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

 Jun 28, 2022
 #1
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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. 

.

 Jun 28, 2022
 #2
avatar+2448 
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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 \(\color{brown}\boxed{11 \over 8}\)

 Jun 28, 2022
 #3
avatar+69 
+2

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

 Jun 28, 2022
edited by ShockFish  Jun 28, 2022
 #4
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Hi Shockfish, nice to see you here smiley

BuilderBoi  Jun 28, 2022
 #5
avatar+69 
+2

Heyyy just wondering if you were here too laugh

ShockFish  Jun 28, 2022

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