16. As a decimal, what is the value of $\frac{1}{9}+\frac{1}{11}$ ?
A 0.10
B 0.20
C 0.2020
D 0.202020
E $0 . \dot{2} \dot{0}$
16. E The sum of $\frac{1}{9}+\frac{1}{11}$ is $\frac{11}{99}+\frac{9}{99}$, that is $\frac{20}{99}$.
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 $\frac{10}{100}$, 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 $\frac{20}{99}=0 . \dot{2} \dot{0}$.
An alternative argument follows.
Let $x=0 . \dot{2} \dot{0}=0.202020 \ldots$ Then $100 x=20.202020 \ldots$.
Subtracting the first equation from the second gives $99 x=20$.

So $0 . \dot{2} \dot{0}=\frac{20}{99}$