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proportion

Relation of a part to the whole (usually expressed as a fraction or percentage). In mathematics two variable quantities x and y are proportional if, for all values of x, y = kx, where k is a constant. This means that if x increases, y increases in a linear fashion.

Direct proportion

If A and B are in direct proportion, then as one grows bigger the other also grows bigger by the same proportion (or ratio). For example, if B doubles then A doubles. This can be written as A ∝ B or A = kB (where k is a constant multiplier). For example, when costing a number of pens:

For every extra pen bought, the cost goes up by the same amount (30p). So the cost of the pens will be proportional to the number bought.

If A is proportional to B there are two things that are true:

the multiplier rule - if A is multiplied by a value then B must be multiplied by the same value.

the graph of A against B is always a straight line through (0,0) and the gradient of the graph is the same as the ratio of A:B.

Inverse proportion

If A is inversely proportional to B, then as B gets bigger A gets smaller by the same factor. For example, if B is increased by a factor of 2 (B doubles) then A decreases by a factor of 2 (A is halved). This can be written as:

A ∝ 1/B or A = k/B

For example, a group of people want to hire a minibus at a charge of £60 a day. If only 1 person uses the minibus it will cost them £60, for 2 people the cost will be £30 each, and for 3 people the cost will be £20 each. A table and graph showing the number of people and cost can be completed:

cost per person ∝ 1/people cost per person = k/people

k = the hire of the bus (£60), so cost per person = 60/people

The shape of the graph is typical of an inversely proportional relationship.

Many laws of science relate quantities that are proportional (for example, Boyle's law).

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