Convert 0.8¯¯¯

Question

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