Representation of `0.bar(2341)` in the form `p //q`, where p and q are integers, `q cancel(=)0`, is

Options

A. `( 781)/( 3330)`
B. `( 1171)/( 4995)`
C. `( 2341)/( 9999)`
D. `( 2339)/( 9990)`

Answer

To represent the repeating decimal \(0.\overline{2341}\) in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the repeating part**: The repeating decimal is \(0.\overline{2341}\). Here, "2341" is the part that repeats.

2. **Set up the equation**: Let \(x = 0.\overline{2341}\).

3. **Multiply by a power of 10**: Since "2341" has 4 digits, multiply \(x\) by \(10^4\) (which is 10000) to shift the decimal point 4 places to the right:
\[ 10000x = 2341.\overline{2341} \]

4. **Subtract the original equation from this new equation**:
\[ 10000x - x = 2341.\overline{2341} - 0.\overline{2341} \]
This simplifies to:
\[ 9999x = 2341 \]

5. **Solve for \(x\)**: Divide both sides by 9999:
\[ x = \frac{2341}{9999} \]

6. **Final representation**: Thus, the representation of \(0.\overline{2341}\) in the form \(\frac{p}{q}\) is:
\[ \frac{2341}{9999} \]

### Conclusion:
The final answer is \(\frac{2341}{9999}\).