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# Proportions to Find Percent, P

## Use proportions to cross - multiply and find percents.

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Proportions to Find Percent, p

Max went berry picking. He picked both raspberries and blackberries, but he had only one pail. Max counts 24 raspberries and 11 blackberries in the scoop he takes from the pail. What percent of Max’s is raspberries and what percent is blackberries?

In this concept, you will learn to use proportions to find the percent, \begin{align*}p\end{align*}.

### Using Proportions to Find the Percent, p

A percent is a part of a whole out of 100.

You can write a percent as a fraction with a denominator of 100. You can use a proportion to figure out a percent.

Look at this proportion: \begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}.

You 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.”

Since percent statements always involve three numbers, given any two of these numbers, you can find the third number using the proportion.

Let’s look at an example.

What percent of 40 is 6?

First, notice that you 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*}You need to fill in the \begin{align*}a\end{align*} and the \begin{align*}b\end{align*} so you can solve for \begin{align*}p\end{align*}, the percent. The words “of 40” let you know that 40 is the base and 6 is the amount out of that base.

Here is your proportion to solve.

\begin{align*}\frac{6}{40}=\frac{p}{100}\end{align*}

Now you can use cross products and solve.

\begin{align*}\begin{array}{rcl} 40p &=& 600 \\ p &=& 15 \end{array} \end{align*}

The answer is 6 is 15% of 40.

Let’s look at another example.

What percent of 300 is 40?

First, use the following percent proportion.

\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}

Next, fill in the given information.

\begin{align*}\frac{40}{300}=\frac{p}{100}\end{align*}

Then, cross-multiply and divide.

\begin{align*}\begin{array}{rcl} 300p &=& 4,000 \\ p &=& 13.3 \% \\ \end{array}\end{align*}

The answer is 40 is 13.3% of 300.

### Examples

#### Example 1

Earlier, you were given a problem about Max and his berries.

24 of his scoop were raspberries and 11 were blackberries. What percent of the scoop were raspberries and what percent were blackberries?

First, the total number of berries in Max’s handful is 24 + 11 = 35.

Next, let’s figure out the percents.

You can start with the raspberries. There are 24 raspberries out of 35 berries in Max’s hand. There is your first proportion.

\begin{align*}\frac{24}{35}=\frac{p}{100}\end{align*}

Now you need to find the percent. You can do so by cross-multiplying and then dividing.

\begin{align*}\begin{array}{rcl} 35p &=& 2,400 \\ p &=& 68.57 \% \end{array}\end{align*}

The answer is 68.57% of Max’s handful berries is raspberries.

Next, let’s look at the blackberries. 11 out of 35 berries are blackberries.

\begin{align*} \begin{array}{rcl} \frac{11}{35} &=& \frac{p}{100} \\ 35p &=& 1,100 \\ p &=& 31.43 \% \end{array}\end{align*}

The answer is 31.43% of Max’s handful of berries is blackberries.

#### Example 2

Jeremy has 25 marbles and 12 of them are cat’s eye marbles. What percent of his marbles are cat’s eye marbles?

You 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*}. You need to find the percent \begin{align*}(p)\end{align*}.

\begin{align*}\begin{array}{rcl} \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{array}\end{align*}

Since \begin{align*}p = 48\end{align*}, the fraction you are looking for is \begin{align*}\frac{48}{100}\end{align*}, which is 48%.

The answer is that 48% of Jeremy’s marbles are cat’s eye marbles.

#### Example 3

What percent of 20 is 2?

First, you can use the following percent proportion.

\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}

Next, you can fill in the given information.

\begin{align*}\frac{2}{20}=\frac{p}{100}\end{align*}Now you can cross multiply and divide.

\begin{align*}\begin{array}{rcl} 20p &=& 200 \\ p &=& 10 \% \end{array}\end{align*}

The answer is 2 is 10% of 20.

#### Example 4

What percent of 30 is 6?

First, you can use the following percent proportion.

\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}

Next, you can fill in the given information.

\begin{align*}\frac{6}{30}=\frac{p}{100}\end{align*}
Now you can cross multiply and divide.

\begin{align*}\begin{array}{rcl} 30p &=& 600 \\ P &=& 20 \% \end{array}\end{align*}

The answer is 6 is 20% of 30.

#### Example 5

What percent of 45 is 15?

First, you can use the following percent proportion.

\begin{align*}\frac{a}{b}=\frac{p}{100}\end{align*}Next, you can fill in the given information.

\begin{align*}\frac{15}{45}=\frac{p}{100}\end{align*}

Now you can cross multiply and divide.

\begin{align*}\begin{array}{rcl} 45p &=& 1,500 \\ p &=& 33.3 \% \\ \end{array}\end{align*}

The answer is 15 is 33.3% of 45.

### Review

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?
11. What percent of 90 is 12?
12. What percent of 75 is 25?
13. What percent of 60 is 12?
14. What percent of 50 is 40?
15. What percent of 88 is 11?

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