$\frac{4}{5} = \frac{12}{15}$ and $-\frac{2}{3} = -\frac{10}{15}$,
so the number halfway between these is $\frac{1}{2}
(-\frac{10}{15}+ \frac{12}{15})$ which is
$\frac{1}{15}$.

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