Variation (Proportion & Inverse Proportion)

If a is proportional to b,
a µ b
and a = kb, where k is a constant
The value of k will be the same for all values of a and b.

If a µ 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


Similarly, if m is proportional to n², m = kn²

Inverse Proportion
If a and b are inversely proportionally to one another,
a µ1/b
\ 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
\ 1 = k/3²
\ k = 9