Question

Convert 0.7¯¯¯ to a fraction.

Show your work.

Answer 1

To convert the repeating decimal \(0.7\overline{7}\) (which can be written as \(0.7777...\)) to a fraction, we will use algebraic manipulation. Here are the steps:

1. **Let \(x\) be equal to the repeating decimal**:
\[
x = 0.7777...
\]

2. **Multiply both sides of the equation by 10**. This shifts the decimal point one place to the right:
\[
10x = 7.7777...
\]

3. **Now, subtract the original equation (\(x\)) from this new equation (\(10x\))**:
\[
10x - x = 7.7777... - 0.7777...
\]

This simplifies to:
\[
9x = 7
\]

4. **Now solve for \(x\)**:
\[
x = \frac{7}{9}
\]

Thus, the repeating decimal \(0.7\overline{7}\) can be expressed as the fraction \(\frac{7}{9}\).