To convert \( 0.02\overline{02} \) (or \( 0.020202...\)) to a fraction, we can use the method of setting up an equation.
Let \( x = 0.020202...\).
To eliminate the repeating decimal, we can multiply both sides of the equation by 100 (since the repeating part is two digits long):
\[ 100x = 2.020202... \]
Now we have two equations:
- \( x = 0.020202... \)
- \( 100x = 2.020202... \)
Next, we subtract the first equation from the second:
\[ 100x - x = 2.020202... - 0.020202... \]
This simplifies to:
\[ 99x = 2 \]
Now, solve for \( x \):
\[ x = \frac{2}{99} \]
Thus, the fraction representation of \( 0.02\overline{02} \) is \( \frac{2}{99} \).
Looking at the given options, none of them match explicitly. However, if we check for an equivalent form of \( \frac{2}{99} \):
If we check:
- \( \frac{1}{5} = 0.2 \) → Not a match
- \( \frac{1}{50} = 0.02 \) → Not a match
- \( \frac{2}{101} \approx 0.01980198 \) → Not a match
- \( \frac{299}{150} \approx 1.9933 \) → Not a match
Seems like the simplest answer we found is \( \frac{2}{99} \), which is not one of the given options.
If these options are supposed to reference different forms or conversions, it’s advisable to verify the fraction or to look for a potential error in the response options.