I was given three different speeds, the times it took to travel at that speed, and I was told to find the average speed without calculating the distance. How?

Sounds like a trick question. Given the speed and times, you can determine the distance traveled for each leg of the trip. Add those distances to get the total distance traveled, and divide by the total time to get the average speed. But if you are prohibited from explivcitly caculating the distances you can instead use a weighted average of velocitoes based on time:

v_1 (\frac {t_1}{t_1+t_2+t_3}) + v_2 (\frac {t_2}{t_1+t_2+t_3} )+ v_3 ( \frac {t_3}{t_1+t_2+t_3} )

Mathematically it's identical.