1MA1 · R1, R11Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 3–6

Compound Measures

Kilometres per hour, grams per cubic centimetre, newtons per square metre: each of these units combines two measurements. They are $\textbf{compound measures}$ , and the three that GCSE tests — speed, density and pressure — all share the same structure, so learning one pattern handles all three.

45 min Ratio & Proportion

The big picture

Each compound measure is one quantity divided by another: speed is distance per unit time, density is mass per unit volume, pressure is force per unit area. The unit tells you the formula — "km per hour" is km ÷ hours.

Most lost marks come from units, not formulas: a time in minutes used in km/h, or a mass in kilograms with a volume in cm³. Edexcel questions frequently include a conversion step deliberately.

What you'll learn

Use speed $=$ distance ÷ time and rearrange it
Use density $=$ mass ÷ volume and pressure $=$ force ÷ area
Convert between units of time and compound units
Solve multi-step problems combining compound measures

Speed

Speed $= \dfrac{\text{distance}}{\text{time}}$ , so distance $=$ speed × time and time $=$ distance ÷ speed.

Times must match the unit of speed. 36 minutes is $\tfrac{36}{60} = 0.6$ hours.

Average speed for a whole journey is total distance ÷ total time.

s = \dfrac{d}{t}

Speed.

Example 1A train travels 45 km in 36 minutes. Find its average speed in km/h.

136 minutes

= 0.6

hours.

2Speed

= 45 \div 0.6

Answer75 km/h

Your turnSame method, new numbers — try it before moving on.

A coach travels 150 km in 2 hours 30 minutes. Find its average speed in km/h.

Your answer

Show worked solution

Solution

12 hours 30 minutes $= 2.5$ hours.
2 $150 \div 2.5 = 60$ km/h.

Answer: $60$ km/h

Tip — Never write 1 h 15 min as 1.15 hours. It is $1\tfrac{15}{60} = 1.25$ hours.

Density

Density $= \dfrac{\text{mass}}{\text{volume}}$ , measured in g/cm³ or kg/m³.

Rearranged: mass $=$ density × volume, volume $=$ mass ÷ density.

\rho = \dfrac{m}{V}

Density.

Example 1Aluminium has density 2.7 g/cm³. Find the mass of a block measuring 2 cm by 4 cm by 5 cm.

1Volume

= 2 \times 4 \times 5 = 40

cm³.

2Mass

= 2.7 \times 40

Answer108 g

Your turnSame method, new numbers — try it before moving on.

A block measures 3 cm by 2 cm by 5 cm and has density 8 g/cm³. Find its mass, in grams.

Your answer

Show worked solution

Solution

1Volume $= 30$ cm³.
2Mass $= 8 \times 30 = 240$ g.

Answer: $240$ g

Pressure

Pressure $= \dfrac{\text{force}}{\text{area}}$ , measured in N/m² (pascals, Pa) or N/cm².

The same force on a smaller area gives a greater pressure — why a knife cuts and snowshoes stop you sinking.

P = \dfrac{F}{A}

Pressure.

Example 1A box weighing 600 N rests on a face of area 0.25 m². Find the pressure on the floor.

P = \dfrac{600}{0.25}

Answer2400 N/m²

Why this works

All three formulas have the form "top ÷ bottom" with the compound unit written "top per bottom". If you remember the unit, you have the formula.

Converting compound units

Convert each part of the unit separately. $72$ km/h $= \dfrac{72\,000 \text{ m}}{3600 \text{ s}} = 20$ m/s.

For area and volume units, the conversion factor is squared or cubed: $1$ m³ $= 1\,000\,000$ cm³, so $3$ g/cm³ $= 3\,000\,000$ g/m³ $= 3000$ kg/m³.

Example 1400 cm³ of a liquid of density 0.8 g/cm³ is mixed with 600 cm³ of a liquid of density 1.1 g/cm³. Find the density of the mixture.

1Masses:

0.8 \times 400 = 320

g and

1.1 \times 600 = 660

2Total mass 980 g, total volume 1000 cm³.

Answer0.98 g/cm³

Your turnSame method, new numbers — try it before moving on.

Convert 72 km/h into m/s.

Your answer

Show worked solution

Solution

1 $72$ km/h $= 72{,}000$ m per hour.
2 $72{,}000 \div 3600 = 20$ m/s.

Answer: $20$ m/s

Tip — The density of a mixture is not the average of the two densities. Find total mass and total volume, then divide.

Think like an examiner

Show unit conversions as a separate step.
Include units in every answer.
For mixtures or multi-stage journeys, work with totals.

Compound measures

\text{speed} = \tfrac{\text{distance}}{\text{time}}

Speed.

\text{density} = \tfrac{\text{mass}}{\text{volume}}

Density.

\text{pressure} = \tfrac{\text{force}}{\text{area}}

Pressure.

Watch out for these

Using 36 minutes as 0.36 hours.

36 minutes is

36 \div 60 = 0.6

hours.

Averaging densities to get the density of a mixture.

Divide total mass by total volume.

Converting 1 m³ to 100 cm³.

100^{3} = 1\,000\,000

cm³.

Mixing kg and g in a density calculation.

Use consistent units throughout.

Stretch yourself

Ella drives 120 km at an average speed of 80 km/h, then 90 km at an average speed of 60 km/h. Find her average speed for the whole journey.

Hint — Find the time for each part first. Average speed is total distance over total time.

Reveal solution

SolutionTime for part 1:

120 \div 80 = 1.5

h. Time for part 2:

90 \div 60 = 1.5

h. Total distance 210 km in 3 h, so the average speed is 70 km/h. (Here the times are equal, so the answer happens to equal the mean of 80 and 60; that is not true in general.)

\blacksquare

Questions students ask

What is a compound measure?

A measure made from two others, such as speed (distance and time) or density (mass and volume).

How do I remember the formulas?

Read the unit: "grams per cubic centimetre" means grams ÷ cm³.

What is a pascal?

One newton per square metre, the SI unit of pressure.

Key takeaways

Speed = distance ÷ time; density = mass ÷ volume; pressure = force ÷ area.
The unit "A per B" tells you the formula A ÷ B.
Convert times to hours, and match units before calculating.
Area and volume conversions square or cube the factor.
Use totals for average speeds and mixed densities.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

A car travels for 2 hours 15 minutes at an average speed of 48 km/h. How far does it travel?[2]
Show mark scheme
- M1 $48 \times 2.25$
- A1 $108$ km
Answer: $108$ km

Question 2Grade 52 marks

A force of 750 N acts on an area of 0.3 m². Work out the pressure.[2]
Show mark scheme
- M1 $750 \div 0.3$
- A1 $2500$ N/m²
Answer: $2500$ N/m²

Question 3Grade 63 marks

300 cm³ of metal A (density 7 g/cm³) and 200 cm³ of metal B (density 9 g/cm³) are combined. Work out the density of the alloy.[3]
Show mark scheme
- P1Masses $2100$ g and $1800$ g
- P1Total mass $3900$ g in $500$ cm³
- A1 $7.8$ g/cm³
Answer: $7.8$ g/cm³

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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