To determine which fraction is equivalent to the repeating decimal [tex]\( 0.\overline{2} \)[/tex], we need to express the repeating decimal as a fraction. Here is the step-by-step process:
1. Set up the equation:
Let [tex]\( x = 0.\overline{2} \)[/tex]. This means that [tex]\( x \)[/tex] is equal to 0.2222... continuing indefinitely.
2. Multiply by 10:
Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 2.2222...
\][/tex]
Now we have two equations:
[tex]\[
x = 0.2222...
\][/tex]
[tex]\[
10x = 2.2222...
\][/tex]
3. Subtract the original equation from the multiplied equation:
Subtract the first equation from the second equation to eliminate the repeating decimal:
[tex]\[
10x - x = 2.2222... - 0.2222...
\][/tex]
Simplifying the left side, we get:
[tex]\[
9x = 2
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides of the equation by 9:
[tex]\[
x = \frac{2}{9}
\][/tex]
Thus, the fraction that is equivalent to the repeating decimal [tex]\( 0.\overline{2} \)[/tex] is [tex]\( \frac{2}{9} \)[/tex].
Looking at the given answer choices, we see that the correct answer is:
A) [tex]\(\frac{2}{9}\)[/tex]
