<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Solve proportions
  • Solve similar figure applications

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

  1. Solve n 3 = 30 .
    If you missed this problem, review [link] .
  2. The perimeter of a triangular window is 23 feet. The lengths of two sides are ten feet and six feet. How long is the third side?
    If you missed this problem, review [link] .

Solve proportions

When two rational expressions are equal, the equation relating them is called a proportion .

Proportion

A proportion    is an equation of the form a b = c d , where b 0 , d 0 .

The proportion is read “ a is to b , as c is to d .”

The equation 1 2 = 4 8 is a proportion because the two fractions are equal. The proportion 1 2 = 4 8 is read “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:

1 teacher 20 students = x teachers 60 students

We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations . We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.

.
Multiply both sides by the LCD, 60. .
Simplify. .
The principal needs 3 teachers for 60 students.

Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.

Solve the proportion: x 63 = 4 7 .

Solution

.
To isolate x , multiply both sides by the LCD, 63. .
Simplify. .
Divide the common factors. .
Check. To check our answer, we substitute into the original proportion.
.
. .
Show common factors. .
Simplify. .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the proportion: n 84 = 11 12 .

77

Got questions? Get instant answers now!

Solve the proportion: y 96 = 13 12 .

104

Got questions? Get instant answers now!

When we work with proportion    s, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?

Solve the proportion: 144 a = 9 4 .

Solution

.
Multiply both sides by the LCD. .
Remove common factors on each side. .
Simplify. .
Divide both sides by 9. .
Simplify. .
Check.
.
. .
Show common factors. .
Simplify. .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the proportion: 91 b = 7 5 .

65

Got questions? Get instant answers now!

Solve the proportion: 39 c = 13 8 .

24

Got questions? Get instant answers now!

Solve the proportion: n n + 14 = 5 7 .

Solution

.
Multiply both sides by the LCD. .
Remove common factors on each side. .
Simplify. .
Solve for n . .
.
Check.
.
. .
Simplify. .
Show common factors. .
Simplify. .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve the proportion: y y + 55 = 3 8 .

33

Got questions? Get instant answers now!

Solve the proportion: z z 84 = 1 5 .

14

Got questions? Get instant answers now!
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask