The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ (on the number line) is

$\text{(A)}\ \frac{1}{4} \qquad \text{(B)}\ \frac{2}{15} \qquad \text{(C)}\ \frac{4}{15} \qquad \text{(D)}\ \frac{53}{200} \qquad \text{(E)}\ \frac{8}{15}$
The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ is simply their [average](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean), which is
\begin{align*} \frac{\frac{1}{5}+\frac{1}{3}}{2} &= \frac{\frac{3}{15}+\frac{5}{15}}{2} \\ &= \frac{\frac{8}{15}}{2} \\ &= \frac{4}{15} \\ \end{align*}
$\boxed{\text{C}}$

For any two fractions that have the property $\frac{a}{b}$ and $\frac{a}{c}$ where a is equal for both fractions;
Set aside the a. Take the average of b and c. For this example, the average of 3 and 5 is 4. Then multiply those numbers. 3*5=15. The number that is exactly in between $\frac{a}{b}$ and $\frac{a}{c}$ will be $\frac{a*((b+c)/2)}{b*c}$.
In this example, it would be $\frac{1((3+5)/2)}{3*5}$, which simplifies to $\frac{4}{15}$.
Therefore, the answer is $\boxed{\text{C}}$
Edit by mathmagical~