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, but the data in statement (2) alone is not sufficient.
• B. If the data in statement (2) alone is sufficient to answer the question, but the data in statement (1) alone is not sufficient.
• 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.
• E. If the data in neither statement (1) nor statement (2) is sufficient to answer the question, and more data is needed.

Hint:

Write Rahul, father, sister and mother's ages in terms of one variable x using the two given ratios, then check if each statement alone lets you solve for x.

Answer 1

The Correct Option is D

Let Rahul's present age be \(6x\) and his father's present age be \(13x\), since these two ages are in the ratio 6 : 13.

The father and the sister are in the ratio 13 : 5. Since the father's age is already taken as \(13x\), the sister's age must be \(5x\), because \(13x : 5x = 13 : 5\).

The mother is naturally older than her daughter, so mother's age minus sister's age = 28, which gives mother's age = \(5x + 28\).

We are asked for father's age minus mother's age, which is \(13x - (5x + 28) = 8x - 28\). So the whole question comes down to finding the value of x.

Statement (1): Rahul : Mother = 1 : 2, so \(\frac{6x}{5x+28} = \frac{1}{2}\). Cross multiplying, \(12x = 5x + 28\), so \(7x = 28\) and \(x = 4\). Putting this back, father minus mother = \(8(4) - 28 = 4\) years. So statement (1) alone pins down a definite answer.

Statement (2): Rahul minus sister = 4. Since Rahul's age \(6x\) is greater than the sister's age \(5x\), we get \(6x - 5x = 4\), so \(x = 4\). This again gives father minus mother = \(8(4) - 28 = 4\) years. So statement (2) alone is also enough on its own.

Since each statement, taken separately, lets us work out x and hence the required difference, either statement alone is sufficient to answer the question. The correct choice is option (4).