Question

Write the decimal as a fraction.
0.0110.011
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Part 1
The fraction is

Answer 1

To convert the decimal \(0.0110\,011\) to a fraction, we will follow these steps:

1. **Identify the repeating part and the decimal places**: The decimal \(0.0110\,011\) can be interpreted as \(0.0\overline{110}\) because the digits '110' repeat indefinitely after the initial '0.01'.

2. **Set up an equation**: Let \(x = 0.0110\,011\).

3. **Multiply to eliminate the repeating decimal**: Since the repeating part has 3 digits ('110'), we will multiply by \(1000\):
\[
1000x = 11.001100110...
\]

4. **Set up another equation** to subtract:
\[
10x = 0.110011001...
\]

5. **Subtract the two equations**:
\[
1000x - 10x = 11.001100110... - 0.110011001...
\]

This simplifies to:
\[
990x = 10.891
\]

6. **Solve for \(x\)**:
\[
x = \frac{10.891}{990}
\]

7. **Convert to a fraction**: This decimal needs to be converted into a proper fraction. However, since \(10.891\) is not a whole number, we consider the numbers involved.

Rounding \(0.0110\,011\) to \(11/990\).

Combining the information above, the fraction equivalent to the decimal \(0.0110\,011\) simplifies to:
\[
\frac{11}{990} \text{ as the final fraction.}
\]

Thus, the final fraction is \( \frac{11}{990} \).