Rates, average speed and compound measures
Read a rate as one quantity per unit of another, then use compatible units to model a situation.
Core and Extended.
Before you begin
- Convert minutes to fractions of an hour.
- Use ratios and divide decimal quantities.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How many hours is 1 hour 30 minutes?
What you will learn
- Calculate with pay, flow and consumption rates.
- Find average speed using total distance and total time.
- Apply given density, pressure and population-density formulas.
Per tells you what to divide by
A rate compares quantities with different units. Pay of $54 for 4.5 hours is 54/4.5 = $12 per hour. A flow of 18 litres in 30 seconds is 0.6 L/s. Dividing the amount by elapsed time creates the rate; multiplying that rate by a new time recovers an amount under a constant-rate assumption.
Fuel consumption can be expressed as kilometres per litre or litres per 100 kilometres. These use different denominators and cannot be compared by simply choosing the smaller number. A car using 6 L for 90 km achieves 15 km/L, equivalent to 100/15 ≈ 6.67 L per 100 km.
Average speed uses the whole journey
Average speed is total distance divided by total elapsed time. Convert 2 hours 15 minutes to 2 + 15/60 = 2.25 hours, not 2.15 hours. A journey of 135 km in that time has average speed 60 km/h. If stops belong to the stated journey time, include them.
The ordinary mean of two speeds is valid for equal time intervals, but generally not for equal distances. Travelling 60 km at 30 km/h takes two hours; another 60 km at 60 km/h takes one hour. Total speed is 120/3 = 40 km/h, not the unweighted average 45 km/h.
Convert units before combining values
To convert km/h to m/s, multiply by 1000 metres per kilometre and divide by 3600 seconds per hour. Thus 72 km/h = 20 m/s. Reversing the conversion multiplies by 3.6. Writing these conversion factors makes their direction visible rather than relying on an unexplained decimal rule.
Unit powers also matter. One cubic centimetre equals one millilitre, while a cubic metre is one million cubic centimetres. A density calculation cannot divide kilograms by cubic centimetres and report g/cm³ without converting mass. Keep the units beside each input until the desired units can be justified.
Use the supplied model and interpret it
Given density = mass/volume, a 156 g sample occupying 60 cm³ has density 2.6 g/cm³. Given pressure = force/area, a 240 N force over 0.08 m² gives 3000 N/m², or 3000 Pa. These formulas define different rates; identify the denominator from the context rather than memorising one operation for every problem.
Population density is population divided by area: 18 000 people in 45 km² gives 400 people/km². It is an average and does not prove that each square kilometre contains exactly 400 people. Models of speed, flow or consumption also average real variation, so state whether constant-rate behaviour is an assumption.
Pause and explain
A journey has equal 60 km legs at 30 and 60 km/h. What is its average speed?
Worked example
A cyclist covers 54 km in 2 hours 15 minutes. Find average speed in km/h and m/s.
Show the worked solution
- Time = 2 + 15/60 = 2.25 hours.
- Average speed = 54/2.25 = 24 km/h.
- Convert: 24 × 1000/3600 = 20/3 m/s ≈ 6.67 m/s.
Answer 24 km/h; 20/3 m/s (about 6.67 m/s)
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A tap fills 45 L in 75 seconds at a constant rate. How much flows in four minutes?
Give me a hint
First find litres per second, then convert four minutes to seconds.
Compare my reasoning
- Rate = 45/75 = 0.6 L/s.
- Four minutes is 240 seconds.
- Volume = 0.6 × 240 = 144 L.
144 L
Look for these in your work
- Matched the time unit to the rate.
- Used the constant-rate assumption explicitly.
Given density = mass/volume, find the density of 210 g occupying 75 cm³. Given pressure = force/area, find the pressure of 360 N over 0.12 m².
Give me a hint
Use the supplied formulas and keep each denominator's units.
Compare my reasoning
- Density = 210/75 = 2.8 g/cm³.
- Pressure = 360/0.12 = 3000 N/m² = 3000 Pa.
- Check: 2.8 × 75 = 210 and 3000 × 0.12 = 360.
2.8 g/cm³; 3000 Pa
Look for these in your work
- Used mass per volume and force per area.
- Checked each result by reversing its division.
A bus travels 80 km in one hour, stops for 30 minutes, then travels 40 km in one hour. Find its average speed for the complete journey and explain why 60 km/h is incorrect.
Give me a hint
Include the stop in the total elapsed time.
Compare my reasoning
- Total distance = 80 + 40 = 120 km.
- Total time = 1 + 0.5 + 1 = 2.5 hours.
- Average speed = 120/2.5 = 48 km/h. Sixty ignores the stop.
48 km/h
Look for these in your work
- Included all elapsed time requested by the context.
- Explained the source of the misleading alternative.
Common mistakes
- Reading 2 hours 15 minutes as 2.15 hours.
- Averaging speeds without checking their time weights.
- Dropping units or mixing squared and cubed conversion factors.
What can you explain now?
Convert 90 km/h to m/s.
Compare with the explanation
25 m/s
Multiply by 1000 and divide by 3600; 90/3.6 = 25.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Explain why average speed must use total distance and total time. Invent a flow-rate question and check its answer by reversing the calculation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources