Question

Convert 0.8¯¯¯
to a rational number.

Answer 1

To convert the repeating decimal \( 0.8\overline{3} \) (where the '3' repeats indefinitely) into a rational number, we can follow these steps:

1. Let \( x = 0.8333\ldots \).
2. To eliminate the repeating part, multiply \( x \) by 10:
\[
10x = 8.3333\ldots
\]
3. Now, subtract the original \( x \) from this new equation:
\[
10x - x = 8.3333\ldots - 0.8333\ldots
\]
This simplifies to:
\[
9x = 8
\]
4. Now, solve for \( x \):
\[
x = \frac{8}{9}
\]

Thus, the repeating decimal \( 0.8\overline{3} \) converts to the rational number \( \frac{8}{9} \).