Question

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.

• A. 5 years
• B. 6 years
• C. 7 years
• D. 8 years
• E. 9 years

Hint:

For family-age problems, anchor ages at event times (births), express all sums in terms of those anchors, and solve the resulting linear equations. The \emph{time gaps between births} are the key shortcuts.

Answer 1

Answer: (E)

Step 1: Set variables at key birth times.
Let the parents’ ages at Hari’s birth be \(F_h\) and \(M_h\). Let \(S = F_h + M_h\).
Let \(d\) be the age gap (Hari is \(d\) years older than Chari).

Step 2: Condition at Chari’s birth.
At Chari’s birth, ages are: Hari \(= d\), Father \(= F_h + d\), Mother \(= M_h + d\).
Given sum \(= 70\): \[ d + (F_h + d) + (M_h + d) = 70 \;\Rightarrow\; S + 3d = 70. \qquad (1) \]

Step 3: Condition at Gouri’s birth.
Chari is \(4\) years older than Gouri \(\Rightarrow\) Gouri is born \(4\) years after Chari. So at Gouri’s birth:
Hari \(= d+4\), Chari \(= 4\), Father \(= F_h + d + 4\), Mother \(= M_h + d + 4\).
Sum of the \emph{four} (excluding newborn Gouri) is \[ (d+4) + 4 + (F_h + d + 4) + (M_h + d + 4) = S + 3d + 16. \] This equals twice the parents’ sum at Hari’s birth, i.e. \(2S\): \[ S + 3d + 16 = 2S \;\Rightarrow\; 3d + 16 = S. \qquad (2) \]

Step 4: Solve the system.
Substitute \(S = 3d + 16\) from (2) into (1): \[ (3d + 16) + 3d = 70 \;\Rightarrow\; 6d + 16 = 70 \;\Rightarrow\; 6d = 54 \;\Rightarrow\; d = 9. \]

Final Answer: \[ \boxed{\text{E. 9 years}} \]