Number Theory

We can solve this problem by defining variables and using a nice little trick. 

First, let's set \(x = 0.031\overline{4}\). we are trying to put x in fractional form. 

 

For that value of x, we have that \(10 x = 0.314\overline{4}\)

 

Now, we subtract x from 10x. We get

\(10x-x = 0.314\overline{4}-0.031\overline{4}\)

 

Since the repeating 4 cancels out, we are left with

\(9 x = 0.283\)

 

Now, we simplfy solve for x. 

\(x = \frac{0.283}{9}\)

\(x=\frac{0.283}{9}\times \frac{1000}{1000}= \frac{283}{9000}\)

 

So, we have

\(0.031\overline{4} = \frac{283}{9000}\)

 

So 283/9000 is our final answer. 

 

Thanks! :)