How to find the average

Joe walks to school at the rate of 3 miles per hour, and jogs home at a rate of 5 miles per hour. If the total time that he took between going to school and going home was 1 hour, what was his average speed.

Distance = d
Walking Speed, dw/dt = 3 mi./hr.
Jogging Speed, dj/dt = 5 mi./hr.
Walking Time = tw
Jogging Time = tj
Total Time, T = 1 hr.

(dw/dt)(tw) = d
3(tw) = d

(dj/dt)(tj) = d
5(tj) = d

tw + tj = 1
tj = 1 - tw

The distance walking and the distance jogging is the same, so

3(tw) = 5(tj)
3(tw) = 5(1 - tw)
3(tw) = 5 - 5(tw)
3(tw) + 5(tw) = 5
8(tw) = 5
tw = 5/8 hr.
tw = 0.625 hr.

d = 3(tw)
d = 3(0.625)
d = 1.875 mi.

Ave. Speed = 2 (d / T)
Ave. Speed = 2 (1.875 / 1)
Ave. Speed = 2 (1.875)
Ave. Speed = 3.75 mi./hr.
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