To convert \( 0.27\overline{27} \) (where the "27" repeats indefinitely) to a rational number, we can use a method involving algebra.
Let \( x = 0.272727...\).
To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 27.272727... \]
Now, we can subtract the original \( x \) from this equation:
\[ 100x - x = 27.272727... - 0.272727... \] \[ 99x = 27 \]
Now, solve for \( x \):
\[ x = \frac{27}{99} \]
Next, we simplify \( \frac{27}{99} \):
Both 27 and 99 can be divided by 9:
\[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, the rational number in simplest form is:
\[ \frac{3}{11} \]
So, the correct response should be \( \frac{3}{11} \).