The present ages of three persons in proportions 4 : 7 : 9. Eight years ago,…

Concept: When several quantities are given in a ratio like a : b : c, express them using a common multiplier x as ax, bx and cx. Any further condition on these quantities (a fixed amount subtracted from each, or a required sum) then becomes a single linear equation in x, which can be solved directly for x and then for the actual quantities.

Application:

1. Let the present ages of the three persons, in the ratio 4 : 7 : 9, be 4x, 7x and 9x years.

2. Eight years ago, their ages were (4x − 8), (7x − 8) and (9x − 8) years.

3. The sum of their ages eight years ago was 56, so (4x − 8) + (7x − 8) + (9x − 8) = 56.

4. Simplify: 20x − 24 = 56.

5. Solve: 20x = 80, so x = 4.

6. Present ages: 4x = 16, 7x = 28, and 9x = 36 years.

Cross-check:

Eight years ago the ages were 8, 20 and 28, which sum to 56 — matching the given condition. Also, 16 : 28 : 36 simplifies to 4 : 7 : 9 (dividing each by 4) — matching the given ratio.