Let \( x = 0.09\overline{9} \), where the overline indicates that the 9 repeats indefinitely.
We can express \( x \) as follows:
\[ x = 0.09999999\ldots \]
To eliminate the repeating decimal, multiply \( x \) by 10:
\[ 10x = 0.99999999\ldots \]
Next, we subtract the original \( x \) from this equation:
\[ 10x - x = 0.99999999\ldots - 0.09999999\ldots \]
This simplifies to:
\[ 9x = 0.9 \]
Now we can solve for \( x \):
\[ x = \frac{0.9}{9} \]
To express \( 0.9 \) as a fraction:
\[ 0.9 = \frac{9}{10} \]
Substituting it back into our equation gives:
\[ x = \frac{\frac{9}{10}}{9} = \frac{9}{10} \times \frac{1}{9} = \frac{1}{10} \]
Therefore, the simplest form of \( 0.09\overline{9} \) as a rational number is:
\[ \boxed{\frac{1}{10}} \]
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