 # The sum of the ages of Anup and his father is 100. `
Question:

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?

Solution:

Let Anup's age be $x$ years.

Therefore, his father's age will be $(100-\mathrm{x})$ years.

When Anup is as old as his father after $(100-2 \mathrm{x})$ years,

Anuj's age $=\left(\frac{100-\mathrm{x}}{5}+100-2 \mathrm{x}\right)$ years $=\frac{600-11 \mathrm{x}}{5}$ years.

Again, when Anup is as old as his father,

Let Anup's age be $x$ years.

Therefore, his father's age will be $(100-\mathrm{x})$ years.

When Anup is as old as his father after $(100-2 \mathrm{x})$ years,

Anuj's age $=\left(\frac{100-\mathrm{x}}{5}+100-2 \mathrm{x}\right)$ years $=\frac{600-11 \mathrm{x}}{5}$ years.

Again, when Anup is as old as his father,

Anuj's age $=\mathrm{x}+8$.

Now,

$\frac{600-11 \mathrm{x}}{5}=\mathrm{x}+8$

or $600-11 \mathrm{x}=5 \mathrm{x}+40$

or $16 \mathrm{x}=560$

or $\mathrm{x}=35$.

Thus, Anup's age $=35$ years

Anup's f ather's age $=100-\mathrm{x}=100-35=65$ years

Anuj's age $=\mathrm{x}+8=35+8=43$ years