`2.bar8768` is equal to

Options

A. `2(4394)/(4995)`
B. `2(292)/(333)`
C. `2(9)/(10)`
D. `2(878)/(999)`

Answer

To solve the problem of finding the value of \( 2.\overline{8768} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the notation**: The notation \( 2.\overline{8768} \) means that the digits "8768" repeat indefinitely. Therefore, we can express this number as:
\[ 2.\overline{8768} = 2 + 0.\overline{8768} \]

2. **Letting \( x \) represent the repeating decimal**: We can set:
\[ x = 0.\overline{8768} \]

3. **Multiplying by a power of 10**: To eliminate the repeating part, we multiply \( x \) by \( 10,000 \) (since "8768" has 4 digits):
\[ 10,000x = 8768.\overline{8768} \]

4. **Setting up the equation**: Now, we can write the equation:
\[ 10,000x = 8768 + x \]

5. **Rearranging the equation**: We can subtract \( x \) from both sides:
\[ 10,000x - x = 8768 \]
\[ 9999x = 8768 \]

6. **Solving for \( x \)**: Now, divide both sides by 9999:
\[ x = \frac{8768}{9999} \]

7. **Adding back to the whole number**: Now, we can substitute back to find \( 2.\overline{8768} \):
\[ 2.\overline{8768} = 2 + \frac{8768}{9999} \]

8. **Combining the terms**: To express this as a single fraction, we can write:
\[ 2 = \frac{2 \times 9999}{9999} = \frac{19998}{9999} \]
Therefore,
\[ 2.\overline{8768} = \frac{19998 + 8768}{9999} = \frac{28766}{9999} \]

9. **Final answer**: Thus, the value of \( 2.\overline{8768} \) is:
\[ \frac{28766}{9999} \]