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Averages

Assume5friendswenttoamovie.Ifthetotalmoneywiththefriendsisequalto1000thenwesaythataveragemoneywitheachpersonisRs.200/ButitisnotnecessarythateachpersonhasRs.200.Somemayhavemoremoneythantheothers,butthetotalmoneyisequaltoRs.1000.LetussayPersonEisrichandheboughtRs.500tothemovie.SothemoneywiththeremainingfriendsisequaltoRs.500.IfEhasnotcometothemovie,theaveragemoneywiththeotherfriendscomesdowntoRs.125.Thatmeans,PersonEhasmoremoneythanthecombinedaveragemoneyofallthefivefriendstogethersohecontributestotheotherfriendsabovetheaveragerequired.Thatmeans,hadpersonEboughtRs.125tothemovietheaveragestandsatRs.125buthehasRs.375morethantheaveragerequiredtomaintainandthisextraamountisdistributedamongallthefriendsequally,sothtatAverageormean:TheMean(Average)ofagroupofnumbersisthesumofthenumbersdividedbythenumberofnumbers:AverageorMean=Sumoftheobservations/Numberofobservations

Iftheaverageofmquantitiesisxandtheaverageageofnotherquantitiesisythentheaverageofallof

themputtogetheris=

Iftheaverageageofmquantitiesisxandtheaverageageofnquantitiesoutofthem(mquantities)isy

thentheaverageoftherestofthequantitiesis=.

Iftheaverageofnnumbersisxandifkisaddedtoorsubtractedfromeachgivennumbertheaverageofnnumbersbecomes(x+k)or(xk)respectively.Intheotherwordsaveragevaluewillbeincreasedordecreasedby

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Iftheaverageofnnumbersisxandifeachgivennumberismultipliedtoordividedbykthentheaverageof

nnumbersbecomeskxor respectively.

Ifapersontravelsadistanceataspeedofxkm/hrandthesamedistanceataspeedofykm/hrthentheaverage

speedduringthewholejourneyisgivenby km/hr.

IfapersoncoversAkmatxkm/hrandBkmatykm/hrandCkmatzkm/hr,thentheaveragespeedincovering

thewholedistanceis km/hr.

Theaverageofn(wherenisanoddnumber)consecutivenumbersisalwaysthemiddlenumberE.g.1,3,5,7,9.TheAverage=Middlenumber=5

Theaveragen(wherenisevennumber)consectuivenumbersistheaverageofthetwomiddlenumbers.

E.g.Averageof(2,4,6,8,10,12)= =7

SolvedExamples

1.Theaverageofagesof10personsinaclubwas32.Whatshouldbetheageofthenewpersonjoiningintheclubsoastoincreasetheaverageby4?Totalageof10persons=10x32=320Totalageof11persons=11x36=396(asthenewaverageis4morethanpersentaverage)Sotheageofthepersonjoiningis=396320=76

Alternatemethod:Iftheageofthenewpersonjoiningtheclubis32thenthereisnochangeintheaverage.Ifthenewaveragehastobe36,thepersonwhoisjoiningmustcontribute4yearstoall11persons.Thatishemusthaveanage44yearsabove32.Sonewaverageis76

2.Theaverageweightoftheteacherandsixstudentsis12kgwhichisreducedby5kgiftheweightoftheteacher

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6

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isexcluded.Howmuchdoestheteacherweigh?Totalageofthestudentsandteachertogether=7x12=84Newaverageafterexcludingteacher=6x7=42Teachersweight=7442=42

Alternatemethod:Teacherhastakenhercontributionof5kgsfromeachofthestudents.Assheiscontributing30kgstoallstudents,oncesheisexcludedthose30kgsremainswithheralongwithoriginalweight12kgs.Soherweightis30+12=42kgs

3.Abatsmanhadacertainaverageofrunsfor16innings.Inthe17thinnings,hemadeascoreof87runstherebyincreasinghisaverageby3.Whatishisaverageafter17innings?Assumehisinitialaverage=XHistotalrunsafter16innings=16XAfterscoring87runshisaveragegotincreasedby3toX+3Sohistotalrunsafter17innings=17x(X+3)Butitwasgiventhatthedifferenceinthetotalscoresafter16inningsand17innings=87ThereforeHisnewaverage=36+3=39

Alternatemethod:His87runsinthe17thinningscontributedtoalltheseventeeninningstoincreasetheaverageby3.Sohemusthavescored17x3=51runsextratomaintaintheaverage.Sohispreviousaveragewillbe8751=36.Presentaverage=39

4.Someconsecutivenaturalnumbers,startingwith1,arewrittenontheboard.Now,oneofthenumberswaserasedandtheaverageoftheremainingnumbersis800/39.Findthenumberwhichwaserased.

Weknowthataverageofnconsecutivenumbesaverage=

Ifthegivennissufficientlylarge,theaveragedoesnotchangemucheventhoughweexcludeoneortwonumbersfromit.Sotheapproximatenumberofobservationsisalmostdoubletotheaverage(Remember:theaverageofconsecutivenumbersalmostliesinthemiddle)

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Theapproximateaverageis800/39=Approx20.Sotheinitialnumbersmaybenearerto40.Inthisquestionitisactually40asfromthedenominatorofthenewaverage800/39.Theinitialnumbersare40.

Sumof40consecutivenumbers=

Sumof39numbers=averagexnumberofobservations= =800

Sothenumberexcluded=820800=20

5.TheaverageweightofthestudentsinfoursectionsA,B,CandDis55kg.TheaverageweightsofthestudentsA,B,CandDare45kg,40kg,75kgand85kgrespectively.IftheaverageweightofthestudentsofsectionAandDtogetheris65kgandthatofBandDtogether55kg,thenwhatistheratioofthenumberofstudentsinAandC?Ifweobservecarefully,thetotalaverageofA,B,C,DisequaltoAverageofBandDtogether.SotheaverageofAandCmustbe55.WeapplyalligationruletodetermineinwhatrationAandCmustbeclubbedtogethertogetanaverageof55.

Soratio=2:1

6.Theaverageageof40studentsinaclassis15years.Iftheageofteacherisalsoincluded,theaveragebecomes16years,findtheageoftheteacher.Ifteacher'sageis15years,thereisnochangeintheaverage.Butteacherhascontributed1yeartoallthestudentsalongwithmaintaininghisageat16.Ageofteacher=Averageageofall+Totalincreaseinage=16+(1x40)=56years

7.Averageageof9membersofaclubis29years.If2morepersonswiththeaverageageof40yearshavebecomethemembersoftheclub,findaverageageofallthe11members.

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g

Averageageof9members=29yearsExcessageof2newmembers=(4029)x2=22years

Therefore,Increaseintheaverageageoninclusionof2newmembers= =2years

Therefore,Averageageof11members=29+2=31years

8.Averageageof18menisdecreasedby1yearwhenonemanwhoseageis49yearsisreplacedbyanewman.Findageofthenewman.Ifthereplacedpersonsageissamewiththeexistingaverage,thereisnochangeintheaverage.Butbyreplacementoveralldecreaseintheageis18x1=18years.Thisisthechangeboughtbythenewman.Ageofnewman=AgeofmanreplacedTotaldecreaseinage=49(1x18)=31years

9.Theaverageweightof12personsis50kg.Onreplacingamanwhoseweightis53kg.withanewman,newaveragebecomes49kg.Theweightofthenewmanis:Ifthereplacedpersonsweightisthesameastheexistingaverage,thereisnochangeintheaverage.Butbyreplacementoveralldecreaseintheageis12x1=12kg.Thisisthechangeboughtbythenewman.Weightofnewman=WeightofmanreplacedTotaldecreaseinweight=531x12=41kg.

10.Theaverageageof12meninagroupisincreasedby2yearswhentwomenwhoseagesare20yearsand22years,arereplacedbynewmembers.Whatistheaverageageofthenewmenincluded?Totalageoftwomenreplaced=20+22=42yearsTotalincreaseinageonreplacement=2x12=24yearsTotalagetwonewpersonsincluded=42+24=66years

Therefore,Averageageofnewpersons= =33years

11.Averageageof7membersofafamilyis29years.Ifpresentageoftheyoungestmemberis5year,findaverageageoftheremainingmembersatthetimeofbirthoftheyoungestmember.Averageage(present)of7members=29years5yearsago,averageageof7memberswas295=24years.Since,theyoungestmemberwasnotborn5yearsago.

Therefore,Averageageofremaining6membersisincreasedby =4years.

Therefore,5yearsago,averageageof6memberswas24+4=28years.

12.Averageweightof8personsis48kg.Ifonemanweighing34kg,isdied,whatisaaverageageoftheremaining7persons.Averageweightof8persons=48kg.Therefore,Excessofaverageweightthantheweightofmandied=48kg.34kg.=14kg.

Therefore,Increaseinweightoftheremaining7persons= =2kg.

Therefore,Weightofremaining7persons=48+2=50kg.

13.Theaverageexpenditureofamanfor10daysisRs.45perday.Ifhisaverageexpenditureforthefirst3daysisRs.52perday,findhisaverageexpenditurefortheremaining7days.Averageexpenditurefor10days=Rs.45Averageexpenditureforfirst3days=Rs.52Therefore,Excessexpenditureforfirst3days=3x(Rs.52Rs.45)=Rs.21

Therefore,Decreaseinaverageexpenditureforremaining7days= =Rs.3

Therefore,Averageexpenditureforremaining7days=453=Rs.42

14.Theaverageweightofthestudentsofaclassis40kg.5newstudentswiththeaverageweightof46kg.havingjoinedtheclass,theaverageweightoftheclassisincreasedby2kg.Findthenumberofstudentsintheclassoriginally?Totalincreaseinweightonincluding5morestudents=5x(4640)=30kg.But,increaseinaverageweight=2kg.

Therefore,Totalstudents(atpresent)= =15students

Therefore,Students(originally)=155=10students

15.Averagetemperaturefrom9thto16thofamonthis andthatfrom10thto17this .Whatisthetemperatureon17th,iftemperatureon9this ?Totaldaysineachcaseare8daysDifferencebetweentemperatureon9thand17th=8xSince,temperaturefor8daysincluding17thismorethanthatof8daysincluding9th.Therefore,Temperatureon17thismorethanthetemperatureon9th.

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Therefore,Temperatureon17th=Temperatureon9th+Differencebetweentemperatures=

16.Theaverageof11observationsis72.Ifaverageoffirst6observationsis70andthatoflast6observationsis71,thenthe6thobservationis:6thobservation=72+6x(7072)+6x(7172)=726x26x1=54

17.Averageexpenditureofapersonforthefirst3daysofaweekisRs.350andforthenext4daysisRs.420.Averageexpenditureofthemanforthewholeweekis:Assumedmean=Rs.350Totalexcessthanassumedmean=4x(Rs.420Rs.350)=Rs.280

Therefore,Increaseinaverageexpenditure=Rs. =Rs.40

Therefore,Averageexpenditurefor7days=Rs.350+Rs.40=Rs.390

18.11friendswenttoahotelanddecidedtopaythebillamountequally.But10ofthemcouldpayRs.60eachasaresult11thhastopayRs.50extrathanhisshare.Findtheamountpaidbyhim.Averageamountpaidby10persons=Rs.60

IncreaseinaverageduetoRs.50paidextrabythe11thmen=Rs. =Rs.5

Therefore,Averageexpenditureof11friends=Rs.60+Rs.5=Rs.65Therefore,Amountpaidbythe11thmen=Rs.65+Rs.50=Rs.115

19.Theaveragemarksobtainedbysomestudentsinanexaminationis54.If20%ofthestudentsgotameanscoreof90marksandthe30%ofthestudentsgotameanscoreof20.Findtheaveragemarksoftheremainingstudents.Remainingstudents=100%20%30%=50%Letremainingstudentsgotameanscoreofxmarks.Then,20%of90+30%of20+50%ofx=54Therefore,18+6+50%ofx=54Therefore,50%ofx=54186=30Therefore,x=30x2=60marks

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MCQ's

1.TheaveragetemperatureofMonday,Tuesday,WednesdayandThursdaywas andthatofTuesday,Wednesday,ThursdayandFridaywas .IfthetemperatureonMondaywas ,thetemperatureofFridaywas:a.b.c.d.CorrectOption:CExplanation:M+T+W+Th=Mondaytemparatureis30.SoT+W+Th=(15230)=122T+W+Th+F=F=(160122)=

2.AshopkeeperearnedRs.504in12days.HisaverageincomeforthefirstfourdayswasRs.40aday.Hisaverageincomefortheremainingdaysis:a.Rs.40b.Rs.42c.Rs.43d.Rs.45CorrectOption:CExplanation:Lettheaverageforremaining8daysbeRs.xaday.Then, or8x=344orx=43Requiredaverage=Rs.43

3.Amanwhosebowlingaverageis12.4takes5wicketsfor26runsandtherebydecreaseshisaverageby0.4.Thenumberofwicketstakenbyhimbeforehislastmatchis:a.85b.78c.72

g

g

g g 4

d.64CorrectOption:AExplanation:Letthenumberofwicketstakenbeforethelastmatch=x.

then

4.TheaverageofweightofthreemenA,BandCis84kg.AnothermanDjoinsthegroupandtheaveragenowbecomes80kg.IfanothermanE,whoseweightis3kg.morethanthatofD,replacesA,thentheaverageweightofB,C,DandEbecomes79kg.TheweightofAis:a.70kgb.72kgc.75kgd.80kgCorrectOption:CExplanation:A+B+C=A+B+C+D=D=(320252)=68andE=(68+3)=71Now,B+C+D+E=B+C+D=(31671)=245kgSo,A=(320245)=75kg

5.TheaverageageofA,B,C,Dfiveyearsagowas45years.ByincludingX,thepresentaverageageofallthefiveis49years.ThepresentageofXis:a.64yearsb.48yearsc.45yearsd.40yearsCorrectOption:CExplanation:As5yearsagoaverageis45,thepresentaverageis50.SosumoftheagesofA,B,C,Dis50x4=200ButgiventhataverageofA,B,C,DandXis49.Sosumoftheirages=49x5=245

4

4

4

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g

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PresentageofX=245200=45years.

6.Theaverageageof24studentsinaclassis10.Iftheteacher'sageisincluding,theaverageincreasesbyone.Theageoftheteacheris:a.25yearsb.30yearsc.35yearsd.40yearsCorrectOption:CExplanation:Sumofthetenstudentsage=24x10Sumofthetenstudentsincludingtheteacher=25x11Ageoftheteacher= =35years

7.Theaverageofmarksobtainedby120candidateswas35.Iftheaverageofmarksofpassedcandidateswas39andthatoffailedcandidateswas15,thenumberofcandidateswhopassedtheexaminationis:a.100b.110c.120d.150CorrectOption:AExplanation:Letthenumberofcandidaeswhopassed=x,Then,

24x=42001800or

x=

8.TheaverageexpenditureofamanforthefirstfivemonthsisRs.120andforthenextsevenmonthsitisRs.130.IfhesavesRs.290inthatyear,hismonthlyaverageincomeis:a.Rs.140b.Rs.150c.Rs.160

g g

g 4 g 4 g

d.Rs.170CorrectOption:BExplanation:Totalincome=Rs. =Rs.1800Averagemonthlyincome

=Rs. =Rs.150

9.Theaveragesalaryof20workersinanofficeisRs.1900permonth.Ifthemanager'ssalaryisadded,theaveragesalarybecomesRs.2000p.m.Whatisthemanager'sannualsalary?a.Rs.24,000b.Rs.25,200c.Rs.45,600d.NoneoftheseCorrectOption:DExplanation:Manager'ssalarypermonth= =Rs.4000Manager'sAnnualsalary=Rs. =Rs.48000

10.Theaverageweightofaclassof40studentsis40kg.Iftheweightoftheteacherisincluded,theaverageweightincreasesby500gms.Theweightoftheteacheris:a.40.5kgb.60kgc.60.5kgd.62kgCorrectOption:CExplanation:Weightoftheteacher= kg=60.5kg

11.Theaverageweightof8menisincreasedby2kgwhenoneofthemen,whoseweightis50kgisreplacedbyanewman.Theweightofthenewmanis:a.52kg

g g

/ g g

g

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b.58kgc.66kgd.68kgCorrectOption:CExplanation:Weightincreased= kg=16kgWeightofnewman=(50+16)kg=66kg

12.Theaverageweightof8personsisincreasedby2.5kg,Whenoneofthemwhoseweightis56kgisreplacedbyanewman.Theweightofthenewmanis:a.66kgb.75kgc.76kgd.86kgCorrectOption:CExplanation:Totalincrease= kg=20kg.Weightofnewman=(50+16)kg=66kg

13.Theaverageageofanadultclassis40years.12newstudentswithanaverageof32yearsjointheclass,therebydecreasingtheaverageby4years.Theoriginalstrengthoftheclasswas:a.10b.11c.12d.15CorrectOption:CExplanation:Letoriginalstrength=x,Then,40x+12 32=(x+12) 36or40x+384=36x+432or4x=48orx=12

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14.Theaverageof6observationsis12.Anewseventhobservationisincludedandthenewaverageisdecreasedby1.Theseventhobservationis:a.1b.3c.5d.6CorrectOption:CExplanation:Seventhobservation=(7 116 12)=5

15.Outoffournumbers,theaverageoffirstthreeis15andthatofthelastthreeis16.Ifthelastnumberis19,thefirstis:a.15b.16c.18d.19CorrectOption:BExplanation:Sumoffournumbers=(15 3+19)=64Sumoflastthreenumbers=(16 3)=48Firstnumber=(6448)=16

16.Theaverageof10numbersiscalculatedas15.Itisdiscoveredlateronthatwhilecalculatingtheaverageonenumber,namely36waswronglyreadas26.Thecorrectaverageis:a.12.4b.14c.16d.18.6CorrectOption:CExplanation:Sumofnumbers=(10 1526+36)=160

Correctaverage=

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17.Theaverageof25resultsis18,thatoffirsttwelveis14andoflasttwelveis17.Thirteenthresultis:a.28b.72c.78d.85CorrectOption:CExplanation:Thirteenthresult=(25 1812 14+12 17)=78

18.Amangoestoaplaceattherateof4kmph.Hecomesbackonabicycleat16kmph.Hisaveragespeedfortheentirejourneyis:a.5km/hrb.6.4km/hrc.8.5km/hrd.10km/hr.CorrectOption:BExplanation:

Averagespeed= km/hr

= km/hr=6.4km/hr.

19.Theaveragetemperatureofthefirstthreedaysis Candthatofthenextthreedaysis C.Iftheaverageofthewholeweekis C,thetemperatureofthelastdayoftheweekis:a.b.c.d.CorrectOption:CExplanation:Lettheseventhdaytemperaturebex.Thentotaltemperatureforthewholeweekis

g g g

45

4 5

g g

g g g 4 g

x=31.5

20.Theaverageageof30studentsinaclassis12years.Theaverageageofagroupof5ofthestudentsis10yearsandthatofanothergroupof5ofthemis14years.Theaverageoftheremainingstudentsis:a.8yearsb.10yearsc.12yearsd.14yearsCorrectOption:CExplanation:Letitbex.Then:

20x=360120or20x=240orx=12

21.Theaverageof8numbersis21.Ifeachofthenumberismultipliedby8,theaverageofthenewsetofnumbersis:a.8b.21c.29d.168CorrectOption:DExplanation:Averageofnewnumbers=

22.Theaverageof50numbersid38.Iftwonumbers,namely45and55arediscarded,theaverageoftheremainingnumbersis:a.36.5b.37.0c.37.5d.37.52CorrectOption:CExplanation:

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g

Totalof50numbers=Totalof48numbers=(1900(45+55)]=1800

Requiredaverage=

23.Theaverageheightof30girlsoutofaclassof40is160cm.andthatoftheremaininggirlsis156cm.Theaverageheightofthewholeclassis:a.158cmsb.158.5cmsc.159cmsd.159.5cmsCorrectOption:CExplanation:

Averageheightofthewholeclass= cms

24.Ifa,b,c,d,earefiveconsecutiveoddnumbers,theiraverageis:a.5(a+4)

b.

c.5(a+b+c+d+e)d.NoneoftheseCorrectOption:DExplanation:Ifthefirstnumberisa,thentheremainingnumbersaa+2,a+4,a+6,a+8

Average:=

25.Theaverageageofthreeboysis15years.Iftheiragesareintheratio3:5:7,theageoftheyoungestboyis:a.9yearsb.15yearsc.18yearsd.21yearsCorrectOption:AExplanation:

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orx=3

Ageofyoungest=3x=9years

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