To determine the fraction equivalent to the repeating decimal [tex]\(0.72727272\ldots\)[/tex], let's go through the steps to convert it to a fraction.
1. Let [tex]\( x \)[/tex] be the repeating decimal. So, [tex]\( x = 0.72727272\ldots \)[/tex].
2. To eliminate the repeating part, multiply [tex]\( x \)[/tex] by 100 (since the repeating block is two digits long):
[tex]\[
100x = 72.72727272\ldots
\][/tex]
3. Now, we have two equations:
[tex]\[
x = 0.72727272\ldots \quad \text{(Equation 1)}
\][/tex]
[tex]\[
100x = 72.72727272\ldots \quad \text{(Equation 2)}
\][/tex]
4. Subtract Equation 1 from Equation 2 to eliminate the repeating decimals:
[tex]\[
100x - x = 72.72727272\ldots - 0.72727272\ldots
\][/tex]
[tex]\[
99x = 72
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 99:
[tex]\[
x = \frac{72}{99}
\][/tex]
6. Now, we simplify the fraction [tex]\(\frac{72}{99}\)[/tex]:
- Find the greatest common divisor (GCD) of 72 and 99. The GCD is 9.
- Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\][/tex]
Therefore, the equivalent fraction of the non-terminating and repeating decimal [tex]\(0.72727272\ldots\)[/tex] is [tex]\(\frac{8}{11}\)[/tex].
So, the correct answer is [tex]\(\frac{8}{11}\)[/tex].
