# 5.6: Mixed Numbers and Improper Fractions

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Pizza Party

At the sixth grade social, the teachers want to provide the students with pizza as a treat. Kids love pizza and the sixth graders aren’t any exception to this rule. The teachers want each student to receive two slices of pizza.

There are 48 students in cluster 6A, and 44 students in cluster 6B.

When Mr. Scott, the teacher in 6B, calls the pizza place. He learns that there are two different options for slicing the pizza. They can either slice it in \begin{align*}8^{ths}\end{align*} or in 10ths. Mr. Scott isn’t sure at first which way is the better way to go. But because he is a teacher, he quickly figures out the math in his head and orders the pizzas cut into \begin{align*}10^{ths}\end{align*}.

If Mr. Scott ordered the pizzas cut into \begin{align*}10^{ths}\end{align*}, how many pizzas did he order?

Would there be any pizza left over for the teachers to have a slice?

If Mr. Scott had ordered the pizzas cut into 8ths, how many pizzas would he have ordered?

Would there be any left over here?

Use what you learn in this lesson on fractions to help you figure out the pizza problem.

Pay attention, after all, there is pizza at stake!!

What You Will Learn

In this lesson you will learn to:

• Measure lengths to a fraction of an inch.
• Rewrite mixed numbers as improper fractions.
• Rewrite improper fractions as mixed numbers.
• Compare and order mixed numbers and improper fractions.

Teaching Time

I. Measure Lengths to a Fraction of an Inch

One of the places that we often see fractions in real life is when we measure different things. Think about using a ruler, sometimes you will have something that measures evenly, meaning that the item measures in whole inches. More often, you will have an item that does not measure evenly. When this happens, you will need to measure the item to a fraction of an inch.

Each whole inch has sixteen lines. This is because one inch is \begin{align*}\frac{16}{16}\end{align*} of an inch long. Count four lines, you are at \begin{align*}\frac{4}{16}\end{align*} or \begin{align*}\frac{1}{4}\end{align*} (a quarter) of an inch. Count to the eighth line, you are at \begin{align*}\frac{8}{16}\end{align*} or \begin{align*}\frac{1}{2}\end{align*} (one half) of an inch. Count to the twelfth line is \begin{align*}\frac{12}{16}\end{align*} or \begin{align*}\frac{3}{4}\end{align*} (three fourths) of an inch.

If you look at this ruler, you will see that the arrow is above a line that does not indicate one whole inch. We need to use our fractions to write this fraction of an inch. If we were to count the lines, we would see that the arrow is above the eighth line. Here is our measurement to a fraction of an inch.

Our answer is \begin{align*}\frac{8}{16}\end{align*} or \begin{align*}\frac{1}{2}\end{align*}.

We can also have an example where we have a whole number measurement and then a fraction of another inch. Let’s look at this example.

Example

To measure this measurement, we can see that we have 1 and a fraction of an inch. If you look you can see that the arrow is above \begin{align*}1 \frac{12}{16}\end{align*} or \begin{align*}1 \frac{3}{4}\end{align*}.

Our answer is \begin{align*}1 \frac{3}{4}\end{align*}.

Now it is time for you to try a few on your own. Simplify the fraction too.

1.

Take a minute to check your work with a peer. Did you simplify the fraction part of the measurement? Did you remember to include the whole number in your answer?

II. Rewrite Mixed Numbers as Improper Fractions

In the last section on measurement, sometimes we ended up with measurements that included whole inches and parts of a whole inch or a fraction of an inch. When we have wholes and parts together, we have a new type of number. It is called a mixed number.

A mixed number is a number that has both wholes and parts in it.

Example \begin{align*}5 \frac{1}{4}\end{align*}

Here we have a mixed number. We have five whole items and one-fourth of a whole.

Now you know how to identify a mixed number. The opposite of a mixed number is an improper fraction.

What is an improper fraction?

An improper fraction is a fraction that has a larger numerator and a smaller denominator.

Huh? What does this mean? Let’s look at an example to better understand what an improper fraction is.

Example

\begin{align*}\frac{12}{5}\end{align*}

Let’s think about what this means. If the denominator tells us how many parts the whole has been divided into, then this whole has been divided into 5 parts. The numerator tells us how many parts of the whole we have in this case, we have twelve parts.

What? If we have twelve out of five parts, then we have MORE than one whole. One whole would be five out of five parts, but we have 12 out of 5 parts. This is where mixed numbers come in.

How do we write a mixed number as an improper fraction?

To write a mixed number as an improper fraction, we want to write a fraction in terms of parts instead of in terms of wholes and parts.

Example

Change \begin{align*}2 \frac{1}{3}\end{align*} to an improper fraction.

To do this, we multiply the whole number times the denominator and add the numerator.

Then we put this over the original denominator.

2 \begin{align*}\times\end{align*} 3 + 1 \begin{align*}=\end{align*} 7

Our original denominator is 3.

Our answer is \begin{align*}2 \frac{1}{3} = \frac{7}{3}\end{align*}.

Notice that the mixed number and the improper fraction are also equivalent.

Change the following mixed numbers to improper fractions.

1. \begin{align*}3 \frac{1}{3}\end{align*}
2. \begin{align*}5 \frac{2}{3}\end{align*}
3. \begin{align*}6 \frac{1}{8}\end{align*}

Take a few minutes to check your work with a neighbor.

III. Rewrite Improper Fractions as Mixed Numbers

We just learned how to write a mixed number as an improper fraction. We can also work the other way around too, we can write improper fractions as mixed numbers.

How do you write an improper fraction as a mixed number?

First, remember that when you write an improper fraction as a mixed number, that you are converting a fraction in all parts to wholes and parts.

Example

\begin{align*}\frac{18}{4}\end{align*} If I have eighteen-fourths, I have eighteen parts and the whole has only been divided into 4 parts. This means that \begin{align*}\frac{4}{4}\end{align*} would be considered a whole.

When the numerator is larger than the denominator, you know that you have more than one whole.

To change an improper fraction to a mixed number, divide the denominator into the numerator. This will tell you the number of wholes.

If there are any left over, this tells you the fraction part.

18 \begin{align*}\div\end{align*} 4 \begin{align*}=\end{align*} 4

But there are 2 left over because 4 \begin{align*}\times\end{align*} 4 = 16 and our numerator is 18. The left over part becomes the numerator over the original denominator.

Our answer is \begin{align*}4 \frac{2}{4}\end{align*} .

Our answer is \begin{align*}4 \frac{1}{2}\end{align*}.

Sometimes, you will have an improper fraction that converts to a whole number and not a mixed number.

Example

\begin{align*}\frac{18}{9}\end{align*} Here eighteen divided by 9 is 2. There isn’t a remainder, so there isn’t a fraction. This improper fraction converts to a whole number.

Now it is time for you to try a few on your own. Be sure your fractions are in simplest form.

1. \begin{align*}\frac{24}{5}\end{align*}
2. \begin{align*}\frac{21}{3}\end{align*}
3. \begin{align*}\frac{32}{6}\end{align*}

Take a few minutes to check your answers. Did you simplify the fraction part of number 3?

IV. Compare and Order Mixed Numbers and Improper Fractions

Now that you have learned how to write mixed numbers and improper fractions and how to convert them back and forth, we can look at how to compare them.

How do we compare a mixed number and an improper fraction?

We compare them by first making sure that they are in the same form. They both need to be mixed numbers otherwise it is difficult to determine which one is greater and which one is less than.

Example \begin{align*}6 \frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{15}{4}\end{align*}

The easiest thing to do here is to convert fifteen-fourths into a mixed number.

\begin{align*}\frac{15}{4} = 3 \frac{3}{4}\end{align*}

Now we know that six and one-half is greater than fifteen-fourths.

Our answer is \begin{align*}6 \frac{1}{2} > \frac{15}{4}\end{align*}.

Try a few of these on your own. Compare the following mixed numbers and improper fractions.

1. \begin{align*}4 \frac{1}{2} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{12}{5}\end{align*}
2. \begin{align*}\frac{16}{3} \ {\underline{\;\;\;\;\;\;\;}} \ \frac{22}{5}\end{align*}
3. \begin{align*}\frac{17}{4} \ {\underline{\;\;\;\;\;\;\;\;}} \ 4 \frac{1}{4}\end{align*}

Take a minute to check your work.

How do we write mixed numbers and improper fractions in order from least to greatest or from greatest to least?

We can work on this task in the same way as with the comparing. First, make sure that all of the terms you are working with are mixed numbers.

Example

Write in order from least to greatest, \begin{align*}\frac{33}{2}, 4 \frac{2}{3}, \frac{88}{11}\end{align*}.

We need to convert thirty-three halves and eighty-eight elevenths to mixed numbers.

\begin{align*}\frac{33}{2} &= 16 \frac{1}{2}\\ \frac{88}{11} &= 8\end{align*}

Our answer is \begin{align*}4 \frac{2}{3}, \frac{88}{11}, \frac{33}{2}\end{align*}.

## Real Life Example Completed

The Pizza Party

Now that you have learned all about mixed numbers and improper fractions, you are ready to answer the questions regarding the pizza party.

Here is the problem once again.

At the sixth grade social, the teachers want to provide the students with pizza as a treat. Kids love pizza and the sixth graders aren’t any exception to this rule. The teachers want each student to receive two slices of pizza.

There are 48 students in cluster 6A, and 44 students in cluster 6B.

When Mr. Scott, the teacher in 6B, calls the pizza place, he learns that there are two different options for slicing the pizza. They can either slice it in \begin{align*}8^{ths}\end{align*} or in 10ths. Mr. Scott isn’t sure at first which way is the better way to go. But because he is a teacher, he quickly figurs out the math in his head and orders the pizzas cut into \begin{align*}10^{ths}\end{align*}.

If Mr. Scott ordered the pizzas cut into \begin{align*}10^{ths}\end{align*}, how many pizzas did he order?

Would there be any pizza left over for the teachers to have a slice?

If Mr. Scott had ordered the pizzas cut into 8ths, how many pizzas would he have ordered?

Would there be any left over here?

First, let’s underline the important information.

There are some facts here to help us with answering the questions.

If each student in each cluster is going to receive two slices of pizza, the first thing we need to do is to multiply the number of students by 2.

48 + 44 \begin{align*}=\end{align*} 92 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 184 slices

If Mr. Scott ordered the pizzas cut into 10, then he would need \begin{align*}\frac{184}{10}\end{align*}. Here is a real life application of improper fractions. We need 184 slices. The pizzas are cut into \begin{align*}10^{ths}\end{align*}, so that means that we need \begin{align*}\frac{184}{10}\end{align*} of pizza.

How many pizzas is that? To figure this out, we turn \begin{align*}\frac{184}{10}\end{align*} into a mixed number.

\begin{align*}184 \div 10 = 18 \frac{4}{10}\end{align*}

He would need 18 pizzas and four slices from another pizza. That is the four-tenths. There would be six slices left over, so there would be enough pizza for the teachers too. Mr. Scott ordered nineteen pizzas.

What if Mr. Scott ordered the pizzas cut into eighths? He would need \begin{align*}\frac{184}{8}\end{align*}. The number of slices did not change, but the way the pizza was sliced did change. How many pizza’s would he need to order if the pizzas were cut into eighths? We need to rewrite the improper fraction to a mixed number.

184 \begin{align*}\div\end{align*} 8 \begin{align*}=\end{align*} 23 pizzas

There wouldn't be any slices left over, so there wouldn’t be any extras for the teachers.

Since Mr. Scott enjoys a good slice of pizza too, he ordered the 19 pizzas that were divided into tenths.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Mixed Number
a number made up of a whole number and a fraction
Improper Fractions
a fraction where the numerator is greater than the denominator
Equivalent
means equal

## Technology Integration

Other Videos:

1. http://www.teachertube.com/members/viewVideo.php?video_id=19595&title=Improper_Fractions_to_Mixed_Numbers_by_Mr_Lee___Chapter_7 – This is a colorful presentation that clearly explains how to convert an improper fraction to a mixed number. You'll have to register at the site to view this video.
2. http://www.teachertube.com/members/viewVideo.php?video_id=106786 – A video on converting mixed numbers and improper fractions using fraction tiles. This is all done visually on the screen. You'll have to register at the site to view this video.

## Time to Practice

Directions: Write each mixed number as an improper fraction.

1. \begin{align*}2 \frac{1}{2}\end{align*}

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

3. \begin{align*}5 \frac{1}{3}\end{align*}

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

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

6. \begin{align*}6 \frac{2}{5}\end{align*}

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

8. \begin{align*}8 \frac{2}{5}\end{align*}

9. \begin{align*}7 \frac{4}{5}\end{align*}

10. \begin{align*}8 \frac{2}{7}\end{align*}

11. \begin{align*}8 \frac{3}{4}\end{align*}

12. \begin{align*}9 \frac{5}{6}\end{align*}

Directions: Convert each improper fraction to a mixed number. Be sure to simplify when necessary.

13. \begin{align*}\frac{22}{3}\end{align*}

14. \begin{align*}\frac{44}{5}\end{align*}

15. \begin{align*}\frac{14}{3}\end{align*}

16. \begin{align*}\frac{7}{2}\end{align*}

17. \begin{align*}\frac{10}{3}\end{align*}

18. \begin{align*}\frac{47}{9}\end{align*}

19. \begin{align*}\frac{50}{7}\end{align*}

20. \begin{align*}\frac{60}{8}\end{align*}

21. \begin{align*}\frac{43}{8}\end{align*}

22. \begin{align*}\frac{19}{5}\end{align*}

23. \begin{align*}\frac{39}{7}\end{align*}

24. \begin{align*}\frac{30}{4}\end{align*}

25. \begin{align*}\frac{11}{7}\end{align*}

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