To convert the repeating decimal [tex]\( 0.\overline{27} \)[/tex] into a fraction in the form [tex]\(\frac{p}{q}\)[/tex], follow these steps:
1. Let [tex]\( x = 0.\overline{27} \)[/tex]:
[tex]\[
x = 0.27272727\ldots
\][/tex]
2. Multiply both sides of the equation by 100 to shift the decimal point two places to the right:
[tex]\[
100x = 27.27272727\ldots
\][/tex]
3. Now, observe that [tex]\( x = 0.27272727\ldots \)[/tex] and [tex]\( 100x = 27.27272727\ldots \)[/tex]. Subtract the original equation (step 1) from this new equation (step 2):
[tex]\[
100x - x = 27.27272727\ldots - 0.27272727\ldots
\][/tex]
[tex]\[
99x = 27
\][/tex]
4. Solve for [tex]\( x \)[/tex] by dividing both sides by 99:
[tex]\[
x = \frac{27}{99}
\][/tex]
5. Simplify the fraction [tex]\(\frac{27}{99}\)[/tex]. The greatest common divisor (GCD) of 27 and 99 is 9. Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{27 \div 9}{99 \div 9} = \frac{3}{11}
\][/tex]
Therefore, the fraction form of the repeating decimal [tex]\( 0.\overline{27} \)[/tex] is:
[tex]\[
\frac{3}{11}
\][/tex]
So, [tex]\( \frac{p}{q} \)[/tex] form of [tex]\( 0.\overline{27} \)[/tex] is [tex]\( \frac{3}{11} \)[/tex].
