# Q. 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?

a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

b) The Question can be answered by using either Statement alone

c) The Question can be answered by both the Statements using together, but cannot be answered using either Statement alone

d) The Question cannot be answered even by using both the Statements together

Correct answer: a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

Given:

– P and Q are two-digit ages

– P = 10a + b

– Q = 10b + a where a and b are digits (1-9)

## Statement I: P is greater than Q

– This means 10a + b > 10b + a

– Simplifies to a > b

– Multiple values possible

– Not sufficient alone

### Statement II: Sum of ages is 11/6 times their difference

– (P + Q) = 11/6 × (P – Q)

– (10a + b) + (10b + a) = 11/6 × [(10a + b) – (10b + a)]

– 11a + 11b = 11/6 × (9a – 9b)

– 66(a + b) = 99(a – b)

– 5b = a

Since a and b must be single digits:

– Only possible solution is a = 5, b = 1

– Therefore P = 51 and Q = 15

Statement II alone is sufficient.

Answer: a) The question can be answered using Statement II alone, but cannot be answered using Statement I alone.