Compound measures

Visual model

Compound measures combine two quantities

massvolumedensity

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

Gold-standard guide

20 mins

What you will learn

Work with density, pressure and other rates.

Use a clear step-by-step method for compound measures.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier ratio, proportion and rates of change skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Density = mass/volume

Step 1

Convert mass to grams

1.62 kg = 1620 g

Step 2

Calculate the volume

Volume = 9 × 6 × 5 = 270 cm³

Step 3

Apply the density formula

Density = mass/volume = $162\frac{0}{2}70$ = 6 g/cm³

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³

Density

$density = mas\frac{s}{volume}.$

Pressure

$pressure = forc\frac{e}{area}.$

Worked example

A rectangular block has mass 1.62 kg and dimensions 9 cm × 6 cm × 5 cm. Find its density in g/cm³.

Convert mass to grams: 1.62 kg = 1620 g.

Calculate the volume: Volume = 9 × 6 × 5 = 270 cm³.

Apply the density formula: Density = mass/volume = $162\frac{0}{2}70$ = 6 g/cm³.

Final answer

Density = 6 g/cm³

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

A rectangular block has mass 1.62 kg and dimensions 9 cm × 6 cm × 5 cm. Find its density in g/cm³.

3 marks4 minscompound-measures-worked

Show solution

Worked solution

1.Convert mass to grams: 1.62 kg = 1620 g.
2.Calculate the volume: Volume = 9 × 6 × 5 = 270 cm³.
3.Apply the density formula: Density = mass/volume = $162\frac{0}{2}70$ = 6 g/cm³.

Final answer

Density = 6 g/cm³

Mark points

M1: convert mass to grams
M1: calculate the volume
M1: apply the density formula
A1: Density = 6 g/cm³

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

Diagnosticrecall

A liquid has density 0.8 g/cm³. Find the mass of 250 cm³.

1 mark2 minscompound-measures-q1

Show solution

Worked solution

1.Spot the skill: Density = mass/volume.
2.Use the convert mass to grams stage first, then calculate the volume.
3.Keep the final answer visible: 200 g.

Final answer

200 g

Mark points

M1: use the correct density = mass/volume. pressure = force/area. speed = distance/time.these follow the same triangle structure.always check units — density in g/cm³ requires mass in grams and volume in cm³.
A1: 200 g

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

Easyprocedure

A force of 120 N acts on an area of 0.3 m². Find the pressure in Pa.

2 marks3 minscompound-measures-q2

Show solution

Worked solution

1.Spot the skill: Density = mass/volume.
2.Use the calculate the volume stage first, then apply the density formula.
3.Keep the final answer visible: 400 Pa.

Final answer

400 Pa

Mark points

M1: use the correct density = mass/volume. pressure = force/area. speed = distance/time.these follow the same triangle structure.always check units — density in g/cm³ requires mass in grams and volume in cm³.
A1: 400 Pa

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

Mediumreasoning

Gold has density 19.3 g/cm³. Find the volume of a 386 g gold bar.

3 marks4 minscompound-measures-q3

Show solution

Worked solution

1.Spot the skill: Density = mass/volume.
2.Use the apply the density formula stage first, then convert mass to grams.
3.Keep the final answer visible: 20 cm³.

Final answer

20 cm³

Mark points

M1: use the correct density = mass/volume. pressure = force/area. speed = distance/time.these follow the same triangle structure.always check units — density in g/cm³ requires mass in grams and volume in cm³.
A1: 20 cm³

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

Hardproblem solving

A 500 g object has density 2 g/cm³. Find its volume.

3 marks5 minscompound-measures-q4

Show solution

Worked solution

1.Spot the skill: Density = mass/volume.
2.Use the convert mass to grams stage first, then calculate the volume.
3.Keep the final answer visible: 250 cm³.

Final answer

250 cm³

Mark points

M1: use the correct density = mass/volume. pressure = force/area. speed = distance/time.these follow the same triangle structure.always check units — density in g/cm³ requires mass in grams and volume in cm³.
A1: 250 cm³

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

Exam-stylemulti-step

A pressure of 50,000 Pa acts over 0.04 m². Find the force.

4 marks6 minscompound-measures-q5

Show solution

Worked solution

1.Spot the skill: Density = mass/volume.
2.Use the calculate the volume stage first, then apply the density formula.
3.Keep the final answer visible: 2000 N.

Final answer

2000 N

Mark points

M1: use the correct density = mass/volume. pressure = force/area. speed = distance/time.these follow the same triangle structure.always check units — density in g/cm³ requires mass in grams and volume in cm³.
A1: 2000 N

Watch out

Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

Grade 9 stretchproblem solving

A force of 180 N acts over an area of 0.012 m². Find the pressure.

4 marks7 minscompound-g9

Show solution

Worked solution

1.Use pressure = force / area.
2.Substitute the values and include units.

Final answer

15,000 Pa

Mark points

M1: use 180 / 0.012
A1: 15,000 Pa

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Compound measures - 2 marksA liquid has density 0.8 g/cm³. Find the mass of 250 cm³.Mark answer

Answer

200 g

2Ratio and sharing - 2 marksTwo people share profit in ratio 5:3. Total profit £640. Find each share.Mark answer

Answer

£400 and £240

3Fractions and ratios - 2 marksA map scale is 1:25,000. Express as a fraction.Mark answer

Answer

$\frac{1}{2}5$ ,000

4Direct and inverse proportion - 3 marksy ∝ 1/x². When x = 2, y = 9. Find y when x = 6.Mark answer

Answer

Mastery check

I can explain the method for compound measures.
I can show clear working without skipping key steps.
I can avoid this mistake: Students mix units — using kg with cm³ gives density in kg/cm³, not g/cm³.Always convert mass and volume into consistent units before dividing.

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