Why is the volume of a steel bar with mass 2340 g and density 7.5 g/cm^3 equal to 312 cm^3 and not 17550 cm^3?

Because density = mass divided by volume, so volume = mass divided by density = 2340 divided by 7.5 = 312 cm^3. Multiplying gives 17550, which is wrong because density is grams per cm^3: the number of cm^3 is found by dividing the total mass by the mass per unit volume.

How do you find the volume of just one part of a composite object when only total mass and density are given?

First find the total volume: total volume = total mass divided by density = 2340 divided by 7.5 = 312 cm^3. Then subtract the known part volume: player = 312 minus 258 = 54 cm^3. A common trap is stopping at 312 and forgetting to subtract.

Can you compare 8.33 m/s and 28.8 km/h directly by looking at the numbers?

No. They are in different units. Convert 28.8 km/h to m/s: 28.8 times 1000 divided by 3600 equals 8 m/s. Now compare 8.33 m/s and 8 m/s. The 8.33 m/s speed is faster. Comparing the raw numbers without converting gives the wrong conclusion that 28.8 is greater.

GCSE Maths Higher · AQA · Compound measures and rates

Density formula direction and units

When density, mass, and volume are all linked by $\text{density} = \text{mass} / \text{volume}$ , the natural instinct when asked for volume is to multiply: mass times density. That gives $2340 \times 7.5 = 17550$ , which is wrong. Density is grams per cm³, so the number of cm³ is found by dividing mass by density: $2340 / 7.5 = 312$ cm³.

A second trap on the same topic is comparing two compound-measure rates without checking their units. Seeing 28.8 km/h next to 8.33 m/s and writing “28.8 is bigger so B is faster” ignores the fact that 28.8 km/h converts to exactly 8 m/s, which is slower than 8.33 m/s. Both traps cost marks at grades 7 to 9.

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How to spot it in your own work

You wrote volume = mass × density and got 17550 cm³ instead of dividing to get 312 cm³.
You found the total volume of a composite object but forgot to subtract the stand volume, stopping at 312 when the answer is 54 cm³.
You left a volume answer in m³ (0.0006) without converting to litres (0.6) as the question required.
You compared rates in different units by their raw numbers, concluding 28.8 km/h > 8.33 m/s is faster without converting 28.8 km/h to 8 m/s first.

An exam question that triggers it

Here is the structure of a typical AQA Higher compound-measures question:

A chess piece is made from a material with density 7.5 g/cm³.

The total mass of the figure (player and stand) is 2340 g.

The stand has a volume of 258 cm³.

Work out the volume of the player piece.

The misconception writes $\text{volume} = 2340 \times 7.5 = 17550$ cm³. The fix: total volume $= 2340 / 7.5 = 312$ cm³, then player $= 312 - 258 = 54$ cm³.

Why students fall for this

Students learn that compound measures link three quantities and that the formula can be rearranged. But when working under pressure they reach for multiplication as the default operation, treating density as if it were a scale factor rather than a ratio. The per-unit reading of the unit g/cm³ (grams per cm³) would resolve the ambiguity, but it is rarely taught explicitly alongside the formula triangle.

The unit-comparison trap has a different root: students learn that bigger numbers mean bigger quantities, and carry that heuristic across to rates without checking whether the units match. 28.8 looks bigger than 8.33, so the conclusion feels obvious. Only converting 28.8 km/h to 8 m/s reveals the reversal.

Both traps are reinforced by partial credit in past papers: a student who sets up the correct formula but divides in the wrong direction often still gets method marks, which masks the error until it matters at the highest grades.

The fix: Formula triangle with per-unit reading, then check units before comparing

Step 1: write the formula and build the triangle. Density = mass / volume. In the triangle, mass sits at the top; density and volume sit at the bottom. Cover the quantity you want and the remaining arrangement gives the operation.

Step 2: read the unit for direction. Density in g/cm³ means grams per cm³. Cover volume: you see mass over density, so $\text{volume} = \text{mass} / \text{density}$ . Cover mass: density times volume.

Step 3: check for extra steps. Read the question carefully. If total volume is found but a part volume is known, subtract. If the answer is in m³ but litres are required, multiply by 1000.

Step 4: check units before comparing rates. If two rates are in different units (m/s vs km/h; g/cm³ vs kg/m³), convert to the same unit first, then compare.

Worked example

NOV23 style: find the player volume from a composite figure.

Total volume.
$\text{volume} = \frac{\text{mass}}{\text{density}} = \frac{2340}{7.5} = 312 \text{ cm}^3$
Trap: $2340 \times 7.5 = 17550$ , wrong operation.
Player volume. $312 - 258 = 54 \text{ cm}^3$ . Trap: stopping at 312 (total, not player).

NOV24 style: volume in litres from a liquid container.

Volume in m³.
$\text{volume} = \frac{537}{895000} = 0.0006 \text{ m}^3$
Convert to litres. $0.0006 \times 1000 = 0.6 \text{ litres}$ . Trap: leaving the answer as 0.0006 m³.

Unit comparison: 8.33 m/s vs 28.8 km/h.

Convert. $28.8 \text{ km/h} = \frac{28.8 \times 1000}{3600} = 8 \text{ m/s}$ .
Compare. 8.33 m/s > 8 m/s. Athlete A is faster. Trap: 28.8 > 8.33 without converting gives the wrong conclusion.

Find out if this is costing you marks

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Common questions

Why is volume = mass / density and not mass × density?: Because density is grams per cm³: each cm³ contributes one density-worth of grams to the total mass. To find how many cm³ are in the total mass you divide: $\text{volume} = \frac{2340}{7.5} = 312 \text{ cm}^3$ . Multiplying gives 17550, which has the wrong units and the wrong magnitude.
How do you find just the player volume when total mass and density are given?: Find the total volume first: $2340 / 7.5 = 312 \text{ cm}^3$ . Then subtract the known stand volume: $312 - 258 = 54 \text{ cm}^3$ . The trap is stopping at 312, which is the whole figure, not the player alone.
Why must you convert units before comparing two rates?: Because the numbers only represent the same thing when they are in the same unit. 28.8 km/h converts to $28.8 \times 1000 / 3600 = 8 \text{ m/s}$ , which is less than 8.33 m/s. Without converting, the comparison of 28.8 against 8.33 gives the wrong direction.

Related misconceptions

Average speed is not the mean of the speedsThe companion compound-measures misconception: for a multi-stage journey use total distance / total time, not the arithmetic mean of the speeds.
Rate from a graph is the gradientReading a compound-measure rate of change from a graph by finding the gradient, connecting the formula approach to the graphical representation.

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