Aptitude .pdf

1. [QuantitativeAptitude]
GeneralFormulaandMethods

2. ALGEBRA:
1.Sumoffirstnnaturalnumbers=n(n+1)/2
2.Sumofthesquaresoffirstnnaturalnumbers=
n(n+1)(2n+1)/6
3.Sumofthecubesoffirstnnaturalnumbers=
[n(n+1)/2]
2
4.
Sumoffirstnnaturaloddnumbers=n
2
5.Average=(Sumofitems)/Numberofitems
ArithmeticProgression(A.P.):
AnA.P.isoftheforma,a+d,a+2d,a+3d,...
whereaiscalledthe'firstterm'anddiscalledthe
'commondifference'
1.n
th
termofanA.P.tn=a+(n-1)d
2.SumofthefirstntermsofanA.P.Sn=n/2[2a+(n-1)d]
orSn=n/2(firstterm+lastterm)
GeometricalProgression(G.P.):
AG.P.isoftheforma,ar,ar2,ar3,...
whereaiscalledthe'firstterm'andriscalledthe
'commonratio'.
1.n
th
termofaG.P.tn=ar
n-1
2.SumofthefirstntermsinaG.P.Sn=a[1-r
n
]/[1-r]

3. PermutationsandCombinations:
1.
n
Pr=n!/(n-r)!
2.
n
Pn=n!
3.
n
P1=n
1.
n
Cr=n!/(r!(n-r)!)
2.
n
C1=n
3.
n
C0=1
4.
n
Cr=
n
Cn-r
5.
n
Cr=
n
Pr/r!
TestsofDivisibility:
1.Anumberisdivisibleby2ifitisanevennumber.
2.Anumberisdivisibleby3ifthesumofthedigitsis
divisibleby3.
3.Anumberisdivisibleby4ifthenumberformedbythe
lasttwodigitsisdivisibleby4.
4.Anumberisdivisibleby5iftheunitsdigitiseither5
or0.
5.Anumberisdivisibleby6ifthenumberisdivisibleby
both2and3.
6.Anumberisdivisibleby8ifthenumberformedbythe
lastthreedigitsisdivisibleby8.
7.Anumberisdivisibleby9ifthesumofthedigitsis
divisibleby9.
8.Anumberisdivisibleby10iftheunitsdigitis0.

4. 9.Anumberisdivisibleby11ifthedifferenceofthesum
ofitsdigitsatoddplacesandthesumofitsdigitsateven
places,isdivisibleby11.
H.C.FandL.C.M:
H.C.FstandsforHighestCommonFactor.Theother
namesforH.C.FareGreatestCommon
Divisor(G.C.D)andGreatestCommonMeasure
(G.C.M).
TheH.C.F.oftwoormorenumbersisthegreatest
numberthatdivideseachoneofthem
exactly.
Theleastnumberwhichisexactlydivisiblebyeachone
ofthegivennumbersiscalledtheir
L.C.M.
Twonumbersaresaidtobeco-primeiftheirH.C.F.is1
H.C.F.offractions=H.C.F.ofnumerators/L.C.Mof
denominators
L.C.M.offractions=G.C.D.ofnumerators/H.C.Fof
denominators
Productoftwonumbers=ProductoftheirH.C.F.and
L.C.M.
PERCENTAGES:
1.IfAisR%morethanB,thenBislessthanAby
R/(100+R)*100
2.IfAisR%lessthanB,thenBismorethanAby
R/(100-R)*100

5. 3.IfthepriceofacommodityincreasesbyR%,then
reductioninconsumption,nottoincreasetheexpenditure
is:R/(100+R)*100
4.IfthepriceofacommoditydecreasesbyR%,thenthe
increaseinconsumption,nottodecreasetheexpenditure
is:R/(100-R)*100
PROFIT&LOSS:
1.Gain=SellingPrice(S.P.)-CostPrice(C.P)
2.Loss=C.P.-S.P.
3.Gain%=Gain*100/C.P.
4.Loss%=Loss*100/C.P.
5.S.P.=(100+Gain%)/100*C.P.
6.S.P.=(100-Loss%)/100*C.P.
RATIO&PROPORTIONS:
1.Theratioa:brepresentsafractiona/b.aiscalled
antecedentandbiscalledconsequent.
2.Theequalityoftwodifferentratiosiscalledproportion.
3.Ifa:b=c:dthena,b,c,dareinproportion.Thisis
representedbya:b::c:d.
4.Ina:b=c:d,thenwehavea*d=b*c.
5.Ifa/b=c/dthen(a+b)/(a–b)=(d+c)/(d–c).

6. TIME&WORK:
1.IfAcandoapieceofworkinndays,thenA's1day's
work=1/n
2.IfAandBworktogetherforndays,then(A+B)'s1
days'swork=1/n
3.IfAistwiceasgoodworkmanasB,thenratioofwork
donebyAandB=2:1
PIPES&CISTERNS:
1.Ifapipecanfillatankinxhours,thenpartoftank
filledinonehour=1/x
2.Ifapipecanemptyafulltankinyhours,thenpart
emptiedinonehour=1/y
3.Ifapipecanfillatankinxhours,andanotherpipecan
emptythefulltankinyhours,thenonopeningboththe
pipes,
thenetpartfilledin1hour=[(1/x)-(1/y)]ify>x
thenetpartemptiedin1hour=[(1/y)-(1/x)]ifx>y
TIME&DISTANCE:
Distance=Speed*Time
1km/hr=5/18m/sec
1m/sec=18/5km/hr
Supposeamancoversacertaindistanceatxkmphand
anequaldistanceatykmph.Then,theaveragespeed
duringthewholejourneyis2xy/(x+y)kmph.

7. PROBLEMSONTRAINS:
1.Timetakenbyatrainxmetreslonginpassingasignal
postorapoleorastandingmanisequaltothetime
takenbythetraintocoverxmetres.
2.Timetakenbyatrainxmetreslonginpassinga
stationaryobjectoflengthymetresisequaltothetime
takenbythetraintocoverx+ymetres.
3.Supposetwotrainsaremovinginthesamedirection
atukmphandvkmphsuchthatu>v,thentheir
relativespeed=u-vkmph.
4.Iftwotrainsoflengthxkmandykmaremovingin
thesamedirectionatukmphandvkmph,whereu>v,
thentimetakenbythefastertraintocrosstheslower
train=(x+y)/(u-v)hours.
5.Supposetwotrainsaremovinginoppositedirections
atukmphandvkmph.Then,their
relativespeed=(u+v)kmph.
5.Iftwotrainsoflengthxkmandykmaremovingin
theoppositedirectionsatukmphandvkmph,thentime
takenbythetrainstocrosseachother=
(x+y)/(u+v)hours.
7.IftwotrainsstartatthesametimefromtwopointsA
andBtowardseachotherandaftercrossingtheytakea
andbhoursinreachingBandArespectively,thenA's
speed:B'sspeed
=√b:√a

8. SIMPLE&COMPOUNDINTERESTS:
Let
Pbetheprincipal,
Rbetheinterestratepercentperannum,and
Nbethetimeperiod.
1.SimpleInterest=(P*N*R)/100
2.CompoundInterest=P(1+R/100)
N
–P
3.Amount=Principal+Interest
LOGARITHMS:
Ifa
m
=x,thenm=logax.
Properties:
1.logxx=1
2.logx1=0
3.loga(xy)=logax+logay
4.loga(x/y)=logax-logay
5.logax=1/logxa
6.loga(x
p
)=p(logax)
7.logax=logbx/logba
Note:Logarithmsforbase1doesnotexist.
AREA&PERIMETER:
Shape Area Perimeter
Circleπ(Radius)2 2π(Radius)
Square(side)
2
4(side)
Rectanglelength*breadth2(length+breadth)

9. 1.Areaofatriangle=1/2*Base*Height
or
2.Areaofatriangle=√(s(s-a(s-b)(s-c))wherea,b,care
thelengthsofthesidesands=(a+b+c)/2
3.Areaofaparallelogram=Base*Height
4.Areaofarhombus=1/2(Productofdiagonals)
5.Areaofatrapezium=1/2(Sumofparallel
sides)(distancebetweentheparallelsides)
6.Areaofaquadrilateral=1/2(diagonal)(Sumofsides)
7.Areaofaregularhexagon=6(√3/4)(side)
2
VOLUME&SURFACEAREA:
Cube:
Letabethelengthofeachedge.Then,
1.Volumeofthecube=a
3
cubicunits
2.SurfaceArea=6a
2
squareunits
3.Diagonal=√3aunits
Cuboid:
Letlbethelength,bbethebreadthandhbethe
heightofacuboid.Then
1.Volume=lbhcubicunits
2.SurfaceArea=2(lb+bh+lh)sq.units
3.Diagonal=√(l
2
+b
2
+h
2
)

10. Cylinder:
Letradiusofthebaseberandheightofthecylinder
beh.Then,
1.Volume=πr
2
hcuunits
2.CurvedSurfaceArea=2πrhsqunits
3.TotalSurfaceArea=2πrh+2πr
2
squnits
Cone:
Letrbetheradiusofbase,hbetheheight,andlbe
theslantheightofthecone.Then,
1.l
2
=√(h
2
+r
2
)
2.Volume=1/3(πr
2
h)cubicunits
3.CurvedSurfaceArea=πrlsqunits
4.TotalSurfaceArea=πrl+πr
2
sq.units
Sphere:
Letrbetheradiusofthesphere.Then,
1.Volume=(4/3)πr
3
cubicunits
2.SurfaceArea=4πr
2
sq.units
Prism:
Volume=(Areaofbase)(Height)