Convert 0.8¯¯¯

Question

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 \( \frac{8}{9} \).