Reversing the digits of the father's age gives the son's age. One year ago,…

Concept

Concept: a two-digit number with tens digit x and units digit y equals 10x + y, and reversing its digits gives 10y + x. When such a number's digit-reversal is linked to another quantity by a ratio condition that applies at a past point in time, set up simultaneous equations in x and y from both conditions, then use the digit constraint (x and y are single digits, with the leading digit from 1 to 9) to pick the unique valid solution.

Application

1. Let the father's present age be 10x + y, where x is the tens digit and y is the units digit.

2. Reversing the digits gives the son's present age: 10y + x.

3. One year ago, the father's age was twice the son's age: (10x + y) – 1 = 2[(10y + x) – 1].

4. Expanding: 10x + y – 1 = 20y + 2x – 2, which simplifies to 8x – 19y = –1, so x = (19y – 1)/8.

5. Since x must be a whole digit from 1 to 9, testing y = 1 through 9 shows that only y = 3 gives an integer x = 7 within that range.

6. So the father's present age is 10(7) + 3 = 73, and the son's present age is 10(3) + 7 = 37.

Cross-check

Cross-check: reversing 73 gives 37, confirming the digit-reversal condition. One year ago the ages were 72 and 36, and 72 is exactly twice 36, confirming the past-age condition too — both hold uniquely for this pair.