asmany people have trouble doing long division, evenwith a pen and paper, if youmasterthismethodpeoplewillthinkyouareagenius.Here’showitworks.Let’ssayyouwanttodivide195by32.HereishowIsetouttheproblem:

Weroundoff32to30bysubtracting2.Thenwedividebothnumbersby10bymovingthedecimalpointoneplacetotheleft,makingthenumberwearedividing19.5andthedivisor3.Nowwedivideby3,whichiseasy,andweadjustforthe−2aswego.Threewon’tdivideinto1sowedividethefirsttwodigitsby3.Threedividesinto19sixtimeswith1

remainder(6×3=18).The1iscarriedtothenextdigitofthedividend(thenumberwearedividing),whichmakes15.

Beforewedivideinto15weadjustforthe2in32.Wemultiplythelastdigitoftheanswer,6,bytheunitsdigitinthedivisor,32,whichis2.

6×2=12Wesubtract12fromtheworkingnumber,15,toget3.Whatwehavedoneismultiplied32by6andsubtractedtheanswertoget3remainder.Howdidwe

dothis?We subtracted 6 times 32 by first subtracting 6 times 30, then 6 times 2. This is really a simple

methodfordoingstandardlongdivisioninyourhead.

Let’stryanotherone.Let’sdivide430by32to2decimalplaces.Wesetouttheproblemlikethis:

Wedividebothnumbersby10toget3and43.Weaddthreezerosafterthedecimalsothatwecancalculatetheanswertotwodecimalplaces.(Alwaysaddonemore0thanthenumberofdecimalplacesrequired.)Ourproblemnowlookslikethis:

Wedivide3into4forananswerof1with1remainder,whichwecarrytothe3,making13.Thenwemultiplytheanswer,1,by−2togetananswerof−2.Takethis2fromourworkingnumber,13,toget11.Howmanytimeswill3divideinto11?Threetimes3is9,sotheansweris3times,with2remainder.

The2remainderiscarriedtothenextdigit,making20.

Weadjustthe20bymultiplying:3×2=−6.Thenwesubtract.20−6=14

Wenowdivide14by3.Threedividesinto14fourtimes,with2remainder.Fouristhenextdigitofouranswer.

Multiplythelastdigitoftheanswer,4,bythe2of32,toget8.Then:20−8=12

Threedividesinto12exactly4timeswithnoremainder.Thenextstepwouldbetomultiply4times2toget8.Wesubtract8fromourremainder,whichis0,soweendupwithanegativeanswer.Thatwon’tdo,so

wedropourlastdigitoftheanswerby1.Ourlastdigitoftheanswerisnow3,with3remainder.

Threetimes2is6,subtractedfrom30leaves24.Twenty-fourdividedby3is8(with0remainder).We need a remainder to subtract from, so we drop the 8 to 7 with 3 remainder. The answer nowbecomes13.437.Becausewewantananswercorrecttotwodecimalplaces,welookatthethirddigitafterthedecimal.Ifthedigitis5orhigherweroundoffupwards;ifthedigitislessthan5weroundoffdownwards.Inthiscasethedigitis7soweroundoffthepreviousdigitupwards.Theansweris13.44,correcttotwodecimalplaces.

EstimatingAnswers

Let’sdivide32into240.Wesettheproblemoutasbefore:32

Webeginbydividingbothnumbersby10.Then,3dividesinto24eighttimes,with0remainder.The0remainderiscarriedtothenextdigit,0,makinganewworkingnumberof0.Weadjustforthe2in32bymultiplyingthelastdigitofouranswerbythe2in32andsubtractingfromourworkingnumber,0.Eighttimes2is16.Zerominus16is−16.Wecan’tworkwith−16sothelastdigitofouranswerwastoohigh.

(We can easily see this, as 8 times 30 is exactly 240.We are not dividing by 30, but by 32. Theansweriscorrectfor30,butclearlyitisnotcorrectfor32.Becauseeachanswerisonlyanestimate,wecanseeinthiscasewewillhavetodropouranswerby1.)Wechangetheanswerto7,so:7×3=21,with3remainder.Ourcalculationlookslikethis:

Wenowmultiply:7×2=14

Thenwesubtractfrom30:30−14=16

Theansweris7with16remainder.Ifwewanttotaketheanswertoonedecimalplace,wenowdivide16by3forthenextdigitofthe

answer.Threedividesinto16fivetimes,with1remainder.The1iscarriedtothenextdigittomake10.Wemultiplythelastanswer,5,bythe2in32togetananswerof10.Subtract10fromourworking

number10togetananswerof0.Thereisnoremainder.Theanswerisexactly7.5.Theproblemnowlookslikethis:

Thisisnotdifficult,butyouneedtokeepinmindthatyouhavetoallowforthesubtractionfromtheremainder.

Reversetechnique—roundingoffupwardsLet’sseehowdirectlongdivisionworksfordividingbynumberswithahighunitsdigit,suchas39.We

wouldround the39upwards to40.Forexample, let’sdivide231by39.Weset theproblemout likethis:

Wedividebothnumbersby10,so231becomes23.1and40becomes4.Fourwon’tdivideinto2,sowe divide 4 into 23. Four divides 5 times into 23 with 3 remainder (4 × 5 = 20).We carry the 3remaindertothenextdigit,1,making31.Ourworklookslikethis:

Wecorrectourremainderfor39insteadof40byadding5timesthe1,whichweaddedtomake40.5×1=531+5=36

Ouransweris5with36remainder.

Remember,theseareonlyestimates.Inthefollowingcasewehaveasituationwheretheestimateisnotcorrect,because39islessthan40.Let’ssaywewanttodivide319by39.Again,weroundoffto40.Ourcalculationwouldlooklikethis:

Wedividebothnumbersby10.

Fourdividesinto31seventimes,with3remainder.Wewritethe7belowthe1in31andcarrythe3remaindertothenextdigit,9,making39.

Wemultiplyouranswer,7,bythe+1toget7.Weadd7to39,givingananswerof46,whichisourremainder.Wecan’thavearemaindergreaterthanourdivisor,so,inthiscase,wehavetoraisethelastdigitofouranswerby1,to8.

Fourtimes8is32,with−1remainder,whichwecarryas−10,andweignorethe9forthemoment.Wemultiply8times+1toget+8,whichweaddtothenextdigit,9,toget17,thenweminusthe10carriedtogetaremainderof7.

Withpractice,thisconceptwillbecomeeasy.

TestyourselfTrytheseproblemsforyourself.Calculatetoonedecimalplace.a)224÷29=b)224÷41=Let’scalculatetheanswerstogether.

Wehavedividedbothnumbersby10.Threedividesinto22seventimes(3×7=21)with1remainder,whichwecarrytothenextdigit,4,tomake14.Wemultiply7×1=7,andaddtothe14toget21.Nowdivide21by3toget7,withnoremaindertocarry.Multiplythelastdigitoftheanswer,7,by1toget7,whichweaddto0forournextworkingnumber.Sevendividedby3is2,with1remainder,whichwecarrytothenextdigit,0,toget10.Wedon’tneedtogoanyfurtherbecauseweonlyneedananswertoonedecimalplace;7.72roundsoffto7.7.Thisisouranswer.Nowforthesecondproblem.

Wedividebothnumbersby10.Fourdividesinto22fivetimes(5×4=20),with2remainder,whichwecarrytothenextdigit,4,tomake24.Nowmultiply5(thelastdigitoftheanswer)by−1toget−5.Subtractthe5from24toget19.Wenowdivide19by4togetananswerof4with3remainder.The3carriedgivesusaworkingnumberof30.Nowmultiplythepreviousdigitoftheanswer,4,by−1toget−4.Subtract4from30toget26.Fourdividessixtimesinto26,andwecanforgettheremainder.Ouranswersofaris5.46.Wecanstopnowbecauseweonlyneedonedecimalplace.Roundedofftoonedecimalplacewehave5.5asouranswer.

Withpractice,thesecalculationscaneasilybedoneinyourhead.Dividethelastfourdigitsofyourtelephonenumberbydifferentnumbersinyourheadforpractice.Trysomeothernumbersaswell.Youwillfinditismucheasierthanyouimagine.

Chapter17

CHECKINGANSWERS(DIVISION)

Castingoutninesisoneofthemostusefultoolsavailableforworkingwithmathematics.Iuseitalmosteveryday.Castingoutninesiseasytouseforadditionandmultiplication.Nowwearegoingtolookathowweusethemethodtocheckdivisioncalculations.

ChangingtomultiplicationWhenwelookedatcastingoutnines tocheckaproblemwithsubtraction,wefoundweoftenhad toreversetheproblemtoaddition.Withdivision,weneedtoreversetheproblemtooneofmultiplication.Howdowedothat?Let’ssaywedivide24by6togetananswerof4.Thereverseofthatwouldbetomultiplytheanswer

bythedivisortogettheoriginalnumberwedivided.Thatis,thereverseof24÷6=4is4×6=24.Thatisnotdifficult.Tochecktheanswertotheproblem578÷17=34,wewouldusesubstitutenumbers.

Substituting,wehave2÷8=7.Thatdoesn’tmakesense.So,wedothecalculationinreverse.Wemultiply:7×8=2.Does7times8equal2withoursubstitutenumbers?

7×8=565+6=111+1=2

Seventimes8doesequal2;ouransweriscorrect.

HandlingremaindersHowwouldwehandle581÷17=34,with3remainder?Wewouldsubtracttheremainderfrom581tomakethecalculationcorrectwithoutaremainder.Wecouldeithersubtracttheremainderfirsttoget578÷17=34,whichistheproblemwealreadychecked,orwedoitinthecastingoutnines,likethis:

(581−3)+17=34Reversed,thisbecomes17×34=581−3.Writinginthesubstitutesweget:

Or,8×7=5−3.8×7=565+6=25−3=2

Theanswercheckscorrectly.Ifyouarenotsurehowtocheckaproblemindivision,tryasimpleproblemlike14÷4=3r2.Reverse the problem, subtracting the 2 remainder from14 to get 3× 4= 14− 2.Then apply the

methodtotheproblemthatyouwanttocheck.

FindingtheremainderwithacalculatorWhenyoucarryoutadivisionwithyourcalculator, itgivesyourremainderasadecimal. Is thereaneasywaytofindoutthetrueremainderinstead?Yes,thereis.Ifyoudivide326by7withacalculator,yougetananswerof46.571428withaneight-

digitcalculator.Whatifyouaretryingtocalculateitemsyouhavetodivideup;howdoyouknowhowmanywillbeleftover?Thesimplewayistosubtractthewholenumberbeforethedecimalandjustgetthedecimalpartof

theanswer.Inthiscaseyouwouldjustsubtract46,whichgives0.571428.Nowmultiplythisnumberbythe number you divided by. For our calculation, we multiply 0.571428 by 7 to get an answer of3.999996.Youroundtheansweroffto4remainder.Whydidn’t the calculator just say4?Because it onlyworkswithnumbers to a limitednumberof

decimalplaces,sotheanswerisneverexact.Multiplyingthedecimalremainderwillalmostalwaysgiveananswerthatisfractionallybelowthe

correct remainder. You will be able to see this for yourself. If you are dividing 326 chairs into 7classroomsatschool,youknowyouwon’thave46ineachroomwith3.999996chairs leftover.Youwouldhave4chairstokeepforspares.

Bonus:Castingtwos,tensandfivesJust as it ispossible tocheckacalculationbycastingout thenines,youcancastoutanynumber tomakeyourcheck.Castingouttwoswillonlytellyouifyouranswershouldbeoddoreven.Whenyoucastouttwos,the

onlysubstitutespossibleare0whenthenumberisevenand1whenthenumberisodd.Thatisnotveryhelpful.WhenIwasinprimaryschoolIsometimescheckedanswersbycastingoutthetens.Allthatdidwas

checkiftheunitsdigitoftheanswerwascorrect.Castingoutthetensmeansyouignoreeverydigitofanumberexcepttheunitsdigit.Again,itisnotveryuseful,butIdiduseitformulti-choicetestswhereacheckoftheunitsdigitwassometimesenoughtorecognisethecorrectanswerwithoutdoingthewholecalculation.Let’stryanexample.Whichoftheseiscorrect?34×72=a)2,345b)2,448c)2,673d)2,254

Multiplyingtwoevennumberscan’tgiveanoddnumberforananswer.(That iscastingout twos.)Thateliminatesa)andc).Tochecktheothertwoanswers,wemultiplytheunitsdigitsofourproblemtogetherandgetananswerof8(4×2=8).Theanswermustendwith8,sotheanswermustbeb).Casting out fives is another option. For the above calculation, the substitute numbers would be

exactlythesame.Butitcanhaveitsusesforcheckingmultiplicationbysmallnumbers.Let’stry7×8=56asanexample.Castingoutfivesweget2×3=1.Wedivideby5and justuse the remainder.Again,weareonlyworkingwith theunitsdigits.Five

dividesonceinto7with2remainder;itdividesonceinto8with3remainderand,justusingthe6of56,

weseethereis1remainder.Does it check?Let’s see:2×3=6 (which is the sameas theunitsdigitof56),and6hasa fives

remainder of 1, just like our check. The problemwith this is that 36, or even 41, would also havecheckedascorrectusingthismethod.Let’strycastingoutsevenstocheck9×8=72.Nineand8havesubstitutesof2and1,and2×1=2.Twoisourcheckanswer.Wecancastoutthe7from72asitrepresentsseventens(7×10),whichleavesuswith2.Ouranswer

iscorrect.Castingoutsevenswasabettercheckthancastingouttwos,tensorfivesbecauseitinvolvedallofthedigitsoftheanswer,butitisstilltoomuchtroubletobeuseful.Castingoutninesmakesfarmoresense,butitisstillfuntoexperiment.

CastingoutnineswithminussubstitutenumbersCanweuseourmethodofcastingoutnines(substitutenumbers) tocheckasimplecalculationlike7times8?Isitpossibletochecknumbersbelow10bycastingoutnines?

7×8=56Thesubstitutesfor7and8are7and8,soitdoesn’thelpusverymuch.Thereisanotherwayofcastingoutninesthatyoumightliketoplaywith.Tocheck7×8=56,you

cansubtract7and8from9togetminussubstitutenumbers.Sevenis9minus2and8is9minus1,sooursubstitutenumbersare−2and−1.Solongastheyare

bothminusnumbers,whenyoumultiplythemyougetaplusanswer,soyoucantreatthemassimply2and1.Let’scheckouranswerusingtheminussubstitutes.

This checks out. Our substitute numbers, 2 and 1, multiplied give us the same answer as oursubstituteanswer,2.Ifyouweren’tsureoftheanswerto7×8youcouldcalculatetheanswerusingthecircles,oryoucan

castouttheninestodouble-checkyouranswer.Youcanhavefunplayingwiththismethod.Let’scheck2×8×16.Theminussubstitutesare7and1.Seventimes1is7.Sevenisourcheckanswer.Therealanswer

addsto7(1+6=7),soouransweriscorrect.Everynumberhastwosubstituteswhenyoucastoutthenines—aplusandaminussubstitute.The

number25hasapositivesubstituteof7(2+5=7)andanegativesubstituteof−2(9−7=2).Let’stryitwith8×8=64.Eightis9minus1,so−1isoursubstitutefor8.Thesubstitutefor64is6+4=10;then,1+0=1.

This only has a limited application but you canplay and experiment.You could use it to check asubtractionwhere thesubstitutefor thenumberyouaresubtracting is larger than thenumberyouaresubtractingfrom.Thisisbecauseminussubstituteschangeaplustoaminusandaminustoaplus.Howdoesthishelp?Let’ssee.Ifwewanttocheck12−8=4,ournormalsubstitutenumbersaren’tmuchhelp.

Becauseusingaminussubstitutechangesthesign—that is,apluschangestominusandaminuschangestoplus—−8hasaminussubstituteof+1.Ourchecknowbecomes3+1=4,whichweseeiscorrect.Ofcourse,wewouldn’tnormallyusethischeckforaproblemlike12–8,butitisanoptionwhenwe

areworkingwithlargernumbers.Forinstance,let’scheck265—143=122.Wefindthesubstitutesbyaddingthedigits:

Oursubstitutecalculationbecomes4−8=5.Ifwefindtheminusequivalentof−8weget+1.So,ourcheckbecomes4+1=5.Wecaneasilyseenowthatouransweriscorrect.Eventhoughyoumightnotusenegativesubstitutenumbersveryoften,theyarestillfuntoplaywith,

and, as you use them, youwill becomemore familiarwith positive and negative numbers.You alsohaveanotheroptionforcheckinganswers.

Chapter18

FRACTIONSMADEEASY

When I was in primary school I noticed thatmany ofmy teachers had problemswhen they had toexplainfractions.Butfractionsareeasy.

WorkingwithfractionsWeworkwithfractionsallthetime.Ifyoutellthetime,chancesareyouareusingfractions(halfpast,aquarter to, a quarter past, etc.).When you eat half an apple, this is a fraction.When you talk aboutfootballorbasketball(half-time,secondhalf,three-quartertime,etc.),youaretalkingaboutfractions.Weevenaddandsubtractfractions,oftenwithoutrealisingitorthinkingaboutit.Weknowthattwo

quartersmakeahalf.Half-timeduringagameisattheendofthesecondquarter.Twoquartersmakeonehalf.Ifyouknowthathalfof10is5,youhavedoneacalculationinvolvingfractions.Youhavehalfof

your10fingersoneachhand.

Sowhatisafraction?Afractionisanumbersuchas½.Thetopnumber—inthiscase1—iscalledthenumerator,andthebottomnumber—inthiscase2—iscalledthedenominator.Thebottomnumber,thedenominator,tellsyouhowmanypartssomethingisdividedinto.Afootball

gameisdividedintofourparts,orquarters.The topnumber, the numerator, tells youhowmanyof these partswe areworkingwith— three-

quartersofacake,orofagame.Writing½ is anotherway of saying 1 divided by 2.Any division problem can be expressed as a

fraction: 6/3means 6 divided by 3; 15/5means 15 divided by 5. Even 5,217 divided by 61 can bewrittenas5,217/61.Wecansay thatany fraction isaproblemofdivision.Two-thirds (2/3)means2dividedby3.Weoftenhavetoadd,subtract,multiplyordividepartsofthings.Thatisanotherwayofsayingwe

oftenhavetoadd,subtract,multiplyordividefractions.Iamgoingtogiveyousomehintstomakeyourworkwithfractionseasier.(Ifyouwanttoreadand

learnmoreaboutfractionsyoucanreadmybookSpeedMathematics.)Hereishowweaddandsubtractfractions.

AddingfractionsIfweareaddingquartersthecalculationiseasy.One-quarterplusone-quartermakestwo-quarters,orahalf.Ifyouaddanotherquarteryouhavethree-quarters.Ifthedenominatorsarethesame,yousimplyaddthenumerators.Forinstance,ifyouwantedtoaddone-eighthplustwo-eighths,youwouldhaveananswerofthree-eighths.Three-eighthsplusthree-eighthsgivesananswerofsix-eighths.Howwouldyouaddone-quarterplusone-eighth?

Ifyouchangethequarterto2/8thenyouhaveaneasycalculationof2/8+1/8.

Itisnotdifficulttoadd1/3and1/6.Ifyoucanseethat1/3isthesameas2/6thenyouarejustaddingsixthstogether.So2/6plus1/6equals3/6.Youjustaddthenumerators.Thiscanbeeasilyseenifyouaredividingslicesofacake.Ifthecakeisdividedinto6slices,and

youeat1piece(1/6)andyourfriendhas2pieces(2/6),youhaveeaten3/6ofthecake.Because3ishalfof6youcanseethatyouhaveeatenhalfofthecake.

Addingfractionsiseasy.Hereishowwewouldadd1/3plus2/5.Thestandardmethodistochangethirdsandfifthsintothesamepartslikewedidwiththirdsandsixths.Here isaneasyway tosolve1/3plus2/5 thatyouprobablywon’tbe taught inschool.Firstly,we

multiplycrosswaysandaddtheanswerstogetthenumeratoroftheanswer.

1×5=53×2=65+6=11

Eleven is the top number (the numerator) of the answer. Now we multiply the bottom numbers(denominators)tofindthedenominatoroftheanswer.

3×5=15Theansweris11/15.

Easy.Hereisanotherexample:

Multiplycrossways.3×7=218×1=8

Weaddthetotalsforthenumerator,whichgivesus29.Thenwemultiplythedenominators:8×7=56

Thisisthedenominatoroftheanswer.Ouransweris29/56.Let’stryonemore.

SimplifyinganswersIneachoftheseexamples,theanswerwegotwasthefinalanswer.Let’stryanotherexampletoseeonemorestepweneedtotaketocompletethecalculation.

Wehavethecorrectanswer,butcantheanswerbesimplified?Ifboththenumeratoranddenominatorareeven,wecandividethemby2togetasimpleranswer.For

example,4/8couldbesimplifiedto½becauseboth4and8aredivisibleby4.Weseewiththeaboveanswerof15/27that15and27can’tbedividedby2,buttheyarebothevenly

divisibleby3(15÷3=5and27÷3=9).Theanswerof15/27canbesimplifiedto5/9.Eachtimeyoucalculatefractions,youshouldgivetheanswerinitssimplestform.Checktoseeifthe

numeratoranddenominatorarebothdivisibleby2,3or5,oranyothernumber.Ifso,dividethemtogivetheanswerinitssimplestform.Forinstance,21/28wouldbecome¾(21÷7=3and28÷7=4).Let’stryanotherexample,2/3+¾.

Inthisexamplethetopnumber(numerator)islargerthanthebottomnumber(denominator).Whenthis is the case, we have to divide the top number by the bottom, and express the remainder as afraction.So,12dividesonetimeinto17with5remainder.Ouransweriswrittenlikethis:15/12

TestyourselfTrytheseforyourself:a)¼+1/3=b)2/5+¼=c)¾1/5=d)¼+3/5=Howdidyougo?Theanswersare:a)7/12b)13/20c)19/20d)17/20

Ifyourmathsteachersaysyoumustusethelowestcommondenominatortoaddfractions,isthereaneasy way to find it? Yes, you simply multiply the denominators and then see if the answer can besimplified.Ifyoumultiplythedenominatorsyoucertainlyhaveacommondenominator,evenifitisn’tthelowest.(Youshouldunderstandthestandardmethodofaddingandsubtractingfractionsaswellasthemethodstaughtinthisbook.)

AshortcutHereisashortcut.Ifthenumeratorsareboth1,weaddthedenominatorstofindthenumeratoroftheanswer(topnumber),andwemultiplythedenominatorstofindthedenominatoroftheanswer.Thesecaneasilybedoneinyourhead.Hereisanexample:

Foryoujusttolookataproblemlikethisandcallouttheanswerwillmakeeveryonethinkyouareagenius!

Here’sanother:ifyouwanttoadd1/3plus1/7,youadd3and7toget10forthenumerator,thenyoumultiply3and7toget21forthedenominator.

TestyourselfTrytheseforyourself.Dothemallinyourhead.a)¼+1/3=b)1/5+¼=c)1/3+1/5=d)¼+1/7=Howdidyougo?Theanswersare:a)7/12b)9/20c)8/15d)11/28

Youshouldhavehadnotroublewiththose.

SubtractingfractionsA similar method is used for subtraction. You multiply the numerator of the first fraction by thedenominator of the second fraction. You then multiply the denominator of the first fraction by thenumeratorofthesecond,andsubtracttheanswer.Youfindthedenominatoroftheanswerintheusualway;bymultiplyingthedenominatorstogether.Here’sanexample:

Whatwedidwasmultiply2×4=8,andthensubtract3×1=3,toget5.Fiveisthenumeratorofouranswer.Wemultiplied3×4=12togetthedenominator.Youcoulddothatentirelyinyourhead.

TestyourselfTrytheseforyourself.Trythemwithoutwritinganythingdownexcepttheanswers.a)½−1/7=b)1/3−1/5=c)1/5−1/7=d)½−1/3=Theanswersare:a)5/14b)2/15c)2/35d)1/6

Ifyoumadeanymistakes,readthesectionthroughagain.Trysomeforyourselfandexplainthismethodtoyourclosestfriend.Tellyourfriendnottotellthe

restofyourclass.

AshortcutJustaswithaddition,thereisaneasyshortcutwhenyousubtractfractionswhenbothnumeratorsare1.Theonlythingtorememberistosubtractbackwards.

Again,theseareeasilydoneinyourhead.Iftheotherkidsinyourclasshaveproblemswithfractionstheywillbereallyimpressedwhenyoujustlookataproblemandcallouttheanswer.

TestyourselfTrytheseforyourself(dotheminyourhead):a)¼−1/5=b)½−1/6=c)¼−1/7=d)1/3−1/7=Theanswersare:a)1/20b)1/3c)3/28d)4/21

Canyoubelievethataddingandsubtractingfractionscouldbesoeasy?

MultiplyingfractionsWhatanswerwouldyouexpect ifyouhadtomultiplyone-halfbytwo?Youwouldhavetwohalves.Howaboutmultiplyingathirdbythree?Youwouldhavethreethirds.Howmuchistwohalves?Howmuchisthreethirds?If you split something into halves and bring the two halves together, howmuch have you got?A

whole.Youcouldsayitmathematicallylikethis:½×2=1.Ifyoucutapieintothirds,andyouhaveall threethirds,youhavethewholepie.Youcouldsayit

mathematicallylikethis:1/3×3=1.

Hereisanotherquestion:whatishalfof12?Theanswerisobviously6.But,whatdidyoudotofindyouranswer?Tofindhalfofanumberyoudivideby2.Or,youcouldsay,youmultipliedbyone-half.Ifyoucanunderstandthat,youcanmultiplyfractions.Hereishowwedoit.Wesimplymultiplythenumeratorstogetthenumeratoroftheanswer,andwe

multiplythedenominatorstogetthedenominatoroftheanswer.Easy!Let’stryittofindhalfof12.

Anywholenumbercanbeexpressedasthatnumberover(dividedby)1.So12isthesameas12/1.Let’stryanother:

Tocalculatetheanswerwemultiply2×4toget8.Thatisthetopnumberoftheanswer.Togetthebottomnumberofouranswerwemultiplythebottomnumbersofthefractions.

3×5=15Theansweris8/15.Itisaseasyasthat.Whatishalfof17?

Multiplythenumerators.

17×1=17Seventeenisthenumeratoroftheanswer.Multiplythedenominators.1×2=2

Twoisthedenominatoroftheanswer.So, the answer is 17 divided by 2, which is 8 with 1 remainder. The remainder goes to the top

(numerator),andthe2atthebottomremainswhereitisfortheanswer.Ouransweris8½.

DividingfractionsIntheprevioussectionwemultiplied17by½tofindhalfof17.Howwouldyoudivide17by½?Let’sassumewehave17orangestodivideamong30children.Ifwecuteachorangeinhalf(divideby½),wewouldhaveenoughforeveryone,plussomeleftover.Ifwecut anorange inhalfweget2pieces for eachorange, sowehad17orangesbut34orange

halves. Dividing the oranges makes the number bigger. So, to divide by one-half we can actuallymultiplyby2,becauseweget2halvesforeachorange.So17/1÷½isthesameas17/1×2/1.To divide by a fraction, we turn the fraction we are dividing by upside down and make it a

multiplication.That’snottoohard!Whosaidfractionsaredifficult?

TestyourselfTrytheseproblemsinyourhead:a)½÷½=b)1/3÷½=c)2/5÷¼=d)¼÷¼=Herearetheanswers.Howdidyougo?a)1b)2/3c)8/5or13/5d)1

ChangingvulgarfractionstodecimalsThefractionswehavebeendealingwitharecalledvulgarfractions,orcommonfractions.Tochangeavulgar fraction to a decimal is easy; you simply divide the numerator by the denominator (the topnumberisdividedbythebottomnumber).Tochange½toadecimalyoudivide1by2.Twowon’tdivideinto1,sotheansweris0andwecarry

the1 tomake10.Thedecimal in theanswergoesbelowthedecimal in thenumberwearedividing.Twodividesinto10exactly5times.Thecalculationlookslikethis:

Here’sanother:one-eighth(1/8)is1dividedby8.Onedividedby8equals0.125.Thatisallthereistoit.Tryafewmoreyourself.

Chapter19

DIRECTMULTIPLICATION

Everywhere I teachmymethods Iamasked,howwouldyoumultiply thesenumbers?Usually Iwillshowpeoplehowtousethemethodsyouhavelearntinthisbook,andthecalculationisquitesimple.Thereareoftenseveralwaystousemymethods,andIdelightinshowingdifferentwaystomakethecalculationsimple.Occasionally I am given numbers that do not lend themselves to my methods with a reference

numberandcircles.WhenthishappensItellpeoplethatIusedirectmultiplication.Thisistraditionalmultiplication,withadifference.

MultiplicationwithadifferenceForinstance,ifIwereaskedtomultiply6times17,Iwouldn’tusemymethodwiththecirclesasIthinkitisnottheeasiestwaytosolvethisparticularproblem.Iwouldsimplymultiply6times10andadd6times7.

6×10=606×7=4260+42=102Answer

Howabout6times27?Sixtimes20is120(6×2×10=120).Sixtimes7is42.Then,120+42=162.Theadditioniseasy:

120plus40is160,plus2is162.Thisismucheasierthanworkingwithpositiveandnegativenumbers.It is easy tomultiply a two-digit numberbyaone-digit number.For these typesofproblems,you

have theoptionofusing60,70and80as referencenumbers.Thismeans that there isnogap in thenumbersupto100thatareeasytomultiply.Let’stryafewmoreforpractice:7×63=

Youcouldusetworeferencenumbersforthis,sowewilltrybothmethods.Firstly,let’susedirectmultiplication.7×60=4207×3=21420+21=441

Thatwasn’ttoohard.Nowlet’suse10and70asreferencenumbers.

63−21=4242×10=420

3×7=21420+21=441

Thecalculationswerealmostidenticalinthiscase,butIthinkthedirectmethodwaseasier.Howabout6×84?6×80=4806×4=24480+24=504

Nowusingtworeferencenumberswehave:

Subtracting36from84weget480.Fourtimes6is24,then480+24=504.Happily,wegetthesameanswer.Thistimethedirectmethodwasdefinitelyeasier,although,again,

thecalculationsendedupbeingverysimilar.Let’stryonemore:8×27.Bydirectmultiplicationwehave:8×20=1608×7=56160+56=216

Nowusingtworeferencenumberswehave:

Thistimethecalculationwaseasierusingtworeferencenumbers.Isitdifficulttousedirectmultiplicationwhenwehaveareferencenumberof60,70or80?Let’stry

itfor67times67.Wewilluse70asourreferencenumber.

We subtract 3 from67 toget 64.Thenwemultiply64byour referencenumber, 70.Seventy is 7times10,sowemultiplyby7andthenby10.Seventimes60is420(6×7=42,thenby10is420).Seventimes4is28.Then,420+28=448,so7times64is448,and70times64is4,480.Multiplythenumbersinthecircles:3×3=9.Then:4,480+9=4,489Answer

Evenifyougobacktousingyourcalculatorforsuchproblems,youhaveatleastproventoyourselfthatyoucandothemyourself.Youhaveacquiredanewmathematicalskill.

Howwouldyouusethismethodtosolve34×76?Onemethodwouldbetouse30asareferencenumberandusefactorsof2×38toreplace76.You

couldalsosolveitdirectlyusingdirectmultiplication.Hereishowwedoit:

Wewillmultiplyfromrighttoleft(asismostcommon),andthentryitagainfromlefttoright.Firstly,webeginwiththeunitsdigits:6×4=24

Wewritethe4andcarrythe2.Thenwemultiplycrosswaysandaddtheanswer.7×4=283×6=1828+18=46

(Toadd18youcouldadd20andsubtract2.)Addthecarried2toget48.Write8andcarry4.Nowmultiplythetensdigits:3×7=21

Addthe4carriedtoget25.Nowwewrite25tofinishtheanswer.Thecalculationwasreallydoneusingtraditionalmethodsbutwritteninoneline.Thefinishedcalculationlookslikethis:

Herearethestepsforthisproblem:6×4=24

7×4=283×6=1828+18+2carried=48

7×3=21+4carried=25

Eachdigitismultipliedbyeveryotherdigit.Thefullworkinglookslikethis:

Solvingfromrighttoleftisprobablyeasiestifyouareusingpenandpaper.Ifyouwanttosolvetheproblemmentallythenyoucangofromlefttoright.Let’stryit.Thirtytimes70is3×7×100.The100representsthetwozerosfrom30and70.3×7=2121×100=2,100

Iwouldsaytomyself,twenty-onehundred.

Nowwemultiplycrossways:3×6and7×4,andthenmultiplytheanswerby10.3×6=187×4=2818+28=4646×10=460

Oursubtotalwas2,100.Tothisweadd400,thenweadd60.2,100+400=2,5002,500+60=2,560

Wearenearlythere.Nowwemultiplytheunitsdigits,andadd.4×6=242,560+24=2,584

Trydoing theproblemyourself inyourhead.Youwill find it iseasier thanyou think.Calculatingfromlefttorightmeanstherearenonumberstocarry.Thecalculationisnotasdifficultasitappears,butyourfriendswillbeveryimpressed.Youjustneed

topracticeshruggingyourshouldersandsaying,‘Oh,itwasnothing.’Ifyoucan’tfindaneasywaytosolveaproblemusingareferencenumberthendirectmultiplication

mightbeyourbestoption.

DirectmultiplicationusingnegativenumbersIdebatedwithmyselfwhetherthissectionshouldbeincludedinthebook.Ifyoufinditdifficult,don’tworryaboutit.Tryitanyway.Butyoumayfindthisquiteeasy.Itinvolvesusingpositiveandnegativenumberstosolveproblemsindirectmultiplication.Ifyoutryitandyouthinkitistoodifficult,forgetit—orcomebacktoitinayear’stimeandtryitagain.Ienjoyplayingwiththismethod.Seewhatyouthink.Hereishowitworks.Ifyouaremultiplyinganumberby79itmaybeeasiertouse80−1asyourmultiplier.Multiplying

by 79 means you are multiplying by two high numbers and you are likely to have high subtotals.Multiplicationby80−1mightbeeasier.Using80−1,8becomes the tensdigitand−1 is theunitsdigit.Youneedtobeconfidentwithnegativenumberstotrythis.Let’shaveago:68×79=

Wesetitoutas:

Webegin bymultiplying the units digits.Eight timesminus 1 isminus 8.Wedon’twrite−8;weborrow10fromthetenscolumnandwrite2,whichisleftoverwhenweminusthe8.Wecarry−1(ten),whichweborrowed,tothetenscolumn.Theworklookslikethis:

Nowwemultiplycrossways.Eighttimes8is64,and6times−1is−6.64−6=58

Wesubtractthe1thatwascarried(becauseitwas−1)toget57.Wewritethe7andcarrythe5.

Forthefinalstepwemultiplythetensdigits.6×8=48

Weaddthe5thatwascarried.48+5=53

Theansweris5,372.

TestyourselfTrytheseforyourself:a)64×69=b)34×78=Herearetheanswers:a)4,416b)2,652Let’sworkthroughthesecondproblemtogether.

Webeginbymultiplying4times−2,whichequals−8.Borrowing10wehave2left.Wecarrythe−10tothenextcolumnas−1(times10).Nowwemultiplycrossways.Eighttimes4is32,plus3times−2is−6.Adding−6isthesameassubtracting6from32,whichgivesus26.Don’tforgetthecarryof−1,soweendupwith25.Writethe5andcarrythe2.Forthefinalstepwemultiplythetensdigits:3×8=24.Weaddthe2whichwecarriedtoget26.Wenowhaveouranswer,2,652.

Thisissimplyamethodyoumightliketoplaywith.Experimentwithproblemsofyourown.

Chapter20

PUTTINGITALLINTOPRACTICE

HowDoIRememberAllofThis?Oftenwhen people read books on high-speedmaths andmathematical shortcuts, they ask, howdo Irememberallofthat?Theyareoverwhelmedbytheamountofinformationandtheysimplysay,Iwillneverrememberallofthatstuff.Imightaswellforgetaboutit.Is there a possibility of this happening with the information in this book? There is always a

possibility,butitisnotlikely.Why?Becausebooksofmathematicalshortcutsthatareeasilyforgottenarejustthat—aseriesofunconnectedshortcutsthathavetobememorisedforspecialoccasions.Andwhenthespecialoccasionoccursweeitherforgettousetheshortcutorwecan’trememberhowtouseit.Thisbookisdifferentbecauseitteachesaphilosophyforworkingwithmathematics.Itteachesbroad

strategiesthatbecomepartofthewaywethink.Puttingthemethodsyouhavelearntinthisbookintopracticewillaffectthewayyouthinkandcalculateinalmosteveryinstance.For instance, our methods of division apply to all division problems — they are not shortcuts

applicabletosomeisolatedcases.Youcanapplythemethodofcheckinganswers(castingoutnines)toalmostanycalculationinarithmetic.Itshouldbecomeanautomaticpartofanycalculation.Weaddandsubtractjustabouteveryday.Themethodsinthisbookapplyeveryday.Ifyouareusingthemethodseveryday,howcanyouforgetthem?Thisbookdoesn’t teachshortcuts—it teachesbasicstrategies.Practise thestrategiesandyouwill

find you are automatically using factors for simple multiplication and division. You will use themethodsforadditionandsubtraction.Myphilosophyhasalwaysbeen, if there isasimplewaytodosomething,whydoitthehardway?Someofmystudentshaveevenaskedmeifusingmylearningmethodsandmathsmethodsisreally

cheating.We have an unfair advantage over the other kids. Is it ethical? I tell them that the betterstudentshavebettermethodsanyway—thatiswhatmakesthembetterstudents.Theysay,buttheotherstudentscoulddoaswellasmeiftheylearntmymethods.Suretheycould.Ifyoufeelstronglyaboutit,teachtheotherkidsyourself,orlendthemyourbook.Mostof themethodstaughtare‘invisible’.That is, theonlydifferenceiswhatgoesoninsideyour

head.Ifsomeonelookedoveryourshoulderwhileyouperformedasubtractioninclass,theywouldn’tseebywatchingyourworkthatyouaredoinganythingdifferenttotherestoftheclass.Thesamegoesfor longdivision,unlessyouareusingthedirect long-divisionmethod.So,everytimeyouperformacalculationyouhavethechoicetousetheoldmethodorusetheeasymethod.Youoftenhearpeoplesaytheyhaveattendedaclassorseminar,themethodsweregreatandeasy,but

theyhaven’tusedanyofitsince.Theyhavewastedtheirmoney.ThatiswhyItellmystudentstomakemymethodspartoftheirlife.Youdon’thavetoperformexerciseseverydayandpractisedrills.Thatisasurewaytogiveup.Justusethemethodseverydayasyouhavetheneedandopportunity.Ifyouhaveanopportunitytouseoneofmymethodsandforgethowitworks,don’tworryaboutit.Justreadthechapteragain,practisethemethodsbysolvingtheexamplesasyoureadthem,andmakesureyouarereadynexttime.Also,itdoesn’thurttoshowoffwhatyoucando.Tellyourfriendsyouknowyourtablesuptothe20

timestableorthe25timestable.Thenhavethemcalloutanycombinationforyoutosolve.Theycancheckyour answerwith a calculator.Even ifyou feelyouare slowgiving theanswer, theywill justthinkthis isonecombinationyoudon’tknowverywell. I thinkmostofyourfriendswouldprefer tothinkyouhavememorisedthetablesthantothinkyoucancalculatetheanswersinaflash.Then,whenyouare finished, ‘confess’ toyour friends thatyoudon’thave themmemorised,youactuallyworkedthemoutastheygavethemtoyou.Theywon’tknowwhethertobelieveyouornot!Itisafunwaytopractiseandtolearnyourtables—andafunwaytoshowoffyourskills.

AdviceForGeniusesIhaveoftensaidthatpeoplethinkyouareveryintelligentifyouareverygoodatmathematics.Peoplewilltreatyoudifferently;yourfriends,yourclassmatesandyourteachers.Everyonewillthinkyouareextrasmart.Hereissomeadviceforhandlingyournewstatus.Firstly,don’texplainhowyoudoeverything.Author IsaacAsimov tellshowheexplainedhowhe

solvedamathematicalproblemtosomeoneservinghiminastore.Whensheheardtheexplanationshesaidscornfully,‘Oh,itwasjustatrick.’Hehadn’tusedthemethodshehadbeentaughtatschool.Itwasalmostlikehehadcheated.Mymethods are not tricks. Sometimes people will introduce me as someone who teaches maths

tricks.Ineverlikethat.‘Tricks’tomeimpliestrickery—youaren’treallydoingwhatyouappeartobe.Also,peoplereactwiththecomment,‘Oh,foramomentIthoughtyouhaddonesomethingclever.’

Theyalsooftenadd,‘Well,anyonecoulddothat,’oncetheyknowhowyoudidit.WhenIamgivingaclassIwillusuallysay,‘That’swhatmakesmymethodssogood.Anyonecandoit!’

AFTERWORDIf youwould like to learnmore ofmymethods then I would recommend you buymy book SpeedMathematics.Ithasmanymoremathematicalstrategiesnotincludedinthisbook.ThisbookwaswrittentoexplainthewayIdomaths.ThesearethemethodsIhavedevelopedand

workedwithovertheyears—someIwastaught,someIworkedoutformyself.Ihaveneverlookedatthesemethodsasaseriesofshortcuts.Ioftenaskthequestioninmybooksandinclasses,howshouldIsubtract9from56?HowdoIsolveit?ItdependsonthemoodIaminatthetime.SometimesIuseonemethod;atothertimesIwilluseanother.Thereisusuallymorethanonewaytogettotheanswer.Ifyouwanttomakethemostofthisbook,usewhatyouhavelearnt.Putwhatyouhavelearntinto

practice.

StudentsPeopleequatemathematicalabilitywithintelligence.Usethemethodsyouhavelearntinthisbookandpeople—yourparents,teachers,classmatesandfriends—willtreatyoulikeyouarehighlyintelligent.Youwillgetareputationforbeingsmart.Thatcanonlybegood.Also,itisafactthatifpeopletreatyouas intelligent you are more likely to act more intelligently. You are likely to tackle more difficultproblems.So,usethemethods.Makethempartofyourlife,partofthewayyouthink.Puttingthisbookinto

practicecanchangethedirectionofyourwholelife.

TeachersTeach these methods in the classroom and you will have high-achieving students and will yourselfbecomeahigh-achievingandsuccessfulteacher.Allofyourstudentswillsucceed.Yourclasswillnotonlyperformbetterbuttheywillalsohaveabetterunderstandingofmathematicalprinciples.Youwillteachmoreinlesstimeandhaveahighlymotivatedclass.Ahighlymotivatedclassisbetterbehavedand will improve everyone’s enjoyment of your classes. Every class will become an adventure.Everyonebenefits.

ParentsEncourageyourchildren to learn thesemethods.Encourage themtoplaywith themethods.Let themshowoffwhattheycando.Neverbecriticalofyourchildrenwhentheymakeamistakeorseemtobeslow learning something. Keep telling your children theywill succeed. Let them know you supportthem.Taketheirmistakesorlackofsuccessasachallenge.Askwhatyoucandotomakethiseasier.Don’tcompareyourchildwithabrotherorsister,oranyoneelse.Compareyourchild’sperformancewithwhatheorshedidinthepast.Lookattheimprovement.Some children in my classes come in thinking they are ‘dumb’. Their self-esteem is low. These

methods are an effectiveway toboost their self-esteem.Youwill find that asyour children improvetheirperformanceinmaths,itwillhaveaflow-oneffect.Theywillimproveinotherareasaswell.

ClassesandTrainingProgramsIhaveconductedclassesandtrainingprogramsaroundtheworld.Ihaveproducedmaterialforteachersandparents to help them to teach thesemethods in the classroomand in the home. I often speak to

home-schoolingparents.Youcanfindoutmoreabouttheseprogramsandmyotherlearningmaterialsbyvisitingmywebsiteatwww.speedmathematics.com.Ialsohavepracticesheetsyoucandownloadfrommywebsite.Youcanemailmeatbhandley@speedmathematics.comifyouhaveanyquestionsaboutthecontents

ofthisbookorifyouwouldliketoarrangeaclass.

AppendixA

USINGTHEMETHODSINTHECLASSROOM

Childrenoftenaskme,‘HowdoIusethemethodsintheclassroom?Won’tmyteacherobjectifIusedifferentmethods?’Itisafairquestion.HereistheanswerIgive.Firstly,manyofthemethodstaughtinthisbookare‘invisible’.Thatis,thedifferenceiswhatyousay

inyourhead.Withadditionandsubtractionproblemsthelayoutandwrittencalculationsarethesame;whatisdifferentiswhatyousaytoyourselfinyourhead.Whenyouaresubtracting8from5andyouborrow10tomakeit8from15;yousaytoyourself8from10is2,plus5is7.Youwritedownthe7.Thestudentswhosubtract8from15writedown7aswell(solongastheydon’tmakeamistake),soanyonelookingoveryourshoulderwouldhavenoideayouareusingadifferentmethod.Thesamegoesforsubtracting3,571from10,000.Yousettheproblemoutthesameaseveryoneelse,

butagain,whatyousayinsideyourheadisdifferent.Yousubtracteachdigitfrom9andthefinaldigitfrom10.No-onelookingatyourfinishedcalculationwouldknowyoudidanythingdifferent.Ifyouwereaskedtosubtract378from613youwouldagainwritetheproblemdownasusual.Then

youcouldfindtheanswerinyourheadbysubtracting400andadding22.613−400=213213+22=235

Using thismethod there isnocarryingorborrowing.You justwrite theanswerand thencheckbycastingnines.Thesamegoesforallmentalcalculation;thatis,allcalculationswhereyouwritenothingdown.No-

oneknowswhatmethodyouuse.Studentsandteachershavealwaysuseddifferentmethods,evenforsuchsimpleproblemsas56minus9,asIpointedoutearlierinthebook.Also,thehigherthegradeyouarein,thelessofaproblemthiswillbe.Nowlet’slookatwhereitcanmakeadifference.Let’ssayyouareaskedtomultiply355by52.You

setouttheproblemastheteacherrequires.

Here is the finished calculation. Can you now make use of what you have learnt in this book?Certainlyyoucan.Youcancastoutninestocheckyouranswer.

Thesubstitutesare4×7=1,whichwecanseeiscorrectbecause4times7equals28,and2plus8is10,whichaddsto1.Hadyou foundyouhadmadeamistake,youcouldcastoutnines to findwhereyoumade it.You

wouldfindthesubstitutenumberforeachpartofthecalculation.Itwouldlooklikethis.

Tocheck,youwouldusethesubstituteslikethis.Firstlyyoumultiplied355by2toget710.Nowyoumultiplythesubstitute.

4×2=8Oursubstituteansweris8,sothispartofthecalculationiscorrect.Next,wemultiplied355by5(or50)toget17,750.Multiplythesubstitutesagain.5×4=20

Thesubstituteansweristhesame,soyoucanacceptthispartofthecalculationascorrect.Nextyouadded710and17,750togetyourfinalanswerof18,460.Weaddthesubstitutesof8+2=

10,and this checkswithyourcalculation. In this caseyouranswer is correctbut, ifyouhadmadeamistake,younotonlywouldhavefoundit,butyouwouldalsoknowwhereyoumadeit.Youwouldknowifthemistakewasinthemultiplication,andwhichpart,orifthemistakewasintheaddition.Iwouldmakethesecheckswithpencilandthenerasethem.Inpractice,Ithinkanyteacherwouldbeintriguedbyyourmethodofchecking.Ihaveneverheardof

thesemethodscausingproblemsforstudentsintheclassroom.Most importantly, once you earn a reputation for beingmathematically gifted, no-one will worry

aboutyourmethods—theywillexpectyoutobedifferent.

AppendixB

WORKINGTHROUGHAPROBLEM

Whatyousayinsideyourheadwhileyouperformmentalmathematicalcalculationsisveryimportant.Youcandoublethetimeittakestosolveaproblembysayingthewrongthings,andhalvethetimebysayingtherightthings.ThatiswhyItellyouwhattosay,forinstance,whenyouaremultiplyingby11inyourhead.Tomultiply24by11,youwouldautomaticallyseethat2plus4islessthan10,soyouwouldlookatthe2andimmediatelystartsaying,‘Twohundredand...’.Whileyouaresayingthatyouwouldseethat2plus4meansyoushouldsay,‘...sixty-...’,andinthesamebreathyouwouldsay,‘...four.’Soyouwouldjustsay,‘Twohundredandsixty-four.’Tomultiply14by15youwouldsimplyadd5to14toget19.Yousayinyourhead,‘Onehundred

andninety . . . ’.Four times5 is20.Adding20gives, ‘Twohundredand ten.’Youwouldsay‘One-ninety,two-ten.’Thelessyousay,thefasteryouwillbe.Practisebyyourself.Youdon’tneedtorecitethewholeprocedure.Youwouldn’tsay,‘Fourteenplusfiveisonehundred

andninety.Nowwemultiplyfour timesfive.Four timesfiveis twenty.Onehundredandninetyplustwenty is twohundredand ten.’Thatwould takeyoumuch longer, and seems like awhole lotmorework.Justsaythesubtotalsandtotalsor,ifyoucan,justtheanswer.Whenyoupractiseanyofthemethods—especiallyifyouwanttoshowoffinfrontofyourfriends

orclass—trytoanticipatewhatiscoming.Ifyouseeyouwillhavetocarry,saythenextnumberwiththecarriednumberalreadyadded.Forinstance,let’smultiply11times84.Youwouldimmediatelyseethat8plus4is12or,atleast,morethan10.Soyouwouldknowtocarry1andstartcallingout,‘Ninehundredand. . .’Youprovidethe2from12togive‘. . . twenty-. . .’,andthensaythe4attheend,giving924.Youwouldcalltheanswerinonebreath:‘Ninehundredandtwenty-four.’Oftenithelpstocallcarrynumberswhattheyare.Thatis,ifyouarecarrying3andthe3represents

300,saythreehundred.Ifitrepresents30,saythirty.Thiscanbeusefulinmultiplicationandaddition.Usually, the biggest hurdle to calling immediate answers is the feeling that the calculation is ‘too

hard’.Practiseforyourselfandyouwillseehoweasyitistogiveimmediateanswersoffthetopofyourhead.Thisappliesifyouaresayingtheansweroutloudorjustcalculatinginyourhead.Practisesomeofthestrategieswithyourclosestfriend—someonewhowon’tembarrassyouifyoumakeamistake.Practisetheproblemsinthisbookinyourhead.Thefirsttimemayseemdifficult,thesecondtimeis

easier,andbythefifthtimeyouwillwonderwhyyoueverthoughttheywerehard.Youwillalsofindthatthiswillbuildyourconcentrationskills.

AppendixC

LEARNTHE13,14AND15TIMESTABLES

Whenyouknowyourmultiplicationtablesuptothe5timestableyoucanmultiplymostproblemsinyourhead.Ifyouknowyour2,3,4and5timestablesupto10timeseachnumber,thenyoualsoknowpartofyour6,7,8and9timestables.Forthe6timestableyouknow2times6,3times6,4times6,and5times6.Thesameappliestothe7,8and9timestables.Asyoulearnyourhighertables,mentalcalculationsbecomemucheasier.It iseasytolearnyour11timestable,andthe12timestableisnotdifficult.Sixtimes12is6times10plus6times2.Sixtimes10is60and6times2is12.Itiseasytoaddtheanswersfor6times12toget72.Thewaytablesusedtobetaughtwasdifficultandboring.Wehaveseenhoweasyitistolearnyour

tablesandhowitcanbedonemuchmorequicklythanintheoldendays.Itisusualtolearntablesuptothe12timestable.Itisnothardtolearnthe13timestableifyouknow

the12timestable.Ifyouknow12threesare36,justaddanother3toget39.Twelvethreesplusonemore3makes13threes.Ifyouknow12foursare48,justaddanother4toget52.Then,whenyouknowyour13timestable,learnthe14timestable.Youhavetwowaystodothis.

Youcanfactorise14to2times7.Thenyoujustdoublethenumberandmultiplyby7,ormultiplyby7anddoubletheanswer.So,6times14wouldbe6times7(42),thendoubledis84.Thatwaseasy.Alsoyoucouldhavedoubledthe6firsttoget12,andmultiply12by7toget84again.Ifyouknowyour12timestable,thatiseasyaswell.Anotheralternativeistosay6times10is60,plus6times4is24.Add60and24fortheanswer,84.Calculatetheseoftenandyouwilllearnyourtableswithouteventrying.Then you will find you can multiply and divide directly by 13, 14 and 15. This will give you anadvantageovertheotherkids.

AppendixD

TESTSFORDIVISIBILITY

Itisoftenusefultoknowwhetheranumberisevenlydivisiblebyanothernumber.Herearesomewaysyoucanfindoutwithoutdoingafulldivision.Divisibilityby2:Allevennumbersaredivisibleby2;thatis,allwholenumbersforwhichthefinal

digitis0,2,4,6or8.Divisibilityby3:Ifthesumofthedigitsisdivisibleby3,thenthenumberisalsodivisibleby3.For

instance,thedigitsumof45is9(4+5=9),whichisdivisibleby3,so45isdivisibleby3.Divisibilityby4:Ifthelasttwodigitsaredivisibleby4,thenumberisdivisibleby4.Thisisbecause

100isevenlydivisibleby4(100=4×25).Thereisaneasyshortcut.Don’tworryhowlongthenumberis;ifthetensdigitiseven,forgetitand

checkiftheunitsdigitisdivisibleby4.(Itmustbezero,4or8.)Ifthetensdigitisodd,carry1totheunitsdigitandcheckifitisdivisibleby4.Thatisbecause20isevenlydivisibleby4(4×5=20).Forinstance,let’scheckif15,476isdivisiblebyfour.Thetensdigit,7,isodd,sowecarry1tothe

unitsdigit,6,tomake16.Sixteenisevenlydivisibleby4,so15,476isevenlydivisibleby4.Is593,768divisibleby4?Thetensdigit,6,isevensowecanignoreit.Theunitsdigit,8,isevenly

divisibleby4,so593,768isevenlydivisibleby4.Divisibilityby5:Five iseasy. If the lastdigitof thenumber is5or0, then thenumber isevenly

divisibleby5.Divisibilityby6:Ifthenumberisevenandthesumofthedigitsisdivisibleby3,thenthenumberis

alsodivisibleby6(because6is2times3).Forinstance,thedigitsumof54is9(5+4=9),whichisdivisibleby3,and54isanevennumber,so54isdivisibleby6.Divisibilityby7:YouwillhavetoreadmybookSpeedMathematics forafullexplanationof this,

buthereisaquickintroduction.Youmultiplythefinaldigitofthenumberby5andaddtheanswertothenumberprecedingit.Iftheanswerisdivisibleby7,thenthenumberisdivisibleby7.Forexample,let’stake343.Thefinaldigitis3.Wemultiply3by5toget15.Weadd15tothenumberinfrontofthe3,whichis34,toget49.Then,49isevenlydivisibleby7,so343isaswell.Divisibilityby8:Ifthefinalthreedigitsareevenlydivisibleby8thenthenumberisevenlydivisible

by8(because1,000=8×125).Herewehaveaneasychecksimilartothecheckfor4.Ifthehundredsdigitisevenitcanbeignored.

Ifitisodd,add4tothefinaltwodigitsofthenumber.Then,ifthefinaltwodigitsaredivisibleby8,thenumberisdivisibleby8.Forexample,isthenumber57,328divisibleby8?Thehundredsdigit,3,isoddsoweadd4tothe

finaltwodigits,28.So,28+4=32.Thirty-twoisdivisibleby8,so57,328isevenlydivisibleby8.Divisibilityby9:Ifthedigitsumaddsto9orisevenlydivisibleby9,thenumberisdivisibleby9.Divisibilityby10:Ifthenumberendswitha0,thenumberisdivisibleby10.Divisibilityby11:Ifthedifferencebetweenthesumoftheevenlyplaceddigitsandtheoddlyplaced

digitsisamultipleof11,thenumberisevenlydivisibleby11.Divisibilityby12: If thedigit sum isdivisibleby3and the last twodigits aredivisibleby4, the

numberisdivisibleby12.Divisibilityby13:Again,afullexplanationisinSpeedMathematics.Therulefor13issimilartothe

rulefor7.Youmultiplythefinaldigitofthenumberby4andaddtheanswertothenumberprecedingit.Iftheanswerisdivisibleby13,thenthenumberisdivisibleby13.Forexample,let’stake91.Thefinaldigitis1.Wemultiply1by4toget4.Weadd4tothenumberinfrontofthe1,whichis9,toget13.And13isevenlydivisibleby13,so91isaswell.Let’stryitagainwith299.Is299evenlydivisibleby13?Itishardtotellwithoutactuallydoinga

division.Wemultiplythefinaldigit,9,by4togetananswerof36.Add36tothe29infrontofthe9toget 65.Youmay recognise that 65 is 13 × 5, but if not, carry out the operation again.We are nowchecking65.Multiply5by4toget20.Add20tothe6in65toget26.Wecanseethat26is2×13,sothereisnodoubtnowthat299isevenlydivisibleby13.Itisoftenusefultoknowifanumberhasfactorsthatcanbeusedtosimplifyafractionortosimplify

amultiplicationordivisionproblem.

AppendixE

KEEPINGCOUNT

Peoplehavekeptcountofthingssincetheearliesttimes.Wekeepcounteveryday.HowmanybooksdoIhave?HowmanyproblemshaveIdone?Howmanygoals?Howmanyruns?Howmanypoints?WhenIwasyoungerIworkedonelectronicequipmentandIhadtokeepcountofhowmanyunitsI

hadrepairedeachday.Ihadagoaltorepairmorethan80unitseachday.Ireached80severaltimesbutIneverpassedit.People often keep count by drawing four vertical strokes and then crossing them outwith a fifth

stroke.Thatwayitiseasytotallytheamountattheendbycountinginfives:5,10,15,20,25,etc.

Some Japanesewomenworked for the same electronics company and they kept count bywritingwhattheysaidwastheJapaneseorChinesesymbolfor5.

AtthetimeIwasusingthewrongsequence,butJackZhangkindlyshowedmethecorrectsequenceforthisbook.ItishowIstillkeepscorewhenIamcountingvariousobjects.Themethodsareusefulwhenyouhavetocountaselectionofitems.Forinstance,red,green,blueandyellowitemsthatareallmixedup.Imightmakealistlikethistokeepcount:

WhenIhavefinishedsortingtheitemsIsimplycountmysymbolsandwritemytotalstotheside.ARussianmethodofcountingistodrawsquaresandthenalinediagonallythroughwhenyoureach

five.Itlookslikethis:

Thesearemethodsyoucanplayandexperimentwith.Seewhichmethodofkeepingcountyoulikebest.

AppendixF

PLUSANDMINUSNUMBERS

NotetoParentsandTeachersThemethod ofmultiplication taught in this book introduces positive and negative numbers tomostchildren.Themethodmakespositiveandnegativetangibleinsteadofanabstractidea.Positivenumbersgoabovewhenyoumultiply;negativenumbersgobelow.Studentsbecomeusedtotheideathatwhenyoumultiply terms thatareboth thesameyougetapositive (plus)answer. If theyaredifferent (oneaboveandonebelow)youhavetosubtract—yougetaminusanswer.Eveniftheydon’tunderstandit,itstillmakessense.Howdoyouexplainpositiveandnegativenumbers?HereishowIliketodoit.Tomeitmakessense

ifyousee‘positive’asmoneypeopleoweyou.Thatismoneyyouhave.‘Negative’ismoneyyouowe,orbills thatyouhave topay.Threebillsof$2 is3 times−2,givingananswerof−$6.Youowe$6.Mathematicallyitlookslikethis:3×2=−6.Now,whatifsomeonetookawaythosethreebillsfor$2?Thatisminus3billsoramountsofminus

$2.Thatmeansyouhave$6morethanbeforethebillsweretakenaway.Youcouldwritethatas:−3×−2=+6.I tell students not to worry about the concept too much. They don’t have to understand it

immediately.ItellthemwewilljustkeepusingtheconceptandIwillkeepexplainingituntiltheydounderstand.Itisnotaracetoseewhocanunderstanditfirst.Itellthestudentsthatunderstandingwillcome.You can give examples of forward speed and headwind. Forward speed is positive; headwind is

negative.Addingandsubtractingpositiveandnegativenumbersisnobigdeal.Youaddpositive;yousubtractnegative.Itisjustamatterofrecognisingwhichiswhich.

AppendixG

PERCENTAGES

WhatisAPercentage?Percentages are important. We are always meeting up with them whether we like them or not.Percentagesareusedinmostsportingcompetitions.Thepercentsignlookslikethis:%.Storesoffersaleswith20%off.Taxesarequotedinpercentages.Wearetoldthereisa6%surcharge

oncertainitems.Mostpeoplearehappytoletsomebodyelsecalculatetheamountforthem,andtheysimplytaketheperson’swordforwhattheypay.Oftenweneedtocalculatewhatwewillhavetopaybeforehand—weneedtobeabletocalculate

percentagesforourselves.Percentagesarereallyfractions:‘percent’actuallymeans‘foreveryhundred’.So50%means50for

every100,or50/100.Youcanseethat50and100arebothdivisibleby50.Fiftydividesonceinto50andtwiceinto100.So50%isthesameas50/100or½.Itmakessensethat50ishalfof100.Ifyoureadthat23%ofpeoplehaveblondhair,itmeansthat23peopleoutofevery100haveblond

hair.Youcouldalsosaythat0.23ofthepopulationhasblondhair,butitiseasierandmorecommontosayitasapercentage.Moneyhasbuilt-inpercentages.Becausethereare100centsinadollar,54centsis54%ofadollar.

CalculatingAQuantityAsAPercentageofAnotherHowdoyoufindwhatpercentageonenumberisofanother?Forinstance,if32peopleatameetingarefemalesandthereare58peopleattending,whatpercentagearefemale?Asafractiontheyare 32/58.Tofindthepercentageyousimplymultiplythisby100.Youcalculatethisasfollows:

32/58(100=3,200/58%3,200÷58=5510/58%

Thiswouldnormallybeexpressedasadecimal:55.17%.So,25boys inacrowdof50wouldmean that50%areboys.Why?Becausehalfareboysand½

times100is50.Wesimplymultiplythefractionby100.On theotherhand, to findapercentageofanumber—for instance,what is20%of2,500—we

multiplythenumberbythepercentagedividedby100.Wewouldsetitoutlikethis:2,500×20/100=

We can cancel the one hundreds, or simply divide 2,500 by 100, which is, of course, 25. Thenmultiply25by20(multiplyby2,thenby10)toget500.

So,summarised,tocalculatethepercentageof17to58,forexample,youwoulddivide17by58andmultiplytheanswerby100.Yourcalculationwouldlooklikethis:

17/58×100=1,700/58=29.3

CalculatingAPercentageofAGivenQuantity

Tocalculate30%of$58youwouldmultiply58by30dividedby100.Because30and100arebothdivisibleby10youwouldmultiply58by3anddivideby10.

58×30/100=1,740/100=17.4or$17.40Ifyouarenotsurehowtodothecalculation,simplifythenumberstoseehowyoudoit.Forinstance,

let’ssayyouwanttofind23%of485andyoudon’tknowwhatyouhavetomultiplyordivide;trythesamecalculationwitheasynumbers.Howdoyoufind50%(½)of10?Youcanseethatyoumultiply10by½oryoumultiplyby50/100.Thenyoucanapplythismethodtofinding23%of485.Yousubstitute23for50and485for10.Youwouldget:485×23/100=111.55

Youcancheckbyestimationbysayingthat23%isalmostaquarter,andaquarterof400is100.Percentages are used in all areas of life: percentage discounts, percentage profits, statistics, sports

resultsandexaminationscores.Wemeetthemeveryday.

AppendixH

HINTSFORLEARNING

Wedon’tallthinkthesamewayandwedon’talllearnthesameway.WhenIwasinteachers’college,oneteachertoldmethatif70%ofhisstudentsunderstoodhisexplanation,theotherstudentsonlyhadthemselves to blame if they didn’t. If most of his students understood, the others should haveunderstoodaswell.Anotherteachertoldme,whenIexplainsomethingIcanexpectonlyabout70%ofmystudentsto

understand.Theydon’tallthinkandlearnthesameway.Ihavetofindotherwaysofexplainingsothattheother30%willunderstandaswell.Thathasbeenmyphilosophy.Ikeepexplainingaprincipleuntileveryoneunderstands.Theproblemisthatastudentwhodoesn’tunderstandtheteacher’sexplanationwillgenerallythinkit

ishisorherownfault.Theythink,Imustbedumb.Theotherkidsunderstand,whycan’tI?I’mnotassmartastheotherkidsorIdon’thaveamathematicalbrain.The same principle applies to learning from books. A book usually has one explanation for each

principletaught.Iftheexplanationdoesn’tsuitthewayyouthinkormakesensetoyou,youareinclinedtothinkitis‘allabovemyhead’.Iamnotsmartenough.Youwouldbewrong.Youneedadifferentexplanation.Ifyouaretryingtolearnsomethingfroma

book, try several. If yourmajor or ‘set’ textbook does the job, that’s great. If you can’t understandsomething,don’tthinkyouarenotsmartenough;tryanotherbookwithadifferentexplanation.Findafriendwhounderstandsitandaskyourfriendtoexplainittoyou.Lookforotherbooksinsecond-handbookshops,askolderstudentsfortheiroldbooks,orgotoyourlibraryandaskforbooksonthesubject.Often,alibrarybookiseasiertounderstandbecauseitisnotwrittenasatextbook.WhenIteachmathematicsandrelatedsubjects,Ialwaysreadtheexplanationgiveninseveralbooks

so that I can find ideas for differentways to teach it in the classroom.Also,when I am teaching aprocedureinmaths,physicsorelectronics,Idoallcalculationsaloud,withallofmythinkingoutloudsoeveryoneunderstandsnotonlywhatIamdoing,butalsohowIamdoingit.Iaskmystudentstodothesamesowecanfollowwhatisgoingoninsidetheirheads.

AppendixI

ESTIMATING

Often,itmakesfarmoresensetoestimatethanitdoestogiveanexactanswer.Someanswerscan’tbegivenwithabsoluteaccuracy;thevalueofpiisalwaysapproximate,asisthevalueofthesquarerootof2.Bothofthesevaluesareusedandcalculatedregularly.Evenpercentagediscountsinyourdepartmentstoreareroundedofftothenearestcentornearest5cents.Whenyouarebuyingpaintorothermaterialsfrom a hardware store you have to estimate. It is a good idea to estimate high to be sure you haveenoughnails,ribbon,orwhateveritisyouarebuying.Anexactamountissometimesnotagoodidea.Weusedtheideaofestimationwhenwelookedatstandardlongdivision.Weroundedoffthedivisor

toestimateeachdigitoftheanswer,thenwetestedeachestimate.IfIambuyingcomputerscreensforaschool,howmuchwill58computerscreenscostmeiftheyare

$399each?Togetaroughestimate,insteadofmultiplying399by58,Iwouldmultiply400by60.So,myestimation is 400×60, or 4×6×100×10,which is 24,000.Because I roundedboth amountsupwardsIwouldsayIactuallyhavetopayabitlessthan$24,000.Ofcourse,whenitcomestimetopay,IwanttopayexactlywhatIowe.Theactualamountis$23,142,butmyinstantestimationtellsmewhatsortofpricetoexpect.IfIamdrivingat100kilometresperhour,howlongwill it takemetodrive450kilometres?Most

studentswouldsay4½hours,butthereareotherfactorstoconsider.WillIneedfuelontheway?Willtherebehold-upsonthefreeway?WillIwanttostopforabreakorhaveamealorsnackontheway?Myestimatemightbe6hours.Also,pastexperiencewillbeafactorinmyestimation.Thegeneralruleforroundingofftoestimateanansweristotrytoroundoffupwardsanddownwards

asequallyasyoucan.Howwouldyouroundoffthefollowingnumbers:123;409;12,857;948;830?Youranswerswoulddependonthedegreeofaccuracyyouwant.ProbablyIwouldroundoffthefirst

numberto125,oreven100.Then:400;13,000;950or1,000;and800or850.IfIamroundingoffinthesupermarketandIwanttoknowifIhaveenoughcashinmypocket,Iwouldroundofftothenearest50centsforeachitem.IfIwerebuyingcarsforacaryardIwouldprobablyroundofftothenearesthundreddollars.How would you estimate the answer to 489 × 706? I would multiply 500 by 700. Because one

numberisroundedoffdownwardsandtheotherupwardsIwouldexpectmyanswertobefairlyclose.700×500=350,000489×706=345,234

Theanswerhasanerrorof1.36%.Thatisprettycloseforaninstantestimate.Estimatinganswersisagoodexerciseasitgivesyoua‘feel’fortherightanswer.Onegoodtestfor

anyanswerinmathematicsis,doesitmakesense?Thatisthemajortestforanymathematicalproblem.

AppendixJ

SQUARINGNUMBERSENDINGIN5

Whenyoumultiplyanumberbyitself(forexample,3×3,or5×5,or17×17)youaresquaring it.Seventeensquared(or17×17)iswrittenas:172.Thesmall2writtenafterthe17tellsyouhowmanyseventeensyouaremultiplying.Ifyouwrote173itwouldmeanthreeseventeensmultipliedtogether,or17×17×17.Now, to square anynumber ending in 5 you simply ignore the 5 on the end and take the number

writteninfront.So,ifwesquare75(75×75)weignorethe5andtakethenumberinfront,whichis7.Add1tothe7toget8.Nowmultiply7and8together.

7×8=56Thatisthefirstpartoftheanswer.Forthelastpartyoujustsquare5.5×5=25

The25isalwaysthelastpartoftheanswer.Theansweris5,625.Why does it work? Try using 70 as a reference number and you will see we are dealing with

somethingwealreadyknow.

Trytheproblemforyourselfusing80asareferencenumber.Itworksoutthesame.So,for1352(135×135)thefrontpartofthenumberis13(infrontofthe5).Weadd1toget14.Now

multiply13×14=182(usingtheshortcutinChapter3).Wesquare5,orjustput25attheendofouranswer.1352=18,225

Forthatweusedeither130or140asareferencenumber.Trysquaring965inyourhead.Ninety-sixisinfrontofthe5.96+1=9796×97=9,312

Put25attheendfortheanswer.9652=931,225

Thatisreallyimpressive.

TestyourselfNowtrytheseforyourself.Don’twriteanything—dothemallinyourhead.a)352

b)852

c)1152

d)9852

Theanswersare:a)1,225b)7,225c)13,225d)970,225

Formore strategies about squaring numbers (and finding square roots), you should readmy bookSpeedMathematics.

AppendixK

PRACTICESHEETS

Pleasephotocopythesesheetsasrequired.

INDEX

13,14and15timestables

Aaddition

—breakdownofnumbers—orderofaddition

Ccalculators

—beatingcastingoutnines

seecheckinganswerssubstitutenumbers

checkinganswers—anysizenumber—castingtwos,tensandfives—fordivision—shortcutfor—substitutenumbers—whyitworks

combiningmethods

Ddecimals

—roundingoffdivisibilityofnumbersdivision

—by9(shortcut)—bynumbersendingin—changingtomultiplication—directlongdivision—estimatinganswers—largernumbers—longdivisionbyfactors—longdivisionmadeeasy—roundingupwards—smallernumbers—usingcircles—withdecimals

doublingnumbers

Eestimating

Ffactorsfractions

—adding—changingtodecimals—dividing—multiplying—simplifyinganswers—subtracting

HhalvingnumbersKkeepingcountLlongdivisionseedivisionMmultiplication

—by5—by9—by11—byfactors—bymultiplesof11—direct—double—explained2—gettingstarted—highernumbers—lowernumbers—numbersabove100—numbersaboveandbelow20—numbersbelow20—numbersintheteens—numbersjustbelow100—ofdecimals—two-digitnumbers

Nnegativenumbersnotestoparentsandteachersnumbersincircles

—multiplying

P

percentages—calculating

positivenumberspracticesheets

Rreferencenumbers

—experimentingwith—numbersabove—numbersaboveandbelow—reasonsforusing—using50—using—usinganynumber—usingdecimals—usingfactorsasdivision—usingfractions99—usingtwonumbers

remainders—findingwithacalculator

rememberingmethodsSsolvingproblemsinyourheadspeedmathematicsmethodsquaringnumbersendingin5substitutenumbers

seecheckinganswerssubstitutenumbers

subtraction—fromapowerof10—numbersaround100—shortcutfor—smallernumbers—writtenmethods

Ttestyourself

Uusingthemethodsinclass

Wworkingthroughproblems