Avg. speed = total distance/total time
distance = speed * t
Let d = distance to work or distance back home...
d = u * t_{1} , where t1 is the time taken
d = v * t_{2} , where t1 is the time taken
d/u = t_{1}
d/v = t_{2}
Total distance = 2*d = 2d
Total time = t_{1} + t_{2} = d/u + d/v
d/u + d/v = (dv + du)/uv
Avg speed for the whole trip = Total distance/total time
2d
--------
(dv+du)/uv
2d * (uv)/ (dv + du)
2duv/ (d(v+u))
The d's cancel:
(2uv)/(u+v)
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