Multiple choice

The average of present ages of members of a family (Father, Mother and son) is 34. 4 years hence, the son's age is half of the mother's age. 10 years ago, the ratio of ages of father and son is 17: 4. Find the sum of ages of father and mother at the time of son's birth.

  1. 45 years

  2. 50 years

  3. 52 years

  4. 48 years

  5. 49 years
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D Correct answer
Explanation

Let F, M, S be current ages. (F+M+S)/3 = 34 => F+M+S = 102. (S+4) = 0.5(M+4) => M = 2S + 4. (F-10)/(S-10) = 17/4 => 4F - 40 = 17S - 170 => 4F = 17S - 130. Substituting M: F + 2S + 4 + S = 102 => F = 98 - 3S. 4(98 - 3S) = 17S - 130 => 392 - 12S = 17S - 130 => 29S = 522 => S = 18. Then M = 40, F = 44. At son's birth (18 years ago), Father was 26, Mother was 22. Sum = 48.

AI explanation

The total present age of the family is 3 times 34, which is 102. Ten years ago the ratio of the father's to son's age was 17:4, so if the son was 4x then, the son is now 4x + 10 and the father is 17x + 10. Four years hence the mother's age will be twice the son's age, so the mother is currently 2(4x + 14) - 4, which equals 8x + 24. The sum of their present ages is 29x + 44 = 102, yielding 29x = 58 and x = 2. The present ages are son 18, mother 40, and father 44, meaning when the son was born the parents' ages summed to (44 - 18) + (40 - 18), which equals 48 years.

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