Question

Convert the following repeating decimal to a fraction in simplest form.
., start overline, 8, end overline
.
8

Answer 1

Let \( x \) be the repeating decimal \( 0.\overline{8} \).

This means that:

\[
x = 0.88888\ldots
\]

To convert this repeating decimal to a fraction, we can set up the following equation:

\[
10x = 8.88888\ldots
\]

Now, we have two equations:

1. \( x = 0.88888\ldots \)
2. \( 10x = 8.88888\ldots \)

Next, we can subtract the first equation from the second:

\[
10x - x = 8.88888\ldots - 0.88888\ldots
\]

This simplifies to:

\[
9x = 8
\]

Now, divide both sides by 9:

\[
x = \frac{8}{9}
\]

Thus, the repeating decimal \( 0.\overline{8} \) is equal to the fraction \( \frac{8}{9} \).

To ensure this is in simplest form, we note that 8 and 9 have no common factors other than 1. Therefore, the fraction \( \frac{8}{9} \) is already in simplest form.

The final answer is:

\[
\boxed{\frac{8}{9}}
\]