6.4: The Percent Equation
Introduction
A Shopping Sale
On Saturday, Taylor’s parents decided to have a big sale at the candy store. They advertised in the local newspapers and on Saturday morning they were ready for a big crowd. Taylor was responsible for counting all of the people who came into the store between 10 am when they opened and Noon.
Taylor sat on a stool at the front of the store and counted people as they came in. She was pleasant and enjoyed seeing all of the different shoppers. She only counted adults or teenagers, not little kids with their parents, since most parents would be doing the buying for the little children. Most people bought something, but a few just came in to look around.
At lunch, she compared her data with her brother Henry, who was helping with the sales at the cash register.
“How many people did you count?” Henry asked.
“I counted 30 adults. How many sales were there?”
“There were 24 sales.”
“That’s not bad. Most people bought something. What percent ended up buying something?” Taylor asked Henry.
Just as Henry was about to answer her, he got called back into the store. Taylor sat there eating her sandwich, thinking about the percentages.
What percent of the people who came into the store made a purchase?
What percent did not purchase anything?
In this lesson, you will learn how to use an equation to solve problems like this one. Pay attention and at the end you will be able to help Taylor with the problem.
What You Will Learn
By the end of this lesson you will be able to demonstrate the following skills.
- Use the Percent Equation \begin{align*}a = p\%\end{align*} \begin{align*}(b)\end{align*} to find part \begin{align*}a\end{align*}.
- Use the Percent Equation to find the percent \begin{align*}p\end{align*}.
- Use the Percent Equation to find the base \begin{align*}b\end{align*}.
- Solve real-world problems involving percents using equations.
Teaching Time
I. Use the Percent Equation \begin{align*}a = p\% \ (b)\end{align*} to find part \begin{align*}a\end{align*}
Think about the proportion that you just learned to find the percent of a number.
\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}
When we used this proportion in problem solving, we multiplied \begin{align*}b\end{align*} and \begin{align*}p\end{align*} and \begin{align*}a\end{align*} by 100. Then we divided the product of \begin{align*}b\end{align*} and \begin{align*}p\end{align*} by 100. Look at this example.
Example
What is 35% of 6?
First, we fill in the proportion.
\begin{align*}\frac{a}{6}=\frac{35}{100}\end{align*}
Next, we multiply and solve for \begin{align*}a\end{align*}.
\begin{align*}100a & = 35(6)\\ 100a & = 210\\ a & = 2.1\end{align*}
Notice that by dividing by 100, we moved the decimal place two places.
Hmmmm. This is the same two places that the percent is represented by. This means that if we changed the percent to a decimal FIRST, we could skip a step and use an equation to find the missing value.
Take a look at the same problem.
Example
What is 35% of 6?
First, change 35% to a decimal.
\begin{align*}35\% = .35\end{align*}
Now we multiply it times 6 (the base), and find \begin{align*}a\end{align*} (the amount). Look at this equation.
\begin{align*}a & = p\%(b)\\ a & = .35(6) \\ a & = 2.1 \end{align*}
Notice that we got the same answer as when we used the proportion. It just simplifies the process.
Let’s look at another example.
Example
What is 25% of 50?
First, change 25% to a decimal.
\begin{align*}25\% = .25\end{align*}
Now use the equation.
\begin{align*}a & = p\%(b)\\ a & =.25(50)\\ a & =12.5 \end{align*}
The answer is 12.5.
6J. Lesson Exercises
Use the equation to find each amount. Include decimals in your answer.
- What is 20% of 16?
- What is 5% of 40?
- What is 15% of 65?
Take a few minutes to check your work.
Write down the equation and how to use it in your notebook. Then continue on to the next section.
II. Use the Percent Equation to Find the Percent \begin{align*}p\end{align*}
We can use the same equation to find the percent of a number. When you look at this problem, you will be given the amount and the base, but the percent will be a mystery. Let’s think about how to do this.
Here is the equation once again.
\begin{align*}a=p\% b\end{align*}
Now let’s look at how to apply this with an example.
Example
What percent of 70 is 14?
First, we have the \begin{align*}a\end{align*} and the \begin{align*}b\end{align*} and we are missing the percent. We can fill the given information into the formula.
\begin{align*}a& =pb \\ 14& =p70 \end{align*}
Now we can figure out what number times seventy is 14. Since the operation is multiplication, we can do the opposite and divide. This is using the inverse operation.
\begin{align*}\frac{14}{70}& =p \\ .2 & = p\end{align*}
Now we change the decimal to a percent.
Our answer is 20%.
Don’t forget to change the decimal to a percent when you are looking for a percent!!
6K. Lesson Exercises
Use the equation to find the missing percent.
- What percent of 50 is 15?
- What percent of 80 is 25?
- What percent of 100 is 12?
Take a few minutes to check your work with a friend.
III. Use the Percent Equation to Find the Base \begin{align*}b\end{align*}
When you know the percent and the amount, you can use the equation to find the base, \begin{align*}b\end{align*}. Let’s look at an example.
Example
3 is 50% of what number?
Remember, that the words “of what number” let you know that you are looking for the base.
Let’s fill the other given information into our equation. First, let’s change the percent to a decimal.
\begin{align*}50 = .50\end{align*}
Now let’s work on the equation.
\begin{align*}3 =.50b\end{align*}
Next, we can use the inverse operation and solve for \begin{align*}b\end{align*}.
\begin{align*}\frac{3}{.50}& =b\\ 6 & = b\end{align*}
The answer is 6.
Let’s look at another example.
Example
9 is 75% of what number?
First, let’s change 75% to a decimal.
75% = .75
Next, let’s fill the given values into the equation.
\begin{align*}9=.75b \end{align*}
Now we can solve for \begin{align*}b\end{align*} using the inverse operation.
\begin{align*}\frac{9}{.75}& =b\\ 12 & = b\end{align*}
Our answer is 12.
6L. Lesson Exercises
Find the base for each problem.
- 8 is 20% of what number?
- 15 is 30% of what number?
- 22 is 40% of what number?
Take a few minutes to check your work with a partner.
IV. Solve Real-World Problems Involving Percents Using Equations
When solving percent problems, you must determine whether you want to find the amount, the base, or the percent.
Example
The Cougars basketball team won 21 games. If that number is 70% of the games that the team played, how many games did they play?
Think: 70% is the percent. 21 is the amount of games that they won. You want to find the base.
First, we can take 70% and write it as a decimal.
\begin{align*}70\% = .70\end{align*}
Next, we can write the given information into an equation and solve it using the inverse operation.
\begin{align*}21 & = .70b\\ \frac{21}{.70}& =b\\ 30 & = b\end{align*}
The team played 30 games in all.
Example
In a hockey game, the Falcons attempted 40 shots on goal and 15% of shots were goals. How many goals did the team make?
Think: 40 shots on goal is the base. 15% is the percent. You want to find the amount of goals made.
First, write the percent as a decimal.
\begin{align*}15\% = .15\end{align*}
Now write the given information into the equation and solve.
\begin{align*}a & = .15(40)\\ a & = 6\end{align*}
The team made 6 goals.
Now let’s go back to the candy store and help Taylor figure out the percentage of shoppers.
Real Life Example Completed
A Shopping Sale
Here is the original problem once again. Use what you have learned to solve this problem.
On Saturday, Taylor’s parents decided to have a big sale at the candy store. They advertised in the local newspapers and on Saturday morning they were ready for a big crowd. Taylor was responsible for counting all of the people who came into the store between 10 am when they opened and Noon.
Taylor sat on a stool at the front of the store and counted people as they came in. She was pleasant and enjoyed seeing all of the different shoppers. She only counted adults or teenagers, not little kids with their parents, since most parents would be doing the buying for the little children. Most people bought something, but a few just came in to look around.
At lunch, she compared her data with her brother Henry, who was helping with the sales at the cash register.
“How many people did you count?” Henry asked.
“I counted 30 adults. How many sales were there?”
“There were 24 sales.”
“That’s not bad. Most people bought something. What percent ended up buying something?” Taylor asked Henry.
Just as Henry was about to answer her, he got called back into the store. Taylor sat there eating her sandwich, thinking about the percentages.
What percent of the people who came into the store made a purchase?
What percent did not purchase anything?
First, notice that we are looking for a percent.
30 is the base and 24 is the amount. Let’s fill in the given information and solve for the percent that did purchase at the store.
\begin{align*}24& =30p\\ \frac{24}{30}& =p\\ .8 & = p\\ 80\% & = p\end{align*}
80% purchased something at the store.
What percent did not purchase something?
100% - 80% = 20%
20% did not purchase anything at the store.
Vocabulary
- Inverse Operation
- the opposite operation.
- Percent
- a part of a whole calculated out of 100.
- Amount
- the part of the whole that “is” out of a base. “Is” is a key word showing amount.
- Base
- the part of the whole that the amount is out of. The phrase “Of what number” let you know that you are looking for the base.
Technology Integration
Khan Academy Taking Percentages
James Sousa, Solving Percent Problems Using the Percent Equation
James Sousa, Example 1: Solve a Percent Problem Using the Percent Equation
James Sousa, Example 2: Solve a Percent Problem Using the Percent Equation
James Sousa, Example 3: Solve a Percent Problem Using the Percent Equation
Other Videos:
- http://www.wonderhowto.com/how-to-solve-percent-equations-315657/ – This is a video on the percent equation and how to solve problems using the percent equation.
Time to Practice
Directions: Use the percent equation to find each amount.
1. What is 20% of 18?
2. What is 10% of 30?
3. What is 5% of 90?
4. What is 12% of 27?
5. What is 18% of 30?
6. What is 50% of 88?
7. What is 75% of 12?
8. What is 75% of 90?
9. What is 22% of 40?
10. What is 25% of 60?
Directions: Use the percent equation to find each percent. You may round your answer to the nearest whole percent if necessary.
11. What percent of 18 is 9?
12. What percent of 20 is 10?
13. What percent of 60 is 15?
14. What percent of 80 is 20?
15. What percent of 25 is 10?
16. What percent of 70 is 35?
17. What percent of 36 is 18?
18. What percent of 100 is 25?
19. What percent of 10 is 5?
20. What percent of 98 is 90?
Directions: Use the percent equation to find each base.
21. 5 is 10% of what number?
22. 15 is 30% of what number?
23. 18 is 20% of what number?
24. 12 is 50% of what number?
25. 15 is 40% of what number?
26. 14 is 20% of what number?
27. 80 is 25% of what number?
28. 60 is 30% of what number?
29. 45 is 40% of what number?
30. 16 is 25% of what number?