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# Decimals

0
219
5

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

Jun 28, 2022

#1
+1

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
+2667
0

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
+66
+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
+2667
0

Hi Shockfish, nice to see you here

BuilderBoi  Jun 28, 2022
#5
+66
+2

Heyyy just wondering if you were here too

ShockFish  Jun 28, 2022