Suppose the path is $x$ miles long. Then she takes $\frac{1}{2}x$ hours to walk up the hill, and another $\frac{1}{4}x$ hours to walk back down. Therefore, in $\frac{3}{4}x$ hours she walks a total distance of $2x$.

This means her average speed is $(2x) \div (\frac{3}{4}x) = \frac{8}{3} = 2\frac{2}{3}$ miles per hour.

*This problem is taken from the UKMT Mathematical Challenges.*