An object has volume 300 cm³ and density 2 g/cm³. What is its mass?

600 g. Mass = density × volume, so 2 × 300 = 600 g. Density is the mass of each single cm³, so 300 cm³ each weighing 2 g comes to 600 g — bigger than the volume number. The common wrong answer 150 divides (300 ÷ 2); that would make a bigger object lighter, which is impossible.

Is 8 m/s the same speed as 28.8 km/h?

Yes. There are 3.6 km/h in every 1 m/s, so 8 × 3.6 = 28.8 km/h. The two are the same speed — you cannot tell by comparing the bare numbers 8 and 28.8, because they are in different units. Always convert to a common unit before comparing rates.

Town P has 24000 people in 80 km² and Town Q has 12000 people in 30 km². Which is more densely populated?

Town Q. Population density = people ÷ area. Town P is 24000 ÷ 80 = 300 people/km²; Town Q is 12000 ÷ 30 = 400 people/km². So Q is denser, even though P has more people. Comparing the raw populations (24000 > 12000) is the trap — you must compare the rate per km².

GCSE Maths Foundation · AQA · Proportion, rates & compound measures

Comparing rates & density: a compound measure carries its unit

Compound-measure questions trip students who treat a rate or density as a bare number detached from its unit. Told an object has volume 300 cm³ and density 2 g/cm³, they divide — $300 \div 2 = 150$ — instead of multiplying, $2 \times 300 = 600$ . The same belief makes them compare 24 s with 28.8 km/h, or pick the town with more people as “denser”.

The thirty-second fix: a compound measure is a rate per single unit. Mass = density × volume, so you multiply for a mass and divide for a volume; and you can never compare two rates until they are in the same unit — convert first, then compare the rate, not the totals.

Already know this is your gap? Skip the diagnostic and jump straight into the targeted lesson.

How to spot it in your own work

You found a mass by dividing — writing $300 \div 2 = 150$ instead of $2 \times 300 = 600$ .
Your mass came out smaller than the volume (for a density above 1), when it should be bigger.
You compared two rates with different units — such as 24 s against $28.8$ km/h — without converting.
You picked the “better” option by a total (fewer litres, more people) instead of the rate (km per litre, people per km²).
You used $\text{mass} = \text{density} \times \text{volume}$ in the wrong direction — multiplying when you needed to divide for a volume.

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Jun22 P1 Q27 shape, calculator):

An object has a volume of 300 cm³ and a density of 2 g/cm³. Work out the mass of the object.

The misconception answer is $300 \div 2 = 150$ — dividing, because “density = mass ÷ volume” is read as an instruction to divide whatever numbers appear. But that makes a bigger object lighter.

Mass = density × volume: $2 \times 300 = 600 \text{ g}$ .

Why students fall for this

Students see “density = mass ÷ volume” and treat the division as the whole recipe — divide the numbers in front of you. But the formula is a relationship between three quantities, not a fixed instruction. Density is the mass of one cm³, so a volume of 300 cm³ at 2 g per cm³ weighs $2 \times 300 = 600$ g. To find a volume instead, you rearrange: $\text{volume} = \text{mass} \div \text{density}$ .

The deeper belief is that a compound measure is just a number with no unit attached. So a speed of 8 m/s and a speed of 28.8 km/h look like “8 versus 28.8”, when in fact $8 \times 3.6 = 28.8$ — they are the same speed. And a town with more people looks denser than a smaller, more crowded one, because the raw count is compared instead of people per km².

AQA Foundation calculator papers exploit every face of this: mass from density and volume (Jun22 P1 Q27), volume from mass and density (Nov24 P2 Q21), a runner’s speed against a stated km/h (Jun22 P3 Q28), and a population-density comparison across different areas (Jun24 P3 Q27).

The fix: A compound measure is a rate per single unit: multiply for a mass, divide for a volume, and convert before comparing

Mass from density and volume: multiply. Mass = density × volume = $2 \times 300 = 600$ g. The answer is bigger than the volume number, because each cm³ adds its own mass. Dividing (150) is the trap.

Volume from mass and density: divide. Rearranged, $\text{volume} = \text{mass} \div \text{density} = 600 \div 2 = 300$ cm³. Same relationship, used the other way.

Convert before you compare, and compare the rate. 8 m/s $\times 3.6 = 28.8$ km/h, so they are equal — bare numbers in different units cannot be compared. And density is a rate: people ÷ area, km ÷ litre — compare those, not the totals.

Worked example

Town P has 24000 people in an area of 80 km². Town Q has 12000 people in an area of 30 km². Which town is more densely populated?

Density is a rate: people ÷ area. Do not compare the populations.
Work out each density.
$\text{Town P} = 24000 \div 80 = 300 \text{ people/km}^2$
$\text{Town Q} = 12000 \div 30 = 400 \text{ people/km}^2$
Compare the rates. $400 > 300$ , so Town Q is more densely populated.

Town Q is denser even though Town P has more people ( $24000 > 12000$ ). The trap is to compare the totals — but density lives in the rate per km². The same move works for speed ( $200 \div 24 \times 3.6 = 30$ km/h) and fuel ( $50 \div 4 = 12.5$ km/L beats $30 \div 3 = 10$ km/L).

Find out if this is costing you marks

The 10-minute diagnostic checks for this pattern (and four others) using AQA-style GCSE Higher items. Free, no signup, anonymous.

Take the diagnostic →

Common questions

Why do I multiply for the mass instead of dividing?: Because density is the mass of one cm³. At 2 g/cm³, every cm³ weighs 2 g, so 300 cm³ weighs $2 \times 300 = 600$ g. Dividing ( $300 \div 2 = 150$ ) would make a bigger object lighter, which is impossible. Mass = density × volume — multiply.
How do I find a volume from a mass and a density, then?: Rearrange the same relationship: $\text{volume} = \text{mass} \div \text{density}$ . A 600 g object at 2 g/cm³ has volume $600 \div 2 = 300$ cm³. Multiply for a mass, divide for a volume — same formula, used the right way round.
Why can’t I just compare the two numbers I’m given?: Because a compound measure carries its unit. 8 m/s and 28.8 km/h look different, but $8 \times 3.6 = 28.8$ — they are the same speed. You must convert both to one unit before comparing; the bare numbers tell you nothing on their own.
The other option used less fuel — isn’t that more efficient?: Not necessarily. Efficiency is a rate: km per litre. Car A goes $30 \div 3 = 10$ km/L; Car B goes $50 \div 4 = 12.5$ km/L. Car B is more efficient even though it used more litres, because it travels further on each one. Compare the rate, not the total.

Related misconceptions

Speed, distance, time & rates: convert the time before you divideThe neighbouring rates misconception — getting the units right inside speed = distance ÷ time before you compute or compare.
Direct vs inverse proportion: more can mean lessThe companion proportion misconception — telling when scaling up makes a quantity grow versus shrink.

← All GCSE Maths Higher misconceptions