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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.

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

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.

Unit price of bracelet is inversely proportional to square of the quantity of the bracelets.</p>
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<p>where <i>k</i> is a constant that we can solve since we know the unit price of the bracelet when 100 quantities are produced.In order to figure out what <i>k</i> is, we first multiply each side by Quantity^2, which gives us:<i>k</i> = Price x Quantity^2We can then plug in the numbers we have, and we get:$2 x 100^2 = 20,000Now that we know the value of <i>k</i>, we can calculate the unit price if we were to produce 500 quantities. We must go back to our original formula of Price = <i>k</i> / Quantity^2 and then plug in our numbers. This gives us:Price = 20,000 / 500^2= $0.</p>
<p>08We now know that it would cost eight cents per bracelet if Chloe decides to produce 500 bracelets.</p>
<h2>Lesson Summary</h2>
<p>In this lesson, you learned what it means when two variables are inversely related or proportional. You learned that two variables are <b>inversely proportional</b> if one variable increases as a result of a decrease in another one. You also learned what happens to the dependent variable (<i>y</i>) if you change the independent variable (<i>x</i>) in inverse relationships: the <i>y</i> variable decreases as the <i>x</i> variable increases, and it increases as the <i>x</i> variable decreases.</p>
<p> We also worked out an example in which we solved various quantities when inverse proportionality is applied. Now you are ready to try some problems on your own on this topic.</p>
<h2>Learning Outcomes</h2>
<p>When this lesson is over, students should be able to:</p>
<li>Define inverse variation and inversely proportional</li>
<li>Write the mathematical definition for inversely proportional variables</li>
<li>Plot <i>x</i> and <i>y</i> inversely proportional variables on a graph</li>
<li>Solve word problems involving inverse proportionality</li>
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