17. Average speed and time calculations

17. Average speed and time calculations - step by step solution procedure and detailed explanation for each problem.

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Below are 10 average speed and time calculation problems, each with a step-by-step solution procedure and detailed explanation. Average speed is calculated using the formula:
Average Speed = Total Distance / Total Time
For problems involving different speeds over segments, the average speed accounts for the total distance and total time, not the arithmetic mean of the speeds. Time calculations often use the formula:
Time = Distance / Speed
or rearrangements of it. These problems cover various scenarios, including uniform motion, multiple segments, and finding distance or time, to provide a comprehensive understanding.
Problem 1: Basic Average Speed
A car travels 120 km in 2 hours. Find the average speed.
Step-by-Step Solution:
  1. Identify the given values:
    • Total distance = 120 km
    • Total time = 2 hours
  2. Use the average speed formula: Average Speed = Total Distance / Total Time
  3. Substitute the values: Average Speed = 120 / 2 = 60 km/h
  4. State the result: The average speed is 60 km/h.
Detailed Explanation: Average speed is the total distance traveled divided by the total time taken. Here, the car covers 120 km in 2 hours, so dividing gives 60 km/h. This assumes constant speed, making it a straightforward application of the formula.
Answer: Average Speed = 60 km/h
Problem 2: Finding Time
A cyclist travels 45 km at an average speed of 15 km/h. How long does it take?
Step-by-Step Solution:
  1. Identify the given values:
    • Distance = 45 km
    • Average Speed = 15 km/h
  2. Use the time formula: Time = Distance / Speed
  3. Substitute the values: Time = 45 / 15 = 3 hours
  4. State the result: The time taken is 3 hours.
Detailed Explanation: To find time, we rearrange the speed formula (Speed = Distance / Time) to isolate time. Dividing the distance (45 km) by the speed (15 km/h) gives 3 hours. This calculation assumes the speed is constant over the entire distance.
Answer: Time = 3 hours
Problem 3: Finding Distance
A train travels at an average speed of 80 km/h for 4 hours. What distance does it cover?
Step-by-Step Solution:
  1. Identify the given values:
    • Average Speed = 80 km/h
    • Time = 4 hours
  2. Use the distance formula: Distance = Speed × Time
  3. Substitute the values: Distance = 80 × 4 = 320 km
  4. State the result: The distance covered is 320 km.
Detailed Explanation: Distance is calculated by multiplying speed by time. Here, 80 km/h for 4 hours yields 320 km. This is a direct application of the formula, assuming constant speed, and is useful for determining how far a vehicle travels in a given time.
Answer: Distance = 320 km
Problem 4: Average Speed Over Two Equal Distances
A car travels 100 km at 50 km/h and another 100 km at 100 km/h. Find the average speed for the entire journey.
Step-by-Step Solution:
  1. Identify the given values:
    • Distance 1 = 100 km, Speed 1 = 50 km/h
    • Distance 2 = 100 km, Speed 2 = 100 km/h
  2. Calculate time for each segment:
    • Time 1 = Distance 1 / Speed 1 = 100 / 50 = 2 hours
    • Time 2 = Distance 2 / Speed 2 = 100 / 100 = 1 hour
  3. Calculate total distance and time:
    • Total distance = 100 + 100 = 200 km
    • Total time = 2 + 1 = 3 hours
  4. Calculate average speed: Average Speed = Total Distance / Total Time = 200 / 3 ≈ 66.67 km/h
  5. State the result: The average speed is approximately 66.67 km/h.
Detailed Explanation: Average speed is not the average of the two speeds (50 + 100)/2 = 75 km/h) because the car spends more time at the slower speed. We calculate the time for each segment, sum the distances (200 km) and times (3 hours), then divide. The result (66.67 km/h) reflects the weighted effect of the slower segment.
Answer: Average Speed ≈ 66.67 km/h
Problem 5: Average Speed Over Two Different Distances
A biker travels 60 km at 30 km/h and then 40 km at 20 km/h. Find the average speed.
Step-by-Step Solution:
  1. Identify the given values:
    • Distance 1 = 60 km, Speed 1 = 30 km/h
    • Distance 2 = 40 km, Speed 2 = 20 km/h
  2. Calculate time for each segment:
    • Time 1 = 60 / 30 = 2 hours
    • Time 2 = 40 / 20 = 2 hours
  3. Calculate total distance and time:
    • Total distance = 60 + 40 = 100 km
    • Total time = 2 + 2 = 4 hours
  4. Calculate average speed: Average Speed = 100 / 4 = 25 km/h
  5. State the result: The average speed is 25 km/h.
Detailed Explanation: The total distance (100 km) and total time (4 hours) are used to find the average speed. Despite different distances and speeds, the times for each segment are equal (2 hours each), making the calculation straightforward. The average speed (25 km/h) lies between the two speeds, weighted by the distances.
Answer: Average Speed = 25 km/h
Problem 6: Average Speed for a Round Trip
A person drives 80 km to a destination at 40 km/h and returns at 80 km/h. Find the average speed for the round trip.
Step-by-Step Solution:
  1. Identify the given values:
    • Distance to destination = 80 km, Speed = 40 km/h
    • Distance back = 80 km, Speed = 80 km/h
  2. Calculate time for each segment:
    • Time to = 80 / 40 = 2 hours
    • Time back = 80 / 80 = 1 hour
  3. Calculate total distance and time:
    • Total distance = 80 + 80 = 160 km
    • Total time = 2 + 1 = 3 hours
  4. Calculate average speed: Average Speed = 160 / 3 ≈ 53.33 km/h
  5. State the result: The average speed is approximately 53.33 km/h.
Detailed Explanation: For a round trip with equal distances, the average speed depends on the time spent at each speed. The slower outbound trip (2 hours) takes longer than the return (1 hour), so the average speed (53.33 km/h) is closer to the slower speed than the arithmetic mean (60 km/h).
Answer: Average Speed ≈ 53.33 km/h
Problem 7: Time to Cover Remaining Distance
A car travels 150 km in 3 hours and needs to cover 90 km more to reach its destination. If it travels at 60 km/h, how long will it take to complete the journey?
Step-by-Step Solution:
  1. Identify the given values for the remaining distance:
    • Remaining distance = 90 km
    • Speed = 60 km/h
  2. Use the time formula: Time = Distance / Speed
  3. Substitute the values: Time = 90 / 60 = 1.5 hours
  4. State the result: The time to cover the remaining distance is 1.5 hours.
Detailed Explanation: The initial 150 km in 3 hours is irrelevant to the remaining distance’s time calculation, as we’re only asked for the time to cover 90 km at 60 km/h. Dividing distance by speed gives 1.5 hours (or 90 minutes). This isolates the calculation for the specified segment.
Answer: Time = 1.5 hours
Problem 8: Average Speed with a Stop
A truck travels 200 km at 50 km/h, stops for 1 hour, then travels 100 km at 50 km/h. Find the average speed for the entire journey.
Step-by-Step Solution:
  1. Identify the given values:
    • Distance 1 = 200 km, Speed 1 = 50 km/h
    • Stop time = 1 hour
    • Distance 2 = 100 km, Speed 2 = 50 km/h
  2. Calculate time for each segment:
    • Time 1 = 200 / 50 = 4 hours
    • Stop time = 1 hour
    • Time 2 = 100 / 50 = 2 hours
  3. Calculate total distance and time:
    • Total distance = 200 + 100 = 300 km
    • Total time = 4 + 1 + 2 = 7 hours
  4. Calculate average speed: Average Speed = 300 / 7 ≈ 42.86 km/h
  5. State the result: The average speed is approximately 42.86 km/h.
Detailed Explanation: The stop time (1 hour) is included in the total time because average speed accounts for the entire journey, including pauses. The total distance (300 km) divided by the total time (7 hours) gives 42.86 km/h, lower than 50 km/h due to the stop.
Answer: Average Speed ≈ 42.86 km/h
Problem 9: Harmonic Mean for Equal Distances
A plane flies 500 km at 250 km/h and returns at 500 km/h. Find the average speed using the harmonic mean formula.
Step-by-Step Solution:
  1. Identify the given values:
    • Distance = 500 km each way, Total distance = 1000 km
    • Speed 1 = 250 km/h, Speed 2 = 500 km/h
  2. Use the harmonic mean formula for equal distances: Average Speed = (2 × S1 × S2) / (S1 + S2)
  3. Substitute the values: Average Speed = (2 × 250 × 500) / (250 + 500)
  4. Calculate: Average Speed = 250000 / 750 = 333.33 km/h
  5. Verify using total distance/time:
    • Time 1 = 500 / 250 = 2 hours
    • Time 2 = 500 / 500 = 1 hour
    • Total time = 2 + 1 = 3 hours
    • Average Speed = 1000 / 3 ≈ 333.33 km/h
  6. State the result: The average speed is approximately 333.33 km/h.
Detailed Explanation: For equal distances, the harmonic mean formula simplifies average speed calculation: (2 × S1 × S2) / (S1 + S2). This gives 333.33 km/h, confirmed by the total distance (1000 km) divided by total time (3 hours). The harmonic mean accounts for the time spent at each speed.
Answer: Average Speed ≈ 333.33 km/h
Problem 10: Time to Catch Up
A car travels at 60 km/h, and 30 minutes later, a motorcycle follows at 90 km/h. How long does it take for the motorcycle to catch up?
Step-by-Step Solution:
  1. Identify the given values:
    • Car speed = 60 km/h
    • Motorcycle speed = 90 km/h
    • Head start = 30 minutes = 0.5 hours
  2. Calculate the car’s head start distance: Distance = 60 × 0.5 = 30 km
  3. Determine relative speed: Relative speed = 90 - 60 = 30 km/h
  4. Calculate time to catch up: Time = Distance / Relative Speed = 30 / 30 = 1 hour
  5. State the result: The motorcycle takes 1 hour to catch up.
Detailed Explanation: The car’s 30-minute head start means it travels 30 km before the motorcycle starts. The motorcycle catches up by closing the 30 km gap at a relative speed of 30 km/h (90 - 60). Dividing the distance by the relative speed gives 1 hour, the time from when the motorcycle starts.
Answer: Time = 1 hour
Summary
These 10 problems cover a range of average speed and time calculations: basic speed, time, and distance; multiple segments with equal or different distances; round trips; stops; harmonic mean for equal distances; and relative speed for catching up. Each problem emphasizes calculating total distance and time accurately, especially when speeds vary, and using appropriate formulas (Distance = Speed × Time or rearrangements). Average speed is sensitive to time spent at each speed, not just the speeds themselves, making careful time calculations critical.

 

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