Question:medium

The present ages of Rahul and his father are in the ratio 6:13. The ratio between the present ages of his father and his sister is 13:5. If the difference between the present ages of his mother and his sister is 28 years, what is the difference between the present ages of his father and his mother?
Statement 1: The ratio of the present ages of Rahul and his mother is 1:2
Statement 2: The difference between the present ages of Rahul and his sister is 4 years

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Express Father's and Sister's ages as multiples of Rahul's age using the two given ratios, then use each statement separately to solve for that multiplier.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
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The Correct Option is D

Solution and Explanation

Step 1: Write every age in terms of Rahul's age.
Since Rahul : Father = 6 : 13, write Father = \(\frac{13}{6}\)Rahul. Since Father : Sister = 13 : 5 and Father is the same value, Sister = \(\frac{5}{13}\)Father = \(\frac{5}{6}\)Rahul. So both Father and Sister can be written purely in terms of Rahul's age, call it \(R\). Mother's age is Sister + 28 = \(\frac{5}{6}R + 28\). We want Father \(-\) Mother = \(\frac{13}{6}R - \left(\frac{5}{6}R+28\right) = \frac{4}{3}R-28\).

Step 2: Use statement (1) alone.
Statement (1) gives Rahul : Mother = 1 : 2, so Mother = \(2R\). Equating with the mother's age found above: \(2R = \frac{5}{6}R+28\), which gives \(\frac{7}{6}R=28\), so \(R=24\). Then Father \(-\) Mother = \(\frac{4}{3}(24)-28 = 32-28=4\) years. One clear value, so statement (1) alone works.

Step 3: Use statement (2) alone.
Statement (2) gives Rahul \(-\) Sister = 4, that is \(R - \frac{5}{6}R = 4\), so \(\frac{1}{6}R=4\), giving \(R=24\). This is the same value of \(R\) as before, so Father \(-\) Mother again comes out to \(\frac{4}{3}(24)-28=4\) years using only statement (2).

Step 4: Conclusion.
Each statement independently pins down Rahul's age and therefore the age difference we want, both giving 4 years. \[ \boxed{\text{Either statement alone is sufficient}} \]
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