To convert the repeating decimal \( 0.27\overline{27} \) to a rational number, we can use the following method:
Let \( x = 0.272727...\).
Then, to eliminate the repeating part, we can multiply by 100 (since two digits are repeating):
\[ 100x = 27.272727... \]
Now, we have:
\[ x = 0.272727... \] \[ 100x = 27.272727... \]
Next, we will subtract the first equation from the second:
\[ 100x - x = 27.272727... - 0.272727... \] \[ 99x = 27 \]
Now, solve for \( x \):
\[ x = \frac{27}{99} \]
Next, we simplify \( \frac{27}{99} \):
Both numbers can be divided by 9:
\[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, the repeating decimal \( 0.27\overline{27} \) can be expressed as the rational number:
\[ \frac{3}{11} \]
So, the correct response is:
\( \frac{3}{11} \)