notes-11-03-2016-maths.pdf · the alligation rule introduction: the word “alligation” is used...

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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…



Contents

1. Preface …………………………………………………………………………………..2

2. Contents …………………………………………………………………………………3

3. Introduction ……………………………………………………………………………4

4. The Alligation Rule? ………………………………………………………………..5

5. Practice Time ………………………………………………………………………….5



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.



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



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



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



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



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?



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?



%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?



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.



%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.



%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



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?



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.



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?



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


mixed with another acid having 40% water; in order to

get an acid having 40% H2SO4.



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


mixed with water; in order to get an acid having 15%

H2SO4.

1 : 1 :

%15:%15

%85

%100%70

2

1

WaterAcid

WaterPureAcidWater



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.



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


:

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



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


:

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.



%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.



25

16

10

21

Re1

Re1

2

Re

Re

Initially

Left

Initially

Left

peatedprocess times the Number of

InitiallyInitially

Left


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?



literWine

Wine

Wine

Hence

WineWine

WineWineBut

WineWine

WaterWine

Wine

moveAdd/WineWine

Originally

Originally

Originally

OriginallyLeft

OriginallyFinally

FinallyLeft

Left


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%?



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



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

notes-11-03-2016-maths.pdf · the alligation rule introduction: the word “alligation” is used...

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