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# 5.8: Use the Percent Equation to Find the Percent

Difficulty Level: At Grade Created by: CK-12
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Practice Percent Equation to find Percent

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Are you a football fan? Take a look at this percent dilemma involving football.

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

Pay attention to this Concept and you will know how to solve this problem by the end of it.

### Guidance

Did you know that you can use the percent equation to find a percent? This means that we will know the value of the part and the whole, a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}, and we will be looking for the percent.

First, look at a proportion and how to go from a proportion to the percent equation.

We can use the proportion ab=p100\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*} to solve percent problems. For almost every problem, we solve the proportion using cross products.

We can also solve problems by using an equation. In this Concept, you will use the same proportion to create a different kind of equation that will help us solve percent problems differently.

ab100aaa=p100=pb=pb100=.01pb\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 p\begin{align*}p\end{align*} by .01 as we will have already accounted for the coefficient of .01 by moving the decimal point.

Now let's apply the percent equation.

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 p\begin{align*}p\end{align*}. Then we know that “of” means multiply. The word “is” means equals. Now we can write the equation.

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

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

32p32p=1832=.5625\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.

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

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.

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

Or

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

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

1012.833=12p12=p\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.

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

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

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.

#### Example A

18 is what percent of 20?

Solution: Missing \end{align*}\begin{align*}90%\end{align*}

#### Example B

5 is what percent of 300?

Solution: Missing \end{align*}\begin{align*}1.6%\end{align*}

#### Example C

60 is what percent of 400?

Solution: Missing \end{align*}\begin{align*}15%\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

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.

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

Divide both sides by 16.

1416.875=p=p\begin{align*}\frac{14}{16} &= p\\ .875 &= p\end{align*}

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

### Vocabulary

Percent
a part of a whole out of 100.

### Guided Practice

Here is one for you to try on your own.

33 is what percent of 50?

Solution

To figure this out, we can use the percent equation.

100a=pb\begin{align*}100a = pb\end{align*}

100(33)=p(50)\begin{align*}100(33)=p(50)\end{align*}

3300=50p\begin{align*}3300 = 50p\end{align*}

330050=p\begin{align*}\frac{3300}{50} = p\end{align*}

Missing \end{align*}\begin{align*}66%\end{align*}

### Practice

Directions: Solve each percent problem by using the percent equation. You may round when necessary.

1. What percent of 600 is 82?
2. What percent of 18 is 17?
3. 150 is what percent of 175?
4. 200 is what percent of 450?
5. 34 is what percent of 70?
6. 12 is what percent of 88?
7. 15 is what percent of 90?
8. 230 is what percent of 600?
9. 334 is what percent of 1000?
10. 2 is what percent of 8?
11. 55 is what percent of 1800?
12. 61 is what percent of 80?
13. 33 is what percent of 90?
14. 78 is what percent of 156?
15. 19 is what percent of 31?

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