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DoverBooksonMathematics
HANDBOOKOFMATHEMATICALFUNCTIONSWITHFORMULASGRAPHSANDMATHEMATICALTABLESEditedbyMiltonAbramowitzandIreneAStegun(0-486-61272-4)ABSTRACTANDCONCRETECATEGORIESTHEJOYOFCATSJiriAdamekHorstHerrlichGeorgeEStrecker(0-486-46934-4)MATHEMATICSITSCONTENTMETHODSANDMEANINGADAleksandrovANKolmogorovandMALavrentrsquoev(0-486-40916-3)REALVARIABLESWITHBASICMETRICSPACETOPOLOGYRobertBAsh(0-486-47220-5)PROBLEMSOLVINGTHROUGHRECREATIONALMATHEMATICSBonnieAverbachandOrinChein(0-486-40917-1)VECTORCALCULUSPeterBaxandallandHansLiebeck(0-486-46620-5)
INTRODUCTIONTOVECTORSANDTENSORSSECONDEDITION-TWOVOLUMESBOUNDASONERayMBowenandC-CWang(0-486-46914-X)FOURIERANALYSISINSEVERALCOMPLEXVARIABLESLeonEhrenpreis(0-486-44975-0)THETHIRTEENBOOKSOFTHEELEMENTSVOL2Euclid(0-486-60089-0)
THETHIRTEENBOOKSOFTHEELEMENTSVOL1EuclidEditedbyThomasLHeath(0-486-60088-2)ANINTRODUCTIONTODIFFERENTIALEQUATIONSANDTHEIRAPPLICATIONSStanleyJFarlow(0-486-44595-X)DISCOVERINGMATHEMATICSTHEARTOFINVESTIGATIONAGardiner(0-486-45299-9)ORDINARYDIFFERENTIALEQUATIONSJackKHale(0-486-47211-6)
METHODSOFAPPLIEDMATHEMATICSFrancisBHildebrand(0-486-67002-3)
BASICALGEBRAISECONDEDITIONNathanJacobson(0-486-47189-6)
BASICALGEBRAIISECONDEDITIONNathanJacobson(0-486-47187-X)
GEOMETRYANDCONVEXITYASTUDYINMATHEMATICALMETHODSPaulJKellyandMaxLWeiss(0-486-46980-8)COMPANIONTOCONCRETEMATHEMATICSMATHEMATICAL
TECHNIQUESANDVARIOUSAPPLICATIONSZAMelzak(0-486-45781-8)MATHEMATICALPROGRAMMINGStevenVajda(0-486-47213-2)
FOUNDATIONSOFGEOMETRYCRWylieJr(0-486-47214-0)
SeeeveryDoverbookinprintatwwwdoverpublicationscom
BOOKSBYAALBERTKLAF
CalculusRefresherforTechnicalMenTrigonometryRefresherforTechnicalMen
ArithmeticRefresher
AAlbertKlaf
Copyrightcopy1964byMollieGKlafAllrightsreserved
ArithmeticRefresherwasfirstpublishedbyDoverPublicationsIncin1964underthetitleArithmeticRefresherforPracticalMen
LibraryofCongressCatalogCardNumber64-18856InternationalStandardBookNumber
9780486141930
ManufacturedintheUnitedStatesbyCourierCorporation21241622wwwdoverpublicationscom
FOREWORD
MyfatherwrotethisArithmeticRefresherforPracticalMenforthemassaudienceofprofessionalsandlaymenwhoarefrequentlyfacedwithnumericalproblemsThebookincludestheknowledgeandpracticalexperiencegatheredduringalifetimeofsearchingcuriosityHecompletedthemanuscriptayearbeforehispassingItisthetestamentofacareerdedicatedtopublicserviceandmathematicalenlightenment
IwishtoexpressmydeepappreciationtomyfatherrsquoscolleagueMrVictorFeigelmanBCEMCEforsolvingthesampleproblemsandcheckingthemanuscriptThanksarealsoduetoMrHaywardCirkerPresidentofDoverPublicationsIncwhowasmyfatherrsquosvaluedfriendaswellashispublisher
ThisbookwastohavebeenoneofaseriesthatbeganwithhisCalculusRefresherforTechnicalMenandprogressedtohisTrigonometryRefresherforTechnicalMenThesucceedingvolumeswillofcourseremainunwrittenButthebesthasbeensaidNowitmustbeusedbythosewhoseektoexperiencethejoyofmathematicsmyfathersodeeplyfelt
FRANKLINSKLAFMD
TableofContents
DoverBooksonMathematicsBOOKSBYAALBERTKLAFTitlePageCopyrightPageFOREWORD
INTRODUCTIONCHAPTERI-ADDITIONCHAPTERII-SUBTRACTIONCHAPTERIII-MULTIPLICATIONCHAPTERIV-DIVISIONCHAPTERV-FACTORSmdashMULTIPLESmdashCANCELLATIONCHAPTERVI-COMMONFRACTIONSCHAPTERVII-DECIMALFRACTIONSCHAPTERVIII-PERCENTAGECHAPTERIX-INTERESTCHAPTERX-RATIOmdashPROPORTIONmdashVARIATIONCHAPTERXI-AVERAGESCHAPTERXII-DENOMINATENUMBERSCHAPTERXIII-POWERmdashROOTSmdashRADICALSCHAPTERXIV-LOGARITHMSCHAPTERXV-POSITIVEANDNEGATIVENUMBERSCHAPTERXVI-PROGRESSIONSmdashSERIESCHAPTERXVII-GRAPHSmdashCHARTSCHAPTERXVIII-BUSINESSmdashFINANCECHAPTERXIX-VARIOUSTOPICSCHAPTERXX-INTRODUCTIONTOALGEBRAAPPENDIXA-ANSWERSTOPROBLEMSAPPENDIXBTABLESINDEXACATALOGOFSELECTEDDOVERBOOKSINALLFIELDSOF
INTEREST
INTRODUCTION
1WhatisarithmeticThescienceofnumberandtheartofcomputation
2WhatisournumericalsystemcalledandwhyisitsocalledItiscalledtheArabicsystembecauseitwasgiventousbytheArabswho
developeditfromtheHindusystem
3WhatisadigitAnywholenumberfrom1through9iscalledadigitThus1234567
89arecalleddigits
4WhatisacipherandwhatisitssymbolThewordldquocipherrdquocomesfromanArabicwordmeaningldquoemptyrdquoandmeans
ldquonodigitrdquoThesymbolforacipheris0
5WhatothercommonlyusedwordsmaybesubstitutedforthewordldquocipherrdquoldquoZerordquoandldquonoughtrdquomaybeusedforldquocipherrdquo
6WhatisthefoundationoftheArabicnumericalsystemThefoundationconsistsoftheninesymbolscalleddigitsmdash1234567
89mdashandonesymbolcalledacipherzeroornought
7WhatisadecimalpointandwhatisitssymbolAdecimalpointisapointthatisusedtoseparatethefractionalpartofa
numberfromawholenumberanditssymbolisadot[]
8Whatismeantbycomputationorcalculation
ComputationorcalculationistheprocessofsubjectingnumberstocertainoperationsThewordldquocalculationrdquocomesfromaLatinwordmeaningldquopebblerdquoasreckoningwasdonewithcountersorpebbles
9HowmanyfundamentaloperationsarethereinarithmeticTherearesixoperationsallgrowingoutofthefirst
Thesixoperationsaredividedintotwogroups(a)threedirectoperationsand(b)threeinverseoperationseachofwhichhastheeffectofundoingoneofgroup(a)
Group(a ) Direct Operat ion Group(b ) InverseOperat ion
1Addition 4Subtraction
2Multiplication 5Division
3Involution 6Evolution
10Whatarethesymbolsfor(a)ldquoequalstordquoorldquoequalsrdquo(b)addition(c)subtraction(d)multiplication(e)division(f)involution(g)evolutionand(h)ldquothereforerdquo(a)Theequalssign[=]meansldquoequalstordquoorldquoequalsrdquo
1+1=2oneplusoneequalstwo
(b)Theplussign[+]meansldquoplusrdquoldquoandrdquoorldquoaddedtordquo
2+2=4twoplustwoequalsfourortwoandtwoequalsfourortwoaddedtotwoequalsfour
(c)Theminussign[mdash]meansldquominusrdquoldquosubtractedfromrdquoorldquofromrdquo
5ndash3=2fiveminusthreeequalstwoorthreesubtractedfromfiveequalstwoorthreefromfiveequalstwo
(d)Themultiplicationsign[times]meansldquomultipliedbyrdquoorldquotimesrdquo
5times3=15fivemultipliedbythreeequalsfifteenorfivetimesthreeequalsfifteen
ldquoTimesrdquomayalsobeindicatedbyadotinthecenterofthelinebetweenthetwonumbers
5bull3=15fivetimesthreeequalsfifteen
(e)Thedivisionsign[divide]meansldquodividedbyrdquo
10divide2=5tendividedbytwoequalsfive
Thesigns or meanldquodividedintordquo
twodividedintotenequalsfive
twodividedintotenequalsfiveThisformisusedinlongdivision
expressedasafractionmeansldquotendividedbytwoequalsfiverdquo
(f)Asmallnumberplacedintheupperright-handcornerofanumberisusedtoindicatethenumberoftimesthenumberistobemultipliedbyitself
25=2times2times2times2times2=32
ReadTwotothefifthpowerequalsthirty-twoTheprocessoffindingapowerofanumberisinvolution
(g)Theradicalsign[radic]meansldquorootofrdquoAfigureisplacedabovetheradictoindicatetheroottakenItisomittedinthecaseofthesquareroot
Itistheinverseoperationofinvolutionandiscalledevolution
Thefifthrootofthirty-twoistwowhichisthenumberthatwhenmultipliedbyitselffivetimeswillgivethirty-two
(h)Thesign[there4]meansldquothereforerdquo
11WhatisthesignificanceofparenthesesenclosingnumbersThepresenceofparenthesesmeansthattheoperationswithintheparentheses
aretobeperformedbeforeanyoperationsoutsideAnumberprecedingparenthesesmeansthatthefinalfigurewithinparenthesesistobemultipliedby
thatnumber
EXAMPLE3(5+2)=21Theoperation(5+2)isperformedfirst=7Then3times7=21Theoperationof3timesisthenperformed
12WhatismeantbyaunitAnyonethingiscalledaunit
13WhatismeantbyanumberAunitorcollectionofunitsiscalledanumber
14WhatismeantbyanintegerwholenumberoranintegralnumberNumbersrepresentingwholeunitsarecalledintegerswholenumbersor
integralnumbers
EXAMPLES128751342659areintegersorwholenumbers
15WhatsymbolsareusedtoexpressnumbersDigitsorfiguresareusedtoexpressnumbers
Thesymbol0=zeroisusedtoexpressldquonodigitrdquo
16HowaredigitsusedtoexpressnumbersinourArabicsystemThevalueofthedigitisfixedbyitspositionstartingfromtherightandgoing
towardstheleft
ThefirstpositionisthatofldquounitsrdquoThenextpositionisthatofldquotensrdquoThethirdpositionisthatofldquohundredsrdquoThesearecalledthethreeldquoordersrdquoAgroupofthreeordersiscalledaperiod
17HowaretheordersandperiodsarrangedintheArabicsystem(Rarelyisthereuseforanynumberlargerthanldquotrillionsrdquo)
18HowdowereadanumberwrittenintheArabicsystemSeparatethenumbersbycommasintoldquoperiodsrdquoorgroupsofthreefigures
beginningattheright
Nowbeginattheleftandreadeachperiodasifitstoodaloneaddingthenameoftheldquoperiodrdquo
EXAMPLE7653460534646(above)
ReadSeventrillionsixhundredandfifty-threebillionfourhundredandsixtymillionfivehundredandthirty-fourthousandsixhundredandforty-six
Notethatthewordldquoandrdquomayinallcasesbeomitted
19WhatistherelationofaunitofanyperiodtothatofthenextlowerperiodTheunitofanyperiod=1000unitsofthenextlowerperiod
EXAMPLE
Onethousand=1000=1000unitsOnemillion=1000000=1000thousandsOnebillion=1000000000=1000millionsOnetrillion=1000000000000=1000billions
20HowwouldyouwriteanumberinfiguresBeginattheleftandwritethehundredstensandunitsofeachldquoperiodrdquo
placingzerosinallvacantplacesandacommabetweeneachtwoperiods
EXAMPLE400536080209
Fourhundredbillionfivehundredthirty-sixmillioneightythousandtwohundrednine
21HowdozerosbeforeorafteranumberaffectthenumberAzeroinfrontofanumberdoesnotaffectit
EXAMPLE0008060=eightthousandsixty
Azeroafteranumbermovesthenumberoneplacetotheleftormultipliesitby10
EXAMPLE8060Nowaddazeroafterthenumberor80600Theeightthousandsixtybecomeseightythousandsixhundred
Twozerosaddedattherightmovesthenumbertwoplacestotheleftormultipliesitby100
EXAMPLE80600Addtwozerosgetting8060000=eightmillionsixtythousandAndsoonwithaddedzeros
ForanothermethodofwritingverylargenumbersseeQuestion534
22Whatarethenamesoftheperiodsbeyondtrillionsuptoandincludingthetwelfthperiod5Trillions6Quadrillions7Quintillions8Sextillions9Septillions10Octillions11Nonillions12Decillions
23HowmaywethinkoftheordersofthesuccessiveperiodsasbeingbuiltupofbundlesoflowerunitsTaketheldquounitsrdquoperiodThelargestdigitthatcanappearintheunitsorderis
9Nowadd1to9anditbecomesabundleoften=10Thismeansdigit1intheldquotensrdquoorderandzerointheunitsorderNotethattheldquotensrdquopositionis10times
theunitsposition
Thelargestnumberthatcanappearintheldquotensrdquoandldquounitsrdquoordersis99Nowadd1to99anditbecomesabundleofonehundred=100Thismeansdigit1intheldquohundredsrdquoorderandzeroinboththetensandunitsorders100mayalsobethoughtofasmadeupof10bundlesofldquotensrdquoNotethattheldquohundredsrdquopositionistentimestheldquotensrdquoposition
NowtaketheldquothousandsrdquoperiodThelargestnumberthatcanappearintheldquounitsrdquoperiodis999Nowadd1to999anditbecomesabundleofonethousand=1000Thismeansdigit1intheldquounitsrdquoorderofthisperiodandzerosintheordersoftheunitsperiodTheldquothousandsrdquopositionistentimestheldquohundredsrdquoposition1000mayalsobethoughtofasmadeupof10bundlesofonehundredsor100bundlesoftens
Thelargestnumberthatcanappearintheldquounitsrdquoorderofthisperiodtogetherwiththeunitsperiodis9999Nowadd1to9999anditbecomesabundleoftenthousand=10000Thismeansdigit1intheldquotensrdquoorderofthisperiodandzerosinalltheotherplaces10000mayalsobethoughtofasmadeupof10bundlesofonethousands100bundlesofonehundredsor1000bundlesoftensTheldquotenthousandsrdquopositionistentimestheldquothousandsrdquoposition
Thelargestnumberthatcanappearinthetensandunitsordersofthisperiodtogetherwithentireunitsperiodis99999Nowadd1to99999anditbecomesabundleofonehundredthousand=100000Thismeansdigit1intheldquohundredsrdquoorderofthisperiodandzeroinalltheotherplaces100000mayalsobethoughtofasmadeupof10bundlesoftenthousands100bundlesofonethousands1000bundlesofonehundredsor10000bundlesoftensTheldquohundredthousandrdquopositionistentimestheldquotenthousandsrdquoposition
FollowasimilarprocedureintheldquomillionsrdquoperiodAdd1to999999gettingabundleofonemillion=1000000Digit1isintheunitsorderofthisperiod1000000maybegottenby10bundlesofonehundredthousands100bundlesoftenthousands1000bundlesofthousands10000bundlesofhundredsor100000bundlesoftensTheldquomillionsrdquopositionistentimestheldquohundredthousandrdquoposition
Add1to9999999gettingabundleoftenmillion=1000000010000000mayalsobegottenby10bundlesofonemillions100bundlesofonehundredthousands1000bundlesoftenthousands10000bundlesofthousands100000bundlesofhundredsor1000000bundlesoftensTheldquotenmillionsrdquopositionistentimestheldquomillionsrdquoposition
Add1to99999999gettingabundleofonehundredmillion=100000000whichmayalsobegottenby10bundlesoftenmillions100bundlesofmillions1000bundlesofonehundredthousands10000bundlesoftenthousands100000bundlesofonethousands1000000bundlesofhundreds10000000bundlesoftens
100000000=10times10000000
Thisprocedurecanbecontinuedtotheotherperiodswhichfollowthisone
NotetherelationofthebundlesAnybundleistentimesthesizeofthebundleonitsrightandonetenththatofabundleatitsimmediateleft
24WhenisadecimalpointusedItisusedtoexpressvalueslessthanone
EXAMPLES
02=twotenthsofoneunit= infractionform
002=twohundredthsofoneunit= infractionform
0002=twothousandthsofoneunit= infractionform
00002=twotenthousandthsofoneunit= infractionform
ForanothermethodofwritingdecimalsseeQuestion536
25Whatarethenamesofthedecimalorfractionalplaces
NotethevalueofthedecimalbecomessmallerandsmallerasyouadvancetotherightAlsothereisnounitsplaceafterthedecimalpointThisreducesthenumberofplacesby1ascomparedwithawholenumber
26HowisadecimalreadReadexactlyasifitwereawholenumberbutwiththeadditionofthe
fractionalnameofthelowestplaceTheabovenumberisreadasldquosixhundredeightymillionfiftyseventhousandninehundredtwenty-threebillionthsrdquoThelowestorsmallestplacehereisbillionths
27WhatistherelationofeachplaceinadecimaltotheplacethatprecedesitEachplaceisone-tenth( )oftheprecedingplaceItisthusaten(10)times
smallerfraction
EXAMPLE
ReadTwohundredforty-seventhousandeighthundredninety-sevenmillionths
28CanyoushowthatzerosaddedafterthelastdigitdonotaffectthevalueofthedecimalEXAMPLE
29HowdoesazeroplacedbeforeadigitaffectthevalueofthedecimalThevalueofadigitisdividedbytenasyoumovefromlefttorightSo
addingazerobeforethedigitmovesthedigitoneplacetotherightandmakesitsvalueonetenthofwhatitwas
EXAMPLE
Addingtwozerosmovesthedigittwoplacestotherightandmakesitsvalueonehundredthofwhatitwas
EXAMPLE
Eachadditionalzeroreducesitsformervaluebyonetenthagain
30HowisanumberreadthatconsistsofawholenumberandadecimalThepointseparatesthewholenumberfromthedecimalThedecimalpointis
readldquoandrdquo
EXAMPLE2451ReadTwenty-fourandfifty-onehundredthsItmayalsobereadTwenty-fourpointfifty-one
Toavoidanypossibilitythatthedecimalpointwillbeoverlookedwrite06insteadof6(=sixtenths)
31HowdowewritedollarsandcentsPlaceadecimalpointbetweenthedollarsandcents$1643=sixteendollars
forty-threecents
Numberstotheleftofthedecimalpointaredollarstotherightofitarecentsinthefirsttwoplaceswithanumberinthethirdplaceasmills$16437=sixteendollarsforty-threecentssevenmills
Note10mills=1cent=$001Thereforeforty-threecentssevenmills=fourhundredthirtysevenmills
Whenthenumberofcentsislessthan10writeazerointhetenthsplaceattherightofthedecimalpoint
$308=threedollarseightcents$310=threedollarstencents
32WhataretheessentialsymbolsintheRomansystemofnumerationInheritedfromtheEtruscanstheRomansystemofnotationusessevencapital
lettersofthealphabetandcombinationsoftheseletterstoexpressnumbers
Abaroveralettermultipliesitsvalueby1000
33Whataretherulesforthevaluesofthesymbolswhenusedincombinations(a)Eachrepetitionofaletterrepeatsitsvalue
EXAMPLES
II=2III=3XX=20XXX=30CCC=300MM=2000
(b)Aletterafteroneofgreatervalueisaddedtoit
EXAMPLES
(c)Aletterbeforeoneofgreatervalueissubtractedfromit
EXAMPLES
(d)Aletterbetweentwolettersofgreatervalueissubtractedfromtheletterwhichfollowsit
EXAMPLES
PROBLEMS1
1Howmanyunitsin379
2Howmanytensin304060
3Howmanytensandunitsin1937467296
4Howmanybundlesofhundredsin300500700900
5Howmanybundlesofhundredstensandunitsin76523448953697765885456798548958842891346738
6Whatis1000calledandhowmanybundlesofhundredsareinit
7Howmanybundlesofthousandshundredstensandunitsaretherein748680909935580325002925762392604087607978503374783959749294
8Whatis10000calledandhowmanybundlesofthousandsareinit
9Howmanybundlesoftenthousandsthousandshundredstensandunitsarein603084695137568453828946563895349569285798975203064595199358349259887229573
10Howmanybundlesofthousandsarein100000andwhatisthisnumbercalled
11Howmanybundlesofhundredthousandstenthousandsthousandshundredstensandunitsarein369243780979703148282297503005386470460007386364117008204951596382245520498287995193579697
12Whatis1000000calledandhowmanybundlesofthousandstenthousandsandtensareinit
13Howmanybundlesofmillionshundredthousandstenthousandsthousandshundredstensandunitsarein1753002752060082852394289594723795000946028017373111427550005830310047328500015590389214237295296086000829307118392862863401
14Whatis1000000000calledandhowmanybundlesofhundredmillionsandthousandsareinit
15Howmanybundlesofbillionshundredmillionstenmillionsmillions
hundredthousandstenthousandsthousandshundredstensandunitsaretherein27392496000140676200170024060104078410751073964325701900800005
16Howwouldyouexpressthefollowinginfiguresusingacommatoseparatetheperiods(a)Fivehundredeighty-four(b)Threehundredseventeen(c)Sixhundredninety-nine(d)Threehundredseven(e)Onethousandfourhundredeighty-three(f)Eightthousandsixty(g)Ninethousandfourhundred(h)Fourteenthousandsixhundredforty(i)Eighty-eightthousandsix(j)Sixty-sixthousandeighteen(k)Threehundredseventhousandtwohundredforty(l)Eightthousandeight(m)Fourthousandninety-nine(n)Seventythousandtwenty-three(o)Sevenhundredninety-fourthousandthree(p)Sixty-twothousandtwohundredthree(q)Twomilliontwohundredeighty-fivethousand(r)Thirty-eightmilliononehundredforty-eightthousand(s)Sevenmilliontwo(t)Sixty-onemillionfifty-eightthousandsix(u)Onehundredtwenty-twobillionseventythousandseven(v)Fivebillionsevenmillioneightthousandninehundrednine(w)Eighteenbilliononemilliontwohundredthreethousandsixteen(x)Tentrilliontwobilliononemillionsevenhundredsix(y)Onehundredmilliontwenty(z)Sixtymillionsixhundredthousandsixhundred
17Howarethefollowingexpressedasdecimals(a)Seventy-threethousandfivehundredeighty-sixhundred-thousandths(b)Eightthousandandeightthousandths(c)Fivetenthsthreetenthstwoandonetenth(d)Sevenandninethousandthstwelvemillionths(e)Twohundredthirty-fivethousandthsfourhundredninety-one
thousandthssixten-thousandthsthreehundredandthreehundredths(f)Fourandsevententhsnineandtwotenthseighty-sixhundredthsfivehundredandfivethousandths
(g)(h)Threehundredsixty-fourthousandfivehundredseventy-fivemillionths(i)Ninehundredeightmillionsixthousandthirty-fourbillionths
18Whatisthenameoftheplaceattherightoftenthsattherightofhundredthsattherightofthousandthsthefourthplacethefifththesixththeseventh
19Howarethefollowingread(a)16005(b)50607(c)00002(d)879375(e)35201(f)865392(g)23441(h)2003487(i)202074(j)20610057(k)30564(l)974356
20Howarethefollowingreadindollarstenthsandhundredthsofadollar(a)$457(b)$555(c)$666(d)$999
21Howis$356356read
22Howarethefollowingreadasdollarsdimesandcentsandasdollarsandcents(a)$652(b)$344(c)$555(d)$975
(e)$444(f)$888
23Howarethefollowingwrittenascentsusingthedollarsign(a)Sixty-sixhundredthsofadollar(b)Eightyhundredthsofadollar(c)Forty-sevenhundredthsofadollar(d)Tenhundredthsofadollar(e)Onedollarandtwentyhundredths(f)Sevendollarsandtwelvehundredths
24Howarethefollowingwrittenindecimalform
(a)
(b)
(c)
(d)
(e)(f)Fivehundredths(g)Fifty-sixten-thousandths(h)Eleventhousandandthirty-sixtenths(i)Fivehundredhundredths(j)Sixhundredforty-threeten-thousandths
25Howmanymillsaretherein(a)$0475(b)$5621(c)$0022(d)$1054(e)$10765(f)$02555(g)$010(h)$04444
26HowarethefollowingexpressedinArabicnotation(a)XI(b)VIII
(c)XX(d)XIV(e)XXX(f)XXXV(g)XL(h)LXXV(i)XVI(j)XCIV(k)LV
(l)DCCC(m)MCMXX(n)LXXXIII(o)(p)XLIX(q)MDCCCXCVI(r)XCV(s)MDLXXXIX(t)MCXLV(u)MCXL(v)CDIX(w)DCIX(x)MDLIV(y)MDLX(z)MDXLVII(arsquo)MMDCCXCII(brsquo)(crsquo)(drsquo)(ersquo)MMMDCCXIX(frsquo)(grsquo)
27HowwouldyouexpressthefollowinginRomannotation(a)12(b)18(c)19(d)43(e)33
(f)28(g)56(h)82(i)76(j)97(k)117()385(m)240(n)512(o)470(p)742(q)422(r)942(s)1426(t)1874(u)5872(v)24764(w)257846(x)1450729(y)4840005(z)10562942
CHAPTERI
ADDITION
34WhyisadditionmerelyashortwayofcountingIfwehavefourapplesinonegroupandfiveinanotherwemaycountfrom
thefirstobjectinonegrouptothelastobjectintheotherandobtaintheresultnineButseeingthat4+5=9underallconditionswemakeuseofthisfactwithoutstoppingtocounteachtimewemeetthisproblem
TheadditionoftwonumbersisthusseentobeaprocessofregroupingWedonotincreaseanythingwemerelyregroupthenumbers
35WhatisourstandardgrouporbundleOurnumbersystemisbasedongroupsorbundlesoften
EXAMPLE9+8=17Twogroupsof9and8areregroupedintoourstandardarrangementof17oronebundleof10and7unitsWhilewesayldquoseventeenrdquowemustthinkldquotenandsevenrdquoorldquo1tenand7unitsrdquo
36WhatisthusmeantbyadditionItistheprocessoffindingthenumberthatisequaltotwoormorenumbers
groupedtogether
37WhatismeantbysumItistheresultobtainedbyaddingnumbers
38Ofthetotalnumberof45additionsoftwodigitsatatimeforalltheninedigitswhichgivesinglenumbersasasumandwhichgivedoublenumbers(a)Thefollowing20pairsresultinone-numbersums
(b)Thefollowing25pairsgivedoublenumbers
39WhatistheruleforadditionWritethenumberssothatunitsstandunderunitstensundertenshundreds
underhundredsetcBeginattherightandaddtheunitscolumnPutdowntheunitsdigitofthesumandcarrytheldquotensrdquobundlestothenextcolumnrepresentingtheldquotensrdquobundlesDothesamewiththiscolumnPutdownthedigitrepresentingthenumberoftensandcarryanyldquohundredsrdquobundlestothehundredscolumnContinueinthesamemannerwithothercolumns
40WhatistheproperwayofaddingAddwithoutnamingnumbersmerelysums
EXAMPLE
41WhatisthesimplestbutslowestwayofaddingColumnbycolumnandonedigitatatimeAddfromthetopdownorfrom
thebottomupeachwayisacheckontheother
EXAMPLE
42WhatisavariationoftheaboveAddeachcolumnseparatelyWriteonesumundertheotherbutseteach
successivesumonespacetotheleftAsubsequentadditiongivesthetotalorsum
EXAMPLE(asabove)
43HowcangroupingofnumbershelpyouinadditionAddtwoormorenumbersatatimetotwoormoreothersinthecolumns
EXAMPLE
44HowisadditionaccomplishedbymultiplicationoftheaverageofagroupWhenyouhaveagroupofnumberswhosemiddlefigureistheaverageofthe
groupthen
sum=averagenumbertimesnumberoffiguresinthegroup
EXAMPLES(a)Of45and6number5=averageofthethree
there4Sum=5times3=15=(4+5+6)(b)Of89and109=average
there4Sum=times3=27=(8+9+10)(c)Of1213and1413=average
there4Sum=13times3=39=(12+13+14)
(d)Of6789and108=averagethere4Sum=8times5=40=(6+7+8+9+10)
(e)Of11121314and1513=averagethere4Sum=13times5=65
Notethatwheneveranoddnumberofequallyspacedfiguresappearsyoucanimmediatelyspotthecenteroneoraverageandpromptlygetthesumofallbymultiplyingtheaveragebythenumberoffiguresinthegroup
45Whatistheprocedureforaddingtwocolumnsatatime
37StartatbottomAdd96to80ofabovethenthe2getting24178Add178tothe20abovethenthe4getting202Add82202tothe
30abovethenthe7getting239=sum
Avariationwouldbetoaddtheunitsofthelineaboveitfirstandthenthetensas
46HowarethreecolumnsaddedatonetimeStartatbottomAddhundredsthentensthenunitsasyoucontinueup
EXAMPLES(a)
(b)
47WhatisaconvenientwayofaddingtwosmallquantitiesbymakingadecimalofoneofthemMakeadecimalofonebyaddingorsubtractingandreversethetreatmentfor
theother
EXAMPLE96+78
Add4to96getting100=decimalnumberSubtract4from78getting74there4Sum=174atonce
48HowmaydecimalizedadditionbecarriedouttoafullerdevelopmentReduceeachnumbertoadecimalAddthedecimalsAddorsubtractthe
increments
EXAMPLE
49Howmaysightreadingbeusedinaddition
Byuseofinstinctyougetanimmediateresult
EXAMPLES
(a)Add26to53
(b)Add67to86
Fixeyesbetweenthetwocolumnswherethedotsareandatonceseea7anda9ora13anda14tomake153Actually70isaddedto9and140to13buteachisdoneinstinctively
50WhatsimplemethodisusedtocheckthecorrectnessofadditionofacolumnofnumbersFirstbeginatthebottomandaddupThenbeginatthetopandadddown
WhenthecolumnsarelongitisoftenbettertowritedownthesumsratherthantocarrytheldquobundlesrdquofromcolumntocolumnPlacesumsinpropercolumns
EXAMPLE
51WhatismeantbyacheckfigureinadditionOnewhichwheneliminatedfromeachnumbertobeaddedandfromthesum
willgiveakeynumberthatmayindicatethecorrectnessoftheadditionThechecknumbers9and11aregenerallyused
52Whataretheinterestingfactsontheuseofthechecknumber9(1)Thefactthattheremainderleftafterdividinganynumberby9isthesame
astheremainderofthesumofthedigitsofthatnumberdividedby9
Ex(a)
Ex(b)
(2)Alsonotethatthesumofthedigitsalonewillgivethesamenumberasaremainderasthedivisionofthenumberby9Thusin(a)6+5+4=15and1+5=⑥In(b)2+6+7+7=22and2+2=④(3)Alsothefactthat9rsquoscanbediscardedwhenaddingthedigitsThusin(a)
6+5+4discard4+5rightawayandtheremainderisagain⑥In(b)2+6+7+7discard2+7butadd6+7=13and1+3=④
53Whatistheprocedureincheckingadditionbytheuseofthecheckfigure9oftencalledldquocastingoutninesrdquo(a)Addthedigitsineachnumberhorizontallyandgeteachremainder
(b)Addthedigitsoftheseremaindersandgetthekeyfigure
(c)Addthedigitshorizontallyoftheanswerandgetthesamekeyfigureiftheansweriscorrect
EXAMPLE
Inpracticeitissufficienttoaddthenumbersmentallytogettheremainders
Notethatall9rsquosanddigitsthataddupto9arediscardedrightawayEachdigitsodiscardedisshownwithadotattheupperrightcorner
54WhyisldquocastingoutninesrdquonotaperfecttestofaccuracyinadditionItispossibletoomitoraddninesorzeroswithoutdetectionAlsofiguresmay
betransposed27isquitedifferentinvaluefrom72althoughthesumofthedigitsisthesame
ThismethodisnotgenerallyrecommendedasapracticaltestinadditionworkbuthasitsgreatestvalueinmultiplicationanddivisionworkHoweveritissometimesusefulasaquickcheckofaddition
55Whataretheinterestingfactsontheuseofthechecknumber11(1)Theremainderleftafterdividinganynumberby11isthesameasthe
remainderleftaftersubtractingthesumofthedigitsintheevenplacesfromthesumofthedigitsintheoddplacesIfthesubtractioncannotbemadeadd11oramultipleofittotheodd-placessum
EXAMPLES
(a)
(b)
(2)ThesameremainderisalsoobtainedbystartingwiththeextremeleftdigitinthenumberandsubtractingitfromthedigittoitsrightWhennecessaryadd11tomakethesubtractionpossibleSubtracttheremainderfromthenextdigitAgainadd11ifnecessaryRepeattheprocessofsubtractionuntilallthedigitsofthenumberhavebeenused
56Whyisthecheckingofadditionworkbytheuseofthecheckfigure11(oftencalledldquocastingoutelevensrdquo)superiortothatofldquocastingoutninesrdquoldquoCastingoutelevensrdquocanindicateanerrorduetotranspositionofdigits
whichisnotpossiblewiththeldquoninesrdquomethod
EXAMPLESupposeournumberis8706
8from(11+7)=1010from(11+0)=1Ifrom6=⑤=Remainder=Checknumber
Nowsupposethetransposednumberis8076
8from(11+0)=3 3from7=44from6=②=Remainder=Checknumber
Thechecknumbersareseentobedifferentandwehaveuncoveredatranspositionofdigits
57Whatistheprocedureincheckingadditionbytheuseofthecheckfigure11(a)Castoutelevensfromeachrowandgeteachremainder
(b)Addtheremaindersandcastoutelevensfromthissumgettingthekeyfigure
(c)CastoutelevensfromtheanswerandgetkeyfigureCompare
EXAMPLE
PROBLEMS
1Countfrom3to99by3rsquos
2Countfrom4to100by4rsquos
3Countfrom6to96by6rsquos
4Countfrom9to99by9rsquos
5Startwith3andcountby2rsquos4rsquos6rsquos8rsquostojustbelow100
6Startwith2andcountby3rsquos5rsquos7rsquos9rsquostojustbelow100
7Startwith9andcountby4rsquos7rsquos9rsquos2rsquostojustbelow100
8Startwith14andcountby6rsquos2rsquos4rsquos8rsquostojustbelow100
9Add269745and983
10Addusingldquocarryoversrdquo
11Add$525$1760$085$175$4565
12Findthesumof
(a ) (b ) (c)
$380865 $987367 $887406
37692 38898 51856
38623 573200 129897
48008 898719 54265
88842 782492 38600
75182 608604 4209
13Whatisthesumof102030bytheaveragemethod
14Whatisthesumof141516bytheaveragemethod
15Whatisthesumof1718192021bytheaveragemethod
16Whatisthesumof3456789bytheaveragemethod
17Whatisthesumof579bytheaveragemethod
18Whatisthesumof131517bytheaveragemethod
19Whatisthesumof1416182022bytheaveragemethod
20Whatisthesumof91215bytheaveragemethod
21Addtwocolumnsatatime
22Addthreecolumnsatatime
23Addthefollowingbythedecimalizingmethod(a)94+75(b)86+69(c)92+48(d)89+52(e)468+982+429(f)346+899+212(g)589+913+165(h)862+791+386
24Addbysightreading(a)27+56(b)21+43(c)32+65(d)49+57(e)68+87(f)76+82
25Agasolinestationownerhad275gallonsleftafterselling632gallonsHowmanygallonsdidhehaveoriginally
26Onepipefromatankdischarges76gallonspersecondwhileanotherpipefromthesametankdischarges16gallonsperminutemorethanthefirstHowmanygallonswillbothpipesdischargeinaminute
27Anautomobiletravels386milesonthefirstdayand416milestheseconddayatwhichtimeitis237milesfromitspointofdestinationWhatisthedistancefromitsstartingpointtoitsdestination
28Asuburbanhousewasbuiltwiththefollowingexpensesmasonry$3565lumber$4850millwork$1485carpentry$3800plumbing$2758painting$679hardware$1508heating$1250andelectricity$687Whatdidthehousecostwhencompleted
29Ifafamilyoftwopersonsspends$135forrent$205forfood$85forclothing$35forfuel$7forlight$22forinsurance$6forcarfare$12forcharityandsaves$18whatistheincomeaftertaxesandotherpayrolldeductions
30Thetwenty-secondofFebruaryishowmanydaysafterNewYearrsquosHowmanydaysfromNewYearrsquostothefourthofJuly
31CheckthefollowingbyfirstaddingupandthenbyaddingdownPlacecheckmarksasproof
32Provethefollowingbyuseofthecheckfigure9
33Provethefollowingbyuseofthecheckfigure11
34Addhorizontallyandvertically
(a)
(b)
CHAPTERII
SUBTRACTION
58WhatissubtractionItisthereverseofadditionSinceweknowthatfiveapples+threeapples=
eightapplesitfollowsreverselythattakingfiveapplesawayfromeightapplesleavesthreeapples
Ortakingthreeapplesawayfromeightapplesleavesfiveapples
8minus5=3 8minus3=5
Aswithadditionsubtractionisthusseentobemerelyaregrouping
group(a)+group(b)=group(c)=8group(c)ndashgroup(a)=3 group(c)ndashgroup(b)=5
59WhymaysubtractionbesaidtobeaformofadditionEx(a)9ndash4=5
Maybethoughtofasldquo4andwhatmake9rdquo4and5make9
Ex(b)16minus9=7
9andwhatmake169and7make16
60Whatthreequestionswillleadtotheprocessofsubtraction(a)Howmuchremains
(b)Howmuchmoreisrequired
(c)Byhowmuchdotheydiffer
In(a)ifBerthas$10andpaysout$6howmanydollarsremainHerethe$6wasoriginallyapartofthe$10
In(b)Berthas$65andwouldliketobuya35-mmcamerathatcosts$89Howmuchmoredoesherequire
In(c)ifBerthas$10andCharleshas$6byhowmuchdotheydifferHerethe$10andthe$6aredistinctnumbers
61Whatarethetermsofasubtraction
IfthesubtrahendwasoriginallyapartoftheminuendthentheansweriscalledtheldquoremainderrdquoIftheminuendandsubtrahendaredistinctnumberstheansweriscalledtheldquodifferencerdquo
62WhyisitsaidthatwecanalwaysaddbutwecannotalwayssubtractSubtractionisnotalwayspossibleItisnotwhenthenumberofthingswhich
wewishtosubtractisgreaterthanthenumberofthingswehave
Ex(a)
Addition5apples+3apples=8applesSubtraction8applesminus3apples=5applesAddition5apples+7apples=12applesSubtraction5applesminus7apples=impossible
ThereexistnonegativeapplesAtbestwecanonlyexpresstherelationas2applesmissing
Ex(b)
7foot-candlesofilluminationminus5foot-candles=2foot-candles
7foot-candlesminus9foot-candlesisimpossiblebecausetherecannotbeanegativeilluminationof2foot-candlesThelimitiszeroilluminationordarkness
Ex(c)Fromanelectriccordof8feetwecancutoff3feetleaving5feetbutwecannotcutoff10feetleavingminus2feetofcord
63WhenisitpossibletosubtractwiththenumberexpressingthesubtrahendgreaterthanthenumberexpressingtheminuendByintroductionoftheconceptofldquodirectionrdquotothequantitiesexpressedby
thenumbersandcallingallnumbersinonedirectionpositivenumbersandnumbersinthereversedirection(fromthestartingpointzero)negativenumbers
Ex(a)
Nowifwestepoff5stepstotherightandthenstepoff7totheleftwelandatminus2
there45minus7=minus2
Ex(b)Ifweletzero=freezingtemperaturethen+5degis5degreesabovefreezingandifitfalls3degreesitwillbe2degreesabovefreezingIfitfalls7degreesitwillbe2degreesbelowfreezingor
Ex(c)Ifzeroislatitudethen+5deglatminus7deglat=minus2deglatThiswouldbeintheSouthernHemisphere
Ifwehave$5inthebankandifwehavecreditwemaybeabletodrawout$7inwhichcase$5minus$7=minus$2overdraftAgainifwehave$10inourpocketandbuysomethingthatcosts$25weareindebtfor$15$10minus$25=minus$15debt
Thenegativenumberisnotaphysicalbutamathematicalconceptionwhichmayormaynothaveaphysicalrepresentationdependingonhowitisapplied
64Whatisthesubtractiontablethatshouldbestudieduntiltheanswerscanbegivenquicklyandcorrectly
SubtractionTable
65Whatistheruleforsubtraction(a)Writethesubtrahendundertheminuendunitsunderunitstensundertens
etc
(b)Beginattherightandsubtracteachfigureofsubtrahendfromthecorrespondingfigureoftheminuendandwritetheremainderunderneath
(c)Ifanyfigureofthesubtrahendisgreaterthantheminuendincreasetheminuendby10(whichuses1unitofthenexthigherorder)andsubtractNowreducetheminuendofthenexthigherorderby1andcontinuetosubtractuntilallthedigitshavebeentakencareof
NotethatyoudonotactuallyaddortakeawayanythingfromthenumberYoumerelyregroupabundlebyunscramblingitandplacingitwiththelowerordertomakethesubtractionpossibleInEx(c)abovewecanseethatwewillneedonethousandsbundletounscrambleto10hundredsonehundredsbundletobecome10tensandonetensbundletobecome10unitsThenumbersthenbecome
66WhatisknownasthemethodofldquoequaladditionsrdquoinsubtractionThemethodisbasedonthefactthatthesamenumbermaybeaddedtoboth
minuendandsubtrahendwithoutchangingthevalueofthedifference
Ex(a)
Ex(b)
ThismethodisquickandsimpleAllyouneedtorememberistoadd1tothenextcolumninthesubtrahendeverytimeyouadd10totheminuendtomakesubtractionpossible
Ex(c)
67WhatisthemodeofthinkingofsubtractionthatiscalledtheAustrianmethodorthemethodofmakingchangeAgooddealofsubtractioninthebusinessworldisconcernedwithmaking
changeItconsistsinbuildingtothesubtrahenduntiltheminuendisreached
Ex(a)
Whensubtractionistobemadepossibleinanycolumnitbecomesamodificationoftheaboveldquoequaladditionrdquomethod
Ex(b)
68HowmaysubtractionbesimplifiedAddorsubtractaquantitytogetamultipleof10Itiseasiertosubtracta
multipleof10fromanotherquantitythantosubtractanyotherdoubledigitnumber
EXAMPLE
Notethattheansweristhesamewhenyouaddorsubtractthesamenumberfromboththeminuendandsubtrahendandthatitiseasiertosubtractwhenthesubtrahendismadeamultipleof10
69HowmaytheabovebeextendedDividethenumbersintocouplesandmakeeachcoupleamultipleof10
(whichisknownasadecimalnumber)
Ex(a)
Ifthesubtrahendinonecoupleislargerthantheminuendtherewillbe1tocarrywhichissubtractedfromthedifferencesofthecouplenextontheleft
Ex(b)
Insubtracting70from52borrowone(hundred)thensubtract1fromthedifferenceof(99ndash40)
Ex(c)
70Howcanthesubtractionoftwo-figurenumbersbedonebysimpleinspectionusingdecimalizationEx(a)
Ex(b)
89minus47=40+9minus7=4298minus36=60+8minus6=6295minus22=70+5minus2=73
71Howcaninvertedorleft-handsubtractionbedoneStartfromtheleftandsubtractnotingwhetherthereisonetocarryfromthe
columnattheright
Ex(a)
Ex(b)
72WhatismeantbythearithmeticalcomplementofanumberAbbreviatedacarithmeticalcomplementistheremainderfoundby
subtractingthenumberfromthenexthighestmultipleof10
EXAMPLE
acof2is10minus2=8acof57is100minus57=43acof358is1000minus358=642acof0358is1000minus0358=0642
73WhatisthesimplestwayofcalculatingtheacofanumberSubtractitsright-handdigitfrom10andeachoftheothersfrom9Thisdoes
awaywithcarryingof1rsquos
EXAMPLEacof68753=31247
Startatleft
6from9=38from9=17from9=25from9=43from10=7
74WhenandhowistheacusedinsubtractionWhenaquantityistobesubtractedfromthesumofseveralothersTo
subtractbymeansoftheacaddtheacofthesubtrahendandsubtractthemultipleof10usedingettingtheac
Ex(a)Subtract9431from9805byac
Nothingisgainedbyuseofacinsosimpleacase
Ex(b)Subtract1284fromthesumof97471283and1292
Ex(c)Frombankdepositsof$22680$34261and$18734deductawithdrawalof$56079togetthenetincrease
75Howdoweproceedtogivechangetoacustomerbytheuseoftheso-calledldquoAustrianmethodrdquoofsubtractionAddfromtheamountofthepurchaseuptothenexthighermoneyunitthen
tothenextandsoonuntilyoureachtheamountofthebilltenderedinpayment
EXAMPLEIfthebillgiveninpaymentis$5andthepurchaseis$238givecustomerthefollowingaschange2centstomake$24010centstomake$25050centstomake$300$2tomake$5
Totalchangeaddsupto$262
76WhatisthebestcheckinsubtractionThesumofremainderandsubtrahendmustequaltheminuendThismeans
wehavetakenawayacertainnumberwenowputitbackandreturntotheoriginalnumberThischeckshouldalwaysbemadeItisdonementally
EXAMPLES
77IsldquocastingoutninesrdquoapracticalcheckinsubtractionItisnotandtoomuchtimemustnotbespentonthismethod
Ex(a)
Itisseenthatthedifferencebetweentheremaindersoftheminuendandsubtrahend=remainderofanswer
Ex(b)
78MaycastingoutofelevensbeusedasacheckYesbutherealsotoomuchtimeshouldnotbedevotedtothismethod
Ex(a)
TaketheminuendStartatleft
TakethesubtrahendStartatleft
Ex(b)
PROBLEMS
Performthefollowingsubtractions
1
2
3
4
5
6
7Ifwesayacertaintreeisinzeropositionandwetake8stepstotherightofthetreewhichwecallthepositivedirectionandthenwestepoff12stepstotheleftwherewillweland
8Ifzeroisfreezingtemperaturewhatdoes+7degmeanWhatdoesminus8degmean
9Ifyourlatitudeiszeroandyoutravelnorthto+11deglatandthensouthwardfor15degwhatwouldbeyourlastposition
10Ifyouhad$85inthebankandyouissuedacheckfor$97whatwouldbeyouroverdraft
11Ifyouhadonly$63andyouwantedtobuya35-mmcamerathatcost$87howmuchwouldyoubeindebt
12Subtract
13Checktheanswerstoproblem12byadditionChecktheanswersbycastingoutninesChecktheanswersbycastingoutelevens
14Whatisthesubtrahendforeachofthefollowingsetsofvalues
15Checktheanswerstoproblem14byadditionandbycastingoutnines
16Usethesimplifiedmethodofsubtractionbymakingthesubtrahendamultipleoften
17Extendthesimplifiedmethodofsubtractiontotwocouplesmakingeachamultipleoftenoradecimalnumber
18Dothefollowingsubtractionsoftwo-figurenumbersbysimpleinspectionusingdecimalization
19Dothefollowingbyinvertedorleft-handsubtraction
20Whatisthearithmeticalcomplementof(a)7(b)69(c)472(d)1282(e)0472(f)79864(g)864348
21(a)Subtract8562from9983byacmethod(b)Subtract46827from87962byacmethod
22Subtract4976fromthesumof84321343and1565byacmethod
23Frombankdepositsof$34276$56259and$13459deductawithdrawalof$63248byacmethod
24Ifa$20billisgiveninpaymentandthepurchaseis$1289whatchangewillthecustomergetusingtheso-calledldquoAustrianrdquomethodofsubtraction
25Ifarailroadcarries2325879passengersoneyearand3874455passengersthefollowingyearwhatistheincrease
26IftheFederalincometaxcollectedoneyearis$67892762945and$71432652982thefollowingyearwhatistheincrease
27(a)Beginwith53andsubtractby2rsquos4rsquos6rsquos8rsquos(b)Beginwith89andsubtractby3rsquos5rsquos7rsquos9rsquos(c)Beginwith74andsubtractby5rsquos7rsquos3rsquos9rsquos
28Amanboughtafarmfor$17500Hekeptittwomonthsduringwhichtimehepaid$43950intaxesand$78275forrepairoffencesHethensolditfor$21500Whatwashisprofit
CHAPTERIII
MULTIPLICATION
79WhatismultiplicationItismerelyasimplifiedformofadditionSupposewehaveeightapplesina
rowandtherearefourrowsWecanaddthemas8+8+8+8=32orwecansaysimply4times8=32Alsoifwehavefourapplesinarowandthereareeightrowsthen
4+4+4+4+4+4+4+4=32or8times4=32
Youseethat4times8=8times4=32Ineachcasethesumis32Whenseveralequalnumbersaretobeaddeditismuchshortertoobtaintheresultbymultiplication
80Whatarethetermsofamultiplication(a)Thenumbertoberepeatediscalledthemultiplicand
(b)Thenumberoftimesthemultiplicandistoberepeatediscalledthemultiplier
(c)Theresultofthemultiplicationiscalledtheproduct
(d)Themultiplicandandthemultiplierarealsoknownasthefactorsoftheproduct
EXAMPLE
81Whatis(a)aconcretenumber(b)anabstractnumber(c)thetypeofnumberofthemultiplierinmultiplication(a)Anumberthatisappliedtoanyparticularobjectiscalledaconcrete
numberExamplesanappleanauto2hoursetc
(b)AnumberthatisnotappliedtoaparticularobjectisanabstractnumberExamples1562
(c)Inmultiplicationthemultiplierisalwaysanabstractnumber
82Whatarethemostusefulproductsthatshouldbecommittedtomemory
MultiplicationTable
83WhenseveralnumbersaremultiplieddoesitmatterinwhatorderthemultiplicationisperformedTheorderofmultiplicationdoesnotmatter
EXAMPLE2times6times4=2times(6times4)=(2times4)times6=48
The2maybemultipliedby6andthisresult(=12)maythenbemultipliedby4toget48orthe6and4mayfirstbemultipliedandthenthe2usedetc
84Whatistheruleinmultiplicationwhen(a)thetwosignsofthenumbersarebothplus[+](b)bothsignsareminus[ndash](c)thetwosignsareunlike(a)Twoplusesproduceaplusproduct
(b)Twominusesproduceaplusproduct
(c)Twounlikesignsproduceaminusproduct
(+4)times(+6)=+24(+4)times(minus6)=minus24(ndash4)times(minus6)=+24(ndash4)times(+6)=minus24
NoteItisnotnecessarytowritetheplusinfrontoftheproduct
85WhatistheeffectuponanumberwhenyoumoveitonetwothreeplacestotheleftintheperiodMovingafigureoneplacetothelefthasthesameeffectasmultiplyingitby
10Example76times10=760Sotomultiplyby10placeazeroattherightofthemultiplicandthusmovingeachdigitoneplacetotheleftandincreasingitsvalue10times
Tomultiplyby100placetwozerosattherightofthemultiplicandExample76times100=7600
Tomultiplyby1000placethreezerosattherightofthemultiplicandetcExample76times1000=76000
86WhatistheruleformultiplyingwheneithermultiplierormultiplicandendsinzerosMultiplythemultiplicandbythemultiplierwithoutregardtothezerosand
annexasmanyzerosattherightoftheproductasarefoundattherightofthemultiplierandmultiplicand
EXAMPLE
87HowisordinarysimplemultiplicationperformedWritethemultiplierunderthemultiplicandplacingtheunitsofthemultiplier
underunitsofmultiplicandandbeginattherighttomultiply
EXAMPLE
Notethattheworkcanbeshortenedbydoingtheldquocarryingrdquomentally
88WhatistheprocedurewhenthenumberstobemultipliedcontainmorethanonedigitEXAMPLE698times457Itwouldnotbeconvenienttosetdown698tobe
added457times
Multiplyingby457isthereforethesameasmultiplyingby7by50andby400andaddingtheresults
(a)Firstmultiply698by7
7times8=56 Write6carry57times9=63+5=68 Write8carry6
7times6=42+6=48 Write48
(b)Thenmultiplyby50Write0inunitscolumnandthenmultiply698by5
5times8=40 Writezerocarry45times9=45+4=49 Write9carry4
5times6=30+4=34 Write34
(c)Thenmultiply698by400Write00andmultiply698by4
4times8=32Write2carry34times9=36+3=39Write9carry3
4times6=24+3=27Write27
Nowaddthethreeresultstoget318986=productOfcourseyoumayomitwritingthezeroswhenyouremembertomovetheproductoneplacetotheleftwhenmultiplyingbythedigitinthetenscolumnandtwoplacestotheleftwhenmultiplyingbythedigitinthehundredscolumnetc
89HowcanthefactthateithernumbermaybeusedasthemultiplierservetoprovideacheckonourmultiplicationEXAMPLE(asabove)ReverseUse698asthemultiplier
90Howcanweextendthemultiplicationtablebeyond12times12bymakinguseofthesmallerproductsby2orby4EXAMPLES
(a)14times13=2times7times13=91times2=182Split14into7times2(b)16times13=2times8times13=104times2=208Split16into8times2(c)18times13=2times9x13=117times2=234Split18into9times2(d)16times16=4times4x16=4times64=256Split16into4times4
91Howcanmultiplicationbytwo-digitnumbersbesimplifiedConvertonetwo-digitnumberintotwoone-digitnumbers
Ex
(a)27times16=27times2times8=54times8=432(b)27times15=27times3times5=81times5=405
92Howcanthemultiplicationoftwo2-digitnumbershavingthesamefigureinthetensplacebesimplified(a)Multiplytheunits
(b)AddtheunitsandmultiplythesumbythetensdigitAnnexazero
(c)MultiplythetensAnnex2zeros
(d)Add(a)+(b)+(c)
EXAMPLES(1)
(2)
(3)
93HowcanmultiplicationbesimplifiedbymultiplyingonefactoranddividingtheotherfactorbythesamequantityEx(a)
Theproductisthesamebecause
Thiscouldalsobedoneas
94WhatcanbedonewhenmultiplicationmaysimplifyoneofthefactorsbutwhentheotherfactorisnotdivisiblebythesamenumberIfmultiplicationofonefactormakesthatfactorsimplerusetheresultasthe
multiplieranddividetheproductbythesamenumberusedtosimplifythemultiplier
Ex(a)45times29
Multiplyfactor45by2getting90Now90times29=2610
Dividethisby2getting
Ex(b)323times35
Notethissimplificationappliestonumbersendingin5upto55togiveprocedureswithintherangeofthemultiplicationtable
Ex(c)271times55
95Whenthetensdigitsarealikeandtheunitsdigitsaddupto10howismultiplicationsimplifiedWritetheproductoftheunitsdigitsIncreaseoneofthetensdigitsby1and
multiplybytheother
Ex(a)
Ex(b)
Ex(c)
96Whentheunitsdigitsarealikeandthetensdigitsaddupto10howismultiplicationsimplifiedWritetheproductoftheunitsdigitsAddunitsdigittoproductoftensdigits
Ex(a)
Ex(b)
Ex(c)
97Whenneitherofabovecombinationsisapplicablehowmayso-calledcrossmultiplicationbeappliedtoadvantageEx(a)
Ex(b)
Ex(c)
98Whentheunitsdigitsare5andthesumofthetensdigitsisevenhowismultiplicationsimplifiedTheproductwillendin25Multiplythetensdigitsandaddhalftheirsum
Ex(a)
Ex(b)
99Whentheunitsdigitsare5andthesumofthetensdigitsisoddhowismultiplicationsimplifiedTheproductwillendin75Multiplytensdigitsandaddhalftheirsum
discardingfraction
Ex(a)
ThismethodmaybeusedwhenthereareonlytwoandnotmorethanthreedigitsineithermultiplierormultiplicandWhendollarsandcentsareinvolvedthetwoenddigitsarecentsanddigitstotheleftaredollars
Ex(b)
Ex(c)
Ex(d)
100Whatismeantbyleft-handmultiplicationorwhatissometimescalledinvertedmultiplicationMultiplyleft-handfiguresfirstandthenthenextandaddtheproducts
Ex(a)
Ex(b)
101Whatismeantbyanaliquot(ălrsquoi-kwŏt)partofanumberItisaquantitywhichcanbeadivisorofanumberwithoutleavinga
remainderItisthereforeafactorofthenumber
Ex(a)5isanaliquotpart(orfactor)of20orof35When20or35isdividedby5thereisnoremainder5isafactorofeithernumber
Ex(b) and25gointo100863and4timesrespectivelyandarealiquotpartsof100orfactorsof100
Ex(c) cent10centand25centarealiquotpartsof$100sincetheyarecontained1210and4timesrespectivelyin$100
102WhatismeantbyafractionalequivalentofanaliquotpartBydefinition
Ex(a) (=aliquotpartof100)100isthebaseThen =fractionalequivalentofthealiquotpartof100( )
Ex(b) (=aliquotpartof100)Then =fractionalequivalentofaliquotpartof100
Itisseenthatthefractionalequivalenthasanumeratorof1andadenominator
whichisthenumberoftimesthatthealiquotpartiscontainedinthegivennumber
103WhenaresomenumbersusefulwhilenotaliquotpartsthemselvesTheyareusefulwhentheyareconvenientmultiplesofaliquotparts
Ex(a) isnotanaliquotpartof100sinceitdoesnotgointo100awholenumberoftimesbut isanaliquotpartof100and is Thefractionalequivalentof is of100
there4 is of100
Ex(b) is Thefractionalequivalentof is
there4 is of100
Ex(c)75is3times25Thefractionalequivalentof25is
there475is of100
104Whataresomeofthealiquotpartsof100andtheirfractionalequivalentsWeknowthatanaliquotpartof100isafactorof100
105Howmayaliquotpartsof100bewrittenasdecimalsAnaliquotpartof100meanssomanyhundredthsandmaybewrittenasa
decimalThebaseis100
EXAMPLE(asabove)
Cipherinfrontofaliquotpart
020405062506660833
Decimalpointinfrontofaliquotpart
125133316662025
106WhyarealiquotpartsusefulincalculationsinvolvingdollarsAliquotpartsof100have100partsastheirbasesAs$100=100centsthenofadollar= centsand ofadollar=20cents
EXAMPLEFindcostof72articleswhenthepriceofoneis16
Ifthepriceofanarticlewereadollarthetotalcostwouldbe$7200butsincethepriceisonly ofadollarthetotalcostis =$1200
107Howmayaliquotpartsof100beusedinmultiplication(a)Tomultiplyby50( of100)Multiplyby100byannexingtwozeros
Thendivideby2tomultiplyby50( of100)
EXAMPLE
(b)Tomultiplyby25( of100)Annextwozerostomultiplyby100Thendivideby4tomultiplyby25( of100)
EXAMPLE
(c)Tomultiplyby20( of100)Annextwozerostomultiplyby100Since20is of100divideby5
EXAMPLE
Inthiscaseitwouldgenerallybeeasiertomultiplydirectly
(d)Tomultiplyby75( of100)Annextwozerostomultiplyby100Sinceof100multiplyby
EXAMPLE
108WhatisthepracticaluseofaliquotpartsinmultiplicationAliquotpartsenableustodispensewithfractionsForourusealiquotpartsare
applicabletobasesofhundredsandotherdecimalnumbers
Ex(a)Whatisthecostof65articlesat$250eachThebasehereis10andis of10Thenaddonezeroanddivideby4
Ex(b)Howmuchwill49itemsat costMultiply49by3=$147andaddtoit
Ex(c)Whatisthecostof38articlesat of$100ButThen
Ex(d)Whatistheresultof37519times125
As125is of1000annexthreezerosanddivideby8Thismultipliesthenumberfirstby1000andthendividesby8tofind125asamultiplier
Alsosince125=(100+25)then
Ex(e)Whatisthecostofeachofthefollowing
109MaythenumberofarticlesandthepricebeinterchangedasameansofsimplifyingaprobleminaliquotpartsYesThus yardsat$315canbechangedto315yardsat
EXAMPLEWhatisthecostof16 yardsofclothat69centayardThiscanbechangedto69yardsat ayard
At$100peryard69yardswouldcost$69
But of$100there4 Ans
110Whatisthecostof1780lboffeedat$1500aton
At1centperlb($100per100lb)1780lbcosts$1780 of$100
there4 costof1780lbat$1500perton
111Howcanwesimplifythemultiplicationby24Multiplyby25byannexingtwozerosanddividingby4Subtracttheoriginal
numberfromtheresult
Ex(a)
Ex(b)Avariation261times124124=(100+24)
Then
112Howcanwesimplifythemultiplicationby26Multiplyby25byannexingtwozerosanddividingby4Addtheoriginal
numbertothis
Ex(a)
Ex(b)
113Howcanwemultiplyanumberby9usingsubtractionEXAMPLE
66492times9=59842866492(10minus1)=664920minus66492
114Howcanwemultiplyby11usingadditionEXAMPLE
Inoneline
Putdown2Addthenextfigure9tothe2Putdown1carry1Then4+1+9=14Putdown4carry1Then6+1+4=11Putdown1carry1Then7+1+6=14Putdown4carry1Then7+1=8
115Howcanwemultiplyby111byusingadditionEXAMPLE
Inoneline
76492times111 Putdown2
Add9+2=11 Putdown1carry1
Add4+9+2+carry1=16 Putdown6carry1
Add6+4+9+1carry=20 Putdown0carry2
Add7+6+4+2carry=19 Putdown9carry1
Add7+6+1carry=14 Putdown4carry1
Add7+1carry=8 Putdown8
116Howcanwesimplifythemultiplicationby8andby7Tomultiplyby8annexazeroandsubtracttwicethenumber
EXAMPLE
Tomultiplyby7annexazeroandsubtract3timesthenumber
EXAMPLE
117Howdowemultiplyby999897orby999998997Annexthepropernumberofzerosandsubtracttherequirednumberoftimes
118WhatismeantbythecomplementofanumberThedifferencebetweenthatnumberandtheunitofanexthigherorder
Ex(a)Complementof7is3becausethedifferencebetween7and10is310isthenexthigherorderof7
Ex(b)Complementof58is42because100minus58is42100isthenexthigherorderof58
119Howiscomplementmultiplicationperformed(a)Findthecomplementofeachnumber
(b)Multiplythecomplementstogether
(c)Subtractoneofthecomplementsfromtheothernumberandmultiplythisby100
(d)Add(b)to(c)
Ex(a)
Multiply92x96 100minus92=8=complement
100minus96=4=complement
8times4=32=productofcomplementsNumber92minus4(=complementof96)=88
88times100=88008800+32=8832Ans
Ex(b)Multiply86times93Complementsare14and7
14times7=98=productofcomplements86minus7=7979times100=7900
7900+98=7998Ans
Ex(c)Multiply942times968Complementsare58and32
Itmaynotpaytousethismethodwiththreefigures
120Howcanwemultiplybyanumberbetween12and20usingonlyonelineintheproductMultiplyasusualbytheunitsfigureofthemultiplierCarryasusualbutalso
addthefigureontherightofthefiguremultipliedThislatteradditiontakescareofthetensfigureofthemultiplier
EXAMPLE
AlltheabovecanbedonementallyofcourseAsyouseebyordinarymultiplicationthemultiplicationofthetensfigure1ofthemultipliermovestheentiremultiplicandoneplacetotheleftandaccountsfortheadditionofthefiguretotherightoftheonebeingmultipliedintheone-lineprocess
121WhatismeantbycrossmultiplicationAmethodofmultiplyingbyanumberofmorethanonedigitwithoutputting
downthepartialproductsThepartialproductsarekeptinmindandonlyonelineresultsastheanswerThesecretistostartwiththeright-handdigitofthemultiplierandcontinuetoprogresstoeachdigitofthemultiplierandaseachisfinishedstartanothertotheleftGettheunitsfirstthenaddupthetenshundredsthousandsetcusingeachdigitofthemultiplierorthemultiplicandAddthecarry-overfigurePuteachproductinitsproperplace
122Whatistheresultof76times64usingcrossmultiplication
123Whatistheresultof847times76usingcrossmultiplication
Thousands7times8+8carry=64Putdown64
124Howcanwecheckamultiplicationbyldquocastingoutninesrdquo(a)Gettheremainderbyaddingdigitsofmultiplicand
(b)Gettheremainderbyaddingdigitsofmultiplier
(c)Multiplyremainders(a)and(b)togetherandgetremainderofthisproduct
(d)Getremainderoftheanswer(orproduct)
Ifremainderof(c)and(d)arealikethemultiplicationisinallprobabilitycorrect
All9digitsorthosewhichaddupto9arediscardedrightaway
EXAMPLE
Remainderofmultiplicand(4)xremainderofmultiplier(3)=12
1+2=③=sameasremainderofanswerorproduct
ThisisnotanabsoluteproofbutonlyatestofthecorrectnessofthemultiplicationThereversingofmultiplierandmultiplicandrequiresmoretimebutitismoreaccuratebecauseiteliminatesthepossibilityoftransposedfiguresorofninesandzerosbeingaddedoromittederroneously
PROBLEMS
1Multiply54by10by100by1000
2Multiply820by10by100by1000
3Multiply1762by10by100by1000
4Multiply631by60
5Multiply45by40by400by4000by400000
6Multiply4700by4by40by400by4000by40000
7Multiply6390by300
8Multiply
(a)870by3600(b)785340by4700(c)98750by400(d)87953by45000(e)48800by78000(f)780000by630(g)387470by4000
9Whatistheproductof
(a)4738multipliedby6(b)892by8(c)953by67(d)628by86(e)438by99(f)673by83(g)768by57(h)4174by647(i)587by756(j)9046by839(k)3490by874(l)5947by638(m)6084by519(n)7493by349(o)9486by305(p)9385by3005(q)3795by803(r)9476by8007(s)2583by7001(t)9434by8002(u)8754by408(v)7004by1371(w)8745by49(x)6354by684(y)2851by1212(z)8172by899
10Multiply
(a)$3885by375(b)$73140by457(c)$87234by741(d)$40010by856(e)$134035by704(f)$465020by708
11Amechanicearns$2885adayWhatwillhispaybeforafive-dayweekForamonthof22days
12If28yardsofcarpetarerequiredforafloorwhatwillbethecostat$925ayard
13OnOctober1Johngotatemporaryjobpaying$82aweekHowmuchdidheearnin23weeks
14Ifitcosts$4065forlaborand$3629formaterialtosprayanacreofvineyard5timeswhatwillbethecosttospray8acres5times
15Thereare21750cubicfeetinthefirst6inchesoftopsoilofanacreofgroundHowmuchwillthissoilweighat80lbpercubicfoot
16Amanbought1124acresoflandat$225anacreHespent$83700forimprovementsandthensold8acresat$450anacre270acresat$535anacre325acresat$380anacre360acresat$660anacreandtherestat$100anacreHowmuchdidhegainorlose
17Ifyoubought$15worthofbooksamonthfor28monthshowmuchwouldyouhavespent
18Joedroveacar400milesat40milesperhourfor20daysHowmanymilesdidhecover
19Whatis(a)14times17(b)16times17(c)18times17(d)16times19Makeuseofthesmallerproductsby2orby4
20Whatis(a)29times18(b)29times15(c)37times16(d)46times14Convertonetwo-digitnumberintotwoone-digitnumbers
21Multiply(a)85times87(b)48times49(c)58times53(d)37times32(e)65times67(ƒ)99times94(g)74times72(h)26times28(i)17times18bythemethodusedwhenthetensfiguresarealike
22Multiply(a)45times16(b) (c)32times18(d) (e)18times18(ƒ)15times16(g) bymultiplyingonefactoranddividingtheotherfactorbythesamequantity
23Multiply(a)35times27(b)237times35(c)117times55(d)42times15(e)89times45by
multiplyingthefactorendingin5tosimplifyitanddividingtheresultsbythesamenumber
24Multiply(a)52times58(b)63times67(c)79times71(d)48times42(e)85times85(ƒ)23times27(g)37times33bythemethodusedwhenunitsaddupto10andtensdigitsarealike
25Multiply(a)63times43(b)75times35(c)94times14(d)47times67(e)58times58(ƒ)84times24(g)26times86bythemethodusedwhenunitsdigitsarealikeandtensdigitsaddupto10
26Multiplybycrossmultiplicationmethodgettinganswerinoneline(a)63times54(b)82times23(c)72times48(d)52times43(e)48times69(ƒ)91times18
27Multiply(a)95times45(b)75times65(c)65times85(d)35times55(e)95times35(ƒ)75times55(g)35times35(h)85times75(i)145times65(j)$135times45(k)$156times75(l)$215times95bysimplifiedmethod
28Multiply(a)87times7(b)92times8(c)64times6(d)657times9(e)49times5(ƒ)432times7byleft-handmultiplication
29Whatpartof100is(a)50(b) (c) (d) (e) (f) (g) (h)(i)
30Whatpartof10is(a)125(b) (c) (d) (e)75(ƒ) (g) (h)
31Whatpartof1is(a)25(b)375(c)625(d)125
32Whatpartof1000is(a)125(b)875(c)625(d)375
33Whatisthecostof84articleswhenthepriceofoneis
34Multiplythefollowingbythealiquot-partmethod
(a) (b) (c)25times5744(d)(e) (ƒ) (g) (h)75times48(i) (j) (k) (l)20times85(m)58times50(n)48times25(o)2840times75
35Whatisthecostof
(a)85articlesat$250eachusingaliquot-partmethod
(b)58articlesat (c)46articlesat(d)36lbat perIb(e)48lbat25cent(ƒ)56lbat(g)24lbat75cent(h) ydat$624peryd(i) ydat72cent
36Whatisthecostof1860lboffeedat$12atonMakeuseofaliquot-partmethod
37Findthecostof72lawnmowersat$125eachusingaliquotpart
38Whatisthecostof48radiosat$6250eachUsealiquot-partmethod
39Multiply(a)32times24(b)68times24(c)242times124(d)57times24usingsimplifiedmultiplicationby24
40Multiply(a)242times26(b)242times26(c)32times26(d)68times26(e)57times26usingsimplifiedmultiplicationby26
41Multiply(a)57384times9(b)58761times9(c)4328times9(d)98989times9(e)847632times9usingsubtractionmethod
42Multiply(a)87583times11(b)9898times11(c)57384times11(d)58761times11(e)4328times11(ƒ)847632times11usingadditionmethod
43Multiply(a)687times8(b)687times7(c)432times8(d)432times7(e)982times8(ƒ)982times7byannexingazeroandsubtractingeithertwiceorthreetimesthenumber
44Multiply(a)687times99(b)687times98(c)687times97(d)982times99(e)982times98(ƒ)982times97byaddingtwozerosandsubtractingtherequirednumberoftimesthenumber
45Multiply(a)84times98(b)94times96(c)86times93(d)79times95(e)82times88(ƒ)982times978byusingcomplementmultiplication
46Multiply(a)37512times16(b)8762times14(c)982times18(d)76582times12(e)8462times13(ƒ)6879times19usingonlyonelineintheproductasshownintextexamples
47Multiply(a)84times76(b)758times84(c)68times47(d)832times59(e)54times132(ƒ)38times78(g)176times42(h)872times74usingcrossmultiplicationandcheckresultsbyldquocastingoutninesrdquo
CHAPTERIV
DIVISION
125WhatismeantbydivisionDivisionistheinverseofmultiplicationAswehaveseenthat
multiplicationismerelyasimplifiedformofadditionwecanconcludethatitsinversedivisioninitssimplestformismerelyrepeatedsubtraction
Ex(a)Whenwemultiply8fourtimesweget8times4=32whichissimplifiedaddition8+8+8+8=32=productNowdividingtheproduct32by8weget4
32minus8=2424minus8=1616minus8=888minus8=0
Wehavesubtracted8successivelyfrom32infourstepstoget
Ex(b)Supposeyouhave972applesandyouwanttodividethemequallyamong324menHowmanyappleswilleachmanreceive
972minus324=648648minus324=324324minus324=0
Countthenumberofsubtractionswhichis3andyouget3applesforeachman
Ex(c)Howmany2rsquosin8Subtract2from8asmanytimesaspossiblenotingthenumberoftimes4astheanswer
126Inwhatotherwaysmaydivisionbethoughtof(a)Divisionproperaspeciesofmeasurementasfindinghowmanytimesone
numberiscontainedinanother
(b)PartitionwhichisdividinganumberintoequalpartsthenumberofsuchpartsbeinggivenThisisimportantwithconcretenumbersandisofnoimportancewithabstractnumbers
Ex(a)Howmanytimesis7containedin35
Ex(b)If3gallonsofmilkyield21ouncesofbutterhowmanyounceswill1gallonyield
Thinkof21ouncesasdividedinto3equalpartswhichwillresultin7ouncesineachpart
127Whatarethetermsofadivision
Dividend=ThenumbertobedividedorseparatedintoequalpartsNumberinfrontofdivisionsign
Divisor=ThenumberofequalpartsintowhichdividendistobeseparatedorthenumberbywhichdividendistobedividedNumberfollowingdivisionsign
Quotient=Resultobtainedbydivision
EXAMPLES
(a)42divide7=6orDividend
(b) or
(c) orDivisor(=7
128WhenthedividendisconcreteandthedivisorisabstractwhatisthequotientThequotientislikethedividend
EXAMPLEIf3gallonsofmilkyield21ouncesofbutterwefindthenumberofouncescontainedin1gallonofmilkbydividing21ouncesby3(notby3gallons)getting7ouncesThedivisorhere(3)isanabstractnumberandtheterm3gallonsservesonlytoindicatethenumberofgroupsintowhich21ouncesistobeseparated
129WhatistheresultwhenboththedividendanddivisorareconcreteThedividendanddivisormustbealikeandthequotientwillbeabstract
EXAMPLE
Sevenouncesgoesinto21ouncesthreetimes
130WhatismeantbyaremainderindivisionWhendivisionisnotexactthepartofthedividendremainingiscalledthe
remainder
EXAMPLE
17divide2=8with1asaremainder
Theremainderisplacedoverthedivisoras here
131WhymaywethinkofdivisionastheprocessoffindingonefactorwhentheproductandtheotherfactoraregivenEXAMPLEIn7times3=21wehavemultiplication
Factor(=7)timesFactor(=3)=Product(=21)
In =7wehavedivision
132HowcanwemakeuseofthefactthatdivisionistheoppositeofmultiplicationEXAMPLEWhatnumbermultipliedby324wouldgive972
Weknowthat324=300+20+4
972=900+70+2
133Ifwewantedtodivide3492meninto4groups
howwouldweproceed
(a)8times4=32or800complete 873(=800+70+3)
4rsquos=3200 4)3492
(b)Subtract3200from3492 -3200 (=4times800)
(c)7times4=28or70times4=280 292
leaves292menstilltobecounted
(d)Subtract280from292 -280 (=4times70)
(e)3times4=12
12leaves12menstilltobecounted
(ƒ)Addingthequotientsweget -12 (=4times3)
800+70+3=873
ThisprocesscanbeshortenedbyomittingthezerosasisdoneinmultiplicationBringdownonlythenumberornumberstobeusedinthenextpartoftheexampleBecarefulinplacingthenumbersdirectlyunderthecolumnsinwhichtheyfirstappeared
Whendividingwithonlyonedigitwemayshortenthestepstillfurtherbyldquothinkingrdquothesubtractionsandcarryingtheremainders
ldquoThinkrdquosubtract8times4=32from34carry2tothe9 making29
ldquoThinkrdquosubtract7times4=28from29carry1to2making12
ldquoThinkrdquosubtract3times4=12from12getting0whichiszeroremainder
134WhatismeantbyldquoshortdivisionrdquoandwhatistheprocessinsimpleformWhenthedivisorissosmallthattheworkcanbeperformedmentallythe
processiscalledshortdivision
EXAMPLEDivide9712by4Writeas
(a)BeginatleftFindhowmanytimesdivisor4iscontainedinthefirstfigureofthedividend
4iscontainedin9twotimeswitharemainder1
(b)Reducethe1tothenextlowerordermaking10whichwith7makes17
4iscontainedin17fourtimeswitharemainder1
(c)Reducethis1tothenextlowerordermaking10whichwith1makes11
4iscontainedin11twotimeswitharemainderof3
(d)Reducethis3tothenextlowerordermaking30whichwith2makes32
4iscontainedin32eighttimeswithnoremainder
135Howdowedivide3762by7usingshortdivision
(a)7isnotcontainedinthefirstfigureofthedividend3and3mustbereducedtothenextlowerordermaking30whichwith7makes37
(b)7iscontainedin37fivetimeswith2remainderReduce2tonextlowerordermaking20whichwith6makes26
(c)7iscontainedin26threetimeswith5remainderReduce5tonextlowerordermaking50whichwith2makes52
(d)7iscontainedin52seventimeswith3remainderwhichiswritten
136HowdoweproceedwithlongdivisionEXAMPLETodivide73158(=Dividend)by534(=Divisor)
(a)Sincedivisorhas3digitstakethefirst3digitsofthedividendandaskhowmanytimesdivisor534iscontainedin731(=first3digitsofdividend)(Usuallyaclueisgivenbytrialofthefirstfigureofdivisorwhichhereis5andfindinghowmanytimesitiscontainedinfirstfiguresofdividendhere7)Divide5into7or1
(b)Writepartialquotient1overthelastfigureof731Here1goesoverthe1of731
(c)Subtract1times534from731getting197andbringdownthe5whichisthenextdigitofthedividendThisresultsinthepartialdividend1975
(d)Dividefirstfigure5ofdivisorinto19(=thefirsttwofiguresofpartialdividend)Writepartialquotient3over5oftheoriginaldividend
(e)Subtract3times534=1602from1975getting373andbringdownthe8whichisthenextdigitofthedividendThisresultsinthepartialdividend3738
(ƒ)Divide5ofdivisorinto37ofpartialdividendWritepartialquotient7over8oforiginaldividend
(g)Subtract7times534=3738from3738ofpartialdividendgettingzeroremainderQuotientistherefore137exact
137WhatdowedowhenthelastsubtractionisnotzeroEXAMPLEDivide73170by534
Theremainder12isexpressedas12overthedivisoror hereThequotientis
Sometimesweplaceadecimalpointafterthelastdigitofthedividendaddzerosandcontinuetheprocessofdivisiontoexpresstheremainderasadecimal
138WhatistheprincipleofthetrialdivisorinlongdivisionEXAMPLEDivide236987by863
(a)Ordinarilytryfirstleft-handdigitofdivisorintothefirsttwodigitsofdividendas8ofdivisorinto23ofdividend
(b)Butwhentheseconddigitofdivisorisnumber5orgreater(6inthiscase)thenincreasethefirstdigitofdivisorby1andtryindividendHeretry9into23
getting2asquotient
(c)Inthenextpartialdividendtry9into64getting7asquotient
(d)Inthefollowingpartialdividendtry9into39getting4asquotient
(e)Remainderhereis Quotientis
139Whatistheruleforlongdivision(a)WritedivisoratleftofdividendwithacurvedlinebetweenthemTakethe
fewestnumberofdigitsatleftofdividendthatwillcontaindivisorandwritethispartialquotientontopovertheright-handdigitofthepartialdividend
(b)Multiplyentiredivisorbythispartialquotientandwritetheproductunderthepartialdividendused
(c)Subtractthisproductandtoremainderannex(bringdown)thenextfigureofdividendforthesecondpartialdividend
(d)Divideasbeforeandcontinueprocessuntilalldigitsofdividendhavebeenusedtomakepartialdividends
(e)Whenthereisaremainderwriteitwiththequotient
140WhatisapureproofofanydivisionMultiplydivisorbyquotientandtothisproductaddtheremainderifanyThe
resultshouldequalthedividend
EXAMPLE
141WhatistheprocedurefordivisionwithUnitedStatesmoneyDivideasinintegralnumberswritingthefirstdigitofthequotientoverthe
right-handdigitofthefirstpartialdividend(Placethedecimalpointinthequotientdirectlyoverthedecimalpointinthedividend)
EXAMPLEDivide$82911by87
142Whatisthequotientofthedivisionof$4536by$027Changethedividendanddivisortocentswhichgives4536centsdividedby
27centsThequotientis168whichisanabstractnumbershowingthenumberoftimes27centgoesinto4536cent
143HowcanfactoringofthedivisorbeusedtoreduceaproblemoflongdivisiontoaseriesofshortdivisionsEXAMPLEDivide27216by432Herethedivisor432canbefactoreddown
farenoughtogiveaseriesofshortdivisionsbythefactorswhichprocedureissubstitutedforthelongdivision
432=Divisor=12times12times3
144WhatistheprocedurefortheabovewhenthereisaremainderEXAMPLEDivide47897by18
Factordivisor18as2times3times3=18Divideby2thenthequotientofthisby3andthequotientofthisby3
Quotientis
Thefirstremainder1remainsunchanged
Theseconddivisionhasaremainder2Asthisdivisionisofonehalfthenumberby3youmultiplytheremainder2by2getting4andaddingthistothe
previousremaindergetting4+1=5
Thenextdivisionisofonesixthofthenumberby3Youthenmultiplythisremainderby6getting12andaddthistotheprevious5getting17whichisthefinalremainder
Itisseenthateachremainderexceptthefirstismultipliedbythefactorsofthedivisionsprecedingitsownandthesumoftheproductsisthetotalremainder
145Whatisthequotientof65349by126usingthefactoring-ofthe-divisormethodDivisor126=2times3times3times7
Quotient=
146Whatistheprocedurefordividingby101001000etcSetoffasmanyfiguresattherightofthedividendasthereareciphersinthe
divisorThefiguresthussetoffaretheremainderTheotherfiguresarethequotient
Ex(a)65divide10=6with5asremainderor (OnecipherindivisorSetoff1figureatrightofdividend)
Ex(b)579divide100=5with79asremainderor (TwociphersindivisorSetoff2figuresatrightofdividend)
Ex(c)
Dividing200by100weget2
Dividing5670by100weget5670
Nowdividing5670by2weget2835(Ans)
Ex(d)
WhenthedivisorendsinoneormorecipherscuttheseoffandalsocutoffanequalnumberoffiguresfromtherightofthedividendThendividebythefiguresremaining
Ex(e)8743divide700=008743divide7=001249
147Howdoweapplytheexcess-of-ninesmethodtoprovethecorrectnessofadivision(a)Getexcessof9rsquosindivisor
(b)Getexcessof9rsquosinquotient
(c)Multiplythesetwoexcessesandgetexcessof9rsquosoftheproduct
(d)Addtothistheexcessof9rsquosinremainderGetexcessofsum
(e)Getexcessof9rsquosindividendandcompare
EXAMPLE
Aquotientmaybeincorrecteventhoughtheexcess-of-ninesmightcheckbutthishappensrarely
148WhatismeantbyanevennumberAnumberdivisibleby2iscalledanevennumberAnevennumbermayend
in2468orinazero
EXAMPLES42547668970areevennumbersEachdividedby2results
in21273834485
149Howcanweknowwhenanumberisdivisibleby3Whenthesumofitsdigitsisdivisibleby3thenumberitselfisdivisibleby3
Ex(a)Number=213Adddigits2+1+3=6Nowsum6isdivisibleby3Thereforenumber213isdivisibleby3Ans=71
Ex(b)Number=531Adddigits5+3+1=9Sum9isdivisibleby3Thereforenumber531isdivisibleby3Ans=177
150Ifwehaveanevennumberanditisdivisibleby3bywhatothernumberisitalsodivisibleThenumberisalsodivisibleby6becauseanevennumberisdivisibleby2
and2times3=6
EXAMPLEGivennumber=162whichisanevennumberAdddigits1+6+2=9whichisdivisibleby3
there4162isdivisibleby6or Ans
151Whenisanumberdivisibleby4Whenitslasttwodigitsaredivisibleby4
EXAMPLE7624Lasttwodigits24aredivisibleby4
152Whenisanumberdivisibleby5Whenitendsin5orzero
Ex(a)
Ex(b)
153Whatnumberoranymultiplesofitcanbedividedby711or13Number1001oranyofitsmultiplescanbedividedby711or13
Ex(a)
Ex(b)
154Whenisanumberdivisibleby8Whenthenumberendsinthreezerosorwhenthelastthreedigitsaredivisible
by8
Ex(a)
Because1000isdivisibleby8whateverprecedesthelastthreefiguresmerelyaddsthatmanythousandsanddoesnotaffectthedivisibilityby8
Ex(b) Nowadd1000getting1136Then
Againadd1000getting2136Then
Ex(c)
Nomatterhowmanyfiguresareplacedinfrontoftheoriginal136thenumberisdivisibleby8
Ex(d)29632Consider divisibleby8
there429632isdivisibleby8getting3704Ans
155Whenisanumberdivisibleby9Whenthesumofitsdigitsisdivisibleby9
Ex(a)Numberis8028Adddigits8+0+2+8=18
Ex(b)Number Adddigits3+8+9+3+4=27and
156Whenisanumberdivisibleby25Whenitendsintwozerosorintwodigitsformingamultipleof25
Ex(a)
Ex(b)
157Whenisanumberdivisibleby125Whenitendsinthreezerosorinthreedigitsformingamultipleof125
Ex(a)
Ex(b)
158Whatisthecriterionforanumberdivisibleby11(a)Whenthesumofeven-placeddigitsequalsthesumofodd-placeddigits
Ex(a)
(b)Whenthedifferencebetweenthesumoftheodd-andeven-placeddigitsisdivisibleby11thenumberitselfisdivisibleby11
Ex(b)
Ex(c)
159Howcanwetellinadvancewhattheremainderwillbewhenthedivisoris9AddthedigitsandthenaddthedigitsofthissumThislastistheremainder
Ex(a)867
Adddigits8+6+7=21(=Sum)Adddigitsofsum2+1=3(=Remainder)
Ex(b)973285
Adddigits9+7+3+2+8+5=34(=Sum)Adddigitsofsum3+4=7(=Remainder)
160Whatisashort-cutwayofdividingby5Multiplyby2andpointoffonedecimalplacetotheleft
Ex(a)23divide523times2=46Pointoffoneplacetoleft46Ans
Topointoffonedecimalplacemeansdividingby10
Ex(b) 832times2=1664Pointoffoneplace1664Ans
161Whatisasimplewayofdividingby25Multiplyby4andpointofftwoplacestotheleft
Ex(a)1394divide251394times4=5576Pointofftwoplaces5576
Topointofftwoplacestotheleftmeansdividingby100
Ex(b)
162Whatisasimplewaytodivideby125Multiplyby8andpointoffthreeplacestotheleft
EXAMPLE7856divide1257856times8=62848Setoffthreeplacestoleftgetting62848
Dividingby1000meanssettingoffthreeplacestotheleft
163Whatistheshort-cutwayofdividingbyanyaliquotpartof100Multiplybytheinvertedfractionrepresentedbythealiquotandpointofftwo
placestotheleft
Ex(a) Invert getting
875times3=2625Pointofftwoplacestoleftgetting2625Ans
Ex(b) Invert getting
Pointofftwoplacestoleftgetting90Ans
Ex(c) Invert getting
12367times6=74202Pointofftwoplacestoleftgetting74202Ans
164Whatisasimplewayofdividingby99(a)Addthetworight-handdigitstotherestofthenumberPutthissumdown
undertheoriginalnumber
(b)Addthetworight-handdigitsofthistotherestofitsnumberandputthisdownundertheothertwo
(c)Keepupthisprocessuntil99oraquantitylessthan99isleftThisistheremainder
(d)Addupthedigitstotheleftexcludingthetworight-handdigitsofeachnumber
Ex(1)
Remainder Add1toquotient7890getting7891
Ex(2)
165Howcanwemakeanumberdivisibleby3(a)AddthedigitsDividethissumby3andgetremainder
(b)Subtractremainderfromoriginalnumber
EXAMPLE13477Adddigits1+3+4+7+7=22
Subtractremainder1from13477getting13476AnsNow
166Howcanwemakeanumberdivisibleby9(a)AddthedigitsDividethissumby9Getremainder
(b)Subtractremainderfromoriginalnumber
EXAMPLE13477Adddigits1+3+4+7+7=22
Subtractremainder4from13477getting13473
167HowdoweobtainanaverageofanumberofitemsDividethesumoftheitemsbythenumberofitemsadded
EXAMPLEFindtheaveragesalesmadebyasalesmanduringtheweekwhenhisdailysalesare
Monday $26860
Tuesday $32985
Wednesday $9745
Thursday $23990
Friday $29670
Numberofitems=5) $123250 (=Sum)
$24650 (=Average)
WeseethatthesalesforMondayTuesdayandFridaywereabovetheaveragewhileforWednesdayandThursdaytheywerebelowaverage
168WhatistheruleforfindingthevalueofoneofanythingAlwaysdividebythatofwhichyouwanttofindthevalueofone
Ex(a)If75bookscost$300whatisthecostof1book
Youwantthecostof1booksodividebythenumberofbooks
Ex(b)Ifadozenhatscost$72whatisthecostof1hat
Youwantthecostof1hatsodividebynumberofhats
Ex(c)Ifapoundofcoffeecosts80centhowmanyouncescanyougetfor10cent
Firstyouwantthenumberofouncesfor1centsodividebycents
For10cent
Ex(d)Ifajeepused16gallonsofgasolineindriving288mileshowmuchdoesitconsumeona486-miletrip
Youwantfirstthenumberofmilesfor1gallonsodividebygallons
then
Ex(e)Ifittakes8minutesforapipetofillatankhowmuchofthetankwillbefilledin1minute
Youwanttheamountfor1minutesodividebyminutes
PROBLEMS
1Howmanytimesis8containedin56
2If3gallonsofmilkyield18ouncesofbutterhowmanyounceswill1gallonyield
3Ifyouhave1048padsofwritingpaperandyouwanttodividethemequallyamong262employeeshowmanypadswilleachonereceive
4Findthequotientsofthefollowingandproveeachbymultiplyingthefactorstogether
(a)6divide2=because2times=6
(b)9divide3=because3times=9
(c)12divide4=because4times=12
(d)18divide9=because9times=18
(e)28divide7=because7times=28
(ƒ)42divide6=because6times=42
(g)48divide8=because8times=48
(h)66divide6=because6times=66
(i)72divide9=because9times=72
(j)84divide7=because7times=84
5If$1ischangedtofive-centpieceshowmanyarethere
6Ifamanearns$16whileaboyearns$6howmuchwilltheboyearnwhilethemanearns$96
7Ifamancanpicktwiceasmuchfruitasaboyand4boysand3menpick5acresoforchardinadaywhatamountofgrounddoeseachcover
8Ifamaneats380gramsofcarbohydrates130gramsofproteinand60gramsoffatseachdayhowmuchdoesheaveragepermeal
9Divide
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
(m) (n) (o) (p)
(q) (r) (s) (t)
(u) (v) (w) (x)
(y) (z)
10Findthequotientof
(a)1607divide19(b)6548divide89(c)3402divide81
(d)3485divide873(e)54963divide863(ƒ)861618divide843
(g)879384divide508(h)938764divide879(i)42896divide269
(j)98641divide679(k)3862847divide76298(l)
(m) (n) (o)
11Ifthereare266pagesinabookandyoucanread38pagesinanhourhowlongwillittakeyoutoreadit
12Findthequotientof
(a)$1836divide12(b)96750divide43(c)$96750divide$43
(d)$43890divide$21(e)$43890divide21cent
13Dividebyfactoringmethod
(a)23112divide108(b)39798divide99(c)35952divide84
14Divide
(a)490divide10(b)487divide10(c)5300divide100(d)15874divide100
(e)385divide10(ƒ)8745divide100(g)490divide20(h)487divide30
(i)5300divide400(j)385divide20(k)8745divide700
(l)697divide1000(m)16720divide800
15Applyexcess-of-ninesmethodtoprovethecorrectnessofthedivisionsofproblem10
16
(a)Is7893divisibleby3(Usingsum-of-digitsmethod)
(b)Is3876divisibleby6(Usingshort-cutmethod)(c)Is3876divisibleby
4(Usinglast-two-digitsmethod)
(d)Is8695divisibleby5(Usingcriterion)
(e)Is14014divisibleby711or13(Usingcriterion)(ƒ)Is7462768divisibleby8(Usingdivisibility-of-last-3-digitsmethod)
(g)Is8658divisibleby9(Usingsum-of-digitsmethod)
(h)Are7800and9864175divisibleby25(Usecriterion)
(i)Are7860000and76375divisibleby125(Usecriterion)
(j)Are3657654and78947divisibleby11(Usecriterion)
17
(a)Whatwillbetheremainderof948divide9(withoutdividingfirst)(b)Canyoutellinadvancetheremainderof864893divide9
18
(a)Divide39by5atoncebyshort-cutmethod(b)Divide482by25byshort-cutmethod(c)Divide6743by125byshort-cutmethod
19Dividethefollowingbyuseofaliquotpartsof100
(a) (b)
(c)
(d) (e)
(ƒ)34560divide5(g) (h)
(i)3475divide25(j)2700divide75(k)1400divide125
20Divide(a)872317divide99(b)867432divide99bysimplemethodshownintext
21Make(a)25694(b)85642divisibleby3bymethodshownintext
22Make(a)25694(b)85642divisibleby9bymethodshownintext
23Ifsixrankingcandidatesonanexaminationhadmarksof921873856807802and791respectivelywhatistheaveragemark
24Sixteenstudentsinaclassinarithmeticmadethefollowinggradesonatest849674938886817781949986716976and84Whatwastheaveragegradeoftheclass
25Anauthorreceivedroyaltiesfromhispublisherduringasix-yearperiodasfollows$89765$91759$89325$99775$114679and$123832Whatistheaverageyearlyroyalty
26Ifyouhaveanappleorchardof2000treesifyouuse4gallonsofsprayingmixtureforeachtreeandyoumix1lbofParisgreenat80centperlbwith150galofwaterwhatwouldbethecostoftheParisgreenfor2sprayingsWhatwouldbethecostpertree
CHAPTERV
FACTORSmdashMULTIPLESmdashCANCELLATION
169WhatisaprimenumberAnumberdivisibleonlyby1anditself
EXAMPLES123571113171923293137etcareprimenumbersEachisdivisibleonlyby1anditself
170WhatisacompositenumberOnethatisdivisiblebyothernumbersinadditionto1anditself
EXAMPLES46810121416183644etcarecompositenumbers
171WhatisafactorofanumberAnexactdivisorofthenumber
Ex(a)2isafactorof6because2isanexactdivisorof6
Ex(b)2346arefactorsof12becauseeachisanexactdivisorof12If3isonefactorof12then4istheotherfactor
172WhatismeantbyfactoringTheprocessofseparatinganumberintoitsfactors
173WhatisaprimefactorAfactorwhichisaprimenumber
Ex
(a)22and3areprimefactorsof12(b)222and3aretheprimefactorsof24
Ofcourse46812arealsofactorsof24butthesearenotprimefactors
174Whatdowecallanumberthathasthefactor2AnevennumberNumbersnotdivisibleby2arecalledoddnumbers
175WhatismeantbyacommondivisororfactorOnethatiscommontotwoormorenumbers
EXAMPLE
4isafactorcommonto12and363isafactorcommonto12and361262arefactorscommonto12and36
Numbersthathavenocommonfactorsaresaidtobeprimetoeachother
176Whatfactsregardingthedivisibilityofnumbersareofassistanceinfactoring(a)2isafactorofallevennumbers
(b)3isafactorwhenthesumofthedigitsisdivisibleby3
(c)4isafactorwhenthetwodigitsattherightarezerosoranumberdivisibleby4
(d)5isafactorwhentheunitsfigureis5orzero
(e)6isafactorofallevennumbersthataredivisibleby3
(f)8isafactorwhenthethreedigitsattherightarezerosoranumberdivisibleby8
(g)9isafactorwhenthesumofthedigitsisdivisibleby9
(h)11isafactorwhenthesumofthedigitsintheevenplacesequalsthesumofthedigitsintheoddplacesorwhenthedifferencebetweenthesetwosumsis11orsomemultipleof11
177HowdowefindtheprimefactorsofanumberDividebyaprimefactorandcontinuetodividebyaprimefactoruntilthelast
quotientisaprimenumber
Ex(a)Whataretheprimefactorsof720
Ex(b)Findtheprimefactorsof7644
178WhatismeantbythegreatestcommondivisororfactorabbreviatedGCDorgcdThelargestdivisororfactorcommontotwoormoregivennumbersisthe
GCD
Ex(a)6isthegreatestcommondivisorof24and30
Ex(b)8isthegreatestcommondivisorof1624and32becauseitisthelargestnumberthatwillexactlydivideeachofthenumbers
179WhatistheruleforfindingtheGCDoftwoormorenumbersSeparatethenumbersintotheirprimefactorsandgettheproductoftheprime
factorsthatarecommontoallthenumbers
Ex(a)
Factors2and3arecommontoboth24and30
there42times3=6=GCD
Ex(b)
Factors222arecommontoallthreenumbers
there42times2times2=8=GCD
180WhatisamoreconvenientmethodoffindingGCDArrangethenumbersasbelowanddividebysomenumberwhichwillexactly
divideeachofthemContinuedoingthisuntilnodivisorcanbefoundtodivideeachlastquotientMultiplyallthecommonfactors
Commonfactorsrarr2times2times2times3=24=GCD
181WhatismeantbyamultipleofanumberItistheproductofthatnumbermultipliedbyaninteger
Ex(a)24isamultipleofnumber12because12multipliedbyaninteger2=24
Ex(b)Whatnumbersaremultiplesof8
2times8=163times8=244times8=32etc
Thus162432etcaremultiplesof8
182WhatismeantbyacommonmultipleoftwoormorenumbersAnumberthatisamultipleofeach
Ex(a)16isacommonmultipleof4and8becauseeitherofthemmultipliedbyaninteger=16
Ex(b)18isacommonmultipleof236and9becauseanyofthesemultipliedbyaninteger=18
183Whatismeantbytheleastcommonmultiple(LCM)oftwoormorenumbersTheleastnumberthatisamultipleofeach
Ex(a)18isacommonmultipleof3and6but12istheleastcommonmultipleof3and6because12isthesmallestnumberwhichcontainseachwithoutaremainder
Ex(b)72isacommonmultipleof69and12but36istheLCMbecauseitisthesmallestnumberwhichcontainseachwithoutaremainder
184Whatisamethodoffindingtheleastcommonmultiple(LCM)of1828and36SeparateeachnumberintoitsprimefactorsMultiplythefactorsusingeach
factorthegreatestnumberoftimesitoccursinanyofthegivennumbersthatarefactored
2doesnotappearasafactormorethantwiceinanynumber
3doesnotappearasafactormorethantwiceinanynumber
7appearsonce
there42times2times3times3times7=252=LCMthatwillcontain1828and36withoutaremainder
185WhatisanothermethodofgettingtheLCMof1828and36Dividethenumbersbyanyprimenumberthatwillexactlydividetwoormore
ofthemAnynumbernotsodivisibleisbroughtdownintactContinuethisprocessuntilnofurtherdivisioncanbemadeMultiplyalldivisorsandthequotientsremainingtogettheLCM
186WhatismeantbycancellationEliminationoffactorsinthedividendanddivisorbeforedividingThe
quotientisnotaffectedbyeliminationoffactorswhicharecommontobothdividendanddivisor
Ex(a)Divide4368by156byfactoringandcancelling
ThesameanswercanbeobtainedbylongdivisionItisnotnecessarytoseparate
thenumbersintotheirprimefactorsThecriteriafordivisibilityofnumbersmaybeusedasshowninquestion176
Ex(b)Compute bymeansofcancellation
Ex(c)Computebycancellation Ans
13isafactorof39and65threeandfivetimesrespectively
Then3iscontainedin105thirty-fivetimes
Theproductoftheremainingfactors5times35=175Ans
Ex(d)Computebycancellation
Findfactorscommontonumbersabovethelineandnumbersbelowthelineandcancelthem
PROBLEMS
1Nametwofactorsof18303681120
2Namethreefactorsof1832455066
3Nameafactorcommonto12and36
4Nameallthefactorsorexactdivisorsof3717
5Makealistofallprimenumbersbelow100
6Makealistofalloddnumbersbelow50
7Separateintoprimefactors45781012131416182124253034
8Separatetheprimecompositeevenandoddnumbersinthefollowing167101112141920212425273334
9Givetheprimefactorsof
(a)310(b)297(c)670(d)741(e)981(f)385(g)2650
(h)1215((i)321(j)1575(k)10935(l)420(m)497
(n)378(o)462(p)2430(q)25344(r)73260(s)599676
(t)273564(u)15625(v)10675(w)12625(x)976
(y)8050(z)3848
10FindtheGCD(greatestcommondivisor)of
(a)68112240(b)2184126147(c)212877
(d)457281(e)4477121(f)1498112(g)248096
(h)284236(i)457281(j)31522679012
(k)144576(l)820697(m)1251751792(n)60043318
(o)125423618163(p)1086905
11Givetwomultiplesof
(a)9and3(b)7and5(c)9and2(d)3and7(e)8and5
(f)6and3(g)8and2(h)92and8(i)36and9
(j)86and4
12FindtheLCM(leastcommonmultiple)of
(a)9and12(b)21and36(c)5and15(d)1215and18
(e)3642and48(f)3918and27(g)51525and35
(h)148135and15(i)324835and70(j)728896and124(k)112255and110
(l)212426and28(m)92142and63
(n)367548and24(o)71456and84(p)2472128and240
13Dividebycancellationmethodoffactorsandprovebylongdivision
(a)38367divide1827(b)52800divide3520(c)90384divide3228
(d)88368divide3682(e)32768divide2048
14Solvebycancellation
(a)3times27times48times81=6times9times54times210(b)81times16times10times12=9times27times2times5(c)8times12times18times32=4times6times9times16(d)42times36times77times22divide11times6times24times21(e)5times30times65times125=15times75times95
15Howmanylbofbutterat55centalbcanbeexchangedfor30dozeggsat66centadoz(Bycancellation)
16Howmanydaysof8hreachwouldoneneedtoworkat$230anhourtopayfor8tonsofcoalat$2760aton(Solvebycancellation)
17If14menearn$725760working27daysof8hourseachat$240anhourhowlongwillittake21menworking8hoursadayatthesameratetoearnthesameamount(Solvebycancellation)
18Ifyoudrove20000milesonnewtiresbeforereplacementandyoupaid$120forthe4newtireswhatwasthetirecostforeach100miles(Solvebycancellation)
CHAPTERVI
COMMONFRACTIONS
187WhatdoesafractionmeanTheLatinfrangeremeansldquotobreakrdquoTheLatinfractusmeansldquobrokenrdquoThus
afractionisabrokenunitorapartofaunitAlsoldquofractionrdquocomesfromthesameLatinrootasthewordldquofragmentrdquomeaningldquoapartrdquoActuallyafractionisanyquantitynumericallylessthanaunit
188WhatarethetermsofafractionEveryfractionhasanumeratorplacedaboveahorizontallineanda
denominatorplacedbelowthelineThedenominatoristhedivisorofthenumerator
EXAMPLE
189WhatisassumedinexpressingfractionaldivisionItisassumedthatallofthepartsintowhichanobjecthasbeendividedareof
exactlyequalsize
190WhatismeantwhenwesaythatathingisdividedequallyintotwopartsandhowisthefractionexpressedTheobjectissaidtobedividedintohalvesTheobjectisdividedintotwo
partsTheobjectorunittobedividedisplacedasthenumeratorofthefractionthenumberofdivisionsisthedenominator
Thus
191Whatismeantby
(a)
(b)
(c)
(d)
192WhatismeantbyaunitfractionWhenthenumeratorofafractionis1itiscalledaunitfractionas
193WhatisavulgarfractionandhowisitclassifiedAvulgarfractionisoneexpressedasadivision
ThedivisorclassifiesthefractionEx(a) isclassifiedasthirdsfromitsdivisor3
Ex(b) isclassifiedastwenty-fifthsfromitsdivisor25
194WhatarethepartsofavulgarfractionandhowisitwrittenThenumeratoristhedividendthedenominatoristhedivisorItiswrittenasa
numeratoraboveanddenominatorbelowashorthorizontalordiagonallineorbar
Ex(a) Numeratortellsusthatonly1ofitsclassisconsidered
Ex(b) Numeratortellsusthat11ofitsclassaretaken
195WhatothermeaninghasthebarinafractionThebarmeansldquodivisionrdquointhesamewayasthesign[divide]
Ex(a)
Ex(b) Bothexpressionsmeanthesamething
Ex(c)
196WhatarethethreewaysinwhichafractionmaybeinterpretedThefraction forexamplemaybethoughtofas(a)3unitsdividedinto2
equalparts
(b)1unitdividedinto2equalpartswith3ofthesepartstakenas3times
(c)Asanindicateddivisionnotyetperformed
EXAMPLESAssume1orunityisaline1inchlong
ThreeunitsdividedintotwoequalpartsEachpart
(b)
(c) canbethoughtofasadivisionnotyetperformed
197Whenweaddupallthefractionalpartsofaunitwhatdowegetasaresult
Wegetthewholeunit
Ex(a)
Ex(b)
Ex(c)
Oranyfractionalexpressionofanumberdividedbyitself=1=unityas
198WhatisasimplefractionOnewhosenumeratoranddenominatorarewholenumbers
EXAMPLE and aresimplefractions
199WhatisacompoundfractionItisafractionofafraction
EXAMPLE of and of arecompoundfractions
200WhatisacomplexfractionOneinwhicheitherthenumeratorordenominatororbotharenotwhole
numbers
Ex(a) Numeratorisnotawholenumber
Ex(b) Denominatorisnotawholenumber
Ex(c) Bothnumeratoranddenominatorarenotwholenumbers
Alltheabovearecomplexfractions
201Whatisaproperfraction
Oneinwhichthenumeratorislessthanthedenominator
EXAMPLE areproperfractionsEachhasavaluelessthanaunitNotethatthenumeratordoesnothavetobe1
202WhatisanimproperfractionOneinwhichthenumeratorequalsorexceedsthedenominatorThefraction
isthusequaltoorgreaterthan1unit
Ex(a)
Ex(b)
203WhatisamixednumberAwholenumberandafractiontakentogether
EXAMPLE aremixednumbers
204HowmayweshortentheprocessoffindingthevalueofanimproperfractionDividethenumeratorbythedenominatorWritethequotientasawhole
numberfollowedbyafractioninwhichtheremainderisexpressedasanumeratoroverthesamedenominator
Ex(a) Thirteengoesinto48threetimeswitharemainderof9 isamixednumber
Ex(b)
205HowdowechangeamixednumberintoanimproperfractionMultiplythewholenumberbythedenominatoraddthenumeratorandplace
thissumoverthedenominator
Ex(a)
Ex(b)
Ex(c)
Thereasoningis
Then =Thisiswhywemultiplythewholenumberbythedenominatorandaddthenumeratortogetthetotalnumberoffifthsinthiscase
206WhathappenstothevalueofafractionwhenwemultiplyordivideboththenumeratorandthedenominatorbythesamenumberThevalueofthefractionisunchanged
Ex(a)
Ex(b)
207WhenisafractionsaidtobereducedtoitslowesttermsWhenthetermsareprimetoeachother
Ex(a) isexpressedinitslowesttermsbecause5and6areprimetoeachother
Ex(b) isnotexpressedinitslowesttermsbecause2isafactorcommontobothnumeratoranddenominator
208HowdowereduceafractiontoitslowesttermsDividebothnumeratoranddenominatorbyacommondivisorandcontinueto
divideuntilallcommondivisorsareeliminatedThisisdonebycancellingthe
commonfactors
Ex(a)
Ex(b)
209HowcanwechangeafractiontohighertermsMultiplybothnumeratoranddenominatorbythesamenumber
Ex(a)Change totwenty-fourths
Multiplybothnumeratoranddenominatorby6
Ex(b)Change tohundredths
Multiplybothnumeratoranddenominatorby5
210Whatmustbedonetofractionsingivingtheanswertoaproblem(a)Reducefractionstolowestterms
EXAMPLE
(b)Reduceimproperfractionstomixednumbers
EXAMPLE
211Howcanweincreasethevalueofafraction(a)Bymultiplyingthenumeratorbyanumbergreaterthan1
EXAMPLE isincreasedto
bymultiplyingnumeratorby2forexample
(b)Bydividingthedenominatorbyanumbergreaterthan1
EXAMPLE isincreasedto
bydividingdenominatorby2forexample
Thevalueofthefractionhasbeendoubledineachcase
EXAMPLEIncreasethevalueof threetimes
212Howcanwedecreasethevalueofafraction(a)Bydividingthenumeratorbyanumbergreaterthan1
EXAMPLE isdecreasedto
bydividingnumeratorby2forexample
(b)Bymultiplyingthedenominatorbyanumbergreaterthan1
EXAMPLE isdecreasedto
bymultiplyingthedenominatorby2forexample
Thevalueofthefractionisreducedone-halfineachcase
EXAMPLEDecrease toone-sixthofitsvalue
213HowdowechangeacompoundfractiontoasimplefractionPlacetheproductofthenumeratorsovertheproductofthedenominators
Ex(a)
Ex(b) of=simplefraction
214HowdowechangeacomplexfractiontoasimplefractionDividethenumeratorbythedenominator
Ex(a)
Ex(b)
215WhatisanothermethodofsimplifyingacomplexfractionMultiplybothnumeratoranddenominatorbyanumberthatdoesnotchange
thevalueofthefraction
EXAMPLE
216WhatistheconditionforaddingorsubtractingoffractionsThefractionsmustallbeofthesameclasswhichmeansthedenominators
mustallbethesame
Addthenumeratorsandplaceoverthecommondenominator
Ex(a)Add and
Ex(b)Ifthereareanywholenumbersaddthemalso
Add
Addwholenumbers1+3+12=16
Addfractions
Then
217WhatistheprocedurewhenthedenominatorsarenotthesameFindtheldquolowestcommondenominatorrdquowhichisthesmallestdenominator
intowhichallwilldivideevenlyThisisthesameastheLCMpreviouslystudied
Ex(a) +Thelowestcommondenominator(LCD)of23and6is6Allthedenominatorsdivideinto6evenly
Now
Ex(b)Add (LCD=20)
Ex(c)Add (LCD=20)MultiplyeachnumeratorbyasmanytimesasthedenominatorgoesintotheLCD
218Whatistheprocedureforsubtractionoffractions(a)Workwithonlytwotermsatatime
(b)Changeamixednumberfirsttoanimproperfractionwhenthemixednumberissmall
(c)FindtheLCD(sameasLCM)
(d)SubtractsmallernumeratorfromlargerPlaceresultoverLCD
(e)Reducetolowestterms
Ex(a)Subtract from (LCD=10)
Ex(b)Subtract from
219Howdowesubtractmixednumberswhentheyarelarge(a)Findthedifferencebetweenthetwofractionsandthenfindthedifference
betweenthewholenumbersBorrow1fromtheminuendtoincreaseitsfractionwhennecessary
Ex(a)
Ex(b)From take Before or canbetakenfrom youmustborrow1or fromtheminuendtomakethefraction Theminuendthenbecomes
220CanawholenumberalwaysbeexpressedinafractionalformYesEXAMPLE Denominatoris1
221InaddingorsubtractingtwofractionshowcanweusecrossmultiplicationtogetthesameresultaswiththeLCDmethod
Ex(a) Cross-multiplynumeratorswithoppositedenominatorstogetnumerator
Multiplydenominatorstogetdenominator
Ex(b)
Ex(c)
222WhatistheprocedureinmultiplyingoneproperfractionbyanotherPlacetheproductofthenumeratorsovertheproductofthedenominators
Ex(a)
Ex(b)
Shortentheworkbycancellationwhenpossible
Ex(c)
Ex(d)
223HowdowemultiplyaproperfractionbyawholenumberEithermultiplythenumeratorordividedenominatorbythewholenumber
Ex(a)
Ex(b)Multiply by11
Ex(c)
Theresultisthesamewhenthemultiplierandmultiplicandareinterchanged
inposition
224WhatistheprocedureformultiplyingonemixednumberbyanotherChangethemixednumberstoimproperfractionsandmultiplyintheusual
waybyplacingtheproductofthenumeratorsovertheproductofthedenominators
Ex(a)
Ex(b)Multiply
225Whatisthefour-stepmethodofmultiplyingonemixednumberbyanother(a)Multiplythefractioninthemultiplierbyeachpartofthemultiplicand
(b)Thenmultiplythewholenumberofthemultiplierbyeachpartofthemultiplicand
(c)Addthepropercolumns
EXAMPLEMultiply
226Howdowemultiplyamixednumberbyaproperfraction(a)Changethemixednumbertoanimproperfractionandmultiplyasusual
(b)Ormultiplythefractionstogetherthenmultiplythewholenumberbythefraction
Ex(a)
Ex(b)Multiply by
Ex(c)Multiply by
Orchangemixednumbertoanimproperfractionfirst Then
227WhatwordisfrequentlyusedinsteadofthemultiplicationsignorthewordldquomultiplyrdquoThewordldquoofrdquo
EXAMPLE
228WhatismeantbythereciprocalofanumberThereciprocalofanumberis1dividedbythenumber
Ex(a)Thereciprocalsof3810and25are and respectively
Since3810and25areequivalentto and respectivelyinfractionformweobtainthereciprocalofafractionbyinvertingthefraction
Ex(b)Thereciprocalsof and are and respectively
229Whentheproductoftwonumbersequals1whatiseachofthetwonumberscalledEachiscalledthereciprocaloftheother
Ex(a) Hence4isthereciprocalof and isthereciprocalof4
Ex(b) Hence isthereciprocalof and isthereciprocalofTogetthereciprocalofafractionweinvertthefraction
230HowcanweshowthattomultiplybythereciprocalofanumberisthesameastodividebythatnumberWehaveseenabovethat Weheremultiplyby toget1
Itisalsotruethat Herewedivideby toget1
But isthereciprocalof
Thereforemultiplyingby isthesameasdividingby
231Howmanytimesare(a) and containedin1(b) and containedin2
(a)
(b)
232IneachcasewhatcanwedowhenwewanttodivideawholenumberbyafractionorafractionbyawholenumberorafractionbyafractionMultiplybyitsreciprocal
EXAMPLEDivide by
Thismeansthat goesinto oneandfour-fifthstimes
233Specificallyhowdowedivideaproperfractionbyawholenumber
Divideitsnumeratorormultiplyitsdenominatorbythewholenumber
Ex(a)Divide by2
Multiplyingthedenominatorbythewholenumberisequivalenttomultiplyingbythereciprocalofthewholenumber
Ex(b)
234HowdowedivideawholenumberbyafractionDividethewholenumberbythenumeratorandmultiplybythedenominator
Ex(a)Divide24by
Ex(b)Divide17by or
Ineachcasethemethodisequivalenttomultiplyingbythereciprocalofthefraction
235HowdowedivideonemixednumberbyanotherChangethemixednumberstoimproperfractionsinvertthedivisorand
multiply(Invertingthedivisorgivesthereciprocalofthedivisor)EXAMPLEDivide by
236HowdowedivideamixednumberbyawholenumberChangemixednumbertoanimproperfractionanddividethenumeratoror
multiplydenominatorbythewholenumber
EXAMPLEDivide by3
Herealsothemethodisequivalenttomultiplyingbythereciprocalofthewholenumber
237WhatisanothermethodtousefortheabovecasewhenthedividendisalargenumberDivideasinwholenumbersandsimplifytheremainingcomplexfraction
EXAMPLEDivide by6
238WhataresomeothermethodsofdividingwholemixednumbersEx(a)Divide482by
Multiplyingbothnumeratoranddenominatorby5doesawaywiththemixednumberinthedivisorbutdoesnotchangethevalueofthefraction
Ex(b)Divide by
TochangetowholenumbersmultiplynumeratoranddenominatorbythecommonmultipleofthedenominatorsofthefractionsLCMhereis12
239WhatisthedifferencebetweenafractionapplicabletoanabstractnumberandoneapplicabletoaconcretenumberThefraction meansthatanabstractunitisdividedinto4equalpartsand3
partsareexpressed
Theexpressionldquo ofadozenrdquoisapplicableto12becausethatisthenumberofunitsinadozenandmaybeexpressedas9
Thefractionldquo ofagallonrdquomaybeexpressedas2quartsbecausethereare4quartsinagallon
240HowdowefindwhatpartthesecondoftwonumbersisofthefirstDividethesecondbythefirst
Ex(a)Whatpartof63is9
Ex(b)Whatpartof74is18
Ex(c)Whatpartof is
Ex(d)Whatpartof is7
241IfyouaregivenanumberthatisacertainfractionofawholehowwouldyoufindthewholeDividethegivennumberbythefraction
Ex(a)6is ofwhatnumber
Ex(b)72is ofwhatnumber
Ex(c)99is ofwhatnumber
Notethatineachcaseyoumultiplybythereciprocalofthefraction
Ex(d)If78is ofthelotwhatisthewholelot
Ex(e)Findthenumberofwhich40is
Ex(f) ofsomeradioequipmentisworth$350Whatisthevalueoftheentirestock
242HowdowetellwhichoneoftwofractionsisthegreaterReducethefractionstotheirlowesttermsbycancellation
GettheLCD(lowestcommondenominator)andchangeeachfractiontohavethisLCDComparenumerators
EXAMPLEWhichofthefollowingisgreater or
(LCD=72times19=1368)
Weseethat792isgreaterThus isgreaterthan middot
243Whatisachain(oracontinued)fractionOneinwhichthedenominatorhasafractionthedenominatorofwhichhasa
fractionthedenominatorofwhichhasafractionetc
EXAMPLE
244WhatchainfractionsareofinteresttousOnlythoseinwhichallnumeratorsare1orunitymdashtheso-calledintegerchain
fractions
245HowisaproperfractionconvertedintoachainfractionWeknowthatdividingbothnumeratoranddenominatorofafractionbythe
samequantitydoesnotchangethevalueofthefraction
DividebothnumeratoranddenominatorbythenumeratorThenumeratorbecomes1andthedenominatorbecomesawholenumberandafractionasaremainder
ConvertthefractionalremainderbydividingbothitstermsbythenumeratorAgainthenumeratorbecomes1andthedenominatorbecomesawholenumberandafractionasaremainder
Continuethisprocessuntilthefractionalremainderhas1asanumerator
EXAMPLEConvert toachainfraction
246HowcantheabovebesimplifiedEachtimedividethepreviousdivisorbytheremainderThequotientsbecome
thedenominatorsofthechainfractionwithunitsfornumeratorsThedenominators11182aretheintegralpartsofthequotients
247HowisachainfractionconvertedtoaproperfractionByinverseprocessstartfromtheendandgoupIntheabovestartwiththe
lastfractionaldenominator
Thenextfractionaldenominatoris
Next
Next
Finally
248OfwhatpracticalusearechainfractionsForonethingtheyenableustofindanotherfractionexpressedinsimpler
terms(smallernumbers)andofavaluenearorveryneartheonewithlargenumbers
EXAMPLEWhatfractionexpressedinsmallernumbersisnearinvalueto
Dividingbothtermsby31weget
expressedasachainfraction
Nowifwerejectthe thefraction willbelargerthan becausethedenominatorwasdecreased
Tocompare with gettheLCDofbothor
157times5=785=LCDThen and
Thus isseentobenearthevalueof
249Whatfractioninsmallertermsnearlyexpresses
Dividenumeratoranddenominatorby3937
isalittlelargerthan butitgivesusaprettygoodideaofitsvalue
250Howcanwegetacloserapproximation
whichissmallerthan003937
Togetstillnearertakethenextpartofthechainfraction
Startfromthebottom
Thisisthenearestfractionto003937unlesswereducetheentirechainfractionwhichwouldgiveus003937itself isonlylargerthan whichisquiteclose
Wethusseethatachainfractioncangiveusaseriesofsuccessiveapproximations
251WhatfeatureofachainfractionmakesitvaluabletousTheapproachtothetruevalueisextremelyrapidItgivesveryrapidly
convergingapproximations
EXAMPLEOfabovevaluesof
Weseethatthesecondapproximationbringsuswithin039percentofitstruevalueVeryrapidindeed
PROBLEMS
1Iftherearefourweeksinamonththreeweeksareequaltowhatpartofthreemonths
2Ifaunitisdividedintotenequalpartswhatisonepartcalled
3Readthefollowing Whatpartofthesefractionsshowsthenumberofpartsintowhichtheunitisdivided
4In whatshowshowmanypartsaretaken
5Whichareproperfractionsimproperfractionsandmixednumbersinthefollowing
(a) (b)
(c) (d)
(e) (f)
6Writeascommonfractionsormixednumbers(a)Twenty-ninetenths(b)Forty-nineelevenths(c)Eightfifteenths(d)Nineone-hundredths(e)Ninety-twoandthree-fourths(f)Onehundredandthirty-fivefifty-sixths(g)Eighty-sevenandninetenths(h)Sixhundredtenths(i)Twenty-threethirty-sevenths(j)Eighteenandsixtwenty-firsts(k)Thirty-oneandseventeennineteenths(l)Onehundredforty-fiveandonehundredthirty-threeonehundredthirty-fifths7
Changetowholeormixednumbers
(a) (b) (c) (d) (e)(f) (g) (h) (i) (j)(k) (l) (m) (n) (o)8Changetoimproperfractions
(a) (b) (c) (d) (e)(f) (g) (h) (i) (j)(k) (l) (m) (n) (o)(p)9
(a)Howmanyfourteenthsinoneunit(b)Howmanyfourteenthsintwounits(c)Howmanyfourteenthsinonehalfunit(d)Doeschanging toitslowerterm changeitsvalue
10Reducethefractionstolowestterms(a) (b) (c) (d) (e)(f) (g)
11Changetohigherterms
(a) to20ths(b) to64ths(c)to84ths(d) to96ths(e) to100ths(f) to24ths
12Findthemissingnumerators
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
13ReducetofractionshavinganLCD
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
14ChangetoimproperfractionsandreducetoLCD
(a) (b)
(c) (d)
15
(a)Increasethevalueof threetimes(b)Increasethevalueof twoandone-halftimes(c)Increasethevalueof fourandone-sixthtimes
16
(a)Decreasethevalueof to thevalue(b)Decreasethevalueof to thevalue(c)Decreasethevalueof to thevalue
17Changetoasimplefraction
(a) of (b) of (c) of(d) of (e) of (f) of
18Changetoasimplefraction
(a) (b) (c) (d) (e) (f)
19Add
(a) (b) (c)(d) (e) (f) (g)(h) (i) (j)(k) (l)
20Subtract
(a) from (b) from (c) from(d) from(e) from (f) from(g) from
21Multiply(a) by (b) by (c) by (d) by4
(e) by12(f)17by (g) by (d) by
(i) by (j) by (k) by (l) by
22Expressthereciprocalsof(a)491135(b)
23Howmanytimesare(a) containedin1(b) containedin2
24Divide(a) by2(b) by3(c)27by(d)19by(e) by(f) by4(g) by7(h)574by(i) by
25Whatpartof(a)72is9(b)86is16(c) is (d) is15(e) is (f) is72(g) is (h) is (i) is (j) is
26(a)8is ofwhatnumber(b)84is ofwhatnumber(c)144is ofwhatnumber
27(a)Findthenumberofwhich60is (b)Five-eighthsofashipmentisworth$430whatisthevalueoftheentireshipment
28Whichfractionhasagreatervalue or
29Express asachain(orcontinued)fraction
30Convert toachainfraction
31
32Whatfractioninsmallernumbersisnearinvalueto
33Whatfractioninsmallertermsnearlyexpressesπ=31416or (Usechain-fractionmethod)34Thewidthofadooropeningis ofitsheightWhatisthewidthwhentheheightis ft
35IfindthatIspent$88whichrepresents ofmytotalallowanceHowmuchdoIhaveleft
36Threecasesofmerchandiseweighing and IbwereshippedThecasesweighed and lbWhatisthetotalweightofthecasesgrossweightandthenetweightofthemerchandise
37Ifalbofbreadhad9sliceshowmanyouncesarethereperslice
38Howmanyreamsofpaperarelistedonthisinvoice andreams
39Ifinatestrunacartraveled26milesin30minuteshowmanymileswillittravelin hoursatthisrate
40Acrateofapplescontaining148appleswasboughtat anappleandsoldat ofthecostWhatwastheprofit
41Twopartnersboughtaparceloflandfor$3600eachpaying Theneach
sold ofhisinteresttoathirdpartyatcostWhatfractionalpartofthetotalinvestmentdoeseachpartynowownandhowmuchiseachworth
42Amanspends ofhissalaryforasuitofclothes foranovercoat forshoesand forahatWhatparthasheleft
43Iftheabovepersonhas$41lefthowmuchhadhetobeginwithandwhatdoeseachitemcost
44Thesidesofanirregularlyshapedyardhavethefollowingmeasurementsyd yd yd ydHowmanyyardsoffencingwillbeneededto
encloseit
45Ifthemineralmatteroftheorgansofthebodyisbones muscles lungs brain howmuchmoremineralmatteristhereinbonethanineachoftheotherorgansgiven
46Ifaboyof10yearsneedsdaily gramsofprotein gramoffatandgramsofcarbohydratesforeachpoundofweighthowmuchofeachwillaboyof10weighing69lbrequire
47Alotis feetwideby feetdeepHowmanyrods( fttoarod)ofwirewillbeneededtofencethelot
CHAPTERVII
DECIMALFRACTIONS
252WhatisdecimaldivisionDivisionofunitsintotenthshundredthsthousandthsetc
EXAMPLES
253WhatisadecimalfractionThepartofaunitobtainedbydecimaldivisionDecimalfractionsareoften
calleddecimalsItisafractionalvalueexpressedintenthshundredthsthousandthsetcThismeansthatthedenominatoris10orsomemultipleof10
254WhatdowecallthedecimalpointTheperiodplacedattheleftoftenthshundredthsetc
EXAMPLES
(threetenths) (sevenhundredths)(fivethousandths)
255Howmaydecimalfractionsbeexpressed(a)Bythepositionofthedecimalpoint
(b)Byadecimaldenominatorintheformofacommonfraction
Ex(a)0207008024017
Ex(b)
256Whatarethenamesofthedecimalplacesandhowaredecimalswritten
EXAMPLES
Toexpresstenthsoneplaceispointedoffas2
Toexpresshundredthstwoplacesarepointedoffas28
Toexpressthousandthsthreeplacesarepointedoffas287
Toexpresstenthousandthsfourplacesarepointedoffas2875
ReadaboveldquoFourandtwohundredeighty-seventhousandfivehundredeighty-threemillionthsrdquo
257HowisadecimalreadThedecimalpointisreadldquoandrdquoReadadecimalexactlyasifitwereawhole
numberandthenaddthefractionalnameofthelowestplace
EXAMPLE5631056923
ReadldquoFiveandsixhundredthirty-onemillionfifty-sixthousandninehundredtwenty-threebillionthsrdquoThelowestdecimalplacehereisbillionths
258WhatistherelationofthenumberoffiguresinadecimaltothenumberofzerosinitsdenominatorwhenexpressedasacommonfractionTheyarethesame
Ex(a)0345hasthreefigurestherefore hasthreezerosinthedenominator
Ex(b)001679hasfivefigurestherefore hasfivezerosinthedenominator
259IsthevalueofadecimalfractionchangedbyaddingoromittingzerosontherightNoEXAMPLE4=40=400Also
Addingzerostotherightdoesnotchangethevalue
260WhatistheeffectondecimalfractionsofmovingthedecimalpointtotheleftMovingthepointoneplacetotheleftdividesthedecimalby10twoplaces
dividesitby100threeplacesdividesitby1000etc
EXAMPLES
Thedecimalpointismovedtotheleftfordivisionby10rsquostomakethedecimalsmaller
261WhatistheeffectofmovingthedecimalpointtotherightMovingthepointoneplacetotherightmultipliesthedecimalby10two
placesby100threeplacesby1000etc
EXAMPLES
Thedecimalpointismovedtotherightformultiplicationby10rsquostomakethedecimallarger
262WhatmustbedonewhenthereisnotasufficientnumberoffiguresinthenumeratortoindicatethedenominatorofadecimalfractionZerosareplacedbetweenthedecimalpointandthefigureorfiguresinthe
numerator
Ex(a)Towriteninehundredthsasadecimalplaceazerobetweenthe9andthedecimalpointotherwisethefractionwouldbeninetenths
Placesufficientzerostotherightofthedecimalpointtomakeupasmanyfiguresinthenumeratorastherearezerosinthedenominatorwhenthefractionalvalueiswrittenasacommonfraction
Ex(b)Towrite notethatthedenominatorhasfivezerosthereforethenumeratormusthavefivefigurestotherightofthedecimalpointItalreadyhastwofiguressoaddthreezerostotherightofthedecimalpointor
263Howaredecimalsclassified(a)Asimpledecimalhasawholenumbertotherightofthedecimalpointas
048386356
(b)Acomplexdecimalhasawholenumberandacommonfractionwrittentotherightofthedecimalpointas
264DoweneedadecimalpointaftereverywholenumberNoThedecimalpointisunderstoodasattherightoftheunitsplace
EXAMPLE6=6=60=600
265HowdowedivideanynumberbyadecimalnumberShiftthedecimalpointoneplacetotheleftforeveryzerointhedivisor
EXAMPLES(a)132divide10=132OnezeroindivisorMove1placetoleft(b)132divide100=132TwozerosindivisorMove2placestoleft(c)132divide10=0132Move1placetoleft(d)132divide100=00132Move2placestoleft
266HowdowemultiplyanynumberbyadecimalnumberShiftthedecimalpointoneplacetotherightforeveryzerointhemultiplier
EXAMPLES
(a)132times10=1320Shift1placetoright(b)132times100=13200Shift2placestoright(c)132times1000=132000Shift3placestoright(d)132times10=132Shift1placetoright(e)132times100=132Shift2placestoright
(f)132times1000=132Shift3placestoright(g)132times10000=1320Shift4placestoright
267WhatisamixednumberindecimalformandhowdowemultiplyanddivideitbyadecimalAnumberthatconsistsofawholenumberandadecimalfractionas132465
Thesamerulesapplyasabove
EXAMPLES
(a)132465times10=132465Move1placetoright(b)132465times100=132465Move2placestoright(c)132465divide10=132465Move1placetoleft(d)132465divide100=132465Move2placestoleft
268HowcanwechangeacommonfractiontoadecimalAnnexzerostothenumeratoranddividebythedenominator
EXAMPLES
(a) or
(b) or
(c) or
(d) or
(e)
(f)
WhentheresultisacomplexdecimaltwoplacesareusuallyfarenoughtocarryoutthedecimalFormostpurposesthreeorfourplaceswillsuffice
269HowcanweextendacomplexdecimalAddzerostothenumeratorofthefractionanddividebythedenominator
Whenthedivisioncomesouteventhefractionistherebyremovedotherwisethedecimalmaybeextendedasmanyplacesasaredesired
Ex(a)Extendthecomplexdecimal
Addthreezerostothenumerator5anddividebydenominator8
Ans=9625Thedivisioncameouteven
Ex(b)Extend to6decimalplaces
Addfourzerostothe5anddivideby12
Ans=394166=sixdecimalplaces
270HowcanweconvertadecimalexpressiontoacommonfractionExpressthedecimalasanumeratoroveradenominatorandreducetolowest
termsThedenominatorisamultipleof10asindicatedbythedecimalpointThenumeratorisawholenumber
Ex(a)Change5toacommonfraction
Thedecimalpointindicates10asthedenominatorThus reducedtolowestterms
Ex(b)
Denominatoris1000Thus
reducedtolowestterms
Ex(c)Change5736toacommonfraction
TherearefourplacestotherightofthedecimalpointthereforetherearefourzerosinthedenominatorThus
271WhatistheprocedureforaddingwholenumbersandsimpledecimalsPlacethenumbersincolumnswiththedecimalpointsdirectlyunderone
anotherandaddintheusualwayThedecimalpointofthesumisdirectlyunderthepointsinthecolumn
EXAMPLEAdd2638745209537283and935
Addingzerosattherightofthedecimaldoesnotaffectthevalue
272WhatistheprocedureforaddingwholenumbersandcomplexdecimalsExtendthecomplexdecimalsthesamenumberofplacesandthenaddinthe
usualway
273WhatistheprocedureforsubtractingsimpledecimalsPlacethedecimalpointinthesubtrahenddirectlyunderthedecimalpointin
theminuendandsubtractasusualThedecimalpointoftheremainderisdirectlyunderthepointsaboveit
EXAMPLESubtract520953from7283
274WhatistheprocedureforsubtractingasimpledecimalandacomplexdecimalExtendtheshortercomplexdecimaluntilthefractionisremovedorthereare
thesamenumberofplacesintheminuendandsubtrahendandthensubtractintheusualway
EXAMPLEFrom subtract
275WhatistheprocedureformultiplyingsimpledecimalsMultiplyintheusualwayandpointoffintheproductasmanyplacesasthere
areplacesinboththemultiplierandmultiplicand
Ex(a)Multiply38by6
Ex(b)
Ex(c)
276WhatistheprocedureformultiplyingcomplexdecimalsExtendthedecimaltoremovethefractionwhenitcanbedoneorchangeto
improperfractions
EXAMPLE
277WhatistheprocedurefordividingonesimpledecimalbyanotherThetermsinadivisionare
(1)Thedivisormustbemadeawholenumberbymovingthedecimalpointtotheextremeright(ortheendofthenumber)Countthenumberofplacesyoumovedthepoint
(2)MovethedecimalpointinthedividendanequalnumberofplacesIfthedividendisawholenumberthenaddasmanyzerosinsteadandplacethepointattheend
(3)Placethedecimalpointinthequotientjustabovethepointinthedividend
Rememberthatadecimalpointisunderstoodaftereverywholenumber
Ex(a)Divide192by06
Sixone-hundredthsiscontainedin192thirty-twohundredtimes
ProofMultiply3200by06(2places)
3200times06=19200(2places)
Ex(b)Whatistheresultofdividing06118by14
Thedecimalpointinthequotientisalwaysdirectlyabovethedecimalpointinthedividend
Ex(c)Divide4030496by478
278WhatistheprocedurefordividingonecomplexdecimalbyanotherChangethecomplexdecimalstosimpledecimalsifpossibleandthendivide
otherwisemultiplybothnumbersbytheLCDofthedenominatorsofthefractionsbeforeyoudivide
EXAMPLEDivide by (LCD=6)
279HowisadecimalnumbershortenedforallpracticalpurposesIfarejectedordiscardeddecimalis5orover1isaddedtothenextfigureto
theleft
EXAMPLE44746143752canbeshortenedto44746144whichisconsideredtobecorrectto4decimalplaces(orfoursignificantfigures)Sincethefifthplacewhichis7isgreaterthan5then1isaddedtothenumbertotheleftofit3whichbecomes4
Nowin44746144thefourthplaceis4Thisislessthan5andisdroppedleaving4474614whichissaidtobecorrecttothreedecimalplaces
447461iscorrectto2decimalplaces44746iscorrectto1decimalplace
280WhatothermethodofdecimalapproximationhasbeeninternationallyapprovedThatofmakingthedecimaleven
Ex(a)48655isshortenedto4866
Thelast5isdroppedand1isaddedtothe5toitslefttomakethedecimaleven
Ex(b)48645isshortenedto4864
Since4isanevennumberyoumerelydropthe5Itisclaimedthatacloseraverageresultisobtainedwhenadecimalismadeeven
281WhatistheleastnumberofsignificantfiguresthatmustbekeptwhenthedecimalispurelyfractionalandcontainsanumberofzerostotherightofthedecimalpointAtleastonesignificantfiguremustbekept
EXAMPLE000072184maybeshortenedto00007
282Whatistheresultof03024times0196correctto2significantfigures
Onecantellatoncethat006iscorrectto2places(byadding1tothe5toget6because9issolarge)
283Whyisittheruletoworkaproblemtoonemoredecimalplacethanweneed
Ithelpsustodeterminewhetherthenextfigurewouldbegreaterorlessthan5andenablesustoknowwhetherornotthefigureweuseissufficientlyaccurate
284Whatcanwedotosimplifythingswhenwewanttogetananswercorrecttotwodecimalplacesinmultiplying4879by3765Thereisnoneedtogothroughthemultiplicationoftheentirenumbers
Ifweweretomultiply5times4(=20)wethusdropalldecimalsandweguessouranswertobesomewhatlessthan20Thisgivesusnodecimalplaces
Now49times38=1862Ifweretainonedecimalplaceinthemultiplierandmultiplicandwegetananswerwithtwodecimalplacesbutwearenotsureofthe62
Soourrulesaystoretainonemoreplacethanrequiredandweget488times377=183976or1840approximatelycorrectto2places
Thecompletemultiplicationwouldbe
4879times3765=18369435
Weseethatthislengthymultiplicationisnotjustified
285WhatisanotherwayofapproximatingthedesiredresultinvolvingdecimalsContractedmultiplicationSincethefigurestotheleftofthedecimalpointare
mostimportant
(1)Multiplyallofthemultiplicandbytheleft-handdigitofthemultiplier
(2)Droprightdigitofmultiplicandandmultiplyremainderbynextfigureofmultiplier
(3)Droptwodigitsofmultiplicandandmultiplyremainderbynextfigureofmultiplier
(4)Continuesuccessivelydroppingonemoredigitofmultiplicandeachtimeyoumultiplybyanotherfigureofthemultiplier
EXAMPLE
286WhatisarecurringdecimalWheninsomecasesofdecimaldivisionthecalculationcanbecarriedon
indefinitelywithrepeatingnumbersorsetsofnumberssuchadecimalisknownasarecurringdecimal
EXAMPLES
(a)(b)(c)
287Howarerecurringcirculatingorrepeatingdecimalsdenoted(a)ByadotovertherecurringfigureThus404means404444etcto
infinity
(b)BydotsplacedoverthefirstandlastfiguresoftherecurringgroupThus
288HowcanweconvertpurerecurringdecimalstofractionsUseninesinthedenominatormdashone9foreverydecimalplaceintherecurring
group
EXAMPLES(a)Recurringdecimal(b)Recurringdecimal (142857times7=999999)
Notethatapurerecurringdecimalisoneinwhichallthedigitsrecur
289HowcanweconvertmixedrecurringdecimalstofractionsInamixedrecurringdecimalthedecimalpointisfollowedbysomefigures
whichdonotrecur
(1)Subtractthenonrecurringfiguresfromallthefiguresandmaketheresultthenumerator
(2)Thedenominatorconsistsofasmanyninesastherearerecurringfiguresfollowedbyasmanyzerosasnonrecurringfigures
EXAMPLES(a)
(b)
(c)
(d)
(e)
290Whyinparticularshouldyouknowthedecimalequivalentsof and
ItisthensimpletofindotherfractionalequivalentsinthisseriesThus
291Howcanwesometimesproduceadecimalequivalentbymultiplyingbothnumeratoranddenominatorbyasuitablenumber
292HowdowefindthewholenumberwhenadecimalpartofitisgivenEx(a)56is8ofwhatnumber
Ex(b)If4ofanumberis64whatisthenumber
293HowisUnitedStatesmoneyrelatedtodecimalfractionsTheunitisthedollarexpressedbythesign$as$15=fifteendollarsDollars
maybedividedintotenthshundredthsandthousandths
294IfaBritishpound(pound)isworth$280andthereare20shillingstothepoundand12pencetotheshillinghowmuchis(a)1shillingworth(b)1pennyworthRememberIfyouwanttogetthevalueofoneunitofanyelementina
problemyoushoulddividebythatelement
(a)Youwanttofindthevalueof1shillingthendividebyshillings
Dividenumeratoranddenominatorby10orwhatisthesamethingmovethedecimalpoint1placetotheleftinnumeratoranddenominator
(b)
295AmanufacturersubmittedabidtotheUnitedStatesgovernmentformilitaryinsigniainthesumof$6839970at31cents millsperdozenHowmanydozenwouldbedelivered
PROBLEMS
1Writeindecimalform(a)Sixtenthsfourtenthstwoandonetenth(b)Sevenandninethousandthsnineandfifty-threethousandthsthreeten-thousandthselevenmillionths
(c)Onehundredfifty-fivethousandthsfourhundredninety-twothousandthssixten-thousandthsthreehundredandfourhundredths
(d)Sixandsevententhseightandtwotenthseighty-sixhundredthsfivehundredandsixthousandths
(e)Fourandthree-eighthshundredthsthirty-sixandfive-seventhsthousandthseightandtwo-thirdsofathousandtheightandfourandtwo-thirdsthousandths
2Writethefollowingfractionsasdecimalfractions
3Read12584062018
4Distinguishbetween0400and000004
5Whatisthedenominatorof45602763expressedinfractionform
6Expressascommonfractions025025002500
7Annexingaciphertoawholenumberincreasesitsvaluehowmanytimes
8Doesannexingaciphertoadecimalaffectitsvalue
9Selectthequantitiesthathavethesamevalueinthefollowing(a)0880088080080(b)04646004600046046004600(c)7387380738000073807380738
10Arrangethefollowinginascendingvalues
260260026260260
11Movethedecimalpointin4soastomakethedecimalsmallerby by
12Movethedecimaltomultiply004by10by100by1000
13Divide246by10by100
14Divide246by10by100
15Multiply246by10by100by1000
16Multiply246by10by100by1000by10000
17Multiply246576by10by100
18Divide246576by10by100
19Changethefollowingtodecimals(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)
20Extendthecomplexdecimals(a)(b)(c) to5places(d) to6places
21Changetocommonfractions(a)6(b)86(c)625(d)1875(e)0125(f)750(g)4765(h)
22Add(a)74866922536245and6286(b)486652366803643986and257(c)3749856309648394and824
23Add(a) and
(b) and(c) and(d) and
24Subtract(a)630842from8394(b)2884from49836(c)49486from23957(d)81564from128096(e)1489736from197134(f)3874from4
25(a)From subtract(b)From subtract(c)From subtract(d)From subtract(e)From subtract
26Multiply(a)49by7(b)054by8(c)845327by58986(d)1232by98736(e)184236by49
27Multiply(a) by(b) by(c) by(d) by(e)6836by
28Divide(a)283by07(b)07229by16(c)5040587by589(d)48735by6486
(e)64575by165(f)9686by136
29Divide(a) by(b) by(c) by(d) by(e) by(f)9957by
30Shorten57857254863tobecorrectto(a)4decimalplaces(b)3decimalplaces(c)2decimalplaces(d)1decimalplace
31Shorten(a)59767(b)59755
32Shorten0000083273totheleastnumberofsignificantfigures
33Findtheresultof04035times0287correctto2significantfigures
34Gettheresultof5987times4876correctto2decimalplacesbyshortenedmultiplication
35Gettheapproximateresultof5987x4876bycontractedmultiplication
36Convertthefollowingrecurringdecimalstofractions(a)(b)(c)(d)(e) (f)(g)(h)(i)
37Whatisthedecimalequivalentof(a) (b) (c) (d) (e) (f) (g) (h)
38(a)78is7ofwhatnumber(b)If6ofanumberis86whatisthenumber(c)81is9ofwhatnumber(d)99is75ofwhatnumber
39IftheBritishpound(pound)isworth$280andthereare20shillingstothepoundhowmucharethreeshillingsworthIfthereare12pencetoashillinghowmuchissixpenceworth
40Ifthetotalcostofashipmentis$7948865at millsperdozenitemswhatisthenumberofdozensintheshipment
41Ifafamilyfoundthatattheendoftheyearithadsaved$455andduringtheyearithadspent29ofitsincomeforfood17forrent25forclothingand21formiscellaneousitemswhatwastheamountofitsincome
42Inacollegetheregistrationwas33inpuresciencecourses26inliberalarts21insocialscienceandtheremainderinengineeringThenumberofstudentsinengineeringwas520WhatisthetotalregistrationofthecollegeHowmanystudentsineachcategory
43Amaninvests22ofhismoneyinbonds32incommonstocks36inrealestateandhehas$3400incashleftoverHowmuchishistotalequityHowmuchhasheineachcategory
44Specificationsforphosphorbronzerequire86copper065tin0007iron002lead0035phosphorusandtheremainderzincHowmanylbofeachelementarerequiredtomake1200lbofphosphorbronze
45Afarmersold8460poundsofapples(eachbushelweighing60lb)for$180abushelWiththeproceedshebought9000lboffertilizerWhatisthe
costofthefertilizerper100lb
46Thedistanceroundawheelis31416timesitsheightWhatisthedistanceroundawheel385feethighRounda32-inchhighwheel
47If100lbofmilkyield5563lbofbutterandagallonofmilkweighs87lbhowmuchbutterwill2gallonsofmilkyield
48Whatisthecostofarailroadticketat$045amileifthedistanceyouaretotravelis475miles
49If6370piecesofcutlerycost$75369tomanufacturewhatisthecostofeachincentsandmills
50Ifyoumade$260onaninvestmentof$4000whatfractionalpartoftheinvestmentdidyoumake
51If2lbofcoffeecost$165howmanylbcanyoubuyfor$2640
52Ifyouboughtsix$1000bondsfor andsoldthemfor (a)whatisthetotalamountpaidforthebonds(b)theamountreceivedforthem(c)theprofit(d)theprofitexpresseddecimallyinthousandths(Note meansoneach$100ofthebondor$96750foreachbond)
53TwoballteamsAandBeachhavingplayed46gameshavearespectivestandingof826and739IfAwinsonly4ofthenext10gamesandBwins6ofthenext10gameshowwilltheclubsstand
CHAPTERVIII
PERCENTAGE
296Whatismeantby(a)percent(b)percentage(a)Percentmeansldquobythehundredrdquothenumberofhundredthsofanumber
InLatinpercentummeansldquobythehundredrdquo
EXAMPLEIfoutof100students30failedinthefinalexaminationthen30percentfailedand70percentpassed
(b)PercentagemeansldquobyhundredthsrdquoandincludestheprocessofcomputingbyhundredthsIndealingwithpercentagewearethusworkingwithdecimalswhosedenominatoris100
EXAMPLE
297Whatisthesymbolusedtorepresentthedenominator100Thetermpercentisexpressedbythesign[]
EXAMPLES(a)(b)
(c)
(d)
(e)(f)(g)
Thepercentsign[]takestheplaceofthefractionlineandthedenominator100
298Inwhatwaysmayagivenpercentoragivennumberofhundredthsofanumberbeexpressed(a)Asawholenumber6(b)Asadecimal06(c)Asafraction
299Whendoweexpressquantitiesaspercentages
Whenwewishtocomparetwoquantitieswhicharenoteasilycommensurableitismoreconvenienttoexpressthemaspercentages
EXAMPLEItisobviousthat4ofaquantityisgreaterthan whileitisnotsoapparentthat268isagreaterproportionof6700than315of8400
300HowdowereduceanumberwrittenwithapercentsigntoadecimalDropthepercentsignandmovethedecimalpointtwoplacestotheleftThis
isequivalenttodividingby100whichisthemeaningofpercentDroppingthemeansdividingby100
EXAMPLES(a)35=35(move2placestolefttodivideby100)(b)135=135(move2placestolefttodivideby100)
301HowdoweconverttoadecimalwhenthepercentisexpressedasanumberandafractionCarryoutthefractioninordertoconvertittoadecimal
EXAMPLES
NoteYoumaycarryoutthefractiondirectlyandaddittothedigitnumbers
302HowcanweconvertawholenumberadecimalfractionafractionoramixednumbertoapercentIneachcasemultiplyby100toannexasign
EXAMPLES
NoteTomultiplyby100movedecimalpoint2placestotherightwheneverthatcanbedonedirectly
303Whatarethepercentequivalentsofverycommonfractions
304WhatpercentofthelargesquareistheshadedpartLargesquarecontains25smallsquares
Shadedpartcontains6smallsquares
Shadedpartis24oflargesquare
305WhatisthemostcommonmethodoffindingagivenpercentofanumberWritethepercentasadecimalandmultiply
Ex(a)Find6of$6700(6=06)Then
Ex(b)Find14of$9751(14=14)
306Whatisanothermethodoffindingagivenpercentofanumber
Find1ofthenumberfirstandthenmultiplybythegivenpercent
Ex(a)Find6of$6700
1of
(Move2placestolefttodivideby100)Then
6of$6700=6times$67=$402
Ex(b)Find4of$1860
1of$1860=$1860there44is4times$1860=$7440
Ex(c)Find of$7000
307WhatisthethirdmethodoffindingagivenpercentofanumberWritethegivenpercentasacommonfractionandmultiply
Thismethodisusefulwhenthegivenpercentistheequivalentofasimplecommonfraction
Ex(a)Find25of$51
Ex(b)Find of$8475
308Whattermsarecommonlyusedinpercentage(a)Thenumberofwhichsomanyhundredthsoracertainpercentistobe
takeniscalledthebase(=B)
(b)Thepercenttobetakenistherate(=R)
(c)Theresultoftheratetimesthebaseisthepercentage(=P)
P(percentage)=R(rate)timesB(base)orP=RtimesB
Ex(a)Findthepercentagewhentherateis4andthebaseis$1860
Ex(b)Find9of50
309WhatistheruleforfindingthepercentagewhenthebaseandratearegivenMultiplythebasebytherateexpressedeitherasadecimaloracommon
fraction
Ex(a)Intestingacertainore25ofitwasfoundtobeironHowmuchironwascontainedin552poundsofore
Ex(b)Suppose27wasironHowmuchironwastherein578poundsofore
27times578lb=27times578=15606lbiron(rate)(base)(percentage)
310WhatistheruleforfindingtheratewhenthepercentageandbasearegivenDividethepercentagebythebasetogettherateSince
Notethatrateisapercentandisafractionoradivision(=acomparisonbetweenpercentageandbase)
Ex(a)$114iswhatpercentof$3800
Dividethequantitybythatwithwhichitisbeingcompared
Ex(b)Aninvestorreceived$38250onaninvestmentof$8500Whatratepercentdidtheinvestmentpay
Youarecomparingthepercentagewiththebase
Ex(c)Amanearns$9000ayearHepays$1800ayearforrentWhatpercentofhissalaryishisrentComparethepercentageof$1800withthebase$9000
311WhatistheruleforfindingthebasewhentherateandthepercentagearegivenDividethepercentagebytherateexpressedeitherasacommonfractionoras
adecimalSince
NoteDividingbythepercentgivesyouthepercentagefor1percent(or1partinahundred)Thenmultiplyingby100givesyouthewholeamount
Ex(a)$435is20ofwhatamount
or
RememberIfyouwanttogetthevalueofoneunitofanyelementintheproblemyoushoulddividebythatelementDividebypercenttofindvalueof1percentTherefore
$2175times100=$2175=valueof100=base
Ex(b)$18720is16ofwhatamount
Ex(c)Whatistheamountofabillif2discountforcashcomesto$285
If$285is2then
or
312Whatismeantby(a)amount(b)differenceinpercentageproblems(a)Amount=base+percentage(b)Difference=basendashpercentage
313Howcanwefindthebasewhentherateandamountaregiven
Ex(a)Therentofanapartmentis$1848peryearandthisisanincreaseof10overthepreviousyearWhatwastherentthepreviousyear
Base=rentpreviousyearAmount=$1848rentthisyear10=rateincrease=10
or
100=base=Rentpreviousyear10=Advancethisyear110=$1848(=Rentthisyear)1=$1848divide110=$1680there4Base=100=100times$1680=$1680
Ex(b)AstorekeepersellsaTVsetfor$270andmakesaprofitof onthetransactionWhatdidtheTVsetcosthim
314Howdowefindthebasewhentherateanddifferencearegiven
Ex(a)Ifthewasteinminingandhandlingcoalamountsto4howmanytonswouldhavetobeminedtoload20carswith30tonseach
Base=tonstobeminedDifference=20times30=600tons
or
100=base=Tonstobemined4=Loss96=600tons1=600divide96=625tons100=625times100=625tonstobemined
Ex(b)Amansellshiscarfor$1500andloses25onthetransactionWhatdidhepayforit
315OnwhatdowealwaysbasethepercentofincreaseinsomequantityItisbasedontheoriginalquantityandnotontheincreasedquantity
Ex(a)Ifthepriceofanewspaperwasraisedfrom5centsto10centswhatwasthepercentofincreaseinpriceTheoriginalpriceis5centsTheincreaseis5cents
Thustherewasa100increaseinprice
Ex(b)Ifatthebeginningoftheyearyouhadabankbalanceof$4500andattheendoftheyearyouhad$5400bywhatpercenthadyourbalanceincreased
316OnwhatdowealwaysbasethepercentofdecreaseinsomequantityItisbasedontheoriginalquantityandnotonthedecreasedquantity
Ex(a)Anewautomobilewhichcost$2200wasworth$1800ayearlaterBywhatpercenthasitdecreasedinvalue
Ex(b)Ifabankdroppeditsinterestratefrom to whatwouldbethepercentofdecreaseintheinterestrate
317Howarepercentslessthan1percentorfractionalpartsof1percentwrittenandusedinbusinessandfinancialmatters
EXAMPLEIfthetaxonahouseisincreasedby whatistheamountofincreaseonahouseassessedat$15750
$15750times0025=$39375=$3938Ans
318HowistheexpressionofldquosomuchperhundredrdquocommonlyusedinbusinessItisusedineachofthefollowingexamples
Ex(a)Whatistheamountofthepremiumona$12000fireinsurancepolicyat55centahundreddollars
120times$55=$66Ans
Ex(b)Abrokerchargesyou$1250per100sharesHowmuchwillitcostyoutobuy500sharesofstock
5times$1250=$6250Ans
Ex(c)Abankruptfirmpaysyou43centonthedollarHowmuchdoyougetwhenyourclaimamountsto$46375
$46375times43=$19941Ans
319Howisthemillusedintaxmatters
EXAMPLEApropertyassessedat$12500istaxedat287millsperdollarHowmuchisthetax
320HowarepercentsaddedsubtractedmultipliedordividedTheyareconvertedtodecimalsfirstandcarriedoutinthesamemanneras
similaroperationsinvolvingdecimals
321IfanumberisincreasedbyacertainpercenttogetanamountwhatpercentmustbesubtractedfromthisamounttogettheoriginalnumberagainTogetbacktotheoriginalnumberadifferentpercentmustbesubtracted
fromtheamount
EXAMPLEIf6of85isaddedtoitweget
06times85+85=51+85=901=Amount
Nowwhatpercentof901mustbesubtractedfromittoget85again
Weseethat51isonly566of901whereas51is6of85theoriginalnumber
322IfBostonhasapopulationof2000000andPhiladelphiais50largerhowmuchsmallerisBostonthanPhiladelphia
(AlsoPhiladelphiais50largerthanBoston)
ThisagainemphasizestherulethatthepercentofincreaseordecreaseofsomequantityisalwaysbasedontheoriginalquantityForBostontheoriginalquantityis2000000andforPhiladelphiaitis3000000
323Ifamanspends30ofhisincomeforrentand10oftheremainderforclotheswhatishissalaryifthelandlordgets$1150morethantheclothier
30ofincome=Rent10ofremainder(100ndash30)=1times7=07=7=Clothes30ofincome=7ofincome+$1150or23ofincome=$1150
there4$1150divide23=$5000=Income
324Amansellshiscartohisfriendandtakesalossof20Hisfriendsellsthecarlatertoathirdpartyfor$1500losing25Howmuchdidtheoriginalownerpayforthecar$1500represents75ofhiscost
$2000represents80oforiginalownerrsquoscost
PROBLEMS
1Whatdoes27meanintermsofpercentage
2Whatpercentof4000is1800
31400iswhatof3600
4Reducetoadecimal
(a)5(b)(c)(d)6(e)75(f)(g)(h)115(i)(j)926(k)003(l)(m)225(n)6(o)250(p)73(q)03(r)(s)(t)(u)60(v)(w)(x)(y)
5Express asdecimalsofapercentandasdecimals
6Expressascommonfractionsinlowestterms(a)1212(b)2525(c)3636(d)7575(e) (f) (g)15015(h)375375(i) 14(j) 05
(k) (l)
7Changetoapercent(a)9(b)6(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)84(n)(o)65(p)(q)8(r)(s)07(t)0425(u)(v)
8Whatpercentofthecircleistheshadedpart
9Whatpercentofthelargesquareistheshadedpart
10Find(a)4of$4800(b)16of$8642(c)6of$8500(d)7of$1940(e) of$6000(f)25of$62(g) of$7625(h) of$1600(i) of1500(j)150of500(k) of7254(l) of6542
11Findtheresultbyfirstfinding1ofthegivennumberinthefollowing(a) of10000(b)4of1600(c) of4000(d) of10000(e) of6000(f)6of7000
12Amanowned960acresoflandHesold ofitHowmanyacresdidhesell
13Amanhad$24000incashHeinvested ofitinbondsand46instocksHowmuchdidheinvestineachandhowmuchmoneyhadheleft
14Intestingacertainore27ofitwasfoundtobeironHowmuchironwascontainedin645lbofore
15Thereare2760votersinacertaintownIf69ofthevotersgotothepollshowmanyvoteswillbecast
16Aninvestorreceived$46050onaninvestmentof$9200Whatratepercentdidtheinvestmentpay
17Amanearns$8000ayearHepays$1600ayearforrentWhatpercentofhissalaryishisrent
18$565is20ofwhatamount
19$23830is18ofwhatamount
20Whatistheamountofabillif2discountforcashcomesto$345
21Whatpercentof(a)138is56(b)495is65(c)9860is1260(d)125is05(e)03is0085(f) is (g)47830is6458(h)2736is5985(i)93is1546(j)66is24(k)107is765(l)1235is05486(m)289is1485
22Findthenumberofwhich(a)360is15(b)459is40(c)56is(d)420is125(e)52is(f)112is(g)306is(h)132is(i)89653is6
23Whatis4of ofanacreofland
24Ifamerchantrsquosscalesweigh14ozforapoundwhatpercentdoesthepurchaserlose
25Whatpercentis of6
266is5ofwhatnumber10ofwhatnumber
278is2ofwhatnumber25ofwhatnumber
28$250is ofwhat ofwhat
29532is105ofwhatnumber90ofwhatnumber
3080is125ofwhatnumber75ofwhatnumber
3195is05ofwhatnumber176ofwhatnumber
32Therentofanapartmentis$1656andthisisanincreaseof12overthepreviousyearWhatwastherentthepreviousyear
33Amansellsarefrigeratorfor$340andmakesaprofitof onthetransactionWhatdidtherefrigeratorcosthim
34Ifthewasteinminingandhandlingcoalamountsto howmanytonswouldhavetobeminedtoload40carswith30tonseach
35Amansellshishousefor$12000andloses12onthetransactionWhatdidthehousecosthim
36Ifthepriceofamagazinewasraisedfrom15centto25centwhatwasthepercentincreaseinprice
37Ifatthebeginningoftheyearyourbankbalancewas$3800andattheendoftheyear$4600bywhatpercenthadyourbalanceincreased
38Anewcarwhichcost$3100wasworth$2700ayearlaterBywhatpercenthaditdecreasedinvalue
39Ifabankdroppeditsinterestratefrom to3whatwouldbethepercentdecreaseintheinterestrate
40Expressinfractionsofapercentandindecimals(a) of1(b) of1(c) of1(d) of1(e) of1(f) of1
41Ifthetaxonahouseisincreasedby whatistheamountofincreaseonahouseworth$14700
42Whatisthepremiumonan$18000fireinsurancepolicyat64centperhundreddollars
43Ifyouarecharged$1250per100sharestobuystockshowmuchwillitcostyoutobuy1200sharesofstock
44Abankruptfirmpaysyou67centonthedollarHowmuchdoyoureceive
whenyourclaimamountsto$58545
45Apropertyassessedat$14500istaxedat243millsperdollarHowmuchisthetax
46If8isaddedto$96toget$10308whatpercentof$10308mustbesubtractedfromittogetbackto$96
47IfuniversityAhasanenrollmentof12000studentsanduniversityBis35largerhowmuchsmallerisuniversityAthanB
48Ifamanspends25ofhisincomeforfoodand12oftheremainderforeducationwhatishissalaryifthelandlordgets$960morethantheschool
49Amansellshishouseandtakesalossof15Thepurchaserlatersellsthehousetoathirdpartyfor$14000losing20Howmuchdidtheoriginalownerpayforthehouse
50Thepriceofeggsdroppedfrom63centadozento56centadozenWhatwasthepercentdecreaseinprice
51Anarticlethatcost$12wassoldfor$16WhatpercentofthecostwasthedifferencebetweenthesellingpriceandthecostWhatpercentofthesellingpricewasthedifferencebetweenthesellingpriceandthecost
52Acollegehadanenrollmentof2600in195022morethanin1940Atthesamerateofincreasehowmanystudentswereenrolledin1960Whatwastheenrollmentin1940
53Whatis(a)64increasedby ofitself(b)45increasedby ofitself(c)054increasedby24ofitself
54Whatnumberincreasedby(a)10ofitselfis462(b) ofitselfis299(c)8ofitselfis3024
55Whatnumberdecreasedby(a) ofitselfis266(b) ofitselfis450(c)7ofitselfis2139
CHAPTERIX
INTEREST
325WhatismeantbyinterestInterestistheamountpaidfortheuseofborrowedmoneyortheamount
receivedfortheuseofmoneyloanedorinvestedInbookkeepingthesegoundertheitemsofinterestcostandinterestearned
326Whatarethethreefactorstoconsiderincalculatinginterest(a)Principal=thesumloanedorthecapitalinvested
(b)Time=durationoftheperiodOneyearisthecustomaryunitoftimeForapartofayearthesubdivisionusedisthemonthortheday
(c)Rate=ratepercent=numberofunitspaiduponeachhundredunitsofborrowedsumTheunitsareexpressedinthemoneyofthecountryconcernedasdollarspoundssterlingfrancsmarkskronerflorinsorpesos
EXAMPLEIf$6arepaidasinterestforeveryhundreddollarsloanedattheendofeachyearthentherate=6per100or6percentor6
Thustherate=theratiooftheinteresttotheprincipalforeachunitoftime
327Howdoweexpressarateofinterest(a)Asanintegraloramixednumberwithapercentsignafterit
EXAMPLE
5=fivepercent=anintegralwithasign
=sixandthree-quarterspercent=amixednumberwithasign
(b)Asadecimalthecorrectwaytowriteit
EXAMPLE
005=fivepercent=
00675=sixandthree-quarterspercent=
328WhatismeantbysimpleinterestInterestcalculatedontheoriginalprincipalforthetimetheprincipalisused
SimpleinterestisnothingmorethanpercentagewithatimeelementinvolvedTheoriginalprincipalremainsconstantandthequantityofinterestforeachunittimeintervalremainsunchanged
EXAMPLE
6intereston$100for1year=$6=simpleinterest06of$100=$66of$100=$6
Thussimpleinterest=apercentagewithatimeelement
329WhatismeantbycompoundinterestItisinterestcalculateduponboththeprincipalandtheinterestwhichhas
alreadyaccruedTheinterestiscompoundedquarterlysemiannuallyorannuallyaccordingtoagreementYoumerelycomputesimpleinterestonthenewprincipalatthevariousperiodsagreedupon
EXAMPLEFindtheinterestfor3yearsat6on$200withinterestcompoundedannually
Forfirstyearinterest=6of$200=06times$200=$12Newprincipal=$200+$12=$212
Forsecondyearinterest=6of$212=06times$212=$1272Newprincipal=$212+$1272=$22472
Forthirdyearinterest=6of$22472=$1348Newprincipal=$22472+$1348=$23820
Originalprincipal=$20000Compoundinterestfor3years=$3820
Notethatthesimpleinterestforthe3yearswouldbe
$200times06times3=$3600
330Whatistheformulaforfiguringsimpleinterest
Interest=principaltimesratetimestimeI=Ptimesrtimest=Prt
EXAMPLEWhatistheintereston$2000at6peryearforahalfyear
331WhatismeantbytheldquoamountrdquoandwhatisitssymbolThesumobtainedbyaddingtheinteresttotheprincipal=amount=S
orS=Principal+Interest=P+IorS=P+PrtsinceI=PrtorS=P(I+rt)sincePisacommonfactorofPandPrt
EXAMPLEIfyouborrowed$500atsimpleinterestfor3yearsat5howmuchwillthecreditorreceiveinall
S=amount=P(1+rt)=$500(1+05times3)
=$500(115)=$575Ans
Creditorwillreceive$575ofwhich$500istheprincipaland$75istheinterest
332Infiguringsimpleinterestforlessthanayearwhatistheruleforestablishing(a)theterminaldays(b)theduedate(a)IntheUnitedStatesweexcludethefirstdayandincludethelastday
EXAMPLEForabankloanmadeJanuary4andfallingdueJanuary27interestwouldbechargedfor23days
(b)Dateofmaturityofaloanisdeterminedbythewordingoftheagreementiftimeisstatedinmonthspaymentisdueonthesamedateofduemonthiftimeisstatedindaysthentheexactnumberofdaysiscountedtogetduedate
EXAMPLEIfinatransactiononJuly5adebtoragreestorepayaloaninfivemonthsthemoneyisdueDecember5Iftheagreementistorun150daystheduedatewouldbeDecember2
NoteGenerallyintheUnitedStatesloansfallingdueonSaturdaySundayoraholidayarepayableonthenextbusinessdayandthisextratimeiscounted
333Howarethemethodsforfiguringsimpleinterestcommonlyreferredto(a)Theordinarymethod(b)Theexactoraccuratemethod(c)Thebankersrsquomethod
Thedifferenceinthesemethodsisinthewaythetimeisfigured
334HowdowefindthetimebytheordinarymethodIntheordinarymethodayearisconsideredtohave12monthsof30days
eachor360days
Thetimeisfoundeasilybycompoundsubtraction
EXAMPLEFindthetimebetweenFebruary81959andMay151957
Year Month Day
1959 2 8
1957 5 15
_____ ____ _____
1 8 23
Borrow1month=30daysandaddittothe8daystomake38days
Subtract15daysfrom38daystoget23days
Borrow1year=12monthsandaddittothe1monthtomake13months
Subtract5monthsfrom13monthstoget8months
Now1957from1958leaves1year
Theresultis1year8monthsand23days
335Howdowefindthetimebytheexactmethod(a)Theactualnumberofdaysineachmonthiscounted
EXAMPLEFindtheexacttimefromMay81958toJanuary121959
May 23days
June 30days
July 31days
August 31days
September 30days
October 31days
November 30days
December 31days
January 12days
249days
(b)UseTable1inAppendixBEachdayoftheyearisindicatedasthetotalnumberofdaysfromJanuary1tothedayinquestioninclusiveFindthenumberoppositethelastdateandfromthissubtractthenumberoppositethefirstdatetogetthenumberofdaysbetweenthedates
EXAMPLEUseabovedatesMay8isthe128thdayDecember31isthe365thdayThen365ndash128=237daysin1958Nowadd12daysinJanuary1959to237daystoget249daysinall
336HowdowefiguretimebythebankersrsquomethodTimeisexpressedinmonthsanddaysorinexactdaysonlyThismethodis
usedtofindthetimeforshortperiods
EXAMPLEWhatisthetimefromJune4toOctober21
FromJune4toOctober4is4monthsFromOctober4toOctober21is17daysAns=4months17days
Or(fromTable1inAppendixB)
October21=294June4=155Ans=294ndash155=139days
The360-dayyearisusedwithexactdays
337Findtheintereston$3000at6fromNovember181958toApril61959(a)bytheordinarymethod(b)bytheexactmethod(c)bythebankersrsquomethod(a)
Year Month Day
1959 4 6
1958 11 18
4 18 =138days
Ayear=12months30dayseachor360days=ordinarymethod
$3000times06times =$69interest=Ordinarymethod
(b)Table1AppendixBNovember18is322nddayoftheyear365ndash322=43daysin1958
April6isthe96thdayoftheyear
Then43+96=139days(exact)
there4$3000times06times =$6855interest=Exactmethod
(c)$3000times06times =$6950interest=Bankersrsquomethod(Exactdaysand360-dayyearareused)
NoteExactmethodproducestheleastinterestofthethreeandthebankersrsquomethodproducesthemost(becausethedenominatorissmallerwhilethenumberofdaysisexact)
338WhatistheconstantrelationshipofexactinteresttoordinaryorbankersrsquointerestbasedonexactnumberofdaysLetN=exactnumberofdays
Then
and
Then
and
Wecanrememberthisbynotingthatexactisalwayslessthanordinaryinterestso
Thereforetogetexactwesubtract ofordinaryfromordinaryTogetordinaryweadd ofexacttoexact
339Whatisthe60-day6percentmethodofcalculatinginterest60daysare ofayear
Theniftheinterestrateis6percentayeartheinterestratefor60daysis
Thereforetofindtheinterestfor60daysat6percentonanyprincipalpointofftwoplacestotheleft
Ex(a)Theintereston$1360for60daysat6is$1360
Now6daysare
Theninterestfor6daysat
Thereforetofindtheinterestfor6daysat6onanyprincipalpointoffthreeplacestotheleft
Ex(b)Theintereston$1360for6daysat6is$136
Ex(c)Findtheintereston$570for66daysat6
340Abusinessmanborrowed$850for75daysat6Howmuchinterestdidhepay
341Howarethealiquotpartsof60usedwhenthetimeisgreaterorlessthan60daysinfindinginterestbythe60-day6methodEXAMPLEWhatarethealiquotpartsof60dayscontainedin(a)49days
(b)58days(c)77days
(a) 30days (b) 30days (c) 60days
15days 20days 15days
4days 6days 2days
49days 2days 77days
58days
58days
342Whatistheintereston$95370for124daysat6
343Whatistheintereston$59860for48daysat6Togetinterestfor30daysfirstget$5986interestfor60daysanddivideby
2
344Howcanwesometimessimplifythe60-day6process(a)Byexchangingtheamountoftheprincipalandthenumberofdays
EXAMPLEFindtheintereston$120for176daysat6Makeit$176for120daysbyexchangingonefortheother
Ans=$352intereston$120for176days
(b)Bydeductingfromtheinterestfor60daystheinterestforthedifferenceintimebetweenthetimegivenand60days
EXAMPLEFindtheintereston$170for50daysat6
345Howdowefindtheinterestatarateotherthan6Firstfindtheinterestat6thentoget
(a)3take oftheinterestat6
(b)4subtract oftheinterestat6
(c) subtract oftheinterestat6
(d)5subtract oftheinterestat6
(e)7add oftheinterestat6
(f) add oftheinterestat6
(g)8add oftheinterestat6
(h)9add oftheinterestat6
EXAMPLEFindtheintereston$790for145daysat andat
346HowcanwemakeuseoftheinterestformulainfindingoneofthefourfactorsmdashinterestprincipalrateandtimemdashwhentheotherthreearegivenWehaveseenthatinterestismerelyapercentageproblemwithatimefactor
or
I(interest)=Prt(principaltimesratetimestime)
Ex(a)Ona360-day-per-yearbasisatwhatratepercentwouldyouhavetoinvest$970for72daystoearn$970interest
Ex(b)Howmuchmoneywouldyouneedtoinvestat5for96daystoearn$1160interest
Ex(c)Howlongwillittaketoearn$1530interestonaninvestmentof$1080at6
347Whatisthe6-day6methodoffindinginterestandwhatisitsprincipalvalueTheinterestfor6daysat6canbefoundbymovingthedecimalpointthree
placestotheleftbecause6is of60
(a)Movedecimalpoint3placestotheleftfor6-dayinterest
(b)Dividethenumberofdaysby6togetthenumberof6-dayunits
(c)Multiplytheresultsoftheabove
Thismethodcanbeusedtochecktheresultofthe60-daymethod
Ex(a)Findtheintereston$300for27daysat6
$30=interestfor6daysat6(move3placestoleft)
there4$30times =$135=interestfor27daysat6
Ex(b)Whatistheintereston$52936for78daysat6
$52936=interestfor6daysat6(move3placestoleft)
there4$52936times13=$688168=$688=interestfor78days
348WhatisthesignificanceofcompoundinterestInsimpleinteresttheprincipalremainsconstantduringthetermofaloan
Incompoundinteresttheprincipalisincreasedbytheadditionofinterestattheendofeachinterestperiodduringthetermofaloan
WhenevertheinterestisaddedtotheprincipalattheendofaperioditissaidtobeconvertedorcompoundedTheprincipalthenbecomeslargeratthebeginningofthesecondperiodthanitwasatthebeginningofthefirstperiodInturntheinterestdueattheendofthesecondperiodislargerthanthatdueattheendofthefirstperiodThisconditioncontinuesforeachsuccessiveperiodduringtheindebtedness
349Whatismeantby(a)compoundamount(b)compoundinterest(c)conversionperiod(d)frequencyofconversion(a)Compoundamount=principal+compoundinterest
(b)Compoundinterest=compoundamountmdashoriginalprincipal
(c)Conversionperiod=intervaloftimeattheendofwhichinterestiscompounded
(d)Frequencyofconversion=numberoftimesayearthattheinterestisconvertedintoprincipal
MostNewYorksavingsbanksconvertinterestfourtimesayearThustheconversionperiodis3monthsandthefrequencyis4
350Whatwill$450amounttointhreeyearsat4ifinterestiscompoundedannually
$45000=principalatbeginningoffirstyear$45000times04=$18=firstyearrsquosinterest
$45000+$18=$468=principalatbeginningofsecondyear$46800times04=$1872=secondyearrsquosinterest
$46800+$1872=$48672=principalatbeginningofthirdyear$48672times04=$1947=thirdyearrsquosinterest
$48672+$1947=$50619=principalorcompoundamountatendofthirdyear
351WhatisashortermethodoffiguringthecompoundamountTheamountatthebeginningofthesecondyearwasseentobeequaltothe
principalatthebeginningofthefirstyearplusoneyearrsquosinterestuponit(seeQuestion350)
$468=$450+$450times04
or
$468=$450(1+04)($450isacommonfactor)
and
$468=$450times104=amountatbeginningofsecondyear
Thustogettheamountforoneyearmultiplytheprincipalby(1+theannualinterestrate)
Theabovemultiplicationsareexpressedinonelineas
$50619=$450times104times104times104
or
$50619=$450times(104)3=amountatcompoundinterest
Thesmallfigure3attheupperright-handsideoftheparenthesisiscalledanexponentandmeansthatthequantityintheparenthesisistobeusedasafactorinmultiplicationthatnumberoftimesInthiscase3correspondstothenumberofyearsforwhichinterestwascomputedandmeansthat(104)istobemultiplied3timesSimilarly(1035)4meansaninterestrateof for4years
352Whatistheformulafortheamountatcompoundinterest
S=amount=$50619(inQuestion351)P=principal=$450(inQuestion351)
r=interestrateperyear=04(inQuestion351)t=numberofyears=3(inQuestion351)
Therefore
S=P(1+r)tS=$450(1+04)3=$450times(104)3=450times1124864
S=$50619
Thusthecompoundamountof$450in3yearswithinterestat4compoundedannuallyis
$50619Ans
353Inordertohave$6000attheendof3yearshowmuchmustyouinvestnowat5compoundedannually
Youmustinvest$518326nowtohave$6000attheendof3yearswhentheinterestiscompoundedannuallyat5
354WhatisusedinactualbusinessandfinancialpracticetosaveagreatdealoftimelaborandcomputationinfiguringcompoundinterestAtablewhichhasbeencomputedgivingtheamountof1(unity)atcompound
interestforvaryingperiodsoftimeandatdifferentratesofinterestThistableiscalledtheldquoCompoundAmountof1rdquo(seeTable2AppendixB)
S=(1+r)t=Formulaforthecompoundamountof1
wheninterestiscompoundedannuallyHereP=1(1+r)tisknownastheaccumulationfactorsincethecompoundamountindicatestheaccumulationofinterest
Accumulationfactortimesanyprincipal=compoundamounttowhichthatprincipalaccumulatesatcompoundinterestduringaspecifiedtime
Youfindinthetablethecompoundamountof1forthepropertime(numberofperiods)andrateandthenmultiplythisfigurebytheprincipalThesymbolforthetimeornumberofperiodsisusuallygivenasnThetablecanbeusedforanydenominationofcurrencysuchaspoundssterlingfrancsmarkslirapesosetcorforanyrequiredunit
Ex(a)Tofindwhat$1willamounttoinoneyearat5entercolumnheadnat1andrunhorizontallyuntilthecolumnheaded5isreachedwhereyouwillfind105
Ex(b)Tofindthecompoundamounton$1for4yearsat5entercolumnnat4andgohorizontallyuntilyoureachthecolumnheaded5whereyouwillfind$121551
Ex(c)Whichisgreater(1)asumofmoneyaccumulatingfor10yearsat2compoundinterestor(2)thesamesumaccumulatingfor5yearsat4compoundinterest
10yearsat2rarr$121899=compoundamountof15yearsat4rarr$121665=compoundamountof1there410yearsat2givesalargercompoundamount
355Whatwould$12000amounttoifinvestedfor7yearsat4compoundedannually
S=$12000times131593=$1579116Ans
(Compoundamountof$1forn=7yearsand4=131593fromtable)
356Whatamountofmoneyinvestedat5fornineyearswouldamountto$589505
(AccordingtoTable2AppendixBcompoundamountof$1for9yearsand5=$155133)
357Ifyoudeposited$1800inabankwhichpays4perannumhowlongwillittakeforthisdeposittogrowto$227758ifinterestiscompoundedannually
RefertoTable2andgodownundercolumnheaded4andyoufind126532isinahorizontallinerunningouttowhere
n=6=t=6yearsAns
Iftheresulthadbeenmoreorlessthan126532thenthetimewouldnothavebeenawholeyearandthetimewouldhavetobeinterpolatedbetweentwointegralyears
358WhatismeantbythenominalrateofinterestWheninterestiscompoundedorconvertedmorethanonceayearthestated
rateofinterestperyeariscalledthenominalrate
EXAMPLEIfasavingsbankpays ondepositscompoundedeveryquarteryearthenominalratewhichyoureceiveis Actuallyyougetalittlemorethan becauseeachbalanceisincreasedateach3-monthintervalbytheinterestaddedtoit
359WhatismeantbytheeffectiveannualrateofinterestRateofinterestactuallyearnedinayear
EXAMPLEHowmuchwill$700amounttoinoneyearifinterestiscompoundedquarterly
Thusarateof4compoundedquarterlyfor1yearwillproducethesameresultasarateof1compoundedannuallyfor4years
Weseethattheoriginal$100earned$406inoneyearThismeans
actuallyearnedduringtheyear
406isknownastheactualoreffectiveannualrate
Thusanominalrateof4compoundedquarterlyisequivalenttoaneffectiverateof406compoundedannuallybecausethesameamountofmoneyis
producedattheendofayear
360WhenarenominalandeffectiveratesequivalentWhentheyproducethesameamountofmoneyattheendofayear
Inabove
Dividebothsidesby$100togetthecompoundamountfor$1
Weseethattheeffectiverate0406isequivalenttothenominalrate04compounded4timesayear
361Whatistheformulashowingtherelationshipbetweenaneffectiverateiandanequivalentnominalraterpcompoundedptimesayear
Inabove
362Whatistheformulaforthecompoundamountof1ataraterp compoundedp timesperannumfort years
Theformulaforthecompoundof1wasshowninQuestion354tobeS=(1+r)twheninterestiscompoundedannually
Toobtainaformulaforthecompoundamountof1ataraterpcompoundedptimesperyear
ismerelysubstitutedfor(1+r)inabovebecauseiandrparetakenasequivalentratesThus
Theexponentpt=thetotalnumberofconversionperiodsduringtheindebtedness
EXAMPLEIf$800isleftondepositfor1yearatanominalrateofcompoundedsemiannuallywhatwillbetheamountattheendoftheyear
363Whatistheruleforuseofcompound-amount-of-1tableswhereinterestiscompoundedatanominalratemorethanonceayear(a)Findvalueofpt=totalnumberofconversionperiodsduringtimeof
indebtedness=nintables
(b)Findrpp=rateperperiod=percentinterestintables
(c)Lookinthecalculatedpercenttablesforthepercentforaquantityinlinehorizontallywiththencolumn(=pt)
EXAMPLEWhatistheamountof1at6compoundedquarterlyfor4years
Lookat gohorizontallyacrossfromn=16andget
126898555Ans
364Amaninvests$8000for12yearsat5compoundedquarterlyWhatamountwillhegetafter12years
Lookat interestforn=48horizontally
S=$8000times181535485=145228388
Thereforehewillreceive
$1452284Ans
PROBLEMS
1(a)Whatpercentof100is6(b)Whatpercentof$1is6cent
(c)If$6ischargedfortheuseof$100whatpercentofthesumloanedisthesumcharged
2Findtheintereston(a)$5for1yearat4at5at6(b)$300for2yearsat2at7at9(c)$400for3yearsat6for2years3monthsat7(d)$1200for1yearat3for3yearsat7for6monthsat8
3Ifyouborrowed$800atsimpleinterestfor4yearsat4howmuchwillthecreditorreceiveattheterminationofthecontractHowmuchwouldtheinterestamountto
4ForabankloanmadeonMarch6andfallingdueonMarch28interestwouldbechargedforhowmanydays
5(a)IfinatransactiononSeptember4adebtoragreestorepayinsixmonthswhenisthemoneydue
(b)Iftheagreementwastorun180dayswhenwouldtheduedatebe
6Findthetimebycompoundsubtractionbetween(a)June141958andAugust281958(b)September121957andJuly181958(c)December141955andMay121958(d)October181954andFebruary61959(e)July291955andMay141959
7FindtheexacttimebetweenthefollowingdatesusingTable1AppendixB(a)May101958andJanuary141959(b)October18andJanuary10(c)July16andNovember11(d)March5andNovember8(e)February161960andJuly71960(rememberthataleapyearhas366days)
8Findtheintereston$2500at5fromOctober171959toMay71960(a)bytheordinarymethod(b)bytheexactmethodand(c)bythebankerrsquosmethodWhichproducestheleastinterestwhichthemost
9Findtheexactintereston$1000fromJanuary12toApril18at3
10Findtheordinaryintereston$6200fromApril6toJuly12at3
11Obtaintheinterestat4on$12000forsixmonthsfromApril15
12Howmuchwill$5000beworth120daysafterApril211960ifinvestedat6ordinaryinterestandwhatistheduedate
13Findtheexactintereston$3800for135daysat
14HowwouldyoufindtheexactinterestgiventheordinaryinterestHowwouldyoufindtheordinaryinterestwhengiventheexactinterest
15Findtheexactinterestwhentheordinaryinterestis(a)$4783(b)$38640(c)$295(d)$1202(e)$29000(f)$375(g)$3479(h)$368(i)$4980
16Findtheordinaryinterestwhentheexactinterestis(a)$328(b)$5490(c)$65860(d)$8136(e)$622(f)$904(g)$22790(h)$446900(i)$6438
17Whatistheprincipalwhichat5for146dayswillyieldanexactinterest$120lessthantheordinaryinterest
18Findtheordinaryandexactintereston$6950fromMay10toAugust23at5
19Findtheinterestfor60daysat6on$1438
20Abusinessmanborrowed$840for75daysat6Howmuchinterestdidhepay
21Whatistheintereston$2470for6daysat6
22Findtheintereston$680for66daysat6
23Whatarethealiquotpartsof60inthefollowing(a)27days(b)75days(c)39days(d)96days(e)40days(f)87days(g)129days(h)105days(i)145days(j)21days(k)126days(l)99days
24Findtheintereston$95370for124daysat6
25Findtheintereston$59890for47daysat6
26Findtheintereston$140for191daysat6(byinterchangingthedaysandprincipal)
27Findtheintereston$180for50daysat6(bydeductingfromtheinterestfor60days)
28Byproperdivisionofdaysfindtheinterestbythe60-day6methodof(a)$697000for156days(b)$386for84days(c)$61775for48days(d)$5900for222days(e)$8749for23days
29FindtheinterestfromApril1toJuly9bythe60-day6methodon$5850
30Byproperdivisionofdaysfindtheinterestbytheappropriatemethodon
(a)$487for142daysat45(b)$653for180daysat(c)$9825for192daysat(d)$3760for164daysat8(e)$217975for105daysat5(f)$470for85daysat(g)$2130for120daysat4(h)$423for129daysat9(i)$3570for75daysat3
31Ona360-day-per-yearbasisatwhatratepercentwouldyouhavetoinvest$860for78daystoearn$840interest
32Howmuchmoneywouldyouneedtoinvestat4for82daystoearn$1290interest
33Howlongwillittaketoearn$1645interestonaninvestmentof$1160at6
34Whatprincipalwillproduce(a)$1870interestat6for72days(b)$835interestat6for126days(c)$14interestat6for96days(d)$1574interestat6for75days
35Inwhattimewill(a)$700produce$14at6(b)$960produce$2235at6(c)$1400produce$11at6(d)$2200produce$84at6
36Atwhatratewill(a)$1400produce$2830in126days(b)$760produce$1160in96days(c)$1680produce$21in75days(d)$3200produce$1820in36days
37Findtheinterestbythe6-day6methodon(a)$300for24days
(b))$150for27days(c)$63842for78days(d)$400for36days(e)$25for66days(f)$500for51days
38Whatwill$550amounttoin3yearsat4ifinterestiscompoundedannually
39Inordertohave$5000attheendof3yearshowmuchmustyouinvestnowat4compoundedannually
40Findthecompoundamounton$1for5yearsat4usingTable2AppendixB
41Whichisgreater(1)asumofmoneyaccumulatingfor8yearsat2compoundinterestor(2)thesamesumaccumulatingfor4yearsat4compoundinterest(usetable)
42Whatwould$10000amounttoifinvestedfor6yearsat compoundedannually
43Whatamountofmoneyinvestedat5for8yearswouldamountto$384140
44Ifyoudeposited$2100inabankwhichpays5perannumhowlongwillittakeforthisdeposittogrowto$2954ifinterestiscompoundedannually
45If$1000isleftondepositfor1yearatanominalrateof4compoundedsemiannuallywhatamountwilltherebebytheendoftheyear
46Whatistheamountof$1at6compoundedquarterlyfor6years(usetable)
47Ifamaninvests$10000for10yearsat5compoundedquarterlywhatamountwillhegetafter10years
48Findthecompoundintereston$2000for8yearsat5compounded(a)annually(b)semiannuallyand(c)quarterly
49Findtheamountof$5placedannuallyfor10yearsat5compoundinterest(usetable)
50Ifinterestat5iscompoundedsemiannuallyfor3yearsitamountstothesameasinterestat compoundedannuallyforhowmanyyears
51Atrustfundof$20000earnsinterestat3ayearcompoundedsemiannuallyWhatwillthefundamounttoin10yearsHowmuchwilltheinterestbeinthattime
CHAPTERX
RATIOmdashPROPORTIONmdashVARIATION
365Whatarethetwowaysofcomparinglikequantities(a)Subtractingthesmallerfromthelargermdashthedifferencemethod
EXAMPLEIfyouare35yearsoldandyoursonis5yearsoldyouare30yearsolderthanyourson(35minus5=30)
(b)Dividingonebytheothermdashtheratiomethod
EXAMPLEYouare7timesasoldasyourson( )
366WhatismeantbyaratioAcomparisonoftwolikequantitiesbydividingonebytheotherAsaratiois
arelationshipoftwoquantitieswemustbespecificandindicatetheorderoftheirrelationship
Ex(a)IfmachineAproduces300unitsperhourwhilemachineBproduces450unitsperhouritisincorrecttosaythattheproductionratioofthesemachinesis WemustsaytheproductionratioofmachineAtothatofmachineBis middot
Ex(b)InQuestion365youmustsaythattheratioofyouragetoyoursonrsquosageis 7andnotthattheratiooftheagesis Youmayalsosaythattheratioofyousonrsquosagetoyoursis
367WhattwotermsaregiveninallratiocalculationsThefirsttermgivenisthenumeratorandiscalledtheantecedentThesecond
termgivenisthedenominatorandiscalledtheconsequent
Ex(a)Whatistherelationbetween4and12
Here4isthefirstterm=antecedentand12isthesecondterm=consequent
Ex(b)Ifonehousecosts$54000andanothercosts$18000theratiobetweenthefirstandsecondhouseis
orratiois3to1Onecoststhreetimestheother
368WhatsymbolisusedtoindicateratioColon[]=ldquotordquo
EXAMPLES
$54000$18000=31412=13(to)(to)(to)(to)
Thecolonisactuallyanabbreviationfor[divide]withthehorizontallineomitted
369Howmayratiosbeexpressed(a)ByasinglewholenumberTheratioof35yearsto5yearsis7(35divide5=7)
(b)AsafractionalnumberTheratioof1ouncetoapoundis
(c)AsadecimalfractionTheratioofonesideofatriangle4incheslongtoasecondside5incheslongis or08
(d)Infractionalformandtreatedlikeafractionmaybereadastheratioof4to5
(e)Withtwodotsseparatingtheterms45meanstheratioof4to5
Notethatwhenaratioisexpressedbyasingleintegralfractionalordecimalnumberthenumber1isthesecondtermoftheratiobutisnotwrittendownTheratioof35to5istheratioof7to1orsimply7
370CantherebearatioofunlikethingsNoThetermsmustbeoflikethingsTherecanbenoratiobetweendollars
andbeansorbetweenhousesandyachtsUnlessthingscanbechangedtosomethingthatmakesthemaliketherecanbenoratioTherecanbearatiobetweenthecostofahouseandthecostofayachtasexpressedindollarsAlsothecomparisonmustnotonlybebetweenquantitiesofthesamekindbutbetweenquantitiesexpressedinthesameunitsWecannotcomparepoundsandinchesfortheyarenotquantitiesofthesamekindandwecannotcomparealengthinincheswithalengthinyardswithoutfirstmakingtheunitsalikethatiswemusteitherreduceyardstoinchesorconvertinchestoyards
371IsaratiodependentupontheunitsofmeasureNoTheratioitselfisalwaysabstractandthetermsmaybewrittenasabstract
numbers
EXAMPLEIftwoboardsare10feetand12feetlongrespectivelytheratioofthefirsttothesecondboardis56whetherweexpresstheirlengthsasinchesfeetoryardsTheunitscanceloutandtheratiois56
372DoesmultiplyingordividingbothtermsofaratiobythesamenumberchangeitsvalueNoEx(a)
Ex(b)
373WhatistherelationbetweenratioandpercentSincearatioisalwaysafractionwemaythinkofapercentasaratioRatios
arefrequentlyexpressedaspercents
EXAMPLEWhenwesay$100is20of$500wemeanthattheratioof
$100to$500is
Problemsinvolvingpercentcanhoweverbesolveddirectlywithoutreferringtoratio
374HowisaratiosimplifiedAratioisalwaysreducedtoitssimplestformPerformtheindicateddivision
andreducetheresultingfractiontoitslowesttermsExpressthefractionasaratio
Ex(a)Ratio simplified
Ex(b)Simplifytheratio
375WhatcanbedoneinordertocomparereadilytwoormoreratiosReducetheratiostosuchformsthatthefirsttermsoftheratiostobe
comparedshallbethesameusually1
Ex(a)Reduce927toaratiohaving1foritsfirstterm
Dividebothtermsby9getting13
Ex(b)Reduce1639toaratiohaving1foritsfirstterm
Dividebothtermsby16getting and
Ex(c)Reduce7849toaratiohaving1foritsfirstterm
Ex(d)Reduce toaratiohaving1foritsfirstterm
Dividebothtermsby
376WhatwouldyoudowhenrequiredtoworkoutacomplicatedratiocontainingfractionspercentsordecimalsSimplifytheratiofirst
(a)Ifthedenominatorsofbothfractionsarealiketheyareintheratiooftheirnumerators
EXAMPLEFindtheratiobetween and
(b)Ifthedenominatorsarenotalikemakethemalikeordividethefirstfractionbythesecondfraction
EXAMPLES(1)Findtheratiobetween and ( )
(2)Findtheratiobetween and
377Howdowedividesomegivennumberinagivenratio(a)Addthetermsoftheratioandmakeitthedenominatorwiththegiven
numberasthenumerator
(b)Multiplythequotientbyeachtermoftheratio
Ex(1)Given65Divide65intheratio23
As65=26+39therefore65isdividedintotwoterms26and39intheratioof23
Ex(2)Ashipmentof1200TVsetsistocontaincolorsetsintheratioof35Howmanyofeachkindarethere
there435=450colorsets750blackandwhitesetsAns
Ex(3)1600booksaretobeallottedtothreeclassesintheratioof479Howmanybookswilleachclassreceive
4+7+9=20=denominator
80times4=320=quotienttimesfirsttermofratio=bookstoclass180times7=560=quotienttimessecondtermofratio=bookstoclass280times9=720=quotienttimesthirdtermofratio=bookstoclass3Total=1600books
there4320560720=479Ans
378Howcanwedivide65intheratio
ReducefractionstoacommondenominatorFirstterm= and =secondtermAddthenumeratorsofthese3+2=5
Divide65by5anduse3and2asnumerators
Firstterm= and =secondterm
there4 Ans
379HowdowesolvearatioprobleminwhichtheratioisnotgivenFirstweassignaratiovalueof1tothegivenbasicquantityWethen
computetheratiovaluesofalltheotherquantitiesbasingourcalculationsonthegivenfactsthusarrivingataratio
ThenweproceedasinQuestion377aswhenratioisgiven
EXAMPLEAcompanybought3trucksThefirstcost timesasmuchasthesecondThethirdcost timesasmuchasthesecondThecompanypaid$30000forthe3trucksHowmuchdiditpayforeach
Addthetermsoftheratio (=denominator=5partsonepartofwhichisthebasictruck)
380Ifthewingspanofaplaneis76ft6inwhatwillthewingspanofamodelhavetobewhentheratioofthelengthofanypartofthemodeltothelengthofthecorrespondingpartoftheactualplaneis172Thelengthofmodelisthus ofthecorrespondinglengthoftheactualplane
or
381Ifabankruptfirmcanpay60centonthedollarandifitsassetsamountto$28000whatareitsliabilitiesPaying60centonthedollarmeansthatitsratioofassetstoliabilities=60
382WhatsellingpriceshouldbeplacedonaTVsetifthecostis$250andthedealeroperatesonamarginof30ofcostAmarginof30ofcost=ratioofmarginonsettoitscost
Thusmarginhere ofcost
Ormargin
there4Sellingprice=$250+$75=$325Ans
383Ifyouallow12ofyourincomeforclothingand21forrent(a)whatistheratioofthecostofrenttothecostofclothing(b)howmuchdoyouspendforrentpermonthwhenyourincomeis$8400peryear(a)
(b) forrent
384Ifatownestimatesthatithastoraise$300000intaxesandtheassessedvaluationofitsrealpropertyis$9000000whatisthetaxrateTaxrate=ratioofamounttoberaisedtoassessedvaluation