Blake volunteered to be part of a "living flag" at the state fairgrounds on Independence Day. He had a small flag and wondered how the organizers would convert the dimensions to accommodate hundreds of people. Blake's flag measured 4 inches by 6 inches. If the length of the living flag was to be 120 feet, what should the width be?

In this concept, you will learn how to write and work with proportions.

### Working with Proportions

A **ratio** represents a comparison between two quantities. **Equivalent ratios** are ratios that are equal. A **proportion** is made up of two equivalent ratios.

An example of a proportion is \begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}

\begin{align*}\frac{2}{4}\end{align*}

andWhen given three parts of a proportion, the fourth can be determined.

Here is an example.

\begin{align*}\frac{1}{2} = \frac{x}{12}\end{align*}

In this proportion, x is unknown.

**Proportional reasoning,** or examining the relationship between two numbers, can be used to determine the value of x.

First, since both denominators are given, begin with those two numbers.

2 and 12

Next, recognize the ratio that is given. In this problem it is

. Think about what can be done to the denominator of the given ratio to make it equal to the denominator of the ratio with the unknown in it. In this case, what can be done to 2 to make it equal to 12.2 x 6 = 12

2 can be multiplied times 6 to equal 12.

Then, multiply both the numerator and denominator of the given ratio times 6.

\begin{align*}\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} = \frac{x}{12}\end{align*}

The answer is 6.

\begin{align*}\frac{1}{2} = \frac{6}{12}\end{align*}

The ratio \begin{align*}\frac{1}{2}\end{align*} is equal to the ratio

### Examples

#### Example 1

Earlier, you were given a problem about Blake and the living flag.

He wondered how his little 4" by 6" parade flag could be used to calculate the width of a giant human flag that has a length of 120 feet.

First, write an equation for the proportion.

\begin{align*}\frac{4}{6}=\frac{x}{120}\end{align*}

Next, work with the denominators.

6 and 120

Then, using the given ratio of

, determine what can be done to 6 to make it equal to 120.6 x 20 = 120

Next, multiply both the numerator and denominator of the given ratio times 20.

The answer is y = 80 feet.

\begin{align*}\frac{4}{6}=\frac{80}{120}\end{align*}

#### Example 2

Use proportional reasoning to solve for x.

\begin{align*}\frac{15}{35}=\frac{x}{7}\end{align*}

First, since both denominators are given, start with those two numbers.

35 and 7

Next, recognize the given ratio,

, and determine what can be done to 35 to make it equal to 7.

Then, divide both the numerator and denominator of the given ratio by 5.

The answer is x = 3

\begin{align*}\frac{15}{35}=\frac{3}{7}\end{align*}

#### Example 3

\begin{align*}\frac{32}{36}=\frac{x}{9}\end{align*}

Solve for x:

First, begin with the given denominators.

36 and 9

Next, using the given ratio of

, determine what can be done to 36 to make it equal to 9.\begin{align*}36\div 4=9\end{align*}

Then, divide both the numerator and denominator of the given ratio by 4.

The answer is x = 8

#### Example 4

\begin{align*}\frac{12}{24} = \frac{4}{z}\end{align*}

Solve for z.

First, begin with the numerators as both are given.

12 and 4

Next, determine what can be done to the numerator 12 in the given ratio of \begin{align*}\frac{12}{24}\end{align*}to make it equal to 4 in the unknown ratio.

Then, divide both sides of the given ratio by 3.

\begin{align*}\frac{12\div 3}{24\div 3}=\frac{4}{8}\end{align*}

The answer is z = 8.

#### Example 5

Solve for y.

\begin{align*}\frac{y}{5} = \frac{12}{20}\end{align*}

First, work with the denominators.

5 and 20

Next, use the denominator, 20, in the given ratio of \begin{align*}\frac{12}{20}\end{align*} and determine what can be done to 20 to make it equal to the denominator, 5, in the unknown ratio.

Then, divide both sides of the given ratio by 4.

The answer is y = 3.

### Review

Tell whether or not each pair of ratios form a proportion.

- \begin{align*}\frac{1}{2} \text{ and } \frac{4}{8}\end{align*}
- \begin{align*}\frac{3}{7} \text{ and } \frac{6}{14}\end{align*}
- \begin{align*}\frac{5}{2} \text{ and } \frac{10}{6}\end{align*}
- \begin{align*}\frac{3}{1} \text{ and } \frac{9}{3}\end{align*}
- \begin{align*}\frac{2}{9} \text{ and } \frac{1.5}{4.5}\end{align*}
- \begin{align*}\frac{4}{9} \text{ and } \frac{8}{10}\end{align*}
- \begin{align*}\frac{1}{4} \text{ and } \frac{5}{20}\end{align*}
- \begin{align*}\frac{3}{4} \text{ and } \frac{9}{10}\end{align*}

Use proportional reasoning to find the value of the variable in each proportion.

- \begin{align*}\frac{1}{4} = \frac{a}{20}\end{align*}
- \begin{align*}\frac{15}{30} = \frac{x}{2}\end{align*}
- \begin{align*}\frac{2}{9} = \frac{n}{63}\end{align*}
- \begin{align*}\frac{z}{7} = \frac{12}{21}\end{align*}
- \begin{align*}\frac{3}{5} = \frac{t}{60}\end{align*}
- \begin{align*}\frac{k}{72} = \frac{5}{12}\end{align*}
- \begin{align*}\frac{x}{32} = \frac{4}{8}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.9.