# 8.22: Problem Solving Plan Proportions

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

**Practice**Problem Solving Plan, Proportions

Takeru Kobayashi is a competitive eater famous for eating hot dogs among other things. Once, he ate 110 hot dogs in 10 minutes at the New York State Fair. How many hot dogs can Kobayashi eat in 1 minute?

In this concept, you will learn the problem solving strategy: use a proportion.

### Problem Solving Plan with Proportions

You can solve ratio problems using **proportions**. A **proportion** is an equation that shows two equivalent ratios. In order to use a proportion, the ratios in the proportion must also compare the same things.

Let’s look at a problem.

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

This problem compares speed per hour; however, it is comparing miles to kilometers. The units are not the same and you cannot use a proportion to solve this problem without converting the units first.

Let’s look at another problem.

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?

The first ratio compares the cheetah’s speed per hour. The second ratio compares a person’s speed per hour. Both ratios can be written as miles per hour. You can use a proportion to solve this problem.

First, write a proportion showing the comparison.

\begin{align*}\frac{\text{cheetah's speed}}{\text{number of hours}}= \frac{\text{person's speed}}{\text{number of hours}}\end{align*}

Next, take the data and fill it into the proportion.

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

The cheetah runs 75 miles per hour. *Per* means “divided by” and “hour” refers to 1 hour. The numerator is 75 and the denominator is 1. 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. The denominator is 3. The person’s speed is unknown so it is represented with the variable \begin{align*}x\end{align*}

Then, solve the proportion using cross products.

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

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

### Examples

#### Example 1

Earlier, you were given a problem about the competitive eater Takeru Kobayashi.

He once ate 110 hot dogs in 10 minutes. To find out how many hot dogs he can eat in 1 minute, write a proportion and solve for \begin{align*}x\end{align*}

First, set up the proportion.

\begin{align*}\frac{\text{hot dogs}}{\text{minutes}}= \frac{\text{hot dogs}}{\text{minutes}}\end{align*}

Next, fill in the given information.

\begin{align*}\frac{110}{10} = \frac{x}{1}\end{align*}

Then, cross multiply and solve for \begin{align*}x\end{align*}

\begin{align*}\begin{array}{rcl}
10x & = & 110(1)\\
10x & = & 110\\
x & = & 11
\end{array}\end{align*}

Takeru Kobayashi can eat 11 hot dogs in 1 minute.

#### Example 2

Use a proportion to solve the following problem.

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

First, set up the proportion.

\begin{align*}\frac{\text{miles}}{\text{minutes}}= \frac{\text{miles}}{\text{minutes}}\end{align*}

Next, fill in the given information.

\begin{align*}\frac{3}{20} = \frac{12}{x}\end{align*}

Now we cross multiply and solve for \begin{align*}x\end{align*}

\begin{align*}\begin{array}{rcl}
3x & = & 20(12)\\
3x & = & 240\\
x & = & 80
\end{array}\end{align*}

The person would run 12 miles in 80 minutes or 1 hour and 20 minutes.

**Use the following information to answer the next three questions:**

**A cheetah runs 75 miles per hour.**

#### Example 3

If you could run twice as fast as a cheetah, how fast could you run?

First, set up the proportion.

\begin{align*}\frac{\text{miles}}{\text{hour}}= \frac{\text{miles}}{\text{hour}}\end{align*}

Next, fill in the given information.

\begin{align*}\frac{75}{1} = \frac{x}{2}\end{align*}

Then, cross multiply and solve for \begin{align*}x\end{align*}

\begin{align*}\begin{array}{rcl}
x & = & 75(2)\\
x & = & 150
\end{array}\end{align*}

You could run 150 miles per hour.

#### Example 4

If you could run half as fast as a cheetah, how fast could you run?

First, set up the proportion.

\begin{align*}\frac{\text{miles}}{\text{hour}}= \frac{\text{miles}}{\text{hour}}\end{align*}

Next, fill in the given information.

\begin{align*}\frac{75}{1} = \frac{x}{0.5}\end{align*}

Then, cross multiply and solve for \begin{align*}x\end{align*}

\begin{align*}\begin{array}{rcl}
x & = & 75(0.5)\\
x & = & 37.5
\end{array}\end{align*}

You could run 37.5 miles per hour.

#### Example 5

If you could run four times as fast as a cheetah, how fast could you run?

First, set up the proportion.

\begin{align*}\frac{\text{miles}}{\text{hour}}= \frac{\text{miles}}{\text{hour}}\end{align*}

Next, fill in the given information.

\begin{align*}\frac{75}{1} = \frac{x}{4}\end{align*}

Then, cross multiply and solve for \begin{align*}x\end{align*}

\begin{align*}\begin{array}{rcl}
x & = & 75(4)\\
x & = & 300
\end{array}\end{align*}

You could run 300 miles per hour.

### Review

Solve each word problem by using a proportion.

- 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?
- If two inches on a map are equal to three miles, how many miles are represented by four inches?
- If eight inches on a map are equal to ten miles, how many miles are 16 inches equal to?
- 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?
- If two inches are equal to twelve feet, how many inches would be equal to 36 feet?
- If four inches are equal to sixteen feet, how many feet are two inches equal to?
- The carpenter chose a scale of 6" for every twelve feet. Given this measurement, how many feet would be represented by 3"?
- If 9 inches are equal to 27 feet, how many feet are equal to three inches?
- If four inches are equal to 8 feet, how many feet are equal to two inches?
- If six inches are equal to ten feet, how many inches are five feet equal to?
- If four inches are equal to twelve feet, how many inches are equal to six feet?
- 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?
- If eight inches are equal to twelve feet, how many inches are equal to six feet?
- How many inches are equal to 20 feet if 4 inches are equal to 10 feet?
- How many inches are equal to 8 feet if six inches are equal to 16 feet?
- Nine inches are equal to twelve feet, so how many inches are equal to 4 feet?
- If a person runs two miles in twelve minutes, how long will it take them to run 4 miles at the same rate?
- A person runs 1 mile in 16 minutes. Given this information, how long will it take him/her to run 3 miles?
- If a person runs two miles in twenty minutes, at what rate does he/she run one mile?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.22.

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In this concept, you will learn the problem solving strategy: use a proportion.

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