<meta http-equiv="refresh" content="1; url=/nojavascript/"> Addition of Fractions ( Read ) | Arithmetic | CK-12 Foundation

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Addition of Fractions with unlike denominators

### Introduction

Do you like to eat pies? How about muffins? Take a look at this dilemma.

Teri and Ren are both in the fifth grade. They are baking pies and muffins for the bake sale. Teri has decided to make two mini strawberry pies and Ren has decided to make strawberry muffins. While Teri works on making her pie crusts, Ren offers to go to the grocery store with his Mom to get the strawberries that they will need.

Teri tells Ren that she needs $\frac{3}{4}$ cup of strawberries for the each pie. Ren know that he will need $\frac{2}{3}$ cup of strawberries for his muffins. In addition, Ren needs an additional  $\frac{1}{4}$  cup to top the muffins.

If strawberries come in pints and there are two cups in one pint, how many pints of strawberries will Ren need to buy at the store?

Ren takes out a piece of paper and a pencil.

This is where you come in. You will learn all about adding fractions and mixed numbers in this Concept.

### Guided Learning

Adding fractions and mixed numbers is as easy as adding whole numbers. The only trick is to make sure that the fractions we are adding have the same denominator.

Imagine adding $\frac{1}{2}$ cup of flour to $\frac{1}{3}$ cup of flour. We know that my new mixture of flour is more than $\frac{1}{2}$ cup of flour and more than $\frac{1}{3}$ cup of flour. We also know that the new mixture is less than 1 cup of flour and greater than $\frac{2}{3}$ cup of flour.

We have to divide the whole into a new number of parts, that is, find a common denominator in order to get a fraction which accurately describes the new amount of flour.

When we use the common denominator of sixths and add the fractions $\frac{3}{6}$ (equivalent fraction of $\frac{1}{2}$ ) to $\frac{2}{6}$ (equivalent fraction of $\frac{1}{3}$ ), we simply add the numerators and keep the denominator the same. If we add $\frac{1}{2}$ cup of flour to $\frac{1}{3}$ cup of flour, we get $\frac{5}{6}$ of a cup of flour.

How do we do this when we add mixed numbers and fractions?

Mixed numbers and fractions can be a little tricky because you are dealing parts and wholes. You can find the sum of them though by keeping in mind that you add parts with parts and wholes with wholes.

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

Here we are going to add a fraction and a mixed number together. You can see that the fractions have different denominators. This is the first thing that we need to change. Both fractions must have the same denominator before we can add them.

To do this, we find the common denominator of 3 and 4. That number is 12. Now we rename the fractions in terms of twelfths. This is where we will create equivalent fractions with denominators of 12.

$\frac{3}{4} &= \frac{9}{12}\\ \frac{1}{3} &= \frac{4}{12}$

Now we can add them. When the denominators are the same, we have to add the numerators only.

$\frac{9}{12}+\frac{4}{12}=\frac{13}{12}$

We can change $\frac{13}{12}$ into the mixed number $1 \frac{1}{12}$ .

Now we had a 2 from the original mixed number. We add this to our sum so far.

The answer is $3 \frac{1}{12}$ .

Here our answer is in simplest form so we leave it alone. Always simplify your answer if possible.

Here are the steps.

3. Add the sum of the fractions to the whole numbers.

Write these steps down in your notebook.

Now it's time for you to practice adding fractions and mixed numbers on your own. Find the sums of these fractions.

#### Example A

$\frac{3}{5}+\frac{2}{3}$

Example B

$\frac{4}{6}+\frac{2}{3}$

#### Example C

$1\frac{3}{4}+\frac{3}{5}$

Now back to the strawberries. Remember that Ren had just taken out a pencil and paper. Here is how Ren began thinking his way through the problem.

First, we will need to find the sum of the strawberries. We will figure out how many cups of strawberries both Teri and Ren will need for their recipes.

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

Next, we need to rename each fraction with a common denominator. We can use 12 as our common denominator.

$\frac{9}{12}+\frac{9}{12}+\frac{8}{12}+\frac{3}{12}$

If we add the fraction parts, we get a sum of $\frac{29}{12}$ . This improper fraction changes to $2\frac{5}{12}$ .

The sum of the strawberries is $2\frac{5}{12}$ .

Now we have to figure out how many pints Ren will need to purchase. There are two cups in a pint. To have enough strawberries, Ren will need to purchase 2 pints of strawberries. There will be some left over, but having some left over is better than not having enough!!

Here is another example.

Donte is making a costume with blue, red, and black fabric. He has $\frac{1}{2}$ yards of blue fabric, $1\frac{2}{3}$ yards of red fabric and $1\frac{4}{5}$ yards of black fabric, how many yards of fabric does Donte have altogether?

Let’s look at the problem carefully and define the values that we know and the value or values that we want to know.

We know that Donte has 3 types of fabric (blue, red, and black) and we know also the lengths of each type of fabric. We want to find out how much fabric Donte has altogether. If we represent this problem in an equation, it would look like this:

Length of blue fabric + length of red fabric + length of black fabric = total length of fabric

Since we know the lengths of the individual colors of fabric, we can rewrite the expression like this:

$\frac{1}{2}+1\frac{2}{3}+1\frac{4}{5}$   = total length of fabric.

If we add the mixed numbers together, we will learn what we want to find out. First, we will add the first two numbers. We use the common denominator of 6 for the fractions and we find the sum of the two mixed numbers:

$\frac{1}{2}+1\frac{2}{3}=1\frac{7}{6}$

Notice that seven-sixths is improper meaning that it is larger than one whole. We can convert this improper fraction to a mixed number.

$1\frac{7}{6}=2\frac{1}{6}$

Now we can this new mixed number, $2\frac{1}{6}$ to the length of the black fabric, $1\frac{4}{5}$ yards.

We use the common denominator of 30 for the fractions and we find the sum of these mixed numbers. $2\frac{1}{6}+1\frac{4}{5}=3\frac{29}{30}$

We can use the exact sum or we can say that Donte has just about 4 yards of fabric.

### Practice Set

Directions : Add the following fractions and mixed numbers.

1. $\frac{3}{7} + \frac{1}{14}$

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

3. $\frac{2}{5}+ \frac{3}{10}$

4. $\frac{1}{9} + \frac{1}{6}$

5. $2 \frac{3}{5} + \frac{17}{20}$

6. $2\frac{1}{6}+\frac{3}{8}$

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

8. $\frac{4}{5}+1\frac{9}{15}$

9. $2 \frac{1}{5} + 4 \frac{14}{15}$

10. $2\frac{4}{6}+1\frac{4}{9}$

11. $2 \frac{2}{3} + 4 \frac{1}{6}$

12. $3 \frac{4}{10} + 6 \frac{1}{9}$

13. $2 \frac{1}{30} + \frac{12}{15}$

14. $\frac{3}{7}+\frac{2}{3}$

15. $\frac{4}{9}+\frac{4}{12}$

### Review

•  Steps for Adding Mixed Numbers
3. Add the sum of the fractions to the whole numbers.