If a "is proportional" to b (which is the same as 'a is in direct proportion with b') then as b increases, a increases. In fact, there is a constant number k with a = kb. We write a ∝ b if a is proportional to b.

The value of k will be the same for all values of a and b and so it can be found by substituting in values for a and b.


If a ∝ b, and b = 10 when a = 5, find an equation connecting a and b.

a = kb (1)

Substitute the values of 5 and 10 into the equation to find k:

5 = 10k

so k = 1/2

substitute this into (1)

a = ½b

In this example we might then be asked to find the value of a when b = 2. Now that we have a formula connecting a and b (a = ½ b) we can subsitute b=2 to get a = 1.

Similarly, if m is proportional to n2, then m = kn2 for some constant number k.

If x and y are in direct proportion then the graph of y against x will be a straight line.

Inverse Proportion (HIGHER TIER)

If a and b are inversely proportionally to one another,

a ∝ 1/b

therefore a = k/b

In these examples, k is known as the constant of variation.


If b is inversely proportional to the square of a, and when a = 3, b = 1, find the constant of variation.

b = k/a2 when a = 3,

b = 1

therefore 1 = k/32

therefore k = 9