# 4.2: Writing and Solving Proportions

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

## Introduction

*Rapid Reading*

Josh is very excited about his book on Mount Everest. He took the book to school and has been reading it every chance he can. In fact, he finished his work in math class so quickly that Ms. Henje made him check his work to be sure that it was accurate. Josh was very excited that it was accurate.

Josh looked at the clock. He still had 18 minutes left to read. Josh opened the book and read about Sir Edmund Hillary and Tenzing Norgay and their first ascent up the mountain on May 29, 1953. He was so engrossed in his reading that he did not even hear the bell ring.

“Time to go,” his friend Evan said tipping the book a little as he went by.

Josh looked up at the clock and then down at this book. He had read 10 pages in 18 minutes. Josh was excited. He picked up the book and thought about silent reading time. He had read 10 pages in 18 minutes so he was sure that he would read more pages during silent reading time since that was a 30 minutes block of time. Josh began to think about the number of pages that he would read during silent reading time.

**Is this true? If Josh reads during silent reading time at the same rate that he did during math class, how many pages will he read? To figure this out, you will need to know how to write a proportion and solve it. This lesson will teach you all that you need to know so you can apply what you learn to this problem at the end of the lesson.**

*What You Will Learn*

In this lesson, you will learn how to complete the following skills.

- Write proportions given verbal descriptions.
- Write and solve proportions using equivalent ratios.
- Write and solve proportions using algebra or cross-products.
- Model and solve real-world problems using proportions.

*Teaching Time*

I. **Write Proportions Given Verbal Descriptions**

In the last lesson, you learned how to identify a ratio. Let’s review for just a moment. **A** *ratio***is a comparison between two quantities or numbers. Ratios can be written in fraction form, with a colon or by using the word “to”.**

Sometimes, you will compare ratios. Sometimes one ratio will be greater than another, and other times they can be **equal or** ** equivalent**. When you have two equal ratios, you have a proportion.

**A** *proportion***is created when two ratios are equal, or we can say that two equal ratios form a proportion.**

We can write a proportion when we know that two ratios are equivalent.

Example

\begin{align*}1 : 2 = 2 : 4\end{align*}

**These two ratios are equivalent. We can say that the two ratios form a proportion.**

Example

*Do these two ratios form a proportion?*

\begin{align*}\frac{3}{4}\end{align*} and 4 : 24

**To figure this out, we have to figure out if the two ratios are equivalent. If they are, then we know that they form a proportion. If not, then they don’t. To figure this out, we can simplify the ratios.**

\begin{align*}& \qquad \ \frac{3}{4} \ is \ in \ simplest \ form.\\ & 4 : 24 \ can \ be \ written \ as \ \frac{4}{24} = \frac{1}{6}\\ & \qquad \qquad \quad \ \ \frac{3}{4} \ne \frac{1}{6}\end{align*}

**These ratios do not form a proportion.**

**These proportions were given to you. You can also write your own proportions.**

**To write a proportion, set two equivalent fractions equal to each other, using the information in the problem.**

For example, if you know the ratio of girls to boys in a class is 2 : 3, and you know there are 24 boys in the class, you can write a proportion in order to find the number of girls in the class. **The most important thing to remember when writing a proportion is to keep the units the same in both ratios.**

\begin{align*}\frac{girls}{boys}: \frac{2}{3} = \frac{x}{24}\end{align*}

**You know the fractions are equivalent because each shows the ratio of girls to boys in the class. The first fraction shows the known ratio of girls to boys. The second ratio shows the known number of boys in the class, 24, and uses a variable to stand for the unknown number of girls.**

Now you are ready to try another example.

Example

*The proportion of red paper to white paper in a stack is 2 to 7. If there are 32 red pieces of paper, what proportion could be used to find the number of pieces of white paper?*

Write the known ratio of red paper to white paper as the first fraction: \begin{align*}\frac{2}{7}\end{align*}.

Now write the second ratio, using \begin{align*}x\end{align*} to stand for the unknown amount. Make sure to keep the units the same as in the first fraction. In this case, the unknown is the amount of white paper, which is in the denominator of the fraction.

\begin{align*}\frac{\text{red paper}}{\text{white paper}} : \frac{2}{7} = \frac{32}{x}\end{align*}

**The proportion \begin{align*}\frac{2}{7} = \frac{32}{x}\end{align*} could be used to find the number of pieces of white paper in the stack.**

**You will learn how to solve proportions in another section, just work on writing them in this section.**

Example

*Jimmy makes $12 per hour. If he makes a total of $84, what proportion could be used to find the number of hours he worked?*

In this case, you are given a unit rate. Write the unit rate as a ratio: \begin{align*}\frac{12}{1}\end{align*}.

Now write the second ratio, using \begin{align*}x\end{align*} to stand for the unknown amount. Make sure to keep the units the same as in the first fraction.

\begin{align*}\frac{dollars}{hours} : \frac{12}{1} = \frac{84}{x}\end{align*}

**The proportion \begin{align*}\frac{dollars}{hours} : \frac{12}{1} = \frac{84}{x}\end{align*} could be used to find the number of hours Jimmy worked.**

Now let’s look at using equivalent ratios with proportions.

II. **Write and Solve Proportions using Equivalent Ratios**

In the last section, you learned how to write proportions to describe situations involving ratios. Now you are going to use that information to write and solve proportions.

Example

*The ratio of teachers to students in a certain school is 2 : 25. If there are 400 students in the eighth-grade class, how many teachers are there?*

**First set up a proportion. The problem gives a ratio of teachers to students, so set up two equivalent ratios comparing teachers to students.**

\begin{align*}\frac{teachers}{students} = \frac{8^{th} \ grade \ teachers}{8^{th} \ grade \ students}\end{align*}

**You can see that we are comparing teachers to students in both ratios. The first one shows the ration in the whole school and the second ratio represents the eighth grade ratios. Next, we fill in the given information.**

\begin{align*}\frac{2}{25} = \frac{x}{400}\end{align*}

**Now use what you know about equivalent ratios to solve the proportion.** Look at the denominators. You know that the first fraction, when the numerator and denominator are multiplied by some number, will equal the second fraction. What number, when multiplied by 25, will equal 400? Since \begin{align*}25 \times 16 = 400\end{align*}, the denominator was multiplied by 16. That means you can multiply the numerator by the same number to find the value of \begin{align*}x\end{align*}.

\begin{align*}2 \times 16 = 32\end{align*}, so \begin{align*}x = 32\end{align*}

**There are 32 teachers in the eighth-grade class.**

Note: You can check that your answer is correct by making sure that the two ratios are equivalent.

\begin{align*}\frac{32}{400} = \frac{8}{100} = \frac{2}{25}\end{align*}

Since the second ratio simplifies to the first, the ratios are equivalent.

**Yes it is. Just remember that what you do to the numerator you have to do the denominator. If you can remember to always apply this rule, then you will create equal ratios.**

Example

*The ratio of apples to bananas at a store is 3 to 8. If there are 90 apples, how many bananas are there?*

**Set up a proportion. Be sure that your units match up. Here apples are in the numerator position and bananas are in the denominator position. Be sure that this stays consistent throughout your work.**

\begin{align*}\frac{apples}{bananas} : \frac{3}{8} = \frac{90}{x}\end{align*}

**Now use what you know about equivalent ratios to solve the proportion.** What number, when multiplied by 3, equals 90? Since \begin{align*}3 \times 30 = 90\end{align*}, the numerator was multiplied by 30. So, multiply the denominator by 30 to find the value of \begin{align*}x\end{align*}.

\begin{align*}8 \times 30 = 240\end{align*}, so \begin{align*}x = 240\end{align*}

**There are 240 bananas at the store.**

Now let’s look at another way to solve proportions.

III. **Write and Solve Proportions using Algebra or Cross-Products**

In the last section you worked on solving proportions. Most of the examples that you completed could be solved by using mental math and what you knew about equal ratios. Sometimes, you won’t be able to use mental math and prior knowledge to solve a proportion. When this happens, you will need to figure out a different strategy.

**Another way of solving a proportion is called cross-multiplying. Here is the rule you should know.**

**If \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}ad = cb\end{align*}.**

**This is also called “the product of the means is equal to the product of the extremes.” The values in the \begin{align*}b\end{align*} and \begin{align*}c\end{align*} positions are called the** *means***, and the values in the \begin{align*}a\end{align*} and \begin{align*}d\end{align*} positions are called the extremes.**

**However, you can just think of multiplying the values that are diagonal to each other, making an \begin{align*}X\end{align*}. After cross-multiplying, you can use algebra to solve for the variable. Let’s apply this information now.**

Example

\begin{align*}\frac{x}{5} = \frac{9}{10}\end{align*}

**With this example, we have been given a proportion that needs solving. To solve it, we can cross-multiply.**

\begin{align*}10(x) = 10x\end{align*}

\begin{align*}9(5) = 45\end{align*}

\begin{align*}10x = 45\end{align*}

**Next, we can solve using algebra. We divide 45 by 10 to find the value of the variable.**

\begin{align*}x = 4.5\end{align*}

**This is our answer.**

Example

\begin{align*}\frac{4}{5} = \frac{16}{x}\end{align*}

**This proportion is written differently because the variable is in a different location. However, we can still solve it by using cross-products.**

\begin{align*}4x &= 80\\ x &= 20\end{align*}

**This is the answer.**

Now let’s look at how we can use proportions to solve real – world problems.

IV. **Model and Solve Real – World Problems using Proportions**

We can apply what we have learned about solving proportions to real-world problems too.

Example

*Amanda read 18 pages in 23 minutes. At this rate, how many pages will she read in 45 minutes?*

**First, set up a proportion.**

\begin{align*}\frac{18}{23} = \frac{x}{45}\end{align*}

**You cannot easily use equivalent fractions to solve this proportion. Cross-multiply to solve the proportion.**

\begin{align*}23x = 18(45)\end{align*}

**Now simplify the equation and solve for \begin{align*}x\end{align*}.**

\begin{align*}23x &= 810\\ x & \approx 35.2\end{align*}

**Amanda will read about 35.2 pages in 45 minutes.**

Example

*At a store, 3 pounds of chicken sells for $13.50. How many pounds of chicken can someone buy for $30?*

**Set up a proportion.**

\begin{align*}\frac{3}{13.50} = \frac{x}{30}\end{align*}

**Cross-multiply and use algebra to solve for \begin{align*}x\end{align*}.**

\begin{align*}13.5x &= 3(30)\\ 13.5x &= 90\\ x & \approx 6.67\end{align*}

**You could buy about 6.67 pounds of chicken for $30.**

**Now let’s go back and work on solving the problem from the introduction.**

## Real-Life Example Completed

*Rapid Reading*

**Here is the original problem once again. Reread it and then write a proportion to show the reading time and pages during both math and silent reading. Finally, solve the proportion to show the number of pages Josh will read during silent reading time. There are two parts to your answer.**

Josh is very excited about his book on Mount Everest. He took the book to school and has been reading it every chance he can. In fact, he finished his work in math class so quickly that Ms. Henje made him check his work to be sure that it was accurate. Josh was very excited that it was accurate.

Josh looked at the clock. He still had 18 minutes left to read. Josh opened the book and read about Sir Edmund Hillary and Tenzing Norgay and their first ascent up the mountain on May 29, 1953. He was so engrossed in his reading that he did not even hear the bell ring.

“Time to go,” his friend Evan said tipping the book a little as he went by.

Josh looked up at the clock and then down at this book. He had read 10 pages in 18 minutes. Josh was excited. He picked up the book and thought about silent reading time. He had read 10 pages in 18 minutes so he was sure that he would read more pages during silent reading time since that was a 30 minutes block of time. Josh began to think about the number of pages that he would read during silent reading time.

*Remember there are two parts to your answer – a proportion and the number of pages Josh will read during silent reading time.*

*Solution to Real – Life Example*

**First, we need to write a proportion to show the comparison between time and pages read.**

**Let’s begin with math class.**

**Josh read 10 pages in 18 minutes. Let’s write the first ratio.**

\begin{align*}\frac{10}{18}\end{align*}

**Next, Josh will read for 30 minutes during silent reading time. We need to figure out the number of pages, so that is our unknown.**

\begin{align*}\frac{x}{30}\end{align*}

**Here is the proportion.**

\begin{align*}\frac{10}{18} = \frac{x}{30}\end{align*}

**We can cross multiply and solve.**

\begin{align*}18x &= 300\\ x &= 16.6\end{align*}

**Josh will read about \begin{align*}16 \frac{1}{2}\end{align*} pages during silent reading time.**

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- Ratio
- a comparison between two quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.

- Equivalent
- means equal.

- Proportion
- formed when two ratios are equivalent. We compare two ratios, they are equal and so they form a proportion.

## Time to Practice

Directions: Solve each proportion using equal ratios.

- \begin{align*}\frac{3}{4} = \frac{x}{12}\end{align*}
- \begin{align*}\frac{5}{6} = \frac{x}{12}\end{align*}
- \begin{align*}\frac{4}{7} = \frac{8}{y}\end{align*}
- \begin{align*}\frac{2}{3} = \frac{12}{y}\end{align*}
- \begin{align*}\frac{4}{5} = \frac{44}{y}\end{align*}
- \begin{align*}\frac{12}{13} = \frac{x}{26}\end{align*}
- \begin{align*}\frac{9}{10} = \frac{81}{y}\end{align*}
- \begin{align*}\frac{6}{7} = \frac{18}{y}\end{align*}
- \begin{align*}\frac{7}{8} = \frac{x}{56}\end{align*}
- \begin{align*}\frac{12}{14} = \frac{36}{x}\end{align*}

Directions: Solve each proportion by using cross – multiplying with algebra. You may round to the nearest tenth when necessary.

- \begin{align*}\frac{3}{5} = \frac{y}{2.5}\end{align*}
- \begin{align*}\frac{6}{7} = \frac{2.5}{y}\end{align*}
- \begin{align*}\frac{4}{5} = \frac{2}{x}\end{align*}
- \begin{align*}\frac{9}{11} = \frac{14}{x}\end{align*}
- \begin{align*}\frac{2}{3} = \frac{5}{y}\end{align*}

Directions: Solve each problem.

- The ratio of fiction to nonfiction books at a library is 5 to 3. If there are 480 nonfiction books, write a proportion that could be used to find \begin{align*}f\end{align*}, the number of fiction books.
- The ratio of cherry trees to apple trees at an orchard is 4 to 9. If there are 184 cherry trees, write a proportion that could be used to find \begin{align*}a\end{align*}, the number of apple trees.
- The ratio of cars to SUVs in a parking lot is 10 to 7. If there are 84 SUVs, write a proportion that could be used to find \begin{align*}c\end{align*}, the number of cars in the lot.

Directions: Write a proportion and use equivalent ratios to solve the following problems.

- Marco makes $25 for every 2 hours he works. If he works for 12 hours, how much will he make?
- Corinne runs 2.8 miles in 30 minutes. If she runs for 150 minutes this week, how many miles will she have run?
- Adam drives 45 miles per hour. If he drives for 3.5 hours, how many miles will he have driven?

Directions: Write a proportion and use cross-multiplying to solve the following problems.

- Marni buys 2.5 pounds of grapefruit for $4.48. To the nearest cent, how much would 6 pounds of grapefruit cost?
- Glenn can make 8 flyers in 35 minutes. How long will it take him to make 50 flyers?
- A store sells 21 pieces of clothing every 45 minutes. How long will it take the store to sell 100 pieces of clothing?
- The basketball team scored 85 points in the last 2 games. How many points can they expect to score after 5 games?