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

Compound Measures

Q: How do I convert a decimal time like 2.7 hours to hours and minutes?

Keep the 2 hours and multiply the decimal part by 60: 0.7 60 = 42, so 2 hours 42 minutes.

A compound measure combines two different units: speed in kilometres per hour, density in grams per cubic centimetre, pressure in newtons per square metre. Each is a rate — one quantity divided by another — and its unit tells you the formula. "Per" means "divide by".

40 min Video by Maths Genie AQA GCSE Maths

The big picture

You do not need to memorise three separate formula triangles. The unit gives it away: km/h is kilometres ÷ hours, so speed = distance ÷ time. Rearranging gives the other two versions.

Most lost marks come from units, not formulas. Minutes must become hours before using km/h, and 45 minutes is 0.75 hours, not 0.45. Converting compound units — such as km/h to m/s — needs both parts converted.

Speed and DensityCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Use speed = distance ÷ time and its rearrangements
Use density = mass ÷ volume and pressure = force ÷ area
Convert between time units and between compound units
Solve problems involving other rates, such as flow rate

Speed, distance and time

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

Convert minutes to hours by dividing by 60: 20 minutes $= \frac{20}{60} = \frac{1}{3}$ hour, and 1 hour 24 minutes $= 1.4$ hours. Average speed uses the total distance and the total time.

Example 1A train travels 150 km in 1 hour 15 minutes. Find its average speed in km/h.

11 hour 15 minutes

= 1.25

hours.

2Speed

= 150 \div 1.25 = 120

Answer

120

km/h

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

A cyclist rides at 18 km/h for 40 minutes. How far do they travel, in km?

Your answer

Show hint

40 minutes is $\frac{40}{60}$ of an hour.

Show worked solution

Solution

140 minutes $= \frac{2}{3}$ hour.
2Distance $= 18 \times \frac{2}{3} = 12$ km.

Answer: $12$ km

Example 2Sara drives 90 km at 60 km/h, then 50 km at 100 km/h. Find her average speed for the whole journey.

1Times:

90 \div 60 = 1.5

h and

50 \div 100 = 0.5

2Total: 140 km in 2 h.

Answer

70

km/h

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

A runner covers 3 km in 15 minutes, then 2 km in 15 minutes. What is the average speed in km/h?

Your answer

Show worked solution

Solution

1Total distance 5 km; total time 30 minutes $= 0.5$ h.
2 $5 \div 0.5 = 10$ km/h.

Answer: $10$ km/h

Tip — Average speed is not the average of the two speeds: $\frac{60 + 100}{2} = 80$ , but Sara's true average is 70 km/h, because she spent longer at the slower speed.

Density and pressure

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

Pressure $= \dfrac{\text{force}}{\text{area}}$ , measured in N/m² (also called pascals). The same force over a smaller area gives a greater pressure — which is why a knife cuts and snowshoes do not sink.

Cover the quantity you want: the others show whether to multiply (side by side) or divide (one above the other).

Example 1A gold bar has volume 40 cm³ and density 19.3 g/cm³. Find its mass.

1Mass

=

density

\times

volume.

19.3 \times 40 = 772

Answer

772

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

A block of wood has mass 540 g and volume 900 cm³. Find its density in g/cm³.

Your answer

Show worked solution

Solution

1Density $= 540 \div 900 = 0.6$ g/cm³.

Answer: $0.6$ g/cm³

Example 2A box weighing 600 N rests on a face measuring 0.5 m by 0.4 m. Find the pressure it exerts.

1Area

= 0.5 \times 0.4 = 0.2

m².

2Pressure

= 600 \div 0.2 = 3000

Answer

3000

N/m²

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

A force of 240 N produces a pressure of 80 N/m². What is the area, in m²?

Your answer

Show worked solution

Solution

1Area $=$ force $\div$ pressure.
2 $240 \div 80 = 3$ m².

Answer: $3$ m²

Converting compound units

Convert each part separately. To change km/h to m/s, change kilometres to metres ( $\times 1000$ ) and hours to seconds ( $\div 3600$ ).

Area and volume units convert with squared and cubed factors: $1$ m² $= 10{,}000$ cm² and $1$ m³ $= 1{,}000{,}000$ cm³.

Example 1Convert 72 km/h to m/s.

172 km/h

= 72{,}000

m in 3600 s.

72{,}000 \div 3600 = 20

Answer

20

m/s

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

Convert 15 m/s to km/h.

Your answer

Show hint

In one hour (3600 s) the object travels $15 \times 3600$ m.

Show worked solution

Solution

1 $15 \times 3600 = 54{,}000$ m per hour.
2 $54{,}000$ m $= 54$ km, so 54 km/h.

Answer: $54$ km/h

Example 2Convert a density of 2.7 g/cm³ to kg/m³.

11 m³

= 1{,}000{,}000

cm³, so the mass of 1 m³ is

2.7 \times 1{,}000{,}000 = 2{,}700{,}000

2{,}700{,}000

= 2700

kg.

Answer

2700

kg/m³

Where 3.6 comes from

Multiplying by 3.6 converts m/s to km/h, because $\frac{3600}{1000} = 3.6$ . It is a useful shortcut, but deriving it from the separate conversions is what earns method marks.

Other rates

Any "per" quantity works the same way: litres per minute, pounds per hour, words per minute. Identify what is divided by what, and keep track of units.

Example 1A tank holds 1,350 litres. Water flows in at 45 litres per minute. How long does it take to fill?

1Time

= 1350 \div 45 = 30

minutes.

Answer

30 minutes

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

A printer prints 24 pages per minute. How many minutes does it take to print 1,080 pages?

Your answer

Show worked solution

Solution

1 $1080 \div 24 = 45$ minutes.

Answer: $45$ minutes

Think like an examiner

State units in the final answer — they are often required for the accuracy mark.
Show time conversions explicitly: "1 h 15 min = 1.25 h".
In multi-step problems (e.g. combining two materials), write what each number represents.

Compound measures

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

km/h, m/s.

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

g/cm³, kg/m³.

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

N/m².

Watch out for these

2 hours 30 minutes

= 2.3

hours.

30 minutes is half an hour:

2.5

hours.

Average speed as the mean of two speeds.

Total distance ÷ total time.

Converting km/h to m/s by dividing by 60.

Multiply by 1000 and divide by 3600.

Using an area in cm² with a force in N to get N/m².

Convert the area to m² first.

Stretch yourself

200 cm³ of metal A (density 8 g/cm³) is melted and mixed with 300 cm³ of metal B (density 5 g/cm³). Find the density of the resulting alloy, assuming no change in total volume.

Hint — Densities do not simply average. Find the total mass and the total volume.

Reveal solution

SolutionMass of A:

8 \times 200 = 1600

g. Mass of B:

5 \times 300 = 1500

g. Total: 3100 g in 500 cm³. Density

= 3100 \div 500 = 6.2

g/cm³.

\blacksquare

Questions students ask

Do I need to learn the formulas?

Yes — speed, density and pressure formulas are not given on the exam. The units remind you: g/cm³ means grams divided by cubic centimetres.

How do I convert a decimal time like 2.7 hours to hours and minutes?

Keep the 2 hours and multiply the decimal part by 60:

0.7 \times 60 = 42

, so 2 hours 42 minutes.

What is the difference between speed and velocity?

At GCSE Maths they are treated the same. In physics, velocity also includes direction.

Key takeaways

"Per" means divide: the unit shows the formula.
Speed = distance ÷ time; density = mass ÷ volume; pressure = force ÷ area.
Convert minutes to hours by dividing by 60.
Average speed = total distance ÷ total time.
Convert compound units one part at a time.

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 204 km at an average speed of 68 km/h. How long does the journey take?[2]
Show mark scheme
- M1 $204 \div 68$
- A13 hours
Answer: 3 hours

Question 2Grade 53 marks

A cylinder of copper has radius 2 cm and height 10 cm. The density of copper is 8.96 g/cm³.

Work out the mass of the cylinder. Give your answer to 3 significant figures.[3]
Show mark scheme
- M1Volume $= \pi \times 2^2 \times 10 = 125.66\ldots$ cm³
- M1Mass $= 8.96 \times$ their volume
- A1 $1130$ g (1125.9…)
Answer: $1130$ g

Question 3Grade 62 marks

A cheetah can run at 30 m/s. A racing car travels at 100 km/h.

Which is faster? Show how you decide.[2]
Show mark scheme
- M1Converts one speed: $30 \times 3600 \div 1000 = 108$ km/h, or $100{,}000 \div 3600 = 27.8$ m/s
- A1The cheetah is faster, with correct comparable figures

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.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A car travels 180 km in 3 hours. Its average speed is…
- 60 km/h
- 540 km/h
- 183 km/h
- 0.017 km/h
Show answer and explanation
Answer: 60 km/h
$180 \div 3$ .
45 minutes as a decimal of an hour is…
- $0.75$
- $0.45$
- $0.8$
- $1.33$
Show answer and explanation
Answer: $0.75$
$45 \div 60$ .
Density = 5 g/cm³ and volume = 12 cm³. Mass = ?
- 60 g
- 2.4 g
- 17 g
- 0.42 g
Show answer and explanation
Answer: 60 g
Mass = density × volume.
Walking at 5 km/h for 1 hour 30 minutes covers…
- 7.5 km
- 6.5 km
- 15 km
- 8 km
Show answer and explanation
Answer: 7.5 km
$5 \times 1.5$ .
A force of 500 N acts on 2.5 m². The pressure is…
- 200 N/m²
- 1250 N/m²
- 502.5 N/m²
- 0.005 N/m²
Show answer and explanation
Answer: 200 N/m²
$500 \div 2.5$ .
Convert 36 km/h to m/s.
- 10 m/s
- 100 m/s
- 0.6 m/s
- 129.6 m/s
Show answer and explanation
Answer: 10 m/s
$36{,}000 \div 3600$ .
60 km at 30 km/h then 60 km at 60 km/h. Average speed?
- 40 km/h
- 45 km/h
- 50 km/h
- 35 km/h
Show answer and explanation
Answer: 40 km/h
120 km in 3 hours.
1 m³ is how many cm³?
- $1{,}000{,}000$
- $1000$
- $100$
- $10{,}000$
Show answer and explanation
Answer: $1{,}000{,}000$
$100^3$ .

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