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# Sums of Fractions with Different Denominators

## Adding equivalent fractions with LCD

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Sums of Fractions with Different Denominators

Have you ever thought of how many layers make up 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 $\frac{1}{4}$ of an inch thick. Then we add the insulation. For this wall we are going to use two different forms of insulation. One is $\frac{3}{4}$ of an inch thick and the other is $\frac{1}{2}$ of an inch thick. Next, we add a $\frac{1}{2}$ inch layer of wall sheathing. Finally we add the siding, that is $\frac{7}{8}$ 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 Concept has all of the information that you will need!

### Guidance

Previously we worked on 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.

$\frac{1}{2} + \frac{1}{4} = \underline{\;\;\;\;\;\;\;\;\;}$ 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.

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

$\frac{1}{2} = \frac{2}{4}$

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

Next, we can add the renamed fractions.

$\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

Our answer is $\frac{3}{4}$ . 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.

#### Example A

$\frac{1}{2} + \frac{2}{6} = \underline{\;\;\;\;\;\;\;\;\;}$

Solution: $\frac{5}{6}$

#### Example B

$\frac{2}{3} + \frac{1}{9} = \underline{\;\;\;\;\;\;\;\;\;}$

Solution: $\frac{7}{9}$

#### Example C

$\frac{4}{5} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}$

Solution: $\frac{17}{15}$ = $1 \frac{2}{15}$

Now let's go back to our original problem.

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.

$\frac{1}{4} + \frac{3}{4} + \frac{1}{2} + \frac{1}{2} + \frac{7}{8}$

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.

$\frac{1}{4} & = \frac{2}{8}\\\frac{3}{4} & = \frac{6}{8}\\\frac{1}{2} & = \frac{4}{8}$

Next, let’s rewrite the expression.

$\frac{2}{8} + \frac{6}{8} + \frac{4}{8} + \frac{4}{8} + \frac{7}{8}$

Now we can add the numerators.

$2 + 6 + 4 + 4 + 7 = 23$

$\frac{23}{8} = 2 \frac{7}{8}''$

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

### Vocabulary

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.

### Guided Practice

Here is one for you to try on your own.

$\frac{2}{7} + \frac{3}{9} = \underline{\;\;\;\;\;\;\;\;\;}$

First we need to find a common denominator. The common denominator for 7 and 9 is 63.

Next, we rename the fractions.

$\frac{2}{7} + \frac{3}{9}$ = $\frac{18}{63} + \frac{21}{63} = \frac{39}{63}$

### Practice

1. $\frac{3}{4} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}$

2. $\frac{6}{7} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}$

3. $\frac{2}{3} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}$

4. $\frac{2}{3} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}$

5. $\frac{1}{2} + \frac{1}{8} = \underline{\;\;\;\;\;\;\;\;\;}$

6. $\frac{3}{6} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}$

7. $\frac{6}{8} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}$

8. $\frac{4}{7} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}$

9. $\frac{4}{5} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}$

10. $\frac{3}{7} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}$

11. $\frac{5}{8} + \frac{2}{3} = \underline{\;\;\;\;\;\;\;\;\;}$

12. $\frac{6}{7} + \frac{1}{3} = \underline{\;\;\;\;\;\;\;\;\;}$

13. $\frac{9}{12} + \frac{1}{6} = \underline{\;\;\;\;\;\;\;\;\;}$

14. $\frac{8}{10} + \frac{1}{2} = \underline{\;\;\;\;\;\;\;\;\;}$

15. $\frac{6}{7} + \frac{3}{4} = \underline{\;\;\;\;\;\;\;\;\;}$

16. $\frac{3}{4} + \frac{2}{6} = \underline{\;\;\;\;\;\;\;\;\;}$

### Vocabulary Language: English

Equivalent Fractions

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
Least Common Multiple

Least Common Multiple

The least common multiple of two numbers is the smallest number that is a multiple of both of the original numbers.
Lowest Common Denominator

Lowest Common Denominator

The lowest common denominator of multiple fractions is the least common multiple of all of the related denominators.
Renaming fractions

Renaming fractions

Renaming fractions means rewriting fractions with different denominators, but not changing the value of the fraction.