Digit-reversed ages with a ratio condition: Mr. Manoj’s age is a two-digit number with digits (a, b). Reversing the digits gives his wife’s age. One-eleventh of the sum of their present ages equals the difference of their ages. If Manoj is older, find the age difference.

Introduction / Context:Two-digit reversal problems convert naturally to digit algebra. If Mr. Manoj's age is 10a + b and his wife's age is 10b + a, we can model the sum and difference, apply the given relation, and solve for permissible digit pairs (a, b) with a > b because Manoj is older.

Given Data / Assumptions:

• Manoj age = 10a + b; wife age = 10b + a, with 1 ≤ a ≤ 9 and 0 ≤ b ≤ 9.
• Manoj older ⇒ a > b.
• (1/11) * (sum) = difference.

Concept / Approach:Compute sum S = 11(a + b) and difference D = 9(a − b). The condition (1/11)S = D reduces to a single linear relation between a and b. Solve for integer digits then compute the age difference.

Step-by-Step Solution:

Verification / Alternative check:Check the condition: sum = 99; (1/11)*99 = 9 equals difference 9. All constraints satisfied.

Why Other Options Are Wrong:10/8 years do not meet the algebraic constraint; “Cannot be determined” is false because the digits are uniquely pinned to 5 and 4 under the given condition.

Common Pitfalls:Forgetting that leading digit a cannot be 0; ignoring the older/younger condition which enforces a > b.

Final Answer:9 years