# 6.2: Adding and Subtracting Fractions with Like Denominators

Difficulty Level: At Grade Created by: CK-12

## Introduction

Measuring for Brackets

Having successfully completed the estimation project, Travis is off to do some more measuring for his uncle. Uncle Larry has told Travis that he needs to make some measurements on a wall in what will be the kitchen. Uncle Larry shows Travis which wall to mark on and hands him a ruler and a pencil.

“I need you to make a small mark at \begin{align*}\frac{1}{8}''\end{align*}, another small mark at \begin{align*}\frac{2}{8}''\end{align*} past the first, and a large mark at \begin{align*}\frac{3}{8}''\end{align*} past the second mark,” says Uncle Larry. “Then continue that pattern across the wall. The most important marks are the large ones, please be sure that those marks are in the correct place. The large marks will indicate where I need to put brackets later.”

“Okay,” says Travis, smiling. He is confident that he knows what he is doing.

Uncle Larry goes off to work on another project and leaves Travis to his work.

“Hmmm,” thinks Travis to himself. “If I write in all of the large marks first, I will be done a lot quicker. Then I can go back and do the small ones. I can add these fractions to figure out at what measurement I need to draw in the large marks.”

Travis has a plan, but will his plan work? If Travis adds up the fractions, at what measurement will the large marks be drawn?

This lesson will teach you all that you need to know to answer each of these questions.

What You Will Learn

In this lesson, you will learn how to do the following:

• Add fractions with like denominators.
• Subtract fractions with like denominators.
• Evaluate numerical expressions involving sums and differences of fractions with common denominators.
• Solve real-world problems involving sums and differences of fractions with common denominators.

Teaching Time

I. Add Fractions With Like Denominators

You have already learned how to add whole numbers and how to add decimals, now you are going to learn how to add fractions. In this lesson, you will learn all about adding fractions with like or common denominators.

What is a like denominator?

A like denominator is a denominator that is the same. This means that the whole has been divided up into the same number of parts. If the denominator of two fractions is a five, then both of those fractions have been divided into five parts. The numerators may be different, but the denominators are the same.

This picture shows two different fractions with like denominators.

Now let’s say that we want to add these two fractions. Because the denominators are common, we are adding like parts. We can simply add the numerators and we will have our new fraction.

\begin{align*}\frac{2}{6} + \frac{4}{6} = \frac{6}{6}\end{align*}

Here it is as a picture.

We combined both of these fractions together to have a fraction we can call six-sixths.

We must simplify or reduce all of our answers. In this example, when we have six out of six parts, we have one whole. You can see that one whole figure is shaded in. We simplify our answer and then our work is complete.

Our final answer is \begin{align*}\frac{6}{6} = 1\end{align*}.

Let’s look at another one. We can work on this one without looking at a picture.

Example

\begin{align*}\frac{2}{8} + \frac{4}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

The first step is to make sure that you have like denominators. In this example, both denominators are 8, so we can add the numerators because the denominators are alike.

Our next step is to add the numerators.

2 + 4 = 6

We put that number over the common denominator.

\begin{align*}\frac{6}{8}\end{align*}

Our last step is to see if we can simplify our answer. In this example, 6 and 8 have the greatest common factor of 2. We divide both the numerator and the denominator by 2 to simplify the fraction.

\begin{align*}\frac{6 \div 2}{8 \div 2} = \frac{3}{4}\end{align*}

Our final answer is \begin{align*}\frac{3}{4}\end{align*}.

Now it is time for you to try a few of these on your own. Be sure that your answer is in simplest form.

1. \begin{align*} \frac{1}{7} + \frac{2}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*} \frac{3}{9} + \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*} \frac{2}{10} + \frac{3}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Take a minute to check your work with a peer. Did you remember to simplify problem number 3?

II. Subtract Fractions with Like Denominators

We can also subtract fractions with like denominators to find the difference between the fractions. As long as the denominators are the same, the fractions are alike, and we can simply subtract the numerators.

Here is an example done with pictures.

\begin{align*}\frac{6}{8} - \frac{3}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

To solve this problem, we simply subtract the numerators. The difference between six and three is three. We put that answer over the common denominator.

Our final answer is \begin{align*}\frac{3}{8}\end{align*}.

We don’t need to simplify this fraction because three-eighths is already in simplest form.

Try a few of these on your own. Simplify the difference if necessary.

1. \begin{align*}\frac{6}{7} - \frac{2}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{5}{9} - \frac{2}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{8}{10} - \frac{4}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

III. Evaluate Numerical Expressions Involving Sums and Differences of Fractions with Common Denominators

Sometimes, we can have a numerical expression that involves both the sums and differences of fractions with common denominators. This means that we will see more than one operation in an expression. We will need to evaluate the expression to find its value.

Example

\begin{align*}\frac{9}{10} - \frac{3}{10} + \frac{1}{10}\end{align*}

To evaluate this expression, we first need to ensure that the fractions all have a same common denominator. In this case, they all have a common denominator of 10.

Next, we work with the numerators. We are going to add or subtract in order from left to right.

9 - 3 = 6 + 1 = 7

Our final step is to put this answer over the common denominator.

\begin{align*}\frac{7}{10}.\end{align*}

Before we can say our answer is finished, we need to see if we can simplify our answer. There isn’t a common factor between 7 and 10 because 7 is prime, so our fraction is in its simplest form.

Our final answer is \begin{align*}\frac{7}{10}\end{align*}.

Let’s look at one more.

Example

\begin{align*} \frac{8}{9} + \frac{4}{9} - \frac{1}{9}\end{align*}

The fractions in this expression all have a common denominator, so we can add/subtract the numerators in order from left to right.

8 + 4 = 12 - 1 = 11

Next, we write this answer over the common denominator.

\begin{align*}\frac{11}{9}\end{align*}

Uh oh! We have an improper fraction. An improper fraction is NOT in simplest form, so we need to change this to a mixed number.

11 \begin{align*}\div\end{align*} 9 \begin{align*}=\end{align*} 1 with two-ninths left over.

Our final answer is \begin{align*}1 \frac{2}{9}\end{align*}.

Evaluate the following numerical expressions. Be sure that your answer is in simplest form.

1. \begin{align*} \frac{6}{7} - \frac{2}{7} + \frac{1}{7}\end{align*}
2. \begin{align*} \frac{3}{4} + \frac{3}{4} - \frac{1}{4}\end{align*}
3. \begin{align*} \frac{7}{8} + \frac{3}{8} - \frac{2}{8}\end{align*}

Check your work. Did you change any improper fractions to mixed numbers?

## Real Life Example Completed

Measuring for Brackets

You have learned all about adding and subtracting fractions with like denominators. Now let’s go back and see how Travis is doing with his measuring.

Having successfully completed the estimation project, Travis is off to do some more measuring for his uncle. Uncle Larry has told Travis that he needs to make some measurements on a wall in what will be the kitchen. Uncle Larry shows Travis which wall to mark on and hands him a ruler and a pencil.

“I need you to make a small mark at \begin{align*}\frac{1}{8}''\end{align*}, another small mark at \begin{align*}\frac{2}{8}''\end{align*} past the first, and a large mark at \begin{align*}\frac{3}{8}''\end{align*} past the second mark,” says Uncle Larry. “Then continue that pattern across the wall. The most important marks are the large ones, please be sure that those marks are in the correct place.”

“Okay,” says Travis, smiling. He is confident that he knows what he is doing.

Uncle Larry goes off to work on another project and leaves Travis to his work.

“Hmmm,” thinks Travis to himself. “If I write in all of the large marks first, I will be done a lot quicker. Then I can go back and do the small ones. I can add these fractions to figure out at what measurement I need to draw in the large marks.”

Travis has a plan, but will his plan work? If Travis adds up the fractions, at what measurement will the large mark be drawn?

First, let’s go back and underline the important information.

For Travis to follow his plan, he needs to add up the fractions to figure out what fraction of an inch should be between the large marks for the brackets.

\begin{align*}\frac{1}{8} + \frac{2}{8} + \frac{3}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

These fractions all have common denominators, so Travis can simply add the numerators.

1 + 2 + 3 = 6

Next, we can put this answer over the common denominator.

\begin{align*}\frac{6}{8}''\end{align*}

Travis needs to make a large mark every six-eighths of an inch. It will be a lot simpler to measure the marks if Travis simplifies this fraction.

\begin{align*}\frac{6}{8} = \frac{3}{4}\end{align*}

Travis needs to make a large mark every \begin{align*}\frac{3}{4}''\end{align*} of an inch. Confident in his calculations, he gets right to work.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Like Denominators
when the denominators of fractions being added or subtracted are the same.
Simplifying
dividing the numerator and the denominator of a fraction by its greatest common factor. The result is a fraction is simplest form.
Difference
the answer to a subtraction problem
Numerical Expression
an expression with multiple numbers and multiple operations
Operation
the four operations in math are addition, subtraction, multiplication and division
Evaluate
to find the value of a numerical expression.

## Time to Practice

Directions: Add the following fractions with common denominators. Be sure your answer is in simplest form.

1. \begin{align*}\frac{1}{3} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}\frac{2}{5} + \frac{2}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}\frac{4}{7} + \frac{2}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}\frac{5}{11} + \frac{4}{11} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}\frac{6}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

6. \begin{align*}\frac{4}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}\frac{3}{4} + \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}\frac{5}{6} + \frac{3}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}\frac{4}{9} + \frac{2}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}\frac{5}{10} + \frac{1}{10} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Find each difference. Be sure that your answer is in simplest form.

11. \begin{align*}\frac{6}{7} - \frac{3}{7} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}\frac{6}{12} - \frac{4}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}\frac{13}{18} - \frac{3}{18} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}\frac{7}{8} - \frac{6}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}\frac{4}{8} - \frac{2}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}\frac{10}{12} - \frac{6}{12} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}\frac{11}{13} - \frac{6}{13} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}\frac{10}{20} - \frac{5}{20} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}\frac{16}{18} - \frac{5}{18} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}\frac{9}{14} - \frac{2}{14} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Evaluate each numerical expression. Be sure your answer is in simplest form.

21. \begin{align*}\frac{1}{4} + \frac{3}{4} - \frac{2}{4}\end{align*}

22. \begin{align*}\frac{8}{9} + \frac{2}{9} - \frac{3}{9}\end{align*}

23. \begin{align*}\frac{7}{8} - \frac{2}{8} + \frac{1}{8}\end{align*}

24. \begin{align*}\frac{10}{12} - \frac{2}{12} + \frac{3}{12}\end{align*}

25. \begin{align*}\frac{15}{20} + \frac{7}{20} - \frac{2}{20}\end{align*}

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