# The sum of the ages of Anup and his father is 100. When Anup is as old as his father now, he will be five times as old as his son Anuj is now. Anuj will be eight years older than Anup is now, when Anup is as old as his father. What are their ages now?

Given:

The sum of the ages of Anup and his father is 100.

When Anup is as old as his father now, he will be five times as old as his son Anuj is now.

Anuj will be eight years older than Anup is now, when Anup is as old as his father.

To do:

We have to find their present ages.

Solution:

Let the age of Anup be $x$ years.

This implies,

Age of Anup’s father $=(100 - x)$ years.

Age of Anuj $= \frac{(100-x)}{5}$ years.

Anup is as old as his father after $(100-2x)$ years.

This implies,

Anuj’s age $=$ Present age of his father $+$ 8

Present age of Anuj $+$ 100 - 2x $=$ Present age of Anup + 8

$\frac{(100 - x)}{5} + (100 - 2x) = x + 8$

$\frac{(100-x)}{5} - 3x = 8 - 100$

$\frac{(100 - x - 15x)}{5} = -92$

$100 - 16x = -460$

$16x=460 + 100$

$x = \frac{560}{16}$

$x = 35$

Therefore,

Present age of Anup is 35 years.

Age of Anup’s father $= 100-x = 100-35 = 65$ years.

The age of Anuj $=\frac{100-x}{5} = \frac{100 - 35}{5} = \frac{65}{5} = 13$ years.