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This article is about proportionality, the mathematical relation. For other uses of the term proportionality, see proportionality (disambiguation).

In mathematics, two quantities are called proportional if they vary in such a way that one of the quatities is a constant multiple of the other, or equivalently if they have a constant ratio.

Definition

More formally, the variable y is said to be proportional (or sometimes directly proportional) to the variable x, if there exists a constant non-zero number k such that

y=k\times x

.

The relation is often denoted:

y\propto x,

and the constant ratio

k=y/x\,

is called the proportionality constant or constant of proportionality of the proportionality relation.

Examples

If you travel at a constant speed, then the distance traveled is proportional to the time spent traveling, with the speed being the constant of proportionality.

The circumference of a circle is proportional to its diameter, with the constant of proportionality equal to π.

On a map drawn to scale, the distance between any two points on the map is proportional to the distance between the two locations the points represent, with the constant of proportionality being the scale of the map.

The amount of force acting on a certain object from the gravity of the Earth at sea level is proportional to the object's mass, with the gravitational constant being the constant of proportionality.

Properties

Since

y=k\times x

is equivalent to

x=(1/k)\times y,

it follows that, if y is proportional to x, with 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.

inverse proportionality

As noted in the definition above two proportional variables are sometime said to be directly proportional. This is done so as to contrast proportionality with inverse proportionality.

Two variables are inversely proportional if one of the variables is directly proportional with the multiplicative inverse of the other, or equivalently if their product is a constant. It follows, that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

y={k \over x}.

For instance, the number of people you hire to shovel sand is (approximately) inversely proportional to the time needed to get the job done.

Exponetial and logarithmic proportionality

A variable y is exponentially proportional to a variable x, if y is directly porportional to the exponential function of x, that is if there exists a non-zero constant k such that

y=ka^{x}.\,

Likewise, a variable y is logarithmically proportional to a variable x, if y is directly porportional to the logarithm of x, that is if there exists a non-zero constant k such that

y=k\log _{a}(x).\,

Experimental determination

To experimentally determine whether two physical quantities are directly proportional, one performs several measurements and plots the resulting points in a Cartesian coordinate system. If the points lie on (or close to) a straight line passing through the origin (0, 0), then the two variables are (probably) proportional, with the proportionality constant given by the line's slope.