`0 . 29bar(56)` when expressed as a vulgar fraction is

Options

A. `2956/1000`
B. `2956/10000`
C. `2927/9900`
D. `2900/9999`

Answer

To convert the repeating decimal \(0.29\overline{56}\) into a vulgar fraction, we can follow these steps:

### Step-by-Step Solution:

1. **Let \(x\) be the repeating decimal**:
\[ x = 0.29\overline{56} \]

2. **Multiply \(x\) by a power of 10 to move the decimal point**:
Since there are two digits before the repeating part (29) and two digits in the repeating part (56), we multiply by \(10000\) (which moves the decimal point four places to the right):
\[ 10000x = 2929.565656\ldots \]

3. **Multiply \(x\) by a power of 10 to isolate the repeating part**:
Now, we multiply \(x\) by \(100\) (which moves the decimal point two places to the right):
\[ 100x = 29.565656\ldots \]

4. **Set up the equation**:
Now we have two equations:
\[ 10000x = 2929.565656\ldots \quad \text{(1)} \]
\[ 100x = 29.565656\ldots \quad \text{(2)} \]

5. **Subtract equation (2) from equation (1)**:
\[ 10000x - 100x = 2929.565656\ldots - 29.565656\ldots \]
This simplifies to:
\[ 9900x = 2900 \]

6. **Solve for \(x\)**:
\[ x = \frac{2900}{9900} \]

7. **Simplify the fraction**:
To simplify \(\frac{2900}{9900}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is \(100\):
\[ x = \frac{29}{99} \]

### Final Answer:
Thus, the repeating decimal \(0.29\overline{56}\) expressed as a vulgar fraction is:
\[ \frac{29}{99} \]