Density, Mass and Volume

Density is a compound measure that links mass and volume. It is a core part of the Edexcel GCSE Mathematics (1MA1) ratio, proportion and rates of change strand. Density questions are practical, often set in real-world contexts, and appear on both Foundation and Higher papers.

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Key Vocabulary

Term	Meaning	SI Unit
Density	Mass per unit volume	g/cm^3 or kg/m^3
Mass	The amount of matter in an object	grams (g) or kilograms (kg)
Volume	The amount of space an object occupies	cm^3 or m^3

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The Density Formula Triangle

The three formulae are:

• Density = Mass / Volume ($D = \frac{M}{V}$D=VM​)
• Mass = Density x Volume ($M = D \times V$M=D×V)
• Volume = Mass / Density ($V = \frac{M}{D}$V=DM​)

Formula	Use When You Need To Find
$D = \frac{M}{V}$D=VM​	Density (given mass and volume)
$M = D \times V$M=D×V	Mass (given density and volume)
$V = \frac{M}{D}$V=DM​	Volume (given mass and density)

Edexcel Exam Tip: You may find it helpful to draw a formula triangle with M at the top and D and V at the bottom. Cover the quantity you want to find, and the remaining two show you the calculation.

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Worked Examples — Finding Density

Worked Example 1

A block of metal has a mass of 540 g and a volume of 200 cm^3. Calculate its density.

• Density = Mass / Volume = 540 / 200 = 2.7 g/cm^3

Worked Example 2

A liquid has a mass of 3.6 kg and occupies a volume of 4,000 cm^3. Find the density in g/cm^3.

• First convert mass: 3.6 kg = 3,600 g
• Density = 3,600 / 4,000 = 0.9 g/cm^3

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Worked Examples — Finding Mass

Worked Example 3

A substance has a density of 8.5 g/cm^3. Find the mass of 20 cm^3 of this substance.

• Mass = Density x Volume = 8.5 x 20 = 170 g

Worked Example 4

Gold has a density of 19.3 g/cm^3. A gold bar has a volume of 50 cm^3. What is its mass in kilograms?

• Mass = 19.3 x 50 = 965 g = 0.965 kg

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Worked Examples — Finding Volume

Worked Example 5

An iron bar has a mass of 1,560 g and a density of 7.8 g/cm^3. Find its volume.

• Volume = Mass / Density = 1,560 / 7.8 = 200 cm^3

Worked Example 6

A plastic toy has a mass of 240 g and a density of 1.2 g/cm^3. Find its volume.

• Volume = 240 / 1.2 = 200 cm^3

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Unit Conversions for Density

Conversion	Method
g/cm^3 to kg/m^3	Multiply by 1,000
kg/m^3 to g/cm^3	Divide by 1,000

Worked Example 7

Convert a density of 2.7 g/cm^3 to kg/m^3.

• 2.7 x 1,000 = 2,700 kg/m^3

Worked Example 8

A liquid has a density of 850 kg/m^3. Convert to g/cm^3.

• 850 / 1,000 = 0.85 g/cm^3

Edexcel Exam Tip: Check which units the question uses. If the mass is in kg and the volume in cm^3, convert one so both are consistent before dividing.

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Density in Context

Worked Example 9

A cuboid has dimensions 10 cm by 8 cm by 5 cm. It is made of wood with a density of 0.6 g/cm^3. Find the mass of the cuboid.

1. Volume = 10 x 8 x 5 = 400 cm^3
2. Mass = Density x Volume = 0.6 x 400 = 240 g

Worked Example 10

A cylindrical candle has a radius of 3 cm and a height of 10 cm. The wax has a density of 0.9 g/cm^3. Find the mass of the candle. (Use pi = 3.14159.)

1. Volume = $\pi r^2 h$πr2h = 3.14159 x 9 x 10 = 282.74 cm^3
2. Mass = 0.9 x 282.74 = 254.47 g (to 2 d.p.)

Worked Example 11

A steel ball bearing has a mass of 33.5 g and a density of 8 g/cm^3. Find the radius of the ball bearing. (Volume of a sphere = $\frac{4}{3}\pi r^3$34​πr3.)

1. Volume = Mass / Density = 33.5 / 8 = 4.1875 cm^3
2. $\frac{4}{3}\pi r^3$34​πr3 = 4.1875
3. r^3 = 4.1875 x 3 / (4 x 3.14159) = 12.5625 / 12.56637 = 0.99970
4. r = cube root of 0.99970 = 1.00 cm (to 2 d.p.)

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Floating and Sinking

An object floats if its density is less than the density of the liquid. It sinks if its density is greater than the liquid.

• Water has a density of approximately 1.0 g/cm^3
• Objects with density < 1.0 g/cm^3 float in water
• Objects with density > 1.0 g/cm^3 sink in water

Worked Example 12

A wooden block has a mass of 450 g and a volume of 600 cm^3. Will it float or sink in water?

• Density = 450 / 600 = 0.75 g/cm^3
• 0.75 < 1.0, so the block will float.

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Practice Problems

1. A metal cube has a mass of 216 g and a volume of 27 cm^3. Find the density. Answer: 8 g/cm^3
2. A liquid has a density of 1.25 g/cm^3. Find the mass of 400 cm^3. Answer: 500 g
3. A stone has a mass of 780 g and a density of 2.6 g/cm^3. Find its volume. Answer: 300 cm^3
4. A cuboid measures 15 cm x 10 cm x 6 cm and has a mass of 2.7 kg. Find its density in g/cm^3. Answer: 3 g/cm^3
5. Convert 7.85 g/cm^3 to kg/m^3. Answer: 7,850 kg/m^3
6. An object has a density of 0.8 g/cm^3. Will it float or sink in water? Answer: Float (0.8 < 1.0)
7. A sphere of radius 5 cm has a density of 3 g/cm^3. Find its mass. (Use pi = 3.14159.) Answer: 1,570.80 g (to 2 d.p.)

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Common Mistakes