Proportionality (mathematics)

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In mathematics, two quantities are proportional if they vary in such a way that one of them is a constant multiple of the other.

Contents

Symbol

The mathematical symbol '∝' is used to indicate that two values are proportional. For example, A ∝ B.

In Unicode this is symbol U+221D.

Direct proportionality

Given two variables x and y, y is (directly) proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that

The relation is often denoted

and the constant ratio

is called the proportionality constant or constant of proportionality.

Examples

  • If an object travels at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.
  • On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.

Properties

Since

is equivalent to

it follows that if y is proportional to x, with (nonzero) proportionality constant k, then x is also proportional to y with proportionality constant 1/k.

If y is proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.

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