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 kmh-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. The period T of the bus service is:

Question

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 kmh-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. The period T of the bus service is: (A) 4.5 min (B) 9 min (C) 12 min (D) 24 min

Tardigrade Admin · Accepted Answer

Let v km h-1 be the constant speed with which the buses ply between the towns A and B. Relative velocity of the bus from A to B with respect to the cyclist = (v - 20) km h-1 Relative velocity of the bus from B to A with respect to the cyclist = (v + 20) km h-1 Distance travelled by the bus in time T(minutes) = vT As per question (v T/v-20)=18 or v T=18 v-18 × 20 and (v T/v+20)=6 or v T=6 v+20 × 6 Equating (i) and (ii), we get 18 v-18 × 20=6 v+20 × 6 or 12 v=20 × 6+18 × 20=480 or v =40 km h -1 Putting this value of v in (i), we get 40 T =18 × 40-18 × 20=18 × 20 or T =(18 × 20/40)=9 min