Convert the following repeating decimal to a fraction in simplest form. \[ .0 \overline{7} \]

To convert the repeating decimal \( 0.\overline{7} \) to a fraction, we can follow these steps: 1. **Let \( x \) be the repeating decimal:** \[ x = 0.77777\ldots \] 2. **Multiply both sides by 10 to shift the decimal point:** \[ 10x = 7.77777\ldots \] 3. **Now, we have two equations:** - \( x = 0.77777\ldots \) - \( 10x = 7.77777\ldots \) 4. **Subtract the first equation from the second:** \[ 10x - x = 7.77777\ldots - 0.77777\ldots \] This simplifies to: \[ 9x = 7 \] 5. **Solve for \( x \):** \[ x = \frac{7}{9} \] Thus, the repeating decimal \( 0.\overline{7} \) can be expressed as the fraction \( \frac{7}{9} \) in simplest form.