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## Add fractions and mixed numbers with like and unlike denominators.

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

Teri and Ren are both in the seventh grade. They are baking pies and muffins for the bake sale. Teri has decided to make a blueberry pie and Ren has decided to make blueberry muffins. While Teri works on making her pie crusts. Ren offers to go to the grocery store with his Mom to get the blueberries that they will need.

Teri tells Ren that she needs 512\begin{align*}5 \frac{1}{2}\end{align*} cups of blueberries for the pie. Plus she will need an additional 14\begin{align*}\frac{1}{4}\end{align*} of a cup for decorating the top of the pie. Ren know that he will need 113\begin{align*}1 \frac{1}{3}\end{align*} cups of blueberries for his pie.

If blueberries come in pints and there are two cups in one pint, how many pints of blueberries 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.

### Guidance

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 12\begin{align*}\frac{1}{2}\end{align*} cup of flour to 13\begin{align*}\frac{1}{3}\end{align*} cup of flour. We know that my new mixture of flour is more than 12\begin{align*}\frac{1}{2}\end{align*} cup of flour and more than 13\begin{align*}\frac{1}{3}\end{align*} cup of flour. We also know that the new mixture is less than 1 cup of flour and greater than 23\begin{align*}\frac{2}{3}\end{align*} 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 6 and add the fractions 36\begin{align*}\frac{3}{6}\end{align*} (equivalent fraction of 12\begin{align*}\frac{1}{2}\end{align*}) to 26\begin{align*}\frac{2}{6}\end{align*} (equivalent fraction of 13\begin{align*}\frac{1}{3}\end{align*}), we simply add the numerators and keep the denominator the same. If we add 12\begin{align*}\frac{1}{2}\end{align*} cup of flour to 13\begin{align*}\frac{1}{3}\end{align*} cup of flour, we get 56\begin{align*}\frac{5}{6}\end{align*} 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.

34+213\begin{align*}\frac{3}{4}+ 2 \frac{1}{3}\end{align*}

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.

3413=912=412

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

912+412=1312\begin{align*}\frac{9}{12}+\frac{4}{12}=\frac{13}{12}\end{align*}

We can change 1312\begin{align*}\frac{13}{12}\end{align*} into the mixed number 1112\begin{align*}1 \frac{1}{12}\end{align*}.

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

The answer is 3112\begin{align*}3 \frac{1}{12}\end{align*}.

Here our answer is in simplest form so we leave it alone. If you can simplify an answer you must do so or the answer is incorrect.

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.

#### Example A

912+2214\begin{align*}9 \frac{1}{2} + 22 \frac{1}{4}\end{align*}

Solution: 3134\begin{align*}31 \frac{3}{4}\end{align*}

#### Example B

213+823\begin{align*}2 \frac{1}{3} + 8 \frac{2}{3}\end{align*}

Solution:11\begin{align*}11\end{align*}

#### Example C

513+17\begin{align*}5 \frac{1}{3} + \frac{1}{7}\end{align*}

Solution:51021\begin{align*}5 \frac{10}{21}\end{align*}

Now back to the blueberries. 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 blueberries. We will figure out how many cups of blueberries both Teri and Ren will need for their recipes.

512+14+113\begin{align*}5 \frac{1}{2}+\frac{1}{4}+1 \frac{1}{3}\end{align*}

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

5612+312+1412\begin{align*}5 \frac{6}{12}+\frac{3}{12}+1 \frac{4}{12}\end{align*}

If we add the fraction parts, we get a sum of 1312\begin{align*}\frac{13}{12}\end{align*}. This improper fraction changes to 1112\begin{align*}1 \frac{1}{12}\end{align*}.

Next we add this to our whole numbers.

The sum of the blueberries is 7112\begin{align*}7 \frac{1}{12}\end{align*}.

Now we have to figure out how many pints Ren will need to purchase. There are two cups in a pint. To have enough blueberries, Ren will need to purchase 4 pints of blueberries. There will be some left over, but having some left over is better than not having enough!!

### Vocabulary

Fraction
a part of a whole
Mixed Number
a whole number and a fraction

### Guided Practice

Here is one for you to try on your own.

Donte is making a costume with blue, red and black fabric. He has 612\begin{align*}6 \frac{1}{2}\end{align*} yards of blue fabric, 323\begin{align*}3 \frac{2}{3}\end{align*} yards of red fabric and 545\begin{align*}5 \frac{4}{5}\end{align*} 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:

612+323+545=\begin{align*}6 \frac{1}{2} + 3 \frac{2}{3} + 5 \frac{4}{5} =\end{align*} 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 mixed numbers. We use the common denominator of 6 for the fractions and we find the sum of the two mixed numbers:

636+346=976\begin{align*}6 \frac{3}{6} + 3 \frac{4}{6} = 9 \frac{7}{6}\end{align*}

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

\begin{align*}9 \frac{7}{6}=10 \frac{1}{6}\end{align*}

Now we can this new mixed number, \begin{align*}10 \frac{1}{6}\end{align*} to the length of the black fabric, \begin{align*}5 \frac{4}{5}\end{align*} yards.

We use the common denominator of 30 for the fractions and we find the sum of these mixed numbers. \begin{align*}10 \frac{5}{30} + 5 \frac{24}{30} = 15 \frac{29}{30}\end{align*}

We can use the exact sum or we can say that Donte has just about 16 yards of fabric.

### Practice

Directions: Add the following fractions and mixed numbers.

1. \begin{align*}\frac{3}{7} + \frac{1}{14}\end{align*}

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

3. \begin{align*}\frac{2}{5}+ \frac{3}{10}\end{align*}

4. \begin{align*}\frac{1}{9} + \frac{1}{6}\end{align*}

5. \begin{align*}2 \frac{3}{5} + \frac{17}{20}\end{align*}

6. \begin{align*}8 \frac{5}{12} + 2 \frac{1}{6}\end{align*}

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

8. \begin{align*}2 \frac{1}{5} + 4 \frac{14}{15}\end{align*}

9. \begin{align*}2 \frac{1}{5} + 4 \frac{14}{15}\end{align*}

10. \begin{align*}12 \frac{1}{5} + 4 \frac{6}{15}\end{align*}

11. \begin{align*}2 \frac{2}{3} + 4 \frac{1}{6}\end{align*}

12. \begin{align*}3 \frac{4}{10} + 6 \frac{1}{9}\end{align*}

13. \begin{align*}2 \frac{1}{30} + \frac{12}{15}\end{align*}

14. \begin{align*}4 \frac{1}{2} + 14 \frac{4}{5}\end{align*}

15. \begin{align*}16 \frac{8}{9} + 14 \frac{9}{12}\end{align*}

### Vocabulary Language: English

Denominator

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
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.
improper fraction

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
inequality

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
LCD

LCD

The least common denominator or lowest common denominator (LCD) of two fractions is the smallest number that is a multiple of both of the original denominators.
LCM

LCM

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both of the original numbers.
Mixed Number

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.