Ten years ago, the age of mother was three times the age of her son. After ten…

Age-relationship problems are solved by anchoring one age at a reference time as a variable, expressing every other age (past, present, future) as that variable shifted by the elapsed years, translating each stated ratio ("three times", "twice") into an equation at the matching time point, and solving for the variable to recover the ages actually asked for.

Let the son's age ten years ago be x. Since the mother's age then was three times the son's, the mother's age ten years ago was 3x.

1. Present age of the son = x + 10; present age of the mother = 3x + 10.

2. Ten years from now, the son's age = x + 20 and the mother's age = 3x + 20.

3. The mother's age ten years from now is twice the son's age then: 3x + 20 = 2(x + 20).

4. Expanding the right side: 3x + 20 = 2x + 40.

5. Solving for x: x = 20.

6. So the present age of the son = x + 10 = 30, and the present age of the mother = 3x + 10 = 70.

7. Ratio of present ages (mother : son) = 70 : 30 = 7 : 3.

Check: ten years ago the son was 20 and the mother 60, and 60 is indeed three times 20. Ten years from now the son will be 40 and the mother 80, and 80 is indeed twice 40 — both conditions hold.

So the ratio of their present ages is 7 : 3.