Let \( x = 0.09\overline{9} \), meaning \( x = 0.09999999\ldots \).
To eliminate the repeating decimal, we can multiply \( x \) by 10:
\[ 10x = 0.99999999\ldots \]
Next, we can set up a second equation for \( x \):
\[ x = 0.09999999\ldots \]
Now, we can subtract the first equation from the second:
\[ 10x - x = 0.99999999\ldots - 0.09999999\ldots \]
This simplifies to:
\[ 9x = 0.9 \]
Next, we can solve for \( x \):
\[ x = \frac{0.9}{9} \] \[ x = \frac{9}{90} \]
Now, we simplify \( \frac{9}{90} \):
\[ \frac{9}{90} = \frac{1}{10} \]
Thus, the rational number in simplest form that represents \( 0.09\overline{9} \) is
\[ \boxed{\frac{1}{10}} \]