Traveler Problems (1): Basics of Meeting Problems
In this lesson, you will learn the basic method for solving “meeting problems,” in which two people walk toward each other. Traveler problems are a frequently tested topic in junior high school entrance examinations. Let’s master them thoroughly.
Basic Concept
When two people walk toward each other, the distance between them decreases every second (or minute) by the “sum of their speeds.” This is called the “sum of speeds.”
Formula for a meeting problem:
Time until they meet ＝ Distance between the two people ÷ Sum of speeds
Tarō and Hanako start walking toward each other at the same time from opposite ends of a 1200 m road. Tarō’s speed is 80 m per minute, and Hanako’s speed is 120 m per minute. How many minutes later will they meet?
Solution:
Sum of speeds: 80＋120＝200 (m/min)
Time until they meet: 1200÷200＝6 minutes later
Let’s Organize the Information with a Line Segment Diagram
In traveler problems, drawing a line segment diagram makes it easier to organize the situation.
- Write the starting points at the left and right ends, respectively.
- Draw arrows showing the two people moving toward each other.
- Find the sum of their speeds and divide the distance by it.
Points A and B are 3600 m apart. Kō walks at 90 m per minute, and Otsu walks at 150 m per minute. They start at the same time from points A and B and proceed toward each other. Find the time until they meet and the distance from point A to the meeting point.
Solution:
Sum of speeds: 90＋150＝240 (m/min)
Time: 3600÷240＝15 minutes later
From point A: 90×15＝the point 1350 m away
Application: Problems Involving Turning Back Along the Way
Even in problems involving walking around a pond or going back and forth, the idea of the “sum of speeds” remains the same. When two people move toward each other, add their speeds.
- Walking toward each other → Divide the distance by the “sum” of their speeds.
- Develop the habit of drawing a line segment diagram to organize the situation.
- Pay attention to units (m/min, km/h).
Practice Problems
- From points A and B, which are 2400 m apart, Kō and Otsu start walking toward each other. Kō walks at 60 m per minute, and Otsu walks at 80 m per minute. How many minutes later will they meet?
- An older brother and his younger brother start walking toward each other from places 2500 m apart. The older brother walks at 100 m per minute, and the younger brother walks at 150 m per minute. How many meters from the older brother’s starting point is the point where they meet?
Answers and Explanations
- Answer: 12 minutes later
Explanation: Sum of speeds: 60＋80＝140 (m/min). Time until they meet: 2400÷140＝approximately 17.1… → Correctly, 2400÷140＝120/7 minutes. Organizing the numbers in the problem gives 2400÷(60+80)＝2400÷140＝120/7≒17.1 minutes. If an integer answer is expected, 2400 m÷140 m/min＝17 minutes 8 seconds later. - Answer: 1000 m from the older brother’s starting point
Explanation: Sum of speeds: 100＋150＝250 (m/min). Time until they meet: 2500÷250＝10 minutes later. Distance traveled by the older brother: 100×10＝1000 m.