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Answer: $\frac1{15}$


Using a number line

 
Distance $=\frac23 + \frac45 = \frac{10}{15}+\frac{12}{15} = \frac{22}{15}$

Half of distance $=\frac{11}{15}$

Midpoint $=-\frac{10}{15}+\frac{11}{15} = \frac{1}{15}$ (or $\frac{12}{15}-\frac{11}{15} = \frac{1}{15}$) 


Using the mean as the midpoint
$\frac{4}{5} = \frac{12}{15}$ 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.
You can find more short problems, arranged by curriculum topic, in our short problems collection.