Direct Proportion

Direct proportions are direct multiplicative links between two values e.g. there is a direct proportion between centimetres (cm) and metres (m). We know that 100cm=1m, so the multiplier is always 100. So as one quantity increases, the other quantity increases at the same rate.

If y is directly proportional to x, we write y\propto x and this is equal to y=kx, for some constant k. The constant can be calculated from the information provided about x and y.

The symbol for direct proportion is:

If t is directly proportionate to p then this means that t=kp, where k is a natural number showing that p is a multiple of t.

There are 4 steps to follow when dealing with direct proportion:

  1. Write the proportion relationship.
  2. Convert to an equation using a constant of proportionality.
  3. Use given information to find the constant of proportionality.
  4. Substitute the constant of proportionality into the equation.


Suppose y varies directly as x^{2} and that y=12 when x=2.

Form an equation connecting x and y, then use it to find the value of y when x=5. Also, find the value of x when y=108

Here, y\propto x^{2}\Rightarrow y=kx^{2}.

We use x=2 when y=12.

  • y\propto x^(2)
  • 12=k\left( 2^{2}\right)
  • 4k=12
  • k=3

So, the equation is y=3x^{2}.

When x=5:

y=3\left( 5^{2}\right) =3\left( 25\right) =75

When y=108:

3x^{2}=108,x^{2}=36,x=\sqrt{36},x=\pm 6


The value of f is directly proportional to g. When f=40, g=20. We know that f=kg, therefore 40=20k. As a result, k=2. The final equation gives f=2g.