To convert the repeating decimal [tex]\( 3.2\overline{4} \)[/tex] into a fraction, follow these steps:
1. Let [tex]\( x = 3.244444\ldots \)[/tex]
2. Multiply both sides of this equation by 10 to shift the repeating part:
[tex]\[
10x = 32.444444\ldots
\][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation to eliminate the repeating part:
[tex]\[
10x - x = 32.444444\ldots - 3.244444\ldots
\][/tex]
[tex]\[
9x = 29.2
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{29.2}{9}
\][/tex]
5. Convert the decimal in the numerator to a fraction. Note that [tex]\( 29.2 \)[/tex] can be written as [tex]\( \frac{292}{10} \)[/tex]:
[tex]\[
x = \frac{292}{10 \cdot 9}
\][/tex]
[tex]\[
x = \frac{292}{90}
\][/tex]
6. Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (2):
[tex]\[
\frac{292 \div 2}{90 \div 2} = \frac{146}{45}
\][/tex]
Hence, the simplified fraction for the repeating decimal [tex]\( 3.2\overline{4} \)[/tex] is:
[tex]\[
\frac{146}{45}
\][/tex]
