Explore a wide range of topics and get answers from experts on IDNLearn.com. Join our knowledgeable community and get detailed, reliable answers to all your questions.

Write the recurring decimal [tex]$0.\dot{8}$[/tex] as a fraction.

[tex]\[
\begin{aligned}
&\text{Let } x = 0.\dot{8} \\
&\text{Then, } 10x = 8.\dot{8} \\
&\text{Subtracting these equations:} \\
&10x - x = 8.\dot{8} - 0.\dot{8} \\
&9x = 8 \\
&x = \frac{8}{9}
\end{aligned}
\][/tex]


Sagot :

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].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.