To convert the repeating decimal [tex]\(3.\overline{2}\)[/tex] into a simplified fraction, follow these detailed steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 3.2222\ldots
\][/tex]
2. Multiply [tex]\( x \)[/tex] by 10 to shift the decimal point to the right, matching the repeating part:
[tex]\[
10x = 32.2222\ldots
\][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation to eliminate the repeating part:
[tex]\[
10x - x = 32.2222\ldots - 3.2222\ldots
\][/tex]
[tex]\[
9x = 29
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{29}{9}
\][/tex]
5. Simplify the fraction if possible. In this case, [tex]\(\frac{29}{9}\)[/tex] is already in its simplest form since 29 is a prime number and does not have any common factors with 9.
Therefore, the repeating decimal [tex]\(3.\overline{2}\)[/tex] as a simplified fraction is:
[tex]\[
\boxed{\frac{29}{9}}
\][/tex]
