If a motorcycle traveling at seventy mile per hour overtakes a car traveling at forty miles per hour that had a three hour head start how far from starting point are the two vehicles?

There are several ways of answering this question:

In all cases, the car, with a three hour head start is 3*40 = 120 miles in front.

Method I:

The motorcycle is travelling at a relative speed of 70-40 = 30 mph. At this speed it has to make up the distance of 120 miles. This will take 120/30 = 4 hours.

At a speed of 70 mph, the motorcycle will have travelled 4*70 = 280 miles.

Method II:

At time t hours from when the motorcycle starts, the car is at 120 + 40*t miles.

At that time the motorcycle is at 70*t miles.

These two must be equal so 120 + 40*t = 70*t

so 120 = 30*t so that t = 4.

Then, as before, the motorcycle is 4*70 = 280 miles from its start.

Method III: ;D

Invoke Zeno's paradox.

When the motorcycle starts, the car is at a point 120 miles ahead. It takes the motorcycle some time to get to that point. During that time the car has moved on a further distance. It then takes the motorcycle some more time to get to that point. But by then the car has moved on by yet another small distance. And so on. How can the motorcycle hope to catch up if, by the time it gets to where the car WAS, the car has moved forward some more?