Amtrak runs many trains on weekday mornings between Washington DC and New York City (Penn Station), and between New York and Washington. Trains run approximately every half hour; the trip takes about 3 and a half hours. If we take these times as exact, and we assume that all trains leave on the hour and on the half hour and travel at the same speed, imagine that you are taking a train from Washington to New York and figure out how many NY to DC trains you will pass as you travel along the way.
Solution
Assume you leave at 9:00. You will be traveling between 9:00 and 12:30, and you will pass every DC-bound train that is traveling during that time. A train that left NYC at 5:30 will arrive in DC just as you are leaving at 9:00 (let's count that one), and a train that is leaving NYC at 12:30 will leave just as you're arriving (let's count that one too). Now, starting with the DC-bound trains starting between 8:30 and 12:30 (including those two times), we count up and see that there are fifteen trains that you will pass. (Extension: At what time will you pass each of these trains?)