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. Which one of the following is correct in respect of the above Question and the Statements?

1. A. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
2. B. The Question can be answered by using either Statement alone
3. C. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
4. D. The Question cannot be answered even by using both the Statements together

Answer: A

Explanation

Let the age of P be 10x + y and the age of Q be 10y + x, where x and y are digits from 1 to 9 (since ages are >10 and Q).**
10x + y > 10y + x => 9x > 9y => x > y.
Possible pairs (x,y) satisfying x>y and x,y are digits (y cannot be 0 as Q would be a single digit):
– If x=2, y=1 => P=21, Q=12. Difference = 9.
– If x=3, y=1 => P=31, Q=13. Difference = 18.
– If x=3, y=2 => P=32, Q=23. Difference = 9.
Since the difference is not unique, Statement I alone is insufficient.

**Statement II: The sum of their ages is 11/6 times their difference.**
P + Q = (11/6) * |P – Q|
(10x + y) + (10y + x) = (11/6) * |(10x + y) – (10y + x)|
11x + 11y = (11/6) * |9x – 9y|
11(x + y) = (11/6) * 9 * |x – y|
x + y = (9/6) * |x – y|
x + y = (3/2) * |x – y|
Since x and y are positive digits, x+y is positive. For the equation to hold, |x-y| must also be positive, implying x ≠ y. Also, for x+y to be a multiple of 3/2, x-y must be even. This means x and y must have the same parity. If x-y is positive, then x > y.
2(x + y) = 3(x – y)
2x + 2y = 3x – 3y
5y = x.
Since x and y are single digits (1-9) and y cannot be 0:
– If y=1, then x=5. This gives P = 51 and Q = 15. (P>Q is satisfied, 51>15).
– If y=2, then x=10 (not a single digit). No other solutions.
So, the only unique pair is (x=5, y=1), which means P=51 and Q=15.
The difference in their ages = P – Q = 51 – 15 = 36.
This uniquely determines the ages and their difference. Thus, Statement II alone is sufficient.

Therefore, the question can be answered by using Statement II alone, but not Statement I alone.