Question: A year hence a father will be 5 times as old as his son. Two years ago the father was 3 times as old as his son will be 4 years hence. Find their present ages.

Solution:

Let the present ages of the father and the son be x years and y years respectively.

According to the question,

Condition I,
A year hence (after) a father will be 5 times as old as his son.
$or, (x +1) = 5(y+1)$
$or, x +1 = 5y +5$
$or, x = 5y +4$

Condition II,
Two years ago, the father was 3 times as old as his son will be 4 years hence.
$or, (x -2) = 3(y +4)$
$or, x -2 = 3y +12$
$or, x = 3y +14$

Put value of x from equation (i) in equation (ii), we get,

$or, 5y +4 = 3y +14$
$or, 5y -3y = 14-4$
$or, 2y = 10$
$or, y = \frac{10}{2}$
$\therefore y = 5$

Put value of y in equation (i), we get,

$or, x = 5×5 +4$
$or, x = 25 +4$
$\therefore x = 29$

So, (x,y) = (29,5)

Therefore, the required age of the father is 29 years and the son is 5 years.