2005 PMWC Problems/Problem T4

Problem

Buses from town A to town B leave every hour on the hour (for example: 6:00, 7:00, …). Buses from town B to town A leave every hour on the half hour (for example: 6:30, 7:30, …). The trip between town A and town B takes 5 hours. Assume the buses travel on the same road. If you get on a bus from town A, how many buses from town B do you pass on the road (not including those at the stations)?

Solution

Without loss of generality, assume that your bus leaves at 12:00 noon. At 11:30 AM, 30 minutes before you leave, the 6:30 AM bus from town B arrives. After that, you pass every bus that leaves town B until 5:00 PM when you arrive. This means you pass every bus from the 7:30 AM bus to the 4:30 PM bus (inclusive) -- this is a total of buses.