Traveling Problems ② Catch-Up Problems
In this lesson, you will learn about "catch-up problems" involving two people moving in the same direction. They are similar to meeting problems, but the way of thinking is slightly different.
Basic Concept
When two people are moving in the same direction, the faster person chases the slower person. The distance between them decreases each minute by the "difference between their speeds."
Formula for catch-up problems:
Time until catching up ＝ Distance between the two people ÷ Difference in speed
The younger brother started first, and the older brother chased him 10 minutes later. The younger brother's speed is 60 m per minute, and the older brother's speed is 100 m per minute. How many minutes after starting will the older brother catch up with the younger brother?
Solution:
When the older brother starts, the younger brother is already 60×10＝600 m ahead
Difference in speed: 100－60＝40 (m/min)
Time until catching up: 600÷40＝15 minutes later
Focus on the "Difference"
The key to catch-up problems is "how many meters ahead the person is" and "how many meters the distance decreases each minute." If you know these two values, you can always solve the problem.
A and B started from the same point at the same time and walked in the same direction. A walks at 80 m per minute, and B walks at 120 m per minute. How many minutes after starting will B be 1200 m ahead of A?
Solution:
Difference in speed: 120－80＝40 (m/min)
Time: 1200÷40＝30 minutes later
When One Person Starts First
A common problem pattern is one in which "one person starts first." Find the distance traveled by the person who started first, and use it as the "distance to catch up."
- Moving in the same direction → Divide the distance by the "difference" in speed
- Distance traveled by the person who starts first＝speed×head start time
- Be sure to distinguish between meeting problems and catch-up problems
Practice Problems
- The older sister started at 70 m per minute. Five minutes later, the younger sister chased her in the same direction at 110 m per minute. How many minutes after starting will the younger sister catch up with the older sister?
- Twelve minutes after Kō started walking at 90 m per minute, Otsu chased him at 150 m per minute. How many meters from the starting point will Otsu catch up with Kō?
Answers and Explanations
- Answer: 8.75 minutes later (8 minutes 45 seconds later)
Explanation: The older sister's head start distance: 70×5＝350 m. Difference in speed: 110－70＝40 (m/min). Time until catching up: 350÷40＝8.75 minutes (8 minutes 45 seconds) later. - Answer: 2700 m from the starting point
Explanation: Kō's head start distance: 90×12＝1080 m. Difference in speed: 150－90＝60 (m/min). Time for Otsu to catch up: 1080÷60＝18 minutes. Distance traveled by Otsu: 150×18＝2700 m.