6.1: Percents and Fractions
Introduction
The Candy Store
Taylor’s family owns a candy store. During her winter vacation, Taylor has the opportunity to work in the candy store and earn some extra money. Since candy is one of her favorite things, she never turns down an opportunity to help out in the store.
The first day that Taylor worked in the store, a family with three small children came in. The little children each wanted gummy bears to eat. Since there weren’t enough gummy bears in the jar, Taylor had to open a new bag of them. The bag said 400 on it.
Taylor took out the bag and began to open it.
“Please give us 25% of the bag,” the mom said, smiling.
Taylor looked at the bag and then back up at the mom. She took out a piece of paper and a pencil.
“You can estimate,” the mom said, still smiling.
Taylor estimated 25% or \begin{align*}\frac{1}{4}\end{align*}
After they had left, Taylor began thinking about how many gummy bears 25% of 400 would be. She picked up the pencil and began to do some figuring.
This lesson is all about percents. To work with percents you have to understand how they relate to parts and fractions. Pay attention and by the end of the lesson, you will be able to do some gummy bear arithmetic.
What You Will Learn
In this lesson, you will learn the how to:
- Recognize percent as a ratio whose denominator is 100.
- Write percents as fractions in simplest form.
- Write fractions as percents.
- Find the percent of a number using fraction multiplication.
Teaching Time
I. Recognize Percent as a Ratio whose Denominator is 100
In Chapter 5, you learned that a ratio is a comparison of two numbers and that a ratio can be written in three ways. For example, if there are 13 red jelly beans and 15 yellow jelly beans in a jar, the ratio of red jelly beans to yellow jelly beans can be written as 13 to 15, 13:15, or \begin{align*}\frac{13}{15}\end{align*}
A percent is a type of ratio. You may have worked with percents before, but now we are going to apply them directly to ratios. Hopefully, you will see percents in a new way.
What is a percent?
A percent is a ratio that compares a number to 100. Percent means “per hundred” and the symbol for percent is %. 100% represents the ratio 100 to 100 or \begin{align*}\frac{100}{100}\end{align*}
If there are 100 jelly beans in a jar and 19 are black, we can say that \begin{align*}\frac{19}{100}\end{align*}
Example
Write a ratio and a percent that describes the shaded part.
The figure shows the ratio of 37 shaded squares to 100 squares.
We can write the ratio as 37 to 100, 37:100, or \begin{align*}\frac{37}{100}\end{align*}
When the denominator of a ratio in fraction form is 100, you can express the ratio as a percent.
\begin{align*}\frac{37}{100}=37\%\end{align*}
The ratio that describes the shaded part is 37 to 100, 37:100, or \begin{align*}\frac{37}{100}\end{align*}
This is a visual way of showing a percent as a ratio. We can use the picture to write the ratio and the percent.
Let’s look at another example that uses a real-life item.
Example
There were 100 questions on a test and Amanda answered 92 of them correctly. What percent did she answer correctly? What percent did she answer incorrectly?
Amanda answered 92 out of 100 questions correctly. We can write this as the ratio \begin{align*}\frac{92}{100}\end{align*}
Since there were 100 questions on the test and Amanda answered 92 correctly, she answered 100 – 92, or 8 out of 100 incorrectly. We can write this as the ratio \begin{align*}\frac{8}{100}\end{align*}
Amanda answered 92% of the questions correctly and 8% incorrectly.
6A. Lesson Exercises
Practice what you have learned by writing a ratio and a percent for each example.
- Karen ate 12 out of 100 blueberries.
- Joey answered 92 questions correctly out of 100 questions on his test.
- Sarah gathered 25 roses out of 100 flowers.
Take a few minutes to check your work with a partner.
II. Write Percents as Fractions in Simplest Form
In the last section you saw how you can write a percent as a ratio. Well, remember that one of the ways of writing a ratio is to write it in fraction form. Fractions can be simplified. Therefore, we can write percents as fractions and reduce them to simplest form.
To write a percent as a fraction, we write the percent as a ratio with a denominator of 100. We can then write the fraction in simplest form, if possible.
Example
Write 80% as a fraction in simplest form.
First, write the percent as a fraction with a denominator of 100.
\begin{align*}\frac{80}{100}\end{align*}
Next, we can simplify this fraction. You can start by canceling a zero in the numerator and one in the denominator.
\begin{align*}\frac{8\bcancel{0}}{10\bcancel{0}}\end{align*}
Now we have eight-tenths to simplify.
\begin{align*}\frac{8}{10}=\frac{4}{5}\end{align*}
Our final answer is that 80% can be written as the fraction \begin{align*}\frac{4}{5}\end{align*}
Yes. Because they are both parts of a whole, we can interchange the way that we write them. A percent can be written as a fraction and a fraction can be written as a percent too.
Let’s look at another example.
Example
Write 12% as a fraction in simplest form.
First, write the percent as a fraction out of 100.
\begin{align*}\frac{12}{100}\end{align*}
Next, we simplify this fraction. The greatest common factor is 4, so we divide the numerator and the denominator by 4.
\begin{align*}\frac{12\div 4}{100 \div 4}=\frac{3}{25}\end{align*}
12% can be written as the fraction \begin{align*}\frac{3}{25}\end{align*}
6B. Lesson Exercises
Write each percent as a fraction in simplest form.
- 18%
- 20%
- 4%
Take a few minutes to check your answers with a friend.
III. Write Fractions as Percents
A fraction can be written as a percent if it has a denominator of 100. Sometimes, you will be given a fraction with a denominator of 100 and sometimes you will have to rewrite the fraction to have denominator of 100 before you write it as a percent. Let’s look at an example.
Example
\begin{align*}\frac{9}{100}\end{align*}
This fraction is already written with a denominator of 100, so we can just change it to a percent.
\begin{align*}\frac{9}{100}=9\%\end{align*}
What do we do if a fraction does not have a denominator of 100?
This is where your work with proportions and equal ratios comes in. Remember that a proportion is two equal ratios. We can write a proportion for a fraction by creating a second fraction equal to the first that has a denominator of 100. Then we solve the proportion. It sounds trickier than it is. Let’s look at an example.
Example
Write \begin{align*}\frac{3}{5}\end{align*}
To start with, notice that the denominator is not 100. Therefore, we need to create a new fraction equivalent to this one with a denominator of 100.
\begin{align*}\frac{3}{5}=\frac{x}{100}\end{align*}
Wow! Here is a proportion. Next, think back to solving proportions. We can cross multiply to find the value of \begin{align*}x\end{align*}
\begin{align*}5x& =300\\
x& =60\\
\frac{3}{5}& =\frac{60}{100}\end{align*}
Now we have a fraction with a denominator of 100. We can write it as a percent.
Our answer is that \begin{align*}\frac{3}{5}\end{align*}
What about if we had an improper fraction?
To work with an improper fraction, you have to think about what that means. An improper fraction is greater than 1, so the percent would be greater than 100%. Sometimes in life we can have numbers that are greater than 100%. Most often they aren’t, but it is important to understand how to work with a percent that is greater than 100.
Example
Write \begin{align*}\frac{9}{4}\end{align*}
First, we write a proportion with a denominator of 100.
\begin{align*}\frac{9}{4}=\frac{x}{100}\end{align*}
Next, we cross multiply to find the value of \begin{align*}x\end{align*}
\begin{align*}4x& =900\\
x& =225\\
\frac{225}{100}& =225\%\end{align*}
Our answer is 225%.
Did you know that you already know some common fraction equivalents for percents? Think of 25 cents, 50 cents, and 75 cents.
25 cents means 25 cents out of a dollar, or 25% of a dollar. Since a quarter is 25 cents, \begin{align*}\frac{1}{4} = 25\%\end{align*}
50 cents means 50 cents out of a dollar, or 50% of a dollar. Since a half dollar is 50 cents, \begin{align*}\frac{1}{2} = 50\%\end{align*}
75 cents means 75 cents out of a dollar, or 75% of a dollar. Since three quarters is 75 cents, \begin{align*}\frac{3}{4} = 75\%\end{align*}
Sometimes, you have fractions that don’t convert easily.
Example
Write \begin{align*}\frac{2}{3}\end{align*} as a percent.
First, set up the proportion.
\begin{align*}\frac{2}{3}=\frac{x}{100}\end{align*}
Next, cross multiply to solve for the value of \begin{align*}x\end{align*}.
\begin{align*}3x& =200\\ x& =66.6\end{align*}
Notice that we end up with a repeating decimal. If we keep dividing, we will keep ending up with 6’s. Therefore, we can leave this percent with one decimal place represented. We just need to round to the tenths place. 66.66 rounds to 66.7.
Our answer is 66.6%.
Sometimes, you will see fractions like this, but you will get used to them and often you can learn the percent equivalent of these fractions by heart.
6C. Lesson Exercises
Write each fraction as a percent.
- \begin{align*}\frac{1}{4}\end{align*}
- \begin{align*}\frac{2}{5}\end{align*}
- \begin{align*}\frac{4}{40}\end{align*}
Take a few minutes to check your work with a peer.
IV. Find the Percent of a Number Using Fraction Multiplication
The table below shows the fraction equivalents for common percents.
\begin{align*}& 5\% && 10\% && 20\% && 25\% && 30\% && 40\% && 50\% && 60\% && 70\% && 75\% && 80\% && 90\%\\ & \frac{1}{20} && \frac{1}{10} && \frac{1}{5} && \frac{1}{4} && \frac{3}{10} && \frac{2}{5} && \frac{1}{2} && \frac{3}{5} && \frac{7}{10} && \frac{3}{4} && \frac{4}{5} && \frac{9}{10}\end{align*}The word “of” in a percent problem means to multiply. If you know the fraction equivalents for common percents, you can use this information to find the percent of a number by multiplying the fraction by the number. If you want to find a part of a whole using a percent, you use multiplication.
Look at the following examples.
Example
Find 40% of 45.
40% of 45 means \begin{align*}40\% \times 45\end{align*}.
Look at the chart. The fraction \begin{align*}\frac{2}{5}\end{align*} is equivalent to 40%.
\begin{align*}40\% \times 45 = \frac{2}{5} \times 45 = \frac{2}{\underset{1}{\cancel{5}}} \times \frac{\overset{9}{\cancel{45}}}{1}=\frac{18}{1}=18\end{align*}
Our answer is that 40% of 45 = 18
What about if the percent is not on the chart?
For a percent that is not on the chart, change the percent to a fraction in simplest form.
Example
Find 85% of 20.
85% of 20 means \begin{align*}85\% \times 20\end{align*}.
\begin{align*}85\% & = \frac{85}{100} = \frac{85 \div 5}{100 \div 5} = \frac{17}{10}\\ 85\% \times 20 & = \frac{17}{20}\times 20 = \frac{17}{\underset{1}{\cancel{20}}}\times \frac{\overset{1}{\cancel{20}}}{1}=\frac{17}{1}=17\end{align*}
The answer is that 85% of 20 is 17.
We can use this in real-life problems all the time. Let’s think about the introduction problem. Now we can use what we have learned to solve it.
Real Life Example Completed
The Candy Store
Here is the original problem once again. Reread it and underline any important information.
Taylor’s family owns a candy store. During her winter vacation, Taylor has the opportunity to work in the candy store and earn some extra money. Since candy is one of her favorite things, she never turns down an opportunity to help out in the store.
The first day that Taylor worked in the store, a family with three small children came in. The little children each wanted gummy bears to eat. Since there weren’t enough gummy bears in the jar, Taylor had to open a new bag of them. The bag said 400 on it.
Taylor took out the bag and began to open it.
“Please give us 25% of the bag,” the mom said, smiling.
Taylor looked at the bag and then back up at the mom. She took out a piece of paper and a pencil.
“You can estimate,” the mom said, still smiling.
Taylor estimated 25% or \begin{align*}\frac{1}{4}\end{align*} of the bag. The family paid and then, very happy with their purchase.
After they had left, Taylor began thinking about how many gummy bears 25% of 400 would be. She picked up the pencil and began to do some figuring.
Taylor wants to figure out how many gummy bears 25% is out of 400.
To do this, she needs to multiply. 25% of 400 means 25% times 400.
We can change 25% to the fraction \begin{align*}\frac{1}{4}\end{align*}.
Here is our new problem.
\begin{align*}\frac{1}{\cancel{4}}\times \frac{\cancel{400}}{1}=\frac{1}{1}\times \frac{100}{1}=100\end{align*}
Our answer is that 25% of 400 is 100. The exact number would have been 100 gummy bears.
“Wow!” thought Taylor, “That is a lot of candy. I hope they remember to brush their teeth.”
Vocabulary
Here are the vocabulary words in this lesson.
- Ratio
- the comparison of two quantities. Ratios can be written in fraction form, using a colon or with the word “to”.
- Percent
- a part of a whole out of 100. It is written using a % sign.
- Proportion
- two equal ratios form a proportion.
- Improper Fraction
- a fraction greater than one where the numerator is larger than the denominator.
Technology Integration
James Sousa, Introduction to Percent
James Sousa, Example 1: Relating Fractions, Decimals, and Percents
James Sousa, Example 2: Relating Fractions, Decimals, and Percents
Other Videos:
- http://www.mathplayground.com/mv_percents.html – This is a basic video by Brightstorm about percents and about finding percents using a proportion.
Time to Practice
Directions: Write each percent as a ratio with a denominator of 100.
1. 10%
2. 6%
3. 22%
4. 41%
5. 33%
6. 70%
7. 77%
8. 19%
Directions: Write each percent as a fraction in simplest form.
9. 12%
10. 10%
11. 5%
12. 25%
13. 40%
14. 60%
15. 90%
Directions: Write each fraction as a percent.
16. \begin{align*}\frac{1}{2}\end{align*}
17. \begin{align*}\frac{1}{4}\end{align*}
18. \begin{align*}\frac{3}{4}\end{align*}
19. \begin{align*}\frac{11}{100}\end{align*}
20. \begin{align*}\frac{1}{5}\end{align*}
21. \begin{align*}\frac{4}{8}\end{align*}
22. \begin{align*}\frac{17}{100}\end{align*}
23. \begin{align*}\frac{125}{100}\end{align*}
24. \begin{align*}\frac{250}{100}\end{align*}
25. \begin{align*}\frac{233}{100}\end{align*}
Directions: Use fraction multiplication to find each percent of a number.
26. 10% of 25
27. 20% of 30
28. 25% of 80
29. 30% of 90
30. 75% of 200