Let \( x = 0.20202020\ldots \), where the digits "20" repeat indefinitely.
To eliminate the repeating part, we can multiply \( x \) by \( 100 \) (since the repeating part consists of two digits):
\[ 100x = 20.20202020\ldots \]
Now, we have two equations:
- \( x = 0.20202020\ldots \)
- \( 100x = 20.20202020\ldots \)
Next, we can subtract the first equation from the second:
\[ 100x - x = 20.20202020\ldots - 0.20202020\ldots \]
This simplifies to:
\[ 99x = 20 \]
Now, solve for \( x \):
\[ x = \frac{20}{99} \]
We need to check if \( \frac{20}{99} \) is in simplest form. The greatest common divisor (GCD) of 20 and 99 is 1, since 20 (which factors to \( 2^2 \times 5 \)) and 99 (which factors to \( 3^2 \times 11 \)) share no common factors.
Thus, \( \frac{20}{99} \) is already in simplest form.
The final answer is:
\[ \boxed{\frac{20}{99}} \]
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