0.23535. _____ = 0.235 can be written as:

Options

A. `235/100`
B. `234/990`
C. `233/990`
D. `234/1000`

Answer

To solve the equation \( 0.23535\ldots = 0.235 \), we need to express \( 0.23535\ldots \) in a more manageable form.

### Step-by-Step Solution:

1. **Identify the repeating decimal**: The number \( 0.23535\ldots \) can be expressed as \( 0.235\overline{35} \), where \( 35 \) is the repeating part.

2. **Let \( x \) be the repeating decimal**:
\[ x = 0.235353535\ldots \]

3. **Multiply by a power of 10**: To eliminate the repeating part, we can multiply \( x \) by \( 1000 \) (since the non-repeating part has 3 digits):
\[ 1000x = 235.353535\ldots \]

4. **Set up the equation**: Now, we can set up the equation:
\[ 1000x = 235 + x \]

5. **Subtract \( x \) from both sides**:
\[ 1000x - x = 235 \]
\[ 999x = 235 \]

6. **Solve for \( x \)**:
\[ x = \frac{235}{999} \]

7. **Convert the fraction to decimal**: Now, we can convert \( \frac{235}{999} \) back to decimal form. Performing the division gives us:
\[ x \approx 0.235235235\ldots \]

8. **Conclusion**: Therefore, we can conclude that:
\[ 0.23535\ldots = \frac{235}{999} \]
and it can also be approximated as \( 0.235 \) when rounded to three decimal places.