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Find a fraction equal to value of the decimal \(0.23\overline{72}\). Give your answer in simplest terms.

 

im pretty sure i have to convert this to some sort of geometric sequence but im not sure how... plz help

 

thank you so much!!\(\)

 Feb 15, 2022
 #1
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There are two standard wasy to do this. 

the GP way is not the easiest but if that is what you think is wanted then lets take a look.

 

\(0.23\bar{72}\\ =0.23 + 0.0072+0.000072 + \dots\\ consider \\ 0.0072+0.000072 + \dots\\ \text{This is a GP}\\ a=0.0072,\quad r=0.01\\ S_\infty=\frac{a}{1-r}\\ S_\infty=\frac{0.0072}{0.99}\\ S_\infty=\frac{72}{9900}=\frac{8}{1100}\\~\\ 0.23\bar{72}=\frac{23}{100}+\frac{8}{1100}=\frac{253+8}{1100}=\frac{261}{1100} \)

 

 

You need to check my working.

 Feb 15, 2022

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