Compound measures

Definition of Compound Measures

Compound measures combine two or more quantities into a single measure, often expressed as a ratio or fraction. They describe how one quantity changes relative to another. For example, speed is a compound measure combining distance and time, density combines mass and volume, and pressure combines force and area.

These measures are essential in many real-life contexts, allowing us to compare and calculate quantities that depend on multiple factors.

Common Compound Measures

Speed is the distance travelled per unit time, written as:

$$\text{Speed} = \frac{\text{distance}}{\text{time}}$$

(Speed is covered in a separate topic, so it won’t be detailed here. See the Speed topic for more details.)

Density is the mass per unit volume of a substance:

$$\text{Density} = \frac{\text{mass}}{\text{volume}}$$

Density tells us how compact a material is. For example, water has a density of about $1000\,\mathrm{kg\,m^{-3}}$, meaning one cubic metre of water has a mass of 1000 kilograms.

Pressure is the force applied per unit area:

$$\text{Pressure} = \frac{\text{force}}{\text{area}}$$

Pressure is measured in pascals (Pa), where $1\,\mathrm{Pa} = 1\,\mathrm{N\,m^{-2}}$. For example, if a force of $10\,\mathrm{N}$ is applied over an area of $2\,\mathrm{m^{2}}$, the pressure is:

$$\frac{10}{2} = 5\,\mathrm{Pa}$$

Calculations with Compound Measures

When working with compound measures, it is important to use consistent units. For example, mass should be in kilograms ($\mathrm{kg}$), volume in cubic metres ($\mathrm{m^{3}}$), force in newtons ($\mathrm{N}$), and area in square metres ($\mathrm{m^{2}}$).

If units are not consistent, convert them before calculating. For example, if volume is given in cubic centimetres ($\mathrm{cm^{3}}$), convert to cubic metres by dividing by $1,000,000$ because:

$$1\,\mathrm{m^{3}} = 100\,\mathrm{cm} \times 100\,\mathrm{cm} \times 100\,\mathrm{cm} = 1,000,000\,\mathrm{cm^{3}}$$

For instance, converting $500\,\mathrm{cm^{3}}$ to cubic metres would be $\frac{500}{1,000,000} = 0.0005\,\mathrm{m^{3}}$.

Solving problems involving compound measures often requires rearranging formulas or combining different units carefully.

For instance, if you know the density and volume of an object, you can find its mass by rearranging the density formula:

$$\text{Density} = \frac{\text{mass}}{\text{volume}} \implies \text{mass} = \text{density} \times \text{volume}$$

Example: A block of metal has a volume of $0.002\,\mathrm{m^{3}}$ and a density of $7800\,\mathrm{kg\,m^{-3}}$. Its mass is:

$$\text{mass} = 7800 \times 0.002 = 15.6\,\mathrm{kg}$$