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avatar+1640 

Convert 0.038\overline{146} to a fraction in simplest form.

 

\(0.038\overline{146}\)

 Feb 8, 2024
edited by Alan  Feb 8, 2024
 #1
avatar+394 
+2

Hello,

To convert \(0.038\overline{146}\) into a fraction, we can use a relatively simple operation, which can be used for solving a variety of problems. We set \(x = 0.038\overline{146}\), but we also see that mutitplying x by 1000 produces a similar repeating decimal, \(1000x = 38.\overline{146}\). We see that these repeating decimals, are both some relatively simple number + the repeating decimal 0.000146146.... To cancel out, we subtract x from 1000x, yielding \(1000x-x=38.\overline{146} - 0.038\overline{146} \), so \(999x=38.108\), and after simplification, x=9527/249750

 Feb 8, 2024
 #2
avatar+129266 
+1

We can find three infinite sums and add these fractions to 38 /1000

 

(1*10^(-4)) /  ( 1 -10^(-3) )  = 1/9990

 

(4*10^(-5)) / (1 -10^(-3)) = 1/24975

 

(6*10^(-6)) / (1-10^(-3))  = 1/166500

 

38/1000 + 1/9990 + 1/24975 + 1/166500  =  9527  / 249750

 

 

cool cool cool

 Feb 9, 2024

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