Average Speed and Advanced Traveler Problems ② Meeting and Catching-Up Problems
We will learn how to think about “meeting problems,” in which two people walk toward each other, and “catching-up problems,” in which they walk in the same direction.
Basics of Traveler Problems
- Meeting problem: time until they meet = initial distance ÷ sum of the two speeds
- Catching-up problem: time until catching up = difference in distance ÷ difference between the two speeds
Example 1 (Meeting Problem): From places 3600 m apart, P walks toward Q at 60 m/min and Q walks toward P at 90 m/min. After how many minutes will they meet?
Solution:
① Every minute, the two people get 60 + 90 = 150 m closer to each other.
② Time until they meet → 3600 ÷ 150 = 24 minutes
Solution:
① Every minute, the two people get 60 + 90 = 150 m closer to each other.
② Time until they meet → 3600 ÷ 150 = 24 minutes
Example 2 (Catching-Up Problem): Q starts in the same direction 10 minutes after P starts. P walks at 50 m/min, and Q walks at 80 m/min. How many minutes after Q starts will Q catch up with P?
Solution:
① Before Q starts, P has gone 50 × 10 = 500 m ahead.
② Every minute, Q gets 80 − 50 = 30 m closer to P.
③ Time until catching up → 500 ÷ 30 ≒ 16.7 minutes (16 minutes 40 seconds)
Solution:
① Before Q starts, P has gone 50 × 10 = 500 m ahead.
② Every minute, Q gets 80 − 50 = 30 m closer to P.
③ Time until catching up → 500 ÷ 30 ≒ 16.7 minutes (16 minutes 40 seconds)
The answer changes completely depending on whether you use the “sum of the speeds” or the “difference between the speeds.” When they move toward each other, use the “sum”; when they move in the same direction, use the “difference.”