To convert the repeating decimal \( 0.09\overline{09} \) to a rational number, we can use algebra.
Let: \[ x = 0.09\overline{09} \]
This means: \[ x = 0.09090909\ldots \]
Next, we can multiply both sides by 100 (since the repeating part is two digits long) to shift the decimal point two places to the right: \[ 100x = 9.09090909\ldots \]
Now we have two equations:
- \( x = 0.09090909\ldots \)
- \( 100x = 9.09090909\ldots \)
Now, we can subtract the first equation from the second: \[ 100x - x = 9.09090909\ldots - 0.09090909\ldots \] \[ 99x = 9 \]
Next, solve for \( x \): \[ x = \frac{9}{99} \]
Now we simplify \( \frac{9}{99} \): \[ \frac{9}{99} = \frac{1}{11} \]
Thus, the repeating decimal \( 0.09\overline{09} \) is equivalent to the rational number: \[ \boxed{\frac{1}{11}} \]