### Learning Outcomes

• Find the common denominator of two or more fractions
• Use the common denominator to add or subtract fractions
• Simplify a fraction to its lowest terms

## Introduction

Before we get started, here is some important terminology that will help you understand the concepts about working with fractions in this section.

• product: the result of  multiplication
• factor: something being multiplied — for  $3 \cdot 2 = 6$ , both $3$ and $2$ are factors of $6$
• numerator: the top part of a fraction — the numerator in the fraction $\dfrac{2}{3}$ is $2$
• denominator: the bottom part of a fraction — the denominator in the fraction $\dfrac{2}{3}$ is $3$

## Math Question Instructions

Many different words are used by math textbooks and teachers to provide students with instructions on what they are to do with a given problem. For example, you may see instructions such as “Find” or “Simplify” in the example in this module. It is important to understand what these words mean so you can successfully work through the problems in this course. Here is a short list of the words you may see that can help you know how to work through the problems in this module.

Instruction Interpretation
Find Perform the indicated mathematical operations which may include addition, subtraction, multiplication, division.
Simplify 1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division

2) Write a mathematical statement in smallest terms so there are no other mathematical operations that can be performed—often found in problems related to fractions and the order of operations

Evaluate 1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division

2) Substitute a given value for a variable in an expression and then perform the indicated mathematical operations

Reduce Write a mathematical statement in smallest or lowest terms so there are no other mathematical operations that can be performed—often found in problems related to fractions or division

When you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using.

The “parts of a whole” concept can be modeled with pizzas and pizza slices. For example, imagine a pizza is cut into $4$ pieces, and someone takes $1$ piece. Now, $\dfrac{1}{4}$ of the pizza is gone and $\dfrac{3}{4}$ remains. Note that both of these fractions have a denominator of $4$, which refers to the number of slices the whole pizza has been cut into. What if you have another pizza that had been cut into $8$ equal parts and $3$ of those parts were gone, leaving $\dfrac{5}{8}$?

How can you describe the total amount of pizza that is left between the two pizzas? There are now two very different fractions and it’s hard to compare the two. You will need a common denominator, technically called the least common multiple. Remember that if a number is a multiple of another, you can divide them and have no remainder.

One way to find the least common multiple of two or more numbers is to first multiply each by $1, 2, 3, 4$, etc.  For example, find the least common multiple of $2$ and $5$.

 First, list all the multiples of $2$: Then list all the multiples of 5: $2\cdot 1 = 2$ $5\cdot 1 = 5$ $2\cdot 2 = 4$ $5\cdot 2 = 10$ $2\cdot 3 = 6$ $5\cdot 3 = 15$ $2\cdot 4 = 8$ $5\cdot 4 = 20$ $2\cdot 5 = 10$ $5\cdot 5 = 25$

The smallest multiple they have in common will be the common denominator for the two! In the example above, that is $10$.

### Example

Describe the amount of pizza left in the examples above using common terms.

To add fractions with unlike denominators, you will need to first rewrite them with common denominators. Then, you know what to do! The steps are shown below.

### Adding Fractions with Unlike Denominators

1. Find a common denominator.
2. Rewrite each fraction using the common denominator.
3. Now that the fractions have a common denominator, you can add the numerators.
4. Simplify by canceling out all common factors in the numerator and denominator.

## Simplifying a Fraction

Often, if the answer to a problem is a fraction, you will be asked to write it in lowest terms. This is a common convention used in mathematics, similar to starting a sentence with a capital letter and ending it with a period. In this course, we will not go into great detail about methods for reducing fractions because there are many. The process of simplifying a fraction is often called reducing the fraction. We can simplify by canceling (dividing) the common factors in a fraction’s numerator and denominator.  We can do this because a fraction represents division.

For example, to simplify $\dfrac{6}{9}$ you can rewrite $6$ and $9$ using the smallest factors possible as follows:

$\dfrac{6}{9}=\dfrac{2\cdot3}{3\cdot3}$

Since there is a $3$ in both the numerator and denominator, and fractions can be considered division, we can divide the $3$ in the top by the $3$ in the bottom to reduce to $1$.

$\dfrac{6}{9}=\dfrac{2\cdot\cancel{3}}{3\cdot\cancel{3}}=\dfrac{2}{3}\cdot1=\dfrac{2}{3}$

Rewriting fractions with the smallest factors possible is often called prime factorization.

In the next example you are shown how to add two fractions with different denominators, then simplify the answer.

### Example

Add $\dfrac{2}{3}+\dfrac{1}{5}$. Simplify the answer.

You can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.

### Example

Add $\dfrac{3}{7}+\dfrac{2}{21}$. Simplify the answer.

In the following video you will see an example of how to add two fractions with different denominators.

You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.

Add $\dfrac{3}{4}+\dfrac{1}{6}+\dfrac{5}{8}$.  Simplify the answer and write as a mixed number.

What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together.

## Subtracting Fractions

When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. Below are some examples of subtracting fractions whose denominators are not alike.

### Example

Subtract $\dfrac{1}{5}-\dfrac{1}{6}$. Simplify the answer.

The example below shows how to use multiples to find the least common multiple, which will be the least common denominator.

### Example

Subtract $\dfrac{5}{6}-\dfrac{1}{4}$. Simplify the answer.

In the following video you will see an example of how to subtract fractions with unlike denominators.