Certainly! Let's convert the repeating decimal [tex]\(0.\overline{2}\)[/tex] into a fraction step-by-step.
1. Let [tex]\( x = 0.\overline{2} \)[/tex]
2. Write the repeating decimal as an equation:
[tex]\[
x = 0.2222\ldots
\][/tex]
3. Multiply both sides of this equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 2.2222\ldots
\][/tex]
4. Now you have two equations:
[tex]\[
\begin{aligned}
x &= 0.2222\ldots \quad \quad \quad (1) \\
10x &= 2.2222\ldots \quad \quad \,\, (2)
\end{aligned}
\][/tex]
5. Subtract the first equation from the second equation:
[tex]\[
10x - x = 2.2222\ldots - 0.2222\ldots
\][/tex]
6. This simplifies to:
[tex]\[
9x = 2
\][/tex]
7. Solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{2}{9}
\][/tex]
So, the repeating decimal [tex]\( 0.\overline{2} \)[/tex] is equal to [tex]\(\frac{2}{9}\)[/tex] as a fraction.
