8.6: Percents, Decimals and Fractions
Introduction
Sweeping and Mopping
Sam’s least favorite part of her afterschool job at the supermarket is closing. When closing the store, the break room has to be swept and mopped. Each of the students who work at the store part-time take turns closing, and every Friday night is Sam’s turn.
On Friday, Sam got her mop and broom and headed up to the break room. It seemed to be even messier than usual.
“Oh no, I will never get done,” Sam sighed, but she picked up the broom and began to sweep.
In just fifteen minutes, Sam had swept four-fifths of the room. She was amazed at how quickly the task was getting done with a little focus and effort.
What fraction of the room does Sam still have to sweep?
What percent of the room has she finished? What percent of the room is still left?
This lesson will teach you all about converting fractions and percents. By the end of the lesson, you will know how to answer these questions.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills.
- Write fractions as percents.
- Write decimals as percents.
- Compare and order fractions, decimals and percents.
- Write fractional or decimal real-world data as percents.
Teaching Time
In our last lesson, you learned how to understand what a percent is. Remember that a percent means a part of a whole, and the whole is 100. You also learned how to take a percent and write it as a decimal and as a fraction. In this lesson, you will learn to work the other way around. Let’s begin by learning how to write fractions as a percent.
I. Write Fractions as a Percent
When writing a percent as a fraction, we can drop the % sign and make that the denominator of 100. Then we take the quantity and write it in the numerator above the 100. This is our fraction.
Example
Write 14% as a fraction.
First, we drop the % sign. % means out of 100, so 100 becomes our new denominator.
14% becomes \begin{align*}\frac{14}{100}\end{align*}.
Yes! We can change the denominator of 100 to a % sign and add that to the quantity in the numerator. Here is what that looks like.
Example
Write \begin{align*}\frac{47}{100}\end{align*} as a percent.
First, drop the 100. Then write 47 with a % sign.
\begin{align*}\frac{47}{100}\end{align*} becomes 47 %
Our answer is 47%.
Write each fraction as a percent.
- \begin{align*}\frac{48}{100}\end{align*}
- \begin{align*}\frac{82}{100}\end{align*}
- \begin{align*}\frac{91}{100}\end{align*}
Go over your answers with a partner.
Not all fractions have a denominator of 100. How do we write a fraction as a percent when the denominator is not 100?
This is the next thing that we need to learn. If a fraction does not have a denominator of 100, to write it as a percent we need to rewrite it as an equal fraction with a denominator of 100.
Here is an example.
Example
Write \begin{align*}\frac{2}{5}\end{align*} as a percent.
To write \begin{align*}\frac{2}{5}\end{align*} as a percent, we must first rewrite it as a fraction with a denominator of 100. Write a proportion to show this comparison.
\begin{align*}\frac{2}{5}=\frac{}{100}\end{align*}
What number was multiplied by 5 to get 100 as a product?
20! So we multiply the numerator by 20 and we will have an equivalent fraction with a denominator of 100.
\begin{align*}2 \times 20 &= 40\\ \frac{2}{5}=\frac{40}{100}&=40\%\end{align*}
Our answer is 40%.
One special fraction to work with is one-third. To convert one-third to a percent is a little tricky because 3 does not divide evenly into 100. Take a look.
\begin{align*}\frac{1}{3}=\frac{}{100}\end{align*}
When completing this problem, we end up with a repeating decimal; the 3’s just continue on and on and on.
.33333333 etc.
To work with this fraction, we call it 33 \begin{align*}\frac{1}{3}\end{align*} %.
Write each fraction below as a percent. Remember you will have to convert each to a fraction with a denominator of 100 first.
- \begin{align*}\frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{5}\end{align*}
- \begin{align*}\frac{3}{4}\end{align*}
Check your work and take a few notes on how to convert a fraction to a percent before moving on.
II. Write Decimals as Percents
In our last lesson you learned to write percents as a decimal. Here is an example.
Example
Write 31% as a decimal.
You will recall that to do this, we drop the % sign and move the decimal point two places in to the left. These two places represent the hundredths place of the decimal.
31% becomes .31
Our answer is .31.
We can also write a decimal as a percent. To do this, we are going to move the decimal point two places to the right and add a percent sign.
Example
Write .14 as a percent.
To do this, we move the decimal point two places to the right because two decimal places represent hundredths and percents are out of 100. Then we add in the % sign.
\begin{align*}\underrightarrow{.14.} = 14\%\end{align*}
Sometimes, you will have a decimal that is written with a zero for the tenths place. We do the same thing to convert to a percent. Move the decimal two places to the right and add a percent sign.
Example
Write .03 as a percent.
.03 becomes 3%
What about a decimal that does not have two decimal places represented?
Example
Write .2 as a percent.
To do this, we move the decimal point two places to the right, which will require adding a zero. Then we can see that two tenths becomes 20 percent.
\begin{align*}\underrightarrow{.20} = 20\%\end{align*}
What about if you have a decimal with more than two places?
When this happens, it is an interesting case, because you have to move the decimal point two places to the right, but you will have a percent that is also a decimal.
Example
.345
.345 becomes 34.5%.
Now it is time for you to practice. Write each decimal as a percent.
- .85
- .09
- .5
Take a few minutes to check your work with a partner. Take a few notes on how to write decimals as percents before moving on to the next section.
III. Compare and Order Fractions, Decimals and Percents
We have already established that fractions, decimals and percents are all related to one another. Because they are related and we can establish equivalents of each, we can also compare each using greater than, less than or equal to. We can also write them in order.
To compare fractions, decimals and percents, we should have them in the same form. If we are comparing a fraction and a percent, we have to write both of them either as fractions or percents so we can figure out which is greater.
Example
Compare 45% and \begin{align*}\frac{4}{5}\end{align*}
To compare these two quantities, first write them in the same form. Let’s change four-fifths to a percent. We do that by writing it as a fraction out of 100, which we can then change to a percent.
\begin{align*}\frac{4}{5}=\frac{80}{100}=80\%\end{align*}
45% is less than 80%.
Our answer is \begin{align*}45\% < \frac{4}{5}\end{align*}.
We can do the same thing when working with decimals and percents.
Example
Compare 18% and .9
To complete this, we have to convert both of these to either percents or decimals. Let’s change .9 to a percent. To do this, we move the decimal point two places to the right.
.9 = 90%
18% is less than 90%.
Our answer is 18% < .9.
Try a few of these on your own. Use greater than >, less than < or = .
- .19 and 19%
- \begin{align*}\frac{2}{5}\end{align*} and 45%
- 56% and 21%
Check your work with a peer.
What about ordering fractions, decimals and percents? When we order a set of numbers or quantities, we write them from least to greatest or from greatest to least. Fractions, decimals and percents are no exception, but it is to order them if they are in the same form.
Example
Write .56, 34%, \begin{align*}\frac{9}{10}\end{align*} and \begin{align*}\frac{1}{2}\end{align*} in order from least to greatest
To do this, we need to write them all in the same form. Let’s convert all of them to percents.
\begin{align*}.56 = 56\%\end{align*}
34% stays the same
\begin{align*}\frac{9}{10} &= \frac{90}{100}=90\%\\ \frac{1}{2} &= \frac{50}{100}=50\%\end{align*}
So we have 56%, 34%, 90% and 50%, now it becomes easy to write them in order.
34%, 50%, 56%, and 90%
Our answer is 34%, \begin{align*}\frac{1}{2}\end{align*}, .56, \begin{align*}\frac{9}{10}\end{align*}.
Take a few notes on ordering fractions, decimals and percents.
Real Life Example Completed
Sweeping and Mopping
Sam is working away and you have been too. Here is the original problem once again. Reread the problem and then use what you have learned to figure out how much of the floor Sam has completed and how much she has left. Be sure to start by underlining the important information.
Sam’s least favorite part of her after school job at the supermarket is closing. When closing the store, the break room has to be swept and mopped. Each of the students who work at the store part-time take turns working on this and every Friday night is Sam’s turn.
On Friday, Sam got her mop and broom and headed up to the break room. It seemed to be even messier than usual.
“Oh no, I will never get done,” Sam sighed, but she picked up the broom and began to sweep.
In just fifteen minutes, Sam had swept four-fifths of the room. She was amazed at how quickly the task was getting done with a little focus and effort.
What fraction of the room does Sam still have to sweep?
What percent of the room has she finished? What percent of the room is still left?
Let’s work through this solution.
If Sam has completed four-fifths of the room, then she has one-fifth left to complete.
What percent of the room has she completed?
To figure this out, we have to figure out what four – fifths is as a percent. To do this, we can figure it out by using an equal ratio out of 100.
\begin{align*}\frac{4}{5}=\frac{80}{100}=80\%\end{align*}
Sam has completed 80% of the room.
What percent of the room does she have left?
You can figure this out two different ways. The first way is to simply subtract 80% from 100%. 100% - 80% (what Sam has completed) \begin{align*}= 20\%\end{align*}
The other way is to convert one-fifth (the amount left) to a percent. We can do this by creating an equal ratio out of 100.
\begin{align*}\frac{1}{5}=\frac{20}{100}=20\%\end{align*}
Since it only took Sam 15 minutes to complete 80% of the room, if she continues with her great effort she will be finished in no time at all.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Percent
- is "per-cent" or "per-hundred", it is a quantity written with a % sign, a part of a whole (100)
- Fraction
- a part of a whole, related to decimals and percents.
- Decimal
- a part of a whole shown by a decimal point, hundredths means two decimal places.
- Equivalent
- means equal
- Compare
- to determine whether a quantity is greater than, less than, or equal to another quantity.
- Order
- to write in order from least to greatest or from greatest to least.
Technology Integration
Khan Academy, Percents, Decimals, and Fractions
James Sousa, An Example Relating Fractions, Decimals, and Percents
James Sousa, An Second Example Relating Fractions, Decimals, and Percents
Other Videos:
http://www.teachertube.com/members/viewVideo.php?video_id=141903 – This video compares fractions, decimals and percents. You'll need to register at the site to view the video.
http://www.mathplayground.com/howto_perfracdec.html – This video shows how to convert fractions and decimals to percents.
http://www.brightstorm.com/d/math/s/algebra/u/pre-algebra/t/percents – This video will help you understand percents and proportions.
Time to Practice
Directions: Write each fraction as a percent.
1. \begin{align*}\frac{4}{100}\end{align*}
2. \begin{align*}\frac{24}{100}\end{align*}
3. \begin{align*}\frac{20}{100}\end{align*}
4. \begin{align*}\frac{76}{100}\end{align*}
5. \begin{align*}\frac{61}{100}\end{align*}
6. \begin{align*}\frac{1}{4}\end{align*}
7. \begin{align*}\frac{3}{4}\end{align*}
8. \begin{align*}\frac{3}{6}\end{align*}
9. \begin{align*}\frac{2}{5}\end{align*}
10. \begin{align*}\frac{4}{5}\end{align*}
11. \begin{align*}\frac{8}{10}\end{align*}
12. \begin{align*}\frac{6}{10}\end{align*}
Directions: Write each decimal as a percent.
13. .31
14. .56
15. .43
16. .08
17. .01
18. .4
19. .6
20. .65
21. .33
22. .19
23. .3
24. .9
25. .11
26. .18
27. .34
28. .99
29. .21
30. .88
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