Ratio and Proportion
1. Ram has Rs.200 and his weight is 75 kg. Find the ratio of the money he had to his weight.
Cannot be defined, as the rati o is defin ed between
two same q uantities in the sam e unit.
2. Ram has Rs.3 and Shyam has 60 paisa. Find the ratio of the money with Ram and Shyam.
R : S = 300 : 60 = 5 : 1
3. The ages of A and B are respectively 21 years and 63 years respectively. Find the ratio of the
ages of A and B.
A : B = 21 : 63 = 1 : 3
4. If one third of A is equal to the three fourth of B, find the ratio of A to B.
A: B =
3 1
: = 9:4
4 3
5. If .25 of A is equal to the .75 of B, then find the ratio of A to B.
A : B = 0.75 : 0.25 = 3 : 1
6. One man adds 3 liters of water to 12 liters of milk and another man 4 liter of water to 10 liters of
milk. What is the ratio of the strengths of the milk in the two mixtures?
Ratio =
12 10
: = 28 : 25
15 14
7. The employer decreases the number of his employees in the ratio 10:9 and increases their
wages in the ratio 11:12. What is the ratio of his two expenditures?
10 × 11 : 9 × 12 = 55 : 54
8. The ratio of the speeds of two cars is 3:2, find the ratio of the time taken by these cars to cover
the equal distances.
1 1
: = 2:3
3 2
9. The ratio of the time taken by three cars to cover a certain distance is 6:2:3. Find the ratio of the
speeds of these cars to cover the same distance.
1 1 1
: : = 1: 3 : 2
6 2 3
10. The same type of work is assigned to three groups of men. The ratio of the persons in the group
is 3:4:5. Find the ratio of the days in which they will complete the work.
1 1 1
: : = 20 : 15 : 12
3 4 5
11. The ratio of the corresponding sides any two rectangles are 4:7. Find the ratio of their areas.
4 2 : 7 2 = 16 : 49
12. The ratio of the diagonals of the two squares is 2:1. Find the ratio of their areas.
2 2 : 12 = 4 : 1
13. The sides of a hexagon are enlarges by 3 times. Find the ratio of the areas of the new hexagon to
old hexagon.
12 : 3 2 = 1 : 3
Compiled By: Anjan
Page 1
Ratio and Proportion
14. If each side of a parallelepiped is doubled, find the ratio of volumes of old to new parallelepiped.
2 3 : 13 = 8 : 1
15. If A, B, C are three quantities in the same unit such that A:B = 5:4 and B:C = 3:2, find:
1) B:C
2) A:C
3) A:B:C.
4) (A+B):(A-C)
A : B : C = 15 : 12 : 8
1) B : C = 12 : 8 = 3 : 2
2) A : C = 15 : 8
3) A : B : C = 15 : 12 : 8
4) ( A + B) : ( A − C ) = (15 + 12) : (15 − 8) = 27 : 7
16. If A, B, C, D are four quantities in the same unit such that A:B = 3:4, B:C = 2:7 and C:D = 1:3, find:
1) A:D
2) B:C
3) B:D
4) B:C:D
5) A:B:C
6) A:B:C:D
7) (B+C):(A+D)
A : B : C : D = 6 : 8 : 28 : 84 = 3 : 4 : 14 : 42
1) A : D = 3 : 42 = 1 : 14
2) B : C = 4 : 14 = 2 : 7
3) B : D = 4 : 42 = 2 : 21
4) B : C : D = 4 : 14 : 42 = 2 : 7 : 21
5) A : B : C = 3 : 4 : 14
6) A : B : C : D = 3 : 4 : 14 : 42
7) ( B + C ) : ( A + D) = (4 + 14) : (3 + 42) = 18 : 45 = 3 : 5
17. 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.
Rs.500
× 3 = Rs.300
5
18. 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.
Rs.160
× 3 = Rs.120
4
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Page 2
Ratio and Proportion
19. The ratio of the money of three persons A, B and C is 2:6:9. If the sum of the shares of A and C is
Rs.121, find the difference of their shares.
Rs.121
× 7 = Rs.77
11
20. The ratio of the money with A and B is 3:4 whereas that of B and C is 2:1. If the difference of the
money with A and C is Rs.200, find the difference of the share of B and C.
A:B:C=6:8:4=3:4:2
Rs.200
× 2 = Rs.400
1
21. An amount is distributed among A, B and C in the ratio 3:5:7. If total share of A and C is Rs.800
more than the share of B, what is the share of C?
Rs.800
× 7 = Rs.1120
5
22. The sum of three numbers P, Q and R is 98. If the ratio between P and Q is 2:3 and that between
Q and R is 5:8, then find Q.
P : Q : R = 10 : 15 : 24
98
× 15 = 30
49
23. The ratio of the money with Rita and Sita is 7:15 and that with Kavita and Sita is 16:7. If Rita has
Rs.490, find the difference between of the money of Sita and Kavita.
R : S : K = 49 : 105 : 240
Rs.490
× 135 = Rs.1350
49
24. 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.
Rs.20
× 8 = Rs.80
2
25. The prices of a scooter and a moped are in the ratio of 9:5. If a scooter cost Rs.4200 more than a
moped, find the price of a moped.
Rs.4200
× 5 = Rs.5250
4
26. A, B and C are partners in a business. In certain year A got 1/3rd of the profit, and B 1/4th of the
profit and C got Rs.5000. What amount does A got as a profit.
1 1 5
C = 1−  +  =
 3 4  12
1 1 5
Hence A : B : C = : :
= 4:3:5
3 4 12
Rs.5000
A=
× 4 = Rs.4000
5
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Ratio and Proportion
27. 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.
180°
× 1 = 5°
36
28. 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.
180° − 110°
× 4 = 20°
14
29. 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.
360°
× 19 = 190°
36
30. The ratio between two numbers 3:4. If each number be increased by 6, the ratio becomes 4:5.
Find the two numbers.
N1 : N 2
3 : 4
4 : 5
1≡ 6
N 1 => 3 × 6 = 18
N 2 => 4 × 6 = 24
31. 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.
N1 : N 2
5 : 1 => 30 : 6
7 : 1 => 28 : 4
2≡6
1≡ 3
24 ≡ 24 × 3 = 72
32. 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.
N1 : N 2
3 : 4 => 6 : 8
7 : 9 => 7 : 9
1≡ 2
14 ≡ 14 × 2 = 28
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Ratio and Proportion
33. 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?
I
E
A : B
3 : 2 => 6:4
5 : 3 => 5:3
1 ≡ Rs.200
2 ≡ 200 × 2 = Rs.400
34. 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.
I
E
R : S
8 : 11
7 : 10
1 ≡ Rs.500
19 ≡ 19 × 500 = Rs.9500
35. 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?
I st : II nd : III rd
2 : 3 : 5
4 : 5 : 7
2 ≡ 20
10 ≡
20
× 10 = 100
2
36. 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.
A
: B : C
1
: 2 : 9
1
: 3 : 8
1 ≡ Rs.500
12 ≡ 500 × 12 = Rs.6000
37. 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.
A
: B : C
6
: 19 : 7 ⇒ 6 : 19 : 7
3
: 10 : 3 ⇒ 6 : 20 : 6
1 ≡ Rs.200
32 ≡ 200 × 32 = Rs.6400
Compiled By: Anjan
Page 5
Ratio and Proportion
38. In 40 liters 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.
M : W
3 : 1 => 6 : 2
2 : 1 => 6 : 3
40 liters
× 1 = 5 liters
8
39. In 60 liters 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.
M : W
7 : 3 => 21 : 9
3 : 7 => 21 : 49
60 lietrs
× 40 = 80 liters
30
40. A mixture contains alcohol and water in the ratio 8:3. On adding 3 liters of water, the ratio of
alcohol to water becomes 2:1. Find the difference of quantities of alcohol and water in the
original mixture.
A : W
8
: 3
=> 8 : 3
2
: 1
=> 8 : 4
3 liters
× 5 = 15 liters
1
41. A mixture contains wine and water in the ratio of 3:2. If 4 liters 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.
Wn : W
3 : 2
=> 3 : 2
1 : 1 => 3 : 3
4 liters
× 5 = 20 liters
1
42. 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.
B : G
2
: 3
=> 4 : 6
5
: 6
=> 5 : 6
200
× 2 = 400 Boys
1
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Ratio and Proportion
43. 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.
1 year
[ −−−−−−−−−−−−−−− ]
Pas t
A:B = 3:4
Pr esent
3
4
4
2×
2
A => 4 ×
:
B =>
:
A P :B P = 5 ×
AP =
3
4
4
3×
2
5×
3
4
:3 × = 5:8
4
2
4160
× 5 = Rs.1600
13
44. 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.
1 year
[ −−−−−−−−−−−−−−− ]
Pas t
Pr esent
L:G = 3:5
3
3
L => 2 ×
:
3×
2
2
5
5
G => 4 ×
:
5×
4
4
A P :B P = 3 ×
AP =
Compiled By: Anjan
3
5
:5 × = 18 :25
2
4
4300
× 7 = Rs. 700
43
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Ratio and Proportion
45. A vessel contains liquids A and B in the ratio 5:3. If 16 liters of mixture are removed and the
same quantity of liquid B is added, the ratio becomes 3:5. What quantity does the vessel hold?
1 st Method :

 Change in
  Change in


≡
 strenght of A   quantity of A 
3 5  16

⇒ - ≡ −
× 5  liters
8 8  8

2
⇒ ≡ 10 liters
8
 Qantity of

8
⇒
 → 1 ≡ 10 × = 40 liters
2
 Mixture in Vessel 
2 nd Method :
 Change in
  Change in


≡

 strenght of B   quantity of B 
3 5 
16

⇒ − ≡  16 −
× 3  liters
8 8 
8

2
⇒ ≡ 10 liters
8
 Qantity of

8
⇒
 → 1 ≡ 10 × = 40 liters
2
 Mixture in Vessel 
46. A bucket contains a mixture of two liquids A and B in the ratio 7:5. If 9 liters of the mixture is
replaced by 9 liters 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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Page 8
Ratio and Proportion
1 st Method :
Change in strenght of A ≡ Change in quantity of A
7
7
 9

−
≡ −
× 7  liters
16 12  12

7
21
⇒
≡
liters
48
4
⇒
⇒ Qantity of Mixture in Vessel → 1 ≡
∴A=
21 48
×
= 36 liters
4
7
36
× 7 = 21 liters
12
2 nd Method :
Change in strenght of B ≡ Change in quantity of B
9
5
9


−
≡ 9 −
× 5  liters
16 12 
12

7
21
liters
⇒
≡
48
4
⇒
⇒ Qantity of Mixture in Vessel → 1 ≡
21 48
×
= 36 liters
4
7
36
× 7 = 21 liters
12
47. Given three quantities in the same unit as 5, 10 and 25. Find the following in proportion to these
quantities:
1) First Proportional
∴A=
x, 5, 10, 25 are in proportion
⇒ 25 x = 5 × 10
⇒x=2
2) Second Proportional
5, x, 10, 25 are in proportion
⇒ 10 x = 5 × 25
⇒ x = 12.5
3) Third Proportional
5, 10 , x, 25 are in proportion
⇒ 10 x = 5 × 25
⇒ x = 12.5
4) Fourth Proportional
5, 10, 25, x are in proportion
⇒ 5 x = 10 × 25
⇒ x = 50
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Page 9
Ratio and Proportion
48. Given two quantities in the same unit as 5 and 25. Find the following in proportion tom these
quantities:
1) First Proportional
25 x = 5 2
⇒ x =1
2) Mean Proportional
5 × 25 = x 2
⇒ x=5 5
3) Third Proportional
5 x = 25 2
⇒ x = 125
49. Divide Rs.1350 into three shares proportional to the numbers 2, 3, and 4.
9 ≡ Rs.1350
1350
⇒1≡
= 150
9
1st = 150 × 2 = Rs.300
2 nd = 150 × 3 = Rs.450
3 rd = 150 × 4 = Rs.600
50. Divide Rs.391 into three parts proportional to the fraction ½, 2/3 and ¾.
1 2 3
: : = 6:8:9
2 3 4
23 ≡ Rs.391
391
⇒1≡
= 17
23
1st = 17 × 6 = Rs.102
2 nd = 17 × 8 = Rs.136
3 rd = 17 × 9 = Rs.153
51. 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.
A : (B + C) = 2 : 9
B : ( A + C ) = 3 : 11
14 ≡ Rs . 1540
11 ≡ Rs . 1540
⇒ 1 ≡
⇒
⇒ 1 ≡
1540
11
A →
1540
2 ≡
11
Compiled By: Anjan
1540
14
⇒ B → 3 ≡
× 2 = Rs . 280
1540
14
× 3 = Rs . 330
And , C = 1540 − ( 280 + 330 ) = Rs .930
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Ratio and Proportion
52. 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.
2 nd : 1st = 4 : 5
1st : 3 rd = 7 : 4
2 nd : 1st : 3 rd = 28 : 35 : 20
83 ≡ 581
⇒1≡
581
=7
83
1st = 35 × 7 = 245
2 nd = 28 × 7 = 196
3 rd = 20 × 7 = 140
53. 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.
Re mainder = 2430 − (5 + 10 + 15) = Rs.2400
12 ≡ Rs.2400
⇒1≡
2400
= Rs.200
12
A = 3 × 200 + 5 = Rs.605
B = 4 × 200 + 10 = Rs.810
C = 5 × 200 + 15 = Rs.1015
54. 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?
4M : 5W : 6 B = 9 × 4 : 8 × 5 : 4 × 6 = 9 : 10 : 6
25 ≡ Rs.425
⇒1≡
425
25
⇒ 5W → 10 ≡
⇒ 1W =
Compiled By: Anjan
425
× 10 = Rs.170
25
170
= Rs.34
5
Page 11
Ratio and Proportion
55. Divide Rs.1320 among 7 men, 11 women and 5 boys such that each woman may 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?
B : W : M = 1 : 3 : (1 + 3) = 1 : 3 : 4
5 B : 11W : 7 M = 1 × 5 : 3 × 11 : 4 × 7 = 5 : 33 : 28
66 ≡ Rs.1320
⇒1≡
1320
= 20
66
7 M → 28 ≡ 20 × 28 = Rs .560
11W → 33 ≡ 20 × 33 = Rs.660
5 B → 5 ≡ 20 × 5 = Rs.100
56. A certain sum of money is divided A, B and C such that for each rupee A has, B has 65 paise and
C has 40 paise. If C’s share is Rs.8, find the sum of money.
A : B : C = 100 : 65 : 40 = 20 : 13 : 8
8 ≡ Rs.8
8
⇒1≡
8
8
⇒ Sum of Money → 41 ≡ × 41 = Rs.41
8
57. 465 coins consist of rupee, 50 paise and 25 paise coins. Their values are in the ratio 5:3:1. Find
the number of each coin.
Values = 5 : 3 : 1
No. of Coins = 5 ×
100
100
100
: 3×
: 1×
= 5:6:4
100
50
25
15 ≡ 465
⇒1≡
465
= 31
15
Rs.1 Coins → 5 ≡ 31 × 5 = 155
50 Paise Coins → 6 ≡ 31 × 6 = 186
25 Paise Coins → 4 ≡ 31 × 4 = 124
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Ratio and Proportion
58. A sum of Rs.11.70 consists of rupees, 50 paise and 5 paise coins in the ratio 3:5:7. Find the
number of each kind of coins.
No. of Coins = 5 : 3 : 1
100
50
5
: 5×
:7×
= 60 : 50 : 7
100
100
100
117 ≡ 1170 Paise
Values = 3 ×
⇒1≡
1170
= 10
117
Values of Rs.1 Coins → 60 ≡ 10 × 60 = 600 Paise
600
=6
100
Values of 50 Paise Coins → 50 ≡ 10 × 50 = 500 Paise
500
∴ 50 Paise Coins =
= 10
50
Values of 5 Paise Coins → 7 ≡ 10 × 7 = 70 Paise
∴ Rs. 1 Coins =
∴ 5 Paise Coins =
Compiled By: Anjan
70
= 14
5
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