To convert the repeating decimal \( 0.\overline{09} \) into a fraction, we can use algebraic methods.
Let \( x = 0.\overline{09} \).
This means: \[ x = 0.090909\ldots \]
Next, we multiply both sides of the equation by 100 (since the repeating part has 2 digits): \[ 100x = 9.090909\ldots \]
Now, we can subtract the first equation from the second: \[ 100x - x = 9.090909\ldots - 0.090909\ldots \] \[ 99x = 9 \]
Now we divide both sides by 99 to solve for \( x \): \[ x = \frac{9}{99} \]
Next, we simplify this fraction: \[ \frac{9}{99} = \frac{1}{11} \quad (\text{by dividing the numerator and the denominator by 9}) \]
Thus, the simplest form of the fraction that represents the repeating decimal \( 0.\overline{09} \) is: \[ \boxed{\frac{1}{11}} \]