Reasoning : Logical Reasoning & Analytical Ability

Q 60 / 325

UPSC CSE Prelims 2024

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?

  1. Statement-I: The age of P is greater than the age of Q.
  2. 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?

EXPLANATION

Correct Option

Let the age of P be represented as 10x + y and the age of Q as 10y + x, where x is the tens digit and y is the units digit. Since the ages are between 10 and 99 years, x and y must be integers from 1 to 9. Also, for P and Q to have distinct ages when digits are interchanged, x ≠ y.

  • Analysis of Statement I: "The age of P is greater than the age of Q."

    • This can be written as: 10x + y > 10y + x
    • Simplifying the inequality: 9x > 9y, which implies x > y.
    • This statement establishes a relationship between the digits, indicating that the tens digit of P's age is greater than its units digit. However, it does not provide specific numerical values for x and y. For example, if (x, y) = (2, 1), P=21, Q=12, difference=9. If (x, y) = (3, 1), P=31, Q=13, difference=18. Since multiple pairs of (x, y) satisfy x > y and result in different age differences, Statement I alone is insufficient to determine a unique difference between their ages.

  • Analysis of Statement II: "The sum of their ages is 11/6 times their difference."

    • The sum of their ages is (10x + y) + (10y + x) = 11x + 11y.
    • The difference of their ages is |(10x + y) - (10y + x)| = |9x - 9y| = 9|x - y|.
    • From the problem context, it is implied that a specific difference exists. If we assume P's age is greater than Q's age (consistent with Statement I, though not strictly required for Statement II alone), then x > y, and the difference is 9x - 9y.
    • According to Statement II: 11x + 11y = (11/6) * (9x - 9y)
    • Divide both sides by 11: x + y = (1/6) * (9x - 9y)
    • Simplify the right side: x + y = (9/6) * (x - y)
    • x + y = (3/2) * (x - y)
    • Multiply by 2: 2(x + y) = 3(x - y)
    • Expand: 2x + 2y = 3x - 3y
    • Rearrange terms: 2y + 3y = 3x - 2x
    • This simplifies to: 5y = x
    • Since x and y are single digits between 1 and 9:
      • If y = 1, then x = 5. This gives P's age as 51 and Q's age as 15. The difference is 51 - 15 = 36.
      • If y = 2, then x = 10, which is not a single digit.
    • Thus, the only possible pair of digits (x, y) is (5, 1). This uniquely determines the ages of P and Q, and consequently their difference. Therefore, Statement II alone is sufficient to answer the question.

Since Statement I alone is insufficient, and Statement II alone is sufficient, the question can be answered by using one of the statements alone, but not the other.

Incorrect Options

Options 2, 3, and 4 are incorrect because Statement II alone is sufficient to determine the difference in ages, while Statement I alone is not sufficient. This aligns with the explanation for Option 1.