Calculating simple speed, distance, & time relationships.

Rate Problems: Speed, Distance, and Time

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Lesson Objectives

  • Understand the relationship between speed, distance, and time.
  • Calculate speed given distance and time using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
  • Calculate distance given speed and time using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \).
  • Calculate time given speed and distance using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
  • Solve word problems involving average speed over multiple segments.

Introduction to Rate Problems

A rate is a ratio that compares two quantities with different units. In everyday life, we encounter rates all the time—kilometers per hour (km/h), miles per hour (mph), meters per second (m/s), and even cost per item. The most common rate problems involve speed, which is the rate at which an object moves. Speed tells us how much distance is covered in a certain amount of time.

Key Formulas for This Lesson
• \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)    (km/h, m/s, etc.)
• \( \text{Distance} = \text{Speed} \times \text{Time} \)
• \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
• \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)
Key Definitions:
• Speed: The rate at which an object covers distance. (e.g., 60 km/h means 60 kilometers per hour).
• Distance: The total length of the path traveled (e.g., kilometers, meters).
• Time: The duration of travel (e.g., hours, minutes, seconds).
• Average Speed: The total distance traveled divided by the total time taken.
• Rate: A ratio comparing two quantities with different units.

Quick Reference: Speed, Distance, Time Relationships

What to FindFormulaUnitsExample
Speed\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)km/h, m/s, mph150 km ÷ 3 h = 50 km/h
Distance\( \text{Distance} = \text{Speed} \times \text{Time} \)km, m, miles60 km/h × 2 h = 120 km
Time\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)h, min, s240 km ÷ 80 km/h = 3 h
Average Speed\( \text{Avg Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)km/h, m/s300 km ÷ 5 h = 60 km/h

Scroll sideways on smaller screens to view the full table.

Calculating Speed

Speed tells us how fast something is moving. It is the distance covered per unit of time. To find speed, divide the distance traveled by the time taken. $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$

Step-by-Step: Finding Speed
1. Identify the distance traveled.
2. Identify the time taken.
3. Divide distance by time: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
4. Include the units (e.g., km/h, m/s).
5. Simplify if needed.
Speed = Distance ÷ Time D S T Cover D with your finger to find: S = D ÷ T   |   D = S × T   |   T = D ÷ S

The speed-distance-time triangle helps you remember the formulas. Cover the quantity you want to find, and the remaining formula appears.

Example 1: Finding Speed
Problem: A car travels 240 kilometers in 3 hours. What is its speed?

Solution:
Step 1: Distance = 240 km, Time = 3 h.

Step 2: Use \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).

Step 3: \( \text{Speed} = \frac{240}{3} = 80 \) km/h.

Answer: The car's speed is 80 km/h.
Example 2: Finding Speed (m/s)
Problem: A runner covers 100 meters in 10 seconds. What is the speed in m/s?

Solution:
Step 1: Distance = 100 m, Time = 10 s.

Step 2: \( \text{Speed} = \frac{100}{10} = 10 \) m/s.

Answer: The runner's speed is 10 m/s.
Watch Out!
Always check the units! Distance might be in kilometers, meters, or miles. Time might be in hours, minutes, or seconds. Make sure the units are consistent before calculating.

Practice for Calculating Speed

  1. A train travels 450 km in 5 hours. What is its speed?
  2. A cyclist covers 60 km in 2 hours. What is the speed in km/h?
  3. A plane flies 1,200 km in 2 hours. What is the speed?
  4. A person walks 15 km in 3 hours. What is the speed?
  5. A car travels 300 km in 4 hours. What is the speed in km/h?

Calculating Distance

If you know the speed and the time, you can find the distance traveled. Distance is speed multiplied by time. $$ \text{Distance} = \text{Speed} \times \text{Time} $$

Step-by-Step: Finding Distance
1. Identify the speed.
2. Identify the time.
3. Multiply speed by time: \( \text{Distance} = \text{Speed} \times \text{Time} \).
4. Include the units (e.g., km, m).
5. Simplify if needed.
Example 3: Finding Distance
Problem: A car travels at 60 km/h for 4 hours. How far does it travel?

Solution:
Step 1: Speed = 60 km/h, Time = 4 h.

Step 2: \( \text{Distance} = 60 \times 4 \).

Step 3: \( \text{Distance} = 240 \) km.

Answer: The car travels 240 km.
Example 4: Finding Distance (m/s)
Problem: A cyclist rides at 8 m/s for 15 seconds. How far does the cyclist travel?

Solution:
Step 1: Speed = 8 m/s, Time = 15 s.

Step 2: \( \text{Distance} = 8 \times 15 \).

Step 3: \( \text{Distance} = 120 \) m.

Answer: The cyclist travels 120 m.
Watch Out!
When multiplying, make sure the time units match the speed units. If speed is in km/h, time should be in hours. If speed is in m/s, time should be in seconds.

Practice for Calculating Distance

  1. A car travels at 80 km/h for 3 hours. How far does it travel?
  2. A train moves at 120 km/h for 2.5 hours. What distance does it cover?
  3. A runner runs at 6 m/s for 20 seconds. How far does he run?
  4. A plane flies at 800 km/h for 1.5 hours. What is the distance?
  5. A boat travels at 25 km/h for 4 hours. How far does it go?

Calculating Time

If you know the speed and the distance, you can find the time taken. Time is distance divided by speed. $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$

Step-by-Step: Finding Time
1. Identify the distance.
2. Identify the speed.
3. Divide distance by speed: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
4. Include the units (e.g., hours, minutes).
5. Simplify if needed.
Example 5: Finding Time
Problem: A car travels 240 km at a speed of 80 km/h. How long does the trip take?

Solution:
Step 1: Distance = 240 km, Speed = 80 km/h.

Step 2: \( \text{Time} = \frac{240}{80} \).

Step 3: \( \text{Time} = 3 \) hours.

Answer: The trip takes 3 hours.
Example 6: Finding Time (minutes)
Problem: A cyclist travels 1,500 meters at a speed of 5 m/s. How long does it take in seconds? In minutes?

Solution:
Step 1: Distance = 1,500 m, Speed = 5 m/s.

Step 2: \( \text{Time} = \frac{1500}{5} = 300 \) seconds.

Step 3: Convert to minutes: \( 300 \div 60 = 5 \) minutes.

Answer: 300 seconds (5 minutes).
Watch Out!
When dividing, make sure the units are consistent. Distance in kilometers divided by speed in km/h gives time in hours. Distance in meters divided by speed in m/s gives time in seconds.

Practice for Calculating Time

  1. A car travels 360 km at 90 km/h. How long does it take?
  2. A train travels 300 km at 100 km/h. What is the travel time?
  3. A runner runs 400 m at 8 m/s. How many seconds does it take?
  4. A plane travels 1,500 km at 600 km/h. What is the flight time?
  5. A boat travels 120 km at 40 km/h. How long does it take?

Average Speed

Average speed is the total distance traveled divided by the total time taken. This is useful when an object travels at different speeds for different parts of a journey. $$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$

Step-by-Step: Finding Average Speed
1. Find the total distance traveled by adding all the distances.
2. Find the total time taken by adding all the times.
3. Divide the total distance by the total time.
4. Include the units (e.g., km/h).
5. Simplify if needed.
Example 7: Average Speed
Problem: A car travels 150 km at 50 km/h, then 120 km at 60 km/h. What is the average speed for the whole trip?

Solution:
Step 1: Total Distance = 150 + 120 = 270 km.

Step 2: Time for first part = \( \frac{150}{50} = 3 \) hours.

Step 3: Time for second part = \( \frac{120}{60} = 2 \) hours.

Step 4: Total Time = 3 + 2 = 5 hours.

Step 5: \( \text{Average Speed} = \frac{270}{5} = 54 \) km/h.

Answer: The average speed is 54 km/h.
Example 8: Average Speed with Different Speeds
Problem: A bus travels 200 km at 80 km/h, then 300 km at 100 km/h, then 150 km at 75 km/h. Find the average speed.

Solution:
Step 1: Total Distance = 200 + 300 + 150 = 650 km.

Step 2: Time1 = \( \frac{200}{80} = 2.5 \) h, Time2 = \( \frac{300}{100} = 3 \) h, Time3 = \( \frac{150}{75} = 2 \) h.

Step 3: Total Time = 2.5 + 3 + 2 = 7.5 h.

Step 4: \( \text{Average Speed} = \frac{650}{7.5} = 86.67 \) km/h (rounded to 2 decimal places).

Answer: The average speed is approximately 86.67 km/h.
Watch Out!
Average speed is NOT the average of the speeds. You must use the total distance and total time. For example, if you travel at 60 km/h for 1 hour and 40 km/h for 1 hour, the average speed is \( \frac{100}{2} = 50 \) km/h, not \( \frac{60+40}{2} = 50 \) km/h (in this case they are the same, but that's not always true!).

Practice for Average Speed

  1. A car travels 180 km at 60 km/h and 240 km at 80 km/h. Find the average speed.
  2. A cyclist rides 80 km at 40 km/h, then 60 km at 30 km/h. What is the average speed?
  3. A train travels 300 km at 100 km/h, then 200 km at 80 km/h. Find the average speed.
  4. A plane flies 1,200 km at 600 km/h, then 800 km at 400 km/h. What is the average speed?
  5. A bus travels 150 km at 50 km/h, then 180 km at 60 km/h, then 120 km at 40 km/h. Find the average speed.

Cumulative Practice Exercises

  1. A car travels 420 km in 6 hours. What is its speed?
  2. A train travels at 120 km/h for 3.5 hours. How far does it go?
  3. A cyclist travels 90 km at 45 km/h. How long does it take?
  4. A runner runs 200 m in 25 seconds. What is the speed in m/s?
  5. A car travels 250 km at 50 km/h, then 200 km at 80 km/h. Find the average speed.
  6. A plane flies 1,000 km at 500 km/h, then 800 km at 400 km/h. Find the average speed.
  7. A train travels 240 km at 80 km/h. How long does it take in hours? In minutes?
  8. A bus travels 180 km at 60 km/h, then 120 km at 40 km/h. Find the average speed.
  9. A cyclist rides 30 km at 15 km/h, then 20 km at 10 km/h. What is the total distance? Total time? Average speed?
  10. A car travels 120 km at 60 km/h, then 180 km at 90 km/h. Find the average speed for the whole journey.
  11. A person walks 5 km at 4 km/h, then 8 km at 5 km/h. What is the average speed?
  12. A boat travels 60 km at 30 km/h, then 40 km at 20 km/h. Find the average speed.
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Solutions to Cumulative Exercises

  1. Step 1: Distance = 420 km, Time = 6 h.
    Step 2: \( \text{Speed} = \frac{420}{6} = 70 \) km/h.
    Answer: 70 km/h.
  2. Step 1: Speed = 120 km/h, Time = 3.5 h.
    Step 2: \( \text{Distance} = 120 \times 3.5 = 420 \) km.
    Answer: 420 km.
  3. Step 1: Distance = 90 km, Speed = 45 km/h.
    Step 2: \( \text{Time} = \frac{90}{45} = 2 \) hours.
    Answer: 2 hours.
  4. Step 1: Distance = 200 m, Time = 25 s.
    Step 2: \( \text{Speed} = \frac{200}{25} = 8 \) m/s.
    Answer: 8 m/s.
  5. Step 1: Total Distance = 250 + 200 = 450 km.
    Step 2: Time1 = \( \frac{250}{50} = 5 \) h, Time2 = \( \frac{200}{80} = 2.5 \) h.
    Step 3: Total Time = 5 + 2.5 = 7.5 h.
    Step 4: \( \text{Average Speed} = \frac{450}{7.5} = 60 \) km/h.
    Answer: 60 km/h.
  6. Step 1: Total Distance = 1,000 + 800 = 1,800 km.
    Step 2: Time1 = \( \frac{1000}{500} = 2 \) h, Time2 = \( \frac{800}{400} = 2 \) h.
    Step 3: Total Time = 2 + 2 = 4 h.
    Step 4: \( \text{Average Speed} = \frac{1800}{4} = 450 \) km/h.
    Answer: 450 km/h.
  7. Step 1: Distance = 240 km, Speed = 80 km/h.
    Step 2: \( \text{Time} = \frac{240}{80} = 3 \) hours.
    Step 3: 3 hours = 3 × 60 = 180 minutes.
    Answer: 3 hours (180 minutes).
  8. Step 1: Total Distance = 180 + 120 = 300 km.
    Step 2: Time1 = \( \frac{180}{60} = 3 \) h, Time2 = \( \frac{120}{40} = 3 \) h.
    Step 3: Total Time = 3 + 3 = 6 h.
    Step 4: \( \text{Average Speed} = \frac{300}{6} = 50 \) km/h.
    Answer: 50 km/h.
  9. Step 1: Total Distance = 30 + 20 = 50 km.
    Step 2: Time1 = \( \frac{30}{15} = 2 \) h, Time2 = \( \frac{20}{10} = 2 \) h.
    Step 3: Total Time = 2 + 2 = 4 h.
    Step 4: \( \text{Average Speed} = \frac{50}{4} = 12.5 \) km/h.
    Answer: Total distance = 50 km, Total time = 4 h, Average speed = 12.5 km/h.
  10. Step 1: Total Distance = 120 + 180 = 300 km.
    Step 2: Time1 = \( \frac{120}{60} = 2 \) h, Time2 = \( \frac{180}{90} = 2 \) h.
    Step 3: Total Time = 2 + 2 = 4 h.
    Step 4: \( \text{Average Speed} = \frac{300}{4} = 75 \) km/h.
    Answer: 75 km/h.
  11. Step 1: Total Distance = 5 + 8 = 13 km.
    Step 2: Time1 = \( \frac{5}{4} = 1.25 \) h, Time2 = \( \frac{8}{5} = 1.6 \) h.
    Step 3: Total Time = 1.25 + 1.6 = 2.85 h.
    Step 4: \( \text{Average Speed} = \frac{13}{2.85} \approx 4.56 \) km/h.
    Answer: Approximately 4.56 km/h.
  12. Step 1: Total Distance = 60 + 40 = 100 km.
    Step 2: Time1 = \( \frac{60}{30} = 2 \) h, Time2 = \( \frac{40}{20} = 2 \) h.
    Step 3: Total Time = 2 + 2 = 4 h.
    Step 4: \( \text{Average Speed} = \frac{100}{4} = 25 \) km/h.
    Answer: 25 km/h.

Conclusion & Summary

Rate problems—especially those involving speed, distance, and time—are used in many real-world situations from planning road trips to analyzing athletic performance. The key is to remember the relationship: speed equals distance divided by time. With this one formula and its rearrangements, you can solve any problem involving uniform motion.

Key Takeaways:
1. Speed: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
2. Distance: \( \text{Distance} = \text{Speed} \times \text{Time} \).
3. Time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
4. Average Speed: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \).
5. Always check units and convert if necessary.

Keep practicing—you'll use these skills to plan trips, compare travel options, and solve real-world problems!

Video Resource

Watch this video for more examples of speed, distance, and time problems.

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