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Chapter 6—Expressions and Equations

6-8 Ratio and Proportion

When you complete the work for this section, you should be able to:
  • Describe the purpose of ratios and demonstrate how to write them.
  • Define proportions and demonstrate how to write them.
  • Demonstrate the procedure known as cross multiplication.
  • Solve simple equations that are expressed as proportions.

 

A First Look at Ratios and Proportions

Definition

A ratio is a comparison of two values. If the values are expressed as a and b, the ratio can be shown as:

a to b     a:b    

where b is not equal (≠) to zero.

Note: In these lessons, we use the fraction notation, , for expressing ratios.

Example

There are 14 guys and 20 girls in the class.

  • The ratio of guys to girls is 14/20 or 7/10.

Examples & Exercises

Expressing Ratios

Use these interactive Examples & Exercises to strengthen your understanding and build your skills:

 

Definition

A proportion is an equation that expresses equality between two ratios:

Where   is one ratio and is the second.

Note:  Proportions such as can be spoken as, "a is to b, as c is to d."

In a proportion such as , a does not necessarily equal c, nor does b necessarily equal d. The ratios, however, are equal.  Consider this simple numerical example: 3/4 = 6/8. The 3 does not equal 6, nor does the 4 equal 8. If you do the division for the two ratios, however, you see that 0.75 = 0.75.  That's obviously true.

Also, if you cross-multiply, you will find the terms are equal.

Cross-Multiplying Proportions

One of the most important and useful characteristics of proportions is the process of cross multiplying.  You can almost say that every time you use proportions, you will be doing a cross-multiplication on some level.

Definition

Cross multiplication is a process performed on proportions where the numerator of each ratio is multiplied by the denominator of the other.

Examples

What happens when a proportion goes bad?

Suppose you see this proportion: 

Is there anything wrong with it? Cross multiply to find out:

3 12 = 4   6

36 = 24 ???  Of course not. This is an false equation. The proportion  is false.

Examples & Exercises

Cross Multiplying Proportions

Use these interactive Examples & Exercises to strengthen your understanding and build your skills:

Proportions are not always expressed with values that happen to be positive integers. Some of the terms may be negative, and some may be decimal fractions. See the following examples.

Examples:

 

Examples & Exercises

Cross Multiplying Proportions

These proportions include some that use negative values and decimal fractions.

 

Solving Proportions

Here is an example of a proportion that includes an unknown value:. This is saying: x is to 12 and 1 is to 4. What is the value of x?

You can probably guess the value of x is 3. But it's important you understand a formal procedure for solving proportions like this.

Step 1. Cross multiply the ratios.

4x = 12

Step 2. Divide both sides of the equal sign by 4, and simplify.

If x truly equals 3 in this proportion, then you should be able to substitute 3 for x in the original problem, and see if the result is a true equation.

Proof Step 1: Substitute 3 for x in the original proportion.

Proof Step 2: Cross multiply and simplify the result.

3 4 = 12 1

12 = 12     Which is a true equation.

This proves that x = 3 in the original problem.

 

Example

Solve this proportion for the value of x.

Step 1: Cross multiply

is the same as 9x = 18

Step 2: Divide both sides by 9 and simplify

x = 2

Solution: Where , x = 2

Check the solution:

Substitute 2 for x in the original proportion.
Cross multiply: 18 = 18
This is a true equation so the solution, x = 2, is correct.

More Examples

In each of the following examples:
  • Cross multiply
  • Divide both sides of the equation by the coefficient of the unknown term
  • Check the answer by substituting it into the original proportion and simplifying the result.
1. Solve

2. Solve

3. Solve

4. Solve

Examples & Exercises

Solving Proportions

Use these interactive Examples & Exercises to strengthen your understanding and build your skills:

David L. Heiserman, Editor

Copyright   SweetHaven Publishing Services
All Rights Reserved

Revised: June 06, 2015