Sandeep's age after six years will be three-seventh of his father's age at…

Concept

In age-ratio problems, represent each person's present age with a variable, then translate every time-shifted statement ("after n years" → age + n; "n years ago" → age − n) into a linear equation. Solving the resulting system of two equations in two unknowns gives the present ages.

Application

Let Sandeep's present age = x years and his father's present age = y years.

1. After six years, Sandeep's age is three-seventh of his father's age at that time: 7(x + 6) = 3(y + 6) ⇒ 7x − 3y = −24 … (i)

2. Ten years ago, the ratio of their ages was 1 : 5: (x − 10) : (y − 10) = 1 : 5 ⇒ 5(x − 10) = (y − 10) ⇒ 5x − y = 40 … (ii)

3. From (ii): y = 5x − 40.

4. Substitute into (i): 7x − 3(5x − 40) = −24 ⇒ 7x − 15x + 120 = −24 ⇒ −8x = −144 ⇒ x = 18.

5. So y = 5(18) − 40 = 50.

Cross-check

Check: six years from now, Sandeep is 24 and his father is 56, and 24 : 56 = 3 : 7, which matches. Ten years ago, Sandeep was 8 and his father was 40, and 8 : 40 = 1 : 5, which matches.

Hence, Sandeep's father's present age is 50 years.