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Preface
The Digi-notes (The Alligation Rule) as presented here is prepared
strictly according to IBPS-PO, IBPS-Clerk, IBPS-SO and other PO/Clerk
level competitive exams. The main aim of the Digi-notes is to develop
the notion of the Alligation rule so that candidate would be able to
handle the quantitative problems of the real world easily and
conveniently and hence in their competitive exams. Candidates are
advised to handle the Digi-notes by keeping in mind that they are
going to LEARN (NOT STUDY) the Alligation rule so that you; yourself
would be able to minimize the steps required to conclude the answer
of the question and hence develop the SHORT TRICKS.
All the best…
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Contents
1. Preface …………………………………………………………………………………..2
2. Contents …………………………………………………………………………………3
3. Introduction ……………………………………………………………………………4
4. The Alligation Rule? ………………………………………………………………..5
5. Practice Time ………………………………………………………………………….5
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The Alligation Rule
Introduction:
The word “Alligation” is used in the sense of mixing of two or
more ingredients. For example if we are mixing milk and
water; we could say it in other words that “We are having an
Alligation of milk and water”
The questions in which we observe the following two points;
we will conclude that the given question could be optimally
solved with the help of the Alligation Rule:
1. The Necessary Condition: A SENSE OF MIXING should be
conveyed from the question. The sense of mixing could
be either real mixing or hypothetical mixing.
2. The Sufficient Condition: The Relative Quantity of the
ingredients should be defined directly or indirectly in the
question as Alligation Rule is always applied on Relative
Quantity (per kg; per litre; percentage; ratio; average
etc); never on Absolute Quantity (Rs.; kg; litre; 100 etc).
Note (Very important): After applying the Alligation Rule; the
ratio obtained are always of the quantity which comes after
“per” i.e.; the quantity from which the division is done or
from which the comparison is made.
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The Alligation Rule:
The rule says that “The ratio in which two ingredients; having
relative values a1 and a2; which are mixed to produce a
mixture (real/hypothetical) having relative value aav; is given
by:
Ensure to subtract in the order in which it is shown here.
Don’t change the order of subtraction; otherwise the final
answer of the question would not be correct.
Practice Time
1. In what proportion must rice at Rs.3.10 per kg be mixed
with rice at Rs.3.60 per kg, so that the mixture is worth
Rs.3.25 per kg?
3:7 Re
15:35
325
360310
21
Ratioquired
RiceRice
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2. There are 65 students in a class. 39 rupees are distributed
among them so that each boy gets 80 paisa and each girl
gets 30 paisa. Find the number of boys in that class.
3935
65 262
5
65
3:2:
30:20
60
8030
/6065
3900 /
BoysAndGirls
BG
BoysGirl
studentsPPaise
StudentsPaise
3. A person travels 285 km in 6 hrs. In two stages. In the
first part of the journey, he travels by bus at the speed of
40 km per hr. In the second part of the journey, he travels
by train at the speed of 55 km per hr. How much distance
did he travel by train?
kmTrainbyTravelledceDisHence
hrsTrain
TrainBus
:
TrainBus
hrkmedAverageSpe
Time
TimeTime
165355 tan
.312
6
;1:15.7:5.7:
5.75.7
5.47
5540
/5.476
285
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4. In a Zoo, there are rabbits and pigeons. If heads are
counted, there are 200 and if legs are counted, there are
580. How many pigeons are there?
11011911
200 ;9:11:
9.0:1.1
9.2
42
/9.2200
580 /
PigeonsThereforeRP
LegsLegs
AnimalsLegsLegs
AnimalsLegs
RabbitsPigeons
5. The present population of a town is 5000. If the number
of the males is increases by 10% while the number of the
females increases by 15%; the population of the town
increases to 5600. Find the numbers of the males:
1. Before the increase
2. After the increase
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600 % 100 12%
5000
% %
10% 15%
12%
3% : 2%
: 3 : 2
3000 1. 3 3000
5
110 2. 3000 3300
100
Overall
Before
After
Population
Male Female
G B
Males
Males
6. Weights of two friends Ram and Shyam are in the ratio of
4:5. Ram’s weight increases by 10% and the total weight
of Ram and Shyam together becomes 82.8 kg, with an
increase of 15%. By what percent did the weight of
Shyam increases?
%19 '
%19
5
4
5
15
515
15
10
%%
5 : 4
WeightsShyaminIncreasePercentage
x
x
haveWe
%:%x
%
x%%
WeightWeight
ShyamRam
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7. A man spends 75% of his income. His income increases by
20% and his expenditure also increases by 10%. Find the
percentage increase in his saving?
%in Saving Increase Percentage
%x
x
x
We have
%x:%
%
%x%
eExpenditurSaving
:
:%%:-penditure Saving:Ex
50
50
3020
3
1
20
10
2010
20
10
31
31757510
8. A man buys two horses for Rs.1350 and sells one so as to
lose 6% and the other so as to gain 7.5% and on the
whole he neither gains nor loses. Find the difference of
the cost price of the horses?
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600.49
1350
750.59
1350
4:5%6:%5.7:
65.7
0
5.76
2
1
21
21
RsHorse
RsHorse
HorseHorse
%:%
%
%%
HorseHorse
9. A trader has 50 kg of rice, a part of which he sells at 10%
profit and the rest at 5% loss. He gains 7% on the whole.
What is the quantity sold at 5% loss?
kgkg
lossatSoldQuantity
%%:PartPart
%:%
%
%%
PartPart
1015
50 %5
1:4312:
312
7
510
21
21
10. If goods are purchased for Rs.450 and 1/3rd be sold
at a loss of 10%. What percent of profit should be taken
on the remainder so as to gain 20% on the whole
transaction?
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%35304022
1
30
20
3020
20
10
2 : 1
3
2
3
11 :
3
1
21
xxx
haveWe
%:%x
%
x%%
PartPart
11. In what proportion must water be mixed with milk
to gain 20% by selling it at cost price?
516
5
6
1
6
5
6
16
5
10
/6
5.%100
%120
/1.
/1./0./1.
/1.
::tater:Spiri W
:
CPCP
LiterRsLiterRs
CP
LiterRsLiterRsLiterRsSP
LiterRsCPLet
SpiritWater
WaterSpirit
WaterSpirit
Spirit
12. How many kg of salt at 42 paisa per kg must a man
mix with 25 kg of salt at 24 paisa per kg so that he may
on selling the mixture at 40 paisa per kg, gain 25% on the
outlay?
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kgalt S
:::Saltalt S
:
SaltSalt
kg ? kg
Paise/kg%%
Paise/kgCPMixture
2045
25
54108
108
32
2442
25
32100125
40
1
21
21
13. A man bought two watches for Rs.1620. He sold one
so as to gain 120% while the other so as to gain 30%; and
gains as much the amount for which he bought the two
watches. Find the difference of the cost price of the
watches.
900.59
1620.
2:7 :
%20%70
%100
%30%120
%1001001620
1620%
21
21
RsRs
Differenceots C
WatchofCostWatchofCost
:
WatchWatch
POverall
14. A man bought two watches each for Rs.2000 and
Rs.3000 respectively. He sold first so as to gain 20% while
the second so as to gain 30%. Find his overall profit
percentage.
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%26
3
2
%20
%30
%20:%30
%
%30%20
3000 2000.
3 : 2
21
x
x
x
xx
x
WatchWatch
Rs.Rs
15. A man bought two watches each for Rs.1620. He
sold one so as to gain 20% while the other so as to gain
30%. Find his overall profit percentage.
%25
1
1
%20
%30
%20:%30
%
%30%20
1600 1600.
1 : 1
21
x
x
x
xx
x
WatchWatch
Rs.Rs
16. A man sold two watches each for Rs.1620. He sold
one so as to gain 20% while the other so as to gain 30%.
Find his overall profit percentage.
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%8.24
12
13
%20
%30
%20:%30
%
%30%20
100%130%
1600. : 100%
120%
1600.
12 : 13
21
x
x
x
xx
x
WatchWatch
RsRs
17. A man sold two watches each for Rs.1620. He sold
one so as to gain 20% while the other so as to lose 20%.
Find his overall profit percentage.
lossP
OR
lossx
x
x
xx
x
WatchWatch
RsRs
Overall %4%4100
20202020%
%4%4
3
2
%20
%20
%20:%20
%
%20%20
100%80%
1600. : 100%
120%
1600.
3 : 2
21
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18. A man sold two commodities each for the same
price, and on the whole he neither gains nor loses. If he
sold one so as to gain 25%; find the percentage profit or
loss on the other commodity?
lossx
xx
x
x
WatchWatch
ss
%3
216%
3
216
125
100
%20
%
%25:%
%0
%%25
100%%x100
: 100%125%
125 : x100
21
19. In what ratio must water be mixed with pure milk
costing Rs.16 per liter so as to get a mixture costing
Rs.7.50 per liter?
15 : 17:
50.7:50.8
50.7
160
MilkWater
MilkWater
20. How much water be mixed with 30 liters of pure
milk costing Rs.16 per liter so as to get a mixture costing
Rs.7.50 per liter?
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literWater
MilkWater
MilkWater
Liter ? Liter
341715
30
15 : 17:
50.7:50.8
50.7
160
30
21. How much water be mixed with 30 liters of pure
milk costing Rs.16 per liter so that on selling the mixture
at Rs.9 per liter; a profit of 20% would be made?
literWater
MilkWater
MilkWater
Liter ? Liter
literperRs CostMixture
341715
30
15 : 17:
50.7:50.8
50.7
160
30
50.7/%100%120
9
22. In what ratio must an acid having 20% H2SO4 be
mixed with another acid having 50% H2SO4; in order to
get an acid having 30% H2SO4.
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1 : 2 :
%10:%20
%30
%50%20
22
214242
AcidAcid
AcidAcidSOHSOH
23. How much an acid having 20% H2SO4 be mixed with
5 liters of an acid having 50% H2SO4; in order to get an
acid having 30% H2SO4.
literAcid
AcidAcid
AcidAcid
Liter ? Liter
SOHSOH
1021
5
1 : 2 :
%10:%20
%30
%50%20
5
1
22
214242
24. Two containers A and B; contains the mixture of milk
and water in the ratio of 3:5 and 4:5 respectively. In what
ratio must the mixture be drawn from the containers A
and B respectively and mixed in a third container C so
that the ratio of the milk and water becomes 8:11?
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63 : 32 :
8
3
19
8:
19
8
9
419
89
4
8
3
BA
BA
QuantityQuantity
VesselVesselMilkMilk
25. Milk and water are mixed in a vessel A as 4:1 and in
vessel B as 3:2. For vessel C, if one takes equal quantities
from A and B, find the ratio of milk to water in C.
3:7)710(:7:10:7:
10
7
10
7
1
1
5
45
3
5
4:
5
3
5
3
5
4
1 : 1
WaterMilkWaterMilkMilk
WaterMilk
Milkx
x
x
haveWe
xx
x
VesselVesselMilkMilk BA
26. In what ratio must an acid having 30% H2SO4 be
mixed with another acid having 40% water; in order to
get an acid having 40% H2SO4.
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1 : 2 :
%10:%20
%40
%60%30
22
214242
AcidAcid
AcidAcidSOHSOH
27. How much an acid having 30% H2SO4 be mixed with
8 liters acid having 40% water ; in order to get an acid
having 50% H2SO4.
literAcid
: :AcidAcid
%:%
%
%%
AcidAcid
liters liter ?
SOHSOH
2431
8
12
1020
40
6030
8
1
21
214242
28. In what ratio must an acid having 30% H2SO4 be
mixed with water; in order to get an acid having 15%
H2SO4.
1 : 1 :
%15:%15
%85
%100%70
2
1
WaterAcid
WaterPureAcidWater
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29. In certain mixture of milk and water; 40% is water.
When 21 liters of mixture is removed and then 21 liters
of water is added; the water becomes 75% in the
mixture. Find :
1. The Final quantity of the mixture.
2. The Initial Quantity of the Mixture
literInitial
literFinal
WaterMixtureduced
WaterPureMixtureduced
Added
Added
Water
36 .2
36127
21 .1
7 : 5 : Re
%35:%25
%75
%100%40
Re
Liter 21
30. In certain mixture of milk ad water; 40% is water.
When 15 liters of mixture is removed and then 21 liters
of water is added; the water becomes 75% in the
mixture. Find :
1. The Final quantity of the mixture.
2. The Initial Quantity of the
Mixture.
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liter. Initial
liter. Final
: ure:Waterduced Mixt
%:%
%
%%
Pure Waterureduced Mixt
Liter
Added
Added
Water
30)2115(362
36127
211
75Re
3525
75
10040
Re
21
31. In a certain quantity of mixture of two liquids A and
B; the ratio of A to B is 7:5. When 9 liters of mixture is
removed and then 9 liters of liquid B is added; the ratio
of A to B becomes 7:9. Find the quantity of A
1. In the Final mixture
2. In the Initial mixture
literABA liter. Initial
literABA liter. Final
: ure:Waterduced Mixt
:
Pure Bureduced Mixt
Liter
Added
Added
B
21712
365:7:362
75.15716
369:7:364
1
91
13Re
48
7
16
716
9
112
5
Re
9
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32. In a certain quantity of mixture of two liquids A and
B; the ratio of A to B is 7:5. When 15 liters of mixture is
removed and then 9 liters of liquid B is added; the ratio
of A to B becomes 7:9. Find the quantity of A
3. In the Final mixture
4. In the Initial mixture
literABA liter. Initial
literABA liter. Final
: ure:Waterduced Mixt
:
Pure Bureduced Mixt
Liter
Added
Added
B
5.24712
425:7:42)915(362
75.15716
369:7:364
1
91
13Re
48
7
16
716
9
112
5
Re
9
33. In a certain quantity of syrup; 5 parts is honey while
3 parts is water. How much PART OF SYRUP must be
replaced with water in order to get a syrup having half
honey and half water.
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%205
1
14
1 ReplacedPart
1 : 4 Re
8
1:
2
12
1
18
3
Re
Part
p:Waterduced Syru
Pure Waterpduced Syru
th
Added
Added
Water
34. A vessel of 80 liters is filled with milk and water. 70%
of milk and 30% of water is taken out of the vessel. It is
found that the vessel is vacated by 55%. Find the initial
quantity of milk and water.
lietrsWater
litersMilk
MilkWater
%:%
%
%%
movedMilk movedWater
305080
50553
80
5:3%25:%15:
2515
55
7030
ReRe
35. A vessel full of pure acid contains 10 liters of it. Out
of which 2 liters are withdrawn; the vessel is then filled
with water. Next 2 liters of mixture are withdrawn and
again vessel is filled up with water. Find the ratio of the
acid left in the vessel with that of the original quantity.
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25
16
10
21
Re1
Re1
2
Re
Re
Initially
Left
Initially
Left
peatedprocess times the Number of
InitiallyInitially
Left
peatedprocess times the Number of
Initially
InitiallyLeft
Pure Acid
Pure Acid
Pure Acid
Pure Acid
Pure Acid
moveAdd/
Pure Acid
Pure Acid
Pure Acid
moveAdd/Pure AcidPure Acid
36. 8 liters are drawn from a cask full of wine and is
then filled with water. This operation is performed three
more times. The ratio of the quantity of the wine now
left in the cask to that of the water is 16:65. How much
wine did the cask hold originally?
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literWine
Wine
Wine
Hence
WineWine
WineWineBut
WineWine
WaterWine
Wine
moveAdd/WineWine
Originally
Originally
Originally
OriginallyLeft
OriginallyFinally
FinallyLeft
Left
peatedprocess times the Number of
Originally
OriginallyLeft
24
8132
818116
81:16:
;
6516:16:
65:16:
Re1
4
Re
37. Rs.1500 is invested in two such parts that if one part
be invested at 6% and the other at 5%, the total interest
in one year from both investments is Rs.85. How much is
invested at 5%?
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500.13
1500.
1 : 2:
3
1:
3
23
17
56
%3
17
100
11500.85.
100
2
21
21
RsRs
Part
PartPart
%
%%
PartPart
rrRs
Rs
trP SI
38. The average of marks obtained by 120 candidates in
a certain examination is 35. If the average mark of passed
candidates is 39 and that of the failed candidates are 15.
What is the number of candidates who passed in the
examination?
10056
120
1 : 5 :
4:20
35
1539
CandidatesPassed
MarksAverageMarksAverage
MarksAverageMarksAverage
FailedPassed
FailedPassed
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39. In three classes X, Y and Z; an algebra test is
conducted. The average score of class X is 83. The
average score of class Y is 76 and the average score of
class Z is 85. The average score of class X & Y is 79 and the
average score of class Y & Z is 81. What is the average
score of the classes X, Y and Z.?
5.81543
585476383 Re
5:4:3:: 5:4 :
4:3 :
;
5 : 4 :
5:4
81
8576
;
4 : 3 :
4:3
79
7683
Averagequired
ZYXZY
YX
Now
ZY
ScoreAverageScoreAverage
Again
YX
ScoreAverageScoreAverage
StudentsStudentsStudents
StudentsStudents
StudentsStudents
StudentsStudents
ZY
StudentsStudents
YX
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