Decimals

Express as a common fraction the reciprocal of $0.\overline{72}$.

I can give you the answer. 

The decimal equals 8 / 11 so its reciprocal would be 11 / 8.  

I can't tell you how to calculate it though.  I guessed and got lucky. 

I guessed 8/9 ... that was too big, so I guessed 8/11 and that was it. 

_.

Let $x = 0. \overline{72}$

This means $100x = 72.\overline{72}$

Subtracting the 2 gives $99x = 72$, meaning $x = {72 \over 99} ={ 8 \over 11}$.

The reciprocal of this, as Guest found, is $\color{brown}\boxed{11 \over 8}$

Just a fun fact, if it repeats every one digit, then the fraction version is [digit] / 9. If it repeats every two digits, then it is [digit] / 99. You can see the pattern here.

⚡⚡⚡

ShockFish