Express `23.overline(43)` in the rational form.

Options

A. `(1)/(9)`
B. `(2320)/(99)`
C. `(14)/(85)`
D. `(2310)/(87)`

Answer

To express \( 23.\overline{43} \) in rational form, we can follow these steps:

### Step-by-Step Solution:

1. **Let \( x \) be the repeating decimal:**
\[ x = 23.43434343\ldots \]

2. **Multiply \( x \) by 100 to shift the decimal point two places to the right:**
\[ 100x = 2343.43434343\ldots \]

3. **Now, we have two equations:**
\[ \text{(1)} \quad x = 23.43434343\ldots \]
\[ \text{(2)} \quad 100x = 2343.43434343\ldots \]

4. **Subtract equation (1) from equation (2):**
\[ 100x - x = 2343.43434343\ldots - 23.43434343\ldots \]
This simplifies to:
\[ 99x = 2343 - 23 \]
\[ 99x = 2320 \]

5. **Now, solve for \( x \):**
\[ x = \frac{2320}{99} \]

6. **Thus, the rational form of \( 23.\overline{43} \) is:**
\[ \frac{2320}{99} \]

### Final Answer:
\[ 23.\overline{43} = \frac{2320}{99} \]