Question

Convert 0.8¯¯¯
to a rational number.(1 point)

Answer 1

To convert the repeating decimal \( 0.8\overline{8} \) (or \( 0.8888\ldots \)) to a rational number, we can use the following steps:

1. Let \( x = 0.8888\ldots \).

2. Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
\[
10x = 8.8888\ldots
\]

3. Now, we have two equations:
\[
x = 0.8888\ldots \quad (1)
\]
\[
10x = 8.8888\ldots \quad (2)
\]

4. Subtract equation (1) from equation (2):
\[
10x - x = 8.8888\ldots - 0.8888\ldots
\]
\[
9x = 8
\]

5. Solve for \( x \):
\[
x = \frac{8}{9}
\]

Thus, the rational number representation of \( 0.8\overline{8} \) is
\[
\boxed{\frac{8}{9}}.
\]