Ratio & Proportion If a recipe for 4 people calls for 2 1 cups of flour, how much flour is needed for a recipe 2 that feeds 15 people? The answer to this question may be found by using ratio and proportion. A ratio is the comparison of two quantities. This comparison may always be written as a fraction of one quantity over the other. Other ways of expressing ratios include using a colon (:), the word to, or the words out of in between the quantities that are being compared. The words out of are only used when the units of the two quantities are the same. Different ways of expressing a ratio are demonstrated here.

EQUIVALENT WAYS OF WRITING THE RATIO 4 GALLONS TO 5 GALLONS 4 gallons 5 gallons

4 gallons to 5 gallons 4 gallons out of 5 gallons 4 gallons : 5 gallons Note: Always include the units that are represented.

Example 1

A recipe for 4 people calls for 6 cups of flour. Write a ratio for this comparison. This ratio may be written as

6 cups flour 4 people

or 6 cups flour : 4 people or 6 cups flour to 4 people. This ratio may be reduced to 3 cups flour : 2 people.

Proportion

The use of the words out of would not be appropriate in this example since the units are different.

A proportion is a statement where two ratios are equal to each other. A proportion is true only if the cross-products of the fraction representations of the ratios are equal. Each cross-product consists of the denominator of one fraction multiplied by the numerator of the other fraction. These facts are summarized here. PROPORTION FACTS A proportion is a statement where two ratios are equal to each other. A proportion is true only when the cross-products are equal.

Example:

Example 2

4 gallons oil 64 gallons gas

1 gallon oil 16 gallons gas

Is the proportion 2 liters oil

4 liters gas

3 liters oil 6 liters gas

is true since 4×16 = 1×64.

true?

The cross-products are 2 × 6 = 12 and 3 × 4 = 12. Since they are equal, this proportion is true.

Example 3

Is the proportion 40 miles 6 hours

7 miles 1 hour

true?

The cross-products are 40 × 1 = 40 and 6 × 7 = 42. Since they are not equal, this proportion is false. Solving Proportions Many times, a part of the proportion is unknown. By multiplying out the cross-products and letting the unknown be represented by (?), it is possible to solve for the unknown. The procedure for solving for the unknown quantity (?) consist of cross-multiplying, and then dividing the known product by the factor that is multiplied by the unknown (?). The procedure is given on the following page.

PROCEDURE FOR SOLVING FOR AN UNKNOWN IN A PROPORTION 1.

Multiply out the cross-products and form an equation by letting the cross products equal each other. The unknown number is represented with (?).

2.

Divide the known product by the factor that is multiplied by (?).

Example 4

Solve the proportion 60 miles 6 hours

5 miles (?) hours

.

The cross-products are 60 × (?) and 6 × 5 . An equation is formed by letting the cross-products equal each other. This results in 60 × (?) = 6 × 5 which results in 60 × (?) = 30. Now, divide the known product, 30, by 60. (?) = 30 ÷ 60 = 0.50 hours Example 5

Solve the proportion 9 cups flour 4 people

(?) cups flour 10 people

.

The cross-products are 4 × (?) and 9 × 10 . An equation is formed by letting the cross-products equal each other. This results in 4 × (?) = 9 × 10 which results in 4 × (?) = 90. Now, divide the known product, 90, by 4. (?) = 90 ÷ 4 = 22.5 cups flour or 22 1 cups flour. 2

Example 6

Solve the proportion 4 quarts : 1 gallon = (?) quarts : 2.5 gallons The fraction form of this proportion is 4 quarts

1 gallon

(?) quarts . 2.5 gallons

An equation is formed by letting the cross-products equal each other. This results in 4 × 2.5 = 1 × (?) which results in 10 = 1 × (?). Dividing 10 by 1 results in (?) = 10 quarts.

Applications Involving Ratio and Proportion At the beginning of this chapter, the problem given was “If a recipe for 4 people calls for 1 2 cups of flour, how much flour is needed for a recipe that feeds 15 people? To solve 2 application problems involving ratio and proportion, first it is necessary to set up a correct proportion with units included. The procedure for solving for an unknown quantity in the proportion can then be used. This procedure is given here. PROCEDURE TO SOLVE APPLICATION PROBLEMS IN RATIO 1.

Set up a correct proportion which includes all units. The units on the left side of the proportion should be the same as the units on the right side. Use (?) to represent the unknown quantity.

2.

Multiply out the cross-products and form an equation with the cross products equal to each other. The unknown number is represented with (?).

3.

Divide the known product by the factor that is multiplied by (?).

Example 7

If a recipe for 4 people calls for 2 1 cups of flour, how much flour is 2 needed for a recipe that feeds 15 people? The correct proportion is

2

1 cups flour 2 4 people

(?) cups flour 15 people

.

The cross-products are 4 × (?) and 2 1 × 15 . 2

The equation with cross-products equal to each other is 4 × (?) = 2 1 × 15. 2

The cross product 2 1 × 15 is equal to 2.5 × 15 = 37.5 . 2

Thus, 4 × (?) = 37.5 . Now, divide the known product, 37.5, by 4. (?) = 37.5 ÷ 4 = 9.375 cups flour. If all of these calculations were done in fraction form, the answer obtained would be 9 3 cups of flour. 8

Example 8

16 gallons of gas are used for each gallon of oil in an outboard motor. If three gallons of gas are added to the tank, how many gallons of oil are added? The correct proportion is 16 gallons gas 1 gallon oil

3 gallons gas (?) gallons oil

.

The cross-products are 16 × (?) and 3 × 1 . The equation with cross-products equal to each other is 16 × (?) = 3 × 1 which results in 16 × (?) = 3. Now, divide the known product, 3, by 16. (?) = 3 ÷ 16 = 0.1875 gallons of oil. This answer may also be represented in fraction form as 3 of a gallon of 16 oil since 3 ÷ 16 = 3 . 16