Teach: Using Equations ③ Speed Problems

Using Equations ③ Speed Problems

Use equations to solve problems involving speed, distance, and time. This type of problem appears often, so learn the method carefully.

Speed Formulas (Review)

　Distance ＝ Speed × Time
　Speed ＝ Distance ÷ Time
　Time ＝ Distance ÷ Speed

When solving speed problems with equations, it is important first to determine what quantities are equal.
　・Are the distances equal?　・Are the times equal?　・Are the speeds equal?

Catching Up and Meeting Problems

In this type of problem, two people start walking separately.
　・Opposite directions (walking toward each other) → The sum of the distances they walk = the total distance
　・The same direction (catching up) → They catch up when the distances they have walked become equal

Example ①
A walks from home to school at 60 m per minute. If the distance from home to school is 1200 m, how many minutes does it take?
Solution:
Let the time be x minutes. Distance = speed × time ＝ 60x (m).
This equals 1200 m, so 60x = 1200 → x = 20 (minutes).
Example ②
Ten minutes after his younger brother left home, his older brother left home and followed along the same road. The younger brother walks at 60 m per minute, and the older brother walks at 80 m per minute. How many minutes after leaving does the older brother catch up with his younger brother?
Solution:
Suppose he catches up x minutes after leaving.
　Younger brother’s walking time = (10 + x) minutes; distance = 60(10 + x) m
　Older brother’s walking time = x minutes; distance = 80x m
The distances are equal when he catches up: 60(10 + x) = 80x
　600 + 60x = 80x → −20x = −600 → x = 30
　The older brother catches up 30 minutes after leaving.

Going and Returning Problems

These problems involve traveling along the same road at different speeds on the way there and back. Use the relationship “time going + time returning = total time.”

Example ③
A person walks from home to the station at 4 km per hour and returns at 6 km per hour. The round trip takes 1 hour. Find the distance from home to the station.
Solution:
Let the distance be x km.
　Time going = x/4 hours; time returning = x/6 hours
Equality: x/4 + x/6 = 1
Multiply both sides by 12: 3x + 2x = 12 → 5x = 12 → x = 12/5 ＝ 2.4 (km)
Key Points
  • Three formulas: distance＝speed×time / speed＝distance÷time / time＝distance÷speed
  • Catching-up problems: the moment when the distances become equal
  • Meeting problems: the sum of the distances equals the total distance
  • Going and returning: form an equality using the sum of the times or distances

Practice Problems

  1. Walking from home to school at 50 m per minute takes 20 minutes. Find the distance from home to school.
  2. A and B start at the same time from points 1500 m apart and walk toward each other. A walks at 60 m per minute, and B walks at 90 m per minute. After how many minutes do they meet?
  3. You ride a bicycle from home to the station at 12 km per hour and return on foot at 4 km per hour. The round trip takes 2 hours. Find the distance from home to the station.

Answers and Explanations

  1. Answer: 1000 m
    Explanation: Distance = speed×time = 50×20 = 1000 m.
  2. Answer: After 10 minutes
    Explanation: If they meet after x minutes, 60x + 90x = 1500 → 150x = 1500 → x = 10.
  3. Answer: 6 km
    Explanation: Let the distance be x km. x/12 + x/4 = 2. Multiply both sides by 12: x + 3x = 24 → 4x = 24 → x = 6.

Math (Grade 7): related lessons