6.2: Percents and Proportions
Introduction
The Mystery Bag of Candy
Taylor’s younger brother Max decided to visit her at the candy store. Max is only seven and can be a handful sometimes, so while Taylor loves to see him, she was a little hesitant to have him in the shop. Plus, what seven year old doesn’t love candy?
Taylor gave Max a small bag to put some candy in. She figured he would take a few pieces, but ended up with a whole bunch of candy.
“How many did you take?” Taylor asked him, looking in the bag.
“I took 40 pieces,” Max said, grinning. “I won’t eat it all now. I will save some for later.”
Taylor looked into the bag. There were 10 candy canes, 16 peanut butter cups and a whole bunch of jelly beans.
She gave Max the bag and watched him walk away chewing.
“I hope I don’t get into trouble for this,” Taylor murmured to herself.
Think about the candy in Max’s bag. It makes for some great math problems. If there are 40 pieces of candy, what percent of Max’s bag is made up of candy canes? What percent of his bag is made up of peanut butter cups?
Once you know this, you will know what percent of the bag is made up of jelly beans. Given that percent, how many jelly beans are in the bag?
All of these questions and ones like them can be answered when you know how to figure out percents using proportions. In this lesson you will learn how to figure out each percent and finally figure out how many jelly beans Max put into his bag.
What You Will Learn
In this lesson, you will execute the following skills.
- Use the proportion \begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*} to find the percent, \begin{align*}p\end{align*}.
- Use the proportion to find part \begin{align*}a\end{align*}.
- Use the cross products property of proportions to find the base \begin{align*}b\end{align*}.
- Solve real-world problems involving percent using proportions.
Teaching Time
I. Use the Proportion \begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*} to Find the Percent \begin{align*}p\end{align*}
First, let’s review what a percent is. A percent is a part of a whole out of 100. We can write a percent as a fraction with a denominator of 100.
We can use a proportion to figure out a percent.
Look at this proportion, \begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}.
We can say that “\begin{align*}a\end{align*} is the amount to the base, \begin{align*}b\end{align*}, and \begin{align*}p\end{align*} is the percent out of 100.”
That may sound a little tricky, but if you get used to thinking in this way, you will find that this is very helpful when solving for one of the missing parts of the proportion. Since percent statements always involve three numbers which may change (the 100 is always the same), given any two of these numbers, we can find the third number using the proportion.
Example
What percent of 40 is 6?
First, thing to notice is that we are looking for the percent. So the \begin{align*}p\end{align*} over 100 is going to stay the same in the proportion.
\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}
We need to fill in the \begin{align*}a\end{align*} and the \begin{align*}b\end{align*} so we can solve for \begin{align*}p\end{align*}, the percent. The words “of 40” lets us know that 40 is the base and 6 is the amount.
Here is our proportion to solve.
\begin{align*}\frac{6}{40}=\frac{p}{100}\end{align*}
Now we can use cross products and solve.
\begin{align*}40p & = 600\\ p & = 15\end{align*}
Our answer is 15%.
Let’s look at another example.
Example
Jeremy has 25 marbles and 12 of them are cat’s eye marbles. What percent of his marbles are cat’s eye marbles?
We can think of this problem as “What percent of 25 is 12?”
12 is the amount \begin{align*}(a)\end{align*} and 25 is the base \begin{align*}(b)\end{align*}. We need to find the percent \begin{align*}(p)\end{align*}.
\begin{align*}\frac{a}{b}& =\frac{p}{100}\\ \frac{12}{25}& =\frac{p}{100}\\ 25p& =1,200\\ \frac{25p}{25}& =\frac{1,200}{25}\\ p& =48\end{align*}
Since \begin{align*}p = 48, our answer is \frac{48}{100}\end{align*}, which is 48%.
The answer is that 48% of Jeremy’s marbles are cat’s eye marbles.
6D. Lesson Exercises
Use the proportion to find each percent.
- What percent of 20 is 2?
- What percent of 30 is 6?
- What percent of 45 is 15?
Take a few minutes to check your work with a neighbor.
II. Use the Proportion to Find Part \begin{align*}a\end{align*}
In the last section we were given the amount and the base. The key words “of” let us know that we had a base and the other number naturally became the amount since we were looking for the percent.
Sometimes, you will be given the base and the percent and you will need to find the amount. You can use the same proportion to figure this out. You just need to fill in the numbers in the correct places and solve.
Let’s look at an example.
Example
What is 25% of 75?
To figure this out, let’s look at what we have been given for information. First, we know the percent so we can fill in that half of the proportion.
\begin{align*}\frac{25}{100}\end{align*}
We have been given the base. “Of 75” lets us know that this is the base. The amount is missing. That is our unknown.
\begin{align*}\frac{a}{75}\end{align*}
Now let’s put it together as a proportion and use cross products to solve for \begin{align*}a\end{align*}.
\begin{align*}\frac{a}{75}& =\frac{25}{100}\\ 100a & = 25(75)\\ 100a & = 1875\\ a & = 18.75\end{align*}
Notice that we moved the decimal point two places to the left when we divided by 100.
Our answer is 18.75 or \begin{align*}18\frac{3}{4}\end{align*}.
Notice that there isn’t a percent sign here because we were looking for an amount, not a percent! This can trip up some students so always pay attention to what you are looking for!
Example
Mr. Green bought both vegetable and flowering plants for his garden. He bought 40 plants and 35% were flowering plants. How many flowering plants did he buy?
We can think of this problem as “What is 35% of 40?”
40 is the base \begin{align*}(b)\end{align*} and 35 is the percent \begin{align*}(p)\end{align*}. We need to find the amount \begin{align*}(a)\end{align*}.
First, let’s set up the proportion.
\begin{align*}\frac{a}{40}=\frac{35}{100}\end{align*}
Now we can use cross products to solve for the amount.
\begin{align*}100a & = 35(40)\\ 100a & = 1400\\ a & = 14\end{align*}
Mr. Green bought 14 flowering plants.
6E. Lesson Exercises
- What is 20% of 30?
- What is 16% of 50?
- What is 22% of 80?
Take a few minutes to check your work with friend. Are your answers accurate?
III. Use the Cross Products Property of Proportions to Find Part \begin{align*}b\end{align*}
You have solved problems where you needed to find the percent. You have solved problems where you needed to find the amount. Now we are going to use cross products to solve for the base.
When you see the words “OF WHAT NUMBER?” you know that you are going to be solving for \begin{align*}b\end{align*}, the base.
Let’s look at an example.
Example
33 is 15% of what number?
Remember that the number following the word “of” is the base. Since there is no number there, we need to find the base \begin{align*}(b)\end{align*}. 33 is the amount \begin{align*}(a)\end{align*} and 15 is the percent \begin{align*}(p)\end{align*}.
\begin{align*}\frac{a}{b}& =\frac{p}{100}\\ \frac{33}{b}& =\frac{15}{100}\\ 15b& =3,300\\ \frac{15b}{15}& =\frac{3,300}{15}\\ b& =220\end{align*}
Our answer is 220.
Example
Six students in Miss Lang’s third period math class got A’s on their math test. This was 24% of the class. How many students are in Miss Lang’s third period math class?
We can think of this problem as “6 is 24% of what number?” First, let’s set up the proportion.
\begin{align*}\frac{6}{b}=\frac{24}{100}\end{align*}
Next, we use cross products to solve for \begin{align*}b\end{align*}.
\begin{align*}24b& =600\\ b& =600 \div 24\\ b & = 25\end{align*}
There are 25 students in the third period math class.
6F. Lesson Exercises
Practice finding the base in the following problems.
- 6 is 25% of what number?
- 12 is 8% of what number?
- 22 is 11% of what number?
Check your answers with a neighbor. Correct any errors and then continue with the lesson.
IV. Solve Real-World Problems Involving Percents Using Proportions
When solving percent problems, you must determine whether you want to find the amount, the base, or the percent. Think about the following key words.
Write these key words in your notebook and then continue with the first example.
Example
There are 25 students attending the field trip to the science museum. 15 of the students are girls. What percent of the attendees are girls?
First, notice that the word “percent” is there without a number. We are looking for the percent. Therefore, we have been given the amount, 15 and the base 25. 15 out of 25 helps us to identify this part of the proportion.
\begin{align*}\frac{15}{25}=\frac{p}{100}\end{align*}
Next, use cross products to solve.
\begin{align*}25p & = 1500\\ p & = 1500 \div 25\\ p & = 60\%\end{align*}
60% of the class is girls.
Now let’s go back and apply what we have learned to our introductory problem.
Real Life Example Completed
The Mystery Bag of Candy
Here is the original problem once again. Reread it and use what you have learned about percents to figure out the questions at the end of the problem.
Taylor’s younger brother Max decided to visit her at the candy store. Max is only seven and can be a handful sometimes, so while Taylor loves to see him, she was a little hesitant to have him in the shop. Plus, what seven year old doesn’t love candy?
Taylor gave Max a small bag to put some candy in. She figured he would take a few pieces, but ended up with a whole bunch of candy.
“How many did you take?” Taylor asked him, looking in the bag.
“I took 40 pieces,” Max said, grinning. “I won’t eat it all now. I will save some for later.”
Taylor looked into the bag. There were 10 candy canes, 16 peanut butter cups and a whole bunch of jelly beans.
She gave Max the bag and watched him walk away chewing.
“I hope I don’t get into trouble for this,” Taylor murmured to herself.
Think about the candy in Max’s bag. It makes for some great math problems. If there are 40 pieces of candy, what percent of Max’s bag is made up of candy canes? What percent of his bag is made up of peanut butter cups?
Once you know this, you will know what percent of the bag is made up of jelly beans. Given that percent, how many jelly beans are in the bag?
First, let’s figure out the percents.
We start with candy canes. There are 10 out of 40. There is our first ratio, and now we need to find the percent.
\begin{align*}\frac{10}{40}& =\frac{p}{100}\\ p & = 25\%\end{align*}
25% of the bag is candy canes.
Next, let’s look at the peanut butter cups. 16 out of 40 are peanut butter cups.
\begin{align*}\frac{16}{40}& =\frac{p}{100}\\ 40p & = 1600\\ p & = 40\%\end{align*}
40% of the bag is peanut butter cups.
Finally, we come to the jelly beans. Now we want to find out how many jelly beans are in the bag out of 40. What is the percent?
Well, we can find it if we think about the two other percents of candy in the bag. 25% is candy canes + 40% is peanut butter cups, so 65% of the bag is filled, leaving 35% for the jelly beans.
Now we can figure out the proportion and solve for the amount, \begin{align*}a\end{align*}.
\begin{align*}\frac{a}{40} & = \frac{35}{100}\\ 100a & = 35(40)\\ 100a & = 1400\\ a & = 14\end{align*}
There are 14 jelly beans in the bag.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Percent
- a part of a whole out of 100
- Proportion
- formed by two equal ratios or two equivalent fractions
Technology Integration
James Sousa, Solving Percent Problems Using the Percent Proportion
James Sousa, Example of Solving a Percent Problem Using a Percent Proportion
James Sousa, Another Example of Solving a Percent Problem Using a Percent Proportion
James Sousa, Another Example of Solving a Percent Problem Using a Percent Proportion
Other Videos:
- http://www.mathvids.com/lesson/mathhelp/986-applications-of-percent – This is a video that shows some applications of percents. It does not use proportions, but is still useful.
- http://teachertube.com/viewVideo.php?video_id=31925&title=percents_and_proportions – This is a video in a power point presentation style that uses proportions and percents. It also includes a brief review of percent concepts.
Time to Practice
Directions: Find each percent using a proportion.
1. What percent of 18 is 9?
2. What percent of 20 is 4?
3. What percent of 28 is 7?
4. What percent of 30 is 6?
5. What percent of 9 is 3?
6. What percent of 36 is 18?
7. What percent of 40 is 8?
8. What percent of 48 is 12?
9. What percent of 50 is 30?
10. What percent of 80 is 60?
Directions: Find each missing amount.
11. What number is 25% of 18?
12. What number is 10% of 20?
13. What number is 45% of 16?
14. What number is 20% of 44?
15. What number is 30% of 100?
16. What number is 25% of 60?
17. What number is 40% of 80?
18. What number is 40% of 60?
19. What number is 50% of 120?
20. What number is 5% of 12?
Directions: Find each missing base.
21. 5 is 10% of what number?
22. 7 is 10% of what number?
23. 10 is 20% of what number?
24. 16 is 40% of what number?
25. 8 is 25% of what number?
26. 14 is 50% of what number?
27. 25 is 5% of what number?
28. 4 is 80% of what number?
29. 18 is 25% of what number?
30. 9 is 3% of what number?