Problem 13 · 2006 AMC 8
Medium
Ratios, Rates & Proportions
closing-speed
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?
Show answer
Answer: D — 11:00 AM.
Show hints
Hint 1 of 2
Two riders heading toward each other close the gap between them at the SUM of their speeds — think of the empty road shrinking, not the riders moving. But first deal with the half-hour head start so you start both clocks together.
Still stuck? Show hint 2 →
Hint 2 of 2
Closing speed: when two things approach each other, add their speeds; the gap divided by that combined speed is the time to meet.
Show solution
Approach: settle the head start, then close the gap at combined speed
- Cassie rides alone from 8:30 to 9:00, half an hour, covering ½ × 12 = 6 miles. So at 9:00 the gap between them is 62 − 6 = 56 miles.
- From 9:00 they ride toward each other, closing the gap at 12 + 16 = 28 mph.
- Time to close 56 miles: 56 ÷ 28 = 2 hours. They meet at 9:00 + 2:00 = 11:00 AM.
- Why add the speeds: in one hour Cassie eats 12 miles of road and Brian eats 16, so 28 miles of gap vanish per hour regardless of where they are. This "combined speed" trick turns every meeting problem into one simple division — just remember to handle any head start first.
Mark:
· log in to save