Four pairs run across the board. In every one, the two cards on the left divide into the answer on the right. Speed = distance ÷ time, density = mass ÷ volume, population density = population ÷ area, fuel economy = distance ÷ fuel.
How to read this board
They are taught in separate subjects, so they look like separate things. But every one of them answers "how much per one": kilometres travelled per hour, grams per cubic centimetre, people per square kilometre, kilometres per litre.
The unit tells you the answer. km/h, g/cm³, people/km², km/L — that slash is the division sign, and it says the left is the numerator and the right is the denominator. When you cannot remember which way round to divide, write the unit first.
Where it trips you up
The places that are easy to get wrong, next to how to check each one on the board.
| Where it trips you up | Why that happens | How the board shows it |
|---|---|---|
| Which one divides which | Speed and density are both built from two quantities, and the formula carries no clue about the order. | Write the unit first: km/h means km ÷ h. The card names carry their units for exactly this reason. |
| They are taught in different places | Speed in maths, density in science, population density in geography. Different rooms hide the shared shape. | All four sit on one board. Read down the column and the form is the same every time. |
| What the number after the division means | "60" on its own says nothing. 60 km/h means 60 km in one hour. | Set the time card to 1. The speed and the distance become the same number. That is what "per one" means. |
| Average speed is not the speed at any moment | 120 km in 2 hours averages 60 km/h, but nothing says you held 60 the whole way. | What the board gives you is the average. Variation along the way is not in this formula. |
| Density is mass ÷ volume, not weight | Everyday speech says "weight", but in science mass is the quantity in kilograms and weight is a force. | The card is named for the mass. g/cm³ reads as grams of mass in one cubic centimetre. |
| Comparing things of different sizes | Totals cannot be compared when the bases differ: a bigger town has more people, which says nothing about how crowded it is. | Change the area and watch the population density. Same population, different density. |
Per-unit quantities exist so that things measured on different scales can be compared on the same footing. Population density, fuel economy, batting averages, hourly rates — all of them divide one amount by the amount being measured per. Put the formulas side by side and the names turn out to be the only thing that differed.
Try this
- Set the time to 1 — the speed and the distance read the same. This is where "per one" lands.
- Double the volume — with the mass unchanged the density halves. Bigger for the same mass means less packed in.
- Use less fuel — the fuel-economy number rises. "Good economy" being the larger number can be read straight off the formula.