Compound measures: speed, density and rates
A compound measure combines two units into one idea: how much of this per unit of that. Getting the units right is usually most of the mathematics.
Part of your national curriculum
- Ratio, proportion and rates of change: Use compound units such as speed, unit pricing and density to solve problems
Lesson overview
What you'll learn in this lesson
Use compound units such as speed, unit pricing and density to solve problems
Key learning points
- • Speed, distance, time
- • Density and mass
- • Unit pricing and best buy
- • Average speed over a journey
This lesson at a glance
- 28 minutes
- 17 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 17
Watch first
Compound measures: speed, density and rates
An English teaching video on this topic from The Organic Chemistry Tutor. Watch it, then scroll down for the lesson.
In a nutshell
Use compound units such as speed, unit pricing and density to solve problems
Learning cycle
Part 2 of 17
Learning cycle 1 of 2
Part 1 · Speed, distance, time
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 17
Learn
Speed, distance, time
Speed = distance ÷ time. A car covering 150 km in 2.5 hours travels at 60 km/h. Rearranging gives distance = speed × time and time = distance ÷ speed. Time in minutes must be converted to hours before dividing, or the answer will be sixty times out.
Explore the idea
Part 4 of 17
Learn
Density and mass
Density = mass ÷ volume, measured in g/cm³ or kg/m³. A block of mass 480 g and volume 60 cm³ has density 8 g/cm³. Comparing densities explains why some objects float, and identical formulas structure pressure as force ÷ area.
Quick check
Part 5 of 17
Quick check
A car travels 150 km in 2.5 hours. Its speed is
Quick check
Part 6 of 17
Quick check
Mass 480 g, volume 60 cm³. Density is
Learning cycle
Part 7 of 17
Learning cycle 2 of 2
Part 2 · Unit pricing and best buy
A short piece of teaching, then a check to make sure it has landed.
Reset break
Part 8 of 17
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Explore the idea
Part 9 of 17
Learn
Unit pricing and best buy
To compare a 750 g pack at £2.40 with a 1.2 kg pack at £3.72, calculate price per 100 g: 32p against 31p. The larger pack is better value, but only just, and best-buy answers must state the comparable rate for both options, not just the winner.
Explore the idea
Part 10 of 17
Learn
Average speed over a journey
Average speed is total distance divided by total time, never the mean of two speeds. Driving 60 km at 60 km/h then 60 km at 30 km/h takes 1 + 2 = 3 hours for 120 km, giving 40 km/h, not 45 km/h.
Quick check
Part 11 of 17
Quick check
Which shows better value?
Quick check
Part 12 of 17
Quick check
Average speed is
Explore the idea
Part 13 of 17
Worked example
Worked answer: a journey of 60 km at 60 km/h then 60 km at 30 km/h. Find the average speed (3 marks)
The first stage takes 60 ÷ 60 = 1 hour and the second takes 60 ÷ 30 = 2 hours, giving 3 hours in total (1). The total distance is 120 km (1). Average speed = 120 ÷ 3 = 40 km/h (1). Averaging 60 and 30 to get 45 km/h ignores the extra time spent at the slower speed.
Challenge round
Part 14 of 17
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 15 of 17
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
Speed = ____ ÷ time.
____ = mass ÷ volume, measured in g/cm³ or kg/m³.
Mastery quiz
Part 16 of 17
Marked quiz
End of lesson quiz: Compound measures: speed, density and rates
4 questions, marked with the reasoning shown. No timer.
1. A car travels 150 km in 2.5 hours. Its speed is
2. Mass 480 g, volume 60 cm³. Density is
3. Which shows better value?
4. Average speed is
Lesson round-up
Part 17 of 17
Lesson round-up
Ready when you are
Quiz score
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Points this lesson
0
Best run
0 in a row
Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 17 · Watch & discover
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Use compound units such as speed, unit pricing and density to solve problems