# 8.8: Problem Solving Strategy: Use a Proportion

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

## Introduction

*The Frog Problem*

Tim loves to read about frogs. While his Mom was grocery shopping, Tim caught sight of a magazine all about frogs. He couldn’t help picking it up and was glad that he had a few dollars in his pocket to buy the magazine.

When Tim got to the car, he was amazed to read that a frog can jump twenty times its body length. That means if a frog is three inches long, it can jump 20 times that far: 5 feet!

3 \begin{align*}\times\end{align*}

“Mom, listen to this,” Tim exclaimed, as he shared his findings.

“Wow Tim, I didn’t know that. You are four feet tall, how far could you jump if you could jump like a frog?”

Tim stopped to think. He wasn’t sure he knew how to figure that out, but he was sure that proportions would be a part of it.

**Do you know how to figure this out? Use the information in this lesson to help Tim hop like a frog.**

*What You Will Learn*

By the end of this lesson you will learn the following skills:

- Read and understand given problem situations.
- Develop and use the strategy: Use a Proportion.
- Plan and compare alternative approaches to solving problems.
- Solve real-world problems involving rates, proportions, or percents using selected strategies as part of a plan.

*Teaching Time*

I. **Read and Understand Given Problem Situations**

This lesson focuses on using a proportion to solve a problem. **To use a proportion to solve a problem certain criteria must be present.** If these criteria are not present, you can’t use a proportion to solve the problem. **This section will help you to recognize the criteria that you will need to solve the problem using a proportion. It will help you to read and understand the problem.**

**What information needs to be present in a problem to solve it using a proportion?**

**To use a proportion in a problem, the problem must have information in both groups. It must also have the same information that is being prepared.** If you think about this it makes perfect sense. A proportion compares two equal ratios-if the information in the problem is different, then two different things are being compared and they are not equal.

Let’s look at an example.

Example

A cheetah can run 75 miles per hour. If you could run three times as fast as a cheetah, how fast would you be able to run?

**This problem compares the same quantities. It compares the cheetah’s speed per hour and the person’s speed per hour. Here is a proportion showing the comparison.**

\begin{align*}\frac{cheetah’s \ speed}{number \ of \ hours}= \frac{Person’s \ speed}{number \ of \ hours}\end{align*}

When you look at these two comparisons, you can see that we are comparing speed with speed. We can use a proportion to solve this problem.

Here is an example where we could not use a proportion.

Example

A car travels 55 miles in two hours. A bus travels 85 km in two hours. Which vehicle traveled a farther distance?

**In this problem our units are not the same. We are comparing hours with hours, but we are comparing miles with kilometers, so the units are different. We could not use a proportion to solve this problem without converting the units first.**

II. **Develop and Use the Strategy: Use a Proportion**

**Let's apply a proportion and use it to solve the cheetah problem.** Let’s look at the example once again.

Example

A cheetah can run 75 miles per hour. If you could run three times as fast as a cheetah, how fast would you be able to run?

**We already wrote the proportion to show what is being compared in this problem. Here is the proportion.**

\begin{align*}\frac{cheetah’s \ speed}{number \ of \ hours}= \frac{Person’s \ speed}{number \ of \ hours}\end{align*}

**Our next step is to take the data and fill it into the proportion.**

\begin{align*}\frac{75}{1}=\frac{x}{3}\end{align*}

Here we wrote in that the cheetah runs 75 miles per hour. *Per* means "divided by", and "hour" refers to only one, so we use one as our denominator and 75 as the speed in the numerator. The person runs three times as fast, so he or she would go as far in 1 hour as a cheetah would in 3 hours. We put 3 in for the denominator. We multiplied the denominator by 3, so the numerator becomes 3(75), and we will use a variable for the person’s speed because we don’t know what it is yet.

**Next we solve the proportion using cross products.**

\begin{align*}x &= 75(3)\\ x &= 225 \ mph\end{align*}

**If a person ran three times as fast as a cheetah, he or she would run 225 mph.**

**That is very fast indeed!!**

III. **Plan and Compare Alternative Approaches to Solving Problems**

We could have solved this problem in another way besides using a proportion. When you read a problem, you should work out the best way to solve it. Often, there is more than one way to solve it.

In this example, we could have used multiplication. Here is the original problem again.

Example

A cheetah can run 75 miles per hour. If you could run three times as fast as a cheetah, how fast would you be able to run?

**We can break it down logically. If a person runs three times as fast as the cheetah, then we can multiply three times the cheetah’s speed to get the person’s speed.**

**75(3) = 225 mph**

**You can see that we still arrived at the same answer! The key is that using a proportion lets you know that you are comparing the correct information.**

**Then you can use cross-products to solve for the missing part of the proportion.**

## Real Life Example Completed

*The Frog Problem*

**Ready? Using what you have just learned, you should be ready to help Tim figure out his frog dilemma. Here is the problem once again. Take a few minutes to reread it and underline any important information.**

Tim loves to read about frogs. While his Mom was grocery shopping, Tim caught sight of a magazine all about frogs. He couldn’t help picking it up and was glad that he had a few dollars in his pocket to buy the magazine.

When Tim got to the car, he was amazed to read that a frog can jump twenty times its body length. That means if a frog is three inches long, it can jump 20 times that far: 5 feet!

3 \begin{align*}\times\end{align*} 20 = 60 inches or 5 feet

“Mom, listen to this,” Tim exclaimed, as he shared his findings.

“Wow Tim, I didn’t know that. You are four feet tall, how far could you jump if you could jump like a frog?”

Tim stopped to think. He wasn’t sure he knew how to figure this out, but he was sure that proportions would be a part of it.

**To start with, let’s write a proportion to compare the frog's length and its jump to Tim’s height and his jump.**

\begin{align*}\frac{frog \ length}{frog \ jump}=\frac{Tim’s \ height}{Tim’s \ jump}\end{align*}

**Now that we have the proportion, we can fill in the information that we know.**

\begin{align*}\frac{3”}{60”}=\frac{4ft}{x}\end{align*}

**Oh, here is our first problem. The frog is in inches and Tim’s height is in feet. Let’s change 4 ft to inches.**

**4 \begin{align*}\times\end{align*} 12 = 48”**

**Now we can solve for the variable which is how far Tim will jump. We do this by solving the proportion.**

\begin{align*}\frac{3”}{60”}& =\frac{48”}{x}\\ 3x &= 2880\\ x &= 960”\end{align*}

**We divided 2880 by three and got 960 inches as our answer. Now we can convert this to feet to show how far Tim jumped.**

\begin{align*} \overset{\ \ 80}{12\overline{)960}}\end{align*}

**If Tim were a frog he could jump 80 feet.**

*If you enjoyed this problem, check out “If You Hopped Like a Frog” by David M. Schwartz-a great picture book of proportions.*

## Technology Integration

Khan Academy, Understanding Proportions

James Sousa, Applications Using Proportions

Other Videos:

http://www.teachertube.com/members/viewVideo.php?video_id=31925 This video begins with the basics of solving problems by writing a proportion and moves into more challenging material. This would be a good video for an advanced student. You'll need to register at the site to access this video.

## Time to Practice

Directions: Solve each word problem by using a proportion.

1. In a diagram for the new garden, one inch is equal to 3 feet. If this is the case, how many feet is the actual garden edge if the measurement on the diagram is 5 inches?

2. If two inches on a map are equal to three miles, how many miles are represented by four inches?

3. If eight inches on a map are equal to ten miles, how many miles are 16 inches equal to?

4. Casey drew a design for bedroom. On the picture, she used one inch to represent five feet. If her bedroom wall is ten feet long, how many inches will Casey draw on her diagram to represent this measurement?

5. If two inches are equal to twelve feet, how many inches would be equal to 36 feet?

6. If four inches are equal to sixteen feet, how many feet are two inches equal to?

7. The carpenter chose a scale of 6” for every twelve feet. Given this measurement, how many feet would be represented by 3”?

8. If 9 inches are equal to 27 feet, how many feet are equal to three inches?

9. If four inches are equal to 8 feet, how many feet are equal to two inches?

10. If six inches are equal to ten feet, how many inches are five feet equal to?

11. If four inches are equal to twelve feet, how many inches are equal to six feet?

12. For every 20 feet of fence, John drew 10 inches on his plan. If the real fence is only 5 feet long, how many inches will John draw on his plan?

13. If eight inches are equal to twelve feet, how many inches are equal to six feet?

14. How many inches are equal to 20 feet if 4 inches are equal to 10 feet?

15. How many inches are equal to 8 feet if six inches are equal to 16 feet?

16. Nine feet are equal to twelve feet, so how many inches are equal to 4 feet?

17. If a person can run 3 miles in 20 minutes, how long will it take the same person to run 12 miles if it is at the same rate?

18. If a person runs two miles in twelve minutes, how long will it take them to run 4 miles at the same rate?

19. A person runs 1 mile in 16 minutes. Given this information, how long will it take him/her to run 3 miles?

20. If a person runs two miles in twenty minutes, at what rate does he/she run one mile?

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