### Instruction: By what common fraction does $0.\overline{81}$ exceed $0.81$?### Response: I notice that $0.\overline{81}$ is a repeating decimal, which means it can be written as a fraction using a simple trick.
If I let x be $0.\overline{81}$, then I can multiply both sides by 100 to get 100x = $81.\overline{81}$.
Then I can subtract x from both sides to get 99x = 81.
This means x = 81/99, which is the fraction form of $0.\overline{81}$.
Now I want to find out how much more this fraction is than 0.81, which is the same as 81/100.
To compare two fractions, I can find a common denominator.
The least common multiple of 99 and 100 is 9900, so I can multiply 81/99 by 100/100 and 81/100 by 99/99 to get equivalent fractions.
This gives me 8100/9900 and 8019/9900.
Now I can subtract the smaller fraction from the larger one to get the difference.
This gives me 8100/9900 - 8019/9900 = 81/9900.