### Proportions

A **proportion** is two ratios that are set equal to each other. Usually the ratios in proportions are written in fraction form. An example of a proportion is . To solve a proportion, you need to **cross-multiply.** The **Cross-Multiplication Theorem**, which allows us to solve proportions using this method, states that if and are real numbers, with and and if , then . Cross-multiplying allows us to get rid of the fractions in our equation. The Cross-Multiplication Theorem has several sub-theorems, called **corollaries.**

**Corollary #1:** If and are nonzero and , then .

*Switch and .*

**Corollary #2:** If and are nonzero and , then .

*Switch* *and* .

**Corollary #3:** If and are nonzero and , then .

*Flip each ratio upside down.*

**Corollary #4:** If and are nonzero and , then .

**Corollary #5:** If and are nonzero and , then .

What if you were told that a scale model of a python is in the ratio of 1:24? If the model measures 0.75 feet long, how long is the real python?

### Examples

#### Example 1

Find the measures of and .

Plug in the lengths of the sides we know.

#### Example 2

In the picture, . Find .

Substitute in the lengths of the sides we know.

#### Example 3

Solve the proportions. Remember, to solve a proportion, you need to **cross-multiply.**

#### Example 4

Your parents have an architect’s drawing of their home. On the paper, the house’s dimensions are 36 in by 30 in. If the shorter length of the house is actually 50 feet, what is the longer length?

To solve, first set up a proportion. If the shorter length is 50 feet, then it lines up with 30 in, the shorter length of the paper dimensions.

#### Example 5

Suppose we have the proportion . Write three true proportions that follow.

First of all, we know this is a true proportion because you would multiply by to get . Using the first three corollaries:

### Review

Solve each proportion.

- Shawna drove 245 miles and used 8.2 gallons of gas. At the same rate, if she drove 416 miles, how many gallons of gas will she need? Round to the nearest tenth.
- The president, vice-president, and financial officer of a company divide the profits is a 4:3:2 ratio. If the company made $1,800,000 last year, how much did each person receive?

Given the true proportion, and and are nonzero, determine if the following proportions are also true.

For questions 13-16, and .

- Find .
- Find .
- Find .
- Find .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.2.

### Resources