To express [tex]\( 0.13\overline{4} \)[/tex] as a fraction [tex]\( \frac{p}{q} \)[/tex]:
1. Let [tex]\( x = 0.13\overline{4} \)[/tex]. This means [tex]\( x \)[/tex] is a decimal with the repeating part "4".
2. To eliminate the repeating decimal, we start by multiplying [tex]\( x \)[/tex] by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 1.3444\overline{4}
\][/tex]
3. To isolate the repeating part, we then multiply [tex]\( x \)[/tex] by 100 to shift the repeating section again:
[tex]\[
100x = 13.4444\overline{4}
\][/tex]
4. Subtract the first equation from the second equation to get rid of the repeating part:
[tex]\[
100x - 10x = 13.4444\overline{4} - 1.3444\overline{4}
\][/tex]
[tex]\[
90x = 12.10
\][/tex]
5. Now solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 90:
[tex]\[
x = \frac{12.10}{90}
\][/tex]
6. To simplify this fraction, notice that it can be written as:
[tex]\[
x = \frac{1210}{9000}
\][/tex]
7. Now, simplify this fraction by finding the greatest common divisor (GCD) of 1210 and 9000 and divide both the numerator and denominator by this GCD.
Upon simplification, you will get:
[tex]\[
x = \frac{112}{825}
\][/tex]
Therefore, the fraction representation of [tex]\( 0.13\overline{4} \)[/tex] is:
[tex]\[
\frac{112}{825}
\][/tex]
