A train goes from P to Q with a speed u km/h, then from Q to R (QR = 2PQ) with a speed 3u km/h, and returns from R to P with a speed $$ \frac{u}{2} $$km/h. What is the average speed (in km/h)of the train for the entire journey starting from P and back to P?
Let's assume the distance between P and Q is 'y' km.
From QR = 2PQ, the distance between Q and R will be 2y km.
Average speed =Â $$\frac{total\ disatnce}{total\ time}$$
=Â $$\frac{3y+3y}{\frac{y}{u}+\frac{2y}{3u}+\frac{3y}{\frac{u}{2}}}$$
=Â $$\frac{6y}{\frac{y}{u}+\frac{2y}{3u}+\frac{6y}{u}}$$
= $$\frac{6y}{\frac{3y}{3u}+\frac{2y}{3u}+\frac{18y}{3u}}$$
= $$\frac{6y}{\frac{23y}{3u}}$$
= $$\frac{18u}{23}$$
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