To determine which fraction represents the repeating decimal [tex]\(0.\overline{12}\)[/tex], we'll follow a systematic method to convert it to a fraction.
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 0.121212\ldots
\][/tex]
2. Multiply both sides of this equation by 100 to shift the decimal point two places to the right (since the repeating block is two digits long):
[tex]\[
100x = 12.121212\ldots
\][/tex]
3. Subtract the original [tex]\( x = 0.121212\ldots \)[/tex] from this new equation:
[tex]\[
100x - x = 12.121212\ldots - 0.121212\ldots
\][/tex]
[tex]\[
99x = 12
\][/tex]
4. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 99:
[tex]\[
x = \frac{12}{99}
\][/tex]
5. Simplify the fraction [tex]\( \frac{12}{99} \)[/tex]:
To do this, find the greatest common divisor (GCD) of 12 and 99, which is 3:
[tex]\[
\frac{12 \div 3}{99 \div 3} = \frac{4}{33}
\][/tex]
So, the fraction that represents the repeating decimal [tex]\(0.\overline{12}\)[/tex] is [tex]\( \frac{4}{33} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\frac{4}{33}}
\][/tex]
