# Direct and Inverse Proportion

These sometimes come up on the exam – so don’t forget about them!

Direct Proportion

If two quantities are in direct proportion then when one increases or decreases the other increases or decreases by the same percentage.

On a graph this is shown with a straight line through the origin: y = kx

Inverse Proportion

As one quantity increases the other one decreases.

The equation and graph for this is y = k/x

Other Proportions

Proportional to can be shown by the symbol $\propto \!\,$.

Examiners might throw in some nasty looking proportions so here’s some examples of what you might expect.

y is proportional to x squared -> y $\propto \!\,$ x²
y is proportional to x cubed -> y
$\propto \!\,$ x³
y is proportional to the square root of x -> y
$\propto \!\,$ √x
y is inversely proportional to x cubed -> y
$\propto \!\,$ 1/x³

In an exam you will probably need to change the proportion into an equation to work something out. To do so just change the proportional sign to an =k – here’s some examples to show you what we mean

$\propto \!\,$ x² becomes y = kx²
$\propto \!\,$ x³ becomes y = kx³
$\propto \!\,$ √x becomes y = k√x
$\propto \!\,$ 1/x³ becomes y = k/x³

So What Sort Of Question Will I Be Asked On This Topic?

Let’s take an example!

Example: y is inversely proportional to x. When x = 12, y = 3. Find the constant proportionality and find x when y = 8.

First write down the proportionality. Here it is inverse proportion so

$\propto \!\,$ 1/x

Replace the $\propto \!\,$ sign with an =k

y = k/x

You can then manipulate the equation like normal

So xy = k

Put in the values from the question

3 x 12 = 36

So k = 36

So the constant proportionality is 36…so now you can find x when y = 8

8x = 36

so x = 4.5