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.
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+2668 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 \(\color{brown}\boxed{11 \over 8}\)
