Question

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

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

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.

Answer 1

The Correct Option is D

Step 1: Set up the ages using a common multiplier.
Rahul : Father = 6 : 13, so let Rahul = \(6k\) and Father = \(13k\) for some positive number \(k\). Father : Sister = 13 : 5 uses the same father value \(13k\), so Sister = \(5k\). The mother's age is not fixed yet, only that Mother \(-\) Sister = 28 years, so Mother = \(5k+28\). We need Father \(-\) Mother = \(13k-(5k+28)=8k-28\), which depends only on \(k\).

Step 2: Check statement (1) alone.
Statement (1) says Rahul : Mother = 1 : 2, so Mother = \(2\times 6k = 12k\). Since we already found Mother = \(5k+28\), setting these equal gives \(12k = 5k+28\), so \(7k=28\) and \(k=4\). Then Father \(-\) Mother \(= 8(4)-28 = 32-28 = 4\) years. This is a single clean number, so statement (1) alone answers the question.

Step 3: Check statement (2) alone.
Statement (2) says Rahul and Sister differ by 4 years, that is \(6k - 5k = 4\) (Rahul being older since \(6k>5k\)), which gives \(k=4\) directly. Then Father \(-\) Mother \(= 8(4)-28=4\) years again. This also gives one clean number using statement (2) alone.

Step 4: Final answer.
Both statements, taken separately, let us solve for \(k\) and hence for Father \(-\) Mother, each giving 4 years. \[ \boxed{\text{Either statement alone is sufficient}} \]