jmc

The average speed is the total distance traveled, namely $1600\text{m}$, divided by the total time of the run. It took Melanie $\frac{400}{5}=80$ seconds to run the first $400$ meters, $\frac{800}{4}=200$ seconds to run the next $800$ meters, and $\frac{400}{8}=50$ seconds to run the final $400$ meters. Thus the total time of her run was $80+200+50=330$ seconds, and so her average speed for the run was $\frac{1600}{330}=\boxed{\frac{160}{33}}$ meters per second.