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.