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# 6.3: Adding and Subtracting Fractions with Different Denominators

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Layers of a Wall

Travis is having a great time working with his Uncle Larry. On his second day of working, Travis and Uncle Larry worked on the layers of a wall.

“When you first look at it you don’t realize that there are many different layers to a wall that add to its thickness,” Uncle Larry tells Travis.

“Well, we start with drywall that is \begin{align*}\frac{1}{4}”\end{align*} thick. Then we add the insulation. For this wall we are going to use two different forms of insulation. One is \begin{align*}\frac{3}{4}”\end{align*} thick and the other is \begin{align*}\frac{1}{2}\end{align*}” thick. Next, we add a \begin{align*}\frac{1}{2}”\end{align*} layer of wall sheathing. Finally we add the siding, that is \begin{align*}\frac{7}{8}”\end{align*} of an inch thick,” Uncle Larry explains.

“Wow, that is pretty thick.” says Travis.

But how thick is it? Travis isn’t sure. Notice that all of these fractions have different denominators. To figure out the thickness of the wall, you will need to know how to add fractions with different denominators.

This lesson has all of the information that you will need!

What You Will Learn

In this lesson, you will learn to execute the following skills:

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

Teaching Time

I. Add Fractions With Different Denominators

In our last lesson, you learned how to add fractions that had the same denominator. When you add fractions with the same denominator, you don’t have to do anything with the denominator, you can just add the numerators. Because the wholes are divided in the same way, they are alike. Therefore, adding these fractions is very simple.

Not all fractions have common denominators. When we have fractions with unlike denominators, we can still add them, but we will need to rename the fractions before we can add them.

How do we add fractions with different denominators?

To add fractions that have different denominators, we have to rename the fractions so that they are alike. We rename them by changing the different denominators of the fractions to common denominators.

Example

\begin{align*}\frac{1}{2} + \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} In this example, we are trying to add halves and fourths. If something is divided into halves, it is divided into two pieces. If something is divided into fourths, it is divided into four pieces.

Here we are trying to add fourths and halves. They are different quantities. You can see that although the bar is the same size, the parts are different sizes. We have to rewrite these fractions so that they have common denominators.

How do we rewrite fractions to have a common denominator?

The first step in doing this is to the find the least common multiple of both of the denominators. This LCM will become the lowest common denominator.

Let’s look at 2 and 4.

First, name the multiples of 2: 2, 4, 6, 8, 10 . . . .

Next, name the multiples of 4: 4, 8, 12, 16

The least common multiple of 2 and 4 is 4.

Our next step is to rewrite each fraction as an equivalent fraction that has four as a denominator.

\begin{align*}\frac{1}{2} = \frac{}{\;4\;}\end{align*} to name one-half in terms of fourths, we need to multiply the numerator and denominator by the same number. 2 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 4, so we multiply the numerator by 2 also. 1 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 2.

\begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}

Our second fraction, \begin{align*}\frac{1}{4}\end{align*}, is already written in terms of fourths so we don’t need to change it at all.

Next, we can add the renamed fractions.

\begin{align*}\frac{2}{4} + \frac{1}{4} = \frac{3}{4}\end{align*}

Our answer is \begin{align*}\frac{3}{4}\end{align*}. This answer is in simplest form, so our work is complete.

As long as you rename fractions with the lowest common denominator, you can add any number of fractions with unlike denominators.

1. \begin{align*}\frac{1}{2} + \frac{2}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}\frac{2}{3} + \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}\frac{4}{5} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Take a few minutes to check your answers with a peer. Number three is a tricky one-did you choose 15 as the LCD? Did you simplify the improper fraction?

II. Subtract Fractions with Different Denominators

Just as we can add fractions with different denominators by renaming them with the lowest common denominator, we can also subtract fractions with different denominators by doing the same thing.

First, remember that to subtract two fractions with different denominators, we rename them with a common denominator.

We do this by finding the least common multiple and then we rename each fraction as an equivalent fraction with that least common multiple as the lowest common denominator.

Example

\begin{align*}\frac{6}{8} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} First, find the least common multiple of 4 and 8. It is 8.

Next, rename each fraction in terms of eighths. Remember that renaming is another way of saying that we create an equivalent fraction in terms of eighths.

\begin{align*}\frac{6}{8}\end{align*} is already in terms of eighths. We leave it alone.

\begin{align*}\frac{1}{4} = \frac{2}{8}\end{align*}

Now we can rewrite the problem and find the difference.

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

We can simplify four-eighths by dividing the numerator and the denominator by the GCF. The GCF is 4.

\begin{align*}\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\end{align*}

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

Yes! The more you practice, the more you will find that this is true!

Subtract the following fractions. Be sure that your answer is in simplest form.

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

Take a few minutes to check your work with a friend.

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

In our last lesson, we worked on evaluating numerical expressions that had multiple operations and multiple fractions in them. This lesson is going to build on that concept, except this time, our fractions are going to have different denominators. We are going to need to rename them with a lowest common denominator before evaluating the expression.

Let’s look at an example.

Example

\begin{align*}\frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*} Right away, you can see that we have THREE different denominators. We need to find the LCM for all three denominators.

Begin by naming the multiples of each number.

2, 4, 6, 8, 10, 12

3, 6, 9, 12

6, 12

All three have the number six in common. This is our lowest common denominator.

Next, we rename all three fractions in terms of sixths, by creating an equivalent fraction for each one.

\begin{align*}\frac{1}{2} & = \frac{3}{6}\\ \frac{1}{3} & = \frac{2}{6}\\ \frac{1}{6}\end{align*}

Notice that one-sixth is already written in terms of sixths, so it can remain the same.

Now we rewrite the problem.

\begin{align*}\frac{3}{6} + \frac{2}{6} - \frac{1}{6}\end{align*}

We can add/subtract in order from left to right.

3 + 2 = 5 - 1 = 4

Our answer is \begin{align*}\frac{4}{6}\end{align*}.

\begin{align*}\frac{4}{6}\end{align*} can be simplified by dividing by the greatest common factor of 2.

Our final answer is \begin{align*}\frac{2}{3}\end{align*}.

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

1. \begin{align*} \frac{4}{5} + \frac{2}{10} - \frac{1}{2}\end{align*}
2. \begin{align*} \frac{4}{8} - \frac{1}{2} + \frac{1}{4}\end{align*}

## Real Life Example Completed

The Layers of a Wall

Now that you have learned about adding and subtracting fractions with different denominators, you are ready to come back to Travis and his fraction dilemma.

Here is the problem once again.

Travis is having a great time working with his Uncle Larry. On his second day of working, Travis and Uncle Larry worked on the layers of a wall.

“When you first look at it, you don’t realize that there are many different layers to a wall that add to its thickness,” Uncle Larry tells Travis.

“Well, we start with drywall that is \begin{align*}\frac{1}{4}”\end{align*} thick. Then we add the insulation. For this wall we are going to use two different forms of insulation. One is \begin{align*}\frac{3}{4}”\end{align*} thick and the other is \begin{align*}\frac{1}{2}”\end{align*} thick. Next, we add a \begin{align*}\frac{1}{2}”\end{align*} layer of wall sheathing. Finally we add the siding, that is \begin{align*}\frac{7}{8}”\end{align*} of an inch thick,” Uncle Larry explains.

“Wow, that is pretty thick.” says Travis.

But how thick is it? Travis isn’t sure.

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

Next, Travis needs to add up all of the fractions to figure out how thick the wall really is. To do this, he needs to write a numerical expression like the ones that we worked on in the last section. The expression looks like this.

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

This expression shows all of the different layers of the wall.

To find a measurement for the thickness of the wall, Travis must add all of these fractions together. To do this, he will need to rename them using the lowest common denominator.

What is the lowest common denominator for 4, 2 and 8? Yes. It is 8.

Rename each fraction in terms of eighths.

\begin{align*}\frac{1}{4} & = \frac{2}{8}\\ \frac{3}{4} & = \frac{6}{8}\\ \frac{1}{2} & = \frac{4}{8}\end{align*}

Next, let’s rewrite the expression.

\begin{align*}\frac{2}{8} + \frac{6}{8} + \frac{4}{8} + \frac{4}{8} + \frac{7}{8}\end{align*}

Now we can add the numerators.

\begin{align*}2 + 6 + 4 + 4 + 7 = 23\end{align*}

\begin{align*}\frac{23}{8} = 2 \frac{7}{8}”\end{align*}

Travis can see that the wall is almost three inches thick.

## Vocabulary

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

Renaming fractions
renaming fractions means rewriting them with a different denominator, but not changing the value of the fraction.
Least Common Multiple
the lowest multiple that two or more numbers have in common.
Lowest Common Denominator
the least common multiple becomes the lowest common denominator when adding or subtracting fractions with different denominators.
Equivalent Fractions
equal fractions. Created by multiplying the numerator and the denominator of a fraction by the same number.

## Technology Integration

Other Videos:

http://www.mathplayground.com/howto_fractions_diffden.html – This is a video on adding fractions with different denominators.

## Time to Practice

1. \begin{align*}\frac{3}{4} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}\frac{6}{7} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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

4. \begin{align*}\frac{2}{3} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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

6. \begin{align*}\frac{3}{6} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}\frac{6}{8} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}\frac{4}{7} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}\frac{4}{5} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}\frac{4}{5} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

Directions: Subtract the following fractions. Be sure that your answer is in simplest form.

11. \begin{align*}\frac{4}{8} - \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}\frac{9}{10} - \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}\frac{10}{10} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}\frac{15}{16} - \frac{2}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}\frac{9}{10} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

16. \begin{align*}\frac{3}{5} - \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

17. \begin{align*}\frac{9}{10} - \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

18. \begin{align*}\frac{20}{30} - \frac{1}{5} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

19. \begin{align*}\frac{18}{19} - \frac{2}{19} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

20. \begin{align*}\frac{4}{6} - \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}

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

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

22. \begin{align*}\frac{6}{9} + \frac{1}{3} - \frac{2}{3}\end{align*}

23. \begin{align*}\frac{4}{5} + \frac{1}{3} - \frac{1}{5}\end{align*}

24. \begin{align*}\frac{8}{9} + \frac{1}{2} - \frac{1}{3}\end{align*}

25. \begin{align*}\frac{3}{4} + \frac{1}{5} - \frac{2}{10}\end{align*}

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