Traveling Problems ③ Applied Problems (Around a Pond and Round Trips)
In this lesson, you will learn applied patterns of traveling problems, such as problems involving walking around a pond and making round trips.
Problems Involving Walking Around a Pond
When two people walk around a pond, there are the following two patterns:
- Opposite directions: They walk toward each other, so use the “sum of their speeds” → meeting problems
- Same direction: One catches up with the other, so use the “difference of their speeds” → catching-up problems
Example 1 (Opposite Directions)
A and B started walking at the same time from the same point in opposite directions around a pond whose circumference is 1800 m. A walks at 80 m per minute, and B walks at 100 m per minute. How many minutes later will they meet for the first time?
Solution:
Sum of speeds: 80＋100＝180 (m/min)
Distance for one lap: 1800 m
Time until they meet: 1800÷180＝10 minutes later
A and B started walking at the same time from the same point in opposite directions around a pond whose circumference is 1800 m. A walks at 80 m per minute, and B walks at 100 m per minute. How many minutes later will they meet for the first time?
Solution:
Sum of speeds: 80＋100＝180 (m/min)
Distance for one lap: 1800 m
Time until they meet: 1800÷180＝10 minutes later
Example 2 (Same Direction)
Under the same conditions, suppose they now start walking in the same direction. How many minutes later will the faster B catch up with the slower A for the first time?
Solution:
Difference of speeds: 100－80＝20 (m/min)
B catches up with A after traveling one lap (1800 m) more than A
Time: 1800÷20＝90 minutes later
Under the same conditions, suppose they now start walking in the same direction. How many minutes later will the faster B catch up with the slower A for the first time?
Solution:
Difference of speeds: 100－80＝20 (m/min)
B catches up with A after traveling one lap (1800 m) more than A
Time: 1800÷20＝90 minutes later
Round-Trip Problems
In problems where one person turns around and comes back, consider the distance traveled as a “round-trip distance.”
Example 3
Point B is 2000 m from point A. Kō walks from A to B and back to A, while Otsu walks from A toward B. Kō walks at 100 m per minute, and Otsu walks at 60 m per minute. They start from point A at the same time. How many minutes after starting will Kō, on the way back, meet Otsu?
Solution:
Time for Kō to reach point B: 2000÷100＝20 minutes
After Kō turns around, they move toward each other
Otsu’s position after 20 minutes: 60×20＝1200 m from point A (in the direction from A to B)
Distance between them after 20 minutes: 2000－1200＝800 m
Sum of speeds: 100＋60＝160 (m/min)
Additional time until they meet: 800÷160＝5 minutes
Total: 20＋5＝25 minutes later
Point B is 2000 m from point A. Kō walks from A to B and back to A, while Otsu walks from A toward B. Kō walks at 100 m per minute, and Otsu walks at 60 m per minute. They start from point A at the same time. How many minutes after starting will Kō, on the way back, meet Otsu?
Solution:
Time for Kō to reach point B: 2000÷100＝20 minutes
After Kō turns around, they move toward each other
Otsu’s position after 20 minutes: 60×20＝1200 m from point A (in the direction from A to B)
Distance between them after 20 minutes: 2000－1200＝800 m
Sum of speeds: 100＋60＝160 (m/min)
Additional time until they meet: 800÷160＝5 minutes
Total: 20＋5＝25 minutes later
Key Points
- Around a pond (opposite directions) → sum of speeds, one lap ÷ sum
- Around a pond (same direction) → difference of speeds, one lap ÷ difference
- For turn-around problems, organize the situation “at the point when one person turns around”
- If asked which meeting it is, consider “time until one meeting × number of meetings”
Practice Problems
- A and B start at the same time in opposite directions around a pond whose circumference is 2400 m. A walks at 120 m per minute, and B walks at 80 m per minute. How many minutes later will they meet for the third time?
- From points P and Q, which are 3000 m apart, Kō walks from P to Q and back to P, while Otsu walks from P toward Q. Kō walks at 150 m per minute, and Otsu walks at 90 m per minute. They start at the same time. How many minutes later will they meet for the second time?
Answers and Explanations
- Answer: 36 minutes later
Explanation: Sum of speeds: 120＋80＝200 (m/min). Time until the first meeting: 2400÷200＝12 minutes later. Time until the third meeting: 12×3＝36 minutes later. *Exact calculation: the third meeting occurs 36 minutes later. - Answer: 25 minutes later
Explanation: Time for Kō to reach point Q: 3000÷150＝20 minutes later. Otsu’s position after 20 minutes, when Kō turns around: 90×20＝1800 m (in the direction from P to Q). Distance between them when Kō turns around: 3000－1800＝1200 m. Sum of speeds: 150＋90＝240 (m/min). First meeting: 20＋1200÷240＝20＋5＝25 minutes later. To find the second meeting, further calculation is necessary.