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A compound measure is built from two units instead of one: miles per hour, grams per cubic centimetre, newtons per square metre. The word doing the work in all of them is per, and it always means the same thing: divided by. Metres per second is metres divided by seconds. Once you have noticed that, the units of a compound measure are telling you its formula, and you never have to remember which way up it goes. That one observation is worth more than any mnemonic in this module.

The compound measures

Read each unit aloud with "per" as "divided by" and the formula appears:

Cover to find time

In this speed (S), distance (D), time (T) formula triangle, which cell do you cover up to get the formula for TIME?

Work out the speed

A runner covers 240 metres in 30 seconds. What is their speed, in m/s?

The triangle, and what it hides

Formula triangles are quick and they are a crutch. Worth knowing both halves of that:

Ninety kilometres

A car travels 90 km in 45 minutes. A student divides 90 by 45, gets 2, and writes "2 km/h". What has gone wrong, and what is the correct answer?

• The division was fine and the units were not. Dividing kilometres by minutes gives kilometres per MINUTE, so the answer is 2 km/min. For km/h the time must be in hours first: 45 minutes is 0.75 hours, and 90 ÷ 0.75 = 120 km/h
• They divided the wrong way round: it should be 45 ÷ 90
• Nothing is wrong. 2 km/h is correct, since 45 minutes is less than an hour
• They should have multiplied by 60 at the end, giving 120 km/h

Units decide everything

Convert first, divide second. Get both quantities into the units the answer is supposed to be in before you touch the calculator. Converting afterwards is where factors of 60 and 1000 go missing. The units tell you the formula. If the answer must be in g/cm³, then it is grams divided by cubic centimetres, and no triangle is needed to work that out. The conversion worth memorising: to go from km/h to m/s, divide by 3.6; to go back, multiply by 3.6. It comes from 1000 metres in a kilometre and 3600 seconds in an hour, and 1000 ÷ 3600 is 1 ÷ 3.6. And sanity-check the size. A person does not run at 200 m/s, a stone is not 0.02 g/cm³, and a car does not travel at 2 km/h. If an answer is absurd, the arithmetic is usually right and a unit is usually wrong.

From km/h to m/s

A cyclist is travelling at 18 km/h. What is that speed in metres per second?

Work out the density

A block has a mass of 60 g and a volume of 20 cm³. What is its density, in g/cm³?

Pressure in the real world

Pressure = force ÷ area, so the same force spread over a different area gives a completely different pressure. Two designs exploit this in opposite directions. A drawing pin. You push with perhaps 20 N. The head is broad, maybe 0.0001 m², so the pressure on your thumb is about 200 000 N/m², which is comfortable. The point is perhaps a thousand times smaller in area, so the pressure there is a thousand times greater, and that is what drives it into the wall. One force, two pressures, because of the two areas. A snowshoe. Here you want the opposite. Your weight is fixed and cannot be reduced, so the only way to lower the pressure on the snow is to increase the area you are standing on. Spread the same force over four times the area and the pressure falls to a quarter, and you stay on the surface. Skis, tractor tyres, camel feet and the flat base of a heavy statue are all the same idea. Sharp knives, ice skates, studs and nails are the idea in reverse.

Elephant or stiletto?

A 4000 kg elephant stands on a wooden floor. A 60 kg woman walks across the same floor in stiletto heels. Which exerts the greater pressure on the floor, and why?

• The stiletto heel, by a wide margin. Her weight is a fraction of the elephant's, and the heel tip has an area perhaps ten thousand times smaller than an elephant's foot. Pressure is force divided by area, and dividing by a very small number produces a very large one
• The elephant, since pressure depends on weight and it is far heavier
• They are about the same, because the elephant's greater weight is cancelled by its greater foot area
• It cannot be decided without knowing the exact areas involved

Plot the position

An object sets off from the start and moves at a steady 2 m/s. On this distance-time graph, with time across and distance up, plot its position after 3 seconds.

What is being measured?

The race rules

In a compound measure the word "per" always means _____, so the units of the answer tell you which way round the formula goes. Speed is distance over time, density is mass over volume, and pressure is force over _____. Average speed is the total distance over the total time, and not the average of the separate speeds. Always convert the quantities into matching units _____ you divide, and check that the size of your answer is _____ before you write it down.