How to Add Fractions with Variables

Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will learn about fractions that have variables in them and variables that have fractions as coefficients. We will learn how to use common denominators to add fractions in both of these scenarios.

Basics

So you need to know how to add fractions with variables? Before we dive in, let's start with some basics over in the shallow area. First off, what are fractions and variables? Let's get these definitions out of the way before we make the big splash!

Fractions can represent a lot of different things such as parts of a whole or ratios. They are typically seen as two whole numbers being divided, such as 4/5 or 7/8. They can be represented side by side but are usually are seen on top of one another. The numerator is the number on top and the denominator is the number on bottom.

Numerator and denominator in a fraction
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You are probably saying, 'Fractions...those are easy!' Well, before we venture to the high dive, let's talk about variables for a minute. Variables are letters that represent unknown values. Common variables in math are x and y, although any letter in the alphabet can be used.

Fractions and Variables

Now that we've got the basics down, we can swim a bit deeper... so how can variables be with fractions? There are two different ways! Variables can be part of the actual fraction such as x/3 or fractions can be coefficients for variables. Coefficients are numbers attached to variables by multiplication. For example, in the term 5x, 5 would be the coefficient.

Types of fractions with variables
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Variables As Part of the Fraction

Okay, now we are ready to dive in! So let's say we want to add 5/x and 3/x. Since both of these fractions have what's called a common denominator, we can just add the numerator. Common denominators allow for fractions to be added when the terms on bottom are the same. Since both fraction have x on the bottom, we can simply add 5 and 3 on top. Thus, our final answer will be 8/x.

Things will not always be that simple though. Sometimes, we will be given denominators that are not the same and will have to find the common denominator. Let's look at the addition of 1/4x and 3/2y to get an example of this. We can see right away that the denominators are not the same. Thus, we have to make them the same, just like when adding regular fractions without variables.


Adding Uncommon Denominators
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The least common denominator here will be 4xy. To get 4xy for the first fraction, we need to multiply by y/y. This will give us y/4xy. Remember, what you multiply the denominator by, you must also multiply the numerator by. For the second fraction, we will multiply by 2x/2x to give us 6x/4xy. Now that we have the same thing on the bottom of both fractions, we can add the numerators. The final answer will be 6x + y / 4xy. The numerator cannot be simplified any more because the two terms have different variables. Also, remember that variables are written in alphabetical order which is why the y term came last.

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