# Constant of proportionality

A constant of proportionality, also referred to as a constant of variation, is a constant value denoted using the variable "k," that relates two variables in either direct or inverse variation.

## Direct variation

Direct variation describes a relationship in which two variables are directly proportional, and can be expressed in the form of an equation as or where y and x are variables, and k is the constant of proportionality. Variables that are directly proportional increase and decrease together; if y increases, x increases at the same rate; if y decreases, x decreases at the same rate.

For example, the number of eggs used is directly proportional to the number of omelets a person can make, and the number of eggs and omelets are related by a constant of proportionality. Given that a recipe requires 2 eggs to make 1 omelet, we can find the constant of proportionality by plugging this into either of the equations above, where y is the number of eggs and x is the number of omelets: Thus, k = 2. What this means is we can determine the number of eggs by multiplying the number of omelets by 2, or the number of omelets by dividing the number of eggs by 2. No matter how many eggs or omelets there are, they will be related by this constant of proportionality. Also, because they have a directly proportional relationship, if we double the number of eggs, we also double the number of omelets; if we halve the number of eggs, we also halve the number of omelets, and so on.

y = kx

# of eggs = 2(# of omelets)

# of omelets # of eggs
1 2
200 400
10,000 20,000

## Inverse variation

Inverse variation describes a relationship in which two variables are indirectly proportional, and can be expressed in the form of an equation as or where y and x are variables, and k is the constant of proportionality. Variables that are inversely proportional have a relationship such that when one variable increases, the other decreases, and vice versa. For example, the number of people performing a task may be inversely proportional to the amount of time it takes to complete the task. If we know that it takes 20 people 15 hours to perform a task, and that the relationship is inversely proportional, we can find the constant of proportionality by multiplying the two:

k = xy = 20 × 15 = 300

The constant of proportionality is therefore 300. Knowing the constant of proportionality between variables allows us to solve certain problems.

Example

If it takes 20 people 15 hours to perform a task, how long will it take 28 people to perform that same task?

We know that the constant of proportionality from above is 300, and we know that there are 28 people instead of 20 people, so plugging the number of people and the constant of proportionality into the equation for inverse variation: It would therefore take 28 people 10.714 hours.