Convert `0.bar(23)` to a vulgar fraction.

Answer

To convert the repeating decimal \(0.\overline{23}\) into a vulgar fraction, we can follow these steps:

### Step-by-Step Solution:

1. **Let \(x\) be the repeating decimal**:
\[ x = 0.\overline{23} \]

2. **Multiply both sides by 100**:
Since the repeating part (23) has 2 digits, we multiply by \(100\):
\[ 100x = 23.\overline{23} \]

3. **Rewrite the equation**:
Now, we can express \(23.\overline{23}\) as:
\[ 100x = 23 + 0.\overline{23} \]
This means:
\[ 100x = 23 + x \]

4. **Subtract \(x\) from both sides**:
To isolate \(x\), we subtract \(x\) from both sides:
\[ 100x - x = 23 \]
Simplifying this gives:
\[ 99x = 23 \]

5. **Solve for \(x\)**:
Now, divide both sides by \(99\):
\[ x = \frac{23}{99} \]

### Final Answer:
Thus, the repeating decimal \(0.\overline{23}\) can be expressed as the vulgar fraction:
\[ \frac{23}{99} \]