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Variation
Proportion
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.
Example
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
If a and b are inversely proportionally to one another,
a ∝ 1/b
∴ a = k/b
In these examples, k is known as the constant of variation.
Example
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
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