Let \( x = 0.36\overline{36} \). This means that \( x \) is a repeating decimal where "36" repeats indefinitely.
To eliminate the repeating part, we can multiply \( x \) by 100 (since there are 2 digits in the repeating part):
\[ 100x = 36.36\overline{36} \]
Next, we can set up the following equation:
\[ 100x = 36.36\overline{36} \ x = 0.36\overline{36} \]
Now, we subtract the second equation from the first:
\[ 100x - x = 36.36\overline{36} - 0.36\overline{36} \]
This simplifies to:
\[ 99x = 36 \]
Now, we solve for \( x \) by dividing both sides by 99:
\[ x = \frac{36}{99} \]
Next, we simplify the fraction \( \frac{36}{99} \). We can find the greatest common divisor (GCD) of 36 and 99, which is 9.
Now, we divide the numerator and the denominator by their GCD:
\[ \frac{36 \div 9}{99 \div 9} = \frac{4}{11} \]
Thus, the simplified form of \( 0.36\overline{36} \) as a fraction is:
\[ \frac{4}{11} \]