Ratio and Proportion

One of the most important applications of fractional equations is ratio and proportion. A ratio is a comparison of two like quantities by division. It is written by separating the quantities by a colon or by writing them as a fraction. To write a ratio, the two quantities compared must be

of the same kind. For example, the ratio of $8 to $12 is written as $8:$12 or. Two unlike quantities cannot be compared by a ratio. For example, 1 inch and 30 minutes cannot form a ratio. However, two different units can be compared by a ratio if they measure the same kind of quantity. For example, 1 minute and 30 seconds can form a ratio, but they must first be converted to the same units. Since 1 minute equals 60 seconds, the ratio of 1 minute to 30 seconds is written 60 seconds:30 seconds, or which equals 2:1 or 2. A proportion is a statement of equality between two ratios. For example, if a car travels 40 miles in 1 hour and 80 miles in 2 hours, the ratio of the distance traveled is 40 miles:80 miles, or and the ratio of time is 1 hour:2 hours, or . The proportion relating these two ratios is:

A proportion consists of four terms. The first and fourth terms are called the extremes of the proportion; the second and third terms are called the means. If the letters a, b, c and d are used to represent the terms in a proportion, it can be written in general form.

Multiplication of both sides of this equation by bd results in the following.

Thus, the product of the extremes of a proportion (ad) equals the product of the means (bc). For example, in the proportion 40 miles:80 miles = 1 hour:2 hours, the product of the extremes is (40 miles)(2 hours) which equals 80 miles-hours, and the product of the means is (80 miles)(1 hour), which also equals 80 miles-hours.

Ratio and proportion are familiar ideas. Many people use them without realizing it. When a recipe calls for 1¹/2 cups of flour to make a serving for 6 people, and the cook wants to determine how many cups of flour to use to make a serving for 8 people, she uses the concepts of ratios and proportions. When the price of onions is 2 pounds for 49 cents and the cost of 3¹/2 pounds is computed, ratio and proportion are used. Most people know how to solve ratio and proportion problems such as these without knowing the specific steps used.

Ratio and proportion problems are solved by using an unknown such as x for the missing term. The resulting proportion is solved for the value of x by setting the product of the extremes equal to the product of the means.

Example 1:

Solve the following proportion for x.

Solution:

5:x = 4:15

The product of the extremes is (5)(15) = 75. The product of the means is (x)(4) = 4x. Equate these two products and solve the resulting equation.

Example 2:

If 5 pounds of apples cost 80 cents, how much will 7 pounds cost?

Solution:

Using x for the cost of 7 pounds of apples, the following proportion can be written.

The product of the extremes is (5)(x) = 5x. The product of the means is (7)(80) = 560. Equate these two products and solve the resulting equation.

The unit of x is cents. Thus, 7 pounds of apples cost 112 cents or $1.12.

Example 3:

A recipe calls for cups of flour to make servings for 6 people. How much flour should be used to make servings for 4 people?

Solution:

Using x for the flour required for 4 people, the following proportion can be written.

The product of the extremes is (6)(x) = 6x.

The product of the means is (4) = 6.

Equate these two products and solve the resulting equation.

The unit of x is cups. Thus, servings for 4 people require 1 cup of flour.

Summary

The important information in this chapter is summarized below.

Linear Equations Summary

There are four axioms used in solving linear equations.

Axiom 1. If the same quantity is added to both sides of an equation, the resulting equation is still true.

Axiom 2. If the same quantity is subtracted from both sides of an equation, the resulting equation is still true.

Axiom 3. If both sides of an equation are multiplied by the same quantity, the resulting equation is still true.

Axiom 4. If both sides of an equation are divided by the same quantity, except 0, the resulting equation is still true.

Axiom 1 is called the addition axiom; Axiom 2, the subtraction axiom; Axiom 3, the multiplication axiom; and Axiom 4, the division axiom.