Ratio, proportion, and why doubling the size is not doubling the material
Scale something up and its parts do not grow at the same rate.
A ratio compares quantities. A proportion says two ratios are equal. Between them they cover recipes, maps, currency, speed, gears, mixing, and most of the arithmetic anyone does after school.
The one method
Nearly every proportion question yields to finding the value of one unit.
If 7 notebooks cost 21 pounds, one costs 3, so 12 cost 36. That is it. The 'unitary method' sounds like a technique from a textbook and is really just the obvious thing, done deliberately.
The main trap is assuming a relationship is proportional when it is not. Two painters take half as long as one; ten do not take a tenth of the time, because they get in each other's way. One kettle boils in three minutes; two kettles do not boil in ninety seconds. Check that doubling the input genuinely doubles the output before you divide.
Where it gets interesting: scaling
Double the length of a cube and almost nobody's first guess about the volume is right.
Length scales by k, area by k squared, volume by k cubed. This is the square-cube law, and it is one of the most consequential facts in the whole of applied mathematics.
Why giants cannot exist
Make a person twice as tall, keeping the same proportions. Their weight depends on volume, so it goes up eight times. The strength of their bones depends on cross-sectional area, so it goes up only four times.
They are now carrying twice as much load per unit of bone strength as before. Scale up further and the bones fail. This is why large animals are not simply enlarged small ones - an elephant's legs are proportionally far thicker than a mouse's, and they have to be.
And why insects cannot be scaled up either
An insect breathes through tubes that rely on gas diffusing in. Oxygen demand scales with volume; the surface available to absorb it scales with area. Scale up and demand outruns supply.
The film-sized giant spider would suffocate long before it could stand. The same reasoning explains why small animals lose heat quickly - heat is generated by volume and lost through surface, and small things have far more surface per unit of volume, which is why a mouse must eat constantly and an elephant does not.
It applies to things too. Pizza: a 16-inch pizza has four times the area of an 8-inch, not twice, so it is usually far better value. Chips are thin because heat enters through the surface. Radiators have fins for the same reason. Once you know to look for area against volume, it turns up everywhere.
Percentages are ratios in disguise
Per cent means per hundred, so 35% is the ratio 35:100. Two things people get wrong:
- Percentage changes do not add. Up 10% then down 10% leaves you at 99%, not 100 - the second 10% is taken from a larger number.
- Percentage of what? A rise from 1% to 2% is a 1 percentage point rise and a 100% increase. Both are true and they sound completely different, which is why the phrasing is chosen carefully in news reports.