# A father is three times as old as his son.

Question:

A father is three times as old as his son. In 12 years time, he will be twice as old as his son. Find the present ages of father and the son.

Solution:

Let the present age of father be x years and the present age of his son be years.

The present age of father is three times the age of the son. Thus, we have

$x=3 y$

$\Rightarrow x-3 y=0$

After 12 years, father's age will be $(x+12)$ years and son's age will be $(y+12)$ years. Thus using the given information, we have

$x+12=2(y+12)$

$\Rightarrow x+12=2 y+24$

$\Rightarrow x-2 y-12=0$

So, we have two equations

$x-3 y=0$

$x-2 y-12=0$

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{(-3) \times(-12)-(-2) \times 0}=\frac{-y}{1 \times(-12)-1 \times 0}=\frac{1}{1 \times(-2)-1 \times(-3)}$

$\Rightarrow \frac{x}{36-0}=\frac{-y}{-12-0}=\frac{1}{-2+3}$

$\Rightarrow \frac{x}{36}=\frac{-y}{-12}=\frac{1}{1}$

$\Rightarrow \frac{x}{36}=\frac{y}{12}=1$

$\Rightarrow x=36, y=12$

Hence, the present age of father is 36 years and the present age of son is 12 years.