Allen, Benedict and Carl started at the same instant from the same point using the same route trying to overtake a fourth cyclist Donald traveling at a constant speed ahead of them. Allen and Benedict each took 10 hours and 4 hours respectively to overtake Donald. Given that Allen, Benedict and Carl each cycled at the constane speed of 4 km/h, 5.5 km/h and 7 km/h respectively throughout the journey, find the time, in hours, that Carl took to overtake Donald.

A=4t represents the distance (km) Allen travels in time t.

B=5.5t similarly represents the distance Benedict travels.

C=7t represents how far Carl travels.

D=dt+h represents how far Donald travels, where d is Donald’s speed in km/hr and h is the headway Donald has because he leaves before the other cyclists.

When t=10 hr, A=4×10=40km, and D=10d+h.

When t=4 hr, B=4×5.5=22km, and D=4d+h.

Therefore we have:

10d+h=40 because A=D when Allen overtakes Donald.

4d+h=22 because B=D when Benedict overtakes Donald.

Subtracting these two equations we get:

6d=18, so d=3km/hr, Donald’s speed.

And h=22-4d=40-10d=10km, the headway at the start.

So D=3t+10. C=7t, so when C=D, 7t=3t+10, 4t=10, t=2.5hr, the time it takes Carl to overtake Donald.

by Top Rated User (711k points)