6. Ratio-Based Age Problems

Age problems almost always use the “x-multiplier” method: if the ratio of two ages is a : b, take the actual ages as ax and bx, then use the additional condition given (past or future age relationship) to form an equation and solve for x.

Example 36: The ratio of the present ages of A and B is 3 : 5. After 6 years, the ratio of their ages becomes 2 : 3. Find their present ages. Let present ages = 3x and 5x. (3x + 6)/(5x + 6) = 2/3 Cross-multiply: 3(3x+6) = 2(5x+6) → 9x + 18 = 10x + 12 → x = 6 Present ages: A = 3×6 = 18 years, B = 5×6 = 30 years. Check: 18:30 = 3:5 ✔. After 6 years: 24:36 = 2:3 ✔.

Example 37: The sum of the ages of A and B is 40 years, and the ratio of their ages is 3 : 5. Find their ages. Total parts = 3+5 = 8. 1 part = 40/8 = 5. A = 3×5 = 15 years, B = 5×5 = 25 years.

Example 38: The ratio of the present ages of a father and his son is 7 : 2. After 8 years, the ratio becomes 9 : 4. Find their present ages. Let ages = 7x, 2x. (7x+8)/(2x+8) = 9/4 Cross-multiply: 4(7x+8) = 9(2x+8) → 28x + 32 = 18x + 72 → 10x = 40 → x = 4 Father = 7×4 = 28 years, Son = 2×4 = 8 years. Check: after 8 years → 36:16 = 9:4 ✔.

Example 39A: Five years ago, the ratio of ages of X and Y was 4 : 5. Ten years from now, the ratio of their ages will be 6 : 7. Find their present ages. Let ages 5 years ago = 4x, 5x. Present ages = 4x+5, 5x+5. Ten years from now: ages = 4x+15, 5x+15. Ratio (4x+15)/(5x+15) = 6/7. Cross-multiply: 7(4x+15) = 6(5x+15) → 28x + 105 = 30x + 90 → 2x = 15 → x = 7.5. Present ages: X = 4(7.5)+5 = 35, Y = 5(7.5)+5 = 42.5. (Check: 5 years ago → 30:37.5 = 4:5 ✔. Ten years hence → 45:52.5 = 6:7 ✔.) Present ages: X = 35 years, Y = 42.5 years.

Age-ratio problems are simply “ratio + linear equation” problems wearing a story costume — once the multiplier x is defined for the ratio at any single point in time, every other time reference in the question (past or future) is just “add or subtract years from both ax and bx,” which keeps the algebra entirely linear and easy to solve.

Example 39B (three-person continued-ratio age problem, additional): The present ages of A, B and C are in the ratio 2 : 3 : 4. If the sum of their present ages is 63 years, find the age of each, five years hence. Total parts = 2+3+4 = 9. 1 part = 63/9 = 7. Present ages: A = 2×7 = 14, B = 3×7 = 21, C = 4×7 = 28. Five years hence: A = 19 years, B = 26 years, C = 33 years.

Example 39C (past-ratio-to-present, reverse direction, additional): The present ratio of ages of X and Y is 3 : 4. Eight years ago, the ratio of their ages was 2 : 3. Find their present ages. Let present ages = 3x, 4x. Eight years ago: (3x−8)/(4x−8) = 2/3. Cross-multiply: 3(3x−8) = 2(4x−8) → 9x − 24 = 8x − 16 → x = 8. Present ages: X = 3×8 = 24 years, Y = 4×8 = 32 years. Check: 8 years ago → 16:24 = 2:3 ✔.