### Instruction: What is the reciprocal of the common fraction form of $.\overline{2}$?### Response: So we have a repeating decimal, let's convert it to a fraction.
Let's call the decimal $x$. So $x=.\overline{2}$.
Let's multiply $x$ by 10. So we get $10x = 2.\overline{2}$.
Let's subtract $x$ from $10x$. So we get $10x - x = 2.\overline{2} - .\overline{2}$ which is the same as $9x = 2$.
Dividing both sides by 9, we get $x = \frac{2}{9}$.
So the reciprocal of the fraction is $\frac{9}{2}$.
And if we divide 9 by 2, we get $4.5$.
And $4.5$ is the same as $4\frac{1}{2}$.