Compound Measures: Speed and Density at KS3 Maths

A compound measure is one that is worked out by combining two or more other measures. Speed combines distance and time; density combines mass and volume. Understanding these measures and their formulas is essential KS3 maths content tested in Year 8 and Year 9.

What are compound measures?

A compound measure is a quantity defined by dividing one measure by another. Because it is a ratio of two quantities, it has a compound unit — two units combined, such as metres per second (m/s) or grams per cubic centimetre (g/cm³).

The most important compound measures at KS3 are:

Measure | Formula | Unit example
Speed | Distance ÷ Time | m/s, km/h, mph
Density | Mass ÷ Volume | g/cm³, kg/m³
Pressure | Force ÷ Area | N/m², Pa

Speed, distance, and time

The formula

Speed = Distance ÷ Time

Rearranged:

Distance = Speed × Time

Time = Distance ÷ Speed

How to use the formula triangle

A useful memory aid: write D (distance) at the top and S × T at the bottom. Cover the quantity you want to find:

Cover D → S × T

Cover S → D ÷ T

Cover T → D ÷ S

Worked example 1: find the speed

A car travels 180 km in 3 hours. Find its average speed.

Speed = Distance ÷ Time = 180 ÷ 3 = 60 km/h

Answer: 60 km/h

Worked example 2: find the distance

A cyclist rides at an average speed of 15 km/h for 2.5 hours. How far does she travel?

Distance = Speed × Time = 15 × 2.5 = 37.5 km

Answer: 37.5 km

Worked example 3: find the time

A train travels 480 km at an average speed of 120 km/h. How long does the journey take?

Time = Distance ÷ Speed = 480 ÷ 120 = 4 hours

Answer: 4 hours

Worked example 4: units must match

A runner completes a 400 m race in 50 seconds. Find his speed in m/s and in km/h.

Speed in m/s: 400 ÷ 50 = 8 m/s

Convert to km/h: 8 m/s × 3600 s/h ÷ 1000 m/km = 28.8 km/h

Answer: 8 m/s or 28.8 km/h

This shows why matching units matters — mixing metres and kilometres or seconds and hours produces wrong answers.

Density, mass, and volume

The formula

Density = Mass ÷ Volume

Rearranged:

Mass = Density × Volume

Volume = Mass ÷ Density

The same triangle approach works: D(ensity) at top, M × V at bottom.

Worked example 5: find density

A block of aluminium has a mass of 540 g and a volume of 200 cm³. Find its density.

Density = Mass ÷ Volume = 540 ÷ 200 = 2.7 g/cm³

Answer: 2.7 g/cm³

(The density of aluminium is indeed approximately 2.7 g/cm³ — a good reality check.)

Worked example 6: find the mass

A piece of iron has a volume of 80 cm³. Iron has a density of 7.87 g/cm³. Find the mass.

Mass = Density × Volume = 7.87 × 80 = 629.6 g

Answer: 629.6 g (approximately 630 g)

Worked example 7: find the volume

A gold ingot has a mass of 1930 g. Gold has a density of 19.3 g/cm³. Find the volume.

Volume = Mass ÷ Density = 1930 ÷ 19.3 = 100 cm³

Answer: 100 cm³

Comparing densities

Density tells you whether an object will sink or float in water (density of water = 1 g/cm³):

Material | Density (g/cm³) | Floats in water?
Cork | 0.24 | Yes
Ice | 0.92 | Yes
Water | 1.00 | (reference)
Aluminium | 2.70 | No
Iron | 7.87 | No
Gold | 19.3 | No

An object floats if its density is less than the density of the liquid it is placed in.

Pressure (extension for Year 9)

Pressure is a third compound measure introduced at the top of KS3 and consolidated at GCSE:

Pressure = Force ÷ Area

Units: pascals (Pa) = newtons per square metre (N/m²).

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

Pressure = 600 ÷ 0.5 = 1200 Pa

The same triangle method applies: Pressure at top, Force × Area at bottom.

Unit conversions with compound measures

Compound measure problems often require unit conversions before applying the formula.

Worked example 8: time in hours and minutes

A bus travels 60 km in 1 hour 30 minutes. Find its average speed in km/h.

Convert the time: 1 hour 30 minutes = 1.5 hours (not 1.3 — a very common error).

Speed = 60 ÷ 1.5 = 40 km/h

Answer: 40 km/h

Worked example 9: density with different units

A liquid has a density of 0.8 g/cm³. Convert this to kg/m³.

1 g/cm³ = 1000 kg/m³ (because 1 g = 0.001 kg and 1 cm³ = 0.000001 m³, so 0.001/0.000001 = 1000).

0.8 g/cm³ = 0.8 × 1000 = 800 kg/m³

Answer: 800 kg/m³

Common mistakes

Mistake 1 — Mixing up time units.
30 minutes is 0.5 hours, not 0.3. Convert all times to decimals of an hour (or all to seconds) before using the formula.

Mistake 2 — Reversing mass and volume in the density formula.
Density = mass ÷ volume (not volume ÷ mass). A denser material has more mass per unit of volume, so density = mass/volume gives a higher number for a heavier material per same volume.

Mistake 3 — Ignoring units in the answer.
Always state the unit (km/h, g/cm³, etc.). A numerical answer without units is incomplete.

Mistake 4 — Using the wrong formula triangle value.
Double-check which quantity you are finding before covering the triangle. Covering D gives the formula for D, not for S.

How compound measures fit the KS3 national curriculum

The Department for Education's KS3 Mathematics Programme of Study requires pupils to "use compound units such as speed, density, unit pricing and others." Speed, distance, and time are listed explicitly, with density introduced as pupils progress through Year 8 and Year 9. All three compound measures are statutory core content for the Key Stage 3 and 4 curriculum, and are examined in GCSE Number.

Frequently asked questions

What is the difference between speed and velocity?

Speed is a scalar — it tells you how fast an object is moving without indicating direction (e.g., 60 km/h). Velocity is a vector — it includes both speed and direction (e.g., 60 km/h north). At KS3 maths you work with speed; velocity is introduced more formally in KS4 physics.

Why does 30 minutes = 0.5 hours and not 0.30 hours?

There are 60 minutes in one hour, so minutes convert to hours by dividing by 60. 30 ÷ 60 = 0.5. Writing 0.30 treats time as if it were decimal (base 10), but clocks work in base 60. Always divide minutes by 60 to convert to a decimal fraction of an hour.

How do I know which quantity to find in a compound measure question?

Read the question carefully and identify which two quantities are given. The quantity not given is the one you calculate. Then select the correct rearrangement of the formula to solve for that quantity.

Can density be less than 1?

Yes. Materials less dense than water (density 1 g/cm³) have a density below 1 g/cm³ — for example, cork at about 0.24 g/cm³ or oil at about 0.8–0.9 g/cm³. These materials float on water. Air has a density of approximately 0.0013 g/cm³.

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