Skill 4 of 11
In Progress


At the bakery, you order a portion of layered birthday cake for 3 €. It is pretty good. You order 3 more. You of course pay three times the price to get three portions, so 9 €.

Obviously, if you order double as many, you pay double as much. Order 3 times as many, and you pay 3 times as much.

  • 1 costs 3 €,
  • 2 cost 6 €,
  • 3 cost 9 €,
  • 4 cost 12 € etc.

Maybe the bakery has a 5-portion discount, where you save 2 €. Then the pattern doesn’t hold beyond five.

  • 5 cost not the expected 15 €, but only 13 €.
  • 6 cost not the expected 18 €, but 16 €

Doubling your number of portions now does not mean doubling the total price.

When the pattern is followed and a double portion means a doubled price, you always pay the same per portion. It is always 3 € per portion, even if you buy three. We could call this phenomenon ‘“per”-portionality’ or better yet: proportionality.1 Let’s even invent a symbol, such as this “doodle”: $\propto$. If ‘the number of portions’ is symbolised $n$, then we can write ‘the price is proportional to $n$’ mathematically as:

$$\text{price}\propto n$$

It can also be written as an equation, of course. Each portion costs 3 €, so the price will be:

$$\text{price}=\underbrace{3\,€/\text{portion}}_{\substack{\text{proportionality}\\ \text{constant}}}\cdot n$$

If the portion-price of 3 € changed depending on $n$ (such as it would with an amount-discount), then there is not proportionality. Only if the 3 € is constant, is $n$ proportional to the total price. Let’s call the ‘3 €/portion’ a proportionality constant.

Writing $\text{price}\propto n$ or writing $\text{price}=3\,\mathrm{€/portion}\cdot n$ thus both imply proportionality, where some proportionality constant is involved. The latter version is of course the most useful one, where we can calculate the values.


  1. Online Etymology Dictionary’ (dictionary), Douglas Harper,