Melanie ran at $5$ m/s for $400$ meters, then ran at $4$ m/s for the next $800$ meters, and finally ran at $8$ m/s for the final $400$ meters. What was her average speed, in meters per second, for the $1600$ meter run?  Express your answer as an improper fraction.
I need to find the average speed, which is the total distance divided by the total time. To find the total time, I need to add the times for each segment of the run. To find the time for each segment, I need to use the formula $t = \frac{d}{s}$, where $t$ is time, $d$ is distance, and $s$ is speed. For the first segment, $t = \frac{d}{s} = \frac{400}{5} = 80$ seconds. For the second segment, $t = \frac{d}{s} = \frac{800}{4} = 200$ seconds. For the third segment, $t = \frac{d}{s} = \frac{400}{8} = 50$ seconds. The total time is $80 + 200 + 50 = 330$ seconds. The total distance is $1600$ meters. The average speed is $\frac{1600}{330} = \frac{160}{33}$ meters per second. 