IMC 2020 Q16

The sum of \(\dfrac{1}{9} + \dfrac{1}{11}\) is \(\dfrac{11}{99} + \dfrac{9}{99}\), that is \(\dfrac{20}{99}\).

Now 20 and 99 do not have any factors in common except 1. So the fraction cannot be simplified.
In option A, we may write 0.10 as the fraction \(\tfrac{10}{100}\), whose denominator is a power of 10. The same is true of options B, C and D. Therefore, when simplified, none of these fractions can have 99 as a denominator.
It is left to the reader to confirm, by division, that \(\dfrac{20}{99} = 0.\dot{2}\dot{0}\).
An alternative argument follows.
Let \(x = 0.\dot{2}\dot{0} = 0.202020....\) Then \(100x = 20.202020...\) .
Subtracting the first equation from the second gives \(99x = 20\).
So \(0.\dot{2}\dot{0} = \dfrac{20}{99}\).