5.7: Changing Decimals to Fractions
Introduction
The Map Disaster
In the game room during the sixth grade social, the map of the world fell off of the wall. Mrs. Jennings was monitoring this group, and she was very glad that no one was hurt. During the event, she put the map to the side to be fixed later.
On Monday, Aaron and Chris asked Mrs. Jennings if they could help Mr. Jones, the custodian, fix the map. Mrs. Jennings said "Certainly!" and gave the boys permission to stay in from recess. Mr. Jones came to the classroom with his tool box.
The first thing he had the boys do was to measure the two holes that the map had fallen from. The holes both measured \begin{align*}1\frac{1}{4}”\end{align*}
While Mr. Jones went to get his power drill, he asked Chris and Aaron to select a peg that would best fit the two holes. Chris and Aaron have three different sized pegs to choose from.
Peg 1 is 1.27” in diameter.
Peg 2 is 1.23” in diameter.
Peg 3 is 1.21” in diameter.
Chris and Aaron are puzzled. They know that they need to select the peg that is the closest to the size of the hole. They just aren’t sure what to choose.
This is where you come in. In this lesson, you will learn all about converting decimals to fractions.
Once you know how to do this, you will be able to convert the peg sizes from decimals to fractions and selecting the correct one will be simple.
What You Will Learn
In this lesson you will learn to do the following:
- Write decimals as fractions.
- Write decimals as mixed numbers.
- Write decimals as equivalent fractions and mixed numbers.
- Describe real-world portion or measurement situations by writing decimals as fractions.
Teaching Time
I. Write Decimals as Fractions
Decimals and fractions are related. They both represent a part of a whole. With a decimal, the part of a whole is written using a decimal point. With a fraction, the part of a whole is written using a fraction bar and has a numerator and a denominator. Because fractions and decimals are related, we can write decimals as fractions.
How do we write decimals as fractions?
We write a decimal as a fraction by using place value. Let’s look at an example written in a place value chart.
Example
.67
Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths | Ten-Thousandths |
---|---|---|---|---|---|---|
. | 6 | 7 |
If we read this fraction out loud, we say, “Sixty-seven hundredths.”
Converting this decimal to a fraction becomes easy because we have the numerator “67” and the denominator “hundredths.”
The fraction is described by reading the decimal.
Our answer is \begin{align*}\frac{67}{100}\end{align*}
Our next step is to see if we can simplify this fraction. In this case, we can’t simplify the fraction so our answer remains the same.
Let’s look at another one.
Example
.5
We can write this decimal in our place value chart.
Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths | Ten-Thousandths |
---|---|---|---|---|---|---|
. | 5 |
We read this decimal as “five tenths.” The numerator is the five and the denominator is the place value of tenths.
Our answer is \begin{align*}\frac{5}{10}\end{align*}
Our next step is to see if we can simplify the fraction. This fraction simplifies to one-half.
Our final answer is \begin{align*}\frac{1}{2}\end{align*}
Now it is time for you to practice. Write the following decimals as fractions in simplest form.
- .8
- .25
- .75
Take a minute to check your work with a peer.
II. Write Decimals as Mixed Numbers
Some decimals represent both a part and a whole. We can take these decimals and write them as mixed numbers. The mixed number and the decimal, which contains a part and a whole, are equivalent because they are both referring to the same amount.
How do we write a decimal as a mixed number?
To write a decimal as a mixed number, we need to have a decimal that has both wholes and parts in it. Here is an example.
Example
4.5
This decimal has four wholes and five tenths. Let’s write this decimal in a place value chart so that we can convert it to a mixed number.
Tens | Ones | Decimal Point | Tenths | Hundredths | Thousandths | Ten-Thousandths |
---|---|---|---|---|---|---|
4 | . | 5 |
We can read this decimal as four and five tenths. The four represents the wholes. The and represents the decimal point. The five is the numerator of the fraction and the tenths represents the denominator.
The answer is \begin{align*}4\frac{5}{10}\end{align*}.
Next, we need to check and see if we can simplify this fraction. In this case, five-tenths can be simplified to one-half.
Our final answer is \begin{align*}4\frac{1}{2}\end{align*}.
Try a few of these on your own. Write each decimal as a mixed number in simplest form.
- 7.8
- 4.45
- 2.25
Take a minute to check your work with a peer.
III. Write Decimals as Equivalent Fractions and Mixed Numbers
When we convert a decimal to a fraction, we are writing two parts that are equivalent or equal. Because of this, we can write more than one equivalent fraction for any single decimal. You will need to think back to our lesson on creating equivalent fractions for this to make sense.
Let’s start with an example.
Example
.75
This decimal can be read as “Seventy-five hundredths.” We know that we can write the fraction by using these words as we read the decimal. The seventy-five is our numerator and the hundredths is our denominator.
\begin{align*}\frac{75}{100}\end{align*}
When we simplify this fraction, we have another equivalent fraction to .75.
\begin{align*}\frac{75}{100}=\frac{3}{4}\end{align*}
Now we can keep on creating equivalent fractions for three-fourths by simply multiplying the same number with the numerator and the denominator. Let’s create another equivalent fraction by multiplying by two.
\begin{align*}\frac{75}{100}=\frac{3}{4}=\frac{6}{8}\end{align*}
We could go on and on. The important thing to notice is that each of these fractions is equivalent to .75, since they are just different forms of the same thing.
How do we write equivalent fractions for decimals that have wholes and parts?
We are going to work with these decimals in the same way, except we will be converting them to mixed numbers and then writing equivalent mixed numbers from there.
Example
4.56
We can write this as a mixed number by reading the decimal. With four and fifty-six hundredths, the four is the whole number, the fifty-six is the numerator and the denominator is the hundredths.
\begin{align*}4\frac{56}{100}\end{align*}
If we simplify the fraction part of this mixed number, we will have another mixed number that is equivalent to the one that we just wrote.
The greatest common factor of 56 and 100 is four. Now we can simplify the fraction part.
\begin{align*}4\frac{56}{100}=4\frac{14}{25}\end{align*}
Now it is time for you to try a few. Write an equivalent fraction or mixed number for each decimal.
- 2.14
- 16.10
- .55
Take a few minutes to check your work.
Real Life Example Completed
The Map Disaster
Now that you have learned all about converting decimals to fractions and mixed numbers, you are ready to help Aaron and Chris hang the map.
Here is the problem once again.
In the game room during the sixth grade social, the map of the world fell off of the wall. Mrs. Jennings was monitoring this group, and she was very glad that no one was hurt. During the event, she put the map to the side to be fixed later.
On Monday, Aaron and Chris asked Mrs. Jennings if they could help Mr. Jones, the custodian, fix the map. Mrs. Jennings said "Certainly!" and gave the boys permission to stay in from recess. Mr. Jones came to the classroom with his tool box.
The first thing he had the boys do was to measure the two holes that the map had fallen from. The holes both measured \begin{align*}1\frac{1}{4}”\end{align*} in diameter.
While Mr. Jones went to get his power drill, he asked Chris and Aaron to select a peg that would best fit the two holes. Chris and Aaron have three different sized pegs to choose from.
Peg 1 is 1.27” in diameter.
Peg 2 is 1.23” in diameter.
Peg 3 is 1.21” in diameter.
Chris and Aaron are puzzled. They know that they need to select the peg that is the closest to the size of the hole. They just aren’t sure what to choose.
First, let’s underline the important information.
For Chris and Aaron to select the correct peg, they are going to have to convert the peg sizes into mixed numbers. Since the measurement of the holes is in mixed number form, if the peg size is in mixed number form we can easily compare.
\begin{align*}\text{Peg} \ 1 &= 1.27 = 1\frac{27}{100}\\ \text{Peg} \ 2 &= 1.23 = 1\frac{23}{100}\\ \text{Peg} \ 3 &= 1.21 = 1\frac{21}{100}\end{align*}
Our holes measured \begin{align*}1 \frac{1}{4}”\end{align*} in diameter. Uh Oh, we can’t figure out which peg is the best choice because the fraction part of the peg sizes is written in hundredths. The fraction part of the hole size is written in fourths.
We can convert the \begin{align*}1\frac{1}{4}”\end{align*} to a denominator of 100.
\begin{align*}1\frac{1}{4}=1\frac{25}{100}\end{align*}
By comparing the sizes, we can see easily now that pegs 1 and 2 are the closest in size to the hole. We need to use critical thinking to decide if one is better than the other. Since peg 1 is actually a little bit bigger than the hole, it would not fit. Our answer is Peg 2. That one is the closest to the size of the holes, without being too big to fit in at all.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Decimal
- a part of a whole written using place value and a decimal point.
- Fraction
- a part of a whole written with a fraction bar dividing the numerator and the denominator.
- Mixed Number
- a number that has a whole number and a fraction.
- Equivalent
- means equal
Technology Integration
Khan Academy Decimals and Fractions
- http://www.teachertube.com/members/viewVideo.php?video_id=9455&title=Converting_Decimals_Video – This is a how to video that teaches how to convert decimals to fractions and fractions to decimals. You'll need to register at the site to view the video.
Time to Practice
Directions: Write each decimal as a fraction. You do not need to simplify them.
1. .67
2. .33
3. .45
4. .27
5. .56
6. .7
7. .98
8. .32
9. .04
10. .07
11. .056
12. .897
13. .372
14. .652
15. .032
Directions: Write each decimal as a mixed number. Simplify the fraction part if possible.
16. 3.5
17. 2.4
18. 13.2
19. 25.6
20. 3.45
21. 7.17
22. 18.18
23. 9.20
24. 7.65
25. 13.11
26. 7.25
27. 9.75
28. 10.10
29. 4.33
30. 8.22
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