Question

Convert 0.8¯¯¯
0.
8
¯
to a rational number.

Answer 1

To convert the repeating decimal \(0.8\overline{8}\) (which means \(0.8888\ldots\)) to a rational number, we can follow these steps:

Let \(x = 0.8888\ldots\).

To eliminate the repeating part, we can multiply both sides of the equation by 10:

\[
10x = 8.8888\ldots
\]

Now we have two equations:

1. \(x = 0.8888\ldots\)
2. \(10x = 8.8888\ldots\)

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

\[
10x - x = 8.8888\ldots - 0.8888\ldots
\]

This simplifies to:

\[
9x = 8
\]

Now, solving for \(x\) gives:

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

Thus, the rational number representation of the repeating decimal \(0.8\overline{8}\) is:

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