Suppose that Amtrak has trains leaving from Washington to New York City and also from New York City to Washington every hour on the hour. Also suppose that the trip from one city to the other takes 4.5 hours and all trains travel at the same speed. If you are on the train from DC to NYC, how many trains going the other way will you pass?

Suppose you leave DC at noon. There are trains on the way from NYC that you will pass, trains that left at:

8:00 (due in DC at 12:30)

9:00

10:00

11:00 and

12:00. That’s 5 so far that you will pass before you reach NYC.

9:00

10:00

11:00 and

12:00. That’s 5 so far that you will pass before you reach NYC.

Now, while you are en route (due in at NYC at 4:30), you will also pass DC-bound trains that left NYC at:

1:00

2:00

3:00

4:00. That’s 4 more. So you’ll pass 9 NYC-DC trains.

2:00

3:00

4:00. That’s 4 more. So you’ll pass 9 NYC-DC trains.