For problems in which two people go around a pond (a circular course), the key is to consider whether they “meet” or “catch up,” using the length of one lap of the pond.
Basics of Traveler Problems on a Pond
- When running in opposite directions: time until meeting = length of 1 lap of the pond ÷ (sum of speeds)
- When running in the same direction: time until catching up = length of 1 lap of the pond ÷ (difference between speeds)
- The second and third meetings also occur repeatedly at the same time intervals.
Example 1 (Opposite Directions): P (60 m/min) and Q (90 m/min) start at the same time in opposite directions around a pond whose one lap is 900 m. How often will they meet?
Solution: 900 ÷ (60 + 90) = 900 ÷ 150 = every 6 minutes
Solution: 900 ÷ (60 + 90) = 900 ÷ 150 = every 6 minutes
Example 2 (Same Direction): This time, the two people start in the same direction around the same pond. How often will Q catch up with P?
Solution: 900 ÷ (90 − 60) = 900 ÷ 30 = every 30 minutes
Solution: 900 ÷ (90 − 60) = 900 ÷ 30 = every 30 minutes
If asked, “How many times will they meet in 30 minutes?”, calculate 30 ÷ 6 = 5 times. Carefully read the problem to determine whether the starting moment (0 minutes) is counted as a meeting.