3.6: Fractions and Decimals
Introduction
Keeping Track of Sales
At the bake sale, the seventh grade has set up three tables full of baked goods. There was so much donated that they also have extra baked goods on reserve. The students have decided to run the bake sale for three days so that they can try to sell their goods during all three lunch periods at school. They hope to raise a lot of money.
Derek and Keisha have been assigned the task of keeping track of sales. They need to keep track of how much of each item is sold. For example, if there are twelve cupcakes and six of them sell, then they could write one-half as the statistic for cupcakes. The students think that if they keep track of each item sold, that they will have a good idea of which items are the big sellers.
The first day goes smoothly. Derek and Keisha each keep track of one table and they split the third table in half. At the end of the sale day, they sit down to compare notes.
Derek has written these amounts down.
Peach pie .10
Cupcakes .75
Brownies .50
Bread .25
Keisha has written these amounts down.
Blueberry pie \begin{align*}\frac{1}{2}\end{align*}
Chocolate cookies \begin{align*}\frac{1}{4}\end{align*}
Blueberry muffins \begin{align*}\frac{1}{3}\end{align*}
Double Fudge Brownies \begin{align*}\frac{3}{4}\end{align*}
Apple pie \begin{align*}\frac{5}{6}\end{align*}
They are surprised to see that one of them has written all of the data in decimals while the other has written all of the data in fraction form. To figure out which items were the biggest sellers they will have to order and combine their data.
Derek needs to write his decimals in order from greatest to least and Keisha needs to write her fractions in order from greatest to least.
This is your task as well. This lesson will teach you how to order fractions and decimals. By the end of the lesson, you will know which items were the top sellers and which ones weren’t.
What You Will Learn
In this lesson, you will learn how to complete the following:
- Write fractions and mixed numbers as terminating decimals.
- Write fractions and mixed numbers as repeating decimals.
- Write decimals as fractions.
- Compare and order decimals and fractions.
Teaching Time
I. Write Fractions and Mixed Numbers as Terminating Decimals
Now that you have mastered fractions and their corresponding operations, it’s time to discover how they relate to decimals. You know that fractions and decimals are related because they are both ways of describing numbers that are part of a whole. In essence, a fraction is simply another way of describing what a decimal describes. They both represent parts of a whole and both can show the same part using different formats. The fraction shows us the part using the fraction bar comparing part to whole and the decimal shows us the part using place value. To start off, we’ll see how fractions can be converted into decimals and how decimals can be converted into fractions.
How do we convert fractions to decimals and decimals to fractions?
First, remember that fractions and decimals are different ways of writing the same thing. Both show us how to represent a part of a whole. Think about how we talk about fractions and decimals because this will become useful as we convert them.
Say this value out loud 0.1. You can say “point one” or “one tenth.” Does the second version sound a little bit familiar? It sounds like the fraction \begin{align*}\frac{1}{10}\end{align*}. It turns out that \begin{align*}0.1 = \frac{1}{10}\end{align*}.
Yes, and a decimal is just another way of writing a fraction.
How do we convert fractions into decimals?
We can easily convert fractions into decimals. You’ve probably noticed by now that a fraction is really a short way of writing a division expression. Writing \begin{align*}\frac{3}{4}\end{align*} is really like writing \begin{align*}3 \div 4\end{align*}. The way that we find out how to write \begin{align*}\frac{3}{4}\end{align*} as a decimal is to go ahead and solve the division problem. Since 4 doesn’t go into 3, we have to expand the number over the decimal point.
\begin{align*}{4 \overline{) {3.0 \;}}}\end{align*}
How many times does 4 go into 3.0? Four goes into 3.0 .7 times.
\begin{align*}& \overset{ \ \ \ \ .7}{4 \overline{ ) {3.0 \;}}}\\ & \underline{-2.8 \ }\\ & \quad \ .2\end{align*}
Be sure when you are writing your quotient above the dividend to keep the original place of the decimal point. Since 4 does not divide evenly into 3.0 and we have a remainder of .2, we can go further to the other side of the decimal point by adding a 0 next to the remainder of .2.
\begin{align*}& \overset{ \ \ \ \ .75}{4 \overline{ ) {3.00 \;}}}\\ & \ \underline{ \ -2.8 \ }\\ & \quad \ \ .20\\ & \ \underline{ \ \ -.20 \ }\\ & \quad \ \ \ \ \ 0 \end{align*}
4 goes evenly into .20 five times, so we have our final answer.
\begin{align*}\frac{3}{4} = 0.75\end{align*}
Example
Convert \begin{align*}\frac{1}{4}\end{align*} to a decimal.
We start this by changing it into a division problem. We will be dividing 1 by 4. You already know that 1 can’t be divided by four, so you will need to use a decimal point and add zeros as needed.
\begin{align*}& \overset{ \ \ \ \ .25}{4 \overline{ ) {1.00 \;}}}\\ & \underline{-8 \quad \ }\\ & \quad \ 20\\ & \ \ \underline{-20 \ }\\ & \qquad 0 \end{align*}
Our answer is .25.
How do we convert mixed numbers to decimals?
When you are working with mixed numbers like \begin{align*}1 \frac{3}{4}\end{align*} for example, it is easiest to simply set the whole number to the side and solve the division problem with the fraction. When you have completed the division problem with the fraction, make sure that you put the whole number back on the left side of the decimal point.
\begin{align*}1 \frac{3}{4} = 1.75\end{align*}
Example
Convert \begin{align*}3 \frac{1}{2}\end{align*} to a decimal.
First, set aside the 3. We will come back to that later.
Next, we divide 1 by 2. Use a decimal point and zeros as needed.
\begin{align*}& \overset{ \ \ \ \ .5}{2 \overline{ ) {1.0 \;}}}\\ & \underline{ \ -10}\\ & \quad \ 0\end{align*}
Now we add in the 3.
Our final answer is 3.5.
3O. Lesson Exercises
Convert each fraction or mixed number to a decimal.
- \begin{align*}\frac{1}{5}\end{align*}
- \begin{align*}\frac{3}{6}\end{align*}
- \begin{align*}4 \frac{4}{5}\end{align*}
Take a few minutes to check your work with a friend.
II. Write Fractions and Mixed Numbers as Repeating Decimals
By now you’ve gotten the hang of converting fractions to decimals. So far, we have been working with what are known as terminating decimals, or decimals that have an end like 0.75 or 0.5.
One reason that we sometimes use fractions instead of decimals is because some decimals are repeating decimals, or decimals that go on forever. If you try to find a decimal for \begin{align*}\frac{1}{3}\end{align*} by dividing, you can divide forever because \begin{align*}\frac{1}{3}\end{align*} written as a decimal \begin{align*}= 0.3333333333 ....\end{align*} It goes on and on. That’s why we usually just simply write a line above the number that repeats. For \begin{align*}\frac{1}{3}\end{align*}, we write: \begin{align*}0.\overline{3}\end{align*}. Let’s check out some examples involving repeating decimals.
Example
Write \begin{align*}\frac{5}{6}\end{align*} using decimals
First, we rewrite \begin{align*}\frac{5}{6}\end{align*} as the division problem \begin{align*}5 \div 6\end{align*}. We already know that we will have to go on the right side of the decimal point, so we are going to begin by dividing 6 into 5.0.
Six goes into 5.0 .8 times, but we have the remainder of .2. Six goes into 0.2 .03 times and we have a remainder of .02. Since 6 always goes into 20 three times, \begin{align*}(3 \cdot 6 = 18)\end{align*} and there will always be a remainder of 2, we can see that it will never evenly divide.
If you keep dividing, you will get 0.83333333333.... forever and ever.
Our final answer is \begin{align*}0.8\overline{3}\end{align*}.
What about mixed numbers?
Well, there are some mixed numbers where the fraction part is a repeating decimal. Let’s look at an example.
Example
Write \begin{align*}2 \frac{2}{3}\end{align*} using decimals.
Just as we did with the terminating decimals, we are going to set the whole number, 2, to the side, and then come back to it when we are ready to add it to the final answer. So, we are simply solving for the decimal equivalent of \begin{align*}\frac{2}{3}\end{align*}. We write the division problem \begin{align*}2.0 \div 3\end{align*}. How many times does 3 go into 2.0? It goes into 2.0 0.6 times.
We have 0.20 as the remainder. How many times does 3 go into 0.20? The answer is 0.06 times.
Are you noticing a pattern here? It is obvious that there will always be a remainder whether we divide 3 into 2.0, 0.2, 0.02, 0.002, or 0.0002 and on and on. Clearly \begin{align*}\frac{2}{3}\end{align*} is a repeating decimal.
For our final answer we write \begin{align*}2.\overline{6}\end{align*}.
3P. Lesson Exercises
Convert each to a repeating decimal.
- \begin{align*}\frac{1}{6}\end{align*}
- \begin{align*}4 \frac{4}{6}\end{align*}
Take a few minutes to check your work with a partner.
III. Write Decimals as Fractions
Now that we have mastered writing fractions as decimals, it will be good to know how to write decimals as fractions.
Consider again the decimal 0.1. We already know that we can say that this number is “one-tenth.” It’s very easy to rewrite decimals as fractions because decimals are already expressed as fractions with a denominator that is a factor of ten.
\begin{align*}0.1 = \frac{1}{10}, 0.01 = \frac{1}{100}\end{align*}
We can also say that \begin{align*}0.86 = \frac{86}{100}\end{align*}.
To convert decimals to fractions, we write the number to the right of the decimal place over a denominator equivalent to the last place value of the decimal number. So, if we have 0.877, we would write \begin{align*}\frac{877}{1000}\end{align*}.
If we have simply 0.6, we can write \begin{align*}\frac{6}{10}\end{align*}, or in simplest terms, \begin{align*}\frac{3}{5}\end{align*}. Always make sure to write your fraction in simplest terms.
Let’s look at an example.
Example
Convert 0.35 to a fraction.
Start by saying the decimal to yourself out loud. To say 0.35 out loud, we can say “35 hundredths,” so we can go ahead and write the fraction down.
\begin{align*}\frac{35}{100}\end{align*}
That’s a big fraction. We want to make our lives a little bit easier, so we will reduce the fraction to simplest terms. This fraction expressed in simplest terms is \begin{align*}\frac{7}{20}\end{align*}.
Our final answer is \begin{align*}\frac{7}{20}\end{align*}.
Example
Convert 2.4 to a mixed number.
Just as we set aside the whole number when converting mixed numbers to decimals, we will set aside the numbers to the left of the decimal point when converting decimals to fractions. So, in this case, we just have to find out what 0.4 is when expressed as a fraction.
We write it directly as the fraction “four tenths” or \begin{align*}\frac{4}{10}\end{align*}. Can we simplify it? You bet. \begin{align*}\frac{4}{10}=\frac{2}{5}\end{align*}.
2.4 expressed as a mixed number \begin{align*}= 2 \frac{2}{5}\end{align*}.
3Q. Lesson Exercises
Convert each decimal to a fraction in simplest form. (Don't forget to return set-aside whole numbers to your final answer.)
- .5
- .67
- 3.21
Take a few minutes to check your answers with a partner. Is your work accurate?
IV. Compare and Order Decimals and Fractions
Eventually, you will be able to convert common fractions to decimals and common decimals to fractions in your head. You already know some of the more common ones like \begin{align*}0.5 = \frac{1}{2}\end{align*}. Knowing this off the top of your head will make it easy for you to compare and order between fractions and decimals. For now, we will practice our newly acquired skills at converting to compare and order. It’s always helpful to check.
Example
Compare \begin{align*}\frac{1}{4}\end{align*} and 0.25 using <, >, or =
To compare a fraction to a decimal or a decimal to a fraction, we will need to convert one of them, so that we can compare a fraction to a fraction or a decimal to a decimal. For this one, we will convert \begin{align*}\frac{1}{4}\end{align*} to a decimal. we divide 1.0 by 4. 4 goes into 1.0 .2 times. There is also a remainder of 0.20 and 4 goes into 0.20 0.05 times. Now we know that \begin{align*}\frac{1}{4}\end{align*} written as a decimal is 0.25.
We compare it as \begin{align*}\frac{1}{4} = 0.25\end{align*}.
Example
Compare \begin{align*}1 \frac{2}{20}\end{align*} and 1.30
Our work in estimating the value of fractions and rounding decimals can be helpful when comparing fractions and decimals because you can look at a fraction or a decimal and quickly have a sense of what the approximate value is. Take a look at the mixed number \begin{align*}1 \frac{2}{20}\end{align*}. Don’t be intimidated by the large denominator, it looks like we can simplify it. Simplify it to \begin{align*}1 \frac{1}{10}\end{align*}.
Now we can take 1.30 and make it a mixed number. 1.30 becomes \begin{align*}1 \frac{3}{10}\end{align*}.
Our final answer is that \begin{align*}1 \frac{2}{20} < 1.30\end{align*}.
When ordering fractions and decimals be sure to use the strategies we just practiced. Make sure that they are in the same form and then order them from least to greatest or from greatest to least. Let’s go back to our original problem now and apply what we have learned.
Real Life Example Completed
Keeping Track of Sales
Here is the original problem once again. Reread the problem and then underline any important information.
At the bake sale, the seventh grade has set up three tables full of baked goods. There was so much donated that they also have extra baked goods on reserve. The students have decided to run the bake sale for three days so that they can try to sell their goods during all three lunch periods at school. They hope to raise a lot of money.
Derek and Keisha have been assigned the task of keeping track of sales. They need to keep track of how much of each item is sold. For example, if there are twelve cupcakes and six of them sell, then they could write one-half as the statistic for cupcakes. The students think that if they keep track of each item sold, that they will have a good idea of which items are the big sellers.
The first day goes smoothly. Derek and Keisha each keep track of one table and they split the third table in half. At the end of the sale day, they sit down to compare notes.
Derek has written these amounts down.
Peach pie .10
Cupcakes .75
Brownies .50
Bread .25
Keisha has written these amounts down.
Blueberry pie \begin{align*}\frac{1}{2}\end{align*}
Chocolate cookies \begin{align*}\frac{1}{4}\end{align*}
Blueberry muffins \begin{align*}\frac{1}{3}\end{align*}
Double Fudge Brownies \begin{align*}\frac{3}{4}\end{align*}
Apple pie \begin{align*}\frac{5}{6}\end{align*}
They are surprised to see that one of them has written all of the data in decimals while the other has written all of the data in fraction form. To figure out which items were the biggest sellers they will have to order and combine their data.
Derek needs to write his decimals in order from greatest to least and Keisha needs to write her fractions in order from greatest to least.
First, Derek needs to write his data in order from greatest to least.
Cupcakes .75
Brownies .50
Bread .25
Peach Pie .10
Next, Keisha needs to write her data in order from greatest to least.
Apple pie \begin{align*}\frac{5}{6}\end{align*}
Double Fudge Brownies \begin{align*}\frac{3}{4}\end{align*}
Blueberry Pie \begin{align*}\frac{1}{2}\end{align*}
Chocolate Cookies \begin{align*}\frac{1}{4}\end{align*}
Blueberry Muffins \begin{align*}\frac{1}{3}\end{align*}
Given this information, the top sellers were the apple pies, the double fudge brownies, the cupcakes, the blueberry pie, and the brownies. Derek and Keisha report their sales to their teacher and they decide to put out more of those items for day two of the bake sale.
Vocabulary
Here is the vocabulary words that are found in this lesson.
- Fraction
- a part of a whole written using a numerator and a denominator and a fraction bar
- Decimal
- a part of a whole written using a decimal point and place value
- Mixed Number
- a number written with a whole number and a fraction.
- Terminating Decimal
- a decimal with an ending.
- Repeating Decimal
- a decimal that does not end but repeats infinitely
Technology Integration
Khan Academy Converting Fractions to Decimals
James Sousa, Converting Fractions to Decimals
James Sousa, Example of Converting Fractions to a Terminating Decimal
James Sousa, Another Example of Converting Fractions to a Terminating Decimal
James Sousa, Example of Converting Fractions to a Repeating Decimal
Other Videos:
http://www.mathplayground.com/howto_fractions_decimals.html – This is a video on how to convert a fraction into a decimal.
Time to Practice
Directions: Write each fraction or mixed number as a decimal.
1. \begin{align*}\frac{3}{5}\end{align*}
2. \begin{align*}\frac{1}{2}\end{align*}
3. \begin{align*}\frac{4}{5}\end{align*}
4. \begin{align*}\frac{3}{4}\end{align*}
5. \begin{align*}\frac{3}{8}\end{align*}
6. \begin{align*}\frac{7}{8}\end{align*}
7. \begin{align*}2 \frac{1}{4}\end{align*}
8. \begin{align*}5 \frac{2}{5}\end{align*}
9. \begin{align*}6 \frac{4}{8}\end{align*}
10. \begin{align*}11 \frac{5}{8}\end{align*}
Directions: Write each fraction or mixed number as a repeating decimal.
11. \begin{align*}\frac{2}{3}\end{align*}
12. \begin{align*}\frac{5}{6}\end{align*}
13. \begin{align*}5 \frac{1}{3}\end{align*}
Directions: Write each decimal as a fraction in simplest form.
14. .25
15. .23
16. .876
17. .5
18. 74
19. .88
20. .987
Directions: Compare the following decimals and fractions using <, >, or =
21. \begin{align*}\frac{1}{3}\end{align*} and .5
22. \begin{align*}\frac{6}{10}\end{align*} and .9
23. .25 and \begin{align*}\frac{1}{10}\end{align*}
24. .16 and \begin{align*}\frac{33}{100}\end{align*}
25. \begin{align*}\frac{2}{5}\end{align*} and .4
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