# 5.3: Using the Percent Equation

**At Grade**Created by: CK-12

## Introduction

*Football Figures*

“You know,” Cameron started to say in study hall. “I think we should shoot for 55 or 60% attendance at football games and not just 50%. I mean think about it, we should all go and support the team. After all, some of us hope to play football at the high school someday!”

The other four kids at the table looked up from their work to discuss the suggestion. Carla was the first to speak.

“I think that’s a good point. I mean if we attend their games, maybe other middle school kids will do the same thing when we are at the high school.”

“Yes, but some kids need a ride or have other things to do,” Jeremy argued.

“Well I’m not saying everyone. I am saying 55 or 60 %,” Cameron said.

“How many is that?” Jeremy asked.

“I can figure it out easy,” Cameron said.

On his paper he wrote these equations:

What is 55% of 380?

What is 60% of 380?

**Before Cameron solves these problems, let’s look at what he wrote. Cameron used the a statement that could be written into a “percent equation”. We can use the percent equation instead of a proportion. The percent equation can be helpful when you look for a percent, a base or a part of the base. Let’s look at how we can use the percent equation before we solve this problem.**

*What You Will Learn*

By the end of this lesson, you will be able to complete the following skills.

- Use the Percent Equation \begin{align*}a = p \% \cdot b\end{align*} to find part \begin{align*}a\end{align*}.
- Use the Percent Equation to find 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*}\underline{a = p \% \cdot b}\end{align*} to find part \begin{align*}\underline{a}\end{align*}**

In the previous lesson, we used the proportion \begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*} to solve percent problems. For most every problem, we solved the proportion using cross products. We can also solve problems by using an equation. In this lesson, we will use the same proportion to create a different kind of equation that will help us solve percent problems differently.

**When we solve the proportion \begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}, we use cross products to find the missing variable. However, even if we leave it in terms of the variables, we can still use cross multiply.**

\begin{align*}\frac{a}{b} &= \frac{p}{100}\\ 100 a &= pb\\ a &= \frac{pb}{100}\\ a &= .01pb\end{align*}

If we change the percent to a decimal by moving the decimal point two places to the left, then there is no need to multiply \begin{align*}p\end{align*} by .01 as we will have already accounted for the coefficient of .01 by moving the decimal point.

**Okay, let’s go through it again. Look at what we just wrote.** We wrote the same thing we just didn’t include values. The variables stayed and we multiplied them. **The key is that if we change the percent to a decimal, then all we have to do is to multiply it by the base and we will be able to figure out the value of \begin{align*}a\end{align*}.**

Let’s look at an example.

Example

*What is 85% of 90?*

**To figure this out, first we change the 85% into a decimal. “OF” is a key word meaning multiply, so we multiply the decimal .85 times 90.**

\begin{align*}.85 \times 90 = 76.5\end{align*}

**This is our answer.**

*Some of you may find that this is much simpler than using a proportion! Either way is correct just be sure that you know what are looking for with each equation.*

Example

What is 7% of 900?

**First, let’s change 7% into a decimal.**

7% = .07

**Next, we multiply it by 900. Notice the key word “of” which means that we multiply.**

\begin{align*}900 \times .07 = 63\end{align*}

**Our answer is 63.**

II. **Use the Percent Equation to Find the Percent \begin{align*}\underline{p}\end{align*}**

In the last section we were looking for the part of the whole that equaled the given percent. Now we are going to use the percent equation to find the percent. This means that we will know the value of the part and the whole, \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, and we will be looking for the percent.

Example

What percent of 32 is 18?

**Let’s look at this problem in some detail. First, we know that we are looking for a percent. We want to use the percent equation to solve this.**

We know that the percent is what is missing, so we can make that \begin{align*}p\end{align*}. Then we know that “of” means multiply. The word “is” means equals. Now we can write the equation.

\begin{align*}32p = 18\end{align*}

Next, we solve for the value of \begin{align*}p\end{align*} by dividing both sides by 32.

\begin{align*}\frac{32p}{32} &= \frac{18}{32}\\ p &= .5625\end{align*}

Now this is the decimal, so we need to convert it to a percent.

\begin{align*}p = 56.25 \%\end{align*}

**This is the answer.**

Example

10 is what percent of 12?

**This problem is worded differently, but we are still looking for a percent. Notice that the “is” is in a different spot, but that still means equals. Let’s write the equation.**

\begin{align*}10 = p12\end{align*}

**Or**

\begin{align*}10 = 12p\end{align*}

Next, we divide both sides by 12 to solve for the value of \begin{align*}p\end{align*}.

\begin{align*}\frac{10}{12} &= \frac{12p}{12}\\ .833 &= p\end{align*}

This is the decimal once again, so we need to convert it to a percent by moving the decimal point.

\begin{align*}83.3 \% = p\end{align*}

**This is the answer.**

*Take a few minutes to copy these key words down in your notebook. Include an example with your notes.*

III. **Use the Percent Equation to Find the Base \begin{align*}\underline{b}\end{align*}**

Sometimes, you will know the percent and the part of the ratio, or part \begin{align*}a\end{align*}, but you will need to find the whole or the base, \begin{align*}b\end{align*}. When this happens, you can use the same key words as before and simply figure out the base by using the percent equation. Let’s look at an example.

Example

78 is 65% of what number?

**Here we know that the word “is” means equals. The numbers may be in a different location, but just pay attention to the key words and you will know what to do. Notice that we have been given the percent and we are missing the “of what number” that is the value of the base. Let’s write the equation.**

\begin{align*}78 = 65\% b\end{align*}

To work with the 65%, it makes sense to convert it to a decimal. We do this by dropping the percent sign and moving the decimal two places to the left.

\begin{align*}78 = .65b\end{align*}

Now we can solve it for the value of \begin{align*}b\end{align*}. Divide both sides of the equation by .65.

\begin{align*}\frac{78}{.65} &= \frac{.65b}{.65}\\ 120 &= b\end{align*}

**This is the answer.**

Example

11 is 77% of what number?

**Once again, pay attention to the key words. You can see that we are once again going to be looking for the value of the base. Let’s write the equation.**

\begin{align*}11 = 77 \% b\end{align*}

Convert the percent to a decimal and solve.

\begin{align*}11 &= .77b\\ \frac{11}{.77} &= b\\ 14.28 &= b\end{align*}

**In this example, you could round to the nearest hundredths place as we did here. Sometimes, you may be asked to round to the nearest tenths place. In that case, the answer would have been 14.3.**

IV. **Solve Real – World Problems Involving Percent using Equations**

Because percents are all around us in the real – world, you will need to know how to use the percent equation to solve many different types of practical problems. Remember the key words that we talked about.

“Of” means multiply

“what percent” means you are looking for a percent-you will need to convert the decimal to a percent at the end of the problem.

“Is” means equals

“Of what number” means the base is missing-it means you look for the whole.

*Write these key words down in your notebooks.*

Now let’s look at applying what we have learned to solve problems.

Example

By mid-September, 50% of the trees lose their leaves. If 850 trees in a grove lost their leaves, how many trees are there in all?

**Let’s start by breaking apart this problem. We have a percent, so we know that we won’t be looking for the percent. We know that 850 trees in a grove lost their leaves, but we don’t know the total number of trees in the grove. The total could be thought of as the whole and this is the base. We are going to be looking for the base.**

**Let’s write the equation.**

\begin{align*}850 = .50b\end{align*}

Now we solve by dividing both sides of the equation by .50.

\begin{align*}\frac{850}{.50} &= \frac{.50b}{.50}\\ 1700 &= b\end{align*}

**There are 1700 trees in the grove.**

Example

In 2007, a local football team won 14 of the 16 regular season games that they played. What percent did they win?

**First, let’s look at which information we have been given. We know that 14 out of 16 were won. The fourteen is the number of games that is the part. The whole of the games is 16 this is the base. We need to find the percent.**

We could say we want to know what percent 14 is of 16. Let’s write the equation.

\begin{align*}14 = 16p\end{align*}

Divide both sides by 16.

\begin{align*}\frac{14}{16} &= p\\ .875 &= p\end{align*}

Now we convert the decimal into a percent by moving the decimal point.

**87.5% is the answer.**

**We can apply this information about percents to the problem from the introduction. Let’s go back and look at that problem once again.**

## Real-Life Example Completed

*Football Figures*

**Here is the original problem once again. First, reread it. Then write two percent equations for each question and solve each. There are four parts to your answer.**

“You know,” Cameron started to say in study hall. “I think we should shoot for 55 or 60% attendance at football games and not just 50%. I mean think about it, we should all go and support the team. After all, some of us hope to play football at the high school someday!”

The other four kids at the table looked up from their work to discuss the suggestion. Carla was the first to speak.

“I think that’s a good point. I mean if we attend their games, maybe other middle school kids will do the same thing when we are at the high school.”

“Yes, but some kids need a ride or have other things to do,” Jeremy argued.

“Well I’m not saying everyone. I am saying 55 or 60 %,” Cameron said.

“How many is that?” Jeremy asked.

“I can figure it out easy,” Cameron said.

On his paper he wrote these equations:

What is 55% of 380?

What is 60% of 380?

*Remember, there are four parts to your answer!*

*Solution to Real – Life Example*

**Now let’s take the two questions and write two equations that we can use to solve those equations.**

**What is 55% of 380? Becomes \begin{align*}x= .55(380)\end{align*}**

**What is 60% of 380? Becomes \begin{align*}x= .60(380)\end{align*}**

**Next, we solve each equation for the part of the whole.**

**55% of 380 = 209 students**

**60% of 380 = 228 students**

**These are our two answers.**

## Vocabulary

Here are the vocabulary words found in this lesson.

- Percent
- a part of a whole out of 100.

## Time to Practice

Directions: Solve each percent problem. You may round your answers to the nearest tenth when necessary.

- How much is 15% of 73?
- What is 70% of 5?
- What is 3% of 4 million?
- What is 18% of 30?
- What is 22% of 56?
- What is 19% of 300?
- What is 21% of 45?
- What is 34% of 250?
- What is 33% of 675?
- What is 30% of 455?
- What percent of 600 is 82?
- What percent of 18 is 17?
- 150 is what percent of 175?
- 200 is what percent of 450?
- 34 is what percent of 70?
- 12 is what percent of 88?
- 15 is what percent of 90?
- 230 is what percent of 600?
- 334 is what percent of 1000?
- 256 is what percent of 800?
- 23 is 9% of what number?
- 10 is 35% of what number?
- 580 is 82% of what number?
- A farmer works on a 3200 acre farm. If he plowed 600 acres, what percent of the farm did he plow?
- In a race across the desert, a buggy has driven 343 miles. This is 45% of the race. How long is the total race?