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Proportions to Find Percent, P

Use proportions to cross - multiply and find percents.

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Use Proportions to Find Percents

Have you ever thought about voting and percents? Take a look at this dilemma.

A senator wants to start a program to encourage more people to vote in his state. In County A, 32,100 people voted. In neighboring County B, 57,800 people voted. Which county needs the program more?

Well, it depends on how many people are in each of the counties.

We can’t compare the polling rates unless you use a percent.

If we know that the first county has a population of 39,150 people and the second county has a population of 81,400 people, we can now find what percent of the people voted.

We are comparing the number of people who voted with the population of each county. We are actually going to find two percents here. Do you know how to do this? Pay attention and you will understand how to complete this task by the end of the Concept.


A percent is a part of a whole that represents a quantity out of 100. Fractions and decimals are also parts of a whole. Sometimes, you will be given information, but not a percent. You will need to know how to figure out the percent. Percents, fractions, decimals and proportions can all help to you solve problems and figure out percents.

You began using proportions to figure out a percent when writing fractions as percents. Remember that proportions involve comparing quantities.

A proportion is a comparison between two equal ratios.

Percents are also written to compare a quantity to 100.

Because both of these are comparing, we can use proportions to help us figure out a percent.

That is a great question.

First, we write the proportion using \begin{align*}a\end{align*} over \begin{align*}b\end{align*}.


This is equal to the percent which is out of 100.


Here is the proportion:

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

Now let's apply this proportion. Take a look at this dilemma.

15 out of 30 is what percent?

To work on this problem, first, we write a ratio comparing our given values to the missing percent.

\begin{align*}\frac{15}{30} = \frac{p}{100}\end{align*}

We know that fifteen is half of thirty, and 50 is half of 100.

Our answer is 50%.

Write each as a percent.

Example A

18 out of 50

Solution:  \begin{align*}36\%\end{align*}

Example B

22 out of 40

Solution:  \begin{align*}55\%\end{align*}

Example C

78 out of 80

Solution:  \begin{align*}97.5\%\end{align*}

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

For each county, we will use the proportion \begin{align*}\frac{a}{b} = \frac{p}{100}\end{align*} where \begin{align*}a\end{align*} is the number of people that voted and \begin{align*}b\end{align*} is the total population.

\begin{align*}& \text{County A} \qquad \qquad \qquad \text{County B}\\ & \ \frac{32100}{39150} = \frac{p}{100} \qquad \qquad \ \frac{57800}{81400} = \frac{p}{100}\\ & 39150p = 32100 \cdot 100 \quad 81400p = 57800 \cdot 100\\ & 39150p = 3210000 \qquad \ 81400p = 5780000\\ & \qquad \ \ p = 82 \% \qquad \qquad \qquad \ p=71 \%\end{align*}

In order to find the percent in each case, we used cross products as we would for any proportion. Now we can see that in County A, 82% of the people voted, while in County B only 71% of the people voted.

The senator should push the program more in County B.


two equal ratios form a proportion.
a part of a whole out of 100.

Guided Practice

Here is one for you to try on your own.

John ran 8 out of 9 miles. What percent of the total miles did he run?


To figure this out, let's first write a proportion so that we can figure out the percent.

\begin{align*}\frac{8}{9} = \frac{p}{100}\end{align*}

Now we can cross-multiply and divide.

\begin{align*}9p &= 800 \\ p &= \frac{800}{9} \\ p &= 88.8\end{align*}

We can round up for our answer.

Our answer is 89%. John ran 89% of the total miles.

Video Review

Use Proportions to Solve Percent Problems


Directions: Find \begin{align*}p\end{align*} in the given problems using cross products. Round to the nearest tenths place.

  1. \begin{align*}\frac{7}{15}=\frac{p}{100}\end{align*}
  2. \begin{align*}\frac{52}{3810}=\frac{p}{100}\end{align*}
  3. \begin{align*}\frac{16}{17}=\frac{p}{100}\end{align*}
  4. \begin{align*}\frac{3}{4}=\frac{p}{100}\end{align*}
  5. \begin{align*}\frac{3}{5}=\frac{p}{100}\end{align*}
  6. \begin{align*}\frac{1}{5}=\frac{p}{100}\end{align*}
  7. A dentist filled cavities in 8 of his 30 patients on Tuesday. What percent had cavities filled?
  8. A florist delivered 18 out of 25 bouquets. What percent was delivered?
  9. The baker sold 3 out of 4 dozen rolls. What percent was sold?
  10. What percent is 85 out of 5000?
  11. What percent is 15 out of 30?
  12. What percent is 88 out of 1200?
  13. What percent is 99 out of 200?
  14. What percent is 100 out of 330?
  15. What percent is 224 out of 5400?

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