Average Speed (Multiple Segments)

Statement

The average speed over a journey with several parts is calculated as the total distance travelled divided by the total time taken:

\[\text{Average speed} = \frac{\text{total distance}}{\text{total time}}\]

This is different from simply averaging the speeds of individual segments. You must always use total distance and total time.

Why it’s true (short reason)

• Speed is defined as distance ÷ time.
• For a journey split into segments, total distance is the sum of all distances.
• Total time is the sum of all times for each segment.
• Dividing total distance by total time gives the true average speed.

Recipe (how to use it)

1. Write down the distance and speed (or time) for each segment of the journey.
2. Calculate the time for each segment (time = distance ÷ speed).
3. Add up all distances to find total distance.
4. Add up all times to find total time.
5. Use the formula: Average speed = total distance ÷ total time.

Spotting it

This formula is used when:

• A journey is broken into parts with different speeds or times.
• The question asks for average speed across the whole journey.
• Speeds cannot just be averaged directly, because times are different for each segment.

Common pairings

• Speed = distance ÷ time (for individual segments).
• Unit conversions (hours to minutes, km to m, etc.).
• Ratio reasoning, when different times are spent at different speeds.

Mini examples

1. Example: A car travels 60 km at 30 km/h, then 60 km at 60 km/h. Times: 2 hours + 1 hour = 3 hours. Total distance = 120 km. Average speed = 120 ÷ 3 = 40 km/h.
2. Example: A cyclist rides 20 km at 10 km/h, then 30 km at 15 km/h. Times: 2 hours + 2 hours = 4 hours. Total distance = 50 km. Average speed = 50 ÷ 4 = 12.5 km/h.

Pitfalls

• Simply averaging speeds (e.g. “(30+60)/2=45”) is usually wrong.
• Forgetting to convert units (minutes to hours, metres to km, etc.).
• Forgetting to add up all times before dividing.
• Mixing up distance and time when calculating segments.

Exam strategy

• Write times clearly under each segment to avoid confusion.
• Always calculate total distance and total time separately.
• Convert everything to consistent units before dividing.
• Check if the result is reasonable (between the lowest and highest segment speeds).

Summary

The average speed over multiple segments is not the average of individual speeds. It must be calculated using total distance and total time. This avoids errors and ensures the average speed reflects the entire journey.