Direct proportion

Direct proportion is the simplest kind of proportion: $y$ is just $x$ multiplied by a fixed number. This guide shows what the form $y = k \times x$ means, why $\frac{y}{x} = k$ says the same thing, and how to find $k$ from a single pair of values. The usual hurdle is not the arithmetic but seeing that the two forms are one rule.

When do I use this?

Whenever one quantity is a fixed multiple of another, like a price and an amount, so that "twice as much of one" means "twice as much of the other". In exam questions it usually shows up as "$y$ is directly proportional to $x$".

The form y = kx

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.

Worked example: the price of milk

Say a litre of milk costs £2. Let's call the price for the total amount of milk $y$, and the number of litres of milk 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

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

$$y = 2 \times x$$

And remember here 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.

The graph is a straight line through the origin. That makes sense: it is a linear function with nothing added on the end, so it crosses the y-axis at 0, and the gradient of the line is $k$.

The other form: y/x = k

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.

Finding k from one pair of values

This second form is the one we use when we are not told $k$. Say we only know that 3 litres of milk cost £6. Then we have an $x$ and a $y$ that belong together, and we put them into $\frac{y}{x} = k$:

$$k = \frac{6}{3} = 2$$

So the constant of proportionality is 2, and the rule is $y = 2x$. Now we can find the price of any amount. For 5 litres:

$$y = 2 \times 5 = 10$$

so 5 litres cost £10. Notice that $\frac{10}{5} = 2$ as well: the ratio between price and amount is the same for every pair, which is exactly what direct proportion promises.

Common mistakes

• "k is where the line 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, the gradient of the line, not a crossing point.
• "y grows when x grows, so it must be proportional." Only if it grows in exactly this way, $y = k \times x$. A taxi at £2 per km plus a £4 start fee is $y = 2x + 4$: it grows with $x$, but it is not proportional, because the ratio $\frac{y}{x}$ is not the same for every trip.
• "y/x = k is a different rule from y = kx." It is the same statement, rearranged. Use whichever form suits the question: $y = k \times x$ to compute $y$, $\frac{y}{x} = k$ to find $k$.

Related

• Back to the pillar: what is proportion?
• The other kind: inverse proportion, where $x \times y$ is constant instead.
• The straight line through the origin is a linear function with $c = 0$.