2.3 Operations with Fractions

Learning Objectives

You will learn to add, subtract, multiply and divide fractions and mixed numbers.
You will be able to model and solve real world problems using simple equations involving sums, differences, products and quotients of fractions and mixed numbers

Introduction

The Blueberries

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 $5 \frac{1}{2}$ cups of blueberries for the pie. Plus she will need an additional $\frac{1}{4}$ of a cup for decorating the top of the pie. Ren know that he will need $1 \frac{1}{3}$ 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. You can figure this out too. In this lesson you will learn all that you need to know to help Ren figure out the blueberry dilemma.

Guided Learning

Add Fractions and Mixed Numbers

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 the 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 6 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. Here is an example.

Example A

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

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, and 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.

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

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

Find a common denominator
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.

Your turn to practice:

$9 \frac{1}{2} + 22 \frac{1}{4}$
$2 \frac{1}{3} + 8 \frac{2}{3}$
$5 \frac{1}{3} + \frac{1}{7}$

Check your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Sums of Fractions or Mixed Numbers

Fractions describe parts of a whole. Because of their accuracy, they are useful in many real-world situations, but especially situations involving measurement. When you confront a situation that requires your math skills, it is helpful to think the problem out before you jump in and try to find an answer. Ask yourself: "What is the problem? What are we trying to find out? What information do we have that will help us solve this problem? What mathematical tools can you use to get the answer?" Let’s look at some real-world situations involving addition of fractions and learn more about problem solving.

Example B

Donte is making a costume with blue, red and black fabric. He has $6 \frac{1}{2}$ yards of blue fabric, $3 \frac{2}{3}$ yards of red fabric and $5 \frac{4}{5}$ 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:

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

$6 \frac{3}{6} + 3 \frac{4}{6} = 9 \frac{7}{6}$

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

$9 \frac{7}{6}=10 \frac{1}{6}$

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

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

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

Subtract Fractions and Mixed Numbers

In real life, we use fractions all the time. Let’s say you are cutting a piece of wood that is $3 \frac{3}{4}$ feet long and you need to cut $\frac{1}{2}$ foot off of the piece of wood. What do you need to do to figure out how much wood you have left, after you make the cut? You guessed it. Subtraction is the key. Subtracting fractions and mixed numbers is a skill that you will use all the time.

Subtract fractions and mixed numbers

If you know how to add fractions, then you already know how to subtract them. The key is to make sure that the fractions that you are subtracting have the same denominator. If the fractions have the same denominator, then subtract the numerators just like you subtract whole numbers and keep the denominator the same in your answer.

Example C

$\frac{6}{9}-\frac{2}{9}$

Notice that the denominators are the same, so we can simply subtract the numerators.

6 - 2 = 4

Our answer is $\frac{4}{9}$ .

If the denominators are not the same, make sure to find the lowest or least common denominator first and then do your subtracting. Think about the first example with sawing wood. If you want to subtract $\frac{1}{2}$ foot from a piece of wood that is $3 \frac{3}{4}$ feet long, you have to find a common denominator first.

We can choose 4 as the least common denominator and rename each fraction in terms of fourths. To do this, we create equivalent fractions. If you use the equivalent fraction $\frac{2}{4}$ for $\frac{1}{2}$ , then you have the same denominator as the fraction in $3 \frac{3}{4}$ .

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

We can also subtract two mixed numbers. We do this in the same way. We subtract the fractions and then subtract the whole numbers.

Example D

$4 \frac{5}{6}-1 \frac{4}{6}$

First, we subtract the fraction parts. These fractions have the same denominator, so we can simply subtract the numerators.

5 - 4 = 1 the fraction here is $\frac{1}{6}$

Next, we subtract the whole numbers.

4 - 1

Our final answer is $3 \frac{1}{6}$ .

Sometimes, when you subtract mixed numbers, you will have to do an extra step. Think about this example.

Imagine you are cutting, or subtracting, $1 \frac{3}{4}$ feet of wood from a piece of wood that is $3 \frac{1}{2}$ feet long. Your subtraction problem looks like this: $3 \frac{1}{2} - 1 \frac{3}{4}$ .

After you find a common denominator, your subtraction problem now looks like this.

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

Take a deep breath and don’t panic! This is where you use your expertise at converting mixed numbers to improper fractions. After you have a common denominator for the fractions, multiply the whole number of the mixed number by the denominator of the fraction. Add this product to the numerator of the fraction.

$3 \frac{2}{4} &= \frac{(3 \times 4)+2}{4}=\frac{14}{4}\\1 \frac{3}{4} &= \frac{(4 \times 1)+3}{4}=\frac{7}{4}$

Your new subtraction problem for the example looks like this.

$\frac{14}{4} -\frac{7}{4}$

Now you simply subtract the numerators and you get $\frac{7}{4}$ . Now you convert this back into a mixed number. Do you remember how to do this?

Don’t forget to rewrite the difference as a mixed number and keep the fraction in lowest terms.

Here are the steps for subtracting mixed numbers.

Subtracting Mixed Numbers:

Make sure the fractions have a common denominator.
If the fraction to the left of the minus sign is smaller than the fraction to the right of the minus sign – convert both mixed numbers into improper fractions
Subtract the improper fractions
Rewrite the difference as a mixed number.

Subtract the following fractions and mixed numbers. Be sure that your answer is in lowest terms.

$\frac{10}{12}-\frac{6}{12}$
$\frac{6}{7}-\frac{3}{4}$
$4 \frac{1}{4}-\frac{3}{4}$

Check your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Sums and Differences of Fractions or Mixed Numbers

Have you begun to realize how useful fractions can be in everyday life? Jorge drinks $\frac{2}{3}$ of a glass of lemonade. Ursula and Andy will meet in $\frac{1}{4}$ of an hour.

In this section, we’ll look at some real-world problems which involve adding and subtracting fractions. When solving real-world problems, it’s important to first define terms. What information does the problem give us? What information does the problem ask us to find out? Once we know where we are and where we want to go, we can figure out how to get there.

Example E

Benito works in a bakery and has baked the world’s longest loaf of cinnamon bread. His loaf measures $11 \frac{5}{8}$ feet. He cuts a piece that measures $1 \frac{1}{2}$ feet long, and gives it to his friend Pamela. He then cuts another piece $2 \frac{2}{3}$ feet long for his friend Serena. How much bread does he have left?

Let’s take careful inventory of the information that the problem gives us. We know that the whole loaf of bread is $11 \frac{5}{8}$ feet long. Pamela gets a piece $1 \frac{1}{2}$ feet long, that is her piece and Serena gets a piece $2 \frac{2}{3}$ feet long. This is the given information.

What do we want to find out? We want to know the length of the bread after he cuts Pamela and Serena’s pieces (loaf after cutting = x ). Let’s write an equation to show the relationship between the values:

Whole loaf – Pamela’s piece – Serena’s piece = loaf after cutting

When we substitute the given values, we have the following equation.

$11 \frac{5}{8} - 1 \frac{1}{2}-2 \frac{2}{3} = x$

Now, we simply solve from left to right. First, find a common denominator between the fractions in $11 \frac{5}{8}$ and $1 \frac{1}{2}$ . Let’s use 8, so we solve $11 \frac{5}{8}-1 \frac{4}{8} = 10 \frac{1}{8}$ .

Next, we can simplify the problem.

$10 \frac{1}{8} - 2 \frac{2}{3} = x$

The lowest common denominator for the fractions is going to be 24. We simplify the problem further.

$10 \frac{3}{24} - 2 \frac{16}{24} = x$

I can already see that I will have to convert the mixed numbers to improper fractions. Simplify again.

$\frac{243}{24}-\frac{64}{24} &= x\\x &= \frac{179}{24}$

Next, we just convert the answer to a mixed number and write in simplest terms.

Solution: $7 \frac{11}{24}$ feet or about $7 \frac{1}{2}$ feet

Multiply Fractions and Mixed Numbers

You have already learned how to add and subtract fractions, but when you have a fraction and you want to figure out a part of that fraction, you need to multiply. Remember, that a fraction is a part of a whole. Sometimes it is tricky to figure out when to multiply fractions when you are faced with a real-world problem. First, let’s learn how to actually multiply fractions and then we can look at applying this to some real-world problems.

Multiplying fractions is always at least a two-step process.

First, you line up two fractions next two each other, and then you are ready to start multiplying.

$\frac{1}{2} \cdot \frac{4}{5}$

Notice that we used a dot to show that we were multiplying.

You will multiply twice. First, multiply the numerators and write the product of the numerators above a fraction bar. Next, multiply the denominators and write that product underneath the fraction bar. You don’t have to find a common denominator. You do, however, have to reduce your answer to simplest terms. We usually think of multiplying as increasing, but don’t be surprised when you get a product that is smaller than one of the factors that you are multiplying.

Let’s try this out.

Example F

$\frac{1}{2} \cdot \frac{4}{5}=\frac{1 \times 4}{2 \times 5}=\frac{4}{10}$

Now we have a fraction called $\frac{4}{10}$ . Is this in simplest form?

That’s right, it isn’t. We can simplify the fraction four-tenths, by dividing the top and the bottom number by the greatest common factor. The greatest common factor of four and ten is two. We divide the numerator and the denominator by two.

$\frac{4}{10}=\frac{4 \div 2}{10 \div 2}=\frac{2}{5}$

Our final answer is $\frac{2}{5}$ .

Multiplying a fraction and a whole number

When you multiply a fraction and a whole number, we have to make the whole number into a fraction. Then multiply across just as you would with two fractions and finally, simplify your answer if possible.

Example

$5 \cdot \frac{1}{2}= \frac{5}{1} \cdot \frac{1}{2}=\frac{5}{2}=2 \frac{1}{2}$

How do we multiply mixed numbers?

Because mixed numbers involve wholes and parts, multiplying mixed numbers requires an extra step. Remember improper fractions? It’s essential that you convert mixed numbers to improper fractions before you multiply. Once you have the mixed numbers in the improper fraction form, multiply the numerators and then multiply the denominators. If you have an improper fraction as your product, convert it back to a mixed number as your final answer.

Let’s look at an example.

$3 \frac{1}{2} \cdot 2 \frac{1}{3}$

First, we convert each to an improper fraction.

$3 \frac{1}{2} &= \frac{7}{2}\\ 2 \frac{1}{3} &= \frac{7}{3}$

Next, we multiply the two improper fractions.

$\frac{7}{2} \cdot \frac{7}{3}=\frac{49}{6}$

Now we can convert this improper fraction to a mixed number.

$\frac{49}{6}=8 \frac{1}{6}$

Our final answer is $8 \frac{1}{6}$ .

Sometimes, when you multiply fractions or mixed numbers, you can end up with very large numbers. When this happens, you can simplify BEFORE multiplying. You simplify on the diagonals by using the greatest common factor of the numbers on the diagonals.

Let’s look at an example.

Example G

$\frac{2}{9} \cdot \frac{18}{30}$

If we look at the numbers on the diagonals, we can see that there are common factors both ways. The greatest common factor of two and thirty is 2. We can divide both by two to simplify them. The greatest common factor of 9 and 18 is 9. We can divide both by 9. Let’s simplify on the diagonals now.

$\xcancel{\frac{2}{9} \cdot \frac{18}{30}} = \frac{1}{1} \cdot \frac{2}{15}$

Now we multiply across for our final answer.

The answer is $\frac{2}{15}$

Multiply. Be sure that your answer is in simplest form.

$\frac{1}{3} \cdot \frac{5}{6}$
$\frac{18}{20} \cdot \frac{4}{9}$
$2 \frac{1}{5} \cdot 3 \frac{1}{2}$

Check your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Products of Fractions or Mixed Numbers

“Let me have about a fourth of that.”

This is an example that would involve multiplying fractions. One of the key words that you will see when working with multiplication and real-world examples is the word “of”. Of is a key word that means multiplication. If you want $\frac{1}{4}$ pound of turkey at the deli, you will ask the butcher to cut $\frac{1}{4}$ times 1 pound $\left[\frac{1}{4} \cdot 1\right]$ .

Let’s look at some other real-world situations involving products of fractions and mixed numbers.

Example H

Dierdre claims that it takes her only $6 \frac{3}{4}$ hours to complete her homework every night. Carlos thinks he can finish his homework in $\frac{2}{3}$ that time. How long does Carlos think it will take him to complete his homework?

We want to know the length of time Carlos thinks he needs to complete his homework.

What’s the relationship of this length of time to the length of time Dierdre requires to finish her homework? If we let D = the amount of time it takes for Dierdre to complete her homework, then we would say that the length of time it takes Carlos to finish his homework is $\frac{2}{3} \cdot D$ . That’s a simple multiplication problem. We solve $6 \frac{3}{4} \cdot \frac{2}{3}$ .

We convert all mixed numbers to improper fractions, $6 \frac{3}{4} = \frac{27}{4}$ which leads to $\frac{27}{4} \cdot \frac{2}{3} = 4 \frac{1}{2}$ .

Carlos thinks that he can complete his homework in $4 \frac{1}{2}$ hours.

Divide Fractions and Mixed Numbers

By now you have a pretty solid understanding of fractions. You can add, subtract and multiply fractions. Of course, it will also be extremely helpful to learn to divide fractions. There are plenty of real-world situations in which you can use your expertise at dividing fractions.

Dividing fractions is a lot like multiplying fractions. In fact, it’s exactly like multiplying fractions! Remember that when you divide two numbers, one number is the dividend and the other number is the divisor. For example, in the division problem $a \div b, a$ is the dividend and is the divisor. To divide two fractions, you simply multiply the dividend by the inverted divisor. How do you invert the divisor? Just flip the fraction over! $\frac{1}{2}$ inverted becomes $\frac{2}{1}, \frac{3}{4}$ inverted becomes $\frac{4}{3}$ . This inverted fraction is also called a reciprocal.

Let’s divide.

Example I

$\frac{6}{8} \div \frac{1}{2}$

In this example, one-half is the divisor. We need to flip the divisor so that we multiply by the reciprocal. Here is a rhyme to help you remember

When dividing fractions, never wonder why

You just flip the second one and then you multiply

Now we rewrite the problem as a multiplication problem.

$\frac{6}{8} \cdot \frac{2}{1}$

Next, we multiply across.

$\frac{12}{8}=1 \frac{4}{8}$

Our last step is to simplify the fraction part of the mixed number.

Our answer is $1 \frac{1}{2}$ .

How do we divide mixed numbers?

Just like with multiplying fractions, if you are dividing mixed numbers, you have to first convert the mixed number to an improper fraction. Remember how multiplying by fractions usually gave us a product that was smaller than one of the factors? Well, with dividing you usually get an answer that is larger than the divisor or the dividend. Let’s look at some examples.

$4 \frac{1}{3} \div 2 \frac{1}{6}$

First, we convert both of these mixed numbers to improper fractions. Let’s rewrite the problem with these numbers.

$\frac{13}{3} \div \frac{13}{6}$

Now we change this to a multiplication problem by multiplying by the reciprocal.

$\frac{13}{3} \cdot \frac{6}{13}$

Next, we can simplify on the diagonals.

$\xcancel{\frac{13}{3} \cdot \frac{6}{13}} = \frac{1}{1} \cdot \frac{2}{1} = 2$

Our answer is 2.

Lesson Exercises

$\frac{5}{10} \div \frac{1}{2}$
$\frac{6}{8} \div \frac{1}{4}$
$6 \frac{1}{4} \div 1 \frac{1}{2}$

Check your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Products and Quotients of Fractions and Mixed Numbers

You know that fractions are all around us in the real world. The beauty of fractions is that they are often more precise than decimals. It is easier to write $\frac{1}{3}$ than .333333333.... Also, fractions are easy to visualize. We can imagine exactly what $\frac{1}{3}$ cup looks like, we just divide 1 cup into 3 equal parts.

You will use your expertise of fractions just as often as you use addition, subtraction and multiplication in everyday life. Let’s look at an example.

Example J

Yeoryia is building a sailboat. She needs 8 planks that are $\frac{3}{4}$ foot in length. She has a piece of wood that is $6 \frac{1}{2}$ feet long. How many planks can she cut from this board and does she have enough to make her sailboat?

Let’s assess the information that the problem has given us. We know that we need planks that are $\frac{3}{4}$ foot in length and we know that we need 8 of them. But, we are not really sure, if we have 8 planks because we are working with a piece of wood that is $6 \frac{1}{2}$ feet long. How many $\frac{3}{4}$ foot planks can be obtained from a $6 \frac{1}{2}$ foot long board? That’s a simple division problem. We set the problem up like this.

Total length of wood $\div$ length of plank needed = number of planks

Let’s plug in the values: $6 \frac{1}{2} \div \frac{3}{4} =$ number of planks.

We convert $6 \frac{1}{2}$ to an improper fraction and we set up the division problem as a multiplication problem with inverted divisor.

$\frac{13}{2} \cdot \frac{4}{3}$

We can cancel out the factors of 2 to get a new multiplication problem, which looks like this.

$\frac{13}{1} \cdot \frac{2}{3}$

We get $\frac{26}{3}$ or $8 \frac{2}{3}$ .

Yeoryia can cut $8 \frac{2}{3}$ planks from the piece of wood that she has. She has enough to make her sailboat.

Remember the dilemma with the blueberries? You now have all the skills that you need to help solve that problem. Let’s go back to the introductory problem and work it through.

Real Life Example Completed

The Blueberries

Reread this problem once again. Then underline any important information before solving it.

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 $5 \frac{1}{2}$ cups of blueberries for the pie. Plus she will need an additional $\frac{1}{4}$ of a cup for decorating the top of the pie. Ren know that he will need $1 \frac{1}{3}$ 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.

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.

$5 \frac{1}{2}+\frac{1}{4}+1 \frac{1}{3}$

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

$5 \frac{6}{12}+\frac{3}{12}+1 \frac{4}{12}$

If we add the fraction parts, we get a sum of $\frac{13}{12}$ . This improper fraction changes to $1 \frac{1}{12}$ .

Next we add this to our whole numbers.

The sum of the blueberries is $7 \frac{1}{12}$ .

Now we have to figure out how many pints Ren will need to purchase. There are two cups in a pint. To have enough 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!!

Review

.

Fraction

A fraction is a part of a whole..

Reciprocal

The reciprocal is also called the multiplicative inverse..

Mixed Number

A mixed number is a number consisting of an integer and a proper fraction.

Greatest Common Factor

The highest number that divides into exactly two or more numbers.

Lowest Common Denominator

The lowest common denominator of two or more fractions is the least common multiple of their

denominators.

Equivalent Fractions

Equivalent fractions are fractions that describe the same part of a whole.

Improper Fractions

An improper fraction is a fraction with a numerator that is greater than the denominator.

James Sousa, Subtraction of Mixed Numbers

Khan Academy Multiplying Fractions

James Sousa, Multiplying Fractions

James Sousa, Example of Multiplying Fractions

Khan Academy Dividing Fractions

James Sousa Dividing Fractions

James Sousa Example of Dividing Fractions

James Sousa Another Example of Dividing Fractions

http://www.mathplayground.com/howto_divide_fractions.html

http://www.mathplayground.com/howto_fractionofanumber.html

http://www.teachertube.com/viewVideo.php?video_id=141252

http://www.mathplayground.com/howto_fractions_diffden.html

Learning Objectives

Introduction

The Blueberries

Teri tells Ren that she needs cups of blueberries for the pie. Plus she will need an additional of a cup for decorating the top of the pie. Ren know that he will need 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. You can figure this out too. In this lesson you will learn all that you need to know to help Ren figure out the blueberry dilemma.

Guided Learning

Add Fractions and Mixed Numbers

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 cup of flour to cup of flour. We know that the new mixture of flour is more than cup of flour and more than cup of flour. We also know that the new mixture is less than 1 cup of flour and greater than 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 (equivalent fraction of ) to (equivalent fraction of ), we simply add the numerators and keep the denominator the same. If we add cup of flour to cup of flour, we get 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. Here is an example.

Example A

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, and create equivalent fractions with denominators of 12.

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

We can change into the mixed number .

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

The answer is .

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

Find a common denominator

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.

Your turn to practice:

Model and Solve Real-World Problems Using Simple Equations Involving Sums of Fractions or Mixed Numbers

Example B

Donte is making a costume with blue, red and black fabric. He has yards of blue fabric, yards of red fabric and 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:

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:

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

Now we can this new mixed number, to the length of the black fabric, yards.

We use the common denominator of 30 for the fractions and we find the sum of these mixed numbers.

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

Subtract fractions and mixed numbers

Example C

Example D

Check your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Sums and Differences of Fractions or Mixed Numbers

Example E

Multiply Fractions and Mixed Numbers

Multiplying fractions is always at least a two-step process.

Example F

Multiplying a fraction and a whole number

Example G

Model and Solve Real-World Problems Using Simple Equations Involving Products of Fractions or Mixed Numbers

Example H

Divide Fractions and Mixed Numbers

Example I

Model and Solve Real-World Problems Using Simple Equations Involving Products and Quotients of Fractions and Mixed Numbers

Example J

Real Life Example Completed

Review

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Video Resources

http://www.mathplayground.com/howto_fractions_diffden.html–

James Sousa, Example of Adding Fractions with Different Denominators

James Sousa, Another Example of Adding Fractions with Different Denominators

James Sousa, Adding Mixed Numbers

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

Imagine adding $\frac{1}{2}$ cup of flour to $\frac{1}{3}$ cup of flour. We know that the 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 can change $\frac{13}{12}$ into the mixed number $1 \frac{1}{12}$ .

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

Donte is making a costume with blue, red and black fabric. He has $6 \frac{1}{2}$ yards of blue fabric, $3 \frac{2}{3}$ yards of red fabric and $5 \frac{4}{5}$ yards of black fabric, how many yards of fabric does Donte have altogether?

$6 \frac{1}{2} + 3 \frac{2}{3} + 5 \frac{4}{5} =$ total length of fabric.

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

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