Subject: Running Through a Train Tunnel

Name: Eugene Chan
Who is asking: Student
Level: Middle

Question:
A man is running through a train tunnel. When he is 2/3 of the way through, he hears a train that is approaching the tunnel from behind him at a speed of 60 mph. Whether he runs ahead or back, he will reach an end of the tunnel at the same time the train reaches that end. At what rate, in miles per hour, is he running? (Assume he runs at a constant rate.)

I think the answer (12 mph) is wrong. Also, I believe it should read 1/3 of the way through, but don't know how to prove it. Could you come up with some way to prove it, please? I would really appreciate it.

You are correct in your questioning of this problem. The way that it is worded it doesn't make sense. If the man is 2/3 of the way through the tunnel, then he can run ahead to the end of the tunnel, a distance of 1/3 of the length of the tunnel, in the time it takes the train to also reach the far end of the tunnel. If instead, he turns and runs back then in the time it takes him to run 1/3 of the length of the tunnel the train will have passed by (over?) him to reach the far end of the tunnel.

If he starts 1/3 of the way through the tunnel as you suggested and runs toward the train then he will reach the beginning of the tunnel, a distance of 1/3 of the length of the tunnel, just as the train reaches the same point. If, instead, he runs away from the train he will have run the same distance, 1/3 of the length of the tunnel, just as the train enters the tunnel. At this instant the man is 2/3 of the way through the tunnel. Since they reach the far end at the same instant, the man will have run 1/3 of the length of the tunnel in the time it takes the train to travel the entire length of the tunnel. Thus the train must be travelling at three times the speed the man can run. Hence the man runs at 20 mph.