To write the recurring decimal [tex]\( 0.\overline{8} \)[/tex] as a fraction, follow these steps:
1. Define the recurring decimal:
Let [tex]\( x = 0.\overline{8} \)[/tex]. This means [tex]\( x = 0.8888\ldots \)[/tex], where the digit '8' repeats indefinitely.
2. Multiply both sides by 10:
Multiply the equation [tex]\( x = 0.\overline{8} \)[/tex] by 10:
[tex]\[
10x = 8.8888\ldots
\][/tex]
3. Set up an equation to eliminate the repeating part:
We now have two equations:
[tex]\[
x = 0.8888\ldots \quad \text{(Equation 1)}
\][/tex]
[tex]\[
10x = 8.8888\ldots \quad \text{(Equation 2)}
\][/tex]
4. Subtract Equation 1 from Equation 2:
By subtracting Equation 1 from Equation 2, we eliminate the repeating part:
[tex]\[
10x - x = 8.8888\ldots - 0.8888\ldots
\][/tex]
Simplifying, we get:
[tex]\[
9x = 8
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by 9:
[tex]\[
x = \frac{8}{9}
\][/tex]
Therefore, the recurring decimal [tex]\( 0.\overline{8} \)[/tex] as a fraction is [tex]\( \frac{8}{9} \)[/tex].
