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

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.

Ready to fix this? The Proportion & rates lesson works through this misconception and the others in Proportion & rates, one altitude at a time.

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, the shape of Jun22 P1 Q27 (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, volume from mass and density, a runner’s speed against a stated km/h, and a population-density comparison across different areas.

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.

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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