A vehicle travels half the distance with speed vvv and the remaining distance with speed 2v2v2v. Its average speed is
v3\frac{v}{3}3v
2v3\frac{2v}{3}32v
4v3\frac{4v}{3}34v
3v4\frac{3v}{4}43v
Average speed vavg=Total DistanceTotal Timev_{avg} = \frac{\text{Total Distance}}{\text{Total Time}}vavg=Total TimeTotal Distance. Let total distance be 2d2d2d. Time t1=d/vt_1 = d/vt1=d/v, t2=d/2vt_2 = d/2vt2=d/2v. vavg=2dd/v+d/2v=2d2d+d2v=2d3d2v=4v3v_{avg} = \frac{2d}{d/v + d/2v} = \frac{2d}{\frac{2d+d}{2v}} = \frac{2d}{\frac{3d}{2v}} = \frac{4v}{3}vavg=d/v+d/2v2d=2v2d+d2d=2v3d2d=34v.
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