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 \begin{align*}5 \frac{1}{2}\end{align*}
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 \begin{align*}\frac{1}{2}\end{align*}
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 \begin{align*}\frac{3}{6}\end{align*}
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.
\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.
\begin{align*}\frac{3}{4} &= \frac{9}{12}\\ \frac{1}{3} &= \frac{4}{12}\end{align*}
Now we can add them. When the denominators are the same, we have to add the numerators only.
\begin{align*}\frac{9}{12}+\frac{4}{12}=\frac{13}{12}\end{align*}
We can change \begin{align*}\frac{13}{12}\end{align*}
Now we had a 2 from the original mixed number. We add this to our sum so far.
The answer is \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.
Adding Mixed Numbers
- Add the fractions.
- Add the whole numbers
- Add the sum of the fractions to the whole numbers
- Be sure that your answer is in simplest form.
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
\begin{align*}9 \frac{1}{2} + 22 \frac{1}{4}\end{align*}
Solution: \begin{align*}31 \frac{3}{4}\end{align*}
Example B
\begin{align*}2 \frac{1}{3} + 8 \frac{2}{3}\end{align*}
Solution:\begin{align*}11\end{align*}
Example C
\begin{align*}5 \frac{1}{3} + \frac{1}{7}\end{align*}
Solution:\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.
\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.
\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 \begin{align*}\frac{13}{12}\end{align*}
Next we add this to our whole numbers.
The sum of the blueberries is \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 \begin{align*}6 \frac{1}{2}\end{align*}
Answer
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:
\begin{align*}6 \frac{1}{2} + 3 \frac{2}{3} + 5 \frac{4}{5} =\end{align*}
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:
\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.
Video Review
- This is a James Sousa video on adding fractions with like denominators.
- This is a James Sousa video on adding fractions with unlike denominators.
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*}