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

direct proportion

Inverse Proportion

As one quantity increases the other one decreases.

The equation and graph for this is y = k/x
inverse proportion (2)

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