Problems based on ages are generally asked in most of the competitive examinations. To solve these problems, the knowledge of linear equations is essential.

In such problems, there may be three situations:

- Age some years ago
- Present age
- Age some years future

Two of these situations are given and it is required to find the third. The relation between the age of two persons may also be given. Simple linear equations are framed and their solutions are obtained.

**Example 1:** The age of father is 4 times the age of his son. If 5 years ago father's age was 7 times the age of his son at that time, what is father's present age?

Let the present age of son be x years.

Then, the present age of father = 4x years.

Given, 5 years ago,

(4x - 5) = 7(x - 5)

4x - 5 = 7x - 35

3x = 30

x = 10

Father's present age = 4x

= 40 years.

**Example 2:** The age of Mr Gupta is four times the age of his son. After 10 years, the age of Mr Gupta will be only twice the age of his son. Find the present age of Mr Gupta’s son?

Let the present age of son be x years.

The present age of father = 4x years.

Given, after 10 years,

(4x + 10) = 2(x + 10)

4x + 10 = 2x + 20

x = 5

So, the present age of Mr Gupta’s son is 5 years.

**Example 3:** 10 years ago Anu's mother was 4 times older than her daughter. After 10 years, the mother will be twice older than the daughter. Find the present age of Anu?

Let the age of Anu 10 years ago be x years.

So, age of mother 10 years ago = 4x.

Given, after 10 years from now, and 20 years from 10 years ago,

(4x + 20) = 2(x + 20)

4x + 20 = 2x + 40

x = 10

Present age of Anu = x + 10

= 20 years.

**Example 4:** The sum of the ages of A and B is 42 years. 3 years back, the age of A was 5 times the age of B. Find the difference between the present ages of A and B?

Let, the age of B, 3 years ago, be x years.

So, age of A, 3 years ago = 5x years.

Given, sum of present ages is 42 years.

(x + 3) + (5x + 3) = 42

6x = 36

x = 6

Present age of A = 5x + 3 = 33 years.

Present age of B = x + 3 = 9 years.

Difference in ages = 33 - 9 = 24 years.

**Example 5:** The sum of the ages of a son and father is 56 years. After four years, the age of the father will be three times that of the son. Find their respective ages?

Let, the age of son, after 4 years, be x years.

So, age of father after 4 years = 3x years.

Given, at present, sum of ages is 56 years.

(x - 4) + (3x - 4) = 56

4x = 64

x = 16

Present age of son = x - 4 = 12 years.

Present age of father = 3x - 4 = 44 years.

**Example 6:** The ratio of the age of father and son at present is 6:1. After 5 years, the ratio will become 7:2. Find the present age of the son?

Let the present age of father and son be 6x and x respectively.

Given, after 5 years,

6x + 5 : x + 5 = 7 : 2

2(6x + 5) = 7(x + 5)

12x + 10 = 7x + 35

5x = 25

x = 5

So, present age of son is 5 years.

**Example 7:** Six years ago Mahesh was twice as old as Suresh. If the ratio of their present ages is 9:5 respectively, what is the difference between their present ages?

Let the present ages be 9x and 5x respectively.

Given, six years ago,

(9x - 6) = 2(5x - 6)

9x - 6 = 10x - 12

x = 6

Difference between present ages = 9x - 5x = 4x

= 24 years.