What is proportion? Direct and inverse proportion

Proportion is a relationship between $x$ and $y$. As we know from functions, $y$ grows by some amount or other when $x$ grows. But when a function grows "proportionally", it grows in a very particular way. There are two kinds of proportion, and each has its own form.

Direct proportion

Direct proportion has the form

$$y = k \times x$$

This means that when $x$ goes up, $y$ goes up by some constant. This value $k$ is what we call the constant of proportionality, or the proportionality factor.

Say a litre of milk costs £2. Let's call the price for the total amount of milk $y$, and the number of litres we buy $x$. Then we can see that when our milk costs £2 per litre, and we write it in the form of direct proportion, we get

$$\text{price of milk} = 2 \times \text{litres of milk}$$

$$y = 2 \times x$$

Remember that $y$ is the price of however much milk we buy, and $x$ is the amount of milk. It says that the price is directly proportional to the amount of milk. $x$ and $y$ are what we call variables, because they vary, that is, they change.

We can also meet direct proportion in the form

$$\frac{y}{x} = k$$

It is really the same thing, we have just moved things around a bit in the original form. As we remember from fractions, a fraction is also a ratio. So here we can say that the ratio between $y$ and $x$ is constant. In direct proportion the ratio between the two variables is constant. The guide on direct proportion works through the example and shows how to find $k$.

Inverse proportion

Inverse proportion is also a relationship between two variables. But it comes in a different form, namely

$$y = \frac{k}{x}$$

It can also come in other forms, if we just rewrite the original form. That could be

$$x \times y = k \quad \text{or} \quad y = k \times \frac{1}{x}$$

Remember, these are just rewrites. We have only moved things around in the original form (the same thing we do in equations when we get $x$ on its own).

The relationship can sound a bit advanced, but it is actually fairly easy to understand. If we are out running at 8 km/h and we have to run 8 km, it takes an hour. If we ran twice as fast, 16 km/h, it would only take us half an hour. Here we would say that the time it takes us is inversely proportional to the speed. The guide on inverse proportion finds the $k$ in that example and shows the graph.

The two forms side by side

Direct proportion, $y = k \times x$, is a linear function with nothing added on the end, so its graph is a straight line through the origin. Inverse proportion, $y = \frac{k}{x}$, is a falling curve: when $x$ goes up, $y$ comes down, just like the time when the speed goes up.

Common misunderstandings

• "Any function where y depends on x is proportional." No. Proportion means $y$ grows in one very particular way: either $y = k \times x$ or $y = \frac{k}{x}$. A taxi fare with a start fee, $y = 2x + 4$, depends on $x$ but is not proportional.
• "Inverse proportion just means y goes down." The defining thing is that the product $x \times y$ stays constant. That $y$ goes down when $x$ goes up is a consequence, not the definition.
• "y/x = k is a different rule from y = kx." It is the same statement, rearranged. Either form tells you the ratio between $y$ and $x$ is constant.
• "k is where the graph crosses the y-axis." In $y = k \times x$ there is no constant added on, so the line goes through the origin. $k$ is what you multiply $x$ by, not a crossing point.

Related guides

• Direct proportion
• Inverse proportion
• Direct proportion is a linear function with $c = 0$, and $\frac{y}{x} = k$ leans on fractions as ratios.