Concept

Inverse Variation

An inverse variation, or inverse proportionality, occurs when two non-zero variables have a relationship such that their product is constant. This relationship is often written with one of the variables isolated on the left-hand side.

$x y = k or y = \frac{k}{x}$

The constant

k

is the constant of variation. When

k = 0,

the relationship is not an inverse variation. In the following example, the constant of variation is

k = 2 .

A graph of a function y=2/x with a point on the graph that can be moved

The constant of variation may be any real number except

0 .

Here are some examples of inverse variations.

Examples of Inverse Variation
Example	Rule	Comment
The gas pressure in a sealed container if the container's volume is changed, given constant temperature and constant amount of gas.	$P = \frac{n R T}{V}$	The variables are the pressure $P$ and the volume $V .$ The amount of gas $n,$ temperature $T,$ and universal gas constant $R$ are fixed values. Therefore, the constant of variation is $n R T .$
The time it takes to travel a given distance at various speeds.	$t = \frac{d}{s}$	The constant of variation is the distance $d$ and the variables are the time $t$ and the speed $s .$

Inverse variation is closely related to other types of variation.

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Concept

Inverse Variation

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