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Express 2.\overline{57} as a base 10 fraction in reduced form.

 Jun 28, 2024
 #1
avatar+1075 
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First, we can use variables to solve this problem in  quite a handy way. First, let's set

\(x = 2.\overline{57}\)

 

For this value of x, we know that

\(100 x= 257.\overline{57}\)

 

Now, we do the really cool trick. We subtract the first equation from the second equation to get

\(100 x - x = 257.\overline{57} - 0.\overline{57}\)

 

Notice that the repeating terms cancel out. Thus, we have

\(99x =257\\ x= \frac{255}{99}\)

 

Dividing by a common factor, we have

\(x= 2 \frac{19}{33}\)

 

Thus, our final answer is \(2.\overline{57} = 2 \frac{19}{33}\)

 

Thanks! :)

 Jun 29, 2024

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