Tony is training for a marathon. A marathon is about 26 miles. Last week, it took him 120 minutes to run 13 miles. If Tony runs at the same speed, how long will it take him to run a full marathon?

In this concept, you will learn to use the cross product to identify proportions.

### Using Cross Products to Identify Proportions

A **proportion** is two equal ratios. Proportions can be written as equations. A **ratio** compares two quantities. One way to check if two ratios are equivalent is to compare the simplest form of each ratio.

Compare the ratios 1:4 and 5:20. Write the ratios as fractions and simplify to compare.

is in simplest form. can be simplified by dividing the numerator and denominator by the

The ratios 1:4 and 5:20 are equivalent ratios.

Another way to check if two ratios are equivalent is to cross multiply and compare the cross products. A **cross product** is the result of multiplying the numerator of one ratio with the denominator of another. If the cross products are equal, then the ratios are proportional.

For ratios

and , if , then and are proportional and can be written as .Let’s use cross products to compare the ratios

and to determine if they are proportional.First, write an equation with the ratios.

Next, cross multiply to find the cross products.

Then, simplify both sides of the equation by multiplying and check if they are equal.

Both sides are equal. The ratios

and are proportional.### Examples

#### Example 1

Earlier, you were given a problem about Tony training for the marathon.

Tony took 120 minutes to run 13 miles. Use cross products to find how long it will take Tony to run 26 miles if he continues at the same speed. (Note: Write the corresponding units in the same location. The minutes are in the numerator and miles are in the denominator.)

First, write an equation with the ratios. Use

to represent the unknown quantity.

Next, cross multiply to find the cross products.

Then, simplify by multiplying 120 and 26.

Divide both sides by 13 to find the unknown quantity.

It will take Tony about 240 minutes or 4 hours to run the marathon.

#### Example 2

Use the cross products to determine if the ratios

and are proportional.First, write an equation with the ratios.

Next, cross multiply to find the cross products.

Then, simplify both sides of the equation by multiplying and check if they are equal.

Both sides are equal. The ratios

and are proportional.#### Example 3

Use the cross products to determine if the ratios

First, write an equation with the ratios.

Next, cross multiply to find the cross products.

Then, simplify both sides of the equation by multiplying and check if they are equal.

18 is not equal to 25.

and are not proportional.#### Example 4

Use the cross products to determine if the ratios

First, write an equation with the ratios.

Next, cross multiply to find the cross products.

Then, simplify both sides of the equation by multiplying and check if they are equal.

Both sides are equal.

and are proportional.#### Example 5

Use the cross products to determine if the ratios

First, write an equation with the ratios.

Next, cross multiply to find the cross products.

Then, simplify both sides of the equation by multiplying and check if they are equal.

112 is not equal to 84.

and are not proportional.### Review

Use the cross products to determine if the following ratios are proportional.