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Revision:Variation

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Variation

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Proportion

If a is proportional to b,

\displaystyle a \varpropto b,

and a = kb, where k is a constant.

The value of k will be the same for all values of a and b.


Example

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

\displaystyle a = kb \quad \quad \quad \quad (1).


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

\displaystyle 5 = 10k,

so \displaystyle k = \frac{1}{2}.


Substitute this into (1):

\displaystyle a = \frac{1}{2}b.


Similarly, if:

m \varpropto n^2,

then:

m = kn^2.


Inverse Proportion

If a and b are inversely proportionally to one another,

\displaystyle a \varpropto \frac{1}{b},

and

\displaystyle a = \frac{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.

\displaystyle b = \frac{k}{a^2}


when a = 3 and b = 1,

\displaystyle 1 = \frac{k}{3^2},

\displaystyle k = 9.


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