A woman says, "If you reverse the digits of my age, the resulting number represents my husband's age. He is older than me, and the difference between our ages is one-eleventh of their sum." What is the husband's age?

Introduction / Context:
This puzzle-type age question involves a two-digit number whose digits are reversed to obtain another age. We know that the husband's age is obtained by reversing the woman's age and that the husband is older. In addition, the difference between their ages is one-eleventh of the sum of their ages. We must use this information to find the husband's age.

Given Data / Assumptions:

- Let the woman's present age be a two-digit number with tens digit a and units digit b.
- Therefore, the woman's age can be written as 10a + b.
- The husband's age is obtained by reversing the digits, so his age is 10b + a.
- The husband is older, which implies b > a.
- The difference between their ages is one-eleventh of the sum of their ages.

Concept / Approach:
We convert the word conditions into algebraic expressions involving a and b. Using the relationship between the difference and the sum of their ages, we form an equation and solve it for a and b. Once we know the digits, we can compute the husband's actual age by forming the reversed number 10b + a.

Step-by-Step Solution:
Step 1: Woman's age = 10a + b, husband's age = 10b + a. Step 2: Difference between their ages = (10b + a) − (10a + b) = 9(b − a). Step 3: Sum of their ages = (10b + a) + (10a + b) = 11(a + b). Step 4: The problem states that the difference is one-eleventh of the sum, so 9(b − a) = (1 / 11) × 11(a + b) = a + b. Step 5: This simplifies to 9b − 9a = a + b ⇒ 8b = 10a ⇒ 4b = 5a. Step 6: Both a and b are digits from 1 to 9 and b > a. The smallest integer solution for 4b = 5a is a = 4, b = 5. Step 7: Woman's age = 10a + b = 10 × 4 + 5 = 45 years. Step 8: Husband's age = 10b + a = 10 × 5 + 4 = 54 years.

Verification / Alternative check:
Check the condition using these ages. Sum of the ages = 45 + 54 = 99 years. Difference between the ages = 54 − 45 = 9 years. One-eleventh of the sum is 99 / 11 = 9 years, which equals the difference. The husband is older, and his age is 54 years, satisfying all conditions.

Why Other Options Are Wrong:
Ages such as 24, 42, 45, and 48 years do not satisfy the specific condition that the difference between the ages is one-eleventh of their sum when combined with a reversed two-digit age for the woman. Only 54 years as the husband's age yields a consistent pair of ages (45 and 54) that meets all requirements.

Common Pitfalls:
Some students may incorrectly set up the difference or the sum, or forget that both ages must be two-digit numbers with reversed digits. Others may attempt random trial and error without using the equation 4b = 5a, which makes the solution straightforward. Careful algebraic translation of the word problem simplifies the puzzle.

Final Answer:
The husband's age is 54 years.