A woman says, "If you reverse my own age, the figures represent my husband's age. He is, of course, senior to me and the difference between our ages is one-eleventh of their sum." The woman's husband age is ?

Options

A. A. 45
B. B. 54
C. C. 56
D. D. 66

Answer

To solve the problem, we need to determine the ages of the woman and her husband based on the clues given. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The woman says that if you reverse her age, it will give her husband's age.
- The husband is older than the woman.
- The difference between their ages is one-eleventh of the sum of their ages.

2. **Setting Up Variables**:
- Let the woman's age be represented as \( W \).
- Let the husband's age be represented as \( H \).
- The relationship between their ages can be expressed as:
\[ H = \text{reverse}(W) \]

3. **Using the Age Difference**:
- According to the problem, the difference between their ages is:
\[ H - W = \frac{1}{11}(H + W) \]

4. **Rearranging the Age Difference Equation**:
- Multiply both sides by 11 to eliminate the fraction:
\[ 11(H - W) = H + W \]
- Expanding this gives:
\[ 11H - 11W = H + W \]
- Rearranging terms leads to:
\[ 10H = 12W \]
- Simplifying gives:
\[ H = \frac{12}{10}W = \frac{6}{5}W \]

5. **Finding Possible Ages**:
- Since \( H \) must be a two-digit number and \( W \) must also be a two-digit number, we can try possible values for \( W \) and calculate \( H \) accordingly.
- The possible ages for \( W \) could be 45, 46, 47, ..., 54 (since reversing must yield a valid two-digit number).

6. **Testing Values**:
- Let's start with \( W = 45 \):
- Reverse of 45 is 54, so \( H = 54 \).
- Difference: \( 54 - 45 = 9 \).
- Sum: \( 54 + 45 = 99 \).
- Check if \( 9 = \frac{1}{11}(99) \):
- \( \frac{99}{11} = 9 \) (True).
- Now check \( W = 54 \):
- Reverse of 54 is 45, so \( H = 45 \) (not valid since \( H \) must be greater).

7. **Conclusion**:
- The only valid pair that satisfies all conditions is \( W = 45 \) and \( H = 54 \).

### Final Answer:
The woman's husband is **54 years old**.