Express 0.23535 ........ in the form of p/q :

Question

Accepted Answer

To express the repeating decimal 0.23535... in the form of a fraction $ \frac{p}{q} $, we need to convert the repeating decimal into a fraction. The given correct answer is Option 1: $ \frac{233}{990} $. Let's justify why this is correct and why the other options are incorrect.

### Explanation for Option 1: $ \frac{233}{990} $

1. **Identify the repeating part**: The decimal 0.23535... has a repeating part of "3535".
2. **Set up the equation**: Let $ x = 0.2353535...$.
3. **Multiply to eliminate the repeating part**: Multiply both sides by 10000 (since the repeating part is 4 digits long) to shift the decimal point:
   $$
   10000x = 2353.53535...
   $$
4. **Subtract to eliminate the repeating part**:
   $$
   10000x - 100x = 2353.53535... - 23.53535...
   $$
   $$
   9900x = 2330
   $$
5. **Solve for $ x $**:
   $$
   x = \frac{2330}{9900}
   $$
6. **Simplify the fraction**:
   $$
   x = \frac{233}{990}
   $$

Thus, the repeating decimal 0.23535... is correctly expressed as $ \frac{233}{990} $, which matches Option 1.

### Explanation for why the other options are incorrect:

- **Option 2: $ \frac{235}{99} $**
  - This fraction does not correctly represent the repeating decimal 0.23535... because the denominator does not account for the four repeating digits. It would be more appropriate for a repeating decimal with two repeating digits.

- **Option 3: $ \frac{2.35}{9.9} $**
  - This is not a valid fraction representation for a repeating decimal. It suggests a division of two decimals rather than a fraction form of a repeating decimal.

- **Option 4: $ \frac{235}{999} $**
  - This fraction would be suitable for a repeating decimal with three repeating digits, but 0.23535... has four repeating digits. Therefore, it does not accurately represent the given repeating decimal.

In conclusion, Option 1: $ \frac{233}{990} $ is the correct representation of the repeating decimal 0.23535..., as it accurately accounts for the repeating sequence and simplifies correctly.