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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).

proportion

In music, the mathematical relationship between the numbers of vibrations of different notes, which are exactly in tune with each other when the ratios between these vibrations are mathematically correct. For instance, a perfect fifth stands in the relation of 2:3 in the number of vibrations of its two notes. The term proportion was also used in early music to designate the rhythmic relationships between one time-signature and another.

proportion

In art, the size, location, or amount of one part or thing compared to another. Some subjects are governed by certain regular proportions. For instance a face consists of a rough oval; the eyes are set half way down the oval on a line that is ‘five eyes’ in width; the eyebrows arc over the whole eye; the bottom of the nose is usually as wide as the gap between the inner corners of the eyes; the mouth lies half way between the end of the nose and the chin, and at rest is as wide as the distance between the pupils of the eyes; the ears stretch from eye-level to just below the nose; and the neck starts from below the ears and gently curves towards the shoulders.

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