Calculating Directly & Inversely Proportional Quantities

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Direct and inverse proportions show how two quantities influence each other. Many everyday phenomena can be understood as direct or inverse proportions. Learn how to recognize and calculate these common proportions in this lesson.

What is a Direct Proportion?

Suppose you want to buy nine pieces of gum, but you aren't sure how much it will cost. You know that one piece of gum costs 10 cents and two pieces cost 20 cents, so you quickly calculate that nine pieces of gum will cost 90 cents. You are able to do this because you determine that the number of pieces of gum you buy and the total cost are directly proportional.

In mathematical terms, two quantities are considered to be directly proportional when both of them increase at the same rate. This means that doubling one causes the other to double as well. We can even write an equation to show exactly how the two quantities are related. For the example above, the mathematical expression relating the number of pieces of gum and the total cost would be:

cost (in cents) = 10 x (number of pieces)

More generally, we can write any direct proportion as:

y = kx

In this equation, y and x are the two directly proportional quantities and k is a proportionality constant that tells you exactly how they are related (like 10 cents/piece above).

On a graph, a direct proportion will look like a straight line and the slope of the line will be equal to the constant k.

direct proportion graph

Let's look another example of a direct proportion.

example 1

In this case, the amount of money you make and the hours you work are directly proportional and the constant of proportionality is your pay rate ($15 per hour).

solution to example 1

What is an Inverse Proportion?

In contrast to a direct proportion, if two quantities are inversely proportional, one of the values will decrease at the same rate that the other increases. The mathematical equation for an inverse proportion can be written in two ways:

y = k/x


xy = k

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