Number Theory

We can apply a handy trick to solve this problem. 

Let's set a variable. Let's set \( x=0.29\overline{6}\)

We want to put x in terms of a fraction. 

 

Now, for that value of x, we have \(10 x = 2.96\overline{6}\)

 

This is important, as now we subtract x from 10x. We get

\(10x-x = 2.96\overline{6}-0.29\overline{6}\)

 

Since the repeating decimal cancels out, we have

\(9 x = 2.67\). 

 

Now, we simply solve for x. We have

\(x = \frac{2.67}{9}\)

\(x=\frac{2.67}{9}\times \frac{100}{100}= \frac{267}{900}\)

\(x = \frac{89}{300}\)

 

Thus, our final answer is 89/300. 

 

Thanks! :)