Maths Book

Part 1

By Abhishek Jain

Topics: 1. Number System 2. HCF and LCM

3. Ratio & proportion 4. Average

5. Percentage 6. Profit & loss

7. Time, Speed & Distance 8. Time & Work

NumberSystemUnderstandingNumbersAmeasurementcarriedout,ofanyquantity,leadstoameaningfulvaluecalledtheNumber.Thisvaluemaybepositiveornegativedependingonthedirectionofthemeasurementandcanberepresentedonthenumberline.

NaturalNumbers(N)Thenumbers1,2,3,4,5…areknownasnaturalnumbers.ThesetofnaturalnumbersisdenotedbyN.Hence,N={1,2,3,4…}.Thenaturalnumbersarefurtherdividedaseven,odd,primeetc.WholeNumbers(W)Allnaturalnumberstogetherwith‘0’arecollectivelycalledwholenumbers.ThesetofwholenumbersisdenotedbyW,andW={0,1,2,3,……}Integers(Z)Thesetincludingallwholenumbersandtheirnegativesiscalledasetofintegers.ItisdenotedbyZ,andZ={–∞,…–3,–2,–1,0,1,2,3,…….∞}.TheyarefurtherclassifiedintoNegativeintegers,Neutralintegersandpositiveintegers.Negativeintegers(Z-)AllintegerslesserthanZeroarecallednegativeintegers.Z-={–1,–2,–3…–∞}

Neutralintegers(Z0)Zeroistheonlyintegerwhichisneithernegativenorpositiveanditiscalledaneutralinteger.Positiveintegers(Z+)AllintegersgreaterthanZeroarecalledpositiveintegers.Z+={1,2,3,……..,∞}RealnumberRealnumbersarethosenumberswhichcanbeeasilyidentified&quantifiedorallnumberswhichcanberepresentedonnumberlinearecalledrealnumber.E.g.–2,–5,6,7.88,9.73,17etc.Realnumbersareoftwotypes: Rationalnumbers&Irrationalnumbers: Fractions(alsoknownasrationalnumbers)canbewrittenasterminating(ending)orrepeatingdecimals(suchas0.5,0.76,or0.333333....).Ontheotherhand,allthosenumbersthatcanbewrittenasnon-terminating,non-repeatingdecimalsarenon-rational,sotheyarecalledthe"irrationals".Exampleswouldbe 2("thesquarerootoftwo")orthenumberpi(π=3.14159...,fromgeometry).Typesofrationalnumbers:Therearetwotypesofrationalnumbers: a.Integers b.FractionsIntegersIntegersaredefinedas:allnegativenaturalnumbers(…..-4,-3,-2,-1and0&allpositivenaturalnumbers(1,2,3,4,5……..)Notethatintegersdonotincludedecimalsorfractions-justwholenumbers.Fractions:Alltherationalnumberswhichareinformof"

#,wherep,q≠ 0andpisnotamultipleofqare

calledfractions.Thenumberonthebottomoffractioniscalleddenominatorandnumberontopofafractioniscalledanumerator.Examples:1.3,1.5,5.6,$

%,&%etc.

Inthefraction,$%,2isthenumeratorand5isdenominator.

IrrationalNumbersAnynumberwhichcannotberepresentedintheform"

#wherepandqareintegersandq≠0

isanirrationalnumber.Onthebasisofnon-terminatingdecimals,irrationalnumbersarenon-terminatingnonrecurringdecimals.Suchas3.4324546345…….isanon-terminating,non-repeatingnumber.Examples:π,√5,√7,etc.Imaginarynumber:-Imaginarynumbersarethosenumberswhichcannotberepresentedonnumberline.E.g. −2, −3,etc.Squarerootsofallnegativenumbersareimaginarynumbers.ClassificationofNumbersi.EvenNumbers:Allnumbersdivisibleby2arecalledevennumbers.E.g.,2,4,6,8,10…Evennumberscanbeexpressedintheform2n,wherenisaninteger.Thus0,–2,−6,etc.arealsoevennumbers.ii.OddNumbers:Allnumbersnotdivisibleby2arecalledoddnumbers.e.g.1,3,5,7,9…Oddnumberscanbeexpressedintheform(2n+1)wherenisanyinteger.Thus–1,−3,−9etc.arealloddnumbers.Addition/Subtractioneven+even=even; even− even=even;even+odd=odd; even− odd=odd;odd+odd=even odd− odd=evenMultiplicationeven× even=even;even× odd=even;odd× odd=odd.MultiplicationofanynumberPositive positive=positivePositive negative=negativeNegative negative=positiveDivisionPositive/positive=positivePositive/negative=negativeNegative/negative=positiveiii.PrimeNumbers:Anaturalnumberthathasnootherfactorsbesidesitselfandunityisaprimenumber.

×

×

×

Examples:2,3,5,7,11,13,17,19…

ImportantObservationaboutprimenumbers:Aprimenumbergreaterthan3,whendividedby6leaveseither1or5astheremainder.Hence,aprimenumbercanbeexpressedintheformof6K±1.Buttheconverseofthisobservationisnottrue,thatanumberleavingaremainderof1or5whendividedby6isnotnecessarilyaprimenumber.Foreg:25,35etcExample1:-Ifa,a+2,a+4areconsecutiveprimenumbers.Thenhowmanysolutions‘a’canhave?Solution:-Noevenvalueof‘a’satisfiesthis.So‘a’shouldbeodd.Butoutofthreeconsecutiveoddnumbers,atleastonenumberisamultipleof3.So,onlypossibilityisa=3andthenumbersare3,5,7.VerifyingwhetherthenumberisprimenumberornotCanbedonebytrialdivision,thatistosaydividingnbyallprimenumbersSmallerthan√𝑛,therebycheckingwhethernisamultipleof𝑚 < 𝑛Example2:-Findwhether161isaprimenumberornot?Solution: 161islittlelessthan13,fromintegersfrom2to13,161isdivisibleby7,hence161isnotaprimenumber.iv.CompositeNumbers:Acompositenumberhasotherfactorsbesidesitselfandunity.e.g.8,72,39,etc.Onthebasisofthisfactthatanumberwithmorethantwofactorsisacompositewehaveonly34compositefrom1to50and40compositefrom51to100.v.PerfectNumbers:AnumberissaidtobeaperfectnumberifthesumofALLitsfactorsexcludingitself(butincluding1)isequaltothenumberitself.

OrThesumofallthepossiblefactorsofthenumberisequaltotwicethenumber.

vi.Co-Primenumbers:Twonumbersare(primeorcomposite)saidtobeco-primetooneanother,iftheydonothaveanycommonfactorotherthan1.e.g.35&12,sincetheybothdon’thaveacommonfactoramongthemotherthan1.Conversionofrecurringdecimalintofractions:

Thepqformofanyrecurring=

The whole number - non-recurring part

As many 9's as number of digits in the recurring part followed by as many0's as digits in the non-recurring part

Example:-Express2.52145145145….asafraction.

Solution:x=2.52145145145…..=2.52145=252145 252 251893

99900 99900−

=

FactorsAdivisorofanintegern,alsocalledafactorofn,isanintegerwhichevenlydividesnwithoutleavingaremainder.Ingeneral,itissaidmisafactorofnfornon-zerointegersmandnifthereexistsanintegerksuchthatn=k×m•1isdivisorofeveryinteger.•Everyintegerisadivisorofitself.•Apositivedivisorofnwhichisdifferentfromniscalledaproperdivisor.•Anintegern>1whoseonlyproperdivisoris1iscalledaprimenumber.Equivalently,onewouldsaythataprimenumberisonewhichhasexactlytwofactors:1anditself.•Anypositivedivisorofnisaproductofprimedivisorsofnraisedtosomepower.•Ifanumberequalsthesumofitsproperdivisors,itissaidtobeaperfectnumberFindingtheNumberofFactorsofanIntegerFirstmakeprimefactorizationofaninteger

p q rn a b c= × × ,wherea,bandcareprimefactorsofnandp,q,andraretheirpowers.Thenumberoffactorsofnwillbeexpressedbyformula(p+1)×(q+1)×(r+1)NOTE:Thiswillinclude1andnitself.Example:Findingthenumberofallfactorsof450?Solution:450=

1 2 22 3 5× × Totalnumberoffactorsof450including1and450itselfis.(1+1)× (2+1)× (2+1)=2×3×3=18FindingtheSumofalltheFactorsofanynaturalnumber.

Firstmakeprimefactorizationofanintegerp q rn a b c= × × ,wherea,bandcareprimefactors

ofnandp,q,andraretheirpowers.Thesumoffactorsofnwillbeexpressedbytheformula:

Example:Findingthesumofallfactorsof450Solution:450=

1 2 22 3 5× ×

Thesumofallfactorsof450is Findingtheproductofallthefactorsofanynaturalnumber.Firstweneedtofindthenumberoffactorsofthatnumber(N)

Let’ssaynumberoffactorsofN=Xthen,productofallthefactorsofN=(𝑁)45

Example:Findproductofallthefactorsof36Solution:36=22×32numberoffactorsare=(2+1)×(2+1)=9

so,productoffactors=3665or364× 36=364×6=answer

PerfectSquareAperfectsquare,isanintegerthatcanbewrittenasthesquareofsomeotherinteger.Forexample16isanperfectsquareof4.Therearesometipsabouttheperfectsquare:•ThenumberofdistinctfactorsofaperfectsquareisALWAYSODD.•ThesumofdistinctfactorsofaperfectsquareisALWAYSODD.•AperfectsquareALWAYShasanODDnumberofOdd-factors,andEVENnumberofEven-factors.•Perfectsquarealwayshasevennumberofpowersofprimefactors.DivisionDivisionistheoperationofdetermininghowmanytimesonequantityiscontainedinanother.Itistheinverseofmultiplication.Formulae:Dividend=(Divisor×Quotient)+RemainderIf15isdividedby6,thenquotientwillbe2andremainderwillbe3.Hence,wecanwriteitas15=6×2+3Here15isdividend,6isdivisor,2isaquotientand3isaremainderIfanumber,whendividedby3leavesremainder1,thenumbercanbewrittenas3x+1.Someimportantfacts1.Thesumoffivesuccessivewholenumbersisalwaysdivisibleby5.2.Thesquareofanoddnumberwhendividedby8leavesaremainderof1.3.Theproductof3consecutivenaturalnumbersisdivisibleby6.4.Theproductof3consecutivenaturalnumbers,thefirstofwhichisanevennumberisdivisibleby24.5.Ifanumberisnotamultipleof3,thenitssquaredividedby3leavesaremainderof1.6.Thelastdigitoftheproductofanynineconsecutivenumbersisalways0.7.Thesumofatwo-digitnumberandanumberformedbyreversingitsdigitsisdivisibleby11.

E.g.28+82=110,whichisdivisibleby118.Athree-digitnumberwhereallthedigitsaresameisdivisibleby37.9.10n-7isdivisibleby3foranynaturaln.

DivisibilityRules

DivisibleBy:

If Examples

2 Thenumberiseven.OrThelast(units)digitofthenumberis0,2,4,6or8

84isdivisibleby2.85isnotdivisibleby2.

3 Thesumofthedigitsofnumberisdivisibleby3.

1248isdivisibleby3.(1+2+4+8=15)346isnotdivisibleby3.(3+4+6=13)

4 Thelasttwodigitsofthenumberisdivisibleby4.Andthenumbershavingtwoormorezerosattheend.

23456isdivisibleby4.(56isdivisibleby4)13000isdivisibleby4.(Twoormorezerosattheend.)

5 Thenumbershaving0or5attheend. 12345isdivisibleby5.(5isthereattheend)1234isnotdivisibleby5.(0or5isnotthereattheend)

6 Thenumberisdivisiblebyboth2and3. 5358isdivisibleby6.(Itisdivisiblebyboth2and3)6782isnotdivisibleby6.(Itisdivisibleby2butnotdivisibleby3)

7 Therulewhichholdsgoodfornumberswithmorethan3digitsisasfollows:

(A)Groupthenumbersinthreefromunitsdigit

(B)Addtheoddgroupsandevengroupsseparately

(C)Thedifferenceoftheoddandevengroupsshouldbe0ordivisibleby

Let’stake85437954.Thegroupsare85,437,954Sumofoddgroups=954+85=1039Sumofevengroups=437Difference=602.Whichisdivisibleby7,hencethenumberisdivisibleby7.

8 Thenumberformedbylastthreedigitsisdivisibleby8.Andthenumbershavingthreeormorezerosattheend.

2056isdivisibleby8.(056isdivisibleby8)13000isdivisibleby8.(Threezerosatlast)

9 Thesumofallthedigitsisdivisibleby9. 5301isdivisibleby9.(5+3+0+1=9whichisdivisibleby9)

10 Thenumberendswithzero. 467590isdivisibleby10.(Itendswithzero)

11 Thedifferencebetweenthesumofdigitsatevenplacesandsumofdigitsatoddplacesis0ordivisibleby11.

10538isdivisibleby11.[(1+5+8)–(0+3)=11whichisdivisibleby11]724867isdivisibleby11.[(7+4+6)–(2+8+7)=0]

12 Thenumberisdivisiblebyboth3and4. 5472isdivisibleby12.(Thenumberisdivisiblebyboth3and4)5475isnotdivisibleby12.(Thenumberisdivisibleby3butnotdivisibleby4)

13 Method:Multiplylastdigitofthenumberby4andaddittotheremainingnumber.Continuethisprocessuntiltwodigitnumberisachieved.Ifthistwodigitnumberisdivisibleby13thenthenumberisdivisibleby13.

182isdivisibleby13.(Multiplylastdigitby4i.e2×4=8.Addittotheremainingnumberi.e18+8=26.26isdivisibleby13so182isdivisibleby13)2145isdivisibleby13(Multiplylastdigitby4i.e5×4=20.Addittotheremainingnumberi.e214+20=234whichisnotatwodigitnumbersorepeattheprocess.Multiplylastdigitof234by4i.e.4×4=16.Addittotheremainingnumberi.e23+16=39whichisdivisibleby13so2145isdivisibleby13)

14 Thenumberisdivisiblebyboth2and7. 7966isdivisibleby14.(Thenumberisdivisiblebyboth2and7)

15 Thenumberisdivisiblebyboth3and5. 3525isdivisibleby15.(Thenumberisdivisiblebyboth3and5)

16 Thenumberformedbylastfourdigitsisdivisibleby16.

41104isdivisibleby16.(Thenumberformedbylastfourdigitsisdivisibleby16)

17 Method:Multiplylastdigitwith5andsubtractitfromtheremainingnumber.Iftheresultisdivisibleby17thentheoriginalnumberisalsodivisibleby17.Repeatthisprocessifrequired.

4029isdivisibleby17(Multiplylastdigitby5i.e.9×5=45.Subtractitfromtheremainingnumber.402–45=357whichisdivisibleby17so4029isdivisibleby17)

18 Thenumberisevenanddivisibleby9. 4428isdivisibleby18.(Itisevenanddivisibleby9)

19 Method:Multiplylastdigitwith2andaddittotheremainingnumber.Iftheresultisdivisibleby19thenoriginalnumberisalsodivisibleby19.Repeatthisprocessifrequired.

1235isdivisibleby19.(Multiplylastdigitwith2i.e5×2=10.Addittotheremainingnumber123i.e123+10=133whichisdivisibleby19so1235isalsodivisibleby19)

Divisibilityruleforanyprimenumbergreaterthan5Ifanysingledigitwritten‘n’timesthenitwillbedivisiblebyprimenumber(P)greaterthan5onlyif‘n’isdivisibleby(P–1)Example:-Whatwillbetheremainderwhen4444……..44timesisdividedby7?Solution:If4isdividedby7,theremainderis4.If44isdividedby7,theremainderis2.If444isdividedby7,theremainderis3Bycheckinglikethis,wecometoknowthat444444isexactlydivisibleby7.Soifwetake6fours,itisexactlydivisibleby7.Similarly12foursisalsoexactlydivisibleby7and42fourswillbeexactlydivisibleby7.Sooutof44,theremainingtwofourswillgivearemainderof2.Nowifweseeallboldnumbersaremultipleof6and6isnothingbut1lessthan7(prime)FUNDA:Howtocalculateremainder,whenanumberisdividedby11,withoutdivision?Step1:Addalltheoddplacenumbers(O)andevenplacenumbers(E)countedfromrighttoleft.Step2:IfO–Eispositive,remainderwillbethedifferencelessthan11.Step3:IfO–Eisnegative,remaindershouldbe(11–difference).Ex.1whatistheremainderwhen2354789341isdividedby11?Sol.Oddplacedigitsum(O)=1+3+8+4+3=19.

Evenplacedigitssum(E)=4+9+7+5+2=27.Difference(D)=19–27=–8Remainder=11–8=3.Ex.2If567P55Qisdivisibleby88;FindthevalueofP+Q.Sol.Thenumberisdivisibleby8means;thenumberformedbythelast3digitsshouldbedivisibleby8whichare55Q.OnlyQ=2satisfythis.Fromthedivisibilityruleof11,(2+5+7+5)–(5+P+6)isdivisibleby11.So8-Pisdivisibleby11.ifP=8,thenonlyitispossible.SoP=8andQ=2.SoP+Q=10.Ex.3Ifthefirst100naturalnumbersarewrittensidebysidetoformabignumberanditisdividedby8.Whatwillbetheremainder?Sol.Thenumberis1234…..9899100Accordingtothedivisibilityruleof8,wewillcheckonlythelast3digits.If100isdividedby8,theremainderis4.PointstorememberThenumber1isneitherprimenorcomposite.1)2istheonlyevennumberwhichisprime.2)(xn+yn)isdivisibleby(x+y),whennisanoddnumber.3)(xn–yn)isdivisibleby(x+y),whennisanevennumber.4)(xn–yn)isdivisibleby(x–y),whennisanoddoranevennumber.5)Thedifferencebetween2numbers(xy)–(yx)willalwaysbedivisibleby9.6)Thesquareofanoddnumberwhendividedby8willalwaysgive1asaremainder.7)Everysquarenumberisamultipleof3orexceedsamultipleof3byunity.8)Everysquarenumberisamultipleof4orexceedsamultipleof4byunity.9)Ifasquarenumberendsin9,theprecedingdigitiseven.10)Ifmandnaretwointegers,then(m+n)!isdivisiblebym!n!

11)(a)n/(a+1)leavesaremainderof1𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛𝑎𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑

12)Productofnconsecutivenumbersisalwaysdivisiblebyn!.FactorialsFactorialofapositiveintegerndenotedbyn!,istheproductofallpositiveintegerslessthanorequalton.Forinstance .TrailingzerosTrailingzerosareasequenceof0'sinthedecimalrepresentation(ormoregenerally,inanypositionalrepresentation)ofanumber,afterwhichnootherdigitsfollow.125000has3trailingzeros;Thenumberoftrailingzerosinthedecimalrepresentationofn!,thefactorialofanonNegativeinteger ,canbedeterminedwiththisformula:

,wherekmustbechosensuchthat .It'seasierifyoulookatanexample.Example:Howmanyzerosareintheend(afterwhichnootherdigitsfollow)of32!?Solution:

(Denominatormustbelessthan32,5$ = 25isless)Hence,thereare7zerosintheendof32!Theformulaactuallycountsthenumberoffactors5inn!,butsincethereareatleastasmanyfactors2,thisisequivalenttothenumberoffactors10,eachofwhichgivesonemoretrailingzero.Findingthenumberofpowersofaprimenumber ,inthe .Theformulais:

...till Example:Whatisthepowerof2in25!?Solution:

UnitdigitUnitissameasfindingremainderby10

Determiningthelastdigitof :

1.Lastdigitof isthesameasthatof ;2.Determinethecyclicitynumber of ;3.Findtheremainder when dividedbythecyclicity;

4.When ,thenlastdigitof isthesameasthatof andwhen ,thenlast

digitof isthesameasthatof ,where isthecyclicityofnumber.5.Integerendingwith0,1,5or6,intheintegerpowerk>0,hasthesamelastdigitasthebase.6.Integersendingwith2,3,7and8haveacyclicityof4.

7.Integersendingwith4(e.g. )haveacyclicityof2.Whennisodd willend

with4andwhenniseven willendwith6.

8.Integersendingwith9(e.g. )haveacyclicityof2.Whennisodd willend

with9andwhenniseven willendwith1.Example:Whatisthelastdigitof ?Solution:Lastdigitof isthesameasthatof .Nowweshoulddeterminethecyclicity

of :1.

17 =7(lastdigitis7) 2.17 =9(lastdigitis9)

3.37 =3(lastdigitis3) 4.

47 =1(lastdigitis1)5.

57 =7(lastdigitis7again!)andsoon…..So,thecyclicityof7is4.Nowdivide39(power)by4(cyclicity),remainderis3.So,thelastdigitof isthesameasthatofthelastdigitof ,isthesameasthatofthelastdigitof ,whichis .Indices:1. am×an=am+n

2. (am)n=amn3. am/an=am–n4. (ab)n=anbn5. a0=16. ( )1/mm a= a

7. Ifam=anthenm=n8. Ifam=bthena=b1/m Basicformulaeforoperationsonnumbers1. (a+b)2=a2+b2+2ab2. (a–b)2=a2+b2–2ab3. (a+b)3=a3+b3+3ab(a+b)4. (a–b)3=a3–b3–3ab(a–b)5. (a2–b2)=(a+b)(a–b)6. (a3–b3)=(a–b)(a2+ab+b2)7. (a3+b3)=(a+b)(a2–ab+b2)8. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)9. a3+b3+c3–3abc=(a+b+c)(a2+b2+c2–ab–bc–ca)10. Ifa+b+c=0,thena3+b3+c3=3abc11. (a+b)2+(a–b)2=2(a2+b2)12. (a+b)2–(a–b)2=4ab13. (a4–b4)=(a–b)(a+b)(a2+b2)SomemorepropertiesonnaturalnumbersSumof1stnNaturalnumberi.e1+2+3+….+n={n(n+1)}/2Sumofsquaresof1stnnaturalnumberi.e12+22+32+…..+n2={n(n+l)(2n+l)}/6

Sumofcubesof1stnNaturalnumberi.e13+23+33+…..+n3=[{n(n+I)}/2]2Sumofsquaresof1stnOddnaturalnumberi.e12+32+52+…..+(2n–1)2={n(2n-1)(2n+I)}/3Sumofsquaresof1stnEvennaturalnumberi.e22+42+62+…..+2n2={2n(n+1)(2n+1)}/3Sumofcubesof1stnOddnaturalnumberi.e13+33+53+…..+(2n-1)3=n2(2n2-1)Sumofcubesof1stnEvennaturalnumberi.e23+43+63+…..+2n3=2[n(n+1)]2Howmanytimesadigitappearsinacertainrange(except0)?Let’sconsideranydigit(Say2)andcheckhowmanytimesitappearsindifferentranges.From1–9,thedigit2isused1timeFrom1–99,thedigit2isused20times.From1–999,thedigit2isused300timesFrom1–9999,thedigit2isused4000timesandsoonSomeOtherPropertiesofNumbersSquares:Thereisadefiniterelationshipbetweentheunitdigitswhensquareofanumberisconsidered,wewillseethatonebyone.1.IfunitdigitofaperfectsquareisIthenten’sdigithastobeEven(e.g.81,121,441etc)2.Ifunitdigitofanumberis2thenitcan’tbeaperfectsquare.3.Ifunitdigitofanumberis3thenitcan’tbeaperfectsquare.4.Ifunitdigitofaperfectsquareis4thenten’sdigithastobeEven(e.g.64,144etc)5.Ifunitdigitofaperfectsquareis5thenten’sdigithastobe2(e.g.25,225,625etc)6.Ifunitdigitofaperfectsquareis6thenten’sdigithastobeOddandmultipleof3.7.Ifunitdigitofanumberis7&8thenitcan’tbeaperfectsquare8.Ifunitdigitofaperfectsquareis9thenten’sdigithastobeEven(e.g.49,169etc)9.Ifunitdigitofaperfectsquareis0thenten’sdigithastobe0(e.g.100,400,900etc)Henceifanumberendwith2,3,7or8thenitcan’tbeperfectsquare.

SolvedExamplesQ1.Whatisthedigitintheunitplaceof3946252+734733?Solution:-3946252+734733Firstwewillfindtheunitdigitof3946252.Weknow,theevenpowerof9resultsin‘1’asitsunitdigit.So,thelastdigitof3946252willbe1.Lastdigitof734733.Bydividing33by4,weget1asaremainder.So,thelastdigitof734733willbesameasof31or3.Lastdigitof3946252+734733=1+3=4Q2.Aisthesmallestintegerwhichwhenmultipliedwith3givesanumbermadeof5’sonly.SumofthedigitsofAisB.SumofthedigitsofBisC.WhatisthevalueofC3?Solution:-Smallestnumbermadeof5’sanddivisibleby3is555.ThusB=5+5+5=15andC=1+5=6So,

3C =216Q3.3762–2362isdivisibleby:Solution:-3762–2362xn–ynisalwaysdivisibleby(x–y)xn–ynisdivisibleby(x+y)whenniseven.So,3762–2362willbedivisiblebyboth(37–23)and(37+23)So,3762–2362willbedivisibleby(37–23)× (37+23)=14× 60=840Q4.LetN=103+104+105+106+107+108+109.ThesumofthedigitsofNis:Solution:-SinceNiswrittenoutasasumofpowersof10,thenNcanbewrittenas1111111000,andsothesumofthedigitsis7.Q5.LetDbeadecimaloftheformD=0.abcdabcdabcd…………,wheredigitsa,b,canddareintegerslyingbetween0and9.Atmostthreeofthesedigitsarezero.BywhatnumberDbemultipliedsothattheresultisanaturalnumber?

Solution:-D=abcd9999 ,

Hence,oncewemultiplyDwith9999wewillgetanaturalnumber.Q6.N=(11111111)2.WhatisthesumofthedigitsofN?Solution:-12=1,112=121,1112=12321so,111111112=123456787654321.Thesumofthesenumbersis64.Q7.Whatistheremainderwhen192319241925isdividedby1924?Solution:-192319241925=(1924–1)someoddpower=xorRemainder=(–1)x=–1orRemainder=1924–1=1923

Q8.N=2×4× 6× 8× 10×…….×100.HowmanyzeroesarethereattheendofN?Solution:-N=250×50!.Thehighestpowerof5in50!is12,thereforetherewillbe12zeroesattheendofN.Q9.Whatistheunit’sdigitof345?Solution:-Weseethatlastdigitsofthepowers31=3,32=9,33=27,34=81,andthenthedigitsstartrepeating.Sothelastdigitof35willbe3,36willbe9,37willbe7andsoon.Thisiscalledcyclicity.Inthiscasethecyclicityis4,asthenumbersstartrepeatingafterthe4thpower.Sowecanfindthelastdigitsofanynumbersraisedtoahighpower.Sincewehavetofind345,wedivide45by4andwegettheremainderas1.Hencethelastdigitof345willbesameas31,or3.Thenumbers2,3,7and8havecyclicity4Q10.Whatisthelastdigitof723x813?Solution:-Divide23by4,wegetremainder3.Hencelastdigitof723willbesameas73or3.Divide13by4,wegetremainderas1;hencelastdigitof813issameas81or8.Sothelastdigitoftheproductis3×8=24,or4Q11.Amanhasacertainnumberofmangoes.HesellshalfofwhathehadandonemoretoA,halfoftheremainderandonemoretoB,halfoftheremainderandonemoretoC,halfoftheremainderandonemoretoD.Thenhehasonly1left.Howmanydidhehaveatthebeginning?Solution:-Insuchsums,itisadvisabletostartfromtheback(trynottousexandformanequation,becauseyouhavefourstepstosolve).BeforeD,hehad(1+1)x2=4.BeforeChehad(4+1)x2=10.BeforeBhehad(10+1)x2=22andbeforeAhehad(22+1)x2=46.Q12.WhatistheminimumnumberbywhichIshoulddivide1250sothatIgetaperfectsquare?Solution:1250=625×2=25×25×2=252×2Thereforetogetperfectsquareweshoulddivideby2.Hence,togetaperfectsquareyoucanalsomultiplyby2.Q13.Whatistherightmostnon-zerodigitofthenumber302720?Solution:Asthegivennumberisamultipleof10,thelast2720digitswillbe0s.Therightmostnon-zerodigitofthenumberwillbetheunitdigitofthenumber32720.Theunitdigitofthepowersof“3”followsacyclicityof"4"In32720as2720isamultipleof"4",thenumberwillhavethesameunitsdigitas34whichis"1".Hence,therightmostnon-zerodigitofthenumberis"1".Q14.When13911+3isdividedby12,whatwouldbetheremainder?Solution:When13isdividedby12theremainderis1→anypowerof13willalwaysgive1astheremainder.Butweadd3,sotheremainderwillbe4Q15.232+233+234+235isdivisiblebySolution:Take232common→232(1+2+4+8)→232(15).Thusthenumberisdivisibleby15.

Q16.Whatistheleastnumberthatmustbesubtractedfrom2000togetanumberwhichisexactlydivisibleby17?Solution:Ondividing2000by17,weget11asremainder.So,requirednumberis11.Q17.Whatistheleastnumberthatmustbeaddedto3000toobtainanumberexactlydivisibleby19?Solution:Ondividing3000by19,weget17asremainderSo,numbertobeadded=19–17=2Q18.Ifthepositiveintegerxleavesaremainderof2whendividedby8,whatwilltheremainderbewhenx+9isdividedby8?Solution:Theeasiestwaytogettheanswerhereistopicknumbers.Pickanumberforxthathasaremainderof2whendividedby8,suchas10.Increasethenumberyoupickedby9.Inthiscase,10+9=19.Nowdivide19by8,whichgivesaremainder3.Q19.230+230+230+230=Solution:230+230+230+230= 302 4× = 30 2 30 2 322 2 2 2+× = = Q20.Anumberisasmuchgreaterthan36asislessthan86.Findthenumber.Solution:Letthenumberbex.Then,x–36=86–xOR2x=86+36=122ORx=61.Hence,therequirednumberis61.Q21.Findanumbersuchthatwhen15issubtractedfrom7timesthenumber,theResultis10morethantwicethenumber.Solution:Letthenumberbex.Then,7x-15=2x+10or5x=25orx=5.Hence,therequirednumberis5.Q22.Thesumofarationalnumberanditsreciprocalis13/6.Findthenumber.Solution:Letthenumberbex.Then,x+(1/x)=13/6OR(x2+1)/x=13/6OR6x2–13x+6=0OR6x2–9x–4x+6=0OR(3x–2)(2x–3)=0Itmeansx=2/3orx=3/2Hencetherequirednumberis2/3or3/2.Q23.Thesumoftwonumbersis184.Ifone-thirdoftheoneexceedsone-seventhoftheotherby8,findthesmallernumber.Solution:Letthenumbersbexand(184-x).Then,(x/3)-((184–x)/7)=8OR7x–3(184–x)=168OR10x=720=>x=72.So,thenumbersare72and112.Hence,smallernumber=72.Q24.Thedifferenceoftwonumbersis11andone-fifthoftheirsumis9.Findthenumbers.Solution:Letthenumberbexandy.Then,x–y=11----(i)and1/5(x+y)=9ORx+y=45----(ii)Adding(i)and(ii),weget:2x=56orx=28.Puttingx=28in(i),weget:y=17.Hence,thenumbersare28and17.

Q25.Ifthesumoftwonumbersis42andtheirproductis437,thenfindtheabsolutedifferencebetweenthenumbers.(S.S.C.2003)Solution:Letthenumbersbexandy.Then,x+y=42andxy=437

x-y= [(𝑥 + 𝑦)$ − 4𝑥𝑦]= [(42)$ − 4×437]= [1764– 1748]= [16]=4.Requireddifference=4.Q26.Thesumoftwonumbersis16andthesumoftheirsquaresis113.Findthenumbers.Solution:Letthenumbersbexand(15-x).Then,x2+(15-x)2=113ORx2+225+x2-30x=113OR2x2-30x+112=0orx2-15x+56=0OR(x-7)(x-8)=0orx=7orx=8.So,thenumbersare7and8.Q27.Theaverageoffourconsecutiveevennumbersis27.Findthelargestofthesenumbers.Solution:Letthefourconsecutiveevennumbersbex,x+2,x+4andx+6.Then,sumofthesenumbers=(27×4)=108.So,x+(x+2)+(x+4)+(x+6)=108or4x=96orx=24.:.Largestnumber=(x+6)=30.Q28.Thesumofthesquaresofthreeconsecutiveoddnumbersis2531.Findthenumbers.Solution:Letthenumbersbex,x+2andx+4.Then,x2+(x+2)2+(x+4)2=2531=>3x2+12x-2511=0=>x2+4x-837=0=>(x-27)(x+31)=0=>x=27.Hence,therequirednumbersare27,29and31.Q29.Oftwonumbers,4timesthesmalleroneislessthan3timesthe1argeroneby5.Ifthesumofthenumbersislargerthan6timestheirdifferenceby6,findthetwonumbers.Solution:Letthenumbersbexandy,suchthatx>yThen,3x-4y=5...(i)and(x+y)-6(x-y)=6OR-5x+7y=6…(ii)Solving(i)and(ii),weget:x=59andy=43.Hence,therequirednumbersare59and43.Q30.Theratiobetweenatwo-digitnumberandthesumofthedigitsofthatnumberis4:1.Ifthedigitintheunit'splaceis3morethanthedigitintheten'splace,whatisthenumber?Solution:Lettheten'sdigitbex.Then,unit'sdigit=(x+3).Sumofthedigits=x+(x+3)=2x+3.Number=l0x+(x+3)=11x+3.11x+3/2x+3=4/1OR1lx+3=4(2x+3)OR3x=9=>x=3.Hence,requirednumber=11x+3=36.Q31.Anumberconsistsoftwodigits.Thesumofthedigitsis9.If63issubtractedfromthenumber,itsdigitsareinterchanged.Findthenumber.Solution:Lettheten'sdigitbex.Then,unit'sdigit=(9-x).

Number=l0x+(9-x)=9x+9.Numberobtainedbyreversingthedigits=10(9-x)+x=90-9x.therefore,(9x+9)-63=90-9xOR18x=144=>x=8.So,ten'sdigit=8andunit'sdigit=1.Hence,therequirednumberis81.Q32.Afractionbecomes2/3when1isaddedtoboth,itsnumeratoranddenominator.and,itbecomes1/2when1issubtractedfromboththenumeratoranddenominator.Findthefraction.Solution:Lettherequiredfractionbex/y.Then,x+1/y+1=2/3=>3x–2y=-1…(i)and,x–1/y–1=1/2or2x–y=1…(ii)Solving(i)and(ii),weget:x=3,y=5therefore,Requiredfraction=3/5.Q33.50isdividedintotwopartssuchthatthesumoftheirreciprocalsis1/12.FindthetwoParts.Solution:Letthetwopartsbexand(50-x).

Then,KL+ K

%MNL=1/12or(%M–OPO)

L(%MNL)=1/12

orx2–50x+600=0or(x–30)(x–20)=0orx=30orx=20.So,thepartsare30and20.Q34.Ifthreenumbersareaddedinpairs,thesumsequal10,19and21.Findthenumbers.Solution:Letthenumbersbex,yandz.Then,x+y=10...(i)y+z=19...(ii)x+z=21…(iii)Adding(i),(ii)and(iii),weget:2(x+y+z)=50or(x+y+z)=25.Thus,x=(25-19)=6;y=(25-21)=4;z=(25-10)=15.Hence,therequirednumbersare6,4and15.Q35.Whatistheleastnumberbywhich825mustbemultipliedinordertoproduceamultipleof715?Solution:825=3 5 5 11715=5 11 13.In825,13ismissing.Restallthefactorsof715arepresentin825Q36.(i)IfX381isdivisibleby11,findthevalueofsmallestnaturalnumberX.(ii)If381Yisdivisibleby9,findthevalueofsmallestnaturalnumberY.Solution:(i)X381isdivisibleby11.Hence,thedivisibilityrulefor11mustbesatisfiedi.e.(X+8)–(3+1)=0oramultipleof11i.e.X+4=11 X=7(ii)381Yisdivisibleby9.Hence,thedivisibilityrulefor9mustbesatisfiedi.e.(3+8+1+Y)mustbedivisibleby9i.e.(12+Y)=18 Y=6

Q37.Whichoftheseistrue?A.3 3isnotarationalnumber.B.If‘a’isrationalnumberand‘n’isaninteger>1,thenanisarationalnumber.C.7isnotthecubeofarationalnumber.Solution:Recallthatrationalnumbersarethosethatcanbeexpressedasratiosi.e.,asa/b(b‡0). 3isirrational.If‘a’isrationalnumber,a2,a3…willberational.Cuberootof7isirrational.HenceallthestatementsaretrueQ38.Whatisthesmallestnumbermust34300bemultipliedwith,tomakeitaperfectsquare?Solution:34300=7 7 7 5 5 2 2=73 52 22.WecanmakeperfectsquareonlyifwehaveevenpowerofalltheprimenumbersNowwehaveevenpowerof5and2butwehave3powersof7nextevennumberafter3is4So,tomakethepower4of7wehavetomultiplynumberwithonemore7So,answeris7Q39.Ifyisapositivenon-zerointeger,findthelargestnumberusingthreey’sandnooperationalsign:a.yyyb.yyyc.(yy)yd.𝑦QR Solution:Consideranypositivenonzerointeger,say,y=3.Nowtryallthegivenoptions,youwillfindthat3&S=327isthelargestnumberQ40.Thenumber6n2+6nfornaturalnisalwaysdivisiblebya.6onlyb.6and12c.12onlyd.18onlye.24onlySolution:Theaboveexpressioncontainsa6,itisalwaysdivisibleby6.Also,n(n+1)isdivisibleby2.Hence,6n(n+1)isalsodivisibleby6 2=12.Thecorrectansweris12and6

Exercise

Q1.Ifnisapositiveinteger,thentheremainderwhen isdividedby7isa)3 b)0 c)5 d)6e)NoneoftheseQ2.Ifa,a+2,anda+4areprimenumbers,thenthenumberofpossiblesolutionsforais:a)1 b)2 c)3 d)nonee)NoneoftheseQ3.Inadivisionsum,thedivisoris12timesthequotientand5timestheremainder.iftheremainderis48,thendividendis?a)4844 b)4488 c)4848 d)4424e)NoneoftheseQ4.Whatisthevalueof*inthenumber17*24sothatitwillbeexactlydivisibleby11?a)5 b)4 c)3 d)2 e)NoneoftheseQ5.Whatistheleastnumberwhichisaperfectsquarebutcontains847asitsfactor?a)5929 b)5595 c)9595 d)5599 e)5549Q6.IfNisaneveninteger,Whichofthefollowingisanoddinteger?a)N+2b)3N–3 c)2N(N)–2N d)N(N)+N+2e)NoneoftheseQ7.xisoddwhileyiseven.Whichoftheseisodd(assumetheseareallintegers)?a)xy b)xy+y c)(x+x)(y+y) d)y/x e)x+yQ8.Whatwillbetheremainderif765432876543218isdividedby9?a)0 b)8 c)4 d)7 e)6Q9.What’svaluefor267494×9999?a)2674682506 b)2674672506 c)2674672509 d)2675672509e)NoneoftheseQ10.Whatisthehighestpowerof7in5000!?(5000!meansfactorial5000×4999×4998……×2×1)a)4998 b)714 c)832 d)816e)NoneoftheseQ11.Whatisthetotalnumberofdifferentdivisorsincluding1andthenumberthatcandividethenumber6400?a)24 b)27 c)54 d)68e)NoneoftheseQ12.Acertainnumberwhensuccessfullydividedby8and11leavesremaindersof3and7respectively.Whatwillberemainderwhenthenumberisdividedbytheproductof8and11,viz88?a)3 b)21 c)59 d)68e)Noneofthese

Q13.Whatisthereminderwhen91+92+93+......+99isdividedby6?a)0 b)3 c)4 d)5e)NoneoftheseQ14.Howmanytimeswillthedigit'0'appearbetween1and1000?a.192 b.292 c.193 d.191e.NoneoftheseQ15.Whatisthetotalnumberofdifferentdivisorsofthenumber7200?a.20 b.4 c.54 d.32e.NoneoftheseQ16.Thepositiveintegernearestto1000&divisibleby2,3,4,5&6is(a)1020 (b)1040 (c)960 (d)1030(e)NoneoftheseQ17.Iftheoperation,^isdefinedbytheequationx^y=2x+y,whatisthevalueofain2^a=a^3(a)0 (b)1 (c)-1 (d)4(e)NoneoftheseQ18.Whenanumberisdividedby36,itleavesaremainderof19.Whatwillbetheremainderwhenthenumberisdividedby12?(a)10 (b)7(c)8(d)9(e)NoneoftheseQ19.Thenumberwhichisformedbywritinganydigit6times(e.g.111111,444444,etc.)isdivisibleby(a)7 (b)11 (c)13 (d)Allofthese(e)NoneoftheseQ20.Howmanytimesdoesdigit6appearwhenyoucountfrom11to100?(a)18 (b)19 (c)20 (d)21(e)NoneoftheseQ21.Convert2.52145145145….intoarationalexpression.

a.253989990 b.

258939900 c.

251399900 d.

25189399900 e.Noneofthese

Q22.Rahulwritesfirsthundredwholenumbers.Howmanytimesdoeshewritezero?Alsofindouthowmanytimeswillnumber‘9’occurinthesetoffirsthundredwholenumbers?a.12,19 b.10,20 c.11,19 d.12,20e.Noneofthese Q23.Whichoffollowingnumbersisexactlydivisibleby99?a.3572403 b.135732 c.913464 d.114345e.NoneoftheseQ24.FindthetotalnumberoffactorsofanumberN=123×132×14a.64 b.192 c.24 d.16e.NoneoftheseQ25.Whichoffollowingnumbersisnotaprimenumber?a.71 b.11 c.119 d.41e.Noneofthese

Q26.LetPbethesetofevenintegers‘a’suchthat125≤ a≤ 275,‘a’isdivisibleby7butnotby11.Howmanyelementsdoes‘P’contain?a.12 b.11 c.10 d.9e.NoneoftheseQ27.Findtheleastnumberwhichwhendividedby6,7,8,9,10leavesremainder1.a.2531 b.2521 c.5041 d.5761e.NoneoftheseQ28.Twonumbers,xandy,aresuchthatwhendividedby6,theyleaveremainders4and5respectivelyFindtheremainderwhen(X2+Y2)isdividedby6.a.2 b.3 c.4 d.5e.NoneoftheseQ29.Pisaprimenumbergreaterthan5.WhatistheremainderwhenPisdividedby6?a.5 b.1 c.1or5 d.4e.NoneoftheseQ30.Whatistheunitdigitof

71317 ?a.1 b.7 c.3 d.9e.NoneoftheseQ31.Anumberwhendividedbyadivisorleavesaremainderof24.Whentwicetheoriginalnumberisdividedbythesamedivisor,theremainderis11.Whatisthevalueofthedivisor?a.13 b.59 c.35 d.37e.NoneoftheseQ32.Whatistheremainderwhen9875347×7435789×5789743isdividedby7?a.3 b.5 c.4 d.2e.NoneoftheseQ33.Whatistheunitdigitof

781 781 781 7811 2 3 ....... 9+ + + + ?a.5 b.3 c.4 d.2e.NoneoftheseQ34.Anumberwhendividedby5and7successivelyleavesremainders2and4respectively.Findtheremainderwhensamenumberisdividedby35.a.20 b.8 c.22 d.27e.NoneoftheseQ35.Whichisthesecondsmallestmultipleof7thatleavesaremainderof9whendividedby19?a.171 b.259 c.218 d.161e.Noneofthese

Q36.Whatwillbetheunitdigitof7142913 ?

a.7 b.9 c.1 d.3e.NoneoftheseQ37.Whatistherightmostnon-zerodigitofthenumber

3430770 ?a.7 b.3 c.9 d.1e.NoneoftheseQ38.Whichofthefollowinghasmostnumberofdivisors?a.99b.101c.176d.182e.Noneofthese

Q39.Ifeachofthethreenonzeronumbersa,b,andcisdivisibleby3,thenabcmustbedivisiblebywhichoneofthefollowingthenumbers?a.27b.81c.6d.12e.NoneoftheseQ40.Ifwewritedownallthenaturalnumbersfrom259to492sidebysidegetaverylargenaturalnumber259260261....491492.Howmany8'swillbeusedtowritethislargenaturalnumbera.32b.43c.52d.53e.NoneoftheseQ41.Whatistheremainderwhen7187isdividedby800?a.143b.243c.343d.443e.NoneoftheseQ42.WhatisthevalueofMandNrespectivelyifM39048458Nisdivisibleby8and11,whereMandNaresingledigitintegers?a.7,8b.8,6c.6,4d.5,4e.NoneoftheseQ43.Ifnisapositiveinteger,whichoneofthefollowingnumbersmusthavearemainderof3whendividedbyanyofthenumbers4,5,and6?a.30n+3b.90n+3c.60n+2d.60n+3e.NoneoftheseQ44.Acertainnumberwhendividedby222leavesaremainder35,anothernumberwhendividedby407leavesaremainder47.Whatistheremainderwhenthesumofthesetwonumbersisdividedby37?a.8b.9c.12d.17e.NoneoftheseQ45.Findtheremainderwhen231isdividedby5a.1b.4c.3d.2e.NoneoftheseQ46.IfN=23×34,M=22×3×5,thenfindthenumberoffactorsofNthatarecommonwiththefactorsofM.a.8b.6c.18d.20e.NoneoftheseQ47.HCFoftwonumbersis2andLCMis42840.Howmanypairsofthese2numbersarepossible?a.8b.16c.32d.4e.NoneoftheseQ48.Findtheremainderwhen289isdividedby89?a.1b.2c.88d.87e.NoneoftheseQ49.Howmanynaturalnumbersbelow660aredivisibleby5and11butnotby3?a.8b.9c.10d.11e.NoneoftheseQ50.Whatistheremainderwhen91+92+93+....+98isdividedby6?a.3b.1c.0d.4e.Noneofthese

Q51.Whatistheunit'sdigitofthenumber747-347a.4b.6c.2d.8e.NoneoftheseQ52.P=1234567891011.............9989991000.findtheremainderwhenPisdividedby16a.4b.0c.8d.2e.NoneoftheseQ53.Theremainderwhenm+nisdividedby12is8,andtheremainderwhenm–nisdividedby12is6.Ifm>n,thenwhatistheremainderwhenmndividedby6?a.1b.3c.5d.2e.NoneoftheseQ54.If0.0ababab….= $

%%,findab.

a.18b.36c.76d.72e.NoneoftheseQ55.Somebirdsweresettledonthebranchesofatree.First,theysatonetoabranchandtherewasonebirdleft.Nexttheysattwotoabranchandtherewasonebranchleft.Howmanybrancheswerethere?a.3b.4c.5d.6e.NoneoftheseQ56.Thenumber23x4534x01isdivisibleby11,howmanyvaluesofxcanbesubstituted?a.1b.4c.9d.10e.NoneoftheseQ57.Supposexandyarethedigitsforwhichx01yisdivisiblebyboth8and9.Whatisthesumofxandy?a.5b.6c.7d.8e.NoneoftheseQ58.Ifthenumber774958A97Bistobedivisibleby72,whataretherespectivevaluesofAandB?a.7and8b.1and6c.5and8d.0and6e.NoneoftheseQ59.Anumberwhendividedby195leavesremainder47.Ifthesamenumberisdividedby15,thentheremainderwillbea.1b.2c.3d.4e.NoneoftheseQ60.Thedifferencebetweenathreedigitnumberandanumberformedbyreversingtheorderofdigitsisalwaysdivisiblebya.3onlyb.9onlyc.11onlyd.9and11e.NoneoftheseQ61.Seemaweighs5/9ofherweightplus20kg.WhatistheactualweightofSeema?a.36kgb.72kgc.27kgd.45kge.NoneoftheseQ62.Anumberconsistsofthreeconsecutivedigits,theoneinunitsplacebeingthegreatestofthethree.Thenumberformedbyreversingthedigitsexceedstheoriginalnumberby22timesthesumofthedigits.Findthenumber.

a.231b.232c.233d.234e.NoneoftheseQ63.RamandShyameachhadgotseveralnumbersofpens.Totalnumberofpenstheyhad,weresuchthatonattemptingtodistributethemamongfivestudents,theyhadtwoleft.Whentheydistributedthemamong3and12,theyagainhadtwolefteachtime.Butwhentheydistributedthepensamong7students,theyhadnoneleft.Iftheyhadpensintheratioof7:6,WhatisthenumberofpensdidRamandShyamhad?a.91,78b.98,84c.35,30d.84,98e.NoneoftheseQ64.Findtheremainderwhen763isdividedby342.a.0b.1c.2d.341e.NoneoftheseQ65.Findtheremainderobtainedwhen76203+21203isdividedby97.a.10b.5c.2d.0e.NoneoftheseQ66.Howmanyzerosarethereattheendoftheproduct33×175×180×12×44×80×66?a.2b.4c.5d.6e.NoneoftheseQ67.Findtheno.ofzerosin752!.a.186b.187c.192d.193e.NoneoftheseQ68.Apositiveintegerwhichwhenaddedto999givesasumwhichisgreaterthanwhenitismultipliedby999.Whichofthefollowingcouldbethevalueofthepositiveinteger?a.2b.1c.3d.4e.NoneoftheseQ69.Whatisthetotalnumberofprimenumber’slessthan100?a.24b.26c.25d.23e.NoneoftheseQ70.If43pisdivisibleby3,thenwhatisthelargestvaluepcantake?a.2b.5c.9d.8e.NoneoftheseQ71.Ifthenumber6pq5isdivisiblebyboth3and5,whichofthefollowingdigitscanreplacepandq?a.9,7b.7,9c.6,5d.both(a)and(b)e.NoneoftheseQ72.Therearefourprimenumberswritteninascendingorder.Theproductofthefirstthreeis1001andthatofthelastthreeis2431.Findthefirstprimenumbera.6b.9c.7d.5e.NoneoftheseQ73.Whatistheremainderwhen152×698isdividedby5a.4b.1c.2d.3e.Noneofthese

Q74.Whatistheremainderwhen75isdividedby4?a.3b.2c.50d.6e.NoneoftheseQ75.Bywhatleastnumber250mustbemultipliedtogetamultipleof15a.3b.5c.15d.25e.Noneofthese

Answers1.B 2.A 3.C 4.B 5.A 6.B 7.E 8.B 9.B 10.C11.B 12.C 13.B 14.D 15.C 16.A 17.B 18.B 19.E 20.B21.D 22.B 23.D 24.B 25.C 26.B 27.B 28.D 29.C 30.C31.D 32.A 33.A 34.C 35.D 36.D 37.C 38.C 39.A 40.D41.C 42.C 43.D 44.A 45.C 46.B 47.B 48.B 49.A 50.C51.B 52.C 53.A 54.B 55.A 56.D 57.D 58.B 59.B 60.D61.D 62.D 63.B 64.B 65.D 66.B 67.B 68.B 69.C 70.D71.A 72.C 73.B 74.A 75.A

HCFandLCMWhatishighestcommonfactor(HCF)andleastcommonmultiple(LCM)?HowdoyoucalculateHCFandLCMoftwoormorenumbers?AreyoulookingforproblemsonHCFandLCM?Thischapterwillanswerallthesequestions.HIGHESTCOMMONFACTOR(HCF)Thelargestnumberthatdividestwoormoregivennumbersiscalledthehighestcommonfactor(HCF)ofthosenumbers.TherearetwomethodstofindHCFofthegivennumbers:PrimeFactorizationMethod-Whenanumberiswrittenastheproductofprimenumbers,thefactorizationiscalledtheprimefactorizationofthatnumber.Forexample,72=2×2×2×3×3=23×32TofindtheHCFofgivennumbersbythismethod,weperformtheprimefactorizationofallthenumbersandthencheckforthecommonprimefactors.Foreveryprimefactorcommontoallthenumbers,wechoosetheleastindexofthatprimefactoramongthegivennumber.TheHCFisproductofallsuchprimefactorswiththeirrespectiveleastIndices.Example:-FindtheHCFof72,288,and1080Solution:72=23×32,288=25×32,1080=23×33×5Theprimefactorscommontoallthenumbersare2and3.Thelowestindicesof2and3inthegivennumbersare3and2respectively.Hence,HCF=23×32=72.Example:-FindtheHCFof36x3y2and24x4y.Solution:36x3y2=22×32×x3×y224x4y=23×3×x4×y.Theleastindexof2,3,xandyinthenumbersare2,1,3and1respectively.HencetheHCF=22×3×x2×y=12x2y.Longdivisionmethod-TofindHCFoftwonumbersbydivisionmethod,wedividethehighernumberbythelowernumber.Thenwedividethelowernumberbythefirstremainder,thefirstremainderbythesecondremainder...andsoon,tilltheremainderis0.ThelastdivisoristherequiredHCF.

Example:-FindtheHCFof288and1080bythedivisionmethod.Solution:

Hence,thelastdivisor72istheHCFof288and1080.CONCEPTOFCO-PRIMENUMBERS:Twonumbersareco-primetoeachotheriftheyhavenocommonfactorexcept1.Forexample,15and32,16and5,8and27arethepairsofco-primenumbers.IftheHCFoftwonumbersN1andN2beH,then,thenumbersleftafterdividingN1andN2byHareco-primetoeachother.Therefore,iftheHCFoftwonumbersbeA,thenumberscanbewrittenasAxandAy,wherexandywillbeco-primetoeachother.SOLVEDPROBLEMSONHCFQ1.Threecompanyofsoldierscontaining120,192,and144soldiersaretobebrokendownintosmallergroupssuchthateachgroupcontainssoldiersfromonecompanyonlyandallthegroupshaveequalnumberofsoldiers.Whatistheleastnumberoftotalgroupsformed?Solution:TheleastnumberofgroupswillbeformedwheneachgrouphasnumberofsoldiersequaltotheHCF.TheHCFof120,192and144is24.Therefore,thenumbersofgroupsformedforthethreecompanieswillbe5,8,and6,respectively.Therefore,theleastnumberoftotalgroupsformed=5+8+6=19.Q2.Thenumbers2604,1020and4812whendividedbyanumberNgivethesameremainderof12.FindthehighestsuchnumberN.Solution:Sinceallthenumbersgivearemainderof12whendividedbyN,hence(2604-12),(1020-12)and(4812-12)arealldivisiblebyN.Hence,NistheHCFof2592,1008and4800.Now2592=25×34,1008=24×32×7and4800=26×3×52.Hence,thenumberN=HCF=24×3=48.Q3.Thenumbers400,536and645,whendividedbyanumberN,givetheremaindersof22,23and24respectively.FindthegreatestsuchnumberN.Solution:NwillbetheHCFof(400-22),(536-23)and(645-24).Hence,NwillbetheHCFof378,513and621.-->N=27.Q4.TheHCFoftwonumbersis12andtheirsumis288.Howmanypairsofsuchnumbersarepossible?Solution:IftheHCFif12,thenumberscanbewrittenas12xand12y,wherexandyareco-primetoeachother.Therefore,12x+12y=288-->x+y=24.Thepairofnumbersthatareco-primetoeachotherandsumupto24are(1,23),(5,19),(7,

17)and(11,13).Hence,onlyfourpairsofsuchnumbersarepossible.Thenumbersare(12,276),(60,228),(84,204)and(132,156).Q5.TheHCFoftwonumbersis12andtheirproductis31104.Howmanysuchnumbersarepossible?Solution:Letthenumbersbe12xand12y,wherexandyareco-primetoeachother.Therefore,12x×12y=31104-->xy=216.Nowweneedtofindco-primepairswhoseproductis216.216=23×33.Therefore,theco-primepairswillbe(1,216)and(8,27).Therefore,onlytwosuchnumbersarepossible.LEASTCOMMONMULTIPLE(LCM)Theleastcommonmultiple(LCM)oftwoormorenumbersisthelowestnumberwhichisdivisiblebyallthegivennumbers.TocalculatetheLCMoftwoormorenumbers,weusethefollowingtwomethods:PrimeFactorizationMethod:Afterperformingtheprimefactorizationofthenumbers,i.e.breakingthenumbersintoproductofprimenumbers,wefindthehighestindex,amongthegivennumbers,ofalltheprimenumbers.TheLCMistheproductofalltheseprimenumberswiththeirrespectivehighestindices.Example:-FindtheLCMof72,288and1080.Solution:72=23×32,288=25×32,1080=23×33×5Theprimenumberspresentare2,3and5.Thehighestindices(powers)of2,3and5are5,3and1,respectively.HencetheLCM=25×33×5=4320.Example:-FindtheLCMof36x3y2and24x4y.Solution:36x3y2=22×32×x3×y224x4y=23×3×x4×y.Thehighestindicesof2,3,xandyare3,2,4and2respectively.Hence,theLCM=23×32×x4×y2=72x4y2.DivisionMethod:TofindtheLCMof72,196and240,weusethedivisionmethodinthefollowingway:

L.C.M.ofthegivennumbers=productofdivisorsandtheremainingnumbers=2×2×2×3×3×10×49=72×10×49=35280.SOLVEDPROBLEMSONLCMExample:-Findthehighestfour-digitnumberthatisdivisiblebyeachofthenumbers24,36,45

and60.Solution:24=23×3,36=22×32,45=32×5and60=23×32×5.Hence,theLCMof24,36,45and60=23×32×5=360.Thehighestfour-digitnumberis9999.9999whendividedby360givestheremainder279.Hence,thenumber(9999-279=9720)willbedivisibleby360.Hencethehighestfour-digitnumberdivisibleby24,36,45and60=9720.Example:-Findthehighestnumberlessthan1800thatisdivisiblebyeachofthenumbers2,3,4,5,6and7.Solution:TheLCMof2,3,4,5,6and7is420.Hence420,andeverymultipleof420,isdivisiblebyeachofthesenumbers.Hence,thenumber420,840,1260,and1680arealldivisiblebyeachofthesenumbers.Wecanseethat1680isthehighestnumberlessthan1800whichismultipleof420.Hence,thehighestnumberdivisiblebyeachoneof2,3,4,5,6and7,andlessthan1800is1680.Example:-Findthelowestnumberwhichgivesaremainderof5whendividedbyanyofthenumbers6,7,and8.Solution:TheLCMof6,7and8is168.Hence,168isdivisibleby6,7and8.Therefore,168+5=173willgivearemainderof5whendividedbythesenumbers.Example:-Whatisthesmallestnumberwhichwhendividedby9,18,24leavesaremainderof5,14and20respectively?Solution:Thecommondifferencebetweenthedivisorandtheremainderis4(9-5=4,18-14=4,24-20=4).NowtheLCMof9,18,and24is72.Now72-4=72-9+5=72-18+14=72-24+20.Therefore,ifwesubtract4from72,theresultingnumberwillgiveremaindersof5,14,and20with9,18,and24.Hence,thenumber=72-4=68.Example:-Anumberwhendividedby3,4,5,and6alwaysleavesaremainderof2,butleavesnoremainderwhendividedby7.Whatisthelowestsuchnumberpossible?Solution:theLCMof3,4,5and6is60.Therefore,thenumberisoftheform60k+2,i.e.62,122,182,242etc.Wecanseethat182isdivisibleby7.Therefore,thelowestsuchnumberpossible=182.

PROPERTIESOFHCFANDLCM1)TheHCFoftwoormorenumbersissmallerthanorequaltothesmallestofthosenumbers.

2)TheLCMoftwoormorenumbersisgreaterthanorequaltothelargestofthosenumbers3)IfnumbersN1,N2,N3,N4etc.giveremaindersR1,R2,R3,R4,respectively,whendividedbythesamenumberP,thenPistheHCFof(N1-R1),(N2-R2),(N3-R3),(N4-R4)etc.4)IftheHCFofnumbersN1,N2,N3...isH,thenN1,N2,N3...canbewrittenasmultiplesofH(Hx,Hy,Hz..).SincetheHCFdividesallthenumbers,everynumberwillbeamultipleoftheHCF.5)IftheHCFoftwonumbersN1andN2isH,then,thenumbers(N1+N2)and(N1-N2)arealsodivisiblebyH.LetN1=HxandN2=Hy,sincethenumberswillbemultiplesofH.Then,N1+N2=Hx+Hy=H(x+y),andN1-N2=Hx-Hy=H(x-y).HenceboththesumanddifferencesofthetwonumbersaredivisiblebytheHCF.6)IfnumbersN1,N2,N3,N4etc.giveanequalremainderwhendividedbythesamenumberP,thenPisafactorof(N1-N2),(N2-N3),(N3-N4)...7)IfListheLCMofN1,N2,N3,N4..allthemultiplesofLaredivisiblebythesenumbers.8)IfanumberPalwaysleavesaremainderRwhendividedbythenumbersN1,N2,N3,N4etc.,thenP=LCM(oramultipleofLCM)ofN1,N2,N3,N4..+R.LCMandHCFoffractionsHCFoffractions= TUVWXYZ[\]^_W]`

aUbWXc\YW[dY^_W]`

LCMoffractions= aUbWXYZ[\]^_W]`

TUVWXc\YW[dY^_W]`

Example:-FindtheLCMandHCFof$%

K$&&%

Ke

Solution: LCM=aUbWX$%&&%

TUVWXK$&Ke=Kg%

h

HCF=TUVWX$%&&%

aUbWXK$&Ke= %

&h

UsageofHCFandLCM:I.Leastnumberleavingremainder‘r’ineachcasewhendividedbyx,yandz.Thus,leastnumberleavingremainder‘r’ineachcasewhendividedby‘x’,‘y’and‘z’=(LCMofx,y,z)+rTheseriesofsuchnumberswillbe(LCMofx,y,z)×n+rExample:-Findtheleastnumberwhichwhendividedby5,6and7leavesaremainder3ineachcase?Solution:Thesmallestnumberwhich,whendividedby5,6and7,leavesremainder3ineachcaseisLCM(5,6,7)=210+3=213II.Leastnumberleavingremainder(x-a),(y-a)and(z-a)whendividedbyx,yandzrespectively.

Thus,leastnumberleavingremainder(x-a),(y-a)and(z-a)whendividedbyx,yandzrespectively=(LCMofx,y,z)–a.Theseriesofsuchnumberswillbe(LCMofx,yandz)×n–a.Example:-Findthesmallestnumberwhichwhendividedby4,5and6leavesremainder2,3and4?Solution:Here,wecanseethat(divisor–remainder)=a,issameforalldivisors,inthiscasea=2.HCFof4,5and6is60soleastnumberisLCM–a=60–2=58III.Tofindthelargestmeasurethatmeasuresx,yandzexactly.Example:-Arectangularpieceofclothhasdimension18m×8m.Whatisleastnumberofequalsquaresthatcanbecutoutofthisclothsuchthatnoclothiswasted?Solution:HCF(18,8)=2So,minimumnumberofsquarethatcanbecutoutofthispiece=18×8/2×2=36Example:-Inaschool437boysand342girlshavebeendividedintoclasses,sothateachclasshasthesamenumberofstudentsandnoclasshasboysandgirlsmixed.Whatistheleastnumberofclassesneeded?Solution:Weshouldhavemaximumnumberofstudentsinaclass.So,wehavetofindtheHCF(437,342)=19HCFisalsofactorofdifferenceofthenumbersSo,numberofclasses=i&g

Kj+ &i$

Kj=23+18=41classes

Example:-Theratiobetweentwonumbersis3:4.IftheirLCMis192,whatisthedifferencebetweenthesquaresofthenumbers?Solution:Letthetwonumbersbe3xand4x.Therefore,theLCMofthesetwonumberswillbe3×4×x=12x.WeknowthattheLCMofthesetwonumbersis192.Equating,weget12x=192orx=16.Hence,thetwonumbersare3×16and4×16.i.e.,48and64Wehavetofindoutthedifferencebetweenthesquaresofthenumbers=642-482=(64+48)×(64-18)=112×16=1792RapidInformationList

S.No

Typeofproblem Approachtotheproblem

1. FindtheGREATESTNUMBERthatwillexactlydividegivennumbers.

Requirednumber=HCFofgivennumbers(greatestdivisor)

2. FindtheGREATESTNUMBERthatwillexactlydividex,yandzleavingremaindersa,bandcrespectively.

Requirednumber(greatestdivisor)=HCFof(x-a),(y-b)and(z-c)

3. FindtheLEASTNUMBERwhichisexactlydivisiblebyx,yandy

Requirednumber=LCMofx,yandz(leastdividend)

StudentoftengetconfusedwhattofindinquestionHCForLCMWheneveryouseejustDIVIDEorDIVIDESinquestionitmeansyouneedtofindHCFAndwheneveryouseeDIVIDEDBYorDIVISIBLEBYitmeansyouneedtofindLCMSolvedExamples:Ex.1.FindtheLCMof70and90[LIC2005]Solution:TofindLCM:

4. FindtheLEASTNUMBERwhichwhendividedbyx,yandzleavestheremaindersa,bandcrespectively.

Itisalwaysobservedthat(x-a)=(y-b)=(z-c)=k(say)∵Requirednumber=(LCMofx,yandz)-k

5. FindtheLEASTNUMBERwhichwhendividedbyx,yandzleavesthesameremainder‘r’ineachcase.

Requirednumber=(LCMofx,yandz)+r

6. FindtheGREATESTNUMBERthatwillexactlydividex,yandzleavessameremainderineachcase.

Requirednumber=HCFof(x-a),(y-b)and(z-c)

7. FindtheHCFof HCFoffractions=

8. FindtheLCMof LCMoffractions=

9. FindtheHCFofdecimalnumbers Step1.FindHCFofgivennumberswithoutdecimalsStep2.IntheHCFmakedecimalpointfromrighttoleftaccordingtothemaximumdecimalplacesamongthegivennumbers.

10. FindtheLCMofdecimalnumbers Step1.FindLCMofgivennumberswithoutdecimalsStep2.IntheHCFmakedecimalpointfromrighttoleftaccordingtothemaximumdecimalplacesamongthegivennumbers.

LCM= Ex.2.FindtheHCFof22,54,108,135,and198[BankClerk2000]Solution:(i)TofindHCF:

Commonfactoris1.So,HCFis1Ex.3.FindtheHCFof24×32×53×7,23×33×52×72and3×5×7×11.[CDS2002]Solution:Letx=24×32×53×7y=23×33×52×72z=3×5×7×11Commonfactorsofx,yandzare3,5,7.∴HighestCommonfactor=3×5×7=105

Ex.4.FindtheLCMof , and [Asst.Grade2005]Solution:Weknowthat

LCMoffractions= TofindLCMoffractions

LCM= LCMof(4,5,7)

LCM=2×2×5×7=140HCFof(5,6,15)5=5×16=2×3×115=3×5×1HCFof(5,6,15)=1

LCM= =140

Ex.5.FindtheHCFof , and [Asst.Grade2004]Solution:Weknowthat

HCFoffractions= Here,

HCF=

HCFof(4,10,20)4=2×210=2×520=2×2×5∴HCF=2LCMof(9,21,63)LCM=7×3×3=63

∴HCF=

Ex.6.FindtheLCMof0.24,1.2and6[SSCGrad,2007]Solution:LCMof0.24,1.2and6MakethesamenumberofdecimalplacesinallthegivennumbersbysuffixingzeroesifnecessaryandthenfindtheHCF/LCMwithoutdecimalpoint.

PutthedecimalpointintheHCF/LCMresultsleavingasmanydigitsonitsrightasthereareineachofthenumbers.Thegivennumberscanbewrittenas1.20,0.24,and6.00NowignoringthedecimalswefindLCMof120,24,and600∴LCM=2×2×2×3×5×5=120×5=600Thus,requiredLCM=600i.e.,6.Ex.7.FindtheHCFandLCMof0.72,0.36and0.45[SBIPO2003]Solution:Thegivennumbercanbewrittenas0.72,3.6and0.45Nowignoringnumbercanbewrittenas72,360and45

(i)TofindHCF∴HCFof(0.72,3.6and0.45)is9TherequiredHCFis0.09

(ii)TofindLCM∴LCM=9×8×5=360

Thus,requiredLCMis3.6Ex.8.TheHCFandLCMoftwonumbersare63and1260respectively.Ifoneofthetwonumbersis315,findtheothernumber.[SSCGrad,2000]Solution:Weknowthat Giventhat,HCF=63;LCM=1260Onenumber(x)=315

y=252∴Othernumberis252.Ex.9.TheLCMoftwonumbersis45timestheirHCF.Ifoneofthenumbersis125andthesumofHCFandLCMis1150,thentheothernumberis?[SBIPO2006]Solution:Giventhat,HCF+LCM=1150;LCM=45(HCF)Onenumber(x)=125Let,HCF+LCM=1150--------1eqn.LCM=45(HCF)----------2eqn.Substituteeqn.2ineqn.1HCF+45HCF=115046HCF=1150

HCF= =>HCF=25LCM=45(25)=>LCM=1125Weknowthat,

Anothernumberis225.Ex.10.Findthegreatestnumberwhichwilldivide772and2778soastoleavetheremainder5ineachcase.[LICADO,2005]Solution:Weknowthat,Tofindthegreatestnumberthatwilldividex,yandzleavingthesameremainderineachcaseandwhenthevalueoftheremainderisrisgiven,thenRequirednumber=H.C.Fof

Giventhatx=772;y=2778;r=5Requirednumber=HCFof(772-5)(2778-5) =HCFof(767,2773)=59Ex.11.Thegreatestnumberwhichondividing1657and2037leavesremainders6and5respectively,is:Solution:Requirednumber=H.C.F.of(1657-6)and(2037-5)=H.C.F.of1651and2032=127.Ex.12.TheL.C.M.oftwonumbersis48.Thenumbersareintheratio2:3.Thensumofthenumberis:Solution:Letthenumbersbe2xand3x.Then,theirL.C.M.=6x.So,6x=48orx=8.

Thenumbersare16and24.Hence,requiredsum=(16+24)=40.Ex.13.Ifthesumoftwonumbersis55andtheH.C.F.andL.C.M.ofthesenumbersare5and120respectively,thenthesumofthereciprocalsofthenumbersisequalto:Solution:Letthenumbersbeaandb.Then,a+b=55andab=5x120=600.

Therequiredsum=1+1=a+b

=55

=11

a b ab 600 120

Ex.14.Threenumberswhichareco-primetoeachotheraresuchthattheproductofthefirsttwois551andthatofthelasttwois1073.Thesumofthethreenumbersis:Solution:Sincethenumbersareco-prime,theycontainonly1asthecommonfactor.Also,thegiventwoproductshavethemiddlenumberincommon.So,middlenumber=H.C.F.of551and1073=29;

Firstnumber=

551

=19;Thirdnumber=

1073

=37.29 29

Requiredsum=(19+29+37)=85.

Exercise1.FindtheL.C.M.of36,48,72and108.a.432 b.428 c.360 d.106e.Noneofthese2.TheH.C.FandL.C.M.ofanytwonumbersare2079and27respectively.Ifoneofthetwonumbersis189,findtheothernumber.a.309b.297c.180d.327e.Noneofthese3.Findthegreatestnumberthatwilldivide296,492and1246leavingremainders8,12and22respectivelya.12b.36c.24d.48e.Noneofthese4.Whichoffollowingisapairofco-primenumber?a.32,52b.16,28c.18,35d.21,42e.Noneofthese5.TheL.C.M.oftwonumbersis252andtheirratiois6:7.Findthenumbers.a.32,49b.16,25c.36,42d.54,63e.Noneofthese6.Aroomis2.47mlongand2.09mbroad.Itisrequiredtopavethefloorwithminimumsquareslabs.Findthenumberofslabsrequiredforfloor.a.143b.361c.142d.165e.Noneofthese7.Findthegreatestnumberthatwilldivide378,330and882leavingremainders3,5and7respectively.a.25b.220c.300d.275e.Noneofthese8.Whichsmallestnumbermustbeaddedto9373sothatitexactlydivisibleby4?a.3b.2c.1d.4e.Noneofthese9.Whichsmallestnumbermustbeaddedto605329sothatitbecomesdivisibleby9?a.3b.2c.5d.7e.Noneofthese10.Sixbellstollingattheintervalof2,4,6,8,10,and12secondsrespectively.In30minuteshowmanytimestheytoll?a.4b.10c.15d.16e.Noneofthese11.FindtheL.C.M.of6/7,12/13,7/15.a.84b.1/84c.13d.1/13e.Noneofthese12.ThesumandthedifferencesoftheL.C.M.andtheH.C.F.oftwonumbersare312and264,respectively.Findthebiggernumberiftheirsumis168.a.108b.92c.96d.100e.Noneofthese13.FindtheH.C.F.of120,140,180and420.

a.15b.20c.25d.30e.Noneofthese14.Findtheleastnumberwhichwhendividedby36,48and54whichwillleavesthesameremainder7.a.432b.436c.437d.439e.Noneofthese15.Findthegreatestnumberwhichondividing304,554and854leavesequalremainder.a.35b.65c.50d.60e.Noneofthese16.Findthegreatestnumberwhichondividing277,757and3307leavesthesameremainder7ineachcase.a.30b.15c.60d.25e.Noneofthese17.FindtheL.C.M.of9/10,12/25,18/35,21/40.a.5/252b.3/1400c.252/5d.1400/3e.Noneofthese18.Findthegreatestnumberthatwilldivide63,45and69soastoleavethesameremainder.a.6b.9c.8d.10e.Noneofthese19.Awholesalesugardealerhas780kilogram,1650kilogramand510kilogramsugarofthreedifferentqualities.Hehastopackitallintoboxesofequalsizewithoutmixing.Findthecapacityofthelargestpossiblebox.a.15b.30c.12d.25e.Noneofthese20.Findtheleastperfectsquarenumberwhichisdivisibleby2,3,4,5,6.a.3600b.900c.1600d.2500e.Noneofthese21.Sixbellscommencetollingtogetherandtollatintervalsof2,4,7,8,10and12secondsrespectively.In30minuteshowmanytimesdotheytolltogether?a.2b.3c.4d.5e.Noneofthese22.Sevenbellsringatintervalsof2,3,4,6,8,9and12minutes,respectively.Theystartedringingsimultaneouslyat6.00inthemorning.Howmanymoretimeswouldallsevenbellsringsimultaneouslytill5.00intheeveningofthesameday?a.6b.7c.8d.9e.Noneofthese23.FindtheLCMof5/2,8/9,11/14a.400b.420c.440d.460e.Noneofthese24.Findthelargestnumberwhichdivides62,132and237toleavethesameremainderineachcasea.35b.25c.45d.55e.Noneofthese25.Anelectricwireissoldonlyinmultiplierof1metreandapersonrequiredseverallengthsofwire,each85cmlong.Toavoidanywastageandtominimizelabour,heshouldpurchase

minimumlengthofa.17mb.17cmc.170cmd.1.7me.Noneofthese26.Findthegreatestnumberthatwilldivide55,127and175,soastoleavethesameremainderineachcasea.12b.18c.24d.30e.Noneofthese27.Findthelargestnumberoffourdigitsexactlydivisibleby12,15,18and27a.9999b.9920c.9720d.9520e.Noneofthese28.Thecircumferencesoftheforeandhindwheelsofacarriagearerespectively.Atanygivenmoment,achalkmarkisputonthepointofcontactofeachwheelwiththeground.Findthedistancetravelledbythecarriagessothatboththechalkmarksareagainonthegroundatthesametimea.217.5mb.2175mc.217.5cmd.2175me.Noneofthese29.Findthegreatestnumberthatwilldivide478and719leavingremainders2and5Respectivelya.236b.238c.240d.119e.Noneofthese30.Acertaincardgamehasaspecialpack.Ifyoudealoutequallytofiveplayerstherearetwocardsleftundealt.Thesameistrueifyoudealtosevenplayers.Ifthecardsaredealttothreeplayersthereisoneleftundealt.Numberofcardsthatcannotbethereinthepackisa.100b.150c.125d.175e.Noneofthese31.TheHCFoftwonumbersis113andtheirLCMis228825.Oneofthenumbersis2825.Findtheothera.9153b.9150c.9163d.9125e.Noneofthese32.Threenumbersareintheratio1:2:3andtheirH.C.F.is12.Thenumbersarea.12,24,36b.1,2,3c.6,12,18d.24,48,72e.Noneofthese33.Theleastnumberoffivedigitswhichisexactlydivisibleby12,15and18isa.10000b.10020c.10800d.10080e.Noneofthese34.Sixbellscommencetollingtogetherandtollatintervalsof2,4,6,8,10and12secondsrespectively.In30minutes,howmanytimesdotheytolltogether?a.10 b.15 c.16 d.14 e.Noneofthese35.Thetrafficlightsatthreedifferentroadcrossingschangeafterevery48sec,72secand108sec respectively. If they all chance simultaneously at 8: 20: 00 hours, then they will againchangesimultaneouslyata.8:27:12hoursb.8:26:06hoursc.8:25:12hoursd.8:27:08hourse.Noneofthese

36.Thegreatestnumberthatwilldivide187,233and279leavingthesameremainderineachcaseisa.23b.46c.1d.92e.Noneofthese37.TheL.C.Mofthreedifferentnumbersis120.WhichofthefollowingcannotbetheirH.C.F?a.12b.24c.48d.8e.Datainadequate38.Thesumoftwonumbersis528andtheirH.C.Fis33.Thenumberofpairsofsuchnumberssatisfyingtheaboveconditionisa.8b.4c.12d.16e.Noneofthese39.Twonumbersareintheratioof15:11.IftheirH.C.Fis13,thenthenumbersarea.195,143b.15,11c.26,13d.Datainadequatee.Noneofthese40.Findtheleastnumberwhichwhendecreasedby4isexactlydivisibleby9,12,15and18a.180b.184c.176d.Cannotbedeterminede.Noneofthese41.Findtheleastnumberwhichwhendividedby12,15,20and24leavesthesameremainder2ineachcasea.120b.122c.118d.Cannotbedeterminede.Noneofthese42.ThesumoftwonumbersisPQandtheirdifferenceis1/7oftheirsum.FindtheirHCFa.7/6(PQ)b.6/7(PQ)c.1/7(PQ)d.1/6(PQ)e.Noneofthese43.TheLCMofthreenumbersis4752andHCFis6.Ifthetwonumbersare48and66,thenfindthethirdnumbera.36b.54c.42d.60e.Noneofthese44.Howmanynumbersbetween200and300whichareexactlydivisibleby6,8,and9?a.2b.3c.4d.1e.Noneofthese45.Therearetwoelectricalwires,oneis9m60cmlongaluminumwireandtheotheris5m12mlongcopperwire.Findthemaximumlengththatcanbeequallycutfromeachwireinsuchawaythatthetotallengthofeachwireisexactlydivisiblebyita.64cmb.64mc.32cmd.128cme.Noneofthese46.Findtheleastnumberofgirls,sothattheycanbearrangedinthegroupof15or20or30a.30b.60c.120d.180e.24047.Findtheleastperfectsquarenumberwhichisexactlydivisibleby3,4,5,6and8a.6400b.14400c.6250d.3600e.Noneofthese48.Findthegreatestnumberlessthan800,whichisdivisibleby8,12,28a.672b.660c.540d.720e.Noneofthese

49.Findtheleastnumberwhichwhendividedby12,15and20leavesremaining4,7and12respectivelya.50b.48c.52d.54e.Noneofthese50.Findtheleastnumberwhichwhendividedby5,6,7and8leavesaremainder3butwhendividedby9leavesnoremaindera.1680b.187c.1683d.3366e.Noneofthese51.Fiveballsbegintotolltogetherandtollatintervalsof24,40,72and120s.Afterwhatintervaltotimewilltheytollagaintogether?a.24min.b.48min.c.12min.d.72min.e.Noneofthese52.Fourwheelsmoving12,24,20and30revolutionsinaminutestartingatacertainpointonthecircumferencedownwards.Afterintervaloftimewilltheycometogetheragaininthesameposition?a.20sb.30sc.1mind.45se.Noneofthese53.Thetrafficlightsatthreedifferentroadcrossingschangeafterevery48.72and108srespectively.Iftheychangesimultaneouslyat9am,atwhattimewilltheychangeagainsimultaneously?a.7min12sb.8minc.7min24sd.7mine.Noneofthese54.Findthesmallestfourdigitnumberthatisexactlydivisibleby8,10and12a.1000b.80c.1080d.900e.Noneofthese55.Findthegreatestfivedigitnumberexactlydivisibleby6,8and12a.99984b.99900c.99800d.99884e.Noneofthese56.Findthegreatestnumberwhichwilldivideby321,428and535exactlya.107b.103c.101d.111e.Noneofthese57.Findthegreatestpossiblelengthwhichcanbeusedtomeasure4m3cm,4m34cmand4m65cmexactlya.31mb.31cmc.3.1md.3.1cme.Noneofthese58.Findtheleastnumberofsquaretilesrequiredtopavethefloorofaroomof4m50cmlongand5m25cmbroada.44b.42c.40d.80e.Noneofthese59.Findthesumofthesmallestfourdigitnumberandthelargestfivedigitnumbera.100999b.109999c.100099d.9999900e.Noneofthese60.Fiveballsbegintotolltogetherandtollrespectivelyatintervalsof6,7,8,9and12.Howmanytimestheywilltolltogetherinonehour,excludingtheoneatthestart?

a.6b.7c.8d.9e.Noneofthese61.Amonghowmanystudents,175bananasand105orangescanbeequallydivided?a.70b.25c.35d.45e.Noneofthese62.FindouttheLCMof234,532,745,1176and256,332,531,724a.256,332,532,745,1176b.256,332,532,724c.256,532,745,1176d.234,332,532,745,1176e.Noneofthese

63.Whatisthesmallestwholenumberthatisexactlydivisibleby a.128b.132c.130d.145e.Noneofthese64.Theleastperfectsquarewhichisdivisiblebyeachofthenumbers12,15,20and24isa.4900b.1600c.3600d.14400e.Noneofthese65.Theleastpositiveintegerwhichleavearemainder2,whendividedbyeachofthenumbers4,6,8,12and16a.98b.50c.48d.52e.Noneofthese66.Thenumberofintegersbetween200and600thataredivisibleby2,3,and7isa.6b.7c.9d.10e.Noneofthese67.Ifthreenaturalnumbers,whoseLCMis360,areintheratio2:3:4,thenthelargestofthemisa.120b.60c.180d.240e.Noneofthese68.TheHCFof168,189and231reducedby8,givesa.11b.13c.17d.21e.Noneofthese69.TheGCDandLCMoftwonumbersare66and384respectively.Ifthefirstnumberisdividedby2,theresultingansweris66.Thesecondnumberisa.190b.192c.120d.90e.Noneofthese70.TheLCMandGCDoftwonumbersare240and16respectively.Ifthetwonumbersareintheratio3:5,thenumbersarea.16,80b.48,80c.12,60d.6,10e.Noneofthese71.TheLCMoftwonumbersis280andtheratioofthenumbersof7:8.Findthenumbersa.35and40b.21and24c.28and32d.32and48e.Noneofthese

Answers

1.A 2.B 3.C 4.C 5.C 6.A 7.A 8.A 9.B 10.D11.A 12.C 13.B 14.D 15.C 16.A 17.C 18.A 19.B 20.B21.A 22.D 23.A 24.A 25.A 26.C 27.C 28.A 29.B 30.B31.A 32.A 33.D 34.C 35.A 36.B 37.C 38.B 39.A 40.B41.B 42.B 43.B 44.A 45.A 46.B 47.D 48.A 49.C 50.C51.B 52.B 53.A 54.C 55.A 56.A 57.A 58.B 59.A 60.B61.C 62.A 63.B 64.C 65.B 66.D 67.A 68.B 69.B 70.B71.A

Ratio&ProportionRatio:Ifaandbaretwoquantitieswiththesameunitssuchthatbisnotequaltozero;thenthequotienta/biscalledtheratiobetweenaandb.Itiswrittenasa:b.Ratiodoesnothaveanyunit.Thequantitiesaandbarecalledtermsoftheratio.a'iscalledantecedent(firstterm)b'iscalledconsequent(secondterm).PropertiesofRatios:#Ifm

"= n

#= U

],thenA:B:C=p:q:r

#IfpA=qB=rC,thenA:B:C=K"= K

#= K

]

#Ifm

n= U

othenmPn

mNn= UPo

UNo

#IfA:B=p:qandB:C=r:sThen,A:B:C=pr:qr:qsProportion:Equalityoftworatiosiscalledproportion.Ifa:b=c:d,wesaya,b,c,dareinproportionora:b::c:d.(1)Thirdproportion:Foranytwonumberaandb,thirdproportion(x)canbefound^p= p

Oorx=p

5

^

(2)Fourthproportion:Foranythreenumbersa,bandc,fourthproportion(x)canbefoundfrom^

p= q

Oorx=p×q

^

(3)Meanproportion:Foranytwonumbersaandb,meanproportion(x)isfoundby^O= O

porx= 𝑎𝑏

Directproportion:-Twovaluesarecalledindirectproportionifonevaluechangesaccordinglywiththesecondvalue.Ifonevalueincreases,secondalsoincreasesandifonedecreases,secondvalueautomaticallydecreases.Examplesofdirectproportion:-1.Timeandwork2.Manandwork3.Numberofarticlesandcostofanarticle4.Workandefficiency5.SpeedanddistanceLettwoquantitiesxandyareindirectlyproportiontoeachotherOr,x∝ ywhereissignofproportionality.

Or,x=kywherekisaconstant.Or,O

Q=k

Indirectorinverseproportion:-Twovaluesarecalledinindirectproportionifoneincreasesandseconddecreasesautomatically.andviceversa.LetxisinverselyproportionaltoythenOr,x∝ K

Q

Or,x=sQ

Or,xy=kwherekisaconstant.Examplesofindirectdirection:-1.Timeandmen2.Menandtime3.Menandefficiency4.SpeedandtimeByusingbothconceptofproportionality,wecansaythatMen×Efficiency×Time =Constant

Work SolvedexamplesEx.1:TheincomesofAandBareintheratio4:3andtheirexpendituresareintheratio3:2.IftheysaveRs.500each,findtheincomeofA?Solution:LettheirincomesbeRs.4xand3xrespectivelyalsotheirexpendituresas3yand2y.Then4x–3y=500 .....(1)&3x–2y=500 ........(2)Solvingequations(1)and(2)wegetx=500andy=500.theincomeofAisRs.2000.Ex.2:Abagcontainsrupee,50p,25pcoinsintheratio5:3:2.IfthetotalamountisRs.70.Findthetotalnumberofcoins?Solution:Letussupposethatnumberofcoinsare5x,3xand2xrespectively.Thentotalmoneywillbe5𝑥×1+3𝑥×K

$+2𝑥×K

i=70

⇒ 28𝑥/4=70⇒ 𝑥=10So,thetotalnumberofcoinswillbe=(5𝑥+3𝑥+2𝑥)=10𝑥=10×10=100

Ex.3:Twonumbersareintheratio3:5,if9isaddedtoeach,thentheyareintheratio21:32.Thesecondnumberis?Solution:Letthenumbersbe3xand5x.Then&OPj

%OPj= $K

&$or32(3x+9)=21(5x+9)

9x=99,x=11Sothenumbersare33and55.Ex.4:TheratioofA’sandB’ssalaryis9:4.IfA’ssalaryisincreasedby15%,thenhistotalsalarybecomesRs.5175.WhatisthesalaryofB?Solution:LetthesalariesofAandBbe9xand4xSo,9𝑥×KK%

KMM=Rs.5175

x=500B’ssalaryis4×500=Rs.2000.Ex.5:Acandoapieceofworkin12days;Bis60%moreefficientthanA.FindthenumberofdaysthatBtakestodothesamepieceofwork?Solution:Weknowthattimeandefficiencyareininverseratiosotheirproductmustbeequal.ProductoftimetakenbyAanditsefficiency=productoftimetakenbyBanditsefficiencyLetA’sefficiencyis100soB’sefficiencywillbe160Then,12× 100=y× 160ory=K$×KMM

KhMory=7.5days

Ex.6:Themonthlysalariesoftwopersonsareintheratioof4:7.IfeachreceivesanincreaseofRs.25inthesalary,theratioisalteredto3:5.Findtheirrespectivesalaries.Solution:Letthesalariesbe4xand7xTherefore,iOP$%

gOP$%=&

%

5(4x+25)=3(7x+25)20x+125=21x+75x=50Therefore,theirsalariesare4(50)&7(50)i.e.,200&350Ex.7:Whatisthevalueof'x'ifitisthemeanproportionalofx-4andx+8?Solution:Asxisthemeanproportionalofx-4andx+8,wehavex-4:x::x:x+8Therefore,ONi

O= O

OPe

orx2=(x-4)(x+8)orx2=x2+4x–32,Therefore,x=8Ex.8:TheratioofA’sandB’ssalaryis9:4.IfA’ssalaryisincreasedby15%,thenhistotalsalarybecomesRs.5175.WhatisthesalaryofB?Solution:LetthesalariesofAandBbe9xand4x9𝑥×KK%

KMM=Rs.5175

x=500B’ssalaryis4×500=Rs.2000.

Ex.9.Whichofthefollowingproportionsiscorrectif12=pq?a.3:q::2:p b.2:q::3:pc.4:q::p:3 d.noneoftheseSolution:12=pqor4×3=p×qor4:q=p:3Ex.10.Ifittakes3painters5daystocompleteajob,howmanydayslongerwillittakeifthereareonly2paintersworking?Solution:Weknowmanisinverselyproportionaltotimesomen×time=constantLetitwilltakexdayswhen2painterareworkingHence3×5=2×𝑥or𝑥=3×5/2=15/2=7.5Ex.11.If1alpha=2betasand1beta=3gammas,howmanyalphasareequalto36gammas?Solution:If1beta=3gammas,Multiplyingbothsidesby12,weget12betas=36gammasWealsoknow2betas=1alphasHence36gammas=12betas=6*2betas=6*1alphas=6alphasHence36gammas=6alphasEx.12.Aquizconsistsoftrueandfalsequestions.Theratioofthenumberoftruequestionstothenumberoffalsequestionsis4:3.Aboutwhatpercentofthequestionsarefalse?Solution:Percentoffalsequestions=(3/7)×100=42.86=43%approx.Ex.13.Jackisnow7timesasoldasJill.If10yearsfromnow,hewillbe3timesasoldasJill,howoldisJacknow?Solution:LetJillbeyyearsold.So,Jackis7yyearsoldnow.Tenyearsfromnow,Jillwillbe(y+10)andJackwillbe(7y+10).So,7y+10=3(y+10)=3y+30.Thus,4y=20ory=5.Jillis5yearsoldandJackis35yearsoldnow.Ex.14.Rs.432isdividedamongstthreeworkersA,BandCsuchthat8timesA'sshareisequalto12timesB'ssharewhichisequalto6timesC'sshare.HowmuchdidAget?Solution:Given8A=12B=6CDividingthisbyLCMof8,12&6i.e.24,weget

8A 12 6 24 24 24

A B C 3 2 4

B C= =

⇒ = =

orA:B:C=3:2:4 SoA’sshare=&

j×432=144

Ex.15.Twocoggedwheelsofwhichonehas32cogsandother54cogs,workintoeachother.Iftheformerturns80timesinthreequartersofaminute,howoftendoestheotherturnin8seconds?Solution:If32cogsturn80timesin45secondThenlet54cogsturnxtimein8second32 45 54 8

80 xx = 24

× ×=

⇒ Ex.16.TheratioofA’sandB’ssalaryis9:4.IfA’ssalaryisincreasedby10%,thenhistotalsalarybecomesRs.4950.WhatisthesalaryofB?Solution:LetthesalariesofAandBbe9𝑥and4𝑥9𝑥×110/100=Rs.4950x=500B’ssalaryis4×500=Rs.2000.Ex.17:Theratiobetweenlargerandsmallernumbersistwelvelessthantheproductofthesetwonumbers.Thesquareofsmallernumberishalfofthelargernumber.Whatisthetotalsumofthesetwonumbers?Solution:Letthesmallerandlargernumbersbexandyrespectively.Accordingtothequestion,Q

O+12=xyandx2=Q

$

Combiningtheaboveequations,wegetx3–x–6=0or(x–2)(x2+2x+3)=0Nowx=2istheonlypositivevalueofx,whereastheothervaluesareimaginaryy=8,Sumofthenumbers=x+y=2+8=10.Problemonages:-ProblemsonagearesimilartothoseofRatioandProportion.Howeverincertainquestions,calculationscanbeshortenedusingtheconceptsofSingaporeMath’s.Forexampletheageofthefather3yearsagowas7timesthatofhisson.Atpresentthefather'sageis5timesthatofhissonthan.findtheagesoffatherandsonweproceedasfollows:F:S7:1<--3YearsAgo5:1<--PresentHereiftheratiosweretheactualagesthenPresentratioshouldbemorethanthatof3yearsago.Alsothesameyearswouldbeaddedtofather'sageandson'sage.Sowhatwedohereiswemaketheratioofpresentagetobemorethanthatof3yearsagoandalsomakethedifferenceoftheratiotobeequal.Todothatwefindthedifferenceoftheratiosandcrossmultiplyas:

F:S7:1<--Differenceis65:1<---Differenceis4Thustheratioofdifferenceis6:4or3:2,crossmultiplyingtheupperratioby2andlowerby3,wehave:F:S14:2<--threeyearsago15:3<--Present-------1:1-----Hereweseethatthedifferenceis1:1whichissameandalsothepresentageismorethanthatof3yearsago.Nowthecalculationisverysimple.15–14=1butweneeddifferenceisequalto3becausethegivenageis3yearsagoSo,weneedtomultiplybothnumeratoranddenominatorby3togetthepresentagesoffatherandsonRequiredanswers,15*3=45years(father)and3*3=9years(son)Theconceptisverysimpleandgivestheansweralmostintuitively.Sometimesthequestionsareaskedlikethis.Thesumofagesofamotherandherdaughteris50years.Also5yearsago,themother'sagewas7timestheageofthedaughter.Whatarethepresentagesofthemotherandthedaughter.Here5yearsago,thesumoftheageswillbe40years.Divide40intheratioof7:1,Thus,1/8×40=5(daughter)7/8×40=35(mother)Somothersagetodayis35+5=40yearsanddaughter'sagetodayis5+5=10years.ShortcutMethods1.IftheageofA‘t’yearsagowas‘n1’timestheageofB,atpresentA’sageisn2timesthatofBthen,

A’spresentageis=

B’spresentageis= 2.ThepresentageofAis‘n1’timesthepresentageofB.If‘t’yearshence,theageofAwouldbe‘n2’timesofB,then

A’spresentage=

B’spresentage= 3.AgeofA‘t’yearsago,was‘n1’timestheageofB.iftyearshenceA’sagewouldbe‘n2’timesofBthen,

A’spresentage=

B’spresentage= 4.ThesumofpresentagesofAandBisSyears.Iftyearsago,theageofAwasntimestheageofBthen,

PresentageofA= years

PresentageofB= 5.ThesumofpresentagesofAandBisSyears.IftyearshencetheageofAwouldbentimestheageofBthen,

PresentageofA= years

PresentageofB= 6.IftheratioofpresentagesofAandBisa:bandtyearshenceitwillbec:d,then,

A’spresentageis=

B’spresentageis= Ifthedifferenceisgiven,then

PresentageofA=

PresentageofB= yearsIfageisgiven,then

A=

B= Ex.1.TheratiobetweenthepresentagesofMandNis5:3respectively.TheratiobetweenM’sage4yearsagoanN’sageafter4yearsis1:1.WhatistheratiobetweenM’sageafter4yearsandN’sage4yearsago?[BankPO2000]Solution:M’sage=5x,N’sage=3x(5x-4):(3x+4)=1:15x–4=3x+42x=8x=4

∴M’spresentage=20years,N’spresentage=12years.∴RequiredRatio=24:8=3:1Ex.2.SixyearsagoJosewastwiceasoldasJoseph.Theratiooftheirpresentageis9:5respectively,whatisthedifferencebetweentheirpresentages?[BankPO2000]Solution:Lettheirpresentagesbe9xand5x.Sixyearsago,

∴Differencebetweentheirpresentages= Ex.3.Sita’smotherwas4timesofSita’sagetenyearsago.Aftertenyears,shewillbetwiceasoldasSita.HowoldisSitatoday?[SSC2002Solution:Takexasmother’spresentage,yasSita’spresentage.Before10years(x-10)=4(y-10)x-10=4y-40x-4y=-30------(1)After10years(x+10)=2(y+10)x+10=2y+20x-2y=10------(2)Bysolving(1)&(2)-2y=-40y=20x=50Sita’sageis20years.Ex.4.AsaidtoB“IamtwiceasoldasyouwerewhenIwasasoldyouarenow.”Sumoftheiragesis42.Findtheirpresentages.[Delhipolice,2007]Solution:Clearlybychoosingoption(b)weseethatAwasequaltoB’sage,i.e.18years,6yearsagoandatthattimeBwas12yearsold.So,A’spresentage,24years,istwiceofB’sthattimeofage,i.e.12years.(Or)LetPresentageofB=xandpresentageofA=42-xThen,18yearsagoA’sage=xandB’sage=x-(42+2x)=3x-42Bygivencondition,A’spresentage=2×(B’sagesomeyearsago)42–x=2(3x-42)x=18∴B’spresentage=18yearsandA’spresentage=24years.Ex.5.Theaverageageof30boysinaclassis15years.Oneboy,aged20years,lefttheclass,buttwonewboyscameinhisplacewhoseagesdifferby5years.Iftheaverageageofalltheboysnowintheclassremains15years,theageoftheyoungernewcomeris:[SSCGrad.2002]

Solution:Lettheagesofthetwonewboysbexandx+5.Bygivencondition,

Ex.6.Mygrandfatherwas8timesolderthanme16yearsago.Hewouldbe3timesmyage,8yearsfromnow.Eightyearsago,whatwastheratioofmyagetothatofmygrandfather?[SSCGrad.2003]Solution:Letmyagebexyears.Then16yearsago,myage=(x-16)yearsMygrandfather’sage=8(x-16)years8yearsfromnow,myage=(x+8)yearsMygrandfather’sage=3(x+8)yearsAlso,3(x-8)-8(x-16)=245x=128

∴Grandfather’spresentage=8(x-16)+16=92.8

Theratiooftheirages8yearsago, Ex.7.ThepresentageofMr.Sanyalis3timestheageofhisson.Sixyearshencetheratiooftheirageswillbe5:2respectively.WhatisthepresentageofMr.Sanyal?[Bank2006]Solution:Letpresentageofson=xyearsPresentageofMr.Sanyal=3xyears

6x+12=5x+306x–5x=30–12x=18years∴PresentageofMr.Sanyal=3×18=54yearsEx.8.PresentagesofSameerandAnandareintheratioof5:4respectively.Threeyearshence,theratiooftheirageswillbecome11:9respectively.WhatisAnand'spresentageinyears?Solution:Letthepresentagesofsameerandanandbe5xyearsand4xyearsrespectively.Then 5x+3/4x+3=11/9

9(5x+3)=11(4x+3) x=6.

Anands'spresentage =4x =24years.

Ex.9.TheratiobetweenthepresentagesofPandQis5:7respectively.IfthedifferencebetweenQ'spresentageandP'sageafler6yearsis2,whatisthetotalofP'sandQ'spresentages?Solution:LetthepresentagesofPandQbe5xyearsand7xyearsrespectively.Then 7x-(5x+6)=2

2x=8 x=4.

Requiredsum =5x+7x =12x =12×4 =48years.

Ex.10.Mybrotheris3yearseldertome.Myfatherwas28yearsofagewhenmysisterwasbornwhilemymotherwas26yearsofagewhenIwasbom.Ifmysisterwas4yearsofagewhenmybrotherwasborn,then,whatwastheageofmyfatherandmotherrespectivelywhenmybrotherwasborn?Solution:Clearly,mymotherwasborn3yearsbeforeIwasbornand4yearsaftermysisterwasborn.So,father'sagewhenbrotherwasborn

=(28+4) =32years.

mother'sagewhenbrotherwasborn

=(26-3) =23years.

Ex.11.Eighteenyearsago,afatherwasthreetimesasoldashisson.Nowthefatherisonlytwiceasoldashisson.ThenthesumofthepresentagesofthesonandthefatherisSolution:LetthepresentagesofFatherandSonbe2xyearsandxyearsrespectively.Then (2x-18)=3(x-18)

2x-18=3x-54 x=54-18 x=36.

Requiredsum =(2x+x) =3x =3x36 =108years.

Ex.12.Thesumoftheagesof5childrenbornattheintervalsof3yearseachis50years.Whatistheageoftheyoungestchild?

Solution:Lettheagesofthechildrenbex,(x+3),(x+6),(x+9)and(x+12)years.Then, x+(x+3)+(x+6)+(x+9)+(x+12)=50

5x=20 x=4.

Ageoftheyoungestchild =4years.Ex.13.Ayesha'sfatherwas38yearsofagewhenshewasbornwhilehermotherwas36yearsoldwhenherbrotherfouryearsyoungertoherwasborn.Whatisthedifferencebetweentheagesofherparents?Solution:Mother'sagewhenAyesha'sbrotherwasborn

=36years.

Father'sagewhenAyesha'sbrotherwasborn

=(38+4)years.=42years.

Requireddifference =(42-36)=6years.

Ex.14.Sneh'sagois1/6thofherfather'sage.Sneh'sfather'sagewillbetwiceofVimal'sageafter10years.IfVimal'seighthbirthdaywascelebratedtwoyearsbefore,thenwhatisSneh'spresentage?Solution:Vimal'sageafter10years =(8+2+10)years

=20years.Sneh'sfatherageafter10years =40yearsSneh'sfatherPresentage =30yearsThereforeSneh'sage =(1/6x30)years

=5years.Ex.15.Afathersaidtohisson,"Iwasasoldasyouareatthepresentatthetimeofyourbirth".Ifthefather'sageis38yearsnow,theson'sagefiveyearsbackwas:Solution:Lettheson'spresentagebexyears.Then,(38-x)=x

2x=38.x=19.Son'sage5yearsback(19-5)=14years.

Ex.16.PresentagesofSameerandAnandareintheratioof5:4respectively.Threeyearshence,theratiooftheirageswillbecome11:9respectively.WhatisAnand'spresentageinyears?Solution:LetthepresentagesofSameerandAnandbe5xyearsand4xyearsrespectively.

Then,5x+3

=11

4x+3 9

9(5x+3)=11(4x+3)45x+27=44x+3345x-44x=33-27x=6.Anand'spresentage=4x=24years.

Ex.17.Sixyearsago,theratiooftheagesofKunalandSagarwas6:5.Fouryearshence,theratiooftheirageswillbe11:10.WhatisSagar'sageatpresent?Solution:LettheagesofKunalandSagar6yearsagobe6xand5xyearsrespectively.

Then,(6x+6)+4

=11

(5x+6)+4 1010(6x+10)=11(5x+10)5x=10x=2.Sagar'spresentage=(5x+6)=16years.

Partnership:-Whentwoormorepersonsagreetorunabusinessjointly,theyarecalledpartnersandthisdealiscalledpartnership.SimplePartnershipWhenallthepartnersinvesttheirmoneyforthesametime,itiscalledsimplepartnership.CompoundPartnershipWhenallthepartnersinvesttheirmoneyforthedifferenttimes,itiscalledcompoundpartnership.1.IfthecapitalsofthreepartnersbeRs.C1,Rs.C2andRs.C3forthesameperiodandthetotalprofitbeRs.P,thensharesofthepartnersintheprofitsare

, and 2.IfthecapitalsofthreepartnersbeRs.C1,C2andRS.C3fortheperiodst1,t2andt3respectivelyandthetotalprofitbeRs.P,thensharesofthepartnersintheprofitsare:

, and 3.Ifinapartnershiptheinvestmentsmadebyfirst,secondandthirdpartnersarex1,x2,x3respectively,thetimeperiodbet1,t2,t3thenratioofprofitsisgivenbyx1t1:x2t2:x3t3.4.If istheratioofinvestmentsandp1:p2:p3betheratioofprofits,theratiooftime

periodsisgivenby

5.Ifp1:p2:p3istheratioofprofitsandt1:t2:t3betheratiooftimeperiods,theratioof

investmentsisgivenby Ex.1.Mr.BurkitstartedabusinessinvestingRs.45000.After3monthsPeterjoinedhimwithacapitalofRs.60000.Afteranother6months,RonaldjoinedthemwithacapitalofRs.90000.Attheendoftheyear,theymadeaprofitofRs.16500.Findtheratiooftheircapitals?[BankPo,2007]Solution:Burkitinvestedhiscapitalfor12months,Peterfor9monthsandRonaldfor3months.So,ratiooftheircapitals=(45000×12):(60000×9):(90000×3)=540000:540000:270000=2:2:1(∵Totalinvestment=Investment×time)Ratiooftheircapitals=2:2:1Ex.2.Mr.SanjaystartedabusinessinvestingRs.25000in1996.In1997heinvestedanadditionalamountofRs.10,000andMr.SureshjoinedhimwithanamountofRs.35000.In1998Mr.SanjayinvestedanotheradditionalamountofRs.10,000andMr.RamjoinedthemwithanamountofRs.35,000.WhatwillbeSuresh’sshareintheprofitofRs.150000earnedattheendofthreeyearsfromthestartofthebusinessin1966?[BankPO2001]Solution:SanjaySureshRam199625,000199735,00035,000199845,00035,00035,000∴Suresh’sshare=$

h×1,50,000 = 𝑅𝑠. 50,000

Ex.3.TheratioofinvestmentsoftwopartnersAandBis11:12andtheratiooftheirprofitis2:3.IfAinvestedthemoneyfor8months,thenthetimeforwhichBinvestedthemoneyis:[SSCGrad2005]Solution:LetA’scapitalbeRs.11xandB’scapitalbeRs.12x.

Ex.4.Rs.1290isdividedbetweenA,B,andC.SothatA’sshareis1K

$timesB’sandB’sshareis

1&itimeC’s.WhatisC’sshare?[CDS2004]

Solution:

∴

∴C’sshare Ex.5.A,BandCinvestRs.4000,Rs.5000andRs.6000respectivelyinabusinessandAgets25%ofprofitformanagingthebusiness,therestoftheprofitisdividedbyA,BandCinproportiontotheirinvestment.Ifinayear,AgetsRs.200lessthanBandCtogether,whatwasthetotalprofitofthatyear?[RRB,2000]Solution:Aftergiving25%ofthetotalprofitamounttoAformanagingthebusiness,therest75%oftotalprofitisdividedamongA,B,andCinproportiontotheirinvestments.In75%oftotalprofit

∴75%oftotalprofit=4x+5x+6x∴Totalprofit=K%O

g%%=20x

∴ShareofA=4x+25%of20x=9xShareofB=5x=5xShareofC=6x=6xGiven,(5x+6x)-9x=200x=100∴Totalprofit=20x=20×100=Rs.2000Ex.6.Inapartnership,AinvestsK

hofthecapitalforK

hofthetime,BinvestsK

&ofthecapitalfor

K&ofthetimeandC,therestofthecapitalforwholetime.FindA’sshareofthetotalprofitofRs.2300[Asst.Grade2003]Solution:

CapitalofC= Letthetotaltimebe12months∴

∴Shareof Ex.7.SumitandRavistartedabusinessbyinvestingRs85000and15000respectively.Inwhatratiotheprofitearnedafter2yearsbedividedbetweenSumitandRavirespectively.Solution:P:Q=85000:15000=17:3Importanttonotetherethatifbothhaveinvestedfordifferentperiodoftimesthenwehadto

multiplywithnumberofmonthstogetthedesiredratio.Ex.8.AandBinvestinabusinessintheratio3:2.If5%ofthetotalprofitgoestocharityandA'sshareisRs.855,thetotalprofitis:Solution:LetthetotalprofitbeRs.100.

Afterpayingtocharity,A'sshare=Rs.

95x3

=Rs.57.5

IfA'sshareisRs.57,totalprofit=Rs.100.

IfA'sshareRs.855,totalprofit=

100x855

=1500.57

Ex.9.A,BandCjointlythoughtofengagingthemselvesinabusinessventure.ItwasagreedthatAwouldinvestRs.6500for6months,B,Rs.8400for5monthsandC,Rs.10,000for3months.Awantstobetheworkingmemberforwhich,hewastoreceive5%oftheprofits.TheprofitearnedwasRs.7400.CalculatetheshareofBintheprofit.Solution:Formanaging,Areceived=5%ofRs.7400=Rs.370.Balance=Rs.(7400-370)=Rs.7030.Ratiooftheirinvestments=(6500x6):(8400x5):(10000x3)=39000:42000:30000=13:14:10

B'sshare=Rs.

7030x14

=Rs.2660.37

Ex.10.AandBstartedapartnershipbusinessinvestingsomeamountintheratioof3:5.CjoinedthenaftersixmonthswithanamountequaltothatofB.InwhatproportionshouldtheprofitattheendofoneyearbedistributedamongA,BandC?Solution:LettheinitialinvestmentsofAandBbe3xand5x.A:B:C=(3xx12):(5xx12):(5xx6)=36:60:30=6:10:5.Ex.11.AandBstartedabusinessinpartnershipinvestingRs.20,000andRs.15,000respectively.Aftersixmonths,CjoinedthemwithRs.20,000.WhatwillbeB'sshareintotalprofitofRs.25,000earnedattheendof2yearsfromthestartingofthebusiness?Solution:A:B:C=(20,000x24):(15,000x24):(20,000x18)=4:3:3.

B'sshare=Rs.

25000x3

=Rs.7,500.10

Ex.12.AmanstartedabusinessinvestingRs.70,000.RakhijoinedhimaftersixmonthswithanamountofRs..1,05,000andSagarjoinedthemwithRs.1.4lakhsafteranothersixmonths.TheamountofprofitearnedshouldbedistributedinwhatratioamongAman,RakhiandSagarrespectively,3yearsafterAmanstartedthebusiness?Solution:Aman:Rakhi:Sagar=(70,000x36):(1,05,000x30):(1,40,000x24)=12:15:16.

Exercise

1.IfA:B=3:4andB:C=8:9,thenfindthevalueofA:B:Ca.3:4:9b.3:8:9c.6:8:9d.2:4:7e.Noneofthese2.Ifa:b=b:cthenratioa4:b4=?a.a3:c2b.a4:c2c.a4:c4d.a2:c2e.Noneofthese3.If^

g= p

j= q

KKthenfinda:b:c

a.3:4:9b.3:8:9c.6:8:9d.7:9:11e.Noneofthese4.IfA=K

iBandB=K

$C,thenfindthevalueofA:B:C.

a.3:4:9b.1:4:8c.6:8:9d.2:4:7e.Noneofthese5.If(a+b):(a-b)=5:3,then(a2+b2):(a2-b2)=?a.13:15b.15:13c.17:15d.15:17e.Noneofthese6.Thetotalnumberofstudentsinaschoolis2140.Ifthenumberofgirlsintheschoolis1200,whatistheratioofthetotalnumberofboystothetotalnumberofgirlsintheschool?a.47:60b.15:13c.60:47d.15:17e.Noneofthese7.Theaverageageofapersonandhissonis30yr.Theratiooftheiragesis4:1respectively.Findtheageofsona.20yrb.16yrc.14yrd.12yre.Noneofthese8.Rahulobtained12marksmorethanthatofTanvi.Iftheratiooftheirmarksis3:4,findthesumoftheirmarksa.96b.84c.7d.12e.Noneofthese9.One-halfofacertainnumberisequalto65%ofthe2ndnumber.Findtheratioof1stto2ndnumbera.13:10b.15:13c.10:13d.15:17e.Noneofthese10.ThetotalweightofSanjayandSureshis120kg.IfSanjayweights30kgmorethanSuresh,whatistheratiooftheweightsofSureshtothatofSanjay?a.3:5b.5:3c.7:5d.5:7e.Noneofthese11.PresentageofMeenais8timesofherdaughter’sage.8yrhence,theratiooftheagesofMeenaandherdaughterwillbe10:3.FindthepresentageofMeenaa.20yrb.32yrc.14yrd.28yre.Noneofthese12.AmitandSureshhaveinvestedintheratioof4:7.IfbothinvestedatotalamountofRs.49500,thenfindtheinvestmentofSuresha.3150b.31500c.1500d.3500e.Noneofthese

Ans:Rs.3150013.ThepresentratioofagesofAandBis4:5.18yrago,thisratiowas11:16.FindthesumofthepresentagesofAandBa.90yrb.75yrc.60yrd.54yre.Noneofthese14.Theratiooftheagesofafathertothatofhissonis5:2.Iftheproductoftheiragesinyris1000,thenfindthefather’sageafter10yra.90yrb.75yrc.60yrd.54yre.Noneofthese15.If70%ofanumberisequaltothree-fifthofanothernumber,whatistheratiobetweenthe1standthe2ndnumber,respectively?a.6:7b.5:3c.7:5d.5:7e.Noneofthese16.Inaclass,thenumberofboysandgirlsisintheratioof4:5.If10moreboysjointheclass,theratioofnumbersofboysandgirlsbecomes6:5.Howmanygirlsarethereintheclass?a.20b.25c.35d.45e.Noneofthese17.Thesumoftheagesofafatherandhissonis100yrnow.5yrago,theirageswereintheratioof2:1.Findtheratiooftheagesoffatherandsonafter10yra.6:7b.5:3c.7:5d.5:7e.Noneofthese18.Thetotalnumberofstudentsinaschoolis8670.Ifthenumberofboysintheschoolis4545,whatwillbetheratioofthetotalnumberofgirlsintheschool?a.303:275b.275:303c.7:5d.5:7e.Noneofthese19.Theaverageageofawomanandherdaughteris16yr.Theratiooftheiragesis7:1.Whatisthewoman’sage?a.28yrb.35yrc.21yrd.14yre.Noneofthese20.TheagesofVaibhavandJagatareintheratioof12:7.After6yr,theratiooftheirageswillbe3:2.Whatisthedifferenceintheirages?a.10yrb.25yrc.30yrd.49yre.Noneofthese21.Theaverageageofamanandhistwotwinsonsbornonthesamedayis30yr.Theratiooftheagesoffathertooneofhissonsis5:2.Whatisthefather’sage?a.90yrb.75yrc.60yrd.50yre.Noneofthese22.AsumofmoneyisdividedamongstA,B,CandDintheratioof3:7:9:13.IftheshareofBisRs.4872,whatwillbethetotalamountofmoneyofAandCtogether?a.8352b.8356c.8350d.8348e.Noneofthese23.Thespeedsofthreecarsareintheratioof2:3:4.Findtheratiobetweenthetimetakenbythesecarstocoverthesamedistancea.6:4:3b.3:8:9c.6:8:9d.2:4:7e.Noneofthese

24.Theratioofanglesofatriangleis3:2:1andthelargestangleis90°.Findthedifferencebetweenthelargestandsmallestanglesa.60°b.90°c.45°d.75°e.Noneofthese25.TheratiobetweenthepresentagesofPunitandSureshis4:5respectively.After7yrPunit’sagewillbe31yr.WhatwasSuresh’sagefouryearsago?a.26yrb.24yrc.18yrd.16yre.Noneofthese26.Inatown,80%ofthepopulationsareadultsofwhichthemenandwomenareintheratioof9:7,respectively.Ifthenumberofadultwomenis4:2lakhs,whatisthetotalpopulationofthetown?a.24yrb.36yrc.48yrd.12yre.Noneofthese27.Mr.Shrimantinherits2505goldcoinsanddividesthemamongsthisthreesons.Bharat,ParatandMarat,inacertainratiooutofthetotalcoinsreceivedbyeachofthem,BharatSells30coins,Paratdonateshis30coinsandMaratlosseshis25coins.Now,theratioofgoldcoinswiththemis46:41:34respectively.HowmanycoinsdidParatreceivefromhisfather?a.800b.600c.850d.900e.Noneofthese28.Rs.600aredividedamongA,BandCsuchthatRs40morethan$

%thofA’sshare,Rs20more

than$gthofB’sshareandRs10morethan j

KgthofC’sshare,allareequal.WhatisB’sshare?

a.250b.300c.270d.280e.Noneofthese29.Perimeterofarectangleis40cmandthelengthandthebreadthareintheratioof3:2respectively.Whatistheareaofrectangleissqcm?a.84b.75c.96d.48e.Noneofthese30.Theexpensiveonrice,fishandoilofafamilyareintheratioof12:17:3.Thepricesofthesearticlesareincreasedby20%,30%and50%,respectively.Findthetotalpercentageincreaseintheexpensesofthefamilya.28K

e%b.28K

j%c.28%

e%d.28&

e%e.Noneofthese

31.SalaryofMr.Xis80%ofthesalaryofMr.YandthesalaryofMr.Zis120%ofthesalaryofMr.X.WhatistheratiobetweenthesalariesofX,Y,Zrespectively?a.33:14:29b.30:28:19c.16:38:19d.20:25:24e.Noneofthese32.Ifthepositionsofthedigitsofatwo-digitnumberareinterchanged,thenumbernewlyformedissmallerthantheoriginalnumberby45.Also,theratioofthenewnumbertotheoriginalnumberis3:8.Whatistheoriginalnumber?a.72b.75c.96d.48e.Noneofthese33.Theratioofwaterandspiritinthemixtureis1:3.Ifthevolumeofthesolutionisincreasedby25%byaddingspiritonly,whatistheresultantratioofwaterandspirit?a.1:3b.4:1c.1:4d.5:7e.Noneofthese

34.Acattakes5leapsforevery4leapsofadog,but3leapsofthedogareequalto4leapsofthecat.Whatistheratioofthespeedsofthecattothattothedog?a.15:34b.14:15c.15:16d.15:17e.Noneofthese35.InthemonthofJanuary,Arun’sincomeandexpenseswereRs.15000andRs9000respectively.Hismonthlyexpensesvarydirectlyasthesquareofhismonthlyincome.Whatishisincomewhenitjustequalshisexpenses?a.24000b.25000c.16000d.20000e.Noneofthese36.TwovesselsPandQcontainmilkandwatermixedintheratiosof5:3and2:3,respectively.Whenthesemixturesaremixedtoformanewmixturecontaininghalfmilkandhalfwater,thenfindtheratioinwhichPandQaremixed?a.1:3b.4:5c.1:4d.5:7e.Noneofthese37.Rs5625aredividedamongstA,BandCsothatAreceivesK

$asmuchasBandCtogether

receiveandBreceiveKiasmuchasAandCtogetherreceives.FindthesumofsharesofAandB

a.4000b.5000c.1000d.3000e.Noneofthese38.Rs710weredividingamongstA,BandCinsuchawaythatAhadRs40morethanBandChadRs30morethanA.HowmuchwasC’sshare?a.200b.250c.270d.300e.Noneofthese39.Theratioof1stand2ndclassestrainfairsbetweentwostationsis3:1andthatofthenumberofpassengerstravellingbetweenthesestationsby1stand2ndclassesis1:50.Ifonaparticularday,Rs2650becollectedfromthepassengerstravellingbetweenthesestations,thenfindtheamountcollectedfrom2ndclasspassengersa.2500b.1500c.1000d.2000e.Noneofthese40.TheratioofthenumberofstudentsstudyinginschoolsA,BandCis5:6:8.Ifthenumberofstudentsineachoftheschoolsisincreasedby30%,25%and25%,respectively,whatwillbethenewratioofthenumbersofstudentsinschoolsA,BandC?a.13:14:29b.13:15:20c.16:38:19d.20:25:24e.Noneofthese41.Inacollege,thenumberofstudentsstudyingHistory,EconomicsandMathematicsareintheratioof4:7:9.IfthenumberofstudentsaHistory,EconomicsandMathematicsbeincreasedby30%,20%and40%respectively,whatwillbethenewratio?a.13:14:29b.13:15:20c.16:38:19d.26:42:63e.Noneofthese42.Ruchi’sfatherwas38yroldwhenshewasbornwhilehermotherwas36yroldwhenherbrotherwasbornwhois4yryoungertoher.Whatisthedifferencebetweentheagesofherparents?a.4yrb.6yrc.8yrd.2yre.Noneofthese

43.Thedifferencebetweenthetwoadjacentanglesofaparallelogramis20°.Whatwouldbetheratiobetweenthesmallerandthelargeranglesoftheparallelogramrespectively?a.1:3b.4:5c.1:4d.5:7e.Noneofthese44.TherearetwotrianglesAandB.TheanglesoftriangleAareintheratioof3:4:5andtheanglesoftriangleBareintheratioof5:6:7.WhatisthedifferencebetweenlargestangleoftriangleAandthesmallestangleoftriangleB?a.25°b.20°c.40°d.15°e.Noneofthese45.Thecapacitiesoftwoglassesaresame.Theyarefilledwith$

&rdpartand$

%thpartofwater,

respectively.Remainingpartofboththeglassedarefilledwithmilk.Ifmixturesofboththeglassesaremixedinabigglass,whatistheratioofwaterandmilkinit?a.8:7b.4:5c.1:4d.5:7e.Noneofthese46.Rs.2186aredistributedamongstA,BandC.IfmoneygiventothemisdecreasedbyRs26,Rs28andRs32respectively,thentheyhavemoneyintheratioof9:13:8.WhatistheamountgiventoA?a.566b.665c.656d.156e.Noneofthese47.Apersongave$

%thpartofhisincometohiseldersonand30%parttohisyoungerson.He

savedhisremainingmoneyinthreetrustsA,BandCintheratioof3:5:2.IfdifferencebetweentheamountsgotbyhisbothsonsisRs2000.HowmuchamounthesavedintrustC?a.1800b.800c.1600d.1200e.Noneofthese48.Inafactory,theratioofthenumbersofemployeesofthreetypesA,BandCis9:13:18andtheirwagesareintheratioof10:7:4.IfnumberofemployeesoftypeCis54andwagesofeveryemployeesoftypeBisRs1400,thenfindthetotalwagesofalltheemployeesa.18000b.36000c.54000d.12000e.Noneofthese49.If(a+b):(b+c):(c+a)=6:7:8anda+b+c=14.FindCa.8b.6c.4d.2e.Noneofthese50.Numberofemployeesinafactorydecreasesintheratioof8:7andsalaryofemployeesincreasesintheratioof5:6.Findweatherthetotalsalarygiventotheemployeesisincreasedordecreasedandinwhatratio?a.21:20deacreasedb.20:21deacreasedc.20:21deacreasedd.20:21increasede.Noneofthese51.If^

&= p

%= q

g,thenfindthevalueof^PpPq

p

a.3b.6c.4d.2e.Noneofthese52.AprofitofRs.7,200istobedistributedamongA,BandCintheratioof3:2:1.ThedifferencebetweentheamountsreceivedbyAandCisa.1800b.3600c.5400d.2400e.Noneofthese

53.Theratioofthepricesofthreedifferenttypesofcarsis4:5:7.IfthedifferencetypesthepricesofcostliestandthecheapestcarisRs.6000,whatisthepriceofthecarofmoderateprice?a.1,00,000b.1,20,000c.2,40,000d.1,50,000e.Noneofthese54.AcertainamountwasdividedbetweenSalimandRahimintheratioof4:3.IfRahim’ssharewasRs.24000,findouttheamounta.18000b.56000c.54000d.24000e.Noneofthese55.karangottwiceasmanymarksinEnglishasinScience.HisTotalmarksinEnglish,ScienceandMathsare180.IftheproportionofhismarksinEnglishandMathsis2:3,whatarehismarksisscience?a.30b.60c.40d.20e.Noneofthese56.Anamountistobedividedamonga,BandCintheratio3:5:6respectively.IfthedifferencebetweenAandB’sshareisRs.600.WhatisC’sshare?a.1800b.4800c.1200d.2400e.Noneofthese57.A’sshareinabusinessis&

ithofB’sshare.BgotaprofileofRs.1,200byinvestingRs.40,000

inthebusiness.WhatwillbetheratioofA’sprofittohisinvestmentinthebusiness?a.18:23b.9:400c.400:9d.23:18e.Noneofthese58.Rs.4800isdistributedamongA,BandCintheratioof6:5:4.ThedifferencebetweensharesofAandCisa.600b.640c.300d.320e.Noneofthese59.Thepriceofascooterandamopedareintheratioof7:4.IfascootercostsRs.3600morethanamoped,whatisthepriceofamoped?a.4600b.4800c.3000d.3200e.Noneofthese60.AstartedwithRs.4500andBjoinedafterwardswithRs.3000.Iftheprofitsattheendofoneyearweredividedintheratio2:1respectively,thenBwouldbea.3monthsb.6monthsc.4monthsd.2monthse.Noneofthese61.Twonumbersareintheratio5:9withtheadditionof9theirratiobecomes16:27.Thensumofthetwonumberswouldbea.144b.99c.55d.199e.Noneofthese62.ThepriceofascooterandaT.Vsetareintheratioof3:2.IfthescootercostsRs.6,000morethantheT.V.set,whatisthepriceoftheT.V.set?a.18000b.48000c.12000d.24000e.Noneofthese63.AnamountofmoneyistobedistributedamongA,BandCintheratioof2:5:7.IfthetotalofA’sandB’sshareisRs.4900whatisthedifferencebetweenB’sandC’sshare?a.1400b.4800c.1200d.2400e.Noneofthese

64.AnamountistobedividedamongA,BandCintheratio3:5:6respectively.IfthedifferencebetweenAandB’sshareisRs.600.WhatisC’sshare?a.1400b.1800c.1200d.2400e.Noneofthese65.Findouttheratiowhosevalueis2/3andtheantecedentis18a.18:27b.9:400c.400:9d.23:18e.Noneofthese66.$

&: %h∷ eg:Whatnumber?

a.KKgb.KM

&c.KM

gd.K$

ge.Noneofthese

67.A:B=2:3,B:C=4:5,C:D=6:7findtheratioofAandD?a.16:27b.16:35c.35:16d.23:18e.Noneofthese68.Twoequalglassesarerespectively1/3and1/4fullofmilk.Theyarethenfilledupwithwaterandthecontentsaremixedinatumbler.Findtheratioofmilkandwaterinthetumblera.18:27b.9:400c.7:17d.23:18e.Noneofthese69.Ifin30litremixtureofmilkandwatertheratioofmilkandwateris7:3,findoutthequantityofwatertomixinordertomakethisratio3:7a.60litresb.15litresc.10litresd.40litrese.Noneofthese70.Inabigtherearecoinsof25paisa,10paisaand5paisaintheratio1:2:3.IfthereareinallRs.30,howmany5paisacoinsarethere?a.140b.150c.120d.240e.Noneofthese71.Shyamhasasisterwhoishalfofhisage.WhenShyamdouble,whatwillbetheratioofhisagetohissister’sage?a.4/3b.5/3c.3/5d.3/4e.Noneofthese72.Onacertainrailwaythefirstandsecondclassfaresare7paisaandepaisaperkilometerrespectively.Amanwhotravelled100kmspentRs.3.40ingoingpartofdistancebyfirstclassandtheinsecondclass.Howmanykilometersdidhetravelinfirstclass?a.14b.15c.12d.10e.Noneofthese73.RatioofA’sagetoB’sfirstequalto4:3.Awillbe26yearsoldafter6years,howoldisBnow?a.14yrsb.15yrsc.12yrsd.10yrse.Noneofthese74.25menwith10boyscandoin6daysasmuchworkas21menwith30boyscandoin5days.Howmanyboysmusthelp40mentodothesameworkin4days?a.14b.15c.12d.10e.Noneofthese75.Acontractortookacontractforbuilding12kilometerroadin15daysandemployed100labouresonthework.After9dayshefoundthatonly5kilometerroadhadbeenconstructed.

Howmanymorelabouresshouldbeemployedtoensurethattheworkmaybecompletedwithinthegiventime?a.144b.105c.120d.110e.Noneofthese76.Whatnumbershas5to1ratiotothenumber10?a.40b.50c.20d.10e.Noneofthese77.15mendoaworkin20days.Inhowmanydayswill20mendothefullwork?a.14b.15c.12d.10e.Noneofthese78.ThreepersonsstartbusinessandmakeprofitofRs.1180.IftheircapitalsareasK

%: Kh: Kehow

shouldtheprofitbedivided?a.480,400,350b.480,450,350c.480,400,300d.450,400,350e.Noneofthese79.A,BandCpurchasedthemangoesinaratio5:3:2.IfthedifferenceofmangoesofAandCis60,findoutthetotalnumberofmangoespurchasedbythema.400b.500c.200d.100e.Noneofthese80.AcompanymakesaprofitofRs.450.Outofthis20%ispaidfortaxesandtherestbedividedamongitspartnersA,BandCinproportionof1:1K

$: 2Findtheshareofeach

a.480,400,350b.80,120,160c.480,400,300d.450,400,350e.Noneofthese81.Agarrisonof2,200menhasprovisionfor16weeksattherateof45gmperdayperman.Howmanymenmustleavesothatthesameprovisionmaylastfor24weeksat33gmperdayperman?a.400b.500c.200d.100e.Noneofthese82.If5mencandoapieceofworkin20days,inhowmanydayswill10menand5boysdothesameworkif1mandoesasmuchworkas2boys?a.4b.5c.2d.8e.Noneofthese83.In21cowseatasmuchas15oxen,howmanycowswilleatasmuchas25oxen?a.34b.35c.32d.30e.Noneofthese84.Therearethreecontainersofequalcapacity.Firstcontainerishalffull,thesecondisonethirdfullandthethirdisempty.Ifallthewaterinthecontainersisdividedequallyamongthecontainers,whatpartofthethirdcontainerwillbefull?a.1/4b.5/8c.5/18d.18/5e.Noneofthese85.Theannualincomesoftwopersonsareintheratio9:7andtheirexpensesareintheratio4:3.IfeachofthemsavesRs.2000peryear,thentheirannualincomesarea.14000and18000b.15000and12000c.12000and15000

d.10000and15000e.Noneofthese86.ApersondividerRs.10,800amonghis3sonsintheratio3:4:5.SecondsonkeptRs.1000forhimself,gaveRs.600tohiswife,anddividedtheremainingmoneyamonghis2daughtersreceiveda.1400b.1500c.1200d.1100e.Noneofthese87.Inaschoolthenumberofboysandthatofthegirlsareintherespectiveratioof2:3.Ifthenumberofboysisincreasedby20%andthatofgirlsisincreasedby10%,whatwillbethenewratioofnumberofboystothatofthegirls?a.8:11b.9:400c.400:9d.23:18e.Noneofthese88.IncomeoftwocompaniesAandBareintheratioof5:8.HadtheincomeofcompanyAbeenmorebyRs.25lakhs,theratiooftheirincomeswouldhavebeen5:4respectively.WhatistheincomeofcompanyB?a.34lakhsb.35lakhsc.40lakhsd.34lakhse.Noneofthese89.Atotalof91boysareseatedinthreerows.Theratiobetweenthenumberofboysseatedinthefirstandthesecondrowis5:2respectivelyandtheratiobetweenthenumberofboysseatedinthesecondandthirdrowis1:3respectively.Howmanyboyswerethereinthesecondrow?a.14b.15c.12d.10e.Noneofthese90.TheratiooftheincomeofAandBasalsothoseofBandCis3:2IfathirdofA’sincomeexceedsafourthofC’sincomebyRs.1000,WhatisB’sincome?a.4000b.1000c.2000d.3000e.Noneofthese91.Asumofmoneyisdividedamong160malesandsomefemalesintheratio16:21.IndividuallyeachmalegetsRs.4andeachfemaleRs.3.Thenumberoffemalesisa.280b.150c.250d.270e.Noneofthese92.Themonthlyincomesoftwopersonsareintheratio4:7,andtheirexpensesareintheratio11:20.IfeachofthemsavesRs.400permonth,thentheirmonthlyincomemustberespectivelya.4800and4200b.4800and8400c.4800and54000d.10000and15000e.Noneofthese93.DivideRs.1250amongA,BandCsothatAgets2/9ofB’sshareandCgets3/4ofA’sshareFindA’s,B’sandC’ssharea.Rs.200,Rs.950,Rs.150b.Rs.200,Rs.900,Rs.150c.Rs.250,Rs.950,Rs.150d.Rs.200,Rs.950,Rs.100e.Noneofthese

94.Rs.180containedinaboxconsistsofone-rupee,50-paisaand25paisacoinsintheratio2:3:4.Whatisthenumberof50-paisecoins?a.280b.150c.250d.120e.Noneofthese95.Thestudentsinthreeclassesareintheratio2:3:5.If20studentsareincreasedineachclass,theratiochangesto4:5:7.Thetotalnumbersofstudentsbeforetheincreasewerea.100b.150c.250d.270e.Noneofthese96.If378coinsofonerupee,50-paisaand25-paisacoinswhosevaluesareintheratioof13:11:7,thenumberof50-paisacoinswillbea.280b.150c.132d.270e.Noneofthese97.Theratioofthenumberofboystothatofgirlsinaschoolis4:1.If75%ofboysand70%ofthegirlsarescholarship-holders,thenthepercentageofstudentswhodonotgetscholarshipis:a.28b.26c.25d.27e.Noneofthese98.AtotalprofitofRs.3,600istobedistributedamongstA,BandCsuchthatA:B::5:4andB:C::8:9.WhatistheshareofC?a.1200b.1500c.1800d.1400e.Noneofthese99.Theratioofnumberboys&girlsinaschoolof720studentsis7:5.Howmanymoregirlsshouldbeadmittedtomaketheratio1:1?a.100b.150c.120d.270e.Noneofthese100.AnnualincomesofA&Bareas4:3&theirexpensesareas3:2.IfeachsavesRs.600perannum,thenannualincomeofAisa.2400b.1500c.2500d.2700e.NoneoftheseAns:2,400101.Aman’scoatcostsseventimesasmuchhisshirt,butthebillforthetwotogetherwasRs.416.Thepriceoftheshirtisa.28b.52c.25d.26e.Noneofthese102.Thespeedsofthreecarsintheratio2:3:4.Theratiobetweenthetimestakenbythreecarstotravelthesamedistanceisa.3:4:6b.4:3:6c.6:4:3d.4:6:3e.Noneofthese103.Fournumbersintheratio1:3:4:7adduptogiveasumof105.Findthevalueofthebiggestnumbera.28b.49c.30d.14e.Noneofthese104.IftheratiooftheagesofMayaandChhayais6:5atpresent,and15yearsfromnow,theratiowillgetchangedto9:8,thenfindMaya’spresentagea.28yrsb.49yrsc.30yrsd.14yrse.Noneofthese

105.IfRs.58isdividedamong150childrensuchthateachgirlandeachboygets25pand50prespectively.Thenhowmanygirlsarethere?a.28b.68c.60d.62e.Noneofthese106.If391bananasweredistributedamongthreemonkeysintheratio1/2:2/3:3/4,howmanybananasdidthefirstmonkeyget?a.128b.149c.103d.104e.Noneofthese107.Iftwonumbersareintheratioof5:8andif9beaddedtoeach,theratiobecomes8:11.Findthelowernumbera.15b.49c.30d.14e.Noneofthese108.Whatnumbermustbetakenfromeachtermofthefraction$g

&%thatitbecome2:3?

a.28b.11c.30d.14e.Noneofthese109.Twentylitresofamixturecontainsmilkandwaterintheratioof5:3.Iffourlitresofthismixtureisreplacedbyfourlitresofmilk,thentheratioofmilktowaterinthenewmixturewillbea.3:7b.3:5c.7:3d.5:3e.Noneofthese110.TheratioofVimal’sageandAruna’sageis3:5andthesumoftheiragesis80years.Theratiooftheiragesafter10yearswillbea.1:2b.3:5c.7:3d.5:3e.Noneofthese111.Theaverageageof12studentsis20years.Iftheageofonemorestudentisincluded,theaveragedecreasesby1,whatistheageofthenewstudent?a.5b.9c.6d.7e.Noneofthese112.PushpaistwiceasoldasRitawas2yearsago.Ifthedifferenceoftheiragesbe2years,howoldisPushpatoday?a.10b.9c.6d.7e.Noneofthese113.JayeshisasmuchyoungertoAnilasheisoldertoPrashant.IfthesumoftheagesofAnilandPrasanthis48years,whatistheageofJayeshage?a.25b.29c.26d.27e.Noneofthese114.TheratiobetweenthepresentagesofMandNis5:3respectively.TheratiobetweenM’sage4yearsagoandN’sageafter4yearsis1:1.WhatistheratiobetweenM’sageafter4yearsandN’sage4yearsago?a.1:2b.3:5c.3:1d.5:3e.Noneofthese115.Whenthesonwillbeasoldasthefathertoday,theirageswilladdupto136years.Whenthefatherwasasoldasthesonistoday,theiragesaddedupto40years.Findthepresentagesofthesonandfatherinyears?a.32,57b.32,56c.15,35d.20,45e.Noneofthese

116.InaCricketEleven,theaverageageofelevenplayersis28years.Outofthese,theaverageagesofthreegroupsofthreeplayerseachare25years,28yearsand30yearsrespectively.Ifinthesegroups,thecaptainandtheyoungestplayerarenotincluded,andthecaptainiselevenyearsolderthantheyoungestplayer,whatistheageofthecaptain?a.25b.35c.26d.27e.Noneofthese117.Averageageofthestudentsinaclassof50is13.Theweightofeachisdirectlyproportionaltotheheightofeachandtheheightisalsofoundtobeindirectproportionwiththeage.Astudentofage11has165cmsofheight,withweightbeing33kgs.Findtheaverageweightoftheclassa.25b.29c.39d.27e.NoneoftheseDirectionsfor(Q.No118&119):Refertothedatabelowandanswerthequestionsthatfollow:TheaverageageofAandBis20years.IfCistoreplaceA,theaverageagewouldbe19andifheweretoreplaceB,theaveragewouldbe21.Also,DishalftheageofC118.AverageageofBandDisa.25b.14c.26d.15e.Noneofthese119.AverageageofA,B,CandDisa.15.5b.19.5c.16.5d.17.5e.Noneofthese120.6friendsA,B,C,D,EandFwhoseagesareinintegershavefollowingrelationbetweentheirages.B’sageis%

htimesthatofA,whereasageofCandDis j

KMtimesthatofB.F’sageare

betweenthatofAandBandEis&$timesoldasBand3yearsoldertoA.Whatistheaverage

weightofallthe6friends?a.13b.10c.12d.15e.Noneofthese121.ThesumoftheagesofthefourmembersoftheRathorefamilyis140years.FiveyearsagotheagesofthefourmembersInder,Sunder,Mrs.RathoreandMr.Rathorewereintheratioof2:3:7:8.AfterhowmanyyearswouldSunderbeasoldashismotheristoday?a.25b.24c.26d.27e.Noneofthese122.ManyyearsagowhenMr.WaughwasaskedastowhoamongSteveandMarkiselder,hesaid,“IntwoyearsfromnowStevewillbetwiceasoldashewasthreeyearsagoandinthreeyearsfromnowMarkwillbethreetimesasoldashewasthreeyearsago.”HowoldisMarktoday,ifMr.Waughwasaskedthequestion27yearsbackandwhoisolderMarkorSteve?a.25yrs,markisolderb.33yrs,markisolderc.32yrs,markisolderd.33yrs,bothareofthesameagee.NoneoftheseDirectionsfor(Q.No123-127):Thesequestionsarebasedonthefollowingdata.Iwasvisitingmyfriend’shouseatTuticorinforthefirsttime.Iknewthathehasfivechildren,twodaughtersandthreesons.ThedaughterswereSonaliandMonaliandthesonswereSurya,SagarandPrithvi.WhenIsteppedintothehouseIwasgreetedbySagarandSonali.Sagarintroduced

himselfandtoldthathewastwiceasoldasSonali.ThenexttomeetmewasMonaliwhosaidthatsheandSonalitogetherweretwiceasoldasSagar.HoweverSagarfeltrelievedwhenSuryawalkedinandintroducedhimselfandsaidthatheandSagarweretwiceasoldastheirsisterstogether.IntheafternoonPrithvicametomyroomwhenwasrelaxingandsaid“SorryUncle,Iwasbusywithmyfriends,sinceitismy21stbirthdaytodayandhencecouldnotmeetyouinthemorning.”WhenIgreetedhimandsaidthatitwasfine,helookedrelievedandsaid,“Youknowuncle,thatthethreeofusSugar,Suryaandmearefivetimesasoldasoursisterstogether.”Sayingthishewentaway.Canyouhelpmefindoutthesolutionforthefollowingquestions?123.WhatistheageofSurya?a.13years10monthsb.10years6monthsc.12yrsd.15yrs6monthse.Noneofthese124.WhatistheageofMonali?a.5years10monthsb.5years3monthsc.12yrsd.5yrs6monthse.Noneofthese125.Whoistheyoungestofthesiblings?a.sonalib.sagarc.suryad.prithvie.Noneofthese126.HowmanyyearsolderthanSonaliisMonali?a.3years6monthsb.3years3monthsc.2yrsd.5yrs6monthse.Noneofthese127.AfterhowmanyyearswouldSuryabetwiceasoldasSagar?a.3years6monthsb.3years3monthsc.2yrsd.5yrs6monthse.Noneofthese128.Averageageofafatherandhistwosonsis27years.Fiveyearsago,theaverageofageofthetwosonswas12years.Ifthedifferencebetweentheagesofthetwosonsis4years,thenthepresentageofthefatherisa.25b.24c.26d.47e.Noneofthese129.Theaverageageof30boysinaclassis15years.Oneboy,aged20years,lefttheclass,buttwonewboyscameinhisplacewhoseagesdifferby5years.Iftheaverageageofalltheboysnowintheclassremains15years,theageoftheyoungernewcomerisa.15b.24c.26d.27e.Noneofthese130.Afathersaidtohisson“Iwasasoldasyouareatpresentatthetimeofyourbirth”.Ifthefather’sageis38yearsnow,theson’sagefiveyearsbackwasa.15b.14c.26d.27e.Noneofthese131.Theaverageageof8personsisincreasedby2years,whenoneofthem,whoseageis24yearsisreplacedbyanewperson.Theageofthenewpersonis

a.25b.24c.26d.27e.Noneofthese132.Harshais40yearsoldandRithis60yearsold.Howmanyyearsagowastheratiooftheirages3:5?a.20b.24c.10d.27e.Noneofthese133.Theaverageageof8menisincreasedby2yearswhentwoofthemwhoseagesare21and23yearsarereplacedbytwonewmen.Theaverageageofthetwonewmenisa.25b.24c.26d.27e.Noneofthese134.Mygrandfatherwas8timesolderthanme16yearsago.Hewouldbe3timesmyage,8yearsfromnow.Eightyearsago,whatwastheratioofmyagetothatofmygrandfather?a.1:2b.1:5c.7:3d.5:3e.Noneofthese135.Agesof‘A’and‘B’areintheratioof2:3respectively.Sixyearshencetheratiooftheirageswillbecome8:11respectively.WhatisB’spresentage?a.20b.24c.10d.27e.Noneofthese136.JayeshistwiceasoldasVijayandhalfasoldasSuresh.IfsumofSuresh’sandVijay’sagesis85yearswhatisJayesh’sageinyears?a.20b.34c.10d.27e.Noneofthese137.SixyearsagoJagannathwastwiceasoldasBadriiftheratiooftheirpresentageis9:5respectively,whatisthedifferencebetweentheirpresentages?a.20b.24c.10d.27e.Noneofthese138.JayeshisasmuchyoungerthanAnilasheisolderthanPrasanth.IfthesumoftheagesofAnilandPrasanthis48years,whatisJayesh’sageinyears?a.20b.24c.10d.27e.Noneofthese139.Theaverageageof30studentsinaclassis15years.If6studentsofthisclasshavetheaverageageof16years,thentheaverageoftheremaining24studentswouldbea.13years10monthsb.10years6monthsc.14yrs9monthsd.15yrs6monthse.Noneofthese140.Thesumoftheagesoftwomenis85.Fiveyearsagothedifferencebetweentheirageswas1/9ofthepresentageoftheelder.Theelder’spresentageisa.45b.24c.10d.27e.Noneofthese141.Theaverageageof10membersofacommitteeisthesameasitwas4yearsago,becauseanoldmemberhasbeenreplacedbyayoungmember.Findhowmuchyoungeristhenewmemberthantheoldmembera.20b.24c.10d.40e.Noneofthese

142.Fatheris5yearsolderthanthemotherandthemother’sageisnowthricethedaughter’sage.Ifthedaughteris10yearsoldnow,whatwasfather’sage,whenthedaughterwasborn?a.25b.24c.10d.27e.Noneofthese143.TheageofAbdulisfivetimesofhisdaughterand5yearsmorethanhiswife.Iftheageofhisdaughteris7years,theageofhiswifeisa.20b.30c.10d.27e.Noneofthese144.10yearsago,theaverageageoffamilyof3memberswas19years.2childrenSonuandMonuhavingbeenborn,averageageofthefamilyissametodate.WhatisthepresentageoftheyoungestmemberofthefamilyassuringthatSonuandMonu’sagesdifferby2years?a.2yrsb.4yrsc.1yrd.3yrse.Noneofthese145.PresentageofAnandis47yearsandhiswife’sageisjust38years.Whentheyweremarried,Anandwasexactlyoneandhalfasoldashiswife.Howmanyyearsagotheyweremarried?a.20b.24c.10d.27e.Noneofthese146.AsaidtoB‘IamtwiceasoldasyouwerewhenIwasoldasyouarenow.”Sumoftheiragesis42.Findtheirpresentagesa.20,24b.24,18c.10,15d.27,35e.Noneofthese147.Mr.SinghhavefivechildrenDolly,Polly,Molly,SollyandLolly.Dollywastheeldestandthechildrenwereborninthesamesequencesuchthattheagedifferencebetweenanytwoconsecutivechildrenisthesame.IfDollyis14yearsold,whatarethepossibleagesofMolly,ifthedifferenceintheagesofthechildrenhasintegralvalues?I.13yearsII.2yearsIII.11yearsa.onlyIb.onlyIIc.onlyIIId.I&IIe.Noneofthese148.Thecaptainofacricketteamof11playersis25yearsoldandthewicketkeeperis3yearsolder.Iftheagesofthesetwoareexcluded,theaverageageoftheremainingplayersis1yearlessthantheaverageofthewholeteam?a.20b.24c.10d.27e.Noneofthese149.Thedifferenceintheage,ofamotherandhersonis24years.Fouryearshencethemother’sagewillbe4timesthatoftheson’sage.Whatisthemother’sage?a.32b.36c.40d.28e.Noneofthese150.Supriyais25yearsyoungerthanhermothertoday.Tenyearsago,Supriyawasone-sixthasoldashermother.HowoldisSupriya’smothernow?a.32b.36c.40d.28e.Noneofthese151.Tenyearsagotheaverageageofafamilyof4memberswas24years.Threechildren

havingbeenborn,theaverageageofthefamilyissametoday.Whatarethepresentagesofchildren,iftwochildrenareidenticaltwinsanddifferbytwoyearsfromtheelderone?a.32,20,20b.36,30,30c.12,10,10d.12,40,40e.Noneofthese152.UtpalandNikilundertookacontract,investingRs.7000andRs.8500respectively.TheyearnedaprofitofRs.9300inthebusiness.WhatwastheshareofNikhil?a.3200b.3600c.4000d.2800e.Noneofthese153.AandBstartedabusinessinvestingRs.20,000andRs.30,000respectively.Awithdrewhiscapitalafter6months.AttheendoftheyearAreceivedRs.2000ashisshareintheprofit.HowmuchwasB’sprofit?a.3200b.3600c.4000d.2800e.Noneofthese154.Ajay,KeshavandVijayinvestedRs.8000,Rs.4000andRs.8,000respectivelyinabusiness.Ajayleftaftersixmonths.Ifafter8monthstherewasaprofitofRs.4005,thenwhatwillbetheshareofKeshav?a.890b.360c.400d.280e.Noneofthese155.RameshstartedabusinessinvestingRs.30,000.Sixmonthslater,YogeshjoinedhiminvestingRs.15,000.IftheymaketheprofitofRs.10,000attheendoftheyear,howmuchshouldtheshareofRameshbe?a.3200b.3600c.4000d.8000e.Noneofthese156.Inafactorytheratioofmaleworkerstofemaleworkerswas5:3.Ifthenumberoffemaleworkerswaslessby40,whatwasthetotalnumberofworkersinthefactory?a.320b.360c.160d.180e.Noneofthese157.Dilip,RamandAvtarstartedashopbyinvestingRs.27000,Rs.81,000andRs.72,000respectively.Attheendofoneyear,theprofitwasdistributed.IfRam’ssharewasRs.36000,whatwasthetotalprofit?a.32000b.36000c.40000d.80000e.Noneofthese158.AnujinvestedRs.80,000andstartedabusiness.Afterfourmonths,GopaljoinedhimwhatwithacapitalofRs.90,000.Ifattheendoftheyear,thetotalprofitearnedbythemwasRs.14,000.WhatwasGopal’sshareinit?a.3200b.6000c.4000d.8000e.Noneofthese159.AstartedabusinessinvestingRs.18,000.AfterthreemonthsBjoinedhiminvestingRs.27,000.Ifattheendoftheyear,thetotalprofitisRs.3,400,whatwillbeA’sshareinit?a.3200b.1600c.4000d.2800e.Noneofthese160.JeewanstartedabusinessinvestingRs.50,000.4monthsafterhestarted.AniljoinedhimwithacapitalofRs.90,000.Ifattheendoftheyear,thetotalprofitinthebusinessisRs.22,000.WhatwouldbeAnil’sshareit?a.12000b.36000c.40000d.80000e.Noneofthese

161.AmalstartedabusinessinvestingRs.20,000.Threemonthslater,KapiljoinedhiminvestingRs.30,000.IftheymakeaprofitofRs.6,800attheendoftheyear,whatwouldbekapil’sshare?a.3600b.6000c.4000d.8000e.Noneofthese162.SubodhstartedabusinessinvestingRs.45000.After4monthsNepaljoinedhiminvestingRs.30,000.IfthetotalprofitearnedbythemattheendoftheyearisRs.13,000.Whatwillbethedifferencebetweentheirshares?a.5000b.6000c.4000d.8000e.Noneofthese163.AstartedwithRs.4500andBjoinedafterwardswithRs.3000.Iftheprofitsattheendofoneyearweredividedintheratio2:1respectively,thenBwouldhavejoinedAforbusinessaftera.3monthsb.6monthsc.4monthsd.8monthse.Noneofthese164.SrikanthandVividhstartedabusinessinvestingamountsofRs185000andRs225000,respectively.IfVividh’sshareintheprofitearnedbythemisRs9000,whatisthetotalprofitearnedbythemtogether?a.25300b.15400c.16400d.8000e.Noneofthese165.AstartsabusinesswithRs.9000andBjoinshimafter6monthswithaninvestmentofRs.45000.WhatwillbetheratiooftheprofitsofAandBattheendofyear?a.1:2b.2:5c.7:3d.5:3e.Noneofthese166.RajanandSajanstartedabusinessinitiallywithRs.14200andRs.15600respectively.IftotalprofitsattheendoftheyearareRs.74500,whatistheRajan’sshareintheprofit?a.25300b.15400c.16400d.35500e.Noneofthese167.Ram,ShyamandKamaltogetherstartabusinessinpartnership.Theratiooftheircapitalsis3:4:7.IftheirannualprofitbeRs.21000,whatwillbekamal’sshareinthisprofit?a.25300b.15400c.16400d.10500e.Noneofthese168.Reena,ReemaandMeeratakeacaronrentandtheyusethatcarfor14h,16hand22h,respectively.WhatamountwillbepaidbyMeera,iftotalrentofthecatisRs.2080?a.300b.840c.820d.880e.Noneofthese169.Pinku,RinkuandTinkudivideanamountofRs.4,200amongstthemselvesintheratioof7:8:6respectively.IfanamountofRs.200isaddedtoeachoftheirshares,whatwillbethenewrespectiveratiooftheirsharesofamount?a.7:8:9b.8:9:7c.9:8:7d.9:7:8e.Noneofthese170.RameshandPriyastartedabusinessinitiallywithRs.5100andRs.6600,respectivelyinvestmentsdonebyboththepersonsarefordifferenttimeperiods.IfthetotalprofitisRs.5460,whatistheprofitofRamesh?

a.2530b.1540c.1640d.datainadequatee.Noneofthese171.A,BandCtogetherstartabusiness.Binvests1/6ofthetotalcapitalwhileinvestmentsofAandCareequal.IftheannualprofitonthisinvestmentisRs.33600,findthedifferencebetweentheprofitsofBandCa.2530b.8000c.8200d.8400e.Noneofthese172.AvinashinvestedanamountofRs.25,000andstartedabusiness.JitendrajoinedhimafteroneyearwithanamountofRs.30,000.AftertwoyearsfromstartingthebusinesstheyearnedtheprofitofRs,46,000.WhatwillbeJitendra’sshareintheprofit?a.25300b.15400c.16400d.17250e.Noneofthese173.Mr.NileshAgarwalopenedaworkshopinvestingRs.40,000.HeinvestedadditionalamountofRs.10,000everyyear.AftertwoyearshisbrotherSureshjoinedhimwithanamountofRs.85,000.ThereafterSureshdidnotinvestanyadditionalamount.OncompletionoffouryearsfromtheopeningofworkshoptheyearnedanamountofRs.1,95,000.WhatwillbeNilesh’sshareintheearning?a.120000b.100000c.110000d.150000e.Noneofthese174.AinvestsRs.3000foroneyearinabusiness.HowmanyBshouldinvestinorderthattheprofitafter1yearmaybedividedintoratioof2:3?a.2530b.8000c.4500d.8400e.Noneofthese175.ThreefriendsA,BandCstartedabusinessbyinvestingamountintheratioof5:7:6respectively.AfteraperiodofsixmonthsCwithdrewhalfoftheamountinvestedbyhim.IftheamountinvestedbyAisRs.40,000andthetotalprofitearnedattheendofoneyearisRs.33,000.WhatisC’sshareinprofit?a.9000b.8000c.8200d.8400e.Noneofthese176.Mr.ShivkumarstartedabusinessinvestingRs.25,000in1996.In1997heinvestedanadditionalamountofRs.10,000andMr.RakeshjoinedhimwithanamountofRs.35,000.In1998Mr.ShivkumarinvestedanotheradditionalamountofRs.10,000andMr.SureshjoinedthemwithanamountofRs.35,000.WhatwillbeRakesh’sshareintheprofitofRs.1,50,000earnedattheendofthreeyearsfromthestartofthebusinessin1996?a.90000b.80000c.50000d.84000e.Noneofthese177.AandBstartedbusinesswithRs.3000andRs.4000respectively.After8monthsawithdrawRs.1000andBadvancesRs.1000more.Attheendoftheyear,theirprofitsamountedtoRs.630.ThentheshareofAisa.200b.220c.240d.300e.Noneofthese178.AandBstartedabusinessjointly.A’sinvestmentwasthricetheinvestmentofBandtheperiodofhisinvestmentwastwotimestheperiodofinvestmentwastwotimestheperiodofinvestmentofB.IfBreceivedRs.4000asprofit,thentheirtotalprofitisa.29000b.28000c.28200d.28400e.Noneofthese

179.AandBarepartnersinabusiness.AcontributesK

iofthecapitalfor15monthsandB

received$&oftheprofit.FindhowlongB’smoneywasused?

a.3monthsb.6monthsc.4monthsd.8monthse.Noneofthese180.A,BandCarepartnersofacompany.DuringaparticularyearAreceivedone-thirdoftheprofit,Breceivedone-fourthoftheprofitandCreceivedtheremainingRs.5,000.HowmuchdidAreceived?a.4000b.8000c.8200d.8400e.Noneofthese181.AbeganabusinesswithRs.4500andwasjoinedafterwardsbyBwithRs.5400.WhendidBjoiniftheprofitsattheendoftheyearweredividedintheratio2:1?a.after3monthsb.after6monthsc.after4monthsd.after8monthse.Noneofthese182.AandBenteredintoapartnershipinvestingRs.16,000andRs.12,000respectively.After3months,AwithdrewRs.5000whileBinvestedRs.5000more.After3moremonths,CjoinsthebusinesswithacapitalofRs.21,000.TheshareofBexceedsthatofC,outofatotalprofitofRs.26,400afteroneyearbya.3600b.3800c.3000d.8400e.Noneofthese183.AandBstartedabusinessjointly.A’sinvestmentwasthricetheinvestmentofBandtheperiodofhisinvestmentwastwotimestheperiodofinvestmentofB.IfBreceivedRs.4000asprofit,thentheirtotalprofitisa.29000b.28000c.28200d.28400e.Noneofthese184.Rs.700isdividedamongA,BandCsothatAreceiveshalfasmuchasBandBhalfasmuchasC.ThenC’sshareisa.200b.220c.400d.300e.Noneofthese185.ManojreceivedRs.6000ashisshareoutofthetotalprofitofRs.9000whichheandRameshearnedattheendofoneyear.IfManojinvestedRs.20,000for6months,whereasRameshinvestedhisamountforthewholeyear,whatwastheamountinvestedbyRamesh?a.3600b.5000c.3000d.4000e.Noneofthese186.YogeshstartedabusinessinvestingRs.45,000.After3months,PranabjoinedhimwithacapitalofRs.60,000.Afteranother6months,AtuljoinedthemwithacapitalofRs.90,000.Attheendoftheyear,theymadeaprofitofRs.20,000.WhatwouldbeAtul’sshateit?a.3600b.5000c.3000d.4000e.Noneofthese187.NirmalandKapilstartedabusinessinvestingRs.9000andRs.12000respectively.After6monthsKapilwithdrewhalfofhisinvestment.Ifafterayear,thetotalprofitwasRs.4,600,whatwasKapil’sshareinit?a.3600b.2300c.3000d.2400e.Noneofthese

188.JayantopenedashopinvestingRs.30,000.Madhujoinedhim2monthslater,investingRs.45,000.TheyearnedaprofitofRs.54,000aftercompletionofoneyear.WhatwillbebeMadhu’sshareofprofit?a.29000b.28000c.30000d.28400e.Noneofthese189.A,B,CentersintoapartnershipinvestingRs.35,000,Rs.45,000andRs.55,000respectively.TherespectivelysharesofA,B,CinanannualprofitofRs.40,500area.Rs.10,500,Rs.13,500,Rs.16,500b.Rs.10,500,Rs.13,500,Rs.16,000c.Rs.10,500,Rs.12,500,Rs.16,500d.Rs.10,000,Rs.13,500,Rs.16,500e.Noneofthese190.Arun,KamalandVinayinvestedRs.8000,Rs.4000andRs.8000respectivelyinabusiness.Arunleftaftersixmonths,Ifaftereightmonths,therewasagainofRs.4005,thenwillbetheshareofKamal?a.800b.890c.850d.820e.Noneofthese191.TwofriendsPandQstartedabusinessinvestingintheratioof5:6.RjoinedthemaftersixmonthsinvestinganamountequaltothatofQ’s.Attheendoftheyear20%profitwasearnedwhichwasequaltoRs.98,000.WhatwastheamountinvestedbyR?a.220000b.210000c.240000d.230000e.Noneofthese192.InabusinessAandCinvestedamountsintheratio2:1whereastheratiobetweenamountsinvestedbyAandBwas3:2.IfRs.1,57,300wastheirprofit,howmuchamountdidBreceive?a.48000b.44000c.42000d.48400e.Noneofthese193.AandBstartedabusinessinpartnershipinvestingRs.20,000andRs.15,000respectively.Aftersixmonths,CjoinedthemwithRs.20,000.WhatwillbeB’sshareinthetotalprofitofRs.25,000earnedattheendof2yearsfromthestartingofthebusiness?a.7500b.6500c.8000d.9000e.Noneofthese194.A,BandCrentapasture.Aputs10oxenfor7months,Bputs12oxenfor5monthsandCputs15oxenfor3monthsforgrazing.IftherentofthepastureisRs.175,howmuchCpaysashisshareofrent?a.25b.35c.45d.50e.Noneofthese195.AandBstartedapartnershipbusinessinvestingsomeamountintheratioof3:5.CjoinedthemaftersixmonthswithanamountequaltothatofB.InwhatproportionshouldtheprofitattheendofoneyearbedistributedamongA,BandC?a.5:10:6b.10:6:5c.6:10:5d.5:6:10e.Noneofthese196.Mr.NileshAgarwalopenedaworkshopinvestingRs.40,000.HeinvestedadditionalamountofRs.10,000everyyear.AftertwoyearshisbrotherSureshjoinedhimwithanamount

ofRs.85,000.ThereafterSureshdidnotinvestanyadditionalamount.OncompletionoffouryearsfromtheopeningofworkshoptheyearnedanamountofRs.1,95,000.WhatwillbeNilesh’sshareintheearning?a.120000b.110000c.140000d.130000e.Noneofthese197.AinvestsRs.3000foroneyearinabusiness.HowmuchBshouldinvestinorderthattheprofitaftermaybedividedintoratioof2:3?a.7500b.4500c.5000d.4000e.Noneofthese198.ThreefriendsA,BandCstartedabusinessbyinvestingamountintheratioof5:7:6respectively.AfteraperiodofsixmonthsCwithdrewhalfoftheamountinvestedbyhim.IftheamountinvestedbyAisRs.40,000andthetotalprofitearnedattheendofoneyearisRs.33,000,whatisC’sshareinprofit?a.7500b.4500c.9900d.4000e.Noneofthese199.Mr.ShivkumarstartedabusinessinvestingRs.25,000in1996.In1997heinvestedanadditionalamountofRs.10,000andMr.RakeshjoinedhimwithanamoutofRs.35,000.In1998Mr.SureshjoinedthemwithanamountofRs.35,000.WhatwillbeRakesh’sshareintheprofitofRs.1,50,000earnedattheendofthreeyearsfromthestartofthebusinessin1996?a.48000b.50000c.45000d.55000e.Noneofthese200.AandBstartedbusinesswithRs.3000andRs.4000respectively.After8monthsAwithdrawsRs.1000andBadvancesRs.1000more.Attheendoftheyear,theirprofitsamountedtoRs.630.ThentheshareofAisa.480b.440c.420d.240e.Noneofthese

Answers1.C 2.D 3.D 4.B 5.C 6.A 7.D 8.B 9.A 10.A11.B 12.B 13.A 14.C 15.A. 16.B 17.B 18.A 19.A 20.A21.D 22.A 23.A 24.A 25.A 26.D 27.C 28.D 29.C 30.A31.D 32.A 33.C 34.C 35.B 36.B 37.D 38.C 39.A 40.B41.D 42.B 43.B 44.A 45.A 46.C 47.D 48.C 49.B 50.D51.A 52.D 53.A 54.B 55.A 56.A 57.B 58.B 59.B 60.A61.A 62.C 63.A 64.B 65.A 66.C 67.B 68.C 69.D 70.B71.A 72.D 73.B 74.D 75.D 76.B 77.B 78.C 79.C 80.B81.C 82.D 83.B 84.C 85.E 86.D 87.A 88.C 89.A 90.D91.A 92.B 93.B 94.D 95.A 96.C 97.B 98.A 99.C 100.A101.B 102.C 103.B 104.C 105.B 106.E 107.A 108.B 109.C 110.A111.D 112.A 113.E 114.C 115.B 116.B 117.C 118.B 119.D 120.C121.B 122.D 123.B 124.B 125.A 126.A 127.A 128.D 129.A 130.B131.E 132.C 133.E 134.B 135.D 136.B 137.B 138.B 139.C 140.A141.D 142.A 143.B 144.D 145.A 146.B 147.B 148.E 149.D 150.C151.C 152.E 153.E 154.A 155.D 156.C 157.D 158.B 159.B 160.A161.A 162.A 163.A 164.C 165.B 166.D 167.D 168.D 169.B 170.D171.D 172.D 173.C 174.C 175.A 176.C 177.C 178.B 179.E 180.A181.E 182.A 183.B 184.C 185.B 186.D 187.B 188.C 189.A 190.B191.B 192.D 193.A 194.C 195.C 196.B 197.B 198.E 199.B 200.D

AverageWheneverweareaskedthemarksscoredbyusinanyexamination,weusuallytellthemarksinpercentage,takingthepercentageoftotalmarksofallsubjects.Thispercentageiscalledaveragepercentage.Generallyspeakingaveragescanbebroadlyclassifiedintotwocategories:1.MathematicalAverages2.Averagesofposition1.MathematicalAveragesArithmeticaverageormean:Theaverageormeanorarithmeticmeanofanumberofquantitiesofthesamekindisequaltotheirsumdividedbythenumberofthosequantities.

Ex:Theaverageof3,10,11,17,and24is Geometricmean:

Geometricmeanof Geometricmeanisusefulincalculatingaveragesofratiossuchasaveragepopulationgrowthrate,averagepercentageincreaseetc.Harmonicmean:

Harmonicmeansof isdenotedby Harmonicmeanisusefulforfindingoutaveragespeedofavehicle,averageproductionperdayetc.2.AveragesofpositionMedian:IfwearrangeNnumbersinorder,increasingordecreasing,themedianisthemiddlenumberifNisodd;andtheaverageofthetwomiddleifNiseven.Mode:Themodeofasetofnumbersisthatvaluewhichoccurswiththegreatestfrequency.Inotherwords,itisthemostcommonvalue.Ingeneral,thenmodemaynotexistorisitexistsitmaynotbeunique.Formulae

1.Average= 2.Supposeamancoversacertaindistanceat kmphandanequaldistanceat kmph.Then,

theaveragespeedduringthewholejourneyis

3.Sumoffirstnnaturalnumber1+2+3+........+n=n(n+1)/24.Sumofsquareoffirstnnaturalnumber12+22+32+42+.....+n2=n(n+1)(2n+1)/65.Sumofcubeoffirstnnaturalnumber13+23+33+.....+n3=[n(n+1)/2]26.Sumoffirstnevennumber2+4+6+.......+2n=n(n+1)7.Sumoffirstnoddnumber1+3+5+7+.......+(2n–1)=n2SomepropertiesofAverage:i)Ifeachobservationisincreasedbya,theaverageoftheseobservationswillalsoincreasebya.ii)Ifeachobservationisdecreasedbya,theaverageoftheseobservationswillalsodecreasebya.iii)Ifeachobservationismultipliedbya,theaverageoftheseobservationswillalsomultiplybya.iv)Ifeachobservationisdividedbya,theaverageoftheseobservationswillalsodividedbya.v)Ifthereare‘n’consecutiveoddnumbersandwewanttofindthedifferencebetweenhighestandlowestnumberthenitwillbe(n-1)×2vi)Ifthereare‘n’consecutiveevennumbersandwewanttofindthedifferencebetweenhighestandlowestnumberthenitwillbe(n-1)×2TheaveragescoreofthreestudentsA,B,andCis50.WhenthescoreofanotherstudentDisaddedtothegroup,theaveragescorebecome47.WhatisthescoreofstudentD?Answer:formostofyou,thescoreofstudentDwouldbe4×47-3×50=38.Forme,thecalculationwouldjustbe47-9=38.SomeofyoumighthaveunderstoodwhatIdid.Letmestartexplainingthroughasimpleexample.Thenweshallextendourexplorationstomorecomplexproblems.

Thetwoimagesaboveshowtwobeambalances,onewithequalarmsandtheotherwiththearmslengthsintheratio1:3.Thedottedlineinbothcasesshowstheaveragevalue.Inthefirstbeambalance,ifyoumoveanyofthepanoneunittowardstheaveragevalue,theotherpanwouldalsomoveoneunitinthedirectionoftheaveragevaluetokeeptheaverageconstant.Forexample,lettheweightsinthetwopansbe50kgand60kg.Theaverageis55kg.Ifyouincreasetheweightin50kgpanbyoneunit(i.e.51)bringingitnearertotheaverage,youwillhavetodecreasetheweightintheotherpanbyoneunit(i.e.59),bringingitoneunit

nearertotheaverage,tokeeptheaverageconstant.Inthesecondbeambalance,ifthearmoflength3moves1unittowardstheaverage,thearmoflength1willhavetomove3unitstowardstheaveragetokeeptheaverageconstant.Forexample,lettheweightinpanof3unitarmlengthbe40kgandtheweightinthepanof1unitarmlengthbe60kg.NowifIincreasetheweightinthefirstpanby1unit,Ishallhavetodecreasetheweightinthesecondpanby3unitstokeeptheaverageconstant.ThesameruleappliesifIammovingapanawayfromtheaverage.Now,understandthis-thesecondbeambalancecanalsoberepresentedwithabeambalanceofequalarmsbutonearmhavingthricetheweightontheotherarm.Savvy?Nowlet’sseetheproblemonceagain.Nowtherearethreeweightsononearm.Theaverageofthefourweightsis47.Tomoveeachofthethreeweight3unitsawayfromtheaverage(50-47)IshallhavetomovetheweightD3×3nineunitsawayfromtheaverage.Therefore,weightD=47-9=38.Q)Fourfriendshaveanaverageweightof68.IfRahimisalsoincludedinthegroup,theaverageweightbecomes72.whatisRahim’sweight?Answer:Sameprocess:therearefourpeopleatadistanceof4unitsfromtheaverageweightof72.Tobalancethem,wewillhavetoplaceapersonat4×4=16unitsfrom72ontheotherside.Therefore,Rahim’sweight=72+16=88.Q)Abatsmaninhis20thinningsmakesascoreof93andtherebyincreaseshisaverageby3.Whatistheaverageafter20innings?Answer:IfyouhaveunderstoodwhatIhavesaidsofar,thenewaverageisnothingbut93-57=36.LetthenewaveragebeA.Therefore,thereare19scoresatadistanceof3unitsfromA.Tobalancethesescoresweneedonescore(whichis93)19×3=57unitsfromAontheotherside.ThereforeA=93-57=36.Ex.1:Findtheaverageofallprimenumbersbetween30and50?Sol:Therearefiveprimenumbersbetween30and50.Theyare31,37,41,43and47.Thereforetherequiredaverage=(31+37+41+43+47)/5=199/5=39.8Ex.2.Findtheaverageoffirst40naturalnumbers?Sol:Sumoffirstnnaturalnumbers=n(n+1)/2;So,sumof40naturalnumbers=(40×41)/2=820.Thereforetherequiredaverage=(820/40)=20.5Ex.3.Findtheaverageoffirst20multiplesof7?Sol:Requiredaverage=7(1+2+3+…….+20)/20or(7×20×21)/(20×2)=(147/2)=73.5

Ex.4.Theaverageoffourconsecutiveevennumbersis27.findthelargestofthesenumbers?Sol:Letthenumbersbex,x+2,x+4andx+6.then,[{x+(x+2)+(x+4)+(x+6)}/4]=27or(4x+12)/4=27orx+3=27orx=24.Thereforethelargestnumber=(x+6)=24+6=30.Ex.5.TherearetwosectionsAandBofaclassconsistingof36and44studentsrespectively.IftheaverageweightofsectionAis40kgandthatofsectionBis35kg,findtheaverageweightofthewholeclass?Sol:Totalweightof(36+44)students=(36×40+44×35)kg=2980kg.Thereforeweightofthetotalclass=(2980/80)kg=37.25kg.Ex:6.Ninepersonswenttoahotelfortakingtheirmeals8ofthemspentRs.12eachontheirmealsandtheninthspentRs.8morethantheaverageexpenditureofallthenine.Whatwasthetotalmoneyspentbythem?Sol:LettheaverageexpenditureofallninebeRs.xThen12×8+(x+8)=9xor8x=104orx=13.Totalmoneyspent=9x=Rs.(9×13)=Rs.117.Ex.7:Ofthethreenumbers,secondistwicethefirstandisalsothricethethird.Iftheaverageofthethreenumbersis44.Findthelargestnumber.Sol:Letthethirdnumberbex.Thensecondnumber=3x.Firstnumber=3x/2.Thereforex+3x+(3x/2)=(44×3)orx=24Solargestnumber=2ndnumber=3x=72.Ex.8:Theaverageof25resultsis18.Theaverageoffirst12ofthemis14&thatoflast12is17.Findthe13thresult.Sol:Clearly13thresult=(sumof25results)–(sumof24results)=(18×25)–(14×12)+(17×12)=450–(168+204)=450–372=78.Ex.9:TheAverageof11resultsis16,iftheaverageofthe1st6resultsis58&thatofthelast63.Findthe6thresult?Sol:6thresult=(58×6+63×6–60×11)=66Ex.10:TheaverageweightofA,B,Cis45Kg.TheaverageweightofA&Bbe40Kg&thatofB,Cbe43Kg.FindtheweightofB.Sol.LetA,B,Crepresenttheirindividualweights.Then,A+B+C=(45×3)Kg=135KgA+B=(40×2)Kg=80Kg&B+C=(43×2)Kg=86KgB=(A+B)+(B+C)–(A+B+C)=(80+86–135)Kg=31Kg.

Ex.11.Theaverageageofaclassof39studentsis15years.Iftheageoftheteacherbeincluded,thentheaverageincreasesby3months.Findtheageoftheteacher.Sol.Totalageof39persons=(39×15)years=585years.Averageageof40persons=15yrs3months=61/4years.Totalageof40persons=((61/4)×40)years=610years.:.Ageoftheteacher=(610–585)years=25years.Ex.12.Theaverageweightof10oarsmeninaboatisincreasedby1.8kgwhenoneofthecrew,whoweighs53kgisreplacedbyanewman.Findtheweightofthenewman.Sol.Totalweightincreased=(1.8×10)kg=18kg.:.Weightofthenewman=(53+18)kg=71kg.Ex.13.Therewere35studentsinahostel.Duetotheadmissionof7newstudents,;theexpensesofthemesswereincreasedbyRs.42perdaywhiletheaverageexpenditureperheaddiminishedbyRs1.Whatwastheoriginalexpenditureofthemess?Sol.LettheoriginalaverageexpenditurebeRs.x.Then,42(x–1)–35x=42or7x=84orx=12.Originalexpenditure=Rs.(35×12)=Rs.420.Ex.14.Abatsmanmakesascoreof87runsinthe17thinningandthusincreaseshisaverageby3.Findhisaverageafter17thinning.Sol.Lettheaverageafter17thinning=x.Then,averageafter16thinning=(x–3).:.16(x–3)+87=17xorx=(87–48)=39.Ex.15.DistancebetweentwostationsAandBis778km.AtraincoversthejourneyfromAtoBat84kmperhourandreturnsbacktoAwithauniformspeedof56kmperhour.Findtheaveragespeedofthetrainduringthewholejourney.Sol.Requiredaveragespeed=((2xy)/(x+y))km/hr=(2×84×56)/(84+56)km/hr=(2×84×56)/140km/hr=67.2km/hr.Ex.16.Inthefirst10oversofacricketgame,therunratewasonly3.2.Whatshouldbetherunrateintheremaining40overstoreachthetargetof282runs?Sol.

Requiredrunrate=

282-(3.2x10)

=250

=6.2540 40

Ex.17.AgrocerhasasaleofRs.6435,Rs.6927,Rs.6855,Rs.7230andRs.6562for5consecutivemonths.HowmuchsalemusthehaveinthesixthmonthsothathegetsanaveragesaleofRs.6500?Sol.Totalsalefor5months=Rs.(6435+6927+6855+7230+6562)=Rs.34009.

Requiredsale=Rs.[(6500x6)-34009]=Rs.(39000-34009)=Rs.4991.

Ex.18.Apupil'smarkswerewronglyenteredas83insteadof63.Duetothattheaveragemarksfortheclassgotincreasedbyhalf(1/2).Thenumberofpupilsintheclassis:Sol.Lettherebe𝑥pupilsintheclass.

Totalincreaseinmarks=

𝑥x1

=𝑥

2 2

𝑥=(83-63)

𝑥=20 𝑥=40.

2 2Ex.19.Alibraryhasanaverageof510visitorsonSundaysand240onotherdays.Theaveragenumberofvisitorsperdayinamonthof30daysbeginningwithaSundayis:Sol.SincethemonthbeginswithaSunday,totherewillbefiveSundaysinthemonth.

Requiredaverage

=

510x5+240x25

30

=855030

=285

Ex.20.Thecaptainofacricketteamof11membersis26yearsoldandthewicketkeeperis3yearsolder.Iftheagesofthesetwoareexcluded,theaverageageoftheremainingplayersisoneyearlessthantheaverageageofthewholeteam.Whatistheaverageageoftheteam?Sol.Lettheaverageageofthewholeteambyxyears.

11x-(26+29)=9(x-1)11x-9x=462x=46x=23.

So,averageageoftheteamis23years.Ex.21.Theaverageageofhusband,wifeandtheirchild3yearsagowas27yearsandthatofwifeandthechild5yearsagowas20years.Thepresentageofthehusbandis:Sol.Sumofthepresentagesofhusband,wifeandchild=(27x3+3x3)years=90years.Sumofthepresentagesofwifeandchild=(20x2+5x2)years=50years.

Husband'spresentage=(90-50)years=40years.Ex.22.InArun'sopinion,hisweightisgreaterthan65kgbutlessthan72kg.HisbrotherdoestnotagreewithArunandhethinksthatArun'sweightisgreaterthan60kgbutlessthan70kg.Hismother'sviewisthathisweightcannotbegreaterthan68kg.Ifallarethemarecorrectintheirestimation,whatistheaverageofdifferentprobableweightsofArun?Sol.LetArun'sweightbyXkg.AccordingtoArun,65<X<72AccordingtoArun'sbrother,60<X<70.AccordingtoArun'smohter,X<=68

Thevaluessatisfyingalltheaboveconditionsare66,67and68.

Requiredaverage=

66+67+68

=

201

=67kg.3 3

Ex.23.Theaverageof20numbersiszero.Ofthem,atthemost,howmanymaybegreaterthanzero?Sol.Averageof20numbers=0.

Sumof20numbers(0x20)=0.Itisquitepossiblethat19ofthesenumbersmaybepositiveandiftheirsumisathen20thnumberis(-a).Ex.24.Theaverageofknumbersis20,andtheaverageof10ofthesenumbersis10.Intermsofk,findtheaverageoftheremainingnumbers.Sol.Thetotaloftheknumbersis20k,andthetotalof10outofthe20numbersis10 10=100.So,thetotaloftheremaining(k–10)numbersis20k–100,andtheiraverageis20(k–5)/(k–10)Ex.25.Theaveragescoreof11studentsis50.Iftheaverageofthefirst6studentsis49andthatofthelast6is52,findthescoreofthe6thstudent.Sol.Totalscoreof11students=50 11=550Totalscoreoffirst6students=6 49=294Totalscoreoflast6students=6 52=312Totalscoreoffirst6andlast6students=312+294=606Scoreofthe6thstudent=606–550=56Ex.26.Outof36studentsinahostel,19left.Duetothis,theexpenditureofthemessdecreasedby$48perday,whiletheaverageexpenditureperheadincreasedby$5.Whatisthesumoforiginalexpenditureandthenewexpenditureperdayofthemess?Sol.Lettheaveragemessexpenditureperstudentbex.Then,totalexpenditure=36x.After19studentsleft,newexpenditure=36x–48Accordingtothequestion&hONie

Kg=(x+5)

36x–48=17x+8519x=133 x=7Sumoforiginalexpenditureandnewexpenditureofmess=36x+36x–48=72x–48=456Ex.27.TheaveragetemperatureforMonday,TuesdayandWednesdaywas55o;theaverageforTuesday,WednesdayandThursdaywas60o;andthetemperatureforThursdaywas56o.FindthetemperatureonMonday.Sol.Totaltemperaturefor(M+T+W)=3´55o=165o----------(1)Totaltemperaturefor(T+W+Th)=3´60o=180o.----------(2)

Onsolving(1)and(2)Th–M=1556–M=15Mon=41o.Ex.28.Ifa,b,c,d,earefiveconsecutiveoddnumbers,findtheiraverageintermsof‘a’.Sol.a,b,c,d,earefiveconsecutiveoddnumbers.Therefore,b=a+2,c=b+2=a+4d=c+2=a+6ande=d+2=a+8Averageofthefivenumbers=(5a+20)/5=a+4.Ex.29.Fouryearsago,theaverageageoffourfriendswas21years.Now,withafifthfriendjoiningthegroup,theiraverageis24years.Whatisthepresentageofthenewfriend?Sol.Totalageofallfriendsincludingnewfriend=24 5=120yearsPresentageofthenewfriend=newtotalofages–oldtotalofages

120–100=20yearsEx.30.Atreeis8.4mtall,anotherishalfitsheightandthethirdoneisitsquarter.Whatistheaverageheightofthetrees?Sol.Heightofthefirsttree=8.4mtallHeightofthesecondtree=½ 8.4=4.2mHeightofthethirdtree=¼ 8.4=2.1mTotalheight=8.4+4.2+2.1=14.7Averageheight=Ki.g

&=4.9m.

Ex.31.TheaveragemonthlyrainfallforayearinTokyois2.7inches.Theaverageforthefirst7monthsis1.1incheslessthantheannualaverage.Ifthetotalrainfallforthenext4monthsis20.8inches,calculatetherainfallinthelastmonth.Sol.Thetotalrainfallforayear=12 2.7=32.4inches.Forthefirst7monthsaveragerainfallis1.1incheslessthantheaverageannualrainfallHence,averagerainfallforthefirst7months=2.7–1.1=1.6inchesTotalrainfallforthefirst7months=7 1.6=11.2Totalrainfallforthelast4months=20.8inchesHence,rainfallinlastmonth=Annualrainfall–[totalrainfalloffirst7andnext4months]=32.4–[11.2+20.8]=32.4–32=0.4inchesEx.32.Averagenumberofmarksscoredby60studentsinaclasswas32.5.Themarksofonestudentwerealteredfrom88to78andmarksofanotherwerealteredfrom13to43.Whatwasthenewaverage?Sol.Totalmarksof60students=60 32.5=1950Decreaseinmarksofthefirststudent=88–78=10Andincreaseinmarksoftheotherstudent=43–13=30Totalvariationinmarks=1950–10+30=1970.

Nownewaverage=KjgMhM

=32.83.

Ex.33.Ashopkeeperearned$504in12days.Hisaverageincomeforthefirst4dayswas$40aday.Hisaverageincomefortheremainingdayswas:Sol.Moneyearnedin12days=$504Moneyearnedinfirst4days=4 40=160Moneyearnedinlast8days=504–160=344Averageforremainingdays=&ii

e=$43.

Ex.34.Ateamofeightmarksmenenteredashootingcompetition.Thebestmarksmanscored85points.Ifhehadscored92points,theaveragefortheteamwouldhavebeen84.Howmanypointsaltogetherdidtheteamscore?Sol.Lettheaveragepointsscoredbytheteamof8marksmenbexTotalpoints=8xNowifbestmarksmanscored92pointsinsteadof85,thenchangeinpoints=92–85=7Newaverage=eOPg

e=84

8x+7=6728x=665Totalpointsscoredbytheteam=8x=665points.Ex.35.IttakesBob2tminutestocompleteformA,whichhaskquestions,and3tminutestocompleteformBwhichhask/2questions.IfBobanswersallthequestions,hisaveragetimetoansweraformBquestionishowmanyminuteslongerthanhisaveragetimetoansweraformAquestion?Sol.kquestionsinformAtake2tminutes.So,onequestioninformAwilltake2t/kminutes.Similarlyk/2questionsinformBtake3tminutes.So,onequestioninformBtakes3t/(k/2)or6t/kminutes.So,thedifferenceintime=6t/k–2t/k=4t/kminutes.Ex.36.Apublisherincursafixedexpenseof$2000towardscomposingandotheractivitiesinprintingabook.Avariablecostof$2.25perbookisincurredtowardspaper,bindingchargesetc.Whatistheaveragecostperbookif40000copiesareprinted?Sol.Fixedexpenses=2000Nowtotalvariablecost=$2.25´40000Totalaveragecost=XdO\cqW`_Pz^]d^p{\qW`_

YW.WXpWWs`

=$MMMP$.%×iMMMiMMMM

= $MMMiMMMM

+2.25= K$M+2.25=$2.30.

Ex.37.Theaveragemonthlysalaryofanofficeconsistingof120clerks,1superintendentandonemanageris$1500.Ifthemanagerandthesuperintendenthavemonthlysalariesof$3000and$2000respectively,whatistheaveragemonthlysalaryoftheclerks?Sol.Totalmonthlysalary=$1500 (120clerks+1suptt.+1manager)=1500 122=$183000TotalsalaryafterSuptt.&Managerleave=183000–(2000+3000)=183000–5000=$178000Nownewmonthlyaveragesalary=KgeMMM

K$M=$1483.34

Ex.33.Threedogsweigh6,12and9kgrespectively.Theweightoftheheaviestdogishowmuchmorethantheaverageweightofthethreedogs?Sol.Weightofthreedogs6,12and9kg.Averageofweightofthreedogs=hPK$Pj

&= $g

&=9kg

Heaviestdogis12kgDifference=12–9=3kgEx.34.Theaverageof3integerss,tand8is12.Whichofthefollowingcannotbethevalueoftheproductst?(a)27(b)52(c)56(d)160Sol.Itisgiventhat(s+t+8)/3=12.Sos+t=36–8=28.Sincealltheanswerchoicesarepositiveintegers,wecanassumethatsandtarepositiveintegers.Nowwewillhavetotryandtheeliminatetheanswerschoiceswhosevaluescanbeequaltos×tsuchthats+t=28.(1)ispossiblebecause27×1=27,(2)isalsopossiblebecause,26×2=52,(5)ispossibleas14×14=196.Also(4)ispossibleas20×8=160.Soweareleftwithchoice(3)onlyanditistherightanswerEx.35.Ann,Jack,CarlandLindaownatotalof43T-shirts.If10oftheT-shirtsbelongtoAnn,whatistheaveragenumberofT-shirtsownedbyeachoftheothers?Sol.ThenumberofTshirtsownedbyJack,CarlandLindais43–10=33.So,theaveragenumberownedbythemis&&

&=11.

Ex.36.Theaveragescoreofacricketerinacertainnumberofinningsis15.Whenhescoredzeroinoneinnings,hisaveragedroppedto13.5.Howmanyinningshavebeenplayedincludingthelastone?Sol.Letthenumberofinnings=xTotalrunsscoredbytheplayer=15xNewnumberofinnings=x+1Alsonewaverage=13.5So,(15x+0)/(x+1)=13.5Numberofinningsplayedincludingthelatestone=9+1=10.Ex.37.Theaverageageofaclassof30studentsandateacherreducesby0.5years,ifweexcludetheteacher.Iftheinitialaverageis14years,findtheageoftheclassteacher.Sol.Averageageof30studentsand1teacher=14years.Totalageof31persons=31×14=434years.Averageageof30students=14-0.5=13.5yearsTotalageof30students=30×13.5=405yearsAgeofteacher=434-405=29years.

Exercise1.Theaveragerunsscoredbytheelevenplayersofacricketteamis60.Iftherunsscoredbythecaptainareneglected,theaverageofrunsscoredbytheremainingplayersincreasesby5.Howmanyrunswerescoredbythecaptain?a.8b.9c.8.5d.10e.Noneofthese2.Inacertaincompany,theaverageproductionby18employeesis400unitspermonth.Ifthetotalproductionpermonthisincreasedby180units,thenwillbetheaverageproductionofeachemployee?a.400b.490c.450d.420e.Noneofthese3.Thenumberofstudentsineachsectionofaschoolis24.Afteradmittingthenewstudents,threenewsectionswerestarted.Nowthetotalnumberofsectionis16andthereare21studentsineachsection.Thenumberofnewstudentsadmittedisa.18b.20c.16d.24e.Noneofthese4.Thetotalofthreeconsecutiveevennumbersis36morethantheaverageofthesethreenumbers.Whichofthefollowingisthesecondofthesethreenumbers?a.18b.20c.16d.24e.Noneofthese

5.Amanwalksfor1 hoursatthespeedin6kmperhour.Thenhewalks10kmin1 hours.Whatishisaveragespeedinkmperhourforthewholejourney?a.8b.9c.6d.10e.Noneofthese6.TheaveragesalaryofPrakash,JayeshandKailashisRs.1100.TheaveragesalaryofPrakashandKailashisRs.900.WhatisJayesh’ssalary?a.1400b.1490c.1500d.1420e.Noneofthese7.Theaverageoftherunsmadeby10playersinacricketteamof11playersis43.Iftheeleventhplayer’srunsarealsoconsidered,theaverageof11playersrunsdecreasesby1.Howmanyrunsdidtheeleventhplayermake?a.32b.20c.16d.24e.Noneofthese8.Theaverageof7numbersis25.Ifeachofthenumbersismultipliedb5,findtheaverageofnewsetofnumbersa.132b.120c.116d.125e.Noneofthese9.Thenumberofstudentsineachsectionofaschoolis24.Afteradmittingthenewstudents,threenewsectionswerestarted.Nowthetotalnumberofsectionsis16andthereare21studentsineachsection.Thenumberofnewstudentsadmittedisa.32b.20c.16d.24e.Noneofthese

10.Theaverageofrunsscoredbythe11playersofacricketteamis60.Iftherunsscoredbythecaptainareneglected,theaverageofrunsscoredbytheremainingplayersincreasesby5.Howmanyrunswerescoredbythecaptain?a.8b.9c.6d.10e.Noneofthese11.Averagescoresof50studentsinaclassis44.Lateron,itwasfoundthatascore23wasincorrectlyrecordedas73.Thecorrectaveragescoreisa.43b.20c.16d.24e.Noneofthese12.Theaverageweightof10oarsmeninaboatisincreasedby2.5kgwhenoneofthecrew,whoweights64kgisreplacedbyanewman.Findtheweightofthenewmana.43b.89c.80d.24e.Noneofthese13.Ifthesumofthreeconsecutiveoddnumbersis213.Findthefirstterm.a.69b.89c.80d.50e.Noneofthese14.Ofthethreenumbers,theaverageofthefirstandthesecondisgreaterthantheaverageofthesecondandthethirdby15.Whatisthedifferencebetweenthefirstandthethirdofthethreenumbera.30b.20c.16d.24e.Noneofthese15.Iftheaverageof‘n’termsisXthenthesumof‘n’terms=a.n/Xb.n2Xc.nXd.X/ne.Noneofthese16.Findtheaverageoffirst6oddmultiplesof4a.143b.189c.180d.124e.Noneofthese17.Astudentwalkstohisschoolat5km/hrandthengoesbackonacycleat15km/hr.Findtheaveragespeedforthewholejourneyinkm/hra.4.3b.8.9c.8d.7.5e.Noneofthese18.Findtheaverageoffirst200naturalnumbers.a.100b.90c.100.5d.90.5e.Noneofthese19.Iwenttoahotelalongwith12friends.IpaidRs.145andalltheotherspaidanequalamount.Intheendwhenwedidsomecalculations,wefoundthattheaveragesumpaidbyallofuswasRs.5morethanwhatwasoriginallypaidbyeachofmyfriends.Howmuchmoneydideachofmyfriendspay?a.100b.90c.80d.110e.Noneofthese20.Thereare20studentsinMr.Talwar’sclass.Onceheconductedanexaminationoutof100.Hethenarrangedthemarksintheascendingorder.HefoundthatRohit,thetopperoftheclassslippedtothetenthposition.Whenhewasaddingthescoresofthelast11studentstheaveragewas64andthatofthetop10was67.Iftheaveragemarkobtainedbyallthestudentsofhisclasswas65,howmanymarksdidRohitscore?

a.75b.90c.74d.73e.Noneofthese21.Mr.Rawatwassittingidleinhishome.Hewasmakingsomecalculationsandhefoundthathisagewas58yearslessthanthesquareoftheageofhisson.Hefurthermadesomecalculationsandmadeastatementthatthesumofthesquares,ofthesumandtheproductofthedigitsinhisage,was100.Findtheageofhissona.8b.9c.8.5d.10e.Noneofthese22.Inarecentsurveyof800socialities,itwasfoundthat3%ofthemattendedonepartyaweekonanaverage.Oftheother97%itwasfoundthathalfattendedanaverageoftwopartiesaweekandtheotherhalfdidnotattendanyparty.Howmanypartyattenderswerethereinanaverageweek?a.800b.900c.850d.1000e.Noneofthese23.Ifaverageof8consecutiveevennumbersis64,whatisthedifferencebetweenthesmallestandthelargestnumbers?a.18b.19c.14d.10e.Noneofthese24.Theaverageof3numbersis27andthatoffirsttwois18.Thethirdnumberisa.42b.45c.48d.36e.Noneofthese25.Ninemenwenttoahotel.8ofthemspentRs.3eachovertheirmealsandtheninthspentRs.2morethantheaverageexpenditureofallthenine.Thetotalmoneyspentbyallofthemwasa.28b.29c.24d.29.25e.Noneofthese26.ThesalarybillforthemonthofMayinrespectof4stenosand2clerkinanofficewasRs.16,000.OnJune1st,onestenoleftandoneclerkwasappointedbywhich,themonthlysalarybillwasreducedbyRs.1,000.Whatisthemonthlysalaryofeachoftheclerk?a.2800b.2900c.2850d.2000e.NoneoftheseDirectionsforQ27&28:Readthegivendataandanswerthequestionsgiven.Abatsmen’saveragescoreforacertainnumberofinningswas21.75perinnings.Heplayed3inningsmoreandscored28,34and37runsrespectively.Thusincreasinghisaverageby1.2527.Howmanyinningsdidheplayinall?a.28b.29c.24d.27e.Noneofthese28.Whatwashistotalscore?a.800b.600c.620d.621e.Noneofthese29.InthemonthofJulyofacertainyear,theaveragedailyexpenditureofanorganizationwasRs.68.Forthefirst15daysofthemonth,theaveragedailyexpenditurewasRs.85andforthelast17days,Rs.51.Findtheamountspentbytheorganizationonthe15thofthemontha.28b.29c.34d.29.25e.Noneofthese30.In1919,W.Rhodes,theYorkshirecricketer,scored891runsforhiscountryatanaverageof

34.27,in1920,hescored949runsatanaverageof28.75,in1921,1329runsatanaverageof42.87andin1922,1101runsatanaverageof36.70.Whatwashisbattingaverageforthefouryears?a.35b.33.50c.35.58d.29.25e.Noneofthese31.InhotelJaysarmin,theroomsarenumberedfrom101to130onthefirstfloor,221to260onthesecondfloorand306to345onthethirdfloor.InthemonthofJune2002,theroomoccupancywas60%onthefirstfloor,40%onthesecondfloorand75%onthethirdfloor.IfitalsoknownthattheroomchargesareRs.200,Rs.100andRs.150oneachofthefloors,thenfindtheaverageincomeperroomforthemonthofJune2002a.88b.88.50c.88.18d.88.9e.Noneofthese32.Asalesmangetsabonusaccordingtothefollowingstructure:IfhesellsarticlesworthRs.XthenhegetsabonusofRs.(x/100-1).InthemonthofJanuary,hissalesvaluewasRs.100,inFebruaryitwasRs.200,fromMarchtoNovemberitwasRs.300foreverymonthandinDecemberitwasRs.1200.Apartfromthis,healsoreceivesabasicsalaryforRs.30permonthfromhisemployer.Findhisaverageincomepermonthduringtheyeara.35b.32.58c.35.58d.29.25e.Noneofthese33.Theweightofabodyascalculatedbytheaverageof7differentexperimentsis53.735g.Theaverageofthefirstthreeexperimentsis54.005gofthefourthis0.004ggreaterthanthefifth,whiletheaverageofthesixthandseventhexperimentwas0.010glessthantheaverageofthefirstthree.Findtheweightofthebodyobtainedbythefourthexperimenta.35b.53.342c.35.58d.29.25e.Noneofthese34.Therearefiveboxesinacargohold.Theweightofthefirstboxis200kgandtheweightofthesecondboxis20%higherthantheweightofthethirdbox,whoseweightis25%higherthanthefirstbox’sweight.Thefourthboxat350kgis30%lighterthanthefifthbox.Findthedifferenceintheaverageweightofthefourheaviestboxesandthefourlightestboxesa.35kgb.70kgc.75kgd.80kge.Noneofthese35.Forquestion34findthedifferenceintheaverageweightoftheheaviestthreeandthelightestthreea.135.5kgb.133.5kgc.133.33kgd.129.25kge.Noneofthese36.19personswenttoahotelforacombineddinnerparty.13ofthemspentRs.79eachontheirdinnerandtherestspentRs.4morethantheaverageexpenditureofallthe19.Whatwasthetotalmoneyspentbythem?a.1500b.1550c.1530d.1535e.Noneofthese37.Theaverageweightof5menisdecreasedby3kgwhenoneofthemweighting150kgisreplacedbyanotherperson.Thisnewpersonisagainreplacedbyanotherpersonwhoseweightis30kglowerthanthepersonhereplaced.Whatistheoverallchangeintheaverageduetothisdualchance?a.10b.9.5c.9.25d.9e.Noneofthese

38.Theweightofpersonispartlyproportionaltotheamountofdailyintakeofcaloriesandpartlyproportionaltohisage.Apersonaged36yrsweighs61kgwhenhisintakeis2500calorieseverydaywhileapersonaged60yrsweighs70gwhenhetakesin1000caloriesperday.Howmanywouldapersonofage24yrsweightwhenhiscalorieintakeis3000perday?a.28b.54c.50d.27e.Noneofthese39.Theaveragescoreofthefirst3studentsinanexaminationisgreaterthanthatofthefirst5studentsby20,butthetotalscoreofthefirst3islessthanthatofthefirst5by70.Findthetotalscoreofthefirst3studentsa.288b.254c.250d.227e.Noneofthese40.Aman’saverageexpenditureforthefirst4monthsoftheyearwasRs.251.25.Forthenext5monthstheaveragemonthlyexpenditurewasRs.26.27morethanwhatitwasduringthefirst4months.IfthepersonspentRs.760inallduringtheremaining3monthsoftheyear,findwhatpercentage(approximate)ofhisannualincomeofRs.3000hesavedintheyeara.8%b.5%c.-5%d.-7%e.Noneofthese41.Therewere42studentsinahostel.Duetotheadmissionof13newstudents,theexpensesofmessincreasebyRs.31perdaywhiletheaverageexpenditureperheaddiminishedbyRs.3.Whatwastheoriginalexpenditureofthemess?a.688b.654c.650d.637e.Noneofthese42.TheaverageweightofboysofAnil’sageandheightis48kgandifAnilweights110%ofaverage,thenAnil’sweightsa.28kgb.54.8kgc.52.8kgd.27kge.Noneofthese43.Theaverageofnconsecutivewholenumberswillalwaysbeawholenumberifa.nisevenb.nisoddc.botha.andb.ispossibled.datainadequatee.Noneofthese44.Theaverageof13resultsis60.Iftheaverageofsevenresultsis57andthatoflastsevenis58,findthe7thresulta.28b.25c.50d.27e.Noneofthese45.Theaveragescoreofacricketerfor10matchesis49.7runs.Iftheaverageforthefirstsevenmatchesare45.Whatistheaverageforthelast3matches?a.60.66b.60c.62.5d.65e.Noneofthese46.NumberBis70morethantwicethenumberA.NumberCis10morethantwicethenumberB.IftheaverageofA,BandCis120.WhatisthevalueofC?a.228b.254c.250d.227e.Noneofthese47.Theaverageoffirstfivemultipliesof9isa.28b.54c.50d.27e.Noneofthese

48.Theaveragemonthlysalaryof30workersinafirmisRs.990.IfthesalaryoftheofficerisaddedtheaverageincreasesbyRs.30.Whatisthemonthlysalaryoftheofficer?a.1928b.1954c.1950d.1920e.Noneofthese49.TheaverageweightofP,QandRis75kg.IftheaverageweightofPandQis60kgandthatofQandRis67kgthentheweightofQisa.28b.29c.50d.27e.Noneofthese50.Theaveragemarksobtainedby35studentswas120.Iftheaverageofpassedcandidateswas130andthatoffailedcandidateswas60,thenumberofcandidateswhopassedtheexaminationa.28b.29c.30d.27e.Noneofthese51.Theaverageof6numbersis32.Ifonenumberisexcludedtheaveragebecomes30.Theexcludednumberisa.42b.54c.50d.27e.Noneofthese52.Theaverageof21numbersis32.Theaverageoffirst11ofthesenumbersis23andtheaverageofthelast11ofthesenumbersis32.Whatisthetwelthnumber?a.28b.54c.50d.datainadequatee.Noneofthese53.Theaveragemarksof50studentsinaclassis60.Theaverageofthosewhopassis75andthosewhofailis50.Howmanystudentsdidfail?a.28b.34c.30d.27e.Noneofthese54.Theaveragemarksof40studentsinasubjectis65.Theaverageofthosewhopassedis75andthosewhofailedis35.Howmanystudentsdidpass?a.28b.54c.50d.27e.Noneofthese55.Theaverageofrunsmadeby11playersinacricketteamis75,Ifthecaptain’srunsisnotconsidered,theaverageoftherunsmadebytheremainingplayersincreasesby4.Howmanyrunsdidcaptainmake?a.28b.35c.36d.27e.Noneofthese56.Theaverageofthreewholenumbersis36.Thesmallestnumberisone-thirdofthelargestnumber.Themiddlenumberistwicethesmallestnumber.Findoutthemiddlenumbera.28b.35c.36d.27e.Noneofthese57.TheaveragemarksinTamil,HistoryandMathsis45.TheaveragemarksinScience,MathsandHistoryis50.IfmarksinScienceare40,whatshouldbethemarksinTamil?a.28b.25c.26d.27e.Noneofthese

58.Theaveragescoreofacricketerfor10matchesis39.9runs.Iftheaverageforthefirstfivematchesis39thenwhatistheaveragescoreforthelastfivematches?a.40b.40.2c.40.5d.40.8e.Noneofthese59.Ofthethreenumber,thefirstistwicethesecondandthesecondisthricethethird.Iftheaverageofthethreenumbersbe20,findthefirstnumbera.28b.35c.36d.27e.Noneofthese60.TheaveragesalaryperheadoftheentirestaffofanofficeincludingtheclerksisRs.60.TheaveragesalaryperheadoftheofficerisRs.400andthatoftheclerksisRs.56.Giventhatthenumberofofficersis13,findthenumberofclerksintheofficea.1128b.1135c.1136d.1127e.Noneofthese61.TheaveragecollectioninaweekofCinemahallisRs.10,504.Theaveragecollectionof6daysexcludingFridayisRs.9,544.WhatisthecollectiononThursday?a.16128b.16135c.16136d.16264e.Noneofthese62.Theaveragemarksof96studentsinaclassis90.Theaveragemarksofthegirlsintheclassis80andtheaveragemarksoftheboysintheclassis100.Whatistherationbetweenthenumberofboysandgirlsintheclass?a.1:2b.1:1c.2:1d.3:2e.Noneofthese63.Theaverageoftwonumbersisx.Ifonenumberisythentheothernumberisa.2x-yb.2y-xc.x-yd.y-xe.Noneofthese64.TheaverageweightofthreemenX,YandZis80kg.AnothermanWjoinsthegroupandtheaverageofthegroupbecomes78kg.IfanothermanVwhoseweightis5kgmorethanofWreplacesX,thentheaverageweightofY,Z,WandVbecomes76kg.TheweightofXisa.88b.85c.86d.87e.Noneofthese65.ThedailyearningsofaAutorickshawdriveraweekareRs106.30,Rs178.40,Rs198.30,Rs167.50,Rs175.00,Rs139.70andRs155.50respectively.Findhisaveragedailyearningfortheweek?a.161.28b.160.35c.160.10d.162.64e.Noneofthese66.13Tablesand8ChairswereboughtforRs5678.IftheaveragepriceofatablebeRs206,whatistheaveragepriceofachair?a.388b.385c.375d.376e.Noneofthese67.TheaveragemonthlyexpenditureofafamilywasRs2100duringfirst3months,Rs2520duringnext4monthsandRs2652duringlast5monthsoftheyear.IfthetotalsavingduringtheyearbeRs.1440,findaveragemonthlyincome?a.2588b.2585c.2590d.2587e.Noneofthese

68.Theaverageweightof40studentsofaclassis55kgs.Oneofthestudentswhoseweightis47kgsisreplacedbyanotherstudentandtheaverageweightwasincreasedby300grams.Findtheweightofthenewstudent.a.59b.60c.61d.62e.Noneofthese69.Theaverageweightof8childrenis21.5kgs.Twoofthemweight20.4kgand22kgsrespectively.Whatistheaverageweightofremainingsix?a.21.5b.21.6c.20.5d.20.6e.Noneofthese70.Therearethreenumbers,thesecondis1/4thofthefirstandthethirdishalfofthesecond.Iftheaverageofthethreenumbersis99,findthesecondnumbera.59b.60c.55d.54e.Noneofthese71.Theaverageofthefirst150naturalnumbersisa.75.5b.70.5c.72.5d.78.5e.Noneofthese72.Whatistheaverageofthecubesofthefirstsixcountingnumbers(i.e.)findtheaverageof13,23,33,43,53,63a.75.5b.73.5c.72.5d.78.5e.Noneofthese73.Theaverageof8numbersis6.Whatisthe9thnumbersothataveragebecomes8.a.24b.25c.26d.27e.Noneofthese74.Abatsmaninhis14thinningsmakesascoreof73runsandtherebyincreaseshisaveragescoreby3.Whatishisaverageafterthe14thinnings?Hehadneverbeen‘notout’?a.34b.35c.36d.37e.Noneofthese75.Theaverageoffirstsixprimenumbersisa.4b.5c.6d.7e.Noneofthese76.Theaveragescoreofacricketerfor10matchesis49.8runs.Iftheaverageforthefirst6matchesis54,findtheaverageforthelast4matchesa.45.5b.43.5c.42.5d.48.5e.Noneofthese77.Theaverageweightof9personsisincreasedby4kgwhenoneofthemwhoseweightis65kgisreplacedbyanewman.Theweightofthenewmanisa.101b.102c.103d.104e.Noneofthese78.Thesumofthreenumbersis175.Iftherationbetweenfirstandsecondbe5:6andthatbetweensecondandthirdbe9:11,thenthesecondnumberisa.56b.57c.55d.58e.Noneofthese

79.Theaveragetemperatureoffirst3daysin32°Candofthenext3daysis29°C.Iftheaverageofthewholeweekis31.5°C,thetemperatureofthelastdayisa.33.5°Cb.37.5°Cc.35.5°Cd.39.5°Ce.Noneofthese80.OnajourneyacrossJaipur,acar’saveragespeedis40km/hrfor50%ofthedistance,30km/hrfor20%ofitand20km/hrfortheremainingdistance.Theaveragespeedforthewholejourneyisa.25km/hrb.29km/hrc.30km/hrd.35km/hre.Noneofthese

Answers

1.D 2.E 3.D 4.A 5.C 6.C 7.A 8.D 9.D 10.D11.A 12.B 13.A 14.A 15.C 16.E 17.D 18.C 19.C 20.C21.D 22.A 23.C 24.B 25.D 26.D 27.D 28.D 29.C 30.C31.C 32.B 33.B 34.C 35.C 36.E 37.D 38.B 39.E 40.C41.E 42.C 43.B 44.B 45.A 46.E 47.D 48.D 49.B 50.C51.A 52.D 53.C 54.E 55.B 56.C 57.B 58.D 59.C 60.E61.D 62.B 63.A 64.B 65.C 66.C 67.C 68.A 69.B 70.D71.A 72.B 73.A 74.A 75.C 76.B 77.A 78.B 79.B 80.B

PercentageTheword"percentage"literallymeans"perhundred"or"foreveryhundred."Therefore,wheneveryoucalculatesomethingasapartof100,thatpartisnumericallytermedaspercentage.Inotherwords,percentageisaratiowhosesecondtermisequalto100.Forexample,1:4canbewrittenas25:100or25%,3:8canbewrittenas37.5:100or37.5%,3:2canbewrittenas150:100or150%,andsoon.IMPORTANTCONCEPTSASSOCIATEDWITHPERCENTAGEBasicformulaofpercentage:

Percentageof:

CommodityPriceIncrease/Decrease:IfthepriceofacommodityincreasesbyR%,thenthereductioninconsumptionsoasnottoincreasetheexpenditureis:

𝑅(100 + 𝑅) ×100 %

IfthepriceofthecommoditydecreasesbyR%,thentomaintainthesameexpenditurebyincreasingtheconsumptionis:

𝑅(100 − 𝑅) ×100 %

Percentageincrease/decrease:Percentageincrease/decreasewhenaquantityaincrease/decreasestobecomeanotherquantitybis

Percentagelessthan/greaterthan:Havealookatthepicturegivenbelow:

YoucanseethatJohnnyistallerthanVicky.WhatwillyouranswerbeifIask(a)BywhatpercentageisJohnnytallerthanVicky?(b)BywhatpercentageisVickyshorterthanJohnny?Answer:

Therefore,tofindthefinalquantityaftera20%increase,wecandirectlymultiplytheoldquantitybyafactorof1.2andgetthenewquantity.Similarly,fora20%decrease,wecanmultiplytheoldquantityby0.8andgetthenewquantity.Thefactorstobemultipliedforvariouspercentageincrease/decreasearegivenbelow:

Thebiggestadvantageofusingthefactorsisthatforsubsequentpercentageincrease/decrease,wejustkeeponmultiplyingthecorrespondingfactorsandgetthefinalquantity.SolvedexamplesEx.1.Therehasbeena30%increaseintheproductionofmilkchocolatesinAmulDairythismonth.Iftheproductionthismonthis9,100boxesofmilkchocolates,whatwasitonemonthago?Sol.Increasedamount=Originalamount×KMMP"

KMM,wherep=30%,Originalamount=Increased

amount=9,1009,100=NxKMMP&M

KMM

N=jKMM×KMMK&M

N=7,000boxes/month.Ex.2.AandBaretwonumberssuchthatB–A=PandB=P%morethanA.FindBintermsofPSol.IfA=100,thenB=100+PandhenceB–A=P,satisfies.Ex.3.a,b,c,dare4numbers,suchthatbis50%ofa,cis25%lessthanbanddis10%morethanc.Ifaisk%higherthand,whatisthevalueofk?Sol.b=^

$

c=&ib

d=KKKMc=KK

KM×&

ib=&&

eMa

Soais^Ncc×100%morethan‘d’

i.e.^NSS|}^SS|}

×100

ig&&×100=142%

Ex.4.Thepopulationofatownincreasedby3%inagivenintervalbutitwouldhavebeen1,500less,iftherehadbeenadecreaseof2%.FindtheoriginalpopulationSol.Earliertherewasanincreaseof3%,insteadifthenthereisadecreaseof2%,itmeansthereisnetchangeof5%inpopulation.Thus5%oftotalpopulation=1,500.

Totalpopulation=30,000.Ex.5.Indiascoredatotalofxrunsin50overs.SriLankatiedthescoresin10%lessovers.IfIndiahadscored30runsmore,thenSriLankawouldhavetieditin50oversatthesamerunrate.FindhowmanyrunswerescoredbyIndia.Sol.AveragerunrateofIndia= O

%M

Avg.runrateofSriLanka= O%M

IfIndiascored(x+30)runs,then,SriLankatiedthematchin50overs.Thatmeans, O

i%×50=x+30

50x=45x+45×30x=i%×&M

%=270.

Ex.6.Thenumberofboysandgirlswhoappearedin“SSCCGL2012”examinationwereintheratio16:9andthenumberofboysandgirlspassingtheexaminationwereintheratio4:3.If75%ofthegirlspassedtheexamination,findthe%ofboyswhopassedtheexaminationSol.Letthenumberofboys&girlsbe16x&9xrespectively.andletboysandgirlspassingare4yand3yrespectively.%ageofgirlspassing=(3y/9x)×100=75⇒y/x=9/4%ofboyspassing=(4y/16x)×100=25(y/x)%.Substitutethevalueofy/x

%ofboyspassing=25×9/4=561/4%.Ex.7.Acollegehasraised75%oftheamountitneedsforanewbuildingbyreceivinganaveragedonationofRs.600fromthepeoplealreadysolicited.Thepeoplealreadysolicitedrepresent60%ofthepeople,thecollegeaskedfordonations.Ifthecollegeistoraiseexactlytheamountneededforthenewbuilding,whatshouldbetheaveragedonationfromtheremainingpeopletobesolicited?Sol.Letxbethetotalnumberofpeoplethecollegewillaskfordonations.Peoplealreadysolicited=0.6x

Amountraisedfromthepeoplesolicited=600×0.6x=360xNow360xconstitutes75%oftheamount.Hence,remaining25%=120xAveragedonationfromremainingpeople=K$MO

M.iO=300.

Ex.8.IfManuwritesatleasthundredlines,JaspreetdoesnotandifJaspreetwrites,Manudoesnot.Eitherofthemwritesatleast150%andatthemost250%ofwhattheotherwrites.Whatisthemaximumnumberoflinesthatthetwocanwrite?Sol.Maximumthatonecanwritewithoutwriting100lines=99.Then,theothercanwrite,amaximumof99×2.5 248lines.Maximumtotal=347lines

Ex.9.FourfriendsA,B,C,Dhavesomecandieswiththem.Bhas10%morecandiesthanA,whereasChas10%morethanBandDhas10%morethanC.ThenumberofcandieswithAandDtogetheris21morethanBandCtogether.HowmanycandiesdoesBhave?Sol.IfAhasxcandies,A xB 1.1xC 1.21xD 1.331xA+D=2.331xB+C–2.31x0.021x=21

x=1,000Bhas1,100candies.Ex.10.Amanearnsx%ontheRs.2,000andy%ontherestofhisincome.IfheearnsRs.700fromRs.4,000andRs.900fromRs.5,000ofincome,findx.Sol.Thetwoequationscanbewritten2000 O

KMM+2000 Q

KMM=700,and

2000 OKMM

+3000 QKMM

=900.Theequationscanbesimplifiedtox+y=35and2x+3y=90.Solvingthesetwoequationssimultaneously,wegetx=15Ex.11.Agamblerhas125coinswithhim.Wheneverhehascoins≥100inhand,helent20%ofthecoins.Whenhehas≤80coins,heborrows25%ofthecoinsinhand.Whatwillbethenumberofcoinsinhishandattheendof29thtransaction?Sol.Hehas125coins(Morethan100).Sohewilllent20%ofthecoins.AftertheIsttransactionremainingcoins=125–125× $M

KMM=100

Againhewilllendout20%ofthecoins.RemainingcoinafterIIndtransaction=80Now,hewillborrow25%coins.No.ofcoinsafterIIIrdtransaction=100.So,ifyouseethataftereveryeventransactionhewillhave80coins,andaftereveryoddtransactionhewillhave100coins.So,after29thtransactionhewillhave100coins.

Ex.12.GauriwenttothestationersandboughtthingsworthRs.25,outofwhich30paisewentonsalestaxontaxablepurchases.Ifthetaxratewas6%,thenwhatwasthecostofthetaxfreeitems?Sol.LettheamounttaxablepurchasesbeRs.x.

Then,6%ofx=30100

x=

30x100

=5.100 6Costoftaxfreeitems=Rs.[25-(5+0.30)]=Rs.19.70

Ex.13.Afruitsellerhadsomeapples.Hesells40%applesandstillhas420apples.Originally,hehad:Sol.Supposeoriginallyhehadxapples.

Then,(100-40)%ofx=420.

60x𝑥=420

100

x=

420x100=700.60

Ex.14.Inacertainschool,20%ofstudentsarebelow8yearsofage.Thenumberofstudentsabove8yearsofageis𝟐

𝟑ofthenumberofstudentsof8yearsofagewhichis48.Whatisthe

totalnumberofstudentsintheschool?Sol.Letthenumberofstudentsbex.Then,Numberofstudentsabove8yearsofage=(100-20)%ofx=80%ofx.

80%ofx=48+2of48

3

80x=80

100x=100.

Ex.15.TwonumbersAandBaresuchthatthesumof5%ofAand4%ofBistwo-thirdofthesumof6%ofAand8%ofB.FindtheratioofA:B.Sol.

5%ofA+4%ofB=2(6%ofA+8%ofB)

3

5A+

4B =

2

6A+

8B

100 100 3 100 100

1A+

1B =

1A+

4B

20 25 25 75

1-1

A=

4-1

B20 25 75 25

1A=

1B

100 75A=100

=4.

B 75 3

Requiredratio=4:3

Ex.16.Inanelectionbetweentwocandidates,onegot55%ofthetotalvalidvotes,20%ofthevoteswereinvalid.Ifthetotalnumberofvoteswas7500,thenumberofvalidvotesthattheothercandidategot,was:Sol.Numberofvalidvotes=80%of7500=6000.

Validvotespolledbyothercandidate=45%of6000

=

45x6000

=2700.100

Ex.17.Threecandidatescontestedanelectionandreceived1136,7636and11628votesrespectively.Whatpercentageofthetotalvotesdidthewinningcandidateget?Sol.Totalnumberofvotespolled=(1136+7636+11628)=20400.

Requiredpercentage= 11628

x100=57%.20400

Ex.18.TwotailorsXandYarepaidatotalofRs.550perweekbytheiremployer.IfXispaid120percentofthesumpaidtoY,howmuchisYpaidperweek?Sol.LetthesumpaidtoYperweekbeRs.z.Then,z+120%ofz=550.

z+120

z=550100

11z=550

5

z=

550x5=250.11

Ex.19.RajeevbuysgoodworthRs.6650.Hegetsarebateof6%onit.Aftergettingtherebate,hepayssalestax@10%.Findtheamounthewillhavetopayforthegoods.Sol.

Rebate=6%ofRs.6650=Rs.

6x6650

=Rs.399.100

Salestax=10%ofRs.(6650-399)=Rs.

10x6251

=Rs.625.10100

Finalamount=Rs.(6251+625.10)=Rs.6876.10

Ex.20.Thepopulationofatownincreasedfrom1,75,000to2,62,500inadecade.TheaveragepercentincreaseofpopulationperyearisSol.Increasein10years=(262500-175000)=87500.

Increase%= 87500

x100 =50%.175000

Requiredaverage= 50

=5%.10

Ex.21.ItcostsRs.5,000topaintthetotalsurfaceofacube.Butinsummer,eachsideexpandsby20%.Whatwillbethecostofpaintingthecubeinthesummer?Sol.Thetotalsurfaceareaofcube=6a2Since,eachsideisincreasedby20%Newside= 1 + $M

KMMa=h

%a

Totalsurfacearea=6×&h$%a2

Sothecostalsoincreasesby&h$%times.

So,answer=&h$%×5,000=Rs.7,200

Ex.22.Ais10%morethanB,andCis10%lesserthanB.IfthedifferencebetweenAandCis18,whatisthevalueofA?Sol. A=10%morethanB=1.1BC=10%lessthanB=0.9BA–C=18 1.1B–0.9B=18

B=90A=99

Ex.23.Insteadoffinding28%ofanumberSaloniincorrectlyfound82%ofthenumber.Shegotananswerthatis540morethantheanswerthatsheissupposedtoget.Whatistheanswersheissupposedtoget?Sol. LetthenumberbekOriginalanswershouldbe0.28kHeransweris0.82kDifference=0.54kItisgivenas540 k=1,000Answer=0.28k=280

Ex.24.Mr.Mirandasaves40%ofhissalaryeverymonth.ButinAprilhisexpensesincreaseby10%andhesavesRs.1,200lessthantheamounthesavesnormally.Whatishissalary(inRs.)?Sol.Letxbethesalary,0.6xbetheexpenses.Thismonth,theexpensesincreaseby10%andsobecome0.66x.Sosavingsdecreaseby0.06x.0.06x=1,200

x=Rs.20,000

Ex.25.Ineveryboxof120bulbs5%aredefective.Ifacompanyneeds2,000non-defectivebulbs,whatistheminimumnumberofboxesithastopurchase?Sol.Eachboxcontains114non-defectivebulbs.Answer=$MMM

KKi=17.4

Sothecompanyhastopurchaseaminimumof18boxestomeetitsrequirementofbulbs.Ex.26.Padmapurchasedaboxofchocolates.Sheate20%ofthemandgave25%toherfriendsfromtheremaining.Again,sheate20%ofitandgave25%toherfriendsfromtheremaining.Thenshegave8chocolatestoaguest.Finally,sheisleftwith35%ofthechocolatesthatsheinitiallyhad.HowmanychocolatesdidPadmaeat?Sol.Assumetheboxcontainsxnumberofchocolates.

1sttime 0.2xeaten

Giventofriend0.2x Leftwithhim=0.6x

2ndtime 0.12xeaten

Tofriend=25%of0.48x=0.12x

Left=0.36x

Remaining=0.36xItisgiven,0.36x-8=0.35xx=800

Sheate800chocolates.0.2x+0.12x=0.32x=256Ex.27.ThecostofanairticketisRs.3,000+tax.Ifadiscountof50%isallowedonthecostoftheairticket(nottax),thecostoftheticketdecreasesby30%.Whatisamountoftaxchargedonthepurchaseoftheticket?Sol.70%of(3,000+tax)=1,500+tax2,100+0.7tax=1,500+tax

tax=2,000Ex.28.Ifastudentscored30%marksinapaper,hefailsby6marksbutifhescores40%markshepassesby2marks.Whatarethemaximummarks?Sol.Samethingsontwodifferentscalesgiven

(Differenceofpercentage)oftotalmarks=(6+2)10%ofmaximummarks=8Maximummarks=80Ex.29.Ifaquantityisincreasedby30%andthendecreasedby50%,byhowmuchpercentagedoesithavetoincreasetobringitbacktotheoriginalquantity?Sol.Assumethequantityas100.

Ifitisincreasedby30%,itwillbe130andifitdecreasesby50%,itwillbecome65.Soithastobeincreasedby35tocometo100again.Soanswer=&%

h%×100 54%.

Ex.30.ThesalaryofapersonisRs.15,000.ButinthenextmonthhegetsabonusofRs.1,000.Hedecidedtosavethebonusandintheprocesssaved10%morethantheamounthesavedthepreviousmonth.Ifthereisnochangeinhisexpensesfromthelastmonth,whatarehismonthlyexpenses?Sol.Rs.1000,increaseshissavingsby10%HispreviousmonthsavingswereRs.10000Soexpenses=15000–10000=Rs.5000Ex.31.xisapositiveinteger.Howmanyvalues‘x’willtakesuchthatx%of250islessthan60?Sol.24%of250is60.xshouldbelessthan24.

Itcanbeanyintegerfrom1to23.So,ittakes23values.Ex.32.Inacertaincinemahall,thereare4kindsoftickets,ACBoxes,Boxes,BalconyandDC.Therelationshipbetweentheratesofticketsis:priceofboxesis20%higherthanDC,DCis40%lessthanACboxesandpriceofbalconyis30%ofthesumofthepriceofDCandACBoxes.IfthesumofthepriceofboxesandACboxesisRs.86thenfindthepriceofthebalcony.Sol.LetsaythepriceoftheACboxes=Rs.xThenthepriceoftheDC=Rs.0.6xThepriceoftheboxes=0.72xAnd,thepriceofthebalcony= &M

KMM(0.6x+x)

=0.48xAnd,0.72x+x=86x= eh

K.g$=50

Sothepriceofbalcony=0.48 50=Rs.24.Ex.33.Inanelectiontherearetwocandidates.Acandidategets40%ofthetotalvotesandlosestheelectionby9,800votes.Findthetotalnumberofvotes.Sol.Winnermustgot60%oftotalvotes%agedifference=20%Andvotesdifference=9800So,totalvotes×20%=9800Totalvotes=49000Ex.34.Theratioofthenumberofmalesandfemalesinavillageis3:2.20%ofthemalesand25%ofthefemalesaregraduates.Thenfindthe%ofthetotalpopulationofthevillagewhoarenotgraduates?

Sol.Letssaythatthemalepopulation=300Letssaythatthefemalepopulation=200GraduateMalepopulation=300× $M

KMM=60

GraduateFemalepopulation=200× $%KMM=50

Noteeducated=500–110=390%ofnoteducated=&jM

%MM×100=78%.

Ex.35.Mr.Rajwasthrowingapartyforhisfriends,andforthatherequired37kgofsweets.Assuming19%wastageandknowingthatthesweetsareavailablein2kgpacketsonly,whatwouldbethecostofthesweetspurchased,ifonepacketcostsRs.15?Sol.HehastobuyXkgsuchthat81%ofX=37kg

X=37×KMMeK

=45.67kg

Hehastobuy46kg(multipleof2kg),i.e.23packetsofsweets.Totalcost=15×23=Rs.345

Ex.36.IfthesalaryofAis20%morethanthatofB,bywhatpercentsalaryofBislessthanthatofA?Sol.LetB'ssalary=100Thus,A'ssalary=100x1.2=120Therefore, $M

K$Mx100=KMM

h=16.66%

Hence,B'ssalaryis16.66%lessthanAEx.37.Asalesman’scommissionis5%onallsalesuptoRs.10,000and4%onallsalesexceedingthisamount.HeremitsRs.31,100totheparentcompanyafterdeductinghiscommission.HissalesisworthSol.Letthetotalsalesamount=RsxThen,thecommission,whichsalesmangotingivenas5%of10000+4%of(x–10000)=x–31,100Bysolving,wegetx=32500Ex.38.InthemonthofJanuary,therailwaypolicecaught4000ticketlesstravelers,inFebruary,thenumberroseby5%.However,duetoconstantvigilbythepoliceandtherailwaystaff,thenumberreducedby5%inMarchandinAprilitfurtherreducedby10%.ThetotalnumberofticketlesstravelerscaughtinthemonthofAprilwasSol.TotalnumberofticketlesstravelersInFebruary=4000+5%=4200InMarch=4200-5%=3990InApril=3990-10%=3591

Exercise1.WhenthepriceofT.V.setswasincreasedby30%,thenumberofT.V.setssolddecreasedby20%.Whatwastheeffectonthesales?a.5%increaseb.7%increasec.4%increased.8%increasee.Noneofthese2.Kiranearns50%ofRaghu’ssalary.Raghuearns80%ofDinesh’ssalary.IfthetotalsalaryofthethreeforamonthisRs.22,000,howmuchdidKiranearnthatmonth?a.5600b.5000c.4000d.4500e.Noneofthese3.Janakispends20%ofhermonthlysalaryforherson’seducation.WhenshewillgetherincrementofRs.170nextmonthsheplanstospendhalfofthatalsoforherson’seducation.ThetotalshewouldspendthenwouldbeRs.645.Whatisherpresentsalary?a.5600b.3200c.2800d.4500e.Noneofthese4.Theclothshophasannouncedareductioninpriceby25%.HowmanykurtaspricedatRs.28shouldonebuytoavailatotalreductionofatleastRs.40?a.5b.6c.4d.7e.Noneofthese5.One-fourthofanumberis10lessthan30percentofthesamenumber.Whatisthatnumber?a.600b.200c.800d.500e.Noneofthese6.Thelengthofarectangleisincreasedby20%.Bywhatpercentshouldthewidthbereducedtomaintainthesamearea?a.15.5b.16.5c.16.66d.18e.Noneofthese7.Whatwillbe40%ofanumberwhose200%is90?a.15.5b.16.5c.16.66d.18e.Noneofthese8.Gopalspends60%ofhissalaryonfooditems,50%oftheremainingonclothesandtransportandsavestheremainingamounteverymonth.IfhisannualsavingsareRs.9,600howmuchmoneydoeshespendonthetransportpermonth?a.2600b.6200c.4800d.datainadequatee.Noneofthese9.Ifone-thirdofone-fourthofanumberis15,whatis30%ofthatnumber?a.56b.54c.58d.50e.Noneofthese10.Ifthree-fourthofanumberis244,whatwillbetheapproximatevalueof136percentofthatnumber?a.650b.250c.850d.450e.Noneofthese11.Whatwillbetheapproximatevalueof161%of841?a.1200b.1350c.1800d.1500e.Noneofthese

12.Whenthepriceofatoywasincreasedby30%,thenumberoftoyssoldwasdecreasedby20%.Whatwastheeffectonthesales?a.6%increaseb.4%deacreasec.4%increased.5%deacreasee.Noneofthese13.30%ofA’ssalaryisequaltothe20%of&

%thofB’ssalary.IfB’ssalaryisRs.2400,whatisA’s

salary?a.960b.200c.800d.500e.Noneofthese14.Inalibrary,40%ofthetotalnumbersofbooksareinEnglish,80%oftherestareinHindiandthebalance300booksareinotherlanguages.Whatisthetotalnumberofbooksinthelibrary?a.2000b.2500c.2800d.2500e.Noneofthese15.Abusinessmaninvests70%inmachinery,20%inrawmaterialandhasRs.16000cashwithhim.Whatisthetotalinvestment?a.1,60,000b.1,20,000c.1,80,000d.1,50,000e.Noneofthese16.Amansaves25%ofhismonthlysalary.Ifonaccountofriseinpricesheistoincreasehismonthlyexpensesby25%heisonlyabletosaveRs.250permonth.Whatishismonthlysalary?a.6000b.4000c.8000d.5000e.Noneofthese17.Atul’smarksinscienceare20lessthan60%ofthetotalmarksobtainedbyhiminscienceandmathstogether.Ifthetotalmarksare250,whatarehismarksinmaths?a.360b.240c.480d.120e.Noneofthese18.Inaselectionprocesscomprisingofwrittentest,grouptasksandinterview,only40%qualifiedinwrittentest,80%ofthosepassedinwrittentestqualifiedingrouptasksand65%ofthesepassedingrouptasksqualifiedforinterview.Ifthenumberofthosequalifiedfinallywasonly78,whatwasthetotalnumberofcandidateswhoappeared?a.450b.225c.375d.325e.Noneofthese19.Acandidateshouldscore45%marksofthetotalmarkstopasstheexamination.Hegets520marksandfailsby20marks.Thetotalmarksintheexaminationarea.2400b.1200c.1800d.1500e.Noneofthese20.Ifthepopulationofthetownis64000anditsannualincreaseis10%,thenitscorrectpopulationattheendof3yearswillbea.85484b.85384c.85184d.85284e.Noneofthese21.Ifx%of1200is20thenxisa.2.5b.2c.1.66d.1.5e.Noneofthese22.Ifyexceedsxby30%thenxislessthanyby

a.23 KK&%b.23 $

K&%c.23 K

KM%d.24 K

K&%e.Noneofthese

23.When80issubtractedfrom80%ofanumber,theresultis80,andthenumberisa.400b.200c.800d.500e.Noneofthese24.Thepriceofanarticleiscutby25%.Torestoreittoitsoriginalprice,thenewpricemustbeincreasedbya.30.5b.33.33c.32.333d.66.66e.Noneofthese25.AandB’ssalariestogetheramounttoRs.4000.Aspends95%ofhissalaryandB,85%ofhis.Ifnowtheirsavingsarethesame,whatisA’ssalary?a.4000b.2000c.8000d.3000e.Noneofthese26.Inanexamination,15,000candidatesappeared.Eachofferedeithermathsorcomputerscienceorboth.If75%offeredmathsand45%computersciencehowmanyofferedboth?a.4000b.2000c.8000d.3000e.Noneofthese27.Areductioninthepriceoforangeenablesapersontopurchase5orangesforRe.1insteadofRs.1.50.Whatisthepercentagereductioninprice?a.33.33%b.20%c.25%d.50%e.Noneofthese28.A’sincomeis‘25%lessthanB’sincomeandB’sincomeis20%lessthanC’sincome.IfC’sincomeisRs.200,thentheincomeofAisa.150b.180c.120d.80e.Noneofthese29.Ramesh’ssalarywasreducedby20%.Inordertohavehissalarybacktooriginalamountitmustberaisedbya.33.33%b.20%c.25%d.50%e.Noneofthese30.Ashop’sreceiptsonthe1stdayoftheweekwereRs.20,000onthesecondday25%morethanthisandonthe3rdday75%ofthesumofthereceiptsofthefirst2days.Whatwastheaverageofthereceiptsduringthisthreedayperiod(inrupees)?a.24000b.26500c.26250d.23000e.Noneofthese31.Amanpays10%ofhissalaryastax.Ifafterspending90%oftheremainder,hehasRs.180leftwithhim,whatishissalarybeforetaxation?a.4000b.2000c.8000d.3000e.Noneofthese32.Theamountofwaterthatshouldbeaddedtoreduce8mlmixercontaining50%alcohol,toamixercontaining40%alcoholisa.4mlb.2mlc.8mld.3mle.Noneofthese33.Amanbymeansofhisfalsebalancedefraudshiscostumerstotheextentof30%bypurchasingthegoodsaswellasbysellingthegoodsfindhisnetprofita.60b.65c.70d.69e.Noneofthese

34.Thesalaryofatypistwasfirstraisedby20%andthenthesamewasreducedby10%.IfhepresentlydrawsRs.1100,whatwashisoriginalsalary?a.1018.52b.1016.52c.1015.52d.1012.52e.Noneofthese35.Inanexamination65%ofthecandidatespassedinMathsand75%passedinScience.If22%failedinboth,findthepasspercentagewhopassedinbothsubjects.a.62%b.20%c.25%d.50%e.Noneofthese36.Inanelection,acandidatesecured42%ofthevotes.Theothercandidatessecured58%anddefeatedhimby480votes.Findthenumberofvotespolleda.4000b.2000c.8000d.3000e.Noneofthese37.Twonumbersare25%and50%lessthanathirdnumber.Expressfirstnumberasapercentageoftheseconda.162%b.120%c.125%d.150%e.Noneofthese38.AmanearnsRs.7000inamonthandspends70%ofhisincome.Howmuchdoeshesaveinayear?a.25,200b.25000c.22,500d.26,500e.Noneofthese39.Thesalaryofaworkerisfirstincreasedby50%andafterwardsreducedby50%.Whatisnetchangeinhissalary?a.33.33%b.20%c.25%d.50%e.Noneofthese40.Twonumbersarerespectively10%and20%morethanathirdnumber,thepercentagethatisfirstofthesecondisa.91.6%b.80%c.75%d.95%e.Noneofthese41.Apersonsaveseveryyear10%ofhisincome,ifhisincomeincreasedeveryyearby5%thenhissavingsincreaseeveryyearbya.5%b.2%c.4%d.6%e.Noneofthese42.If40%ofanumberisequalto240.Findthenumbera.500b.600c.800d.750e.Noneofthese43.IfA%ofanumberisequaltoB%ofsecondnumber,thentherationbetweenfirstandsecondnumberisa.A:Bb.B:Ac.(A+B):Ad.(A+B):Be.Noneofthese44.Whatis60%of30%of900grams?a.163gmsb.164gmsc.170gmsd.162gmse.Noneofthese45.If40%of7K

$=x%of80.Whatisthevalueofx?

a.3.5b.3.25c.3.75d.4.5e.Noneofthese

46.Whatratepercentis1minute48secondstoanhour?a.5%b.2%c.4%d.6%e.Noneofthese47.Iftwonumbersarerespectivelyx%andy%morethanathirdnumber,thefirstasapercentofsecondisa.KMMPO

KMMPQ×100%b.KMMPO

KMMNQ×100%c.KMMNO

KMMPQ×100%d.KMMNO

KMMNQ×100%

e.Noneofthese48.Twonumbersare20%and40%lessthanathirdnumber.Thenfirstnumberaspercentageofthesecondis…..a.133.33%b.120%c.125%d.150%e.Noneofthese49.Adding12%ofxtoxisequivalenttomultiplyingxbywhat?a.133.33%b.120%c.112%d.150%e.Noneofthese50.Subtracting21%ofxfromxisequivalenttomultiplyingxbywhata.0.79b.0.5c.0.25d.0.85e.Noneofthese51.IfA’ssalaryis50%lessthanthatofB,thenhowmuchpercentisB’ssalarymorethanthatofA?a.100%b.120%c.125%d.150%e.Noneofthese52.Ifthepriceofacommodityincreasesbyx%,thenreductioninconsumption,soasnottoincreasetheexpenditureisa. O

KMMNO×100 %b. (KMMNO)

KMMPO×100 %c. (KMMNO)

KMMNO×100 %d. O

KMMPO×100 %

e.Noneofthese53.Ifthepriceofcoffeeseedsisincreasedby50%,findbyhowmuchpercentahouseholdermustreducetheconsumptionofcoffeesoasnottoincreasetheexpenditure?a.33.33%b.20%c.25%d.50%e.Noneofthese54.Ifthepriceofricefallsdownby20%byhowmuchpercentmustahouseholderincreasehisconsumption,soasnottodecreaseexpenditureonthisitem?a.33.33%b.20%c.25%d.50%e.Noneofthese55.Thepriceofcommodityisdiminishedby40%anditsconsumptionincreasesby16%findtheeffectontherevenue.a.3%increaseb.2%increasec.5%decreased.4%decreasee.Noneofthese56.A%ofBisB%of?a.Ab.Bc.(A+B)d.(A-B)%e.Noneofthese

57.Thepriceofaradioiscutby20%.Torestoreittotheformervaluethenewpricemustbeincreasedbya.33.33%b.20%c.25%d.50%e.Noneofthese58.Inanexamination65%ofthestudentspassedinScience,68%passedinMathsand25%failedinbothsubject.Findthepasspercentage?a.50%b.58%c.60%d.75%e.Noneofthese59.Inanexamination75%ofthestudentspassedinEnglishand55%passedinScienceand33%failedinboththesubjects.Findthepasspercentage?a.50%b.58%c.63%d.75%e.Noneofthese60.IfA%failedinsubject1,andB%isfailedinsubject2,andC%failedinboththesubject.Findthepasspercentagea.[100-(A-B-C)]%b.[100+(A+B-C)]%c.[100-(A-B+C)]%d.[100-(A+B-C)]%e.Noneofthese61.Areductionof40%inthepriceoforangeswouldenableapersontoget2dozenmoreforRs.5.Findthereducedpriceperdozena.Re.4b.Re.1c.Re.2d.Re.3e.Noneofthese62.Twonumbersare40%and44%lessthanathirdnumber.Howmuchpercentisthesecondnumberlessthanthefirst?a.5%b.6.5%c.6.66%d.6.33%e.Noneofthese63.Thesalaryofapersonisfirstincreasedby20%andthenreducedby20%whatisnetchangeinhissalary?a.3%increaseb.4%increasec.5%decreased.4%decreasee.Noneofthese64.Ifthepricehasfallenby25%,whatpercentofitsconsumptionbeincreasedsothattheexpendituremaybethesameasbefore?a.33.33%b.20%c.25%d.50%e.Noneofthese65.ThesalaryofArunisfirstdecreasedby20%andthenisincreasedby30%.Findtheneteffect?a.3%increaseb.4%increasec.5%decreased.4%decreasee.Noneofthese66.SellingpriceofaT.V.isincreasedby10%.Asaresultofit,thesaleisdecreasedby20%.Findtheeffectonsalesa.13%increaseb.14%increasec.12%decreased.14%decreasee.Noneofthese67.IfY’ssalaryis20%morethanX’sandZ’ssalaryis10%lessthanY’s.HowmanypercentZ’ssalaryismorethanX’s?a.5%b.8%c.6%d.7%e.Noneofthese

68.Y’sincomeis30%morethanX’sandZ’sincomeis40%morethanY’s.HowmuchpercentisZ’ssalaryismorethanX’s?a.50%b.85%c.82%d.90%e.Noneofthese69.PriceofTeaisincreasedby50%.Byhowmuch,newpricemustbereducedtorestoreittotheoriginalprice?a.33.33%b.20%c.25%d.50%e.Noneofthese70.PriceofCoffeeisreducedby25%byhowmuchnewpricemustbeincreasedbytorestoreittooriginalprice?a.33.33%b.20%c.25%d.50%e.Noneofthese71.When70issubtractedfrom70%ofanumber,theresultis70.Findthenumbera.400b.800c.250d.200e.Noneofthese72.Thepriceofcookinggashasincreasedby50%thepercentageofreductionthatafamilyshouldeffectintheuseofgassoasnottoincreasetheexpenditureonthisaccountisa.33.33%b.20%c.25%d.50%e.Noneofthese73.Thepriceofoilisincreasedby25%.Iftheexpenditureisnotallowedtoincrease,theratiobetweenthereductioninconsumptionandoriginalconsumptionisa.5:1b.1:5c.3:2d.2:3e.Noneofthese74.Thepopulationofavillageis1600in1997.Itincreases20%seatsannually.Howmanystudentswillbetherein1999?a.2300b.2100c.2302d.2306e.Noneofthese75.xissixtimesaslargeasy.Thepercentagebywhichyislessthanxisa.83.33%b.80%c.82%d.85%e.Noneofthese76.Onelitreofwaterisevaporatedfrom20litresofsolutioncontaining5%salt.Percentageofsaltintheremainingsolutionisa.33.33%b.20%c.25%d.50%e.Noneofthese77.Whichnumberis70%lessthan90?a.30b.33c.27d.36e.Noneofthese78.Anumberexceeds30%ofitselfby60.Thenumberisa.85%

eb.85%

gc.85i

gd.85&

ge.Noneofthese

79.Whatpercentof5%of10%a.33.33%b.20%c.25%d.50%e.Noneofthese80.Subtracting8%ofAfromAisequivalenttomultiplyingAbyhowmuch?a.0.92b.0.5c.0.25d.0.8e.Noneofthese

Answers1.C 2.C 3.C 4.B 5.B 6.C 7.D 8.D 9.B 10.D11.B 12.C 13.A 14.B 15.A 16.B 17.D 18.C 19.B 20.C21.C 22.A 23.B 24.B 25.D 26.D 27.A 28.C 29.C 30.C31.B 32.B 33.D 34.A 35.A 36.D 37.D 38.A 39.C 40.A41.A 42.B 43.B 44.D 45.C 46.C 47.A 48.A 49.C 50.A51.A 52.D 53.A 54.C 55.D 56.A 57.C 58.B 59.C 60.D61.B 62.C 63.B 64.A 65.B 66.C 67.B 68.C 69.A 70.A71.D 72.A 73.B 74.E 75.A 76.B 77.C 78.B 79.D 80.A

Profit&Loss

CostPrice:Theprice,atwhichanarticleispurchased,iscalleditscostprice,abbreviatedasC.P.SellingPrice:Theprice,atwhichanarticleissold,iscalleditssellingprices,abbreviatedasS.P.ProfitorGain:IfS.P.isgreaterthanC.P.,thesellerissaidtohaveaprofitorgain.Loss:IfS.P.islessthanC.P.,thesellerissaidtohaveincurredaloss.IMPORTANTFORMULAE

1. Gain=(S.P.)-(C.P.)2. Loss=(C.P.)-(S.P.)3. LossorgainisalwaysreckonedonC.P.4. GainPercentage:(Gain%)

Gain%=Gainx100

C.P.

5. LossPercentage:(Loss%)

Loss%=

Lossx100

C.P.6. SellingPrice:(S.P.)

SP=

(100+Gain%)xC.P

100

7. SellingPrice:(S.P.)

SP=

(100-Loss%)xC.P.

100

8. CostPrice:(C.P.)

C.P.=

100xS.P.

(100+Gain%)

9. CostPrice:(C.P.)

C.P.=

100xS.P.

(100-Loss%)

10. Ifanarticleissoldatagainofsay35%,thenS.P.=135%ofC.P.11. Ifanarticleissoldatalossofsay,35%thenS.P.=65%ofC.P.12. Whenapersonsellstwosimilaritems,oneatagainofsayx%,andtheotherataloss

ofx%,thentheselleralwaysincursalossgivenby:Loss%=

CommonLossandGain%

2 =

x

2.

10 1013. Ifatraderprofessestosellhisgoodsatcostprice,butusesfalseweights,then

Gain%=

Errorx100

%.(TrueValue)-(Error)Ifshopkeepermarkstheitembym%(ofCP)andgivesd%discountthen, Profit%=𝒎− 𝒅− 𝒎×𝒅

𝟏𝟎𝟎

Solvedexamples

Ex.1.OnsellinganarticleforRs.240,atraderincursalossof4%.Atwhatamountmusthesellthearticleforsothathemakesaprofitof10%?Sol.S.P.=Rs.240Loss=4%

0.96C.P=240 C.P=250Nowtraderwantstoearn10%.ThusnewS.P=250 1.1=Rs.275.ShortCutmethod: �K

KMMPOK= �$

KMMPO$whereS1=Rs.240,x1=–4%,x2=10%

SolvingwegetS2=Rs.275.Ex.2.MohansoldtwobooksatRs.140each.Hemadeaprofitof20%onsellingoneofthebooks,andincurredalossof20%onsellingtheother.Whatistheoutcomeofsellingboththebooks?Sol.LetthecombinedcostpriceofbotharticlesbeRs.Y.SellingpriceofbotharticlesisRs.280.Overallloss%= O5

KMM%=(400/100)=4%(wherex=20)

0.96Y=280Y=Rs.291.66

Overallloss=291.66–280=Rs.11.66=Rs.12(appx.)Ex.3.IftheSPof12articles=CPof10articles,whatistheloss%?Sol.Thereisalossof2articlesforevery12articlesTherefore%loss=2/12 100=16.66%.Ex.4.IfatoyissoldforRs.10.80,therewouldbealossof10%.Atwhatpriceshoulditbesoldtoearnaprofitof20%?Sol.Accordingtothegivencondition:0.9x=10.8(wherexisthecostpriceofthetoy)

x=12Sellingpricetogain20%=1.212=14.4Ex.5.AcycleissoldforRs.880atalossof20%.Atwhatpriceshoulditbesoldsothatthere

isagainof10%?Sol.S.P.=Rs.880Loss=20%LetC.P.bex.Then0.8x=880

x=Rs.1100Nowgain=10%∴NewS.P.=1.1 1100=Rs.121Ex.6.BysellingatransistorforRs.572,ashopkeeperearnsaprofitequivalentto30%ofitscostprice.Finditscostprice.Sol.LetC.P.=xGivenS.P.=572Profit= &M

KMMx=0.3x

C.P.+Profit=S.P.x+0.3x=5721.3x=572x=Rs.440Ex.7.Amilkmanbuys10equalsizedmilkdrums.IfhesellsmilkatRs.5perlitre,helosesRs.200butifhesellsitatRs.6alitre,hewillgainRs.150.Whatisthecapacityofeachdruminlitres?Sol.Lettotalmilkinonevesselbexlitre.Milkin10vesselswillbecome10x.ForRs.5perlitre,totalmilkcost=10x 5=50xForRs.6perlitre,totalmilkcost=10x 6=60xFromthegiveninformation,wecansaythat60x–50x=350

10x=350x=35litres

Ex.8.Abusinessmanmarkshisgoodsinsuchawaythatevenafterallowing12.5%discountonacashpurchasehegains20%.IfthecostpriceisRs.140,themarkedpriceisSol.12.5%(1/8)discountisonthemarkedprice(M.P)

S.P=7/8M.P………(1)AlsoC.P=Rs.140Profit=20%

S.P.=1.2 140=168………(2)From(1)and(2)7/8M.P=168 M.P=Rs.192.Ex.9.Adealermarksarticlesatapricethatgiveshimaprofitof30%.6%ofthegoodswerelostinafire,24%weresoiledandhadtobesoldathalfthecostprice.Iftheremainderweresoldatthemarkedprice,whatpercentageprofitorlossdidthedealerearn?Sol.Letsaytotalarticle=x

Letsaythecostofeacharticle=Rs.100Markedprice=130Sellingpriceofsoiledarticle(24%)=x× $i

KMM×50=12x

Sellingpriceofrestofthearticle=x× gM

KMM×130=91x

Totalsellingprice=91x+12x=103xProfit= &O

KMMO×100=3%.

Ex.10.Fromabatchofironore,80%ironcanbeextracted.ThecostoftheoreisRs.0.30perkgandthecosttoproduceironfrom100kgoforeisRs.40.IfeachkgofironissoldatRe.1,whatistheprofitpercentage?Sol.Ifwetake100kgore,itcostsRs.30andproductioncostisRs.30.Fromthiswewillget80kgofironSo,toget80kgofiron,Rs.70isrequired.Bysellingthis80kgironatRe.1perkg,itcomesRs.80Sopercentageprofit=eMNgM

gM×100=14.28%

Ex.11.Ashopkeeperisabletomakeaprofitof25%evenafterallowingadiscountof25%.Howmuchprofitwillhebeabletomakebynotallowinganydiscount?Sol.LetxbetheM.P.andybetheC.P.So,ifhegives25%discount,hisS.P.willbe0.75xSincehegot25%profit,hisS.P.shouldbe1.25y∴0.75x=1.25y x=%

&y

66.66%morethanyi.e.theM.P.is66.66%morethantheC.P.Ex.12.Abusinessmanpurchasedanarticleandmarkeditspriceas30%more.Afterallowing10%discount,heearnedRs.1,020asprofit.Whatisthecostpriceofthearticle?Sol.LetC.P.ofthearticlebeRs.x.So,M.P.willbecome1.3x.Aftergiving10%discount,hisS.P.=0.9×1.3x.=1.17x

Profit=0.17xButtheprofitisgivenas1,020,∴0.17x=1,020

x=Rs.6,000Ex.13.Amanpurchasedapacketof10pensandgot2pensfreewithit.Hemarkedtheprice20%morethantheC.P.Ifhedidnotallowanydiscountandsoldallthe12pensatthisprice,whatwashisprofitpercentage?Sol.AssumeC.P.ofeachpenisRe.1.So,C.P.isRs.10( hepurchased10pens).

M.P.ofeachpenisRs.1.20andhehas12pensnow.So,hewillget12´(1.20)=Rs.14.4Thepercentageprofit=Ki.iNKM

KM×100=44%

Ex.14.Thecostpriceof20articlesisthesameasthesellingpriceofxarticles.Iftheprofitis25%,thenthevalueofxis:Sol.LetC.P.ofeacharticlebeRe.1C.P.ofxarticles=Rs.x.S.P.ofxarticles=Rs.20.Profit=Rs.(20-x).

20-xx100=25

x2000-100x=25x

125x=2000x=16.

Ex.15.Inacertainstore,theprofitis320%ofthecost.Ifthecostincreasesby25%butthesellingpriceremainsconstant,approximatelywhatpercentageofthesellingpriceistheprofit?Sol.LetC.P.=Rs.100.Then,Profit=Rs.320,S.P.=Rs.420.NewC.P.=125%ofRs.100=Rs.125NewS.P.=Rs.420.Profit=Rs.(420-125)=Rs.295.

Requiredpercentage= 295

x100 =1475

%=70%(approximately).420 21

Ex.16.ThepercentageprofitearnedbysellinganarticleforRs.1920isequaltothepercentagelossincurredbysellingthesamearticleforRs.1280.Atwhatpriceshouldthearticlebesoldtomake25%profit?Sol.LetC.P.beRs.x.

Then,1920-x

x100=x-1280

x100x x

1920-x=x-12802x=3200x=1600

RequiredS.P.=125%ofRs.1600=Rs.

125x1600

=Rs2000.100

Ex.17.Sampurchased20dozensoftoysattherateofRs.375perdozen.HesoldeachoneofthemattherateofRs.33.Whatwashispercentageprofit?Sol.

CostPriceof1toy=Rs.

375

=Rs.31.2512

SellingPriceof1toy=Rs.33

So,Gain=Rs.(33-31.25)=Rs.1.75

Profit%=

1.75x100

=28

%=5.6%31.25 5

Ex.18.WhenaplotissoldforRs.18,700,theownerloses15%.Atwhatpricemustthatplotbesoldinordertogain15%?Sol.85:18700=115:x

x=

18700x115

=25300.85

Hence,S.P.=Rs.25,300.Ex.19.Atradermixes26kgofriceatRs.20perkgwith30kgofriceofothervarietyatRs.36perkgandsellsthemixtureatRs.30perkg.Hisprofitpercentis:Sol.C.P.of56kgrice=Rs.(26x20+30x36)=Rs.(520+1080)=Rs.1600.S.P.of56kgrice=Rs.(56x30)=Rs.1680.

Gain= 80

x100=5%.1600

Ex.20.AsoldawatchtoBatagainof5%andBsoldittoCatagainof4%.IfCpaidRs.1,092forit,thepricepaidbyAisSol.GivenCpaidRs.1,092LetApaidRs.XThenamountpaidbyBtoA=Rs.1.05XAmountpaidbyCtoB=1.04×1.05X=1,092X=Rs.1,000ShortCutmethod:Fromoptions:LetApaidRs.1,000ThenBpaidtoA=1,000× %

KMM+1,000=Rs.1,050

Moreover,CpaidtoB=1,050+ 1050× iKMM

=Rs.1092.Itsatisfiesthegivencondition.Ex.21.AsellsanarticletoBataprofitof20%.BsellsittoCataprofitof15%.CpaidRs.190morethanitcostA.WhatprofitdidAmake?Sol.LetthearticlecostRs.XtoAA’ssellingprice=B’scostprice=X×1.2B’ssellingprice=C’scostprice=X×1.2×1.15Now,X×1.2×1.15–X=190

1.38X–X=1900.38X=190X=KjM×KMM

&e=500

A’sprofit=20%of500=Rs.100

Ex.22.Atradesman,bymeansofafalsebalance,defrauds10%inbuyinggoodsand10%inselling.What%doeshegainonhisoutlaybyhisdishonesty?Sol.Assumethetradesmanbuy100gmforRs.100.Sincehedefrauds10%,hewillget110gmforRs.100.Whileselling,hewillgiveonly90gmforRs.100.So,on110gm,hecanearnKMM×KKM

jM=122.22,so,theprofitis22.22%

Ex.23.Aclothmerchantsaysthatduetoslumpinthemarket,hesellstheclothat10%loss,butheusesafalsemetre–scaleandactuallygains15%.Findtheactuallengthofthescale.Sol.Pleasenotethatinthisquestionheisgainingandlosingatthesametime.Itmeanshissellingpriceandcostpricewillbedifferent.Nowheisgaining15%ontheactualscaleheisusing(falsescale)andheisshowinghislossonthetruescaleofonemetre.Letfalsescalelength=Lcm.Accordingtothegivenconditions1.15L=0.9×100.L= jM

KK%×100=78.25cm.

Ex.24.Apharmaceuticalcompanyproduces6,000bottlesofcoughsyrupforRs.132,000.Itdistributed20%ofthebottlestothedoctorsasasampleandsold2/3oftheremainingbottlesat25%discount.Theremainingbottlesweresoldtoshopkeepersat10%discountontheprintedpriceofthebottlethatwas50%morethanthecostprice.Findtheprofit/loss.Sol.Costprice=K&$MM

hMMM=Rs.22

SellingPrice=22×K%MKMM

=331,200bottlesaregivenforfree.Sellingpriceofrestofthebottles=4,800×$

&× g%

KMM×33=79,200

SPoftheremaining=1,600×33× jMKMM

=47,520TotalS.P.=126,720Loss(%)= %$eM

K&$MMM×100=4%

Ex.25.Afruitvendorpurchased100applesfromwholesalemarketatRs.50each.If10ofthemweredamaged,whatpriceshouldbechargedontheremainingapplestoearn20%profit?Sol.Hisinvestment=100×50=5,000Ifhewantstoget20%profit,hehastoearnRs.6,000on90apples.So,eachapplecostshMMM

jM=Rs.66.66

Ex.26.TherearetwotypesofironoreAandB.OreAhas80%ironandBhas60%iron.Thecostof1kgoforeAisRs.1.50andoforeBisRs.1.20.(Assumenootherproductexceptironwillcomefromtheore).Iftheprocessingcostisnegligibleandthereisnoothercost,whichofthetwooresismoreprofitabletoprocess?Sol.100kgoforeAgives80kgofironandcostsRs.150.100kgoforeBgives60kgofironandcostsRs.120.

So,Aisbetter.Ex.27.Byselling12mangoes,afruitsellergainsaprofitequaltosellingpriceof2mangoes.Whatishisprofitpercent?Sol.ConsiderS.P.ofamangoasRe.1P=2,S.P.=12,C.P.=10Profit= $

KM×100=20%

Ex.28.Ifmilkmanmixes1litrepurewaterinevery4litreofpuremilkandsolditatthecostprice.Whatistheprofitpercentearnedbyhim?Sol.Find��

U�=%

imaking25%profit

Ex.30.Byselling12articles,ashopkeepergainsaprofitequaltocostpriceof2articles.Whatishisprofitpercent?Sol.Assume,CPofonearticleisRe.1.So,byselling12articles(worthRs.12)hecanmakeaprofitofRs.2So,profit%= $

K$×100=16.66%

Ex.31.Amanufacturerestimatesthatoninspection,12%ofthearticlesproducedwillberejected.Heacceptsanordertosupply22,000articlesatRs.7.50each.Heestimatestheprofitonhisoutlayincludingthemanufacturingofrejectedarticles,tobe20%.Findthecostofmanufacturingeacharticle.Sol.Whenthemanufacturermakes100articles,hesellsonly88.Revenues=88×7.5=Rs.660Butprofitbeing20%ofoutlay,totaloutlaycostof100articlesisRs.550.SothecostofperarticleisRs.5.50Ex.32.Therewouldbe10%loss,ifatoyissoldatRs.10.80perpiece.Atwhatpriceshoulditbesoldtoearnaprofitof20%?Sol.Accordingtothegivencondition:0.9x=10.8(wherexisthecostpriceofthetoy)

x=12Sellingpricetogain20%=1.2×12=14.4Ex.33.Asweetmakermade620sweetsatacostof80paisepersweet.Outofthese,120sweetsgotspoiledandhesoldrestofthesweetsinsuchaway,thatwhenhesold400sweets,hewouldhavemadeaprofitof50%onthetotalcostof620sweets.Findhisactualprofit%onthetotaloutlay.Sol.Costofthesweets=620×0.80=Rs.496Letthesellingpricepersweet=Rs.xSellingpriceof400sweets=400x400x=496×K%M

KMM

x=K%M×ijhKMM×iMM

=Rs.1.86Sellingpriceof500sweets=1.86×500=930.

Profit(%)=i&iijh

×100=87.5Ex.34.RajgopalanpurchasedasecondhandTVforRs.4,800andspent20%ofthecostonitsrepairs.IfhewantstoearnRs.1,440asnetprofitonit,howmuchpercentageshouldheaddtothepurchasepriceoftheTV?Sol.Repaircost=0.2×4,800=960ToearnnetprofitofRs.1,440heshouldalsorecoverhisrepaircost.ItmeanshehastoaddRs.2,400(1,440+960)tohispurchasepricetoearn1,440asnetprofit.%markup=(2400/4800)×100=50%.Ex.35.Awholesalersells30pensforthepriceof27penstoaretailer.Theretailersellsthepensattheirmarkedprice.Findthe%gainfortheretailer.Sol.AssumethemarkedpriceofonepenisRe.1.Sotheretailergets30pensforRs.27andhesellsthoseforRs.30.Soprofit= &

$g×100=11K

j%

Ex.36.Amansellsanarticleatagainof10%.Ifhehadboughtitat10%lessandsolditforRs.132less,hewouldhavestillgained10%.ThecostpriceofthearticleisSol.Firstprofit=10%SP=CP+ KM

KMMofCP=1.1CP…..(1)

Newcostprice=0.9CPNewsellingprice=S.P–132Now,S.P.-132=1.1×0.9CP……(2)From(1)and(2)1.1CP–132=0.99CP0.11CP=132CP=Rs.1,200Alternatemethod:LetCP1=Rs.100SP1=110NewCP2=90SP2=99DifferenceinSP=SP1–SP2=11(insecondcase)thenoriginalCP=100.IfheissellingfarRs.132lessthenCP=KMM

KK×132=Rs.1,200

Exercise

1.Kamaleshpurchased120reamsofpaperatRs.100perreamandtheexpenditureontransportwasRs.480.Hehadtopayanoctroldutyof50paiseperreamandthecooliechargeswereRs.60.Whatshouldhechargeperreamtogain40%?a.147b.127c.117d.137e.Noneofthese2.OnsellinganarticleatRs.530,thegainis20%morethanthelossincurredonsellingitatRs.475.Inordertogain20%,thesellingpricewillbea.400b.200c.800d.600e.Noneofthese3.Themarkedpriceofashirtandtrousersareintheratioof1:2.Theshopkeepergives40%discountontheshirt.Ifthetotaldiscountonbothis30%,thediscountofferedonthetrousersisa.33.33%b.20%c.25%d.50%e.Noneofthese4.Ifacommissionof10%isgivenonthemarkedpriceofabook,thepublishergains20%.Ifthecommissionisincreasedto20%,thegainpercentisa.6.66%b.5%c.8%d.6.5%e.Noneofthese5.Themarkedpriceofanarticleis20%morethanthecostpriceandadiscountof20%isgivenonthemarkedprice.Inthiskindofsale,thesellera.33.33%b.20%c.25%d.50%e.Noneofthese6.AshopkeepersellsabookwhosemarkedpriceisRs.30atadiscountof15%andgivesapencostingRs.1.50freewitheachbook.Eventhenhemakesaprofitof20%.Hiscostpriceperbookisa.30b.20c.25d.50e.Noneofthese7.AmerchantpurchasesacalculatorforRs.500andfixesthelistpriceinsuchawaythatafterallowingadiscountof10%,heearnsaprofitof20%.Findthelistpriceofthecalculatora.666.66b.333.33c.450d.500e.Noneofthese8.Ramanlalashopkeeperearnsaprofitof33.33%aftersellinganarticleat20%discountontheprintedprice.Findtheratioofthecostpriceandtheprintedofthearticlea.5:3b.1:5c.3:5d.5:1e.Noneofthese9.Atraderhassomewatchesinhisstock.Hemarkshiswatches20%abovethecostprice.Hesoldhalfthestockatmarkedprice,onequarteratadiscountof20%onthemarkedpriceandtherestatadiscountof40%onthemarkedprice.Whatishisgainpercent?a.3%b.2%c.5%d.7%e.Noneofthese10.AsoldawatchtoBat40%gainandBsoldittoCatalossof20%.IfCboughtthewatchfor

Rs.432,atwhatpricedidApurchaseit?a.383.33b.328c.385.71d.380e.Noneofthese11.Ashopkeeperallowsadiscountof10%onthemarkedpriceofanitembutchargesasalestax8%onthediscountprice.IfthecustomerpaysRs.729asthepriceincludingthesalestax,thenwhatisthemarkedpriceoftheitem?a.750b.600c.500d.450e.Noneofthese12.ThemarkedpriceofawatchisRs.1600.Theshopkeepergivessuccessivediscountsof10%andx%tothecustomer.IfthecustomerpaysRs.1224forthewatch,findthevalueofx.a.75b.10c.12d.15e.Noneofthese13.Adealersoldthree-fourthofhisarticlesatagainof24%andtheremainingatthecostpricepercentageofgaininthewholetransactionisa.18b.10c.12d.15e.Noneofthese14.OnaRs.10000paymentorder,apersonhaschoicebetweenthreesuccessivediscountsof10%,10%and30%andthreesuccessivediscountsof40%,5%and5%.Bychoosingthebetterone,hecansavea.275b.210c.125d.255e.Noneofthese15.Atradermarkedthesellingpriceofanarticleat10%abovethecostprice.Atthetimeofselling,heallowscertaindiscountandsuffersalossof2%.Thediscountallowedisa.10KM

K&%b.10𝟏𝟎

𝟏𝟏%c.10 j

KK%d.9KM

KK%e.Noneofthese

16.Dueto25%fallintherateofeggs,onecanbuy2dozeneggsmorethanbeforebyinvestingRs.200.Whatwasoriginalrateperdozen?a.Rs$MM

&b.Rs$MM

jc.Rs𝟐𝟎𝟎

𝟔d.RsKMM

he.Noneofthese

17.AshopkeeperpurchasedsomebooksfromapublicationworthRs.750.Becauseofsomereasons,hehadtoselltwo-fifthpartofthebookatalossof15%.Onwhichgainheshouldsellhisrestofthebookssothathegetsneithergainnorloss?a.10%b.20%c.25%d.50%e.Noneofthese18.Thecostpriceofanitemistwo-thirdofitssellingprice.Whatisthegainorlosspercentonthatitem?a.33.33b.20c.25d.50e.Noneofthese19.Abooksellersellsabookatagainof20%.Ifhehadboughtitat20%lessandsellandsolditforRs.6less,hewouldhavegained20%.TheCPofthebookisa.15b.20c.25d.50e.Noneofthese20.Byselling32orangesforRs.30amanloss25%.HowmanyorangesshouldbesoldforRs.24soastogain20%inthetransaction?a.15b.20c.25d.16e.Noneofthese

21.Atoyissoldat10%profit.HaditbeensoldforRs.10less,theprofitmaybe5%only.Findthecostprice.a.200b.600c.500d.400e.Noneofthese22.AnarticlecostingRs.2500issoldat20%loss.Findthesellingpricea.3000b.2000c.4000d.5000e.Noneofthese23.IfsellinganarticleforRs.880causes10%lossonthesellingprice.Finditscostpricea.896b.869c.968d.986e.Noneofthese24.Findthemarkedpricewhensellingprice=2880anddiscount=4%a.3000b.2000c.4000d.5000e.Noneofthese25.Findtherateofdiscountwhenmarkedprice=Rs.250andsellingpriceRs.235a.3%b.2%c.4%d.6%e.Noneofthese26.ApersonpurchasedaToyat25%discountonitsmarkedpricebutsolditatthemarkedprice.Findthegainpercent?a.33.33%b.20%c.25%d.50%e.Noneofthese27.Apersonallows4%discountonthemarkedpriceofhisgoodsandstillearnsaprofitof20%.FindthecostpriceforhismarkedatRs.850?a.860b.680c.780d.880e.Noneofthese28.Amanmarkshisgoodsatsuchapricethathecandeduct87.5%cashandyetmake20%profit.WhatisthemarkedpriceofanarticlewhichcostminimumhimRs.140a.190b.189c.192d.195e.Noneofthese29.AmansellsT.V.toBatagainof10%andBcellsittoCatagainof5%.IfCpaysRs.4620foritwhatdiditcosttoA?a.3000b.2000c.4000d.5000e.Noneofthese30.Amansells2toysatRs.616each.Ononehegained12%andontheotherhelost12%.Findhistotalgainorlossinthetransaction.a.Rs.18lossb.Rs.16lossc.Rs.18profitd.Rs.16profite.Noneofthese31.Amanboughtathingandsolditatagainof8%.Ifhehadboughtitfor10%lessandsolditforRs.240lesshewouldhavemadeaprofitof16.66%.Findthecostpriceofthethinga.860b.800c.780d.880e.Noneofthese32.GainorLossisalwayscountedon…..a.sellingpriceb.markedpricec.costpriced.datainadequatee.Noneofthese

33.TheretaillistpriceofaradioisRs.210.Thesellerallowsadiscountof5%forcashandstillmakesaprofitof14%onhisoutlay.Themanufacturer’spricetotheretaileris26%higherthanthecostprice?a.125b.139c.150d.110e.Noneofthese34.AfanissoldforRs.924.HaditbeensoldforRs.940therewouldhavebeenanadditionalprofitof2%onthecostprice.Findthecostpriceofthefana.860b.820c.800d.880e.Noneofthese35.Thedifferenceinsellingatoyatprofit8%and12%isRs.24.Findthecostpriceoftoya.860b.680c.620d.600e.Noneofthese36.Apersonbuys40pencilsfromawholesaleratthemarkedpriceof36pencils.Ifthepersonsellsatthemarkedprice,findhisprofitpercenta.11$

j%b.11𝟏

𝟗%c.11 K

KM%d.10K

j%e.Noneofthese

37.Singlediscountequivalenttoaseriesofdiscountsof20%and30%isa.44%b.40%c.45%d.50%e.Noneofthese38.AradiowhoseTagpriceisRs.7000isavailableat2successivediscountsof20%and25%.Findthesingleequivalentdiscounta.44%b.40%c.45%d.50%e.Noneofthese39.ThecostofapicturealbumisRs.500itisavailableattwosuccessivediscountsof15%and20%.Findtheamountpaidbythecustomera.300b.200c.340d.360e.NoneoftheseAns:Rs.34040.Findsinglediscountrateequivalenttoaseriesofdiscounts20%,25%and30%a.58%b.40%c.45%d.50%e.Noneofthese41.Ifashopkeeperallowsx%discountonthemarkedpriceofanarticlethenthesellingpriceofthearticle=Ans:(100-x)%ofmarkedprice42.Apersonallows30%discountonallhisarticles,thenumberofthearticlessoldisincreasedby30%.Findtheeffectontotalsale?Ans:9%decrease43.AchairwhichcostsRs.800wassoldatalossof5%.FinditssellingpriceAns:Rs.760

44.Isoldapenatalossof25%onmyoutlayExpressthislossasapercentageofthesellingpricea.33.33%b.20%c.25%d.50%e.Noneofthese45.Kabirbuysanarticlewith25%discountontheoriginalprice.Hemakesaprofitof10%bysellingitatRs.660.Whatistheoriginalpriceofthearticle?Ans:Rs.80046.Vishnubought170kgofsugarattherateofRs.5.80perkgandmixeditin130kgsugarpurchasedatRs.5.30perkg.HesoldthemixturewithaprofitofRs.161.TherateofthemixtureperkgwasAns:Rs.6.1247.Subhashbought20kgofteaattherateofRs.30perkgand30kgattherateofRs.25perkg.HemixedthetwoandsoldthemixtureattherateofRs.22.50perkg.Whatwashislossinthistransaction?Ans:Rs.22548.TheamountoflossincurredbysellinganarticleforRs.800is50%oftheamountgainedifitissoldforRs.1,400.Atwhatpriceshoulditbesoldtohave2%profit?Ans:Rs.120050.ThelossincurredonsellinganarticleforRs.270isasmuchastheprofitmadewhenitissoldat10%profit.Whatisthecostpriceofthearticle?Ans:Rs.30051.Ifthesellingpriceof40articlesisequaltothecostpriceof50similararticles,whatisthepercentgainorloss?a.33.33%profitb.20%lossc.25%lossd.50%profite.Noneofthese52.SukumarChowdhuryboughtpapersheetsforRs.3600andspentRs.100ontransport.PayingRs.300,hehad300boxesmadeofthem,whichhesoldatRs.16each.Whatishisprofitpercent?Ans:2053.Sohanlalpurchased120reamsofpaperattherateofRs.100perream.TheexpenditureontransportwasRs.480.Hehadtopayanoctroidutyof50paiseperreamandthecooliechargeswereRs.60.Whatshouldbethesellingpriceofeachreamifhewantsaprofitof20%?Ans:Rs.12654.Neerajboughtacarforacertainsumofmoney.Hespent10%ofthecostofrepairsandsoldthecarforaprofitofRs.11,000.Howmuchdidhespendonrepairsifhemadea20%profit?Ans:Rs.500055.BysellinganarticleforRs.469,Maheshgained15%.Whatwasthecostpriceofthearticle?

Ans:Rs.40056.Afruitvendorpurchased35dozenapplesatRs.10perdozen.Outofthis2dozengotrottenandhesold28dozenatRs.12perdozen.RemainingapplesweresoldbysoldbyhimatRs.15perdozen.Whatprofitdidhemakeinthisbargain?Ans:Rs.6157.Therewouldbe10%lossifriceissoldatRs.5.40perkg.Whatpriceperkgshoulditbesoldtoearnaprofitof20%?Ans:Rs.7.2058.Dilipprepared50showpiecesofsandalwood.ThecostofproductionofeachpiecewasRs.10.ApartfromthishehastopayRs.5perpieceascommissiontothesellingagentandRe.1perpiecetotheworkshopassistant.Whatshouldbethesellingpriceofeacharticleifhewishedtosellthemat30%profit?Ans:Rs.20.8059.Samirpurchased24kgofwheatattherateofRs.6.50perkgand26kgofwheatattherateofRs.8.00perkg.Hemixedthetwoandsoldthemixture.Approximatelyatwhatpriceperkgshouldhesellthemixturetomake30percentprofitinthistransaction?Ans:Rs.9.5060.IfthechairsboughtatpriceRs.300toRs.450aresoldatpricesrangingfromRs.400toRs.525,whatisthemaximumpossibleprofitthatmightbemadeinselling10chairs?Ans:Rs.225061.Thesellingpriceof50articlesisequaltothecostpriceofthesame60articles.Whatisthepercentprofitorloss?a.33.33%profitb.20%profitc.25%profitd.50%profite.Noneofthese62.Ashopkeeperpurchases18kgofsugarattherateofRs.4.25perkgand22kgattherateofRs.7.00perkg.HemixesthetwoandsellsthemixtureattherateofRs.5.35perkg.Whatwillbehisgainorlossinthetransaction?Ans:Rs.16.50loss63.InafactorythereareAgradeandBgradeemployees.Eachemployeecontributesasmanyrupeesasthenumberofemployeesinhisgrade.TheemployercontributesRs.618.IfthetotalcontributionisRs.839andtotalcontributionmadebygradeBemployeesisRs.25,howmanyemployeesarethereinthefactory?Ans:1964.AshopkeeperearnsaprofitofRs.60bysellinganarticle.HadhesoldthesamearticleatRs.520,theprofitpercentwouldhavebeen30.Atwhatpricedidhesellthearticle?Ans:Rs.460

65.Prabhatbought20kgofriceattherateofRs.8.50perkgand30kgofriceattherateofRs.9.75perkg.Hemixedthetwoforsellingthemixture.Approximatelyatwhatpriceperkgshouldhesellthemixturetoget25%profitonthecostprice?Ans:Rs.10.5066.BysellinganarticleSheetalearnedaprofitequaltotheK

iofthepriceheboughtit.Ifhe

solditatRs.375,whatwasthecostprice?Ans:Rs.30067.ThepercentprofitmadewhenanarticleissoldforRs.78istwiceasmuchaswhenitissoldforRs.69.Whatisthecostpriceofthearticle?Ans:Rs.6068.Pujaribought20quintalsofriceatRs.250perquintaland30quintalsatRs.280perquintal.HemixedthetwoandsoldthemixtureforRs.270aquintal.Whatwashisgaininthedeal?Ans:Rs.10069.Prakashboughtasuitcasewith25%discountontheprice.Atwhatpriceshouldhesellthesuitcaseifhewantstomakeaprofitof25%onhispurchaseprice?Ans:Can’tbedetermined70.Agrainmerchantbought50kgofwheatattherateofRs.7perkgand20kgattherateofRs.8perkg.HemixedbothandsoldattherateofRs.10perkg.Whatishistotalprofit?Ans:Rs.19071.TheprofitearnedbysellingawatchforRs.850isasmuchasthelossincurredwhenitissoldforRs.650.Whatisthecostpriceofthewatch?Ans:Rs.75072.Ashopkeepergivesadiscountof25%onthemarkedpriceofanarticleandearnsaprofitof20%onhisoutlay.Ifhehadnotgivenanydiscount,thentheprofitonhisoutlaywouldhavebeenAns:60%73.Afruit-sellersellsmangoesattherateofRs.50for10mangoes.Thenumberofmangoespurchasedbythefruit-sellerforRs.50inordertogetaprofitof60%isAns:1674.BysellinggoodsforRs.380,‘A’loses5percentonhisoutlay.Thepriceatwhich‘A’mustsellthemtogain10percent,wouldbeAns:Rs.44075.IsoldmytableforRs.144earningprofitpercentequaltothecostpriceofthetablenumerically.ThecostpriceofthetableisAns:Rs.80

76.ApersonbuysabookforRs.200andsellsitforRs.225.Whatwillbehisgainpercent?Ans:12.5%77.AcalculatorisboughtforRs.350andsoldatagainof15%.Whatwillbethesellingpriceofcalculatorinrupees?Ans:402.5078.Ifthecostpriceis95%ofthesellingprice.Whatistheprofitpercent?Ans:5.2679.Theownerofacellphoneshopchargeshiscustomer28%morethanthecostprice.IfthecustomerpaidRs.8960forthecellphone,whatwasthecostpriceofthecellphone?Ans:Rs.700080.SellingpriceofanarticleforRs.2220andthepercentprofitearnedis20%.Whatisthecostpriceofthearticle?Ans:Rs.185081.TheprofitearnedaftersellingapairofshoesforRs.2033isthesameasthelossincurredaftersellingthesamepairofshoesforRs.1063.Whatisthecostpriceoftheshoes?Ans:Rs.154882.Amansellsanarticleataprofitof40%.Ifhehadboughtitat40%lessandsoldforRs.5less,hewouldhavegained50%.Findthecostpriceofthearticle.Ans:Rs.1083.Charupurchasedadinnersetat &

KMofitssellingpriceandsolditat10%morethanitsSP.

FindthegainpercentAns:1084.RajansoldanarticleforRs.6000atalossof25%.FindthecostpriceAns:Rs.800085.Theownerofafurnitureshopchargeshiscustomer18%morethantheCP.IfacustomerpaidRs.10207foradiningtable,finditsoriginalpriceAns:Rs.865086.MeenapurchasedtwofanseachatRs.1200.Shesoldonefanatthelossof5%andotheratthegainof10%.FindtotalgainoflosspercentAns:2.5%profit87.KapilsoldtwobikesforRs.50000each.Ononehegains14%andontheother,helosses14%,findhisgainorlosspercentinthiswholetransactionAns:1.96%loss

88.Ifthecostpriceof20articlesisequaltothesellingpriceof15articles.FindtheprofitpercentAns:33.33%89.Atradesmanmarkshisgoodsat20%abovethecostpricebutallowspurchasersadiscountof5%,Whatprofitpercentdoeshemake?Ans:14%90.ProfitearnedbysellinganarticleforRs.1450issameasthelossincurredbysellingthearticleforRs.1280.Whatisthecostpriceofthearticle?Ans:Rs.136591.IfamanwantstosellhischairforRs.720hegets25%loss.Togain25%,heshouldsellifforAns:Rs.120092.TheSPofanarticleaftertwosuccessivediscountsof10%and5%onmarkedpriceisRs.171.FindthemarkedpriceAns:Rs.20093.ThedifferencebetweentheCPandSPofanarticleisRs.240.IftheprofitisRs.20%,thesellingpriceisAns:Rs.144094.PankalpurchasedanitemforRs.7500andsolditatthegainof24%.Fromthatamounthepurchasedanotheritemandsolditatthelossof20%.Whatishisoverallgain/loss?Ans:LossofRs.6095.SumitpurchasedanitemforRs.4000andsolditatthegainof35%.Fromthatamount,hepurchasedanotheritemandsolditatthelossof20%.Whatishisoverallgain/loss?Ans:GainofRs.320

Time,Speed&distance

GeneralInformationTheterms‘Time’and‘Distance’arerelatedtothespeedofamovingobject.GeneralFormulae1.Speed:Itisobtainedbydividingthedistancecoveredbytheobject,bythetimeittakestocoverthatdistance.Thus,Speed=od`_^Yq\_]^z\{{\c

�d[\_^s\Y

Distance=Speed×TimeTime=od`_^Yq\

�"\\c

2.ConversionofUnits:Onekilometer/hour= KMMM[_`.

hM×hM`\q.= %

Kem/sec.

Onemeter/second=Ke%km/hr.

Thus,𝑥km/hr= 𝑥× %Ke

m/sec.

And,𝑥m/sec= 𝑥× Ke%

km/hr3.(i)Ifthetimetakenisconstant,thedistancetravelledisproportionaltothespeed,thatis,morethespeed;morethedistancetravelledinthesametime.(ii)Ifthespeedisconstant,thedistancetravelledisproportionaltothetimetakenthatismorethedistancetravelled;morethetimetakenatthesamespeed.(iii)Ifthedistancetravelledisconstant,thespeedisinverselyproportionaltothetimetaken,thatis,morethespeed;lessthetimetakenforthesamedistancetravelled.4.IftwopersonsAandBstartatthesametimefromtwopointsPandQtowardseachotherandaftercrossingtheytakeT1andT2hoursinreachingQandPrespectively,

thenm�``"\\c

n�``"\\c= �$

�K

5.Ifabodytravelsd1,d2,d3,…..dnmeterswithdifferentspeedss1,s2,s3,…..,snm/sec.intimeT1,T2,T3,….,Tnsecondsrespectively,thentheaveragespeedofthebodythroughoutthejourneyisgivenby𝑉 = �W_^{cd`_^Yq\_]^z\{{\c

�W_^{_d[\_^s\Y= cKPc$Pc&P...PcY

�KP�$P�&P⋯P�Y

6.IftheratioofthespeedsofAandBisa:b,thentheratioofthetimetakenbythemtocoverthesamedistanceisK

^: Kp𝑜𝑟𝑏: 𝑎

SomeImportantShortcutFormulasRule1:Ifsomedistanceistravelledatxkm/hrandthesamedistanceistravelledatykm/hrthentheaveragespeedduringthewholejourneyisgivenby

Rule2:Ifapersontravelsacertaindistanceatxkm/hrandreturnsatykm/hr,ifthetimetakentothewholejourneyisThoursthentheonewaydistanceisgivenby

Rule3:IftwopersonsAandBstarttheirjourneyatthesametimefromtwopointsPandQtowardseachotherandaftercrossingeachothertheytakeaandbhoursinreachingQandPrespectively,then

Rule4:IfthesamedistanceiscoveredattwodifferentspeedsS1andS2andthetimetakentocoverthedistanceareT1andT2,thenthedistanceisgivenby

Ex.1.Apersoncrossesa600mlongstreetin5minutes.Whatishisspeedinkmperhour?Sol.

Speed=

600m/sec.5x60

=2m/sec.Convertingm/sectokm/hr(seeformulassection)

=

2x18

km/hr5

=7.2km/hr.Ex.2.Anaeroplanecoversacertaindistanceataspeedof240kmphin5hours.Tocoverthesamedistancein1𝟐

𝟑hours,itmusttravelataspeedof:

Sol.Distance=(240x5)=1200km.Speed=Distance/TimeSpeed=1200/(5/3)km/hr.[Wecanwrite1$

&hoursas5/3hours]

Requiredspeed= 1200x3 =720km/hr.

5Ex.3.Walkingatthree-fourthofhisusualspeed,amancoversacertaindistanceintwohoursmorethanthetimehetakestocoverthedistanceathisusualspeed.Thetimetakenbyhimtocoverthedistancewithhisusualspeedis:[SSC2000]

Sol.Newspeed=&iofusualspeed

Oldspeed=o�

Newtime=oldtime(T)+2Newspeed= od`_^Yq\

�\�_d[\

34×

𝐷𝑇 =

𝐷𝑇 + 2

3(T+2)=4T3T+6=4TT=6hoursAlternatemethod:LetustaketimetakenbyactualspeedasThours.Newspeed=&

iofusualspeed

Newtimetaken=i&ofnormaltime

Herenewtime=T+2(T+2)=i

&T

3T+6=4TT=6hoursEx.4.TheratiobetweenthewalkingspeedsofAandBis2:3andtherefore,Atakes10minutesmorethanBtoreachthedestination.IfAhadwalkedatdoublethespeed,hewouldhavecoveredthedistanceinSol.LetspeedofAandBbe2xand3xm/srespectively.TimetakenbyAandBtocoverthesamedistanceis3xand2xrespectively.3x–2x=10x=10Atakes30minutes.Whenthespeedisdoubled,Atakes15minutesEx.5.Twocyclistsmeeteachotherat10a.m.onFerozepurroad.Aftertheirmeeting,oneofthemproceededinthenorthdirectionandtheotherproceededintheeastdirection.Exactlyatnoon,theywere60kmapart.Ifthedifferencebetweentheirspeedsis6km/hr,findthespeedofthefastercyclist.Sol.Letspeedofthesloweronebexkm/hr.So,speedofthefasteronewillbe(x+6)km/hr.Thedistancetravelledin2hris2xand2(x+6)km.

FromthePythagorasTheorem:(2x)2+[2(x+6)]2=(60)24x2+4(x2+36+12x)=36008x2+48x+144=3600x2+6x–432=0Bysolvingthis,x=18km/hrSo,thespeedofthefastercyclistis24km/hr.Ex.6.Abullock-cartovertakestwopersonswalkinginthesamedirection,inavillage,attherateof3kmphand5kmphandpassesthemcompletelyin12and18secondsrespectively.Calculatethelengthofthebullock-cart(inmetres).Sol.Lengthofbullock-cart=Differenceofspeed×[Z{_d"{dq^_dWYWX_d[\

cdXX\]\Yq\WX_d[\

=2× %Ke×K$×Ke

h=20m

Ex.7.Amancancompleteajourneyin10hrs.Hetravelsthefirsthalfofthejourneyatthespeedof21km/hrandthesecondhalfatthespeedof24km/hr.Findthetotaldistanceofthejourneyinkilometres.Sol.Firsthalfspeed=21km/hrSecondhalfspeed=24km/hrTotaltime=10hrsAveragespeed=$×$K×$i

$KP$i=$×$K×$i

i%=KK$

%=22.4km/hr

Distancecovered=S×T=22.4×10=224kmEx.8.Amanwantstocover50kmonhisbicycle.Hisspeedis12.5km/hr.Afterevery12.5km,hetakesrestfor20minutes.Howlongwillhetaketocoverthewholedistance?Sol.Speed=12.5km/hrDistancetobecovered=50kmTimerequired= %M

K$.%×10=KMM

K$%=4hr

Restintervals=20min×3=60min=1hr(Afterthethirdstoppage,hewillreachhisdestination)Totaltime=4+1=5hrEx.9.Acartakes5hourstocoveradistanceof300km.Whatistherequiredspeedofthecarinkm/hrtocoverthesamedistancein4/5thoftheprevioustime?Sol.Theearlierspeed=300km/5hror60km/hrTheearliertime=5hrThenewtime=i

%thofprevioustime=i

%×5=4hr

Nowhehastocover300kmin4hr. Thenewspeed=&MM

i=75km/hr.

Ex.10.Amanrodeacertaindistanceatthespeedof25km/hrandwalkedbackatthespeedof4km/hr.Thewholejourneytookhim5hours48minutes.Whatdistancedidhecoverinall?Sol.Letthetotaldistancebe2xkm.

Then(x/25)+(x/4)=548/60=54/5=29/529x/100=29/5 x=20km.Totaldistancecovered=2x=40km

Ex.11.Amantravelsacertaindistanceatthespeedof20km/hrandreturnstothesamepointatthespeedof30km/hr.HisaveragespeedduringthewholejourneyisSol.Averagespeed=$×$M×&M

%M=K$M

%=24km/hr

Ex.12.Astartedhisjourneyonbicycleat7.30a.m.ataspeedof8km/hr.Bstartedfromthesamepointhalfanhourlater,butataspeedof10km/hr.AtwhattimedidBovertakeA?Sol.WhenBstarts,Awouldhavecovered=8 ½=4kminhalfanhour.RelativespeedforB=10–8=2km/hrTimetakenbyBtoovertakeA=i

$=2hr.

BovertookAat8+2=10a.m.Ex.13.Acartravelsadistanceof840kmatauniformspeed.Ifthespeedofthecarisincreasedby10km/hr,ittakestwohourslesstocoverthesamedistance.Whatistheoriginalspeedofthecar?Sol.Lettheoriginalspeedofthecarbexkm/hrAccordingtothegivencondition[840/(x+10)-(840/x)]=2Aspergivenoptions,letxbe60km/hrTimetaken=eiM

hM=14hr

Now,increasedspeedshallbe=60+10=70km/hrTime=eiM

gM=12hr.

Differenceintime=14–12=2hrEx.14.ThespeedofAandBisintheratio:2:3.Atakes10minutesmorethanthetimetakenbyBtoreachthedestination.IfAhadwalkedatdoublethespeed,hewouldhavecoveredthesamedistanceinSol.Given,bothofthemaretravellingthesamedistance.Speedwillbeinverselyproportionaltotime.AandBwilltake3xminand2xminrespectivelytocoverthesamedistance.

Also3x–2x=10 x=10Awilltake30min.tocoverthedistance.

IfAdoubleshisspeed,thentimetakenwillbe15min.Ex.15.Atraincoversacertaindistancein50minutesatthespeedof48km/hr.Whatshouldbethespeedofthetraintocompletethejourneyin40minutes?Sol.Given40:50::48km/hr:xkm/hrWherexisthespeedofthetrainTimetakenis40min.

x=%M×iMiM

=60km/hr

Ex.16.Excludingstoppages,thespeedofabusis45kmphandincludingstoppagesitis36kmph.Forhowmanyminutesdoesthebusstopperhour?Sol.Buswilltravel45kmwithoutstoppagesand36kmwithstoppagesinonehourDifferenceof9km’stimeiswastedinstoppages.Timewastedinstoppagesperhour= j

i%=K

%hours=12minutes.

Ex.17.Ashipwentonavoyage.Afterithadtravelled180miles,aplanestartedwith10timesthespeedoftheship.Findthedistance(inmiles)whentheywillmeetfromstartingpoint.Sol.Letspeedoftheship=xandTimetraveledaftertraveling180mbutNow,accordingtothequestion,180+xt=10xtOnsolving,wegetxt=20Totaldistancetraveled=180+20=200m.

Ex.18.Aballmovesat120meterspersecondandstrikesanobjectinthreeseconds.Ifitmovesat100meterspersecond,howlongdoesittaketostrikethesameobject?Sol.Totaldistancecoveredbyball=120 3=360mNow,speedofball=100m/sTimetaken=&hM

KMM=Ke

%m/s.

Ex.19.Tonystartedforhisofficeataspeedof40kmph.Ifhetravelsatthisspeed,hecanreachhisofficeexactlyontime.Butaftertravelling20km,herealizedthathehadforgottenanimportantdocumentbackathome.So,hewentbackhomeataspeedof60kmph,collectedthedocumentandagainstartedforhisofficeataspeedof80kmphandreachedhisofficeexactlyontime.Whatisthedistancebetweenhishomeandhisoffice?Sol.Ifthedistanceis‘d’km,heneeds c

iMℎ𝑟𝑠toreachhisoffice.

So, ciM= $M

iM+ $M

hM+ c

eM

d=$MM&=66$

&km

Ex.20.Brownstarted2hoursafterJohnstartedandovertakeshimin4hours.IfthedifferencebetweenthespeedofJohnandBrownis20kmph,whatisthedistancetravelledbyBrown,beforemeetingJohn?Sol.BrownovertakesJohnin4hours.Thismeansthathecancover4×20=80kminthistime.Thatmeansin2hoursbeforeBrownstartsJohnhascovered80km.So,John’sspeedis40kmphandthatofBrownis60kmph,So,thedistancetravelledbyBrown=60×4=240km.Ex.21.Amanwantstocover50kmonhisbicycle.Hecovers12.5kmph.Afterevery12.5km,hetakesrestfor20minutes.Howlongwillhetaketocoverthewholedistance?Sol.Speed=12.5km/hr

Distancetobecovered=50kmTimerequired= %M

K$.%×10=%MM

K$%=4hours

Restintervals=20min×3=60min=1hour(afterthirdstoppagehewillreachhisdestinationandtherewillbenostoppageafterthat)Totaltime=4+1=5hours.Ex.22.AtrainstartsfromDelhiat1:20p.m.andreachesIndoreat5:50p.m.SecondtrainstartsfromIndoreat1:50p.m.andreachesDelhiat7:10p.m.Findtheratioofspeedsofthetwotrains.Sol.Timetakenbythefirsttrain=4hr30min=4.5hrTimetakenbythesecondtrain=5hr20min=Kh

&hr

Ratiooftimetaken=j$:Kh

&

⇒Ratiointheirspeeds=Kh&:j$=32:27

ProblemontrainsI. Whentraincrossesapoleor

anystandingthingwithnegligiblelength

𝑳𝒕𝑺𝒕

𝐿_ =lengthoftrain𝑆_ =speedoftrain

II. Whentraincrossesarunningmaninsamedirection

𝑳𝒕𝑺𝒕 − 𝑺𝒎

𝐿_ =lengthoftrain𝑆_ =speedoftrain𝑆[ =speedofman

III. Whentraincrossesarunningmaninoppositedirection

𝑳𝒕𝑺𝒕 + 𝑺𝒎

𝐿_ =lengthoftrain𝑆_ =speedoftrain𝑆[ =speedofman

IV. Whentraincrossesaplatformoranystandingthingwithsomelength

𝑳𝒕 + 𝑳𝒑𝑺𝒕

𝐿_ =lengthoftrain𝑆_ =speedoftrain𝐿" =lengthofplatform

V. Whentraincrossesanotherrunningtraininsamedirection

𝑳𝒕𝟏 + 𝑳𝒕𝟐𝑺𝒕𝟏 − 𝑺𝒕𝟐

𝐿_K =lengthoftrain1𝐿_$ =lengthoftrain2𝑆_K =speedoftrain1𝑆_$ =speedoftrain2

VI. Whentraincrossesanotherrunningtraininoppositedirection

𝑳𝒕𝟏 + 𝑳𝒕𝟐𝑺𝒕𝟏 + 𝑺𝒕𝟐

𝐿_K =lengthoftrain1𝐿_$ =lengthoftrain2𝑆_K =speedoftrain1𝑆_$ =speedoftrain2

RelativeSpeedCase1:Twobodiesaremovinginoppositedirectionsatspeed𝑉K&𝑉$respectively.Therelativespeedisdefinedas𝑉] =𝑉K+𝑉$Case2:Twobodiesaremovinginsamedirectionsatspeed𝑉K&𝑉$respectively.Therelativespeedisdefinedas𝑉] =|𝑉K+𝑉$|Ex.1.Atraintravelingat72kmphcrossesaplatformin30secondsandamanstandingontheplatformin18seconds.Whatisthelengthoftheplatforminmeters?Sol.Whenthetraincrossesamanstandingonaplatform,thedistancecoveredbythetrainisequaltothelengthofthetrain.However,whenthesametraincrossesaplatform,thedistancecoveredbythetrainisequaltothelengthofthetrainplusthelengthoftheplatform.Theextratimethatthetraintakeswhencrossingtheplatformisonaccountoftheextradistancethatithastocover=lengthoftheplatform.Therefore,lengthoftheplatform=speedoftrain×extratimetakentocrosstheplatformLengthofplatform=72kmph×12secondsConverting72kmphintom/sec,weget72kmph= %

Ke×72=20m/sec

Therefore,lengthoftheplatform=20×12=240meters.Ex.2.Atraintravelingat100kmphovertakesamotorbiketravelingat64kmphin40seconds.Whatisthelengthofthetraininmeters?Sol.Whenatrainovertakesanotherobjectsuchasamotorbike,whoselengthisnegligiblecomparedtothelengthofthetrain,thenthedistancetraveledbythetrainwhileovertakingthemotorbikeisthesameasthelengthofthetrain.Thelengthofthetrain=distancetraveledbythetrainwhileovertakingthemotorbike

=relativespeedbetweenthetrainandthemotorbike×timetakenInthiscase,asboththeobjectsi.e.,thetrainandthemotorbikearemovinginthesamedirection,therelativespeedbetweenthem=differencebetweentheirrespectivespeeds=100-64=36kmph.Distancetraveledbythetrainwhileovertakingthemotorbike=36kmph×40seconds.Thefinalanswerisgiveninmetersandthespeedisgiveninkmphandthetimeinseconds.Soletusconvertthegivenspeedfromkmphtom/sec.1kmph=5/18m/secTherefore,36kmph=36×5/18=10m/sec.Relativespeed=10m/sec.Timetaken=40seconds.Therefore,distancetraveled=10×40=400metersEx.3.Atrain125mlongpassesaman,runningat5km/hrinthesamedirectioninwhichthetrainisgoing,in10seconds.Thespeedofthetrainis:Sol.

Speedofthetrainrelativetoman= 125

m/sec10

= 25

m/sec.2

= 25

x18

km/hr2 5

=45km/hr.Letthespeedofthetrainbexkm/hr.Then,relativespeed=(x-5)km/hr.

x-5=45 x=50km/hr.Ex.4.Thelengthofthebridge,whichatrain130metreslongandtravellingat45km/hrcancrossin30seconds,is:Sol.

Speed= 45x5

m/sec = 25

m/sec.18 2

Time=30sec.Letthelengthofbridgebexmetres.

Then,130+x

=25

30 22(130+x)=750x=245m.

Ex.5.Twotrainsrunninginoppositedirectionscrossamanstandingontheplatformin27secondsand17secondsrespectivelyandtheycrosseachotherin23seconds.Theratiooftheirspeedsis:Sol.Letthespeedsofthetwotrainsbexm/secandym/secrespectively.Then,lengthofthefirsttrain=27xmetres,andlengthofthesecondtrain=17ymetres.

27x+17y=23

x+y27x+17y=23x+23y4x=6y

x=3.

y 2Ex.6.Twotrainsofequallengtharerunningonparallellinesinthesamedirectionat46km/hrand36km/hr.Thefastertrainpassestheslowertrainin36seconds.Thelengthofeachtrainis:Sol.Letthelengthofeachtrainbexmetres.Then,distancecovered=2xmetres.Relativespeed=(46-36)km/hr

= 10x5m/sec

18

= 25

m/sec9

2x=25

36 92x=100x=50.

Ex.7.Twotrainsaremovinginoppositedirections@60km/hrand90km/hr.Theirlengthsare1.10kmand0.9kmrespectively.Thetimetakenbytheslowertraintocrossthefastertraininsecondsis:Sol.Relativespeed=(60+90)km/hr

= 150x5

m/sec18

= 125

m/sec.3

Distancecovered=(1.10+0.9)km=2km=2000m.

Requiredtime= 2000x3

sec=48sec.125

Ex.8.Ajoggerrunningat9kmphalongsidearailwaytrackin240metresaheadoftheengineofa120metreslongtrainrunningat45kmphinthesamedirection.Inhowmuchtimewillthetrainpassthejogger?Sol.Speedoftrainrelativetojogger=(45-9)km/hr=36km/hr.

=36x5

m/sec18

=10m/sec.Distancetobecovered=(240+120)m=360m.

Timetaken= 360

sec =36sec.10

Ex.9.A270metreslongtrainrunningatthespeedof120kmphcrossesanothertrainrunninginoppositedirectionatthespeedof80kmphin9seconds.Whatisthelengthoftheothertrain?Sol.Relativespeed=(120+80)km/hr

= 200x5

m/sec18

= 500

m/sec.9

Letthelengthoftheothertrainbexmetres.

Then,x+270

=500

9 9x+270=500x=230.

Ex.10.Twotrains,each100mlong,movinginoppositedirections,crosseachotherin8seconds.Ifoneismovingtwiceasfasttheother,thenthespeedofthefastertrainis:Sol.Letthespeedoftheslowertrainbexm/sec.Then,speedofthefastertrain=2xm/sec.Relativespeed=(x+2x)m/sec=3xm/sec.

(100+100)=3x

824x=200

x=25

.3

So,speedofthefastertrain=50m/sec

3

= 50

x18

km/hr3 5

=60km/hr.Ex.11.Amanstealsacarat1:30pmanddrivesat40kmph.At2pm,theownerstartschasinghiscarat50kmph.Atwhattimewillhecatchtheman?Sol.Distancecoveredbythethiefin1hour=40kmDistancecoveredinK

$ℎ=20km

Now,timetakentocatchthethief= od`_^Yq\]\{^_dz\`"\\c

Relativespeed=50–40=10kmph(Botharemovinginsamedirection)Timetakentocatchthethief=$M

KM=2hours.

Time=4p.m( 2p.m.+2hours)

Ex.12.Atrainisgoingat48kmph.Apassengercountsthetelegraphpostsonthelineashepassesthem.Ifthepostsare50metresapart,howmanypostswillhecrossperminute?Sol.Speedoftrain=48kmph=48×5/18m/sec=ie×%×hM

Kem/min.=800m/min

Numberofposts=800/50=16Ex.13.Findthetimetakenbytwotrains,one210mlongandtheother270mlong,tocrosseachother,iftheyarerunningatspeedsof46kmphand54kmphrespectivelyinthesamedirection.Sol.Relativespeed=54–46=8kmph.Distancetobecovered=210+270=480mTime=0.480/8=0.06hours.=216s.Ex.14.TwotrainsstartatthesametimefromstationAandstationBandproceedtowardseachotherattherateof16kmand21kmperhourrespectively.Whentheymeet,itisfoundthatonetrainhastravelled60kmmorethantheother.ThedistancebetweenthetwostationsisSol.Distancebetweentwostations=c `�P`5

`��`5

=60 $KPKh$KNKh

=60×&g%=444km.

Ex.15.TwopersonsstartfromAandBwiththespeedof25kmphand49kmphrespectivelytowardseachother.Aftertheycrosseachother,thepersonfromBcovers145kmtoreachA.Whatisthedistance(inkm)coveredbyA&B?Sol.LetthetotaldistancebetweenAandBbex.Sinceboththepersonsarestartingatthesametimethereforetimetakenbythemtomeetwillbesame.Ratioofspeed=ratiosofdistance

25:49::145:(x–145).Therefore,x=429.2km.Ex.16.Brownstarted2hoursafterJohnstartedandovertakeshimin4hours.IfthedifferencebetweenthespeedofJohnandBrownis20kmph,whatisthedistancetravelledbyBrown,beforemeetingJohn?Sol.BrownovertakesJohnin4hours.Thismeansthathecancover4×20=80kminthistime.Thatmeansin2hoursbeforeBrownstartsJohnhascovered80km.So,John’sspeedis40kmphandthatofBrownis60kmph,So,thedistancetravelledbyBrown=60×4=240km.Ex.17.Atrainoflength120metreovertakesanother100meterlongtrainin8seconds.Ifthespeedoftheslowertrainis27kmph,whatisthespeedofthefastertrain?Sol.

t=8sec---(1)V1=27km/h=27x %

Kem/s=7.5m/s

Speedoffastertrain=V2Netspeed=V2-V1--(2)From(1)&(2)V2-V1=

$$Me=27.5

V2=35m/s=35xKe%km/h=126km/hr

Ex.18.Atraincrossesaplatformin60seconds,travellingatthespeedof54kmph.Ifthelengthoftheplatformis500metres,whatisthelengthofthetrain?Sol.t=60secSpeed=54kmph=15m/sl1=500l2=?15={�P{5

hM=%MMP{5

hM

l2=900-500=400mEx.19.Agoodstrainrunsatthespeedof72kmphandcrossesa250mlongplatformin26seconds.Whatisthelengthofthegoodstrain?Sol.

Speed= 72x5

m/sec =20m/sec.18

Time=26sec.Letthelengthofthetrainbexmetres.

Then,x+250

=2026

x+250=520x=270.

Ex.20.Twotrainsoflength115mand110mrespectivelyrunonparallelrails.Whenrunninginthesamedirectionthefastertrainpassesthesloweronein25seconds,butwhentheyarerunninginoppositedirectionswiththesamespeedsasearlier,theypasseachotherin5seconds.Findthespeedofeachtrain?Sol.Letthespeedoftrainsbexm/secandym/secrespectively.Whentheymoveinsamedirectiontheirrelativespeedis:x-yWhentheymovesinoppositedirectiontheirrelativespeedis:x+y

Onsolvingtwoequationsx=27m/sandy=18m/secEx.21.Twotrains110mand100minlengthrespectivelyarerunninginsamedirections,oneattherateof100km/hrandtheotherattherateof64km/hr.Atwhattimefastertrainwillclearothertrainfromthemomenttheymeet?Sol.Sincetrainsarerunninginsamedirection,sorelativespeed=100-64=36km/hr=[36*(5/18)]=10m/secTotallengthtobetravelled=110+100=210m

Ex.22.Atrainpassesastandingpoleontheplatformin5secondsandpassestheplatformcompletelyin20seconds.Ifthelengthoftheplatformis225meters.Thenfindthelengthofthetrain?Sol.LetthelengthofthetrainisxmeterSospeedoftrain=(x/5)m/secAlsospeedoftrain=(225+x)/20m/sec

Ex.23.Aftertravellingacertaindistance,atraindevelopsasnaganddecreasesitsspeedtohalfitsoriginalspeedandreachesitsdestination45minuteslate.Hadthesnagoccurred30kmfurtheron,itwouldhavereacheditsdestination15minutesearlier.Whatisthespeedofthetrain?Sol.LettheinitialspeedbeS1andthelaterspeedbeS2.Whenspeeddecreasesby½theusualspeed,thenforanyconstantdistance,timetakenwouldbetwicetheusual.Incondition(I),theusualtimetakentotravelCB=tCB=45minutesandincondition(II),theusualtimetakentotravelDB=tDB=45–15=30minutes.Sinceincondition(I)andcondition(II),thetimetakentotravelDBisthesame,theusualtimetakentotravelCD=tCD=tCB–tDB=45–30=15minutes.TheusualtimetakentotravelCD,tCD=15min.

Theusualspeed=Uo_��

= &M���}kmph=120kmph

Theusualspeedis120kmph.Ex.24.TwotrainsstartatthesametimefromstationsAandDandproceedtowardseachotherat36kmphand42kmphrespectively.Whentheymeet,itisfoundthatonetrainhastravelled48kmmorethantheother.Thedistancebetweenthetwostations(inkm)isSol.LettrainfromAcoveradistanceofxkmandtrainfromDcoverx+48km.Time= O

&h= OPie

i$

42x=36x+1728

x=Kg$eh

=288Totaldistance=288+288+48=624kmEx.25.AtrainstartsfromXtowardsYwhichisatadistanceof55kmataspeedof40kmperhour.Afterrunningacertaindistance,itincreasesitsspeedto50kmperhourandreachesYin1hour15minutesafterleavingX.Afterwhattimedidthetrainchangeitsspeed?Sol.Letthetrainincreaseitsspeedaftercoveringthedistance'a'km.Accordingtothequestion

1560 =

𝑎40 +

55 − 𝑎50

⇒250=5a+55×4–4a⇒a=30kmTimetocover'a'=&M

iM=0.75hour=45min

BoatsandstreamsDownstream/Upstream:Inwater,thedirectionalongthestreamiscalleddownstream.And,thedirectionagainstthestreamiscalledupstream.Importantfacts:1)Ifthespeedofaboatinstillwaterisukm/hrandthespeedofthestreamisvkm/hr,then:Speeddownstream=(u+v)km/hr.Speedupstream=(u-v)km/h2)Ifthespeeddownstreamisakm/hrandthespeedupstreamisbkm/hr,then:Speedinstillwater=K

$(𝑎 + 𝑏)km/hr

Rateofstream=K$(𝑎 − 𝑏)km/hr

SomeShortcutMethodsRule1:Amancanrowcertaindistancedownstreamint1hoursandreturnsthesamedistanceupstreamint2hours.Ifthespeedofstreamisykm/h,thenthespeedofmaninstillwaterisgivenby

Rule2:Amancanrowinstillwateratxkm/h.Inastreamflowingatykm/h,ifittakeshimthourstorowtoaplaceandcomeback,thenthedistancebetweentwoplacesisgivenby

Rule3:Amancanrowinstillwateratxkm/h.Inastreamflowingatykm/h,ifittakesthoursmoreinupstreamthantogodownstreamforthesamedistance,thenthedistanceisgivenby

Rule4:Amancanrowinstillwateratxkm/h.Inastreamflowingatykm/h,ifherowsthesamedistanceupanddownthestream,thenhisaveragespeedisgivenby

Ex.1.Amancanrowaboatinupstreamat6kmphanddownstreamat12kmph.Findtheman'srateinstillwaterandtherateofcurrent.Sol.

Ex.2.Amancanrow15km/hrinstillwater.Ittakeshimtwiceaslongtorowupstreamastorowdownstream.Findtherateofstream?Sol.Letman'srateinupstreambexkm/hr,thenhisrateindownstreamis=2xkmph

Sorateindownstream=2x=20km/hr

Ex.3.Amancanswim6km/hrinstillwater.Ifriver'srateis2km/hr,ittakestheman3hourstorowtoaplaceandback.Howfaristheplace?Sol.Man'srateinupstream=(6-2)km/hr=4km/hrMan'sstreamrateindownstream=(6+2)km/hr=8km/hr

Letthedistanceoftheplacebexkmthen,

Ex.4.Aboatcovers24kmupstreamand36kmdownstreamin6hoursanothertimeitcovers36kmupstreamand24kmdownstreamin13/2hours.Findouttherateofcurrent?Sol.Lettherateofupstreambexkmph,rateofdownstreambeykmph,then

Onaddingbothequation:

Onsubtractingbothequation:

Onsolvingbothfinalequationswegetx=8andy=12,now

Ex.5.Ariverflowsat1.5kmphandaboatmanwhocanrowat2.5kmphonstillwater,takes7.5hourstogoacertaindistanceupthestreamandreturnagaintothestartingpoint.Findthetotaldistancecoveredbytheboatman.Sol.Thespeedsoftheboatmanupanddownthestreamare1kmphand4kmphrespectively.Ifthedistancecoveredeachway=xkm,totaltimetogoupandreturn=O

K+ O

i=7.5hr.

i.e.%ix = K%

$⇒x=K%

$× i%=6km

Thedistancecoveredbothwaysis12kmEx.6.Aboyrowsaboatupstreamforadistanceof9mileswhenthestreamisflowingat2m/h.Then,herowsbackwiththecurrent.Thewholetriptakes6hours.Whatwillbetheboy'srowingspeedinstillwater?Sol.LetXm/hbethespeedoftheboatinstillwater.Accordingtothegivenconditionj

ON$+ j

OP$=6

2X2–6X–8=0 X2–3X–4=0(X–4)(X+1)=0 X=4orX=–1(inadmissible)

Therequiredspeedis4m/hEx.7.Amanrows5km/hrinstillwater.Iftheriverisflowingat1km/hr,ittakeshim75minutestorowtoaplaceandback.Howfaristheplace?Sol.Letthedistanceoftheplacebexkm.Accordingtothegivencondition:

Oh+ O

i=75/60

4(2x+3x)=605x=15x=3km.

Ex.8.AboatmantakeshisboatupstreamfromplaceAtoplaceB,whereAB=21km.Then,hereturnstoplaceA.Hetakes10hrinall.Thetimetakenbyhim7kmdownstreamisequaltothetimetakenbyhim3kmupstream.Findthevelocityoftheriver.Sol.LetthespeedsupstreamanddownstreambeSuandSdrespectively.21/Su+21/Sd=107/Sd=3/SuSolvingthetwoequationssimultaneously,wegetSd=7kmphSu=3kmphLetthespeedoftheboatandthecurrentbexkm/hrandykm/hrrespectively.x+y=7x–y=3Solvingthetwoequationssimultaneously,wegetx=5km/hrandy=2km/hr.Ex.9.Thespeedofaboatinstillwateris10m/sandthespeedofthestreamis6m/s.Iftheboatismovingupstreamandbackdownstream,whatistheratioofthetimetakentocoveraparticularstretchofdistanceineachdirection?Sol.Giventhatthedistancetravelledineachdirectionisthesame,andtheeffectivespeedsareintheratio(10–6):(10+6)=1:4, Thetimetakentotravelupstreamanddownstreamisintheratio(1/1):(1/4)=4:1.Ex.10.Amotorboatcantravelat10km/hrinstillwater.Ittravelled91kmdownstreamandthenreturned,taking20hoursaltogether.Whatistherateofflowoftheriver?Sol.Letspeedofthestreambexkm/hr91/(10–x)+91/(10+x)=2020/(100–x2)=20/91

100–x2=91 x2=9x=3km/h

Ex.11.Aboattakes6hourstotravelfromAtoBupstream.Iftherivercurrentflowsattherateof2kmph,howlongwilltheboattaketotravelfromBtoA(downstream),thedistancebeing36km?Sol.Upstermst=6hrV1=xV2=2km/hrdistance=36V1-V2=

&hh=6

V1=8km/hr

DowntermsV1=8V2=2net=V1+V2=10km/hrDistance=36,T=&h

KM=3.6hrs

Exercise1.Rajahastoleaveofficeexactlyafter20minutestocatchatrainwhichleavesstation6.40p.m.andits5.45p.m.now.Howmuchtimedoesittaketotravelfromofficethestation?Ans:35minutes2.BalustartedinacarfromCochintoTrivandrumat11a.m.Hetravelledat60kmperhourandcoveredhalfthedistanceintwohoursAtwhatuniformspeedshouldhetraveltoreachTrivandrumat4.00p.m.?Ans:40km/hour3.ThechargesofhiredcarareRs.3perkmforthefirst60km.Rs.5perkmforthenext60kmandRs.10perkmforthefurtherjourney.IfthebalanceamountleftoverwithAtulis K

KMless

thanwhathepaidtowardsthechargesofthehiredcarfortravelling200km.howmuchmoneydidhehaveinitiallywithhim?Ans:Rs.12164.‘A’travelsacertaindistanceataspeedofxkmperhourandanequaldistanceatthespeedofykmperhour.Thentheaveragespeedof‘A’forthewholejourney(inkm)perhourisAns:2xy/(x+y)5.Iascendamountainwithaspeedof3kmperhour.ThenIimmediatelyreturnfromthesummitandrundownwardswithaspeedof6kmperhour.ThenmyaveragespeedfortheroundtripisAns:4km/hour6.‘A’and‘B’arewalkinginthesamedirection.‘A’whoisanaheadwalksattherateof3.5kmanhourwhile‘B’walksattherateof4kmperhour.Thedistancebetweenthenis6km.The‘B’overtakes‘A’inAns:12hours7.Amanridingacycleat12kmperhourcanreachavillagein4.5hours.Ifheisdelayedby1.5houratthestart,theninordertoreachhisdestinationintime,heshouldridewithaspeedofAns:18km/hour8.Aboygoestoschoolwithaspeedof3km/hourandreturnswithaspeedof2km/hour.Ifhetakes5hoursinall,thenthedistance(inkm/hour)betweenthevillageandtheschoolis:Ans:6

9.Aboywalkstoschoolattherateof3kmperhourandreachesschoolat7.35a.m.Thenhewalksat5kmperhourandreachesat7.25a.m.Ifhisschooltimeis7.30a.m.howfarishisschoolfromhishouse?Ans:1.25km10.Acaristravellingisattherateof50kmperhourwhereastrainismovingat20meterspersecond.TheirrespectivespeedsareintheratioAns:25:3611.Athiefstealsascooteranddrivesof60kmperhour.Thepolicemandiscoversthetheftafter15minutesandstartchasinghimattherateof75kmperhourhismotor-cycle.Afterwhattimethethiefcaughtbythepoliceman?Ans:onehour12.Acarcoversadistanceof690kmin30h.Whatistheaveragespeedofthecar?Ans:23km/h13.Abusisrunningwithauniformspeedof37km/h.Whatdistancewillbecoveredbybusin8h?Ans:296km14.Apersonridingabikecrossesabridgewithaspeedof54km/h.Whatisthelengthofthebridgeifhetakes4mintocrossthebridge?Ans:3600m15.Amancoveredadistanceof12kmin90minbycycle.Howmuchdistancewillhecoverin3hifheridesthecycleatauniformspeed?Ans:24km16.Amancompletes30kmofajourneyat6km/handtheremaining40kmofthejourneyin5h.FindtheaveragespeedforthewholejourneyAns:7km/h17.Acertaindistanceiscoveredatacertainspeed.Ifhalfofthedistanceiscoveredindoubletime,theratioofthetwospeedsisAns:4:118.Movingh

gofitsusualspeedatrainis10minlate.Finditsusualtimetocoverthejourney

Ans:60min19.AcarcoversadistancefromtownAtotownBatthespeedof58km/handcoversthedistancefromtownBtotownAatthespeedof52km/h.Whatistheapproximateaveragespeedofthecar?Ans:55km/h

20.AcarreachedRaipurfromsonagarhin35minwithanaveragespeedof69km/h.Iftheaveragespeedisincreasedby36km/h,howlongwillittaketocoverthesamedistance?Ans:23min21.Astudentwalksfromhishouseat2.5km/handreacheshisschoollateby6min.Nextday,heincreaseshisspeedby1km/handreaches6minbeforeschooltime.Howfaristheschoolfromhishouse?Ans:g

ikm

22.JohnstartedfromAtoBandvinodfromBtoA.IfthedistancebetweenAandBis125kmandtheymeetat75kmfromA.WhatistheratioofJohn’sspeedtothatofvinod’sspeed?Ans:3:223.Amancovershalfonhisjourneyat6km/handtheremaininghalfat3km/h.FindhisaveragespeedAns:4km/h24.ThespeedsofAandBareintheratioof3:4.Atakes20minmorethanthetimetakenbyBtoreachadestination,inwhattimedoesAreachthedestination?Ans:1.33h25.AistwiceasfastasBandBisthriceasfastasC.ThejourneycoveredbyCin56minwillbecoveredbyAinAns:9.33min26.Theratioofspeedsofatrainandacaris16:15respectivelyandabuscoveredadistanceof480kmin8h.Thespeedofthebusin&

i𝑡ℎofthespeedoftrain.Whatdistancewillbecovered

bycarin6h?Ans:450km27.Theratiobetweenthespeedsoftwobusesis5:3.Ifthe1stbusruns400kmin8h,thenfindthespeedofthe2ndbus.Ans:30km/h28.Ifsohailwalksfromhishometoofficeat16km/h,heislateby5min.Ifhewalksat20km/h,hereaches10minbeforetheofficetime.FindthedistanceofhisofficefromhishouseAns:20km29.Acarcoversadistanceof200kmin2h40minwhereasajeepcoversthesamedistancein2h.Whatistheratiooftheirspeeds?Ans:3:430.Apersongoesfromonepointtoanotherpointwithaspeedof5km/handcomesbackto

startingpointwithaspeedfor3km/h.Findtheaveragespeedforthewholejourney.Ans:3.75km/h31.Abustravelsattherateof54km/hwithoutstoppagesandittravelsat45km/hwithstoppages.Howmanyminutesdoesthebusstoponanaverageperhour?Ans:10min32.Acartravelsadistanceof75kmatthespeedof25km/h.Itcoversthenext25kmofitsjourneyatthespeedof5km/handthelast50kmofitsjourneyatthespeedof25km/h.Whatistheaveragespeedofthecar?Ans:15km/h33.Acarrunsatthespeedof40km/hwhennotservicedandrunsat65km/hwhenserviced.Afterservicingthecarcoversacertaindistancein5h.Howmuchapproximatetimewillthecartaketocoverthesamedistancewhennotserviced?Ans:8h34.Anexpresstraintravelledatanaveragespeedof100km/h,stoppingfor3minafterevery75km.Alocaltraintravelledataspeedof50km/h,stoppingfor1minafterevery25km.Ifthetrainsbegantravellingatthesametime,howmanykilometersdidthelocaltraintravelinthetimeinwhichtheexpresstraintravel600km?Ans:307.5km35.Ifyoutravel39kmataspeedof26km/h,another39kmataspeedof39km/handagain39kmataspeedof52km/h.Whatisyouraveragespeedfortheentirejourney?Ans:37km/h36.AandBaretwostations1000kmapart.AtrainstartsfromAandmovestowardsBat40km/h.AnothertrainstartsfromBatthesametimeandmovestowardsAat60km/h.HowfarfromAwilltheycrosseachother?Ans:400km37.XandYstartwalkingtowardseachotherat8amatthespeedsof3km/hand4km/h,respectively.Theywereinitially17.5kmapart.Atwhattimedotheymeet?Ans:10.30am38.AtrainleavesManipurat6amandreachesDispurat10am.AnothertrainleavesDispurat8amandreachesManipurat11:30am.Atwhattimedothetwotrainscrosstrainscrosseachother?Ans:8:56am39.Theaveragespeedofacaris75km/h.Thedriverfirstdecreasesitsaveragespeedby40%andthenincreasesitby50%.Whatisthenewaveragespeednow?Ans:67.5km

40.Acarrunsatthespeedof50km/hwhennotservicedandrunsat60km/h,whenserviced.Afterservicing,thecarcoversacertaindistancein6h.Howmuchtimewillthecartaketocoverthesamedistancewhennotserviced?Ans:7.2h41.Acardrivercoversadistancebetweentwocitiesataspeedis60km/handonthereturnhisspeedis40km/h.Hegoesagainfromthe1sttothe2ndcityattwicethatoriginalspeedandreturnsathalftheoriginalreturnspeed.Findhisaveragespeedfortheentirejourney.Ans:40km/h42.ThedistancebetweentwostationsXandYis450km.AtrainLstartsat6pmfromxandmovestowardsyatanaveragespeedof60km/h.AnothertrainMstartsfromyat5:20pmandmovestowardsxatanaveragespeedof80km/h.Howfarfromxwillthetwotrainsmeetandatwhattime?Ans:170km,8:50pm43.TwocarsAandBarerunningtowardseachotherfromtwodifferenceplaces88kmapart.IftherationofthespeedsofthecarsAandBis5:6andthespeedofthecarBis90km/h,afterwhattimewilltheymeeteachother?Ans:32min44.Ifa200mlongtraincrossesaplatformofthesamelengthasthatofthetrainin20seconds,thenthespeedofthetrainisAns:72km/h45.Atrain150mlongisrunningataspeed90kmph.TimetakenbythetraintocrossatreeisAns:6sec46.A100mlongtraincrossesastationarymanin10seconds.Whatisthespeedofthetrain?Ans:36km/h47.TwotrainsAandB,travellinginoppositedirections,crosseachotherinsixseconds.ThespeedoftrainAis126km/h,whilethatoftrainBis90km/h.IfthelengthtrainAis160metres,whatisthelengthoftrainB?Ans:200m48.Atraincrossesastationarypersonin3secondswhileittakes10secondstocrossaplatform,travellingatsamespeed.Ifthelengthoftheplatformis210meters,whatisthelengthofthetrain?Ans:90m49.A225mlongtraincrossesanother150mlongtrainin12.5sec.Ifoneofthetrainstravels@72km/h,whatisthespeedoftheother?Ans:36km/h

50.Whatisthespedofatrainwhichovertakesamanwalkingataspeedof5km/hin30seconds,ifthetrainis274metreslong?Ans:37.88km/h51.Twotrains(X,Y)startfromAandBrespectively,towardseachother.AftertheymeetatpointC,thetrainXtakestwohourstoreachBwhilethetrainYtakes4.5hourtoreachA.IfthedistancebetweenplacesAandBis450kms,howfarshouldCbefromA?Ans:270km52.150mlongtraintakes10stopassamanwhoisgoinginthesamedirectionat2km/h.Whatisthespeedofthetraininkm/h?Ans:5653.AtrainstartsfromstationXattherateof60km/handreachesstationYin45minutes.Ifthespeedisreducedby6km/h.howmuchmoretimewillthetraintaketoreturnfromstationYtostationX?Ans:5min54.Atraintakes18secondstopasscompletelythroughtastation162mlongand15secondsthroughanotherstation120mlong.ThelengthofthetrainisAns:90m55.Atrain100metrelongcompletelypassesamanwalkinginthesamedirectionat6km/hin5secondsandacartravellinginthesamedirectionin6seconds.Atwhatspeedwasthecartravelling?Ans:18km/h56.Howmanysecondswilla500metreslongtraintaketocrossamanwalkingwithaspeedof3km/hinthedirectionofthemovingtrainifthespeedofthetrainis63km/h?Ans:3057.Atrainrunsattherateof120km/h.ThespeedinmetrespersecondisAns:33.3358.Atrainrunningatthespeedof72km/hcrossesa260metreslongplatformin23seconds.Whatisthelengthofthetraininmetres?Ans:20059.TrainAleavesMumbaiCentralstationforLucknowat11a.m.runningatthespeedof60km/h.TrainBleavesMumbaiCentralstationforLucknowbythesamerouteat2p.m.onthesamedayrunningatthespeedof72km/h.Atwhattimewillthetwotrainsmeeteachother?Ans:5amonthenextday60.Atrainrunningatspeedof120km/hcrossesasignalpostin15seconds.Whatisthelength

ofthetraininmetres?Ans:50061.TwotrainsstartfromPandQrespectivelyandtraveltowardseachotherataspeedof50km/hand40km/hrespectively.Bythetimetheymeet,thefirsttrainhastravelled100kmmorethanthesecond.ThedistancebetweenPandQisAns:900km62.Atrainrunningat g

KKofitsownspeedreachedaplacein22hours.Howmuchtimecouldbe

savedifthetrainranatitsownspeed?Ans:8h63.Atrainis125mlong.Ifthetraintakes30secondstocrossatreebytherailwayline,thenthespeedofthetrainisAns:15km/h64.Atrainpassestwobridgesoflengths800mand400min100secondsand60secondsrespectively.ThelengthofthetrainisAns:200m65.Atrain800metreslongisrunningatthespeedof78km/h.Ifitcrossesatunnelin1minute,thenthelengthofthetunnel(inmetres)isAns:50066.Twotrainsone160mandtheother140mlongarerunninginoppositedirectionsonparallelrails,thefirstat77km/handtheotherat67km/h.Howlongwilltheytaketocrosseachother?Ans:7.5s67.Twotrainsstartatthesametime,onefromNewDelhitoAgra,theotherfromAgratoNewDelhi,iftheyarriveattheirrespectivedestinationsinonehourandfourhoursrespectivelyafterpassingeachother,howmuchfasterisonetrainrunningthantheother?Ans:2times68.AandBaretwostations500kmapart.AtrainstartsfromAandmovestowardsBattherateof20km/h.AnothertrainstartsfromBatthesametimeandmovestowardsAattherateof30km/h.HowfarfromAwilltheycrosseachother?Ans:200m69.Apassengertrain,movingattherateof90km/hovertakesagoodstrain,2timesaslongandmovingonparallellinesattherateof50km/handpassesitcompletelyin54seconds.Howlongwouldthepassengertraintaketopasscompletelythroughastation400metersinlength?Ans:24s70.TwotrainsforBombayleaveDelhiat6amand6.45amandtravelat100km/hand136

km/hrespectively.HowmanykilometersfromDelhiwillthetwotrainsbetogether?Ans:283.33km71.Twotrains,CalcuttaMailandBombayMail,startatthesametimefromstationsCalcuttaandBombayrespectivelytowardseachother.Afterpassingeachother,theytake12hoursand3hourstoreachBombayandCalcuttarespectively.IftheCalcuttaMailismovingatthespeedof48km/h,thespeedoftheBombayMailisAns:96km/h72.Withoutstoppage,atraintravelsacertaindistancewithanaveragespeedof60km/h,andwithstoppage,itcoversthesamedistancewithanaveragespeedof40km/h.Onanaverage,howmanyminutesperhourdoesthetrainstopduringthejourney?Ans:20min/h73.Arailwaypassengercountsthetelegraphspolesontherailashepassesthem.Thetelegraphpolesareatadistanceof50metres.Whatwillbehiscountin4hours,ifthespeedofthetrainis45km/h?Ans:360074.TwotrainsAandBstartsimultaneouslyintheoppositedirectionsfromtwopointsAandBandarriveattheirdestinationsin9and4hoursrespectivelyaftertheirmeetingeachother.AtwhatratedoesthesecondtrainBtravelifthefirsttraintravelsat80km/h?Ans:120km/h75.TwotrainsfromHowrahleaveMuzaffarpurat8.30amand9.00amrespectivelyandtravelat60km/hand70km/hrespectively.HowmanykilometersfromMuzaffarpurwillthetwotrainsmeet?Ans:180kmDirectionsfor(Q.No76&77):Answerthefollowingquestionsonthebasisoftheinformationgivenbelow:(i)TrainsAandBtravellingonthesamerouteheadingtowardsthesametraindestination.TrainBhasalreadycoveredadistanceof220kmsbeforethetrainstarted(ii)Thetwotrainsmeeteachotherafter11hoursafterthestartoftrainA(iii)Hadthetrainsbeentravellingtowardseachother,(fromadistanceof220km)theywouldhavemetafteronehour76.WhatisthespeedoftrainBinkm/h?Ans:10077.WhatisthespeedoftrainAinkm/h?Ans:12078.Atrainmeetsanaccidentaftermoving30km.Itsspeedthencomesdowntoi

%thofthe

originaloneandconsequentlyruns45minslate.Hadtheaccidenttakenplace18kmfarther

away,itwouldhavebeen36minlate.FindtheoriginalspeedofthetrainandthedistancebetweenthetwostationsAns:30km/h,120kmDirectionsfor(Q.No79&80):Readthefollowingandanswerthequestionsthatfollow.TheKalingaExpressstartedfromPatnatoTataat7pmataspeedof60km/h.Anothertrain,theRajadhaniExpress,startedformTatatoPatnaat4amnextmorningataspeedof90km/h.ThedistancebetweenPatnatoTatais800km79.HowfarfromTatawillthetwotrainsmeet?Ans:156km80.Atwhattimewillthetwotrainsmeet?Ans:5:44am81.Atrain600mlongcrossesapolein12s.Whatisthespeedofthetraininkm/h?Ans:180km/h82.Atrainmovingwithaspeedof25m/scrossesastandingmanin16s.Whatisthelengthofthetrain?Ans:400m83.Atrain130mlongpassesabridgein20smovingwithaspeedof90km/h.FindthelengthofthebridgeAns:370m84.Atrain250mlongcrossesaplatformofequallengthin50s.FindthespeedofthetrainAns:10m/s85.Atrainspeedspastapoleis10sandspeedspastaplatformof300min25s.ItslengthinmetresisAns:200m86.Atrain100mlongcompletelypassesamanwalkinginthesamedirectionat2km/hin9sandacartravellinginoppositedirectionin3s.Atwhatspeedwasthecartravelling?Ans:78km/h87.Twotrainsofequallengtharerunningonparallellinesinthesamedirectionat45km/hand35km/h.Thefastertrainpassestheslowertrainin27s.ThelengthofeachtrainisAns:300m88.Amaninatrainnoticesthathecancount21telephonepostsin2min.Iftheyareknowntobe50mapart,thenatwhatspeedisthetraintravelling?Ans:30km/h

89.Atrainoflength150mtakes40stocrossatunneloflength300m.ThespeedofthetrainisAns:40.5km/h90.Ifatrainrunsat40km/hour,itreachesitsdestinationlateby11minutes.Butifitrunsat50km/hour,itislateby5minutesonly.Thecorrecttime(inminutes)forthetraintocompletethejourneyisAns:1991.Atrainovertakestwopersonswhoarewalkinginthesamedirectioninwhichthetrainisrunning,attherateof2kmphand4kmphandpassesthemcompletelyin9and10secondsrespectively.Thelengthofthetrain(inmetres)Ans:5092.Twotrainsoflength70mand80marerunningatspeedsof68km/hrand40km/hrrespectivelyonparalleltracksinoppositedirections.Inhowmanysecondswilltheypasseachother?Ans:593.Amanrows6km/hinstillwater.Iftheriverisrunningat3km/h,ittakeshim45minutestorowtoplaceandback.Howfaristheplace?Ans:1.6875km94.Aboatgoes15kmupstreamin80minutes.Thespeedofthestreamis5km/h.ThespeedofboatinstillwaterisAns:16.25km/h95.Inastream,BliesinbetweenAandCsuchthatitisequidistantfrombothAandC.AboatcangofromAtoBandbackin6h30minuteswhileitgoesfromAtoCin9h.HowlongwouldittaketogofromCtoA?Ans:4h96.Inastreamthatisrunningat2km/h,amangoes10kmupstreamandcomesbacktothestartingpointin55min.FindthespeedofthemaninstillwaterAns:22km/h97.Amotorcyclewhosespeedinstillwateris15km/hgoes30kmdownstreamandcomesbackinatotal4hours30min.DeterminethespeedofthestreamAns:5km/h98.Twoswimmers,atoppositeendsofa90-footpool,starttoswimthelengthofthepool,oneattherateof3feetpersecond,theotherat2feetpersecond.Theyswimbackandforthfor12minutes.Allowingnolossoftimeattheturns,findthenumberoftimestheypasseachotherAns:20

99.TwoportsAandBare300kmapart.TwoshipsleaveAforBsuchthatthesecondleaves8hoursafterthefirst.TheshipsarriveatBsimultaneously.Findthetimetheslowershipspentonthetripifthespeedofoneofthemis10km/hhigherthanoftheotherAns:20h100.Inonehour,aboatgoes11kmalongthestreamand5kmagainstthestream.Thespeedoftheboatinstillwater(inkm/hr)isAns:8101.Amancanrowupstreamat8kmphanddownstreamat13kmph.ThespeedofthestreamisAns:2.5km/hr102.Amanrowsdownstream32kmand14kmupstream.Ifhetakes6hourstocovereachdistance,thenthevelocity(inkmph)ofthecurrentAns:1.5103.Aboatrunningdownstreamcoversadistanceof16kmin2hourswhileforcoveringthesamedistanceupstream,ittakes4hours.Whatisthespeedoftheboatinstillwater?Ans:6km/hr104.Aboatmangoes2kmagainstthecurrentofthestreamin1hourandgoes1kmalongthecurrentin10minutes.Howlongwillittaketogo5kminstationarywater?Ans:1hr15min105.Amancanrowthree-quartersofakilometeragainstthestreamin11.25minutes.Thespeed(inkm/hr)ofthemaninstillwaterisAns:5106.Amantakestwiceaslongtorowadistanceagainstthestreamastorowthesamedistanceinfavourofthestream.Theratioofthespeedoftheboat(instillwater)andthestreamisAns:3:1107.Aboatrunningupstreamtakes8hours48minutestocoveracertaindistance,whileittakes4hourstocoverthesamedistancerunningdownstream.Whatistheratiobetweenthespeedoftheboatandspeedofthewatercurrentrespectively?Ans:cannotbedetermined108.Ifaboatgoes7kmupstreamin42minutesandthespeedofthestreamis3kmph,thenthespeedoftheboatinstillwaterisAns:13km/hr109.Aman’sspeedwiththecurrentin15km/hrandthespeedofthecurrentis2.5km/hr.The

man’sspeedagainstthecurrentisAns:10km/h110.Ifamanrowsattherateof5kmphinstillwaterandhisrateagainstthecurrentis3.5kmph,thenthemanratealongthecurrentis:Ans:6.5kmph111.Aboatcantravelwithaspeedof13km/hrinstillwater.Ifthespeedofthestreamis4km/hr,findthetimetakenbytheboattogo68kmdownstreamAns:4hours112.Speedofaboatinstandingwateris9kmphandthespeedofthestreamis1.5kmph.Amanrowstoaplaceatadistanceof105kmandcomesbacktothestartingpoint.ThetotaltimetakenbyhimisAns:24hours113.Thespeedofaboatinstillwateris15km/hrandtherateofcurrentis3km/hr.Thedistancetravelleddownstreamin12minutesisAns:3.6km114.Amancanrowat5kmphinstillwater.Ifthevelocityofcurrentis1kmphandittakeshim1hourtorowtoaplaceandcomeback,howfaristheplace?Ans:2.4km115.Aboattakes19hoursfortravellingdownstreamfrompointAtopointBandcomingbacktoapointCmidwaybetweenAandB.Ifthevelocityofthestreamis4kmphandthespeedoftheboatinstillwateris14kmph.WhatisthedistancebetweenAandB?Ans:180km116.Amancanrow9.33kmphinstillwaterandfindsthatittakeshimthriceasmusttimetorowupthanastorowdownthesamedistanceintheriver.ThespeedofthecurrentisAns:4.66km/hr

117.Aboatcoversacertaindistancedownstreamin1hour,whileitcomesbackin1 hours.Ifthespeedofthestreambe3kmph,whatisthespeedoftheboatinstillwater?Ans:15kmph118.Amotorboat,whosespeedis15km/hrinstillwatergoes30kmdownstreamandcomesbackinatotalof4hours30minutes.Thespeedofthestream(inkm/hr)isAns:5119.Thespeedofaboatinstillwateris10km/hr.Ifitcantravel26kmdownstreamand14kmupstreaminthesametime,thespeedofthestreamisAns:3km/hr

120.Aboattakes90minuteslesstotravel36milesdownstreamthantotravelthesamedistanceupstream.Ifthespeedoftheboatinstillwateris10mph,thespeedofthestreamisAns:2mph121.Amanrowstoaplace48kmdistancein14hours.Hefindsthathecanrow4kmwiththestreaminthesametimeas3kmagainstthestream.TherateofthestreamisAns:1km/hr122.Aboatcovers24kmupstreamand36kmdownstreamin6hourswhileitcovers36kmupstreamand24kmdownstreamin6.5hours.ThevelocityofthecurrentisAns:2mph123.Athisusualrowingrate,Rahulcantravel12milesdownstreaminacertainriverin6hourslessthanittakeshimtotravelthesamedistanceupstream.Butifhecoulddoublehisusualrowingrateforhis24-mileroundtrip,thedownstream12mileswouldthentakeonlyonehourlessthantheupstream12miles.Whatisthespeedofthecurrentinmilesperhour?Ans:2.66124.Amancanrowupstreamat7km/handdownstreamat11km/h.Whatistheman’srateinstillwater?Ans:9km/h125.Amancanrowupstreamat4km/hranddownstreamat7km/h.Whatistherateofthecurrent?Ans:1.5km/h126.Amanrows25kmdownstreamand20kmupstreamtaking5heachtime.Whatisthevelocityofthecurrent?Ans:0.5km/h127.Amancanrow9km/hinstillwater.Ittakeshimtwiceaslongtorowupastorowdowntheriver.FindtherateofthestreamAns:3km/h

Time&WorkInratioandproportion,welearntthattherearetwotypesofproportions:DirectproportionandIndirectproportion.Iffournumbersa,b,canddareindirectproportion,thenwewriteitasa:b::c:d

andfromtherule:

Q)If4pencilscost$10,thenwhatwill6pencilscost?Solution:NumberofpencilsandtheirCostarequantitiesthataredirectlyproportional.So,wecanframetheexpression“4isto10issameas6istoWhatPriceP?i.e.4:10::6:P,then4×P=10×6,soP=(10×6)/4=15So,6pencilswillcost$15Now,considerthefollowingexample:Q)2mencompleteaworkin1day,then1manalonewillfinishthesameworkinhowmanydays?Solution:BetweenNumberofmenandAmountofwork,thereisnotadirectvariation,ratherthereisanindirectproportion.Tofinishasameamountofwork,MoremenwilltakeLesstime,whileFewermentakeMoretimeNow,fromtheRatioandProportionlesson,weknowthatiffourquantitiesa,b,canddareinindirectproportion,thenwecansettherelationas:“avariestobindirectlyascvariestod”denotedasbelow:a:b::d:c,andfromtherule“productofmeansisequaltoproductofextremes”,wewrite:

a×c=b×dNow,if2mentake1day,then1manwilltakewhattimecanbesetupasbelow:2:1::N:1,sothatweget2×1=N×1,soN=2days.Q)If3menfinishaworkin4hours,then4menwillfinishthesameworkinwhattime?Solution:Since,NumberofmenandTimetakentofinishtheworkareinverselyproportional,wecanwrite:3isto4issameasNisto4,whereNistherequirednumberofdays.3:4::N:4.Fromtherule,wewrite:3×4=4×N,So,N=3days.1.AVeryImportantFormula:1.Tofinishasameamountofwork,ifM1mentakeD1daysandM2mentakeD2days,thenM1×D1=M2×D2Letusnowapplythisformulatotheabovequestion:M1=3,D1=4,M2=4D2=?3×4=4×D2So,D2=32.Again,tofinishthesameamountofwork,ifM1mentakeD1daysworkingH1hoursaday,andM2mentakeD2daysworkingH2hoursaday,thenwecanwrite:M1×D1×H1=M2×D2×H2Q)Tofinishasameamountofwork,if4mentake6daysworking5hoursaday,thenhowmanydayswill6mentakeworking4hoursaday?Solution:Usingtheformula:M1×D1×H1=M2×D2×H24×6×5=6×D2×4Weget:D2=5days.2.AnotherVeryImportantConcept:One-dayWorkConcept:Tofinishasameamountofwork,amantakes4dayswhileawomantakes6days,eachworkingalone.Ifthetwopersonsworktogether,inwhattimewilltheycompletethework?Solution:Mantakes4days,Womantakes6days,So,workingtogether,bothtake4+6,i.e.10days.ThisisOBVIOUSLYWRONG.Why?Ofcourse,twopersonsjoininghandstofinishoffaworkwillnaturallytakelesstimethanthat

takenbyeachoneofthem.Hereagain,theconceptofindirectproportioncomesintoplayonNumberofmenandTimetakentocompleteasameamountofwork.Then,whatistheway?Heregoesthesolution:Ifthemancompletestheworkin4days,thenineachoneofthe4days,hewillcomplete1/4thofthetotalwork.(ItislegitimatelyassumedinTimeandWorkquestionsthatworkdoneineachdayisasamepartofthewhole)So,theManwillcompletea¼thonthefirstday,another¼thonthe2ndday,athird1/4thonthe3rdday,andthefinal¼thonthe4thday:So,thetotalamountofwork(denotedbynumber“1”)oneachofthe4daysis:¼+¼+¼+¼=1So,therewegettoseeoneveryimportantconceptinTimeandWork:IfapersoncompletesaloneaworkinNdays,theninonedaythepartoftheworkdoneis1/NNote:“Becarefultonotethatthefractionforone-dayworkisareciprocalofthenumberfortimetakentofinishthewholeworkPlainlyspeaking,one-dayworkandnumberofdaysarenumbersreciprocalsofeachother”(Fraction¼forone-dayworkisreciprocalof4,thenumberfortimetakentofinishthewholework)ThisistheconceptofOne-DayWork:Now,applyingthisone-dayworkconcepttotheaboveexample:WorkcompletedinOnedaybytheMan=1/4,Woman=1/6,Man&Womantogether=1/4+1/6=5/12Therefore,numberofdaysbothtaketocompletethewholeworkis12/5AnotherVeryImportantFormula:IfonepersonAtakes“x”daystocompleteaworkaloneandanotherpersonBtakes“y”daystocompletethesameworkalone,thenthenumberdaysbothAandBtakeworkingtogetheris:xy/(x+y)Applyingthisformulatotheabovequestion,weseethemanandwomanwilltogethertake:4×6/(4+6)=24/10=12/5days.AVeryImportantFormula:IfthreepersonsA,BandCtake“x”,“y”and“z”daysrespectivelytocompleteaworkworkingalone,thenthenumberofdaystakenbyallthreeworkingtogetheris:xyz/(xy+yz+xz)Q)Workingattheirownindividualrates,threepersonsfinishasameworkin4,5and6daysrespectively.Inwhattimewilltheycompletetheworktogether?Solution:Applytheaboveformula:(4×5×6)/(4×5+5×6+4×6)=120/74=60/37days.

Ex.1.If9menworking6hoursadaycandoaworkin88days.Then6menworking8hoursadaycandoitinhowmanydays?Sol.Fromtheaboveformulai.e(m1*t1/w1)=(m2*t2/w2)so(9*6*88/1)=(6*8*d/1)onsolving,d=99days.Ex.2.If34mencompleted2/5thofaworkin8daysworking9hoursaday.Howmanymoremanshouldbeengagedtofinishtherestoftheworkin6daysworking9hoursaday?Sol.Fromtheaboveformulai.e(m1*t1/w1)=(m2*t2/w2)so,(34*8*9/(2/5))=(x*6*9/(3/5))sox=136mennumberofmentobeaddedtofinishthework=136-34=102menEx.3.If5womenor8girlscandoaworkin84days.Inhowmanydayscan10womenand5girlscandothesamework?Sol.Giventhat5womenisequalto8girlstocompleteaworkso,10women=16girls.Therefore10women+5girls=16girls+5girls=21girls.8girlscandoaworkin84daysthen21girls—————?Answer=(8*84/21)=32days.Therefore10womenand5girlscanaworkin32daysEx.4.WorkerAtakes8hourstodoajob.WorkerBtakes10hourstodothesamejob.HowlongittakebothA&B,workingtogetherbutindependently,todothesamejob?Sol.A'sonehourwork=1/8.B'sonehourwork=1/10(A+B)'sonehourwork=1/8+1/10=9/40BothA&Bcanfinishtheworkin40/9daysEx.5.Acanfinishaworkin18daysandBcandothesameworkinhalfthetimetakenbyA.Then,workingtogether,whatpartofthesameworktheycanfinishinaday?Sol.GiventhatBalonecancompletethesameworkindays=halfthetimetakenbyA=9daysA'sonedaywork=1/18B'sonedaywork=1/9(A+B)'sonedaywork=1/18+1/9=1/6Ex.6.AistwiceasgoodaworkmanasBandtogethertheyfinishapieceofworkin18days.InhowmanydayswillAalonefinishthework.Sol.ifAtakesxdaystodoaworkthenBtakes2xdaystodothesamework=>1/x+1/2x=1/18

=>3/2x=1/18=>x=27days.Hence,Aalonecanfinishtheworkin27days.Ex.7.Acandoacertainworkin12days.Bis60%moreefficientthanA.HowmanydaysdoesBalonetaketodothesamejob?Sol.RatiooftimetakenbyA&B=160:100=8:5SupposeBalonetakesxdaystodothejob.Then,8:5::12:x=>8x=5*12=>x=15/2days.Ex.8.Acandoapieceofworkn7daysof9hourseachandBalonecandoitin6daysof7hourseach.Howlongwilltheytaketodoitworkingtogether82/5hoursaday?Sol.Acancompletetheworkin(7*9)=63daysBcancompletetheworkin(6*7)=42days=>A'sonehour'swork=1/63andB'sonehourwork=1/42(A+B)'sonehourwork=1/63+1/42=5/126Therefore,Bothcanfinishtheworkin126/5hours.Numberofdaysof82/5hourseach=(126*5/(5*42))=3daysEx.9.AtakestwiceasmuchtimeasBorthriceasmuchtimetofinishapieceofwork.Workingtogethertheycanfinishtheworkin2days.Bcandotheworkalonein?Sol.SupposeA,BandCtakex,x/2andx/3hoursrespectivelyfinishtheworkthen1/x+2/x+3/x=1/2=>6/x=1/2=>x=12So,Btakes6hourstofinishthework.Ex.10.Xcando¼ofaworkin10days,Ycando40%ofworkin40daysandZcando1/3ofworkin13days.Whowillcompletetheworkfirst?Sol.WholeworkwillbedonebyXin10*4=40days.WholeworkwillbedonebyYin(40*100/40)=100days.WholeworkwillbedonebyZin(13*3)=39daysTherefore,Zwillcompletetheworkfirst.ComplexProblemsonTimeandWork="#6b8e23">Ex.11.AandBundertaketodoapieceofworkforRs600.Aalonecandoitin6dayswhileBalonecandoitin8days.WiththehelpofC,theycanfinishitin3days,Findtheshareofeach?Sol.C'soneday'swork=(1/3)-(1/6+1/8)=1/24Therefore,A:B:C=Ratiooftheironeday'swork=1/6:1/8:1/24=4:3:1A'sshare=Rs(600*4/8)=300B'sshare=Rs(600*3/8)=225C'sshare=Rs[600-(300+225)]=Rs75

Ex.12.Acandoapieceofworkin80days.Heworksatitfor10days&thenBalonefinishestheremainingworkin42days.InhowmuchtimewillAandB,workingtogether,finishthework?Sol.WorkdonebyAin10days=10/80=1/8Remainingwork=(1-(1/8))=7/8Now,workwillbedonebyBin42days.WholeworkwillbedonebyBin(42*8/7)=48daysTherefore,A'soneday'swork=1/80B'soneday'swork=1/48(A+B)'soneday'swork=1/80+1/48=8/240=1/30Hence,bothwillfinishtheworkin30days.Ex.13.P,QandRarethreetypistswhoworkingsimultaneouslycantype216pagesin4hoursInonehour,RcantypeasmanypagesmorethanQasQcantypemorethanP.Duringaperiodoffivehours,RcantypeasmanypagesasPcanduringsevenhours.Howmanypagesdoeseachofthemtypeperhour?Sol.LetthenumberofpagestypedinonehourbyP,QandRbex,yandzrespectively.Thenx+y+z=216/4=54—————1z-y=y-x=>2y=x+z———–—25z=7x=>x=5x/7—————3Solving1,2and3wegetx=15,y=18,andz=21Ex.14.RonaldandElanareworkingonanassignment.Ronaldtakes6hourstotype32pagesonacomputer,whileElantakes5hourstotype40pages.Howmuchtimewilltheytake,workingtogetherontwodifferentcomputerstotypeanassignmentof110pages?Sol.NumberofpagestypedbyRonaldinonehour=32/6=16/3NumberofpagestypedbyElaninonehour=40/5=8Numberofpagestypedbybothinonehour=((16/3)+8)=40/3Timetakenbybothtotype110pages=110*3/40=8hours.Ex.15.TwoworkersAandBareengagedtodoawork.Aworkingalonetakes8hoursmoretocompletethejobthanifbothworkingtogether.IfBworkedalone,hewouldneed41/2hoursmoretocompetethejobthantheybothworkingtogether.Whattimewouldtheytaketodotheworktogether.Sol.(1/(x+8))+(1/(x+(9/2)))=1/x=>(1/(x+8))+(2/(2x+9))=1/x=>x(4x+25)=(x+8)(2x+9)=>2×2=72=>x2=36=>x=6Therefore,AandBtogethercandotheworkin6days.

Ex.16.AandBcandoaworkin12days,BandCin15days,CandAin20days.IfA,BandCworktogether,theywillcompletetheworkinhowmanydays?Sol.(A+B)'soneday'swork=1/12;(B+C)'soneday'swork=1/15;(A+C)'soneday'swork=1/20;Addingweget2(A+B+C)'soneday'swork=1/12+1/15+1/20=12/60=1/5(A+B+C)'sonedaywork=1/10So,A,B,andCtogethercancompletetheworkin10days.Ex.17.AandBcandoaworkin8days,BandCcandothesameworin12days.A,BandCtogethercanfinishitin6days.AandCtogetherwilldoitinhowmanydays?Sol.(A+B+C)'soneday'swork=1/6;(A+B)'soneday'swork=1/8;(B+C)'soneday'swork=1/12;(A+C)'soneday'swork=2(A+B+C)'soneday'swork-((A+B)'sonedaywork+(B+C)'sonedaywork)=(2/6)-(1/8+1/12)=(1/3)-(5/24)=3/24=1/8So,AandCtogetherwilldotheworkin8days.Ex.18.AcandoacertainworkinthesametimeinwhichBandCtogethercandoit.IfAandBtogethercoulddoitin10daysandCalonein50days,thenBalonecoulddoitinhowmanydays?Sol.(A+B)'soneday'swork=1/10;C'soneday'swork=1/50(A+B+C)'soneday'swork=(1/10+1/50)=6/50=3/25Also,A'soneday'swork=(B+C)'soneday'sworkFromiandii,weget:2*(A'soneday'swork)=3/25=>A'soneday'swork=3/50B'soneday'swork=(1/10-3/50)=2/50=1/25Balonecouldcompletetheworkin25days.Ex.19.AisthriceasgoodaworkmanasBandthereforeisabletofinishajobin60dayslessthanB.Workingtogether,theycandoitin:Sol.RatiooftimestakenbyAandB=1:3.Ifdifferenceoftimeis2days,Btakes3daysIfdifferenceoftimeis60days,Btakes(3*60/2)=90daysSo,Atakes30daystodothework=1/90A'soneday'swork=1/30;B'soneday'swork=1/90;(A+B)'soneday'swork=1/30+1/90=4/90=2/45

Therefore,A&Btogethercandotheworkin45/2daysTopEx.20.Acandoapieceofworkin80days.Heworksatitfor10daysandthenBalonefinishestheremainingworkin42days.InhowmuchtimewillA&B,workingtogether,finishthework?Sol.WorkDonebyAn10days=10/80=1/8Remainingwork=1-1/8=7/8Now7/8workisdonebyBin42daysWholeworkwillbedonebyBin42*8/7=48days=>A'soneday'swork=1/80andB'soneday'swork=1/48(A+B)'soneday'swork=1/80+1/48=8/240=1/30Hencebothwillfinishtheworkin30days.Ex.21.45mencancompleteaworkin16days.Sixdaysaftertheystartedworking,somoremenjoinedthem.Howmanydayswilltheynowtaketocompletetheremainingwork?Solution:M1*D1/W1=M2*D2/W2=>45*6/(6/16)=75*x/(1-(6/16))=>x=6daysEx.22.Ais50%asefficientasB.CdoeshalftheworkdonebyA&Btogether.IfCalonedoestheworkin40days,thenA,BandCtogethercandotheworkin:Sol.A'soneday'swork:B'sonedayswork=150:100=3:2LetA's&B'soneday'sworkbe3xand2xdaysrespectively.ThenC'soneday'swork=5x/2=>5x/2=1/40=>x=((1/40)*(2/5))=1/100A'soneday'swork=3/100B'soneday'swork=1/50C'soneday'swork=1/40So,A,BandCcandotheworkin131/3days.Ex.23.Acanfinishaworkin18daysandBcandothesameworkin15days.Bworkedfor10daysandleftthejob.InhowmanydaysAalonecanfinishtheremainingwork?Sol.B's10day'swork=10/15=2/3Remainingwork=(1-(2/3))=1/3Now,1/18workisdonebyAin1day.Therefore1/3workisdonebyAin18*(1/3)=6days.Ex.24.Acanfinishaworkin24days,Bn9daysandCin12days.B&Cstarttheworkbutareforcedtoleaveafter3days.TheremainingworkdonebyAin:Sol.(B+C)'soneday'swork=1/9+1/12=7/36WorkdonebyB&Cin3days=3*7/36=7/12Remainingwork=1-(7/12)=5/12

Now,1/24workisdonebyAin1day.So,5/12workisdonebyAin24*5/12=10daysEx.25.XandYcandoapieceofworkn20daysand12daysrespectively.Xstartedtheworkaloneandthenafter4daysYjoinedhimtillthecompletionofwork.Howlongdidtheworklast?Sol.workdonebyXin4days=4/20=1/5Remainingwork=1-1/5=4/5(X+Y)'soneday'swork=1/20+1/12=8/60=2/15Now,2/15workisdonebyXandYinoneday.So,4/5workwillbedonebyXandYin15/2*4/5=6daysHenceTotaltimetaken=(6+4)days=10daysEx.26.Adoes4/5ofworkin20days.HethencallsinBandtheytogetherfinishtheremainingworkin3days.HowlongBalonewouldtaketodothewholework?Sol.WholeworkisdonebyAin20*5/4=25daysNow,(1-(4/5))i.e1/5workisdonebyA&Bindays.WholeworkwillbedonebyA&Bin3*5=15days=>B'soneday'swork=1/15-1/25=4/150=2/75So,Balonewoulddotheworkin75/2=37½days.Ex.27.AandBcandoapieceofworkin45daysand40daysrespectively.TheybegantodotheworktogetherbutAleavesaftersomedaysandthenBcompletedtheremainingworkn23days.ThenumberofdaysafterwhichAlefttheworkwasSol.(A+B)'soneday'swork=1/45+1/40=17/360WorkdonebyBin23days=23/40Remainingwork=1-(23/40)=17/40Now,17/360workwasdoneby(A+B)in1day.17/40workwasdoneby(A+B)in(1*(360/17)*(17/40))=9daysSo,Aleftafter9days.Ex.28.Acandoapieceofworkin10days,Bin15days.Theyworkfor5days.TherestofworkfinishedbyCin2days.IftheygetRs1500forthewholework,thedailywagesofBandCare:Sol.PartofworkdonebyA=5/10=1/2PartofworkdonebyB=1/3PartofworkdonebyC=(1-(1/2+1/3))=1/6A'sshare:B'sshare:C'sshare=1/2:1/3:1/6=3:2:1A'sshare=(3/6)*1500=750B'sshare=(2/6)*1500=500C'sshare=(1/6)*1500=250A'sdailywages=750/5=150/-B'sdailywages=500/5=100/-C'sdailywages=250/2=125/-DailywagesofB&C=100+125=225/-

Ex.29.Aalonecancompleteaworkin16daysandBalonecancompletethesamein12days.StartingwithA,theyworkonalternatedays.Thetotalworkwillbecompletedinhowmanydays?Sol.(A+B)'s2dayswork=1/16+1/12=7/48workdonein6pairsofdays=(7/48)*6=7/8remainingwork=1-7/8=1/8workdonebyAon13thday=1/16remainingwork=1/8–1/16=1/16on14thday,itisB'sturn1/12workisdonebyBin1day.1/16workisdonebyBin¾day.Totaltimetaken=13¾days.Ex.30.A,BandCcandoapieceofworkin20,30and60daysrespectively.InhowmanydayscanAdotheworkifheisassistedbyBandConeverythirdday?Sol.A'stwoday'swork=2/20=1/10(A+B+C)'soneday'swork=1/20+1/30+1/60=6/60=1/10Workdonein3days=(1/10+1/10)=1/5Now,1/5workisdonein3daysTherefore,Wholeworkwillbedonein(3*5)=15days.Ex.31.Sevenmencancompleteaworkin12days.Theystartedtheworkandafter5days,twomenleft.Inhowmanydayswilltheworkbecompletedbytheremainingmen?Sol.7*12mencompletetheworkin1day.Therefore,1man's1day'swork=1/847men's5dayswork=5/12=>remainingwork=1-5/12=7/125men's1day'swork=5/845/84workisdonbythemin1day7/12workisdonebythemin((84/5)*(7/12))=49/5days=94/5days.Ex.32.12mencompleteaworkin9days.Aftertheyhaveworkedfor6days,6moremenjoinedthem.Howmanydayswilltheytaketocompletetheremainingwork?Sol.1man's1daywork=1/10812men's6dayswork=6/9=2/3remainingwork=1–2/3=1/318men's1dayswork=18/108=1/61/6workisdonebythemin1daytherefore,1/3workisdonebythemin6/3=2days.Ex.33.Aman,awomanandaboycancompleteajobin3,4and12daysrespectively.Howmanyboysmustassist1manand1womantocompletethejobin¼ofaday?

Sol.(1man+1woman)'s1dayswork=1/3+1/4=7/12Workdoneby1manand1womenn1/4day=((7/12)*(1/4))=7/48Remainingwork=1-7/48=41/48Workdoneby1boyin¼day=((1/12)*(1/4))=1/48Therefore,Numberofboysrequired=((41/48)*48)=41daysEx.34.12mencancompleteapieceofworkin4days,while15womencancompletethesameworkin4days.6menstartworkingonthejobandafterworkingfor2days,allofthemstoppedworking.Howmanywomenshouldbeputonthejobtocompletetheremainingwork,ifitistobecompletedin3days.Sol.oneman'soneday'swork=1/48onewoman'soneday'swork=1/606men's2day'swork=((6/48)*2)=¼Remainingwork=3/4Now,1/60worksdonein1dayby1woman.So,¾workwillbedonein3daysby(60*(3/4)*(1/3))=15woman.Ex.35.Twelvechildrentakesixteendaystocompleteaworkwhichcanbecompletedby8adultsin12days.Sixteenadultsleftandfourchildrenjoinedthem.Howmanydayswilltheytaketocompletetheremainingwork?Sol.onechild'sonedaywork=1/192;oneadult'soneday'swork=1/96;workdonein3days=((1/96)*16*3)=1/2Remainingwork=1–½=1/2(6adults+4children)'s1day'swork=6/96+4/192=1/121/12workisdonebythemin1day.½workisdonebythem12*(1/2)=6daysEx.36.Sixteenmencancompleteaworkintwelvedays.Twentyfourchildrencancompletethesameworkin18days.12menand8childrenstartedworkingandaftereightdaysthreemorechildrenjoinedthem.Howmanydayswilltheynowtaketocompletetheremainingwork?Sol.oneman'soneday'swork=1/192onechild'soneday'swork=1/432Workdonein8days=8*(12/192+8/432)=8*(1/16+1/54)=35/54Remainingwork=1-35/54=19/54(12men+11children)'s1day'swork=12/192+11/432=19/216Now,19/216workisdonebythemin1day.Therefore,19/54workwillbedonebythemin((216/19)*(19/54))=4daysEx.37.Twenty-fourmencancompleteaworkin16days.Thirty-twowomencancompletethesameworkintwenty-fourdays.Sixteenmenandsixteenwomenstartedworkingandworkedfor12days.Howmanymoremenaretobeaddedtocompletetheremainingworkin2days?Sol.oneman'soneday'swork=1/384

onewoman'soneday'swork=1/768Workdonein12days=12*(16/384+16/768)=12*(3/48)=3/4Remainingwork=1–¾=1/4(16men+16women)'stwoday'swork=12*(16/384+16/768)=2/16=1/8Remainingwork=1/4-1/8=1/81/384workisdonen1dayby1man.Therefore,1/8workwillbedonein2daysin384*(1/8)*(1/2)=24menEx.38.4menand6womencancompleteaworkin8days,while3menand7womencancompleteitin10days.Inhowmanydayswill10womencompleteit?Sol.Let1man's1day'swork=xdaysand1woman's1day'swork=yThen,4x+6y=1/8and3x+7y=1/10.Solvingthesetwoequations,weget:x=11/400andy=1/400Therefore,1woman's1day'swork=1/400=>10womenwillcompletetheworkin40days.Ex.39.Oneman,3womenand4boyscandoapieceofworkin96hours,2menand8boyscandoitin80hours,2men&3womencandoitin120hours.5Men&12boyscandoitinhowmanyhours?Sol.Let1man's1hour'swork=x1woman's1hour'swork=y1boy's1hour'swork=zThen,x+3y+4z=1/96———–(1)2x+8z=1/80———-(2)adding(2)&(3)andsubtracting(1)3x+4z=1/96———(4)From(2)and(4),wegetx=1/480Substituting,weget:y=1/720andz=1/960(5men+12boy)'s1hour'swork=5/480+12/960=1/96+1/80=11/480Therefore,5menand12boyscandotheworkin480/11or437/11hours.

Exercise1.AtakestwiceasmuchtimeasBandCtakesthriceasmuchtimeasBtofinishawork.Workingtogether,theycanfinishtheworkin12days.FindthenumberofdaysneededforAtogotheworkaloneAns:442.6boyscancompleteapieceofworkin16h.Inhowmanyhourswill8boyscompletethesamework?Ans:123.AisthriceasgoodaworkmanasBandthereforeisabletofinishajobin30dayslessthanB.Howmanydayswilltheytaketofinishthejobworkingtogether?Ans:11K

i

4.5menstartworkingtocompleteaworkin15days.After5days,10womenareaccompaniedbythemtocompletetheworkinnext5days.Iftheworkistobedonebywomenonly,whencouldtheworkbeover,if10womenhavestartedit?Ans:15days5.AandBtogethercancompleteaworkin3days.Theystartedtogetherbutafter2days,Bleftthework.Iftheworkiscompletedafter2moredays,Balonecoulddotheworkinhowmanydays?Ans:66.90menareengagedtodoapieceofworkin40daysbutitisfoundthatin25days,$

&workis

completed.Howmanymenshouldbeallowedtogooffsothattheworkmaybefinishedintime?Ans:157.36workmenareemployedtofinishacertainworkin48days.Butitisfoundthatin24daysonly$

%workisdone.Howmanymoremenmustbetakentofinishtheworkintime?

Ans:188.If12menor18womencandoapieceofworkin14days.Howlongwill8menand16womentaketofinishthework?Ans:9days9.20mencompleteone-thirdofaworkin20days.Howmanymoremenshouldbeemployedtofinishtherestofworkin25moredays?Ans:1210.3mencandoapieceofworkin6days.5womencandothesameworkin18days.If4menand10womenworktogether,thenhowlongwillittaketofinishthework?

Ans:3days11.3mencandoapieceofworkin18days.6childrencanalsodothatworkin18days.4menand4childrentogetherwillfinishtheworkinhowmanydays?Ans:912.Aalonecancompleteaworkin12daysandBalonecancompletethesameworkin24days.InhowmanydayscanAandBtogethercompletethesamework?Ans:8days13.MeeshugetsRs.110foreverydaythatheworks.IfheearnsRs.2750inamonthof31days,forhowmanydaysdidhework?Ans:25days14.10menand8womentogethercancompleteaworkin5days.Workdonebyonewomaninadayisequaltohalftheworkdonebyamaninoneday.Howmanydayswillittakefor4menand6womentocompletethatwork?Ans:10days.15.Ifonemanortwowomenorthreeboyscanfinishaworkin88days,thenhowmanydayswilloneman,onewomenandoneboytogethertaketofinishthesamework?Ans:48days16.AandBcancompleteaworkin12daysand18days,respectively.Abeginstodotheworkandtheyworkalternatelyoneatatimeforonedayeach.Thewholeworkwillbecompletedinhowmanydays?Ans:14K

&

17.Itisgiventhat16menworking18hadaycanbuildawall36malong,4mbroadand24mhighin20days.Howmanymenwillberequiredtobuildawall64mlong,6mbroadand18mhighworking12hadayin16days?Ans:6018.Acontractistobecompletedin92daysand234menweresettowork,eachworking16haday.After66days,i

goftheworkiscompleted.Howmanyadditionalmenmaybeemployedso

thattheworkmaybecompletedintime,eachmannowworking18haday?Ans:16219.12enginesconsume30metrictonsofcoalwheneachisrunning18haday.Howmuchcoalwillberequiredfor16engines,eachrunning24haday,itbeinggiventhat6enginesofformertypeconsumeasmuchas8enginesoflattertype?Ans:40tones20.Agarrisonof6600menhadprovisionsfor64daysattherateof1700gperhead.Attheend

of14days,reinforcementarrivesanditwasfoundthattheprovisionwilllast34daysmoreattherateof1650gperhead.Whatisthestrengthofthereinforcement?Ans:340021.6mencandoapieceofworkin12dayswhile8womencandothesameworkin18days.Thesameworkcanbedoneby18childrenin10days.4men,12womenand20childrenworktogetherfor2days.Ifonlymenhavetocompleteremainingworkin1day,thenfindtherequirednumberofmen.Ans:3622.Acertainnumberofpersonscandigatrench200mlong,100mbroadand20mdeepin20days.Thesamenumberofpersonscandiganothertrench40mbroadand30mdeepin60days.Findthelengthofthe2ndtrenchAns:1000m23.Ifacertainnumberofworkmencandoapieceofworkin50h,inhowmanyhourswillanothersetofanequalnumberofmen,doapieceofwork,twiceasgreat,supposingthat4menofthe1stsetcandoasmuchworkin1has6menofthe2ndsetdoin1h?Ans:15024.Ifmmenworkingmhperday,candomunitsofworkinmdays,thennmenworkingnhperdaywouldbeabletocompletehowmanyunitsofworkinndays?Ans:Y

S

[525.Acontractorundertooktodoacertainpieceofworkin18days.Heemployedcertainnumberofmenbut12ofthembeingabsentfromthevery1stday,therestcouldfinishtheworkin30days.FindthenumberofmenoriginallyemployedAns:3026.Thereissufficientfoodfor800menfor62days.After56days,560menleavetheplace.Forhowmanydayswilltherestofthefoodlastfortherestofthemen?Ans:20days27.Rashmicandoapieceofworkin16days.Ravinacandothesameworkin12i

%dayswhile

Gitikacandoitin32days.AllofthemstartedtoworktogetherbutRashmileavesafter4days.Ravinaleavesthejob3daysbeforethecompletionofthework.Howlongtheworklast?Ans:9days28.Sitacancompleteaworkin10days.Geetais25%moreefficientthanSitaandRitais60%moreefficientthanGeeta.Howmanydayswilltheytakentocompletethatwork?Ans:6%

gdays

29.AandBcandoapieceofworkin9days18days,respectively.AstheywereIII,theycoulddo90%and72%oftheirefficiencyrespectively.Howmanydayswilltheytaketocompletethe

worktogether?Ans:2K

g

30.Acando&

ipartofaworkin18days.Bcando$

&partofthelaneworkin12daysandCcan

do60%ofthelaneworkin24days.IfA,BandCdotheworkalonefor4days,6daysand8days,respectively.Whatpercentofworkisremaining?Ans:30%31.Xcando20%ofaworkinaday,Ycando35%ofthesameworkinadayandX,YandZtogethercando50%ofthesameworkinaday.HowmuchpercentofworkcanbedonebyZinaday?Ans:5%32.A,BandCcandoapieceofworkin18,24and36days,respectively.Theyworkalternatively,firstdayC,SeconddayBandthirddayA,fourthdayCandsoon.Howmanydayswillbeneededtocompletetheworklikethisway?Ans:2433.Eightchildrenand12mencompleteacertainpieceofworkin9days.Eachchildtakestwicethetimetakenbyamantofinishthework.Inhowmanydayswill12menfinishthesamework?Ans:1234.Onemanandonewomencandoworkin10days.Twomencandotheworkin8days.Inhowmanydaysonewomanalonecandothesamework?Ans:26$

&

35.AandBcantogetherdoaworkin24days.Bcompletesone-thirdoftheworkin12days.InhowmanydayscanAcompletetheremainingwork?Ans:4836.12mentake18daystocompleteaworkwhereas12childrentake27daystocompletethesamework.Howmanydayswill10menand12childrentogethertaketocompletethesamework?Ans:12days37.IfAandBcandoapieceofworkin15days,BandCin20dayswherasAandCcandoitin30days,thenA,BandCtogethercandoitinhowmanydays?Ans:13K

&days

38.TwopipesAandBcanfillacisternindividuallyin20minutesand30minutesrespectively.AthirdwastepipeCcanemptythecisternin1hour.IfboththepipesAandBandthewastepipeCareopenedsimultaneouslyandinitiallythecisternisempty,thenthecisternwillbefilledin

Ans:15minutes39.A50metrelongand35metrewidepondwasdugby250labourersin18daysunderthefoodforworkprogramme.Thenumberofdaysinwhicha70metrelongand40metrewidepondhavingthesamedepthcanbedugby3000labourersisAns:2440.AandBcandoseparatelyapieceofworkin20and30daysrespectively.TheyworktogetherforsometimeandthenBstops.IfAcompletestherestoftheworkin10days,thenBhasworkedforAns:6days41.‘A’takesonehourtodigapitofK

$× K$× K$metresdimension,While‘B’takesoneanda

quarterhour.Ifworkersworkfor8hoursperdayand72pitsarerequiredtobedug,then‘A’and‘B’togetherwillcompletetheworkinAns:5days42.Afarmerploughingacertainareaoflandattherateof10m2perhourfinishedhisworkat6pm.Ifheploughsattherateof12m2perhour,hefinisheshisworkat4p.m.ThetotalsquaremetreoflandhewasploughingwasAns:120m243.‘A’completes$

jofadiggingworkonthefirstdayand&

goftheremainingworkonthesecond

day.Hehastodig8cubicmetresonthethirddaytocompletethework.ThetotalcubicmetresdiggingtobedonewasAns:18cu.m44.Atankoflength,breadthandheight2m,3.75mand4.73mrespectivelyisfilledbyatapintwohours.TimeneededtofillupatankhavingdimensionsdoubleoftheabovewillbeAns:16hours45.XandYtogetherfinishaworkin8hours.IfXalonecanfinishitin12hours,thenYalonecanfinishthesameinAns:24hours46.Ifxmencandoapieceofworkin8daysand(x+4)mencoulddothesameworkin6days,thenxisequaltoAns:1247.Acandoapieceofworkin24days.IfBis60%moreefficientthanA,thenthenumberofdaysrequiredbyBtodothesameworkisAns:1548.IfAandBcandoapieceofworkin10and15daysrespectively,thentheamountofwork

theycantogetherdoin3daysisAns:K

$

49.Amanand3womencandoapieceofworkin20days.Ifawomancando50%oftheworkthatamancando,inhowmanydaysawomanwilldothejobwhenengagedalone?Ans:100days50.If5mentake15hourstoharvestacrop,then7womenwilldothesamejobinhowmanyhours,giventhatamandoes40%moreworkascomparedtotheworkdonebyawoman?Ans:15hours51.AandBcandoaworkin2days.IfBalonecandothesameworkin6days,thenAalonecandotwiceasgreatworkinAns:6days52.Atapcanfillatubwithwaterin6hours,andanothertapattachedtothetubcanemptyin10hours,thenthetimetofillthetubwhenboththetapsareopenedatatimeisAns:15hours53.40mencanbuildawall20mhighin15days.Thenumberofmenrequiredtobuildasimilarwall25mhighin6days,willbeAns:12554.PipeAcanfillatankin4hoursandpipeBcanemptyitin6hours.Ifbothofthemareopenedatthesametimewhenthetankisempty,thennumberofhoursrequiredtofillthetankwillbeAns:1255.Acertainjobisassignedtoagroupofmentodoin20days.But12mendidnotturnupforthejobandtheremainingmendidthejobin32days.TheoriginalnumberofmeninthegroupwasAns:3256.14personscancompleteaworkin16days.8personsstartedtheworkin12daysaftertheystartedthework8morepersonjoinedthem.Howmanydayswilltheytaketocompletetheremainingwork?Ans:8days57.15mencancompleteaworkin10dayswhile20boyscancompletethesameworkin15days.Howmanydayswill10menand10boystogethertaketocompletethesamewok?Ans:1058.14workerscanmake1400toysin5days.Onedayaftertheystartedthework14moreworkersjoinedthem.Howmanydayswilltheytaketocompletetheremainingwork?Ans:2

59.14mencancompleteaworkin12days.4daysaftertheystartedthework,2moremenjoinedthem.Howmanydayswilltheytaketocompletetheremainingwork?Ans:760.RohanandSunilseparatelycancompleteaworkin8hoursand4hoursrespectively.Howmuchtimewilltheytakewhenworkingtogether?Ans:2$

&hours

61.Ganesh,RamandSohantogethercandoaworkin16days.IfGaneshandRamtogethercandothesameworkin24daysthen,howlongwilltakeSohanalonetodothesamework?Ans:48days62.AandBtogethercandoaworkin8days.IfAalonecandoitin12days,theninhowmanydayscanBalonedoit?Ans:2463.AandBtogethercandoaworkin24days.BalonedoesitsK

&partin12days.Howlongwill

Aalonetaketodotheremainingwork?Ans:4864.Ram,DilipandShekharcanCompleteaworkin20days.IfRamandDiliptogethercancompletethesameworkin30days,thenhowlongwillShekartaketocompleteit?Ans:6065.AandBseparatelycancompleteaworkin6daysand3daysrespectively.Iftheyworktogether,theninhowmanydayswilltheycompletethework?Ans:266.Acertainworkcanbedoneinacertaintimeby25men,with5menless,itcouldhavebeendonein3daysmore.Inwhattimecanitbedoneby40men?Ans:7K

$days

67.Acertainworkcanbedoneinacertaintimeby36men,buthadtherebeen9menmore,itcouldhavebeendonein5daysless.Inhowmanydayscan20mendoit?Ans:45days68.Awallcanbebuiltbyacertainnumberofmasonsin27days.Butitrequires15menlesstofinishitin30days.Howmanymasonswillbuilditin50days?Ans:8169.AandBcandoaworkin12and20daysrespectively.If,aftertheyworkatittogetherforsometime,Cwhocandothewholeworkin10days,finishesitin2days,howlongdidAandBwork?

Ans:6days.70.90menareengagedtodoapieceofworkin40days,butitisfoundthatin25days2/3workincompleted.Howmanymenshouldbeallowedtogooffsothattheworkmaybefinishedintime?Ans:1571.8menwith2boyscandoapieceofworkin7days,which25menwith13boyscandoin2days.Whatfractionofthewholecanbedonein1dayby11menwith20boys?Ans:1/372.Onemanworksfor5hoursadayfor10daysandcompletesapieceofwork.Oneboyworksfor8hoursfor20daystocompletethesamejob.Whatistheratiooftheirefficiencies?Ans:16:573.AisthriceasgoodaworkerasB&takes8dayslesstodoapieceofworkthanB.InhowmanydayscanBdothework?Ans:12days74.AandBcandoapieceofworkin12days,BandCin15daysandCandAin20days.Whentheyworktogether,thetimerequiredwillbeAns:10days75.If60gunsfiring5roundsin6minkill350menin1K

ihours,howmanygunsfiring7roundsin

9minwillkill980menin25minatthesamerate?Ans:240guns76.If7menor8womenor10boyscanfinishapieceofworkin24daysof9hours,findhowmanymenwiththehelpof4womenand5boyscanfinishitin18daysof6hours?Ans:7men77.Acaneat10burgersin2daysandBcaneat20burgersin1day.Forhowlongwill250burgerslastifbothstarteatingthemtogether?Ans:10days78.If10mencancompleteapieceofworkin30days,howmanymenarerequiredtocompletethesamein25days?Ans:1279.10mencancompleteapieceofworkin18days.After6days,5moremenjoined.Inhowmanydaystheremainingworkwouldbecompleted?Ans:1280.15personscanfill35boxesin7days.Howmanypersonsarerequiredtofill65boxesin5days?

Ans:3981.If30menworking7hoursadaycandoapieceofworkin18days,inhowmanydayswill21menworking8hoursadaydothesamepieceofwork?Ans:22K

$

82.If3menor4womencanploughafieldin43days,howlongwill7menand5womentaketoploughit?Ans:12days83.AandBcantogetherfinishaworkin20days.Theyworkedatitfor20daysandthenBleft.TheremainingworkdonebyAalonein20moredays.AalonecanfinishtheworkinAns:60days84.AdoeshalfassuchworkasBinthree-fourthofthetime.Iftogethertheytake18daystocompletethework,howmuchtimeshallBtakentodoit?Ans:30days85.Acandoacertainjobis12days.Bis60%moreefficientthanA.Thenumberofdays,ittakesBtodothesamepieceofworkisAns:7K

$

86.IfRamesh,SureshandHarishcandoapieceofworkin15days,10daysand6daysrespectively,howlongwilltheytaketodoit,ifallthethreetogethercompletetheremainingwork?Ans:3days87.Twelvemencancompleteaworkin8days.Threedaysaftertheystartedtheworkand3moremenjoined.Inhowmanydayswillallofthemtogethercompletetheremainingwork?Ans:488.AandBcanseparatelytodoapieceofworkin20daysand15daysrespectively.Theyworkedtogetherfor6days,afterwhichBwasreplacedbyC.Iftheworkwasfinishedinnext4days,thenthenumberofdaysinwhichCalonecoulddotheworkwillbeAns:4089.AdoeshalfasmuchasBandCdoeshalfasmuchworkasAandBtogether.IfCalonecanfinishtheworkin40days,thentogetherallwillfinishtheworkinAns:13K

&days

90.A,BandCcandoapieceofworkin36,54and72daysrespectively.TheystartedtheworkbutAleft8daysbeforethecompletionoftheworkwhileBleft12daysbeforethecompletion.ThenumberofdaysforwhichCworkedisAns:24