Q73. 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?

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 <100). We need to find the absolute difference |P-Q|. **Statement I: The age of P is greater than the age of Q (P > 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.