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?

### Proportions

A **proportion** is when two ratios are set equal to each other.

**Cross-Multiplication Theorem:** Let and be real numbers, with and . If , then .

The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by or because it is the same number divided by itself .

**Proof of the Cross-Multiplication Theorem:**

Think of the Cross-Multiplication Theorem as a shortcut. Without this theorem, you would have to go through all of these steps every time to solve a proportion. The Cross-Multiplication Theorem has several sub-theorems that follow from its proof. The formal term is *corollary.*

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

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

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

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

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

#### Solving Proprotions

Solve the proportions.

To solve a proportion, you need to * cross-multiply*.

a)

b)

c)

#### Real-World Application

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 your parents’ house is actually 50 feet, what is the longer length?

Set up a proportion. If the shorter length is 50 feet, then it will line up with 30 in. It does not matter which numbers you put in the numerators of the fractions, as long as they line up correctly.

So, the dimension of your parents’ house is 50 ft by 60 ft.

#### Writing True Proportions

Suppose we have the proportion . Write down the other three true proportions that follow from this one.

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

If you cross-multiply all four of these proportions, you would get for each one.

#### Python Problem Revisited

The scale model of a python is 0.75 ft long and in the ratio 1:24. If is the length of the real python in ft:

The real python is 18 ft long.

### Examples

#### Example 1

In the picture, .

Find the measures of and .

This is an example of an *extended* proportion. Substituting in the numbers for the sides we know, we have . Separate this into two different proportions and solve for and .

#### Example 2

In the picture, . Find .

Substituting in the numbers for the sides we know, we have

#### Example 3

If in the picture above, find .

### 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?
- Many recipes describe ratios between ingredients. For example, one recipe for paper mache paste suggests 3 parts flour to 5 parts water. If we have one cup of flour, how much water should we add to make the paste?
- A recipe for krispy rice treats calls for 6 cups of rice cereal and 40 large marshmallows. You want to make a larger batch of goodies and have 9 cups of rice cereal. How many large marshmallows do you need? However, you only have the miniature marshmallows at your house. You find a list of substitution quantities on the internet that suggests 10 large marshmallows are equivalent to 1 cup miniatures. How many cups of miniatures do you need?

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

For questions 15-18, and .

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

Explain why is a valid proportion. HINT: Cross-multiply and see if it equals .*Writing*Explain why is a valid proportion. HINT: Cross-multiply and see if it equals .*Writing*

### Review (Answers)

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