Let's solve the problem of writing [tex]\(0.\overline{12}\)[/tex] as a fraction step-by-step.
1. Let [tex]\( x = 0.\overline{12} \)[/tex]. This means [tex]\( x \)[/tex] is the repeating decimal [tex]\( 0.121212\ldots \)[/tex].
[tex]\[
x = 0.\overline{12}
\][/tex]
2. Multiply both sides of the equation by 100 to shift the decimal point two places to the right.
[tex]\[
100x = 12.\overline{12}
\][/tex]
3. Now we have two equations:
[tex]\[
x = 0.\overline{12}
\][/tex]
[tex]\[
100x = 12.\overline{12}
\][/tex]
4. Subtract the first equation from the second equation to eliminate the repeating part.
[tex]\[
100x - x = 12.\overline{12} - 0.\overline{12}
\][/tex]
5. Simplify the left-hand side and the right-hand side of the equation.
[tex]\[
99x = 12
\][/tex]
6. Solve for [tex]\( x \)[/tex] by dividing both sides by 99.
[tex]\[
x = \frac{12}{99}
\][/tex]
7. Simplify the fraction [tex]\(\frac{12}{99}\)[/tex] by finding the greatest common divisor (GCD) of 12 and 99. The GCD of 12 and 99 is 3.
[tex]\[
\frac{12 \div 3}{99 \div 3} = \frac{4}{33}
\][/tex]
So [tex]\( 0.\overline{12} \)[/tex] is equal to [tex]\( \frac{4}{33} \)[/tex].
