To convert the repeating decimal [tex]\(0.2\overline{3}\)[/tex] to a fraction, we will use algebraic manipulation. Follow these steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 0.23333\ldots
\][/tex]
2. Multiply [tex]\( x \)[/tex] by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 2.3333\ldots
\][/tex]
3. Now multiply [tex]\( x \)[/tex] by 100 to shift the repeating part:
[tex]\[
100x = 23.3333\ldots
\][/tex]
4. Subtract the equation [tex]\( 10x = 2.3333\ldots \)[/tex] from the equation [tex]\( 100x = 23.3333\ldots \)[/tex]:
[tex]\[
\begin{aligned}
100x - 10x &= 23.3333\ldots - 2.3333\ldots \\
90x &= 21
\end{aligned}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{21}{90}
\][/tex]
6. Simplify the fraction [tex]\(\frac{21}{90}\)[/tex]:
Both 21 and 90 can be divided by their greatest common divisor (GCD), which is 3:
[tex]\[
\frac{21 \div 3}{90 \div 3} = \frac{7}{30}
\][/tex]
Hence, the repeating decimal [tex]\( 0.2\overline{3} \)[/tex] converted to a fraction is:
[tex]\[
\boxed{\frac{7}{30}}
\][/tex]
