SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 28
Average speed is total distance divided by total time. It is not usually the simple average of two speeds. Multi-stage journeys make this distinction visible because different stages may cover different distances or take different amounts of time.
This guide develops rate structure, average rate, average speed, total-distance/total-time reasoning, equal-distance and equal-time contrasts, stoppages, unit conversion, tables, graphs, estimation and common weighted-average errors. It extends Rate, Speed and Unit Conversion into multi-stage reasoning.
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1. Rate compares two different quantities
A rate has a numerator quantity and denominator quantity: kilometres per hour, dollars per item, litres per minute or words per minute.
Speed is a rate comparing distance with time.
2. Average speed uses totals
Average speed = total distance ÷ total time.
Worked example
A journey covers 120 km in 2 h and then 60 km in 1 h.
Total distance = 180 km.
Total time = 3 h.
Average speed = 60 km/h.
3. Simple averaging of speeds can fail
A vehicle travels 60 km at 30 km/h and then 60 km at 60 km/h.
The simple average of 30 and 60 is 45 km/h, but this is not the actual average speed.
Time for first stage = 2 h. Time for second = 1 h.
Total distance = 120 km; total time = 3 h.
Average speed = 40 km/h.
4. Equal time is a special case where the arithmetic mean works
If a vehicle travels 1 h at 30 km/h and 1 h at 60 km/h:
Distances are 30 km and 60 km.
Total distance = 90 km; total time = 2 h.
Average speed = 45 km/h.
The arithmetic mean works because the time weights are equal.
5. Equal distance produces a different average
For equal distances at speeds u and v, the slower stage takes longer and therefore influences the average more strongly.
Example
100 km at 50 km/h takes 2 h; 100 km at 100 km/h takes 1 h.
Average speed = 200/3 ≈ 66.7 km/h, not 75 km/h.
6. Stoppage time matters if it belongs to the total journey time
A bus travels 90 km in 1.5 h, stops for 0.5 h, then travels 60 km in 1 h.
Total distance = 150 km.
Total elapsed time = 3 h.
Average speed over the full elapsed journey = 50 km/h.
Boundary
If the question asks for average speed only while moving, exclude stoppage time. Read the wording carefully.
7. Average rate is broader than average speed
A machine processes 120 items in 3 h and 180 items in 2 h.
Total items = 300.
Total time = 5 h.
Average processing rate = 60 items/h.
8. Units must match before totals are combined
A stage lasting 45 minutes must be converted to 0.75 h before combining with speeds in km/h.
Example
72 km/h for 45 min covers 72×0.75 = 54 km.
9. Distance, speed and time form one reversible relationship
d = vt.
v = d/t.
t = d/v.
These are rearrangements of one structure rather than three unrelated formulas.
10. Multi-stage journeys should be organised before calculation
| Stage | Distance | Time | Speed |
|---|---|---|---|
| 1 | 80 km | 1 h | 80 km/h |
| 2 | 90 km | 1.5 h | 60 km/h |
Total distance = 170 km, total time = 2.5 h, average speed = 68 km/h.
A table prevents stage information from being mixed incorrectly.
11. When distance is missing, derive it from speed and time
Stage 1: 50 km/h for 2 h → 100 km.
Stage 2: 80 km/h for 1.5 h → 120 km.
Total distance = 220 km; total time = 3.5 h.
Average speed ≈ 62.9 km/h.
12. When time is missing, derive it from distance and speed
Stage 1: 90 km at 60 km/h → 1.5 h.
Stage 2: 120 km at 80 km/h → 1.5 h.
Equal times here mean average speed = (60+80)/2 = 70 km/h.
13. Average speed must lie between the stage speeds when all times are positive
If stage speeds are 40 km/h and 70 km/h, the average speed over the moving stages should lie between 40 and 70 km/h.
An answer of 85 km/h would be impossible under this two-stage model.
Reasonableness check
The average is weighted by time spent at each speed.
14. More time at the slower speed pulls the average downward
Suppose a vehicle travels 1 h at 80 km/h and 3 h at 40 km/h.
Total distance = 80+120=200 km.
Total time = 4 h.
Average speed = 50 km/h, closer to 40 because more time was spent there.
15. More distance at a speed does not directly determine the weighting
Average speed weights speeds through time, because distance = speed×time and the final denominator is total time.
This is why equal-distance journeys do not use equal speed weights.
16. Round-trip problems often reveal the trap
A cyclist travels 30 km out at 15 km/h and 30 km back at 30 km/h.
Outward time = 2 h.
Return time = 1 h.
Total distance = 60 km; total time = 3 h.
Average speed = 20 km/h.
17. Average speed and average velocity are different concepts
Average speed uses total distance. Average velocity uses displacement divided by total time.
For a round trip returning to the start, displacement is zero, so average velocity is zero while average speed is positive.
Treat formal velocity work according to the learner’s course; the distinction helps prevent later confusion.
18. Distance-time graphs encode rate visually
On a distance-time graph, gradient represents speed when distance is plotted vertically and time horizontally.
A steeper straight segment indicates greater speed.
A horizontal segment indicates no change in distance: the object is stationary.
19. Total journey average can be recovered from the graph endpoints and elapsed time
If a distance-time graph ends at 180 km after 3 h and began at zero distance, average speed over the full interval is 180/3 = 60 km/h.
Intermediate slopes may differ, but the total average uses the complete distance and time.
20. Common average-rate errors
| Error | Why it fails | Repair prompt |
|---|---|---|
| Averages 30 and 60 to get 45 for equal distances | Times are unequal | What are total distance and total time? |
| Ignores stoppage time | Elapsed-time denominator incomplete | Does the question include the stop? |
| Adds hours and minutes directly | Units mismatch | Can all times be expressed in one unit? |
| Uses final-stage distance only | Total distance not accumulated | What distance was covered across every stage? |
| Average lies above every stage speed | Model or arithmetic error | Can a time-weighted average exceed all positive stage speeds? |
21. Practice laboratory
- Travel 120 km in 2 h and 60 km in 1 h. Find average speed.
- Travel 60 km at 30 km/h and 60 km at 60 km/h. Find average speed.
- Travel 1 h at 40 km/h and 1 h at 70 km/h. Find average speed.
- Travel 2 h at 50 km/h and 1 h at 80 km/h. Find total distance and average speed.
- Travel 100 km at 50 km/h and 150 km at 75 km/h. Find average speed.
- A bus travels 80 km in 1 h, stops 30 min, then travels 40 km in 0.5 h. Find average speed over full elapsed time.
- A machine produces 180 units in 3 h and 160 units in 2 h. Find average rate.
- Convert 36 minutes to hours.
- At 90 km/h for 36 min, find distance.
- Travel 45 km at 30 km/h and 45 km at 90 km/h. Find average speed.
- Explain why the arithmetic mean of two speeds works for equal times.
- Explain why it can fail for equal distances.
- A round trip is 20 km each way, out at 20 km/h and back at 40 km/h. Find average speed.
- A distance-time graph ends at 240 km after 4 h. Find average speed from start if the start distance was zero.
22. Explained answers
1. 180/3=60 km/h.
2. Times 2 h and 1 h; 120/3=40 km/h.
3. (40+70)/2=55 km/h.
4. Distance=100+80=180 km; time=3 h; average=60 km/h.
5. Times 2 h and 2 h; total 250 km in 4 h=62.5 km/h.
6. Total distance=120 km; total time=2 h; average=60 km/h.
7. 340/5=68 units/h.
8. 0.6 h.
9. 90×0.6=54 km.
10. Times 1.5 h and 0.5 h; total 90 km in 2 h=45 km/h.
11. Equal times give equal weights in total distance.
12. Equal distances create unequal times, so the speeds have unequal time weights.
13. Times 1 h and 0.5 h; total 40 km in 1.5 h=26.7 km/h approximately.
14. 240/4=60 km/h.
23. Complete mixed problem
A fictional delivery route has three stages:
- 72 km at 48 km/h;
- a 30-minute stop;
- 90 km at 60 km/h.
Stage 1 time = 72/48 = 1.5 h.
Stop = 0.5 h.
Stage 2 time = 90/60 = 1.5 h.
Total distance = 162 km.
Total elapsed time = 3.5 h.
Average speed over the full elapsed route = 162/3.5 ≈ 46.3 km/h.
Check: because the stop contributes time but no distance, the average can be lower than both moving speeds.
24. Teaching average rate through totals first
Ban the phrase “average the speeds” until the learner has written total distance and total time.
Use equal-time and equal-distance pairs side by side so the learner sees when the arithmetic mean does and does not work.
Changed-case test
Keep the two speeds fixed but double the time spent at the slower speed. The average should move closer to the slower speed. This reveals the weighting mechanism.
25. Questions students often ask
Why can’t I always average two speeds?
Because average speed depends on total distance and total time, and the time spent at each speed may differ.
When does the arithmetic mean work?
For two stages with equal time durations.
Do stops count?
They count if the question asks for average speed over the full elapsed journey.
Can average speed be below both moving speeds?
Yes, if stoppage time is included because time passes while distance does not increase.
26. Return path and sources
Average-rate reasoning depends on unit discipline and total-quantity control. Revisit Rate, Speed and Unit Conversion for core rate structure and Coordinates, Linear Graphs and Relationships for graph interpretation.
Official curriculum reference: MOE Secondary Syllabus Directory. Exact average-rate and graph depth varies by subject level and school.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Separate the stages, standardise units, reconstruct missing distance or time, total the journey, divide totals, estimate the plausible range and return the average to its stated time interval.
