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Maths · Multiplicative reasoning

Compound measures

Use the formulae for speed, density and pressure, and convert between the units of compound measures.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    How many metres in a kilometre?

    Show answerHide answer

    1000

  2. 2

    How many seconds in an hour?

    Show answerHide answer

    3600

  3. 3

    Work out \(240 \div 3\).

    Show answerHide answer

    80

  4. 4

    How many cm³ in a litre?

    Show answerHide answer

    1000

  5. 5

    What is the area of a rectangle 20 cm by 10 cm?

    Show answerHide answer

    200 cm²

Learning Objectives

  1. 1Use \(\text{speed} = \dfrac{\text{distance}}{\text{time}}\).
  2. 2Use \(\text{density} = \dfrac{\text{mass}}{\text{volume}}\).
  3. 3Use \(\text{pressure} = \dfrac{\text{force}}{\text{area}}\).
  4. 4Convert between units for compound measures.

The Three Formulae

Know these and their units.

  • Speed

    Formula: \(\dfrac{\text{distance}}{\text{time}}\). Typical units: km/h, m/s

  • Density

    Formula: \(\dfrac{\text{mass}}{\text{volume}}\). Typical units: g/cm³, kg/m³

  • Pressure

    Formula: \(\dfrac{\text{force}}{\text{area}}\). Typical units: N/m², Pa

Speed

A train travels 240 km in 3 hours. Find its average speed.

Show the solutionHide the solution
  1. 1 Speed \(= \dfrac{\text{distance}}{\text{time}}\) \(\dfrac{240}{3}\)
  2. 2 Work it out 80

Answer80 km/h

Density

A block of metal has mass 450 g and volume 60 cm³. Find its density.

Show the solutionHide the solution
  1. 1 Density \(= \dfrac{\text{mass}}{\text{volume}}\) \(\dfrac{450}{60}\)
  2. 2 Work it out 7.5

Answer7.5 g/cm³

Pressure

A force of 600 N acts on an area of 0.5 m². Find the pressure.

Show the solutionHide the solution
  1. 1 Pressure \(= \dfrac{\text{force}}{\text{area}}\) \(\dfrac{600}{0.5}\)
  2. 2 Work it out 1200

Answer1200 N/m² (1200 Pa)

Changing Units

Change 90 km/h into metres per second.

Show the solutionHide the solution
  1. 1 90 km is 90 000 m \(90 \times 1000\)
  2. 2 1 hour is 3600 s \(90\,000 \div 3600\)
  3. 3 Work it out 25

Answer25 m/s

Rearranging a Formula

Aluminium has density 2.7 g/cm³. Find the mass of a piece with volume 40 cm³.

Show the solutionHide the solution
  1. 1 Mass \(=\) density \(\times\) volume \(2.7 \times 40\)
  2. 2 Work it out 108

Answer108 g

Watch the Units

Units must match before you divide.

  • Time

    Change minutes to hours before dividing by km per hour: 30 minutes is 0.5 hours.

  • Area

    1 m² is 10 000 cm², so pressure in N/m² needs area in m².

  • Density

    1 g/cm³ is 1000 kg/m³.

Which Metal?

A mystery block has mass 540 g and volume 200 cm³. Work out its density and use the table to decide which metal it is: aluminium 2.7 g/cm³, iron 7.9 g/cm³, copper 8.9 g/cm³.

1. Divide mass by volume.

2. Compare with the table.

A good answer shows: Density \(= 540 \div 200 = 2.7\) g/cm³, so the block is aluminium.

Can I...?

  1. 1Use \(S = D \div T\) in all three forms.
  2. 2Use \(D = M \div V\) in all three forms.
  3. 3Use \(P = F \div A\) in all three forms.
  4. 4Convert km/h to m/s.
  5. 5Convert minutes to hours.
  6. 6Use the correct units in an answer.
  7. 7Choose the right formula.
  8. 8Rearrange a formula.

Summary & Exam Focus

  • Speed \(=\) distance \(\div\) time.
  • Density \(=\) mass \(\div\) volume.
  • Pressure \(=\) force \(\div\) area.
  • Convert units before calculating.

Exam focus

A block of metal has weight 40 N. Its largest face has area 0.02 m². Work out the pressure on the ground when it rests on this face. (2 marks) (2 marks)

Pressure is force divided by area, so the smaller the area, the greater the pressure. Make sure the area is in m² if the answer is in N/m².

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Compound measure
A measure made from two other measures, such as speed.
Speed
Distance travelled per unit of time.
Density
Mass per unit of volume.
Pressure
Force per unit of area.
Newton (N)
The unit of force.
Pascal (Pa)
One newton per square metre.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    A train travels 240 km in 3 hours. Work out its average speed.

    Show answerHide answer

    Model answer

    \(240 \div 3 = 80\) km/h.

    Mark scheme

    • \(240 \div 3\) — M1
    • 80 — A1
  2. Question 2 Non-calculator 2 marks

    A block of metal has a mass of 450 g and a volume of 60 cm³. Work out the density of the metal.

    Show answerHide answer

    Model answer

    \(450 \div 60 = 7.5\) g/cm³.

    Mark scheme

    • \(450 \div 60\) — M1
    • 7.5 — A1
  3. Question 3 Non-calculator 2 marks

    Change a speed of 90 km/h into metres per second.

    Show answerHide answer

    Model answer

    \(90 \times 1000 \div 3600 = 25\) m/s.

    Mark scheme

    • \(90 \times 1000 \div 3600\) — M1
    • 25 — A1
  4. Question 4 Calculator 4 marks

    The diagram shows a solid block with a weight of 40 N. The block rests on a table. Work out the pressure on the table when the block rests (a) on its largest face and (b) on its smallest face.

    A cuboid block 20 cm by 10 cm by 5 cm with a weight of 40 newtons resting on a table.
    Show answerHide answer

    Model answer

    (a) Largest face: \(0.2 \times 0.1 = 0.02\) m², so pressure \(= 40 \div 0.02 = 2000\) N/m². (b) Smallest face: \(0.1 \times 0.05 = 0.005\) m², so pressure \(= 40 \div 0.005 = 8000\) N/m².

    Mark scheme

    • (a) Area 0.02 m² — M1
    • (a) 2000 — A1
    • (b) Area 0.005 m² — M1
    • (b) 8000 — A1
  5. Question 5 Calculator 3 marks

    A car travels at an average speed of 60 km/h. Work out how long it takes to travel 150 km. Give your answer in hours and minutes.

    Show answerHide answer

    Model answer

    \(150 \div 60 = 2.5\) hours, which is 2 hours 30 minutes.

    Mark scheme

    • \(150 \div 60\) — M1
    • 2.5 hours — A1
    • 2 hours 30 minutes — A1
  6. Question 6 Calculator 2 marks

    Aluminium has a density of 2.7 g/cm³. Work out the mass of a piece of aluminium with a volume of 40 cm³.

    Show answerHide answer

    Model answer

    \(2.7 \times 40 = 108\) g.

    Mark scheme

    • \(2.7 \times 40\) — M1
    • 108 — A1

Quick check

  1. A car travels 150 km in 3 hours. What is its speed?

    1. A450 km/h
    2. B153 km/h
    3. C50 km/h
    4. D0.02 km/h
    Show answerHide answer

    C: 50 km/h

    \(150 \div 3 = 50\) km/h.

  2. A rock has mass 300 g and volume 100 cm³. What is its density?

    1. A0.33 g/cm³
    2. B3 g/cm³
    3. C30 g/cm³
    4. D400 g/cm³
    Show answerHide answer

    B: 3 g/cm³

    \(300 \div 100 = 3\) g/cm³.

  3. A force of 50 N acts on 2 m². What is the pressure?

    1. A25 N/m²
    2. B100 N/m²
    3. C0.04 N/m²
    4. D52 N/m²
    Show answerHide answer

    A: 25 N/m²

    \(50 \div 2 = 25\) N/m².

  4. What is 36 km/h in m/s?

    1. A3.6 m/s
    2. B36 m/s
    3. C100 m/s
    4. D10 m/s
    Show answerHide answer

    D: 10 m/s

    \(36 \div 3.6 = 10\).

  5. Which change makes pressure larger for the same force?

    1. AA larger area
    2. BA smaller area
    3. CMore time
    4. DA larger volume
    Show answerHide answer

    B: A smaller area

    A smaller area gives a greater pressure.

  6. A journey takes 45 minutes at 40 km/h. How far is it?

    1. A40 km
    2. B1800 km
    3. C30 km
    4. D53 km
    Show answerHide answer

    C: 30 km

    45 minutes is 0.75 hours, so \(40 \times 0.75 = 30\) km.

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