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Family Ages Equation Problem

XAT 2017 · Quantitative Ability & Data Interpretation · Hard · Ages

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Hari's family consisted of his younger brother (Chari), younger sister (Gouri), and their father and mother. When Chari was born, the sum of the ages of Hari, his father and mother was 70 years. The sum of the ages of four family members, at the time of Gouri's birth, was twice the sum of ages of Hari's father and mother at the time of Hari's birth. If Chari is 4 years older than Gouri, then find the difference in age between Hari and Chari.

  1. A.

    5 years

  2. B.

    6 years

  3. C.

    7 years

  4. D.

    8 years

  5. E.

    9 years

Answer

E

Explanation

Let current ages be F,M,H,C,GF, M, H, C, G. Given G=C−4G = C - 4. When Chari was born (CC years ago): (F−C)+(M−C)+(H−C)=70  ⟹  F+M+H−3C=70(F - C) + (M - C) + (H - C) = 70 \implies F + M + H - 3C = 70 ---(1)

When Gouri was born ((C−4)(C-4) years ago): Ages of F,M,H,CF, M, H, C were F−(C−4),M−(C−4),H−(C−4),C−(C−4)=4F - (C-4), M - (C-4), H - (C-4), C - (C-4) = 4. Sum = F+M+H+C−4(C−4)=F+M+H−3C+16F + M + H + C - 4(C-4) = F + M + H - 3C + 16. At Hari's birth (HH years ago), sum of FF and MM ages was (F−H)+(M−H)(F - H) + (M - H). So, F+M+H−3C+16=2(F+M−2H)=2F+2M−4HF + M + H - 3C + 16 = 2(F + M - 2H) = 2F + 2M - 4H. Using (1), 70+16=2(F+M−2H)=86  ⟹  F+M−2H=43  ⟹  F+M=43+2H70 + 16 = 2(F + M - 2H) = 86 \implies F + M - 2H = 43 \implies F + M = 43 + 2H.

Substitute F+MF+M into (1): 43+2H+H−3C=70  ⟹  3H−3C=27  ⟹  H−C=943 + 2H + H - 3C = 70 \implies 3H - 3C = 27 \implies H - C = 9. Difference in age between Hari and Chari is 9 years.

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