To convert the repeating decimal \(0.27\overline{27}\) to a rational number, we will denote it as \(x\).
Let:
\[ x = 0.27272727\ldots \]
To eliminate the repeating decimal, we can multiply \(x\) by \(100\) (since the repeating part has two digits):
\[ 100x = 27.27272727\ldots \]
Now we can set up a system of equations. We have:
- \(x = 0.27272727\ldots\)
- \(100x = 27.27272727\ldots\)
Next, we subtract the first equation from the second:
\[ 100x - x = 27.27272727\ldots - 0.27272727\ldots \]
This simplifies to:
\[ 99x = 27 \]
Now, we can solve for \(x\):
\[ x = \frac{27}{99} \]
Now we should simplify the fraction \(\frac{27}{99}\):
The greatest common divisor (GCD) of \(27\) and \(99\) is \(9\). Therefore, we divide both the numerator and the denominator by \(9\):
\[ \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \]
Thus, \(0.27\overline{27}\) as a rational number in simplest form is:
\[ \frac{3}{11} \]
So, the final answer is:
\(\frac{3}{11}\)