Proportionality and scale

Two quantities are proportional if they change by the same factor: twice as much of the one gives twice as much of the other.

Buns	Price
1	12 kr
3	36 kr
5	60 kr
10	120 kr

Divide the price by the number: always 12. That number is called the constant of proportionality, and it is the price per bun.

The relationship is written y = 12x — and as a graph it becomes a straight line through the origin.

Not proportional:

Hours	Taxi cost
0	50 kr
1	350 kr
2	650 kr

Here it costs 50 kr before the car has moved at all. The relationship is linear (y = 300x + 50) but not proportional — the line does not go through the origin.

The test is simple: is y/x the same for every row? Then it is proportional.

Three uses of the same idea:

Use	Question	Calculation
The unit price	what does one cost?	price / number
Scale	how large in reality?	the length on the map · the scale number
A recipe	how much for 6 people?	amount · (6 / the number in the recipe)

A scale of 1:50 000 means that 1 cm on the map is 50 000 cm in reality, that is, 500 metres. The unit conversion is where most of the errors arise, not the multiplication.

Inverse proportionality is when the product is constant instead of the quotient:

People	Hours
1	12
2	6
3	4
6	2

The product is always 12. More people, less time. The relationship is y = 12/x — a hyperbola, not a line.

A warning: inverse proportionality rarely holds in reality. Twelve people do not do the job in one hour, and a thousand people do not do it in 43 seconds. Models hold within a range.

Why AI cares: a linear model with one variable is a proportional or a linear relationship. price = 2400·area is proportional; price = 2400·area + 150000 is linear but not proportional. The constant 150000 is called the intercept and is precisely the taxi's starting charge.

Decide whether the relationship is proportional. Divide y by x for each row and see whether the quotient is the same.

#	Relationship	Proportional?
1	2 kg of potatoes 30 kr, 5 kg 75 kr
2	A mobile plan: 99 kr/month + 2 kr/GB
3	A square's perimeter and its side
4	A square's area and its side
5	Distance and time at a constant speed
6	The number of workers and the time for a job

The key:

1. Yes — 15 kr/kg in both cases.
2. No — 99 kr is paid even at 0 GB. Linear, not proportional.
3. Yes — the perimeter is 4s.
4. No — the area is s². Double the side and you get four times the area.
5. Yes — s = v·t.
6. No — inversely proportional.

Number 4 is the most important to understand. It is why a pizza twice the size contains four times as much pizza, and why a model with images twice the size needs four times as much computation. Areas and volumes do not grow proportionally with lengths.

Compute it yourself: a pizza 30 cm in diameter against two of 20 cm — which gives the most? (The answer: the large one, 707 cm² against 628 cm².)