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# Proportionality

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.

References:

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