$0.23535 ....... = 0.2\overline{35}$ can be written as:

Q: $0.23535 ....... = 0.2\overline{35}$ can be written as:

To understand why Option 3, $\frac{233}{990}$, is the correct representation of the repeating decimal $0.2\overline{35}$, let's break down the conversion process from a repeating decimal to a fraction. ### Explanation for Option 3: 1. **Identify the repeating part**: The decimal $0.2\overline{35}$ indicates that the digits "35" repeat indefinitely after the initial "2". This can be expressed as $0.2353535...$. 2. **Set up the equation**: Let $x = 0.2353535...$. 3. **Eliminate the repeating part**: - Multiply $x$ by 1000 to shift the decimal point three places to the right: \[ 1000x = 235.353535... \] - Multiply $x$ by 10 to shift the decimal point one place to the right: \[ 10x = 2.353535... \] 4. **Subtract the two equations**: \[ 1000x - 10x = 235.353535... - 2.353535... \] \[ 990x = 233 \] 5. **Solve for $x$**: \[ x = \frac{233}{990} \] Thus, the repeating decimal $0.2\overline{35}$ is correctly represented by the fraction $\frac{233}{990}$, which is Option 3. ### Explanation for why other options are incorrect: 1. **Option 1: $\frac{235}{100}$**: - This fraction represents the decimal $2.35$, which is not a repeating decimal and does not match $0.2\overline{35}$. 2. **Option 2: $\frac{234}{990}$**: - Simplifying $\frac{234}{990}$ does not yield the repeating decimal $0.2\overline{35}$. Instead, it represents a different decimal value. 3. **Option 4: $\frac{234}{1000}$**: - This fraction represents the decimal $0.234$, which is a terminating decimal and does not match the repeating decimal $0.2\overline{35}$. In conclusion, Option 3, $\frac{233}{990}$, is the only fraction that accurately represents the repeating decimal $0.2\overline{35}$.

Question

Accepted Answer

To understand why Option 3, $\frac{233}{990}$, is the correct representation of the repeating decimal $0.2\overline{35}$, let's break down the conversion process from a repeating decimal to a fraction.

### Explanation for Option 3:

1. **Identify the repeating part**: The decimal $0.2\overline{35}$ indicates that the digits "35" repeat indefinitely after the initial "2". This can be expressed as $0.2353535...$.

2. **Set up the equation**: Let $x = 0.2353535...$.

3. **Eliminate the repeating part**:
   - Multiply $x$ by 1000 to shift the decimal point three places to the right: 
     $$
     1000x = 235.353535...
     $$
   - Multiply $x$ by 10 to shift the decimal point one place to the right:
     $$
     10x = 2.353535...
     $$

4. **Subtract the two equations**:
   $$
   1000x - 10x = 235.353535... - 2.353535...
   $$
   $$
   990x = 233
   $$

5. **Solve for $x$**:
   $$
   x = \frac{233}{990}
   $$

Thus, the repeating decimal $0.2\overline{35}$ is correctly represented by the fraction $\frac{233}{990}$, which is Option 3.

### Explanation for why other options are incorrect:

1. **Option 1: $\frac{235}{100}$**:
   - This fraction represents the decimal $2.35$, which is not a repeating decimal and does not match $0.2\overline{35}$.

2. **Option 2: $\frac{234}{990}$**:
   - Simplifying $\frac{234}{990}$ does not yield the repeating decimal $0.2\overline{35}$. Instead, it represents a different decimal value.

3. **Option 4: $\frac{234}{1000}$**:
   - This fraction represents the decimal $0.234$, which is a terminating decimal and does not match the repeating decimal $0.2\overline{35}$.

In conclusion, Option 3, $\frac{233}{990}$, is the only fraction that accurately represents the repeating decimal $0.2\overline{35}$.