# SOLVED PROBLEMS ON AGES

Problem 1 :

If Azhagan is 50 years old and his son is 10 years old, then the simplest ratio between the age of Azhagan to his son is

(A) 10 : 50     (B) 50 : 10     (C) 5 : 1     (D) 1 : 5

Solution :

Ratio between the age of Azhagan to his son is

= 50 : 10

= ⁵⁰⁄₁₀ ¹⁰⁄₁₀

= 5 : 1

The correct answer choice is option (C).

Problem 2 :

Saran is 6 times as old as his son Sankar. After 4 years, he will be 4 times as old as his son. What are their present ages?

(A) 30, 5     (B) 36, 6     (C) 48, 8     (D) 24, 4

Solution :

Let x be the present age of the son.

Then the present age of Saran is 6x.

After 4 years,

age of Saran = 6x + 4

age of sankar = x + 4

Given : After 4 years, Saran will be 4 times as old as his son Sankar.

6x + 4 = 4(x + 4)

6x + 4 = 4x + 16

2x = 12

x = 6

Present age of Saran = 6(6) = 36 years

Present age of Sankar = 6 years

The correct answer choice is option (B).

Problem 3 :

The sum of the present ages of a father and his son is 60 years. Six years ago, father's age was five times the age of the son. After 6 years, what will the the son's age be?

(A) 12 years     (B) 14 years     (C) 18 years     (D) 20 years

Solution :

Let f and s be the present ages of father son.

Given : The sum of the present ages of a father and his son is 60.

f + s = 60 ----(1)

Given : Six years ago, father's age was five times the age of the son.

f - 6 = 5(s - 6)

f - 6 = 5s - 30

f = 5s - 24

Substitute f = 5s - 24 into (1).

5s - 24 + s = 60

6s - 24 = 60

6s = 84

s = 14

The present age of the son is 16 years.

After 6 years, the age of the son will be

= 14 + 6

= 20 years

The correct answer choice is option (D).

Problem 4 :

The age of a man is 4 times the sum of the ages of his two sons. Ten years hence, his age will be double of the sum of the ages his sons. The father's present age is

(A) 50 years    (B) 55 years    (C) 60 years    (D) 65 years

Solution :

Let f be the present age of father and s be the sum of present ages of two sons.

Given : The age of a man is 4 times the sum of the ages of his two sons.

f = 4s ----(1)

Given : Ten years hence, the age of father will be double of the sum of the ages his sons.

f + 10 = 2(s + 10 + 10)

Here, there are two 10's added to s. Because there are two sons.

f + 10 = 2(s + 20)

f + 10 = 2s + 40

f = 2s + 30 ----(2)

From (1) and (2),

4s = 2s + 30

2s = 30

s = 15

Substitute s = 15 into (1).

f = 4(15)

f = 60

The present age of father is 60 years.

The correct answer choice is option (C).

Problem 5 :

At present, the ratio between the ages of Arun and Deepak is 4 : 3. After 6 years, Arun’s age will be 26 years. What is the age of Deepak at present?

(A) 12 years    (B) 15 years    (C) 19½ years    (D) 21 years

Solution :

Ratio between the present ages of Arun and Deepak is

= 4 : 3

From the above ratio,

present age of Arun = 4x

present age of Deepak = 3x

After 6 years,

age of Arun = 4x + 6

Given : After 6 years, Arun’s age will be 26 years.

Then, we have

4x + 6 = 26

4x = 20

x = 5

Age of Deepak at present :

= 3(5)

= 15 years

The correct answer choice is option (B).

Problem 6 :

A is younger than B. Seven times the difference between their ages is equal to sum of their ages. Ratio of B's age after 12 years and A's present age is 5 : 3, then what is A's present age?

(A) 30 years    (B) 36 years    (C) 42 years    (D) 48 years

Solution :

Let a and b be the present ages of A and B.

Given : A is younger than B. Seven times the difference between their ages is equal to sum of their ages.

7(b - a) = a + b

7b - 7a = a + b

6b = 8a

3b = 4a ----(1)

Given : Ratio of B's age after 12 years and A's present age is 5 : 3.

(b + 12) : a = 5 : 3

3(b + 12) = 5a

3b + 36 = 5a

From (1), substitute 3b = 4a.

4a + 36 = 5a

36 = a

The present age of A is 36 years.

The correct answer choice is option (B).

Problem 7 :

If the ratio of the ages of son and father in 2015 and 2023 are 1 : 4 and 3 : 8 respectively, then the sum of the ages of son and father in 2010 is

(A) 40     (B) 30     (C) 35     (D) 45

Solution :

Given : Ratio of the ages of son and father in 2015 is

= 1 : 4

From the above ratio,

age of son in 2015 = x

age of father in 2015 = 4x

Then, we have

age of son in 2023 = x + 8

age of father in 2023 = 4x + 8

Ratio of the ages of son and father in 2023 is

= (x + 8) : (4x + 8) ----(1)

Given : Ratio of the ages of son and father in 2023 is

= 3 : 8 ----(2)

From (1) and (2),

(x + 8) : (4x + 8) = 3 : 8

8(x + 8) = 3(4x + 8)

8x + 64 = 12x + 24

40 = 4x

10 = x

age of son in 2015 = x = 10

age of father in 2015 = 4x = 40

age of son in 2010 = 10 - 5 = 5

age of father in 2010 = 40 - 5 = 35

Sum of the ages of son and father in 2010 is

= 5 + 35

= 40

The correct answer choice is option (A).

Problem 8 :

If the ratio of father's age to son's age is 4 : 1 and the product of their ages is 196, then the ratio of their ages after 5 years will be

(A) 3 : 1     (B) 10 : 3     (C) 11 : 4     (D) 14 : 5

Solution :

Given : Ratio of father's age to son's age is

4 : 1

From the above ratio,

father's age = 4x

son's age = x

Given : Product of their ages is 196.

(4x)(x) = 196

4x2 = 196

x2 = 49

x = 7

At present,

father's age = 4(7) = 28 years

son's age = 7 years

After 5 years,

father's age = 28 + 5 = 33 years

son's age = 7 + 5 = 12 years

Ratio of their ages after 5 years will be

= 33 : 12

= 11 : 4

The correct answer choice is option (C).

Problem 9 :

The ages of Vivek and Sumit are 2 : 3. After 12 years, their ages will be in the ratio 11 : 15. The age of Sumit is

(A) 32 years    (B) 42 years    (C) 48 years    (D) 56 years

Solution :

Given : The ages of Vivek and Sumit are 2 : 3.

From the above ratio,

present age of Vivek = 2x

present age of Sumit = 3x

After 12 years,

age of Vivek = 2x + 12

age of Sumit = 3x + 12

After 12 years, the ratio between their ages will be

= (2x + 12) : (3x + 12) ----(1)

Given : After 12 years, the ratio betweern their will be

= 11 : 15 ----(2)

From (1) and (2),

(2x + 12) : (3x + 12) = 11 : 15

15(2x + 12) = 11(3x + 12)

30x + 180 = 33x + 132

48 = 3x

16 = x

Present age of Sumit :

= 3x

3(16)

= 48 years

The correct answer choice is option (C).

Problem 10 :

The ratio of the ages of the father and the son at present is 19 : 5. After 4 years, the ratio will become 3 : 1. What is the sum of the present ages of the father and the son?

(A) 40     (B) 42     (C) 48     (D) 52

Solution :

Given : The ratio of the ages of the father and the son at present is

= 19 : 5

From the above ratio,

present age of father = 19x

present age of son = 5x

After 4 years,

age of father = 19x + 4

age of son = 5x + 4

After 4 years, the ratio between their ages will be

= (19x + 4) : (5x + 4) ----(1)

Given : After 4 years, the ratio of their ages will become

= 3 : 1 ----(2)

From (1) and (2),

(19x + 4) : (5x + 4) = 3 : 1

1(19x + 4) = 3(5x + 4)

19x + 4 = 15x + 12

4x = 8

x = 2

present age of father = 19x = 19(2) = 38 years

present age of son = 5x = 5(2) = 10 years

Sum of the present ages of the father and the son :

= 38 + 10

= 48

The correct answer choice is option (C).

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