In this problem, we need to convert $0.\overline{7}$ (a repeating decimal) into a fraction by solving an equation.

Step-by-Step Solution:

1. Let $x = 0.\overline{7}$. This repeating decimal is the same as writing $0.77777...$ with $7$ repeating indefinitely.

2. Multiply both sides of the equation $x = 0.\overline{7}$ by 10 to shift the decimal point: $10x = 7.\overline{7}$

3. Now we have two equations: $x = 0.\overline{7}$ $10x = 7.\overline{7}$

4. Subtract the first equation from the second: $10x - x = 7.\overline{7} - 0.\overline{7}$ $9x = 7$

5. Divide both sides by 9 to isolate $x$: $x = \frac{7}{9}$

Therefore, $0.\overline{7}$ can be written as the fraction $\frac{7}{9}$.

Would you like further details on any of the steps or have any questions?

Additional Questions for Further Exploration

1. How can repeating decimals other than $0.\overline{7}$ be converted to fractions?
2. What is the difference in approach for converting non-repeating decimals to fractions?
3. Can you use this method to convert $0.\overline{3}$ to a fraction?
4. Why does subtracting the equations help in isolating $x$?
5. How does this approach apply to repeating decimals with more than one digit, like $0.\overline{23}$?

Tip

When converting a repeating decimal to a fraction, multiply by a power of 10 that matches the number of repeating digits. This helps align the decimal parts for subtraction.