Compound Measures

A compound measure is built from two other measures by dividing one by another. The everyday example is speed — how far you go for each unit of time:

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

Measured in miles per hour (mph), the "per" is the division: distance shared out over the time taken. The same relationship rearranges two ways, so any one of the three can be found from the other two:

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

A handy memory aid is the formula triangle: write 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).

Other compound measures follow exactly the same pattern. Density is mass shared over volume, and pressure is force shared over area:

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

Each one rearranges with its own formula triangle, just like speed. The one thing to watch is the units: g/cm³ for density, mph for speed, and so on — the "per" tells you which measure is divided by which.

A compound measure divides one quantity by another:

\text{speed} = \text{distance} \div \text{time} (e.g. mph);
\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.