UPSC CSE 2024 · PYQ · Data Sufficiency / Ages · hard
A Question is given followed by two Statements I and II. Consider the Question and the Statements. Age of each of P and Q is less than 100 years but more than 10 years. If you interchange the digits of the age of P, the number represents the age of Q. Question: What is the difference of their ages? Statement-I: The age of P is greater than the age of Q. Statement-II: The sum of their ages is 11/6 times their difference.
A.The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone✓ Correct
B.The Question can be answered by using either Statement alone
C.The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
D.The Question cannot be answered even by using both the Statements together
Explanation
Let P = 10a + b and Q = 10b + a (digits interchanged). Difference = 9(a − b); Sum = 11(a + b). Statement-I (P > Q) only tells us a > b but doesn't fix the difference. Statement-II says Sum = (11/6) × Difference: 11(a + b) = (11/6) × 9|a − b|, giving 11(a+b) = (33/2)|a−b|, so 2(a+b) = 3|a−b|. If a > b: 2(a+b) = 3(a−b) → 2a + 2b = 3a − 3b → a = 5b. So b = 1, a = 5 gives P=51, Q=15, diff = 36. If b > a: 2(a+b) = 3(b−a) → b = 5a, giving P=15, Q=51, diff = 36. In both cases the difference is 36, so Statement-II alone gives a unique difference of 36. Statement-I alone does not. Hence answerable using one statement alone (II) but not the other (I).
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