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.

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We can write the mathematical definition of inversely proportional as seen in figure 1.

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:

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:Chloe came up with a new design for a bracelet and is now interested in producing the bracelet in large quantities so that she can sell them on her online store. Her initial calculations suggest that the cost of producing one bracelet varies inversely as the square of the number of the bracelets made. If Chloe makes 100 bracelets, then it would cost \$2 per bracelet. What would be the unit price of the bracelets if Chloe decides to manufacture 500 bracelets?In order to find the solution, we must first write the equation that describes the relationship between the quantities of bracelets and the unit price of the bracelet.

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