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# Proportions in Percent Equations

## Solve word problems using cross products

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Use Proportions to Solve Percent Problems

The student council decided to conduct a survey to see how many middle school students attend football games on Friday nights. They asked the students when they entered the stadium what grade they were in and then recorded the results. The student council members did this for three weeks to get a good estimate on how many middle school students attend the games. They found that 42% of the 850 students who attended were from the middle school. How many students does this represent?

In this concept, you will learn to use proportions to solve percent problems.

### Proportions

Proportions can be used to compare any equal quantity. Remember that proportions are created when two ratios are equal. Because of this, proportions can be used to find any missing piece. If you know a percent, you can also find a part of the whole.

Let’s look at an example.

At Big Town Middle School, approximately 37% of the students participate in athletic programs. If there are 968 students at the school, how many uniforms do they need to purchase?

Although 37% is a useful number, it does not tell you how many students need uniforms. You know that for every 100 students, 37 will need a uniform. But that is still not enough information to make a purchase.

First, if there are 968 students at the school, use the proportion \begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*} to find the missing value. In this case you are looking for \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100}\\ \frac{a}{968} &=& \frac{37}{100} \end{array}\end{align*}

Next, use cross products to help isolate \begin{align*}a\end{align*}. In other words, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{a}{968} &=& \frac{37}{100}\\ 100a &=& 968 \times 37\\ 100a &=& 35816 \end{array}\end{align*}

Then, divide by 100 to find the number of uniforms to purchase.

\begin{align*}\begin{array}{rcl} 100a &=& 35816\\ \frac{100a}{100} &=& \frac{35816}{100}\\ a &=& 358.16 \end{array}\end{align*}

Therefore, the team needs to purchase 358 uniforms.

Since you can use a proportion to solve for any missing variable, you can also find the whole if you know a percent and its corresponding part.

Let’s look at another example.

At Big Town High School, approximately 58% participate in athletic programs. If 1670 students are involved in an athletics program, then how many students are there in the school?

First, if there are 1670 students at the school, use the proportion \begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*} to find the missing value. In this case you are looking for \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100}\\ \frac{1670}{b} &=& \frac{58}{100} \end{array}\end{align*}

Next, use cross products to help isolate \begin{align*}b\end{align*}. In other words, cross multiply.

\begin{align*}\begin{array}{rcl} \frac{1670}{b} &=& \frac{58}{100}\\ 58b &=& 1670 \times 100\\ 58b &=& 167000 \end{array}\end{align*}

Then, divide by 58 to find the number of students in the school.

\begin{align*}\begin{array}{rcl} 58b &=& 167000\\ \frac{58b}{58} &=& \frac{167000}{58}\\ b &=& 2879.3 \end{array}\end{align*}

Therefore there are 2879 students in the school.

### Examples

#### Example 1

Earlier, you were given a problem about the student football crowd.

First, you know that 42% of the students at the games are in middle school. Write the percent as a ratio.

\begin{align*}42\%= \frac{42}{100}\end{align*}

Next, there are 850 students attending the games. Write the proportion to start solving the problem.

\begin{align*}\begin{array}{rcl} \frac{a}{850} &=& \frac{42}{100}\\ 100a &=& 42 \times 850\\ 100a &=& 35700 \end{array}\end{align*}

Then, divide by 100 to solve for \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} 100a &=& 35700\\ \frac{100a}{100} &=& \frac{35700}{100}\\ a &=& 357 \end{array}\end{align*}

The answer is \begin{align*}a = 357\end{align*}.

Therefore, 357 middle school students attend the game.

#### Example 2

A forest range discovered that 25% of the trees in his area were infected with a parasite. If there are 3060 trees in his area, how many trees are infected?

First, if there are 968 students at the school, use the proportion \begin{align*}\frac{a}{b}= \frac{p}{100}\end{align*} to find the missing value. In this case you are looking for \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{a}{b} &=& \frac{p}{100}\\ \frac{a}{3060} &=& \frac{25}{100} \end{array}\end{align*} Next, use cross products to help isolate \begin{align*}a\end{align*} .

\begin{align*}\begin{array}{rcl} \frac{a}{3060} &=& \frac{25}{100}\\ 100a &=& 3060 \times 25\\ 100a &=& 76500 \end{array}\end{align*}

Then, divide by 100 to find the number of trees infected.

\begin{align*}\begin{array}{rcl} 100a &=& 76500\\ \frac{100a}{100} &=& \frac{76500}{100}\\ a &=& 765 \end{array}\end{align*}

Therefore, there are 765 of the 3060 trees infected.

#### Example 3

Solve for \begin{align*}b\end{align*} in the proportion: \begin{align*}\frac{9}{b} = \frac{20}{100}\end{align*}

First, use cross products to help isolate \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{9}{b} &=& \frac{20}{100}\\ 20a &=& 9 \times 100\\ 20a &=& 900 \end{array}\end{align*}

Next, divide by 20 to solve for \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} 20b &=& 900\\ \frac{20b}{20} &=& \frac{900}{20}\\ b &=& 45 \end{array}\end{align*}

The answer is \begin{align*}b = 45\end{align*}.

#### Example 4

Solve for \begin{align*}a\end{align*} in the proportion: \begin{align*}\frac{a}{10}=\frac{40}{100}\end{align*}

First, use cross products to help isolate \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{a}{10} &=& \frac{40}{100}\\ 100a &=& 40 \times 10\\ 100a &=& 400 \end{array}\end{align*}

Next, divide by 100 to solve for \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} 100a &=& 400\\ \frac{100a}{100} &=& \frac{400}{100}\\ a &=& 4 \end{array}\end{align*}

The answer is \begin{align*}a = 45\end{align*}.

#### Example 5

Solve for \begin{align*}a\end{align*} in the proportion: \begin{align*}\frac{a}{3}= \frac{18}{100}\end{align*}

First, use cross products to help isolate \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{a}{3} &=& \frac{18}{100}\\ 100a &=& 3 \times 18\\ 100a &=& 54 \end{array}\end{align*}

Next, divide by 100 to solve for \begin{align*}a\end{align*}.

\begin{align*}\begin{array}{rcl} 100a &=& 54\\ \frac{100a}{100} &=& \frac{54}{100}\\ a &=& 0.54 \end{array}\end{align*}

The answer is \begin{align*}a = 0.54\end{align*}.

### Review

Solve the following proportions for the \begin{align*}a\end{align*} value.
1. \begin{align*}\frac{a}{23} = \frac{85}{100}\end{align*}

2. \begin{align*}\frac{a}{1500} = \frac{7}{100}\end{align*}

3. \begin{align*}\frac{a}{5} = \frac{61}{100}\end{align*}

4. \begin{align*}\frac{a}{4} = \frac{75}{100}\end{align*}

5. \begin{align*}\frac{a}{5} = \frac{40}{100}\end{align*}

6. A small car company sold 65,000 cars last year. Ninety-five percent of those cars had airbags. How many cars had airbags?

Solve the following proportions for the \begin{align*}b\end{align*} value. Round to the nearest tenths place.

7. \begin{align*}\frac{88}{b} = \frac{22}{100}\end{align*}

8. \begin{align*}\frac{600}{b} = \frac{74}{100}\end{align*}

9. \begin{align*}\frac{1}{b} = \frac{85}{100}\end{align*}

10. \begin{align*}\frac{2}{b} = \frac{66}{100}\end{align*}

11. \begin{align*}\frac{3}{b} = \frac{75}{100}\end{align*}

12. A recent government survey shows that in New City, people spend 35% of their monthly income on rent. If average rent is \$780, what is the average income?

Use the proportion \begin{align*}\frac{a}{b} = \frac{p}{100}\end{align*} to solve the following problems.

13 A fluorescent light bulb uses 35% as much energy as an incandescent bulb. If an incandescent uses 75 watts, how much does the fluorescent use?

14. An average human weighs 160 pounds. An elephant weights 1500% more. How much does the elephant weigh?

15. An average elephant eats about 350lbs of food per day. Using your calculation from problems #14, what percent of its own weight does it consume each day?

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### Vocabulary Language: English

Percent

Percent means out of 100. It is a quantity written with a % sign.

Proportion

A proportion is an equation that shows two equivalent ratios.