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 age of the woman is:

Options

A. 23 years
B. 34 years
C. 45 years
D. 56 years

Explanation

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Define Variables
Let the woman's age be represented as a two-digit number where:
- The unit digit is \( x \)
- The ten's digit is \( y \)

Thus, the woman's age can be expressed as:
\[ \text{Woman's Age} = 10y + x \]

### Step 2: Represent Husband's Age
According to the problem, if the woman's age is reversed, it represents her husband's age. Therefore, her husband's age can be expressed as:
\[ \text{Husband's Age} = 10x + y \]

### Step 3: Set Up Inequality
Since the husband is older than the woman, we have:
\[ 10x + y > 10y + x \]
This simplifies to:
\[ 9x > 9y \quad \Rightarrow \quad x > y \]

### Step 4: Set Up the Age Difference Equation
The problem states that the difference between their ages is one-eleventh of their sum. Therefore, we can write:
\[ (10x + y) - (10y + x) = \frac{1}{11} \left((10x + y) + (10y + x)\right) \]

### Step 5: Simplify the Equation
The left-hand side simplifies to:
\[ 10x + y - 10y - x = 9x - 9y = 9(x - y) \]
The right-hand side simplifies to:
\[ \frac{1}{11} (11x + 11y) = x + y \]
Thus, we have:
\[ 9(x - y) = x + y \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 9x - 9y = x + y \quad \Rightarrow \quad 9x - x = 9y + y \quad \Rightarrow \quad 8x = 10y \]
This simplifies to:
\[ \frac{x}{y} = \frac{10}{8} = \frac{5}{4} \]
This means:
\[ 4x = 5y \quad \Rightarrow \quad 4x - 5y = 0 \]

### Step 7: Finding Possible Values
Since \( x \) and \( y \) are digits (0-9), we can find integer solutions for \( x \) and \( y \):
- Rearranging gives \( y = \frac{4}{5}x \)
- Possible pairs of \( (x, y) \) that satisfy \( 4x = 5y \) and \( x > y \) can be checked.

### Step 8: Testing Values
1. If \( y = 4 \), then \( x = 5 \) (valid since \( 5 > 4 \)).
- Woman's Age = \( 10(4) + 5 = 45 \)
- Husband's Age = \( 10(5) + 4 = 54 \)
- Difference = \( 54 - 45 = 9 \)
- Sum = \( 54 + 45 = 99 \)
- Check: \( 9 = \frac{1}{11} \times 99 \) (True)

2. Testing other values like \( y = 3 \) gives \( x = 4 \) (not valid since \( 4 < 3 \)).

### Conclusion
The only valid solution is:
\[ \text{The age of the woman is } 45 \text{ years.} \]