Let \( x = 0.09\overline{09} \) (the bar indicates that the "09" repeats indefinitely). We can express \( x \) mathematically as follows:
\[ x = 0.09090909\ldots \]
To eliminate the repeating decimal, we can multiply \( x \) by \( 100 \) (since the repeating part is 2 digits long):
\[ 100x = 9.09090909\ldots \]
Now, we have two equations:
- \( x = 0.09090909\ldots \) (1)
- \( 100x = 9.09090909\ldots \) (2)
Next, we will subtract equation (1) from equation (2):
\[ 100x - x = 9.09090909\ldots - 0.09090909\ldots \]
This simplifies to:
\[ 99x = 9 \]
Now, we can solve for \( x \):
\[ x = \frac{9}{99} \]
Now we simplify \(\frac{9}{99}\):
The greatest common divisor (GCD) of 9 and 99 is 9. Thus, we divide the numerator and denominator by 9:
\[ x = \frac{9 \div 9}{99 \div 9} = \frac{1}{11} \]
Therefore, the rational number in simplest form that corresponds to \( 0.09\overline{09} \) is:
\[ \boxed{\frac{1}{11}} \]
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