Compound Measures

Look at a car's dashboard: miles per hour. A recipe card: calories per serving. A weather app: millimetres of rain per hour. Each of these squashes two measurements into one by dividing — and that little word "per" is doing all the work. It means "for each," and it always signals a division.

These are compound measures: measurements built from two others. Speed is distance shared over time. Density is mass shared over volume. Pressure is force shared over area. Being able to juggle them is a genuine real-life superpower — it's the maths behind driving, cooking, diving, and every science lab you'll ever stand in.

The key trio

Three compound measures cover almost everything you'll meet. Each is one quantity divided by another:

\text{speed} = \frac{\text{distance}}{\text{time}} \qquad \text{density} = \frac{\text{mass}}{\text{volume}} \qquad \text{pressure} = \frac{\text{force}}{\text{area}}

Every one rearranges the same way, so any one of the three parts can be found from the other two. For speed:

\text{distance} = \text{speed} \times \text{time} \qquad \text{time} = \frac{\text{distance}}{\text{speed}}

A handy memory aid is the formula triangle: distance on top, speed and time below. Cover the one you want and the triangle shows what to do — cover distance and you see speed beside time (multiply); cover time and you see distance over speed (divide).

The formula triangle

Put your finger over the quantity you are after. What is left tells you whether to multiply (two side by side) or divide (one over another).

Worked example 1 — find a speed

A cyclist rides 45 km in 3 hours. What is their average speed?

Speed is on the "cover" list as distance ÷ time: \text{speed} = \dfrac{45}{3} = 15 km/h.
Check the units: km ÷ h gives km/h. That matches — good sign.

Worked example 2 — rearrange to find time

A train covers 240 km at 80 km/h. How long does it take?

Cover time in the triangle: distance over speed.
\text{time} = \dfrac{240}{80} = 3 hours.

Units check: km ÷ (km/h) = hours. Perfect.

Worked example 3 — density

A metal block has mass 600 g and volume 75 cm³. What is its density?

\text{density} = \dfrac{\text{mass}}{\text{volume}} = \dfrac{600}{75} = 8 g/cm³.
That's about the density of iron — heavier than aluminium (2.7), lighter than gold (19.3).

Worked example 4 — convert the units

A car travels at 20 m/s. What is that in km/h?

In one second it goes 20 m, so in one hour (3600 s) it goes 20 \times 3600 = 72000 m.
That's 72000 \div 1000 = 72 km, so 72 km/h.

Shortcut: to turn m/s into km/h, multiply by 3.6 (20 \times 3.6 = 72). To go back, divide by 3.6.

Each measure rearranges with its own formula triangle, just like speed — density puts mass on top, pressure puts force on top. The pattern never changes: the thing before "per" goes on top, and "per what" goes underneath.

A compound measure divides one quantity by another:

\text{speed} = \text{distance} \div \text{time} (e.g. km/h);
\text{density} = \text{mass} \div \text{volume} (e.g. g/cm³);
\text{pressure} = \text{force} \div \text{area} (e.g. N/m²);
each rearranges via its formula triangle — cover the one you want;
always watch the units: the "per" shows what is divided by what.

Two traps catch almost everyone with compound measures:

Match your units first. Mixing minutes with hours, or metres with kilometres, gives nonsense answers. If a journey is 30 minutes, that's 0.5 hours before you divide — not 30. Convert everything to consistent units before you calculate.
Get the division the right way up. Speed is distance ÷ time, not time ÷ distance. Flipping it is the classic slip. Let the units check you: if you want km/h, you must divide km by h — any other combination is wrong.

Weight isn't the answer — density is. A pebble sinks because it's denser than water (more mass packed into each cm³). A steel ship is mostly hollow air inside its hull, so its overall density — total mass ÷ total volume, including all that air — is less than water, and up it floats. Hammer that same steel into a solid ball and it drops like a stone.

Compound measures are quietly everywhere: a speedometer reading mph, a recipe's "per serving," a scuba diver watching the pressure climb about one extra atmosphere for every 10 m of depth, a materials scientist comparing which alloy is lightest for its strength. Master "per" and you can read the real world in numbers.