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Two towns A and B are connected by a
regular bus service with a bus leaving in either direction every T minutes. A
man cycling with a speed of 20 km h^{-1} in the direction A to B
notices that a bus goes past him every 18 min in the direction of his motion,
and every 6 min in the opposite direction. What is the period T of the bus
service and with what speed (assumed constant) do the buses ply on the road?

Answer:

Let V be the speed of the bus running between
towns A and B. Speed of the cyclist, v = 20 km/h

Relative speed of the bus moving in the
direction of the cyclist = V - v = (V - 20) km/h

The bus went past the cyclist every 18 min
i.e., 16/80 h (when he moves in thedirection of the bus).

Distance covered by the bus =

…
(i)

Since one bus leaves
after every T minutes, the distance travelled by the bus will be equal to

…
(ii)

Both equations (i) and (ii) are equal.

Relative speed of the
bus moving in the opposite direction of the cyclist = (V + 20) km/h

Time taken by the bus to go past the cyclist

From equations (iii) and (iv), we get

Substituting the value of V in equation (iv),
we get

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