1995 AJHSME Problems/Problem 25

Problem

Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?

Solution

Say you are on the Houston-bound train that left at 12:30 in the afternoon, looking out the window to see how many buses you pass. At 12:45 pm, the Dallas bus that left at 8:00 am is 4:45 away (note-- a:b means a hours, b minutes) from Dallas, and therefore 15 minutes from Houston. Your bus is also 15 minutes from Houston, so you are delighted at the first bus you have passed. At 1:15 pm, the 9:00 am Dallas bus meets you again, being 4:15 away from Dallas and therefore 45 minutes from Houston. After a while you might notice that the buses meet you in 30 minute intervals after 12:45 pm; indeed, the bus that left an hour later than the most recently passed bus has traveled 30 minutes closer, and so have you, for a total of an hour closer, and when you passed the most recent bus, the bus after it had indeed left an hour later. You decide to predict how many buses you pass by finding all times that are 30 minute intervals from 12:45. They are 12:45 1:15, 1:45 2:15, 2:45 3:15, 3:45 4:15, 4:45 5:15, arrive at 5:30, Hence you predict that you will pass 10 buses. After a bit of betting with other passengers, a tour bus passes by at 4:20, throwing off the amount. Uh-Oh!!