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The Digi-notes (Ratio and Proportion) 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 Ratio and Proportion 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 Ratio/Proportion 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. What is Ratio?..…………………………………………………………………………..4

5. The Fundamental Property of Ratio….…………………………………………5

6. AnImportantResult……………………………………………………………………..5

7. Questions.…………………………………………………………………………………..5

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Introduction:

The Ratio could be perceived as a special case of percentage. All the

concept related to approaching the of the Ratio questions could be

easily correlated with the approach what we had already developed in

the percentage Digi-notes (Specially the application of the Unitary

Method). Like percentage; ratio is extensively used in the Data Analysis.

So you are advised to go through the chapter thoroughly keeping in

mind to learn how we are approaching the question.

What is Ratio?:

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Mathematically: Let a and b be the two quantities.

Then ratio of a and b is defined as the fractionb

a .

This fraction is simply denoted as a : b when we are talking about the ratio.

And is is read as “a is to b”.

Practically: Since with the help of the fractions we are making

comparison between the quantities, therefore ratio is the comparison between two quantities.

The fundamental Property of Ratio: As the multiplication or division by the same number to the

numerator and denominator doesn’t affect a faction; in the same way the multiplication or division of each term of ratio by the same non-zero number doesn’t affect the ratio.

An Important Result:

It is to be noted that the individual ratios to be combined:

Should have one quantity in common.

And the common quantity of the Ist ratio should be at the second place and that of the IInd ratio at the first place.

Questions:

st

nd

I Ratio A : B a : b

II Ratio B : C c : d

Then combined Ratio A : B : C ac : bc : bd

Page 6 of 27

Que: If A, B, C are three quantities such that A:B = 5:4 and B:C = 3:2,

find:

1) B : C 2) A : C 3) (A + B) : (A - C)

Sol: We have 𝐴 ∶ 𝐵 ∶ 𝐶 = 5 × 3 ∶ 3 × 4 ∶ 4 × 2 = 15 ∶ 12 ∶ 8

Hence

1) B ∶ C = 12 ∶ 8 = 3 ∶ 2

2) A ∶ C = 15 ∶ 8

3) (A + B): (A − C) = (15 + 12): (15 − 8) = 27 ∶ 7

Que: The sum of the money with A and B is Rs.500. If the ratio of the

money they had is 3:2, find the share of A.

Sol:

𝐴 =𝑅𝑠.500

(3+2)× 3 = 𝑅𝑠. 300

Page 7 of 27

Que: The ratio of the money with A and B is 7:3. If the difference of the

money they had is Rs.160, find the share of B.

Sol:

𝐵 =𝑅𝑠. 160

(7 − 3)× 3 = 𝑅𝑠. 120

Que: A sum of money is divided between two persons in the ratio of

3:5. If the share of one person is Rs.20 less than that of the other, find

the sum.

Sol:

𝐴 =𝑅𝑠. 20

(5 − 3)× (5 + 3) = 𝑅𝑠. 80

Que: The ratio of the shares of three persons A, B and C is 7:5:11. If the

difference of the shares of A and C is Rs.16000; find the total amount of

money with B and C.

Sol:

𝐴 =𝑅𝑠. 16000

(11 − 7)× (7 + 11) = 𝑅𝑠. 72000

Que: The ratio of the shares of three persons A, B and C is 12:9:11. The

total amount of the money with A and C is Rs.5600 more than the

money with B; find the difference of the amount of the money with B

and C.

Sol:

𝐴 =𝑅𝑠.5600

(12+11−9)× (11 − 9) = 𝑅𝑠. 800

Page 8 of 27

Que: The ratio of the incomes of four persons A, B, C and D is 5:3:9:4.

The sum of the incomes of A and C is 84000; find the difference of the

incomes of B and D?

Sol:

𝐴 =𝑅𝑠. 84000

(5 + 9)× (4 − 3) = 𝑅𝑠. 6000

Que: The ratio of the angles of a triangle is 10:11:15. Find the difference

of the second largest and the smallest angle of the triangle.

Sol:

Required Differenec =1800

(10 + 11 + 15)× (11 − 10) = 50

Que: The ratio of any two angles of a triangle is 5:9. If the third angle is

measured to be 110 degree, then find the difference of the other two

angles.

Sol:

Required Differenec =1800−1100

(9+5)× (9 − 5) = 200

Que: The ratio of the four angles of a quadrilateral is 9:10:14:3. Find the

sum of the second smallest and second largest angle of the

quadrilateral.

Sol:

Required Differenec =3600

(9+10+14+3)× (10 − 9) = 100

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Ques: P, Q and R are three numbers. The ratio of P and Q is 2:3 and that

of Q and R is 2:1. If the sum of P and R is 980; find Q.

Sol:

We have, 𝑃: 𝑄: 𝑅 = 4 ∶ 6 ∶ 3

Hence; 𝑄 =980

(4+3)× 6 = 840

Que: The salary of Amit is 3/4th of the salary of Raju; while the salary of

Raju is 25% less than the salary of Sohan. The difference of the salaries

of Amit and Sohan is Rs.9800; find 142

7% of the sum of the salaries of

Raju and Sohan.

Sol:

A : R = 3 : 4

R : S = 75% : 100% = 3 : 4

Hence A : R : S = 9 : 12 : 16

Required Sum = {9800

16−9× (12 + 16)} ×

1

7= 𝑅𝑠. 5600

Que: Rs.5600 is divided among three persons A, B and C in such a way

that; A received 2/5th of the total share of B and C together. Find:

1) Share of A

2) Share of B

3) Share of C

4) Total share of A and B

5) Total share of B and C

6) Total share of C and A

Page 10 of 27

Sol:

: 2 :5

:

56001 2 .1600

7

2

3

4

56005 5 .4000

7

6

A B C

Hence

A Rs

CND

CND

CND

B C Rs

CND

Que: Rs.15600 is divided among three persons A, B and C in such a way

that; A received 2/3rd of the total share of B and C together while B

received 3/7th of the total share of A and C together. Find the share of

C.

Sol:

: 2:3

: 3:7

:

156002 .6240

515600

3 .468010

.15600 .6240 .4680 .4680

A B C

and B A C

Hence

A Rs

and B Rs

Hence

C Rs Rs Rs Rs

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Que: Divide Rs.1540 among A, B, C such that A shall receive 2/9 as

much as B and C together and B shall receive 3/11 as much as A and C

together.

Sol:

: 2 : 9

: 3 :11

:

15402 .280

11

15600 3 .330

14

.1540 .280 .330 .930

A B C

and B A C

Hence

A Rs

and B Rs

Hence

C Rs Rs Rs Rs

Que: Rs.2575 is divided among three persons A, B and C in such a way

that if each of their shares be reduced by Rs.25; the ratio of their

remainders would be 2 : 3 : 5. Find the sum of the shares of B and C.

Sol:

2575 25 25 25

8 25 25 .205010

B C Rs

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Que: The ratio between two numbers 3:4. If each number be increased

by 6, the ratio becomes 4:5. Find the two numbers.

Sol:

1 2

1

2

:

3 : 4

4 : 5

1 6

3 3 6 18

4 4 6 24

N N

N

N

Que: The ratio between two numbers 5:1. If each number be decreased

by 6, the ratio becomes 7:1. Find the difference between the numbers.

Sol:

1 2

1 2

5 1 30 6

7 1 28 4

2 6

1 3

24 24 3 72

N : N

: :

: :

N ~ N

Que: The ratio between two numbers 3:4. If each number be increased

by 2, the ratio becomes 7:9. Find the sum of the two numbers.

Page 13 of 27

Sol:

1 2

1 2

3 4 6 8

7 9 7 9

1 2

14 14 2 28

N : N

: :

: :

N N

Que: The incomes of A and B are in the ratio 3:2 and their expenditures

are in the ratio 5:3. If each saves Rs.200, what is the difference of their

incomes?

Sol:

3 2 6 4

5 3 5 3

1 200

2 200 2 400

A : B

I : :

E : :

Rs.

A ~ B Rs.

Que: The incomes of Ram and Shyam are in the ratio 8:11 and their

expenditures are in the ratio 7:10. If each of them saves Rs.500, then

find the sum of their Income.

Sol:

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8 11

7 10

1 500

19 19 500 .9500

R : S

I :

E :

Rs.

R S Rs

Que: The students in the three classes are in the ratio 2:3:5. If 20

students are increased in each class the ratio changes to 4:5:7. What

was the total number of students in the three classes before the

increase?

Sol:

:

2 : 3 : 5

4 : 5 : 7

2 20

2010 10

2 100

st nd rd

st nd rd

I : II III

I II III

Que: An amount of money is distributed among A, B and C in the ratio

1:2:9. If C gives Rs.500 from his share to B, the new ratio becomes

1:3:8. Find the total amount of money.

Sol:

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1 2 9

1 3 8

1 500

12 500 12

6000

A : B : C

: :

: :

Rs.

A B C

Rs.

Que: An amount of money is distributed among A, B and C in the ratio

6:19:7. If C gives Rs.200 from his share to B, the new ratio becomes

3:10:3. Find the total amount of money.

Sol:

:

6 : 19 : 7 6:19:7

3 : 10 : 3 6:20:6

1 .200

32 200 32

.6400

A : B C

Rs

A B C

Rs

Que: In 40 litres mixture of milk and water the ratio of milk and water is

3:1. Find the quantity of water to be added to make this ratio 2:1.

Sol:

3 1 6:2

2 1 6:3

40 1 5

8

M : W

:

:

litersliters

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Que: In 60 litres mixture of milk and water the ratio of milk to water is

7:3. Find the quantity of water to be added to make the ratio of milk to

water 3:7.

Sol:

7 3 21:9

3 7 21:49

60 40 80

30

M : W

:

:

lietrsliters

Que: A mixture contains alcohol and water in the ratio 8:3. On adding 3

litres of water, the ratio of alcohol to water becomes 2:1. Find the

difference of quantities of alcohol and water in the original mixture.

Sol:

8 3 8:3

2 1 8:4

3 5 15

1

A : W

:

:

litersliters

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Que: A mixture contains wine and water in the ratio of 3:2.If 4 litres of

water is added to the mixture, wine and water in the mixture become

equal. Find the sum of the quantities of wine and water in the original

mixture.

Sol:

3 2 3:2

1 1 3:3

4 5 20

1

Wn : W

:

:

litersliters

Que: The ratio of number of boys and girls in a school is 2:3. If 200 boys

are additionally admitted in the school the ratio changes to 5:6. Find

the difference of the number of boys and girls in the school originally.

Sol:

2 3 4:6

5 6 5:6

200 2 400

1

B : G

:

:

Boys

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Que: The ratio of A’s and B’s income last year was 3:4. The ratio of their

own incomes of last year and this year is 4:5 and 2:3 respectively. If the

total sum of their present incomes is Rs.4160, then find the present

income of A.

Sol:

1 year

[------------------------]

Past Present

3 4

3 34 : 5

4 44 4

2 : 32

A: B :

A

B

2

3 45 3 5 8

4 24160

5 160013

P P

P

A : B : :

A Rs.

Que: One year ago, the ratio between Laxman’s and Gopal’s salaries

was 3:5. The ratio of their individual salaries of last year and present

year are 2:3 and 4:5 respectively. If their total salaries for the present

year are Rs.4300, find the difference of the salaries of Laxman and

Gopal.

Page 19 of 27

Sol:

1 year

[--------------------------]

Pr

3 5

3 32 : 3

2 25

4 : 4

Past esent

L :G :

L

G

5

54

3 53 5 18 25

2 44300

7 70043

P P

P

A : B : :

A Rs.

Que: A vessel contains liquids A and B in the ratio 5:3. If 16 litres of

mixture are removed and the same quantity of liquid B is added, the

ratio becomes 3:5. What quantity does the vessel hold?

Sol:

3 5 165

8 8 8

210

8

8

1 102

40

liters

liters

Qantity of

Mixture in

Vessel

liters

Question: A bucket contains a mixture of two liquids A and B in the

ratio 7:5. If 9 litres of the mixture is replaced by 9 litres of liquid B, then

the ratio of the two liquids become 7:9. How much of the liquid A was

there in the mixture?

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Sol:

7 7 97

16 12 12

7 21

48 4

21 48

14 7

36

367 21

12

liters

liters

Qantity of

Mixture in

Vessel

liters

A liters

Proportion:

Let a , b, c and d are the four quantities.

If we calculated the ratios a : b and c : d in that order and find that

dcba ::

Then we say that a, b, c and d are in proportion in that order.

Hence equality of ratios is called proportion. Here

a is called the first proportional.

b is called the second proportional.

c is called the third proportional.

d is called the fourth proportional.

Since we have

Means

of

Extremes

of

bcad

d

c

b

a

c:da:b

ProductProduct

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Continued Proportion:

Let a , b and c are the three quantities.

If we calculated a : b and b : c in that order and find that

cbba ::

Then we say that a, b and c are in continued proportion in that

order. That’s why it is called continued proportion because b is in

continuation in the equality of the ratios. Here

a is called the first proportional.

b is called the mean/second proportional.

c is called the third proportional.

Since we have

Mean

Square of

Extremes

of

bad

c

b

b

a

b:ca:b

Product

2

Que: Given three quantities in the same unit as 5, 10 and 25. Find the

following in proportion to these quantities:

1) First Proportional

210525

25 ,10 ,5 ,

xx

proportioninarex

2) Second Proportional

5.1225510

25 ,10 , ,5

xx

proportioninarex

Page 22 of 27

3) Third Proportional

51225510

25105

.xx

oportion are in pr, x, ,

4) Fourth Proportional

5025105

,25 ,10 ,5

xx

proportioninarex

Question: Given two quantities in the same unit as 5 and 25. Find the

following in proportion tom these quantities:

5) First Proportional

1525 2 xx

6) Mean Proportional

55255 2 xx

7) Third Proportional

125255 2 xx

Question: Divide Rs.1350 into three shares proportional to the numbers

2, 3, and 4.

Page 23 of 27

Sol:

600.41503

450.31502

300.21501

1509

13501

1350.9

Rs

Rs

Rs

Rs

rd

nd

st

Que: Divide Rs.391 into three parts proportional to the fraction ½, 2/3

and ¾.

Sol:

153.9173

136.8172

102.617 1

1723

3911

391.23

9:8:64

3:

3

2:

2

1

Rs

Rs

Rs

Rs

rd

nd

st

Question: Divide 581 into three parts such that 4 times of the first may

be equal to 5 times the second and 7 times the third.

Page 24 of 27

Sol:

1407203

1967282

2457351

783

5811

58183

20:35:283:1:2

4:73:1

5:41:2

rd

nd

st

rdstnd

rdst

stnd

Que: Divide Rs. 2430 among three persons A, B, C such that if their

shares be diminished by Rs.5, Rs.10 and Rs.15 respectively, the

remainders shall be in the ratio 3:4:5.

Sol:

1015.152005

810.102004

605.52003

200.12

24001

2400.12

2400.)15105(2430Re

RsC

RsB

RsA

Rs

Rs

Rsmainder

Que: Rs.425 is divided among 4 men, 5 women and 6 boys such that the

share of a man, a woman and a boy may be in the ratio of 9:8:4. What

is the share of a woman?

Page 25 of 27

Sol:

34.5

1701

170.1025

425105

25

4251

425.25

6:10:9

64:58:496:5:4

RsW

RsW

Rs

BWM

Que: Divide Rs.1320 among 7 men, 11 women and 5 boys such that

each womanmay have 3 times as much as a boy, and a man as much as

a woman and a boy together. Find how much each person receive?

Sol:

: : 1: 3 : (1 3) 1: 3 : 4

5 :11 : 7 1 5 : 3 11: 4 7

5 : 33 : 28

66 .1320

13201 20

66

7 28 20 28 .560

11 33 20 33 .660

5 5 20 5 .100

B W M

B W M

Rs

M Rs

W Rs

B Rs

Que: A certain sum of money is divided A, B and C such that for each

rupee A has, B has 65 paisa and C has 40 paisa. If C’s share is Rs.8, find

the sum of money.

Page 26 of 27

Sol:

41.418

841

8

81

8.8

8:13:2040:65:100::

RsMoneyofSum

Rs

CBA

Ques: 465 coins consist of rupee, 50 paisa and 25 paisa coins. Their

values are in the ratio 5:3:1. Find the number of each coin.

Sol:

5 : 3:1

100 100 100. 5 : 3 :1

100 50 25

5 : 6 : 4

15 465

4651 31

15

. 1 5 31 5 155

50 6 31 6 186

25 4 31 4

Values

No of Coins

Rs Coins

Paise Coins

Paise Coins

124

Que: A sum of Rs.11.70 consists of rupees, 50 paisa and 5 paisa coins in

the ratio 3:5:7. Find the number of each kind of coins.

Page 27 of 27

Sol:

. 5 : 3 :1

100 50 53 : 5 : 7

100 100 100

60 : 50 : 7

117 1170

11701 10

117

60 10 60 600

. 1

600 . 1 6

100

50

No of Coins

Values

Paise

Values ofPaise

Rs Coins

Rs Coins

Values of

Pa

50 10 50 500

500 50 10

50

7 10 7 70

5

70 5 14

5

Paiseise Coins

Paise Coins

Values ofPaise

Paise Coins

Paise Coins