Q. 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.

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 Paper

**Explanation:**

## Age of each of P and Q is less than 100 years

Let’s think through this step-by-step:

1) We’re told that P and Q’s ages are between 10 and 100 years, and that interchanging the digits of P’s age gives Q’s age.

2) Let’s consider Statement I: “The age of P is greater than the age of Q.”

This alone doesn’t give us enough information to calculate the exact difference in their ages.

3) Now, let’s look at Statement II: “The sum of their ages is 11/6 times their difference.”

This is more promising. Let’s see if we can solve it:

Let x be P’s age and y be Q’s age.

x + y = 11/6 * (x – y)

6(x + y) = 11(x – y)

6x + 6y = 11x – 11y

17y = 5x

y = 5x/17

Now, remember that y is formed by interchanging the digits of x.

The only two-digit numbers that satisfy this condition and the equation y = 5x/17 are:

x = 51 and y = 15

Therefore, using Statement II alone, we can determine that the difference in their ages is 51 – 15 = 36 years.

4) Statement I isn’t needed to solve the problem, and it doesn’t provide enough information on its own to solve it.

Therefore, the correct answer is indeed a) The Question can be answered by using one of the Statements alone (Statement II), but cannot be answered using the other Statement alone (Statement I).