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! :)