Problem 25 · 1995 AJHSME
Stretch
Ratios, Rates & Proportions
relative-motioninterval-overlap
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 5 hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass on the highway (not in the station)?
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Answer: D — 10.
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Hint 1 of 3
You don't have to track positions or speeds. The only thing that lets two buses meet on the road is that they are on the road AT THE SAME TIME — so this is really a question about overlapping time windows, not motion.
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Hint 2 of 3
Pin down ONE Houston-bound bus and its 5-hour window. Then an oncoming bus is passed exactly when its own 5-hour window overlaps that one — so count the oncoming departure times whose trips overlap.
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Hint 3 of 3
An oncoming bus already on the road when yours starts still counts (you pass it later on), and one that starts before you finish counts too. So look both BACKWARD and forward from your window — don't just count buses that leave during your trip.
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Approach: forget speed — count oncoming buses whose road-time overlaps yours
- The freeing insight: two buses on the same highway pass each other if and only if they share the road at the same moment. Speeds and exact meeting points never matter — only whether their time-on-road windows overlap. So pick one bus and compare windows.
- Pin your Houston-bound bus: say it leaves Dallas at 12:00 and (5-hour trip) arrives Houston at 17:00, so it owns the road-window 12:00–17:00.
- A Dallas-bound bus that left Houston at time t owns the window (t, t+5). It overlaps yours when it hasn't already arrived (t + 5 > 12:00, i.e. t > 7:00) AND it has already left (t < 17:00). So the meeting condition is simply 7:00 < t < 17:00.
- Houston buses leave on the half hour, so the departures in that range are 7:30, 8:30, 9:30, … , 16:30 — that's 10 buses (one every hour across a 10-hour span).
- The trap this catches: only counting buses that leave during your own 12:00–17:00 trip gives 5 or 6 and misses the ones already underway when you start — that's why the overlap test must reach back to 7:00. Sanity check: the early 7:30 bus reaches Dallas at 12:30 (just after you set out, so you do pass it on the road), and the late 16:30 bus is still rolling when you arrive — both genuine highway meetings, not station ones.
- Why this transfers: whenever you're asked 'how many of these cross paths,' translate each traveler into a time interval and count overlapping intervals — the geometry of who-is-where dissolves into simple interval arithmetic.
Another way — draw it as a distance–time picture:
- Sketch time across the bottom and distance (Dallas at the bottom, Houston at the top) up the side. Your bus is one slanted line going up; every Dallas-bound bus is a slanted line going down, starting on the half hours.
- Two lines crossing = a pass. Your line spans the 5-hour width, and a down-line crosses it exactly when it starts within 5 hours before you finish and ends within 5 hours after you begin — the same 7:00-to-17:00 window. Counting the crossing lines gives 10.
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