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Inversely Proportional: Definition, Formula & Examples

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  • 0:00 What Is Inversely…
  • 1:29 Graphing Inverse Relationships
  • 3:00 Example: A…
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Lesson Transcript
Instructor: Neelam Mehta

Neelam has taught variety of math and science subjects. She has masters' degrees in Chemical Engineering and Instructional Technology.

In this lesson, you will understand the definition of the term 'inversely proportional' and be able to write the algebraic expressions for inverse variations. You will also be able to apply the inverse variation relationships to solve different types of problems.

What Is Inversely Proportional?

In math, quantities can change when you change another quantity. When two quantities or variables are connected, we say that there is a relationship between the two. Variables can have one of three relationships or variations: direct, inverse and joint.

In this lesson, we focus on understanding the definition of inverse variation: if one quantity increases as a result of decrease in another quantity or vice versa, then the two quantities are inversely proportional. We can write the mathematical definition of inversely proportional as seen in figure 1.

Figure 1: y is the variable that is inversely proportional to variable x raised to the n power, and k is a nonzero number.
Definition of Inversely Proportional

Say we have n = 1, then the definition can be simplified and written as: y=k/x, where 'y' is inversely proportional to 'x'.

If x is raised to the second power, then we say that y is inversely proportional to the square of x or cube of x if raised to the third power, and so on. The value of n can be a fraction as well such as ½ power. When you have an exponent as 1/2, it is also known as the square root. In this instance, we would say that y is inversely proportional to the square root of x, and we would write it in the following way:


Let's get a better understanding of what inversely proportional means by plotting value of x and y for different values of n:


Graph of

Take a look at the plot when y is inversely proportional to x with a constant k of 50. Notice that as the value of x increases, the value of y decreases. 50 is divided by the increasing value of x, resulting in smaller and smaller values of y. This happens because x is in the denominator. Often times you will end up with a relationship that is inversely proportional if the independent variable is in the denominator.

Let's look at what happens if you have x to the second power in the denominator rather than first power. Notice that, initially, as the value of x increases, the value of y decreases very rapidly, but then the decline is slower in comparison to the beginning of the graph. However, the trend is still the same as before: as the value of x goes up, the value of y goes down.

Graph of

If you were to plot y as a function of 50 divided by square root of x, even then this trend will still stay the same. An increase in value of x will result in the decrease in value of y, or the other way around. Basically, when one variable goes in one direction, the other variable usually goes in the opposite direction. This is the reason why this type of relationship is called inversely proportional.

Now that we have a better understand of this relationship, let's see how we can apply it to solve problems.

Example: A Manufacturing Problem

Let's look at an example of how this would work in the real world.

Example:

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