# Rational Expressions and Equations

## Solve Proportion and Similar Figure Applications



### Solve Proportions

When two rational expressions are equal, the equation relating them is called a proportion.
The equation  is a proportion because the two fractions are equal. The proportion  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:
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.
Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.
When we work with proportions, 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?
To solve applications with proportions, we will follow our usual strategy for solving applications. But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match and the units in the denominators must match.
In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos is proportional to the number of dollars. If two quantities are related by a proportion, we say that they are proportional.


### Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.
For example, the two triangles in  are similar. Each side of  is 4 times the length of the corresponding side of .
This is summed up in the Property of Similar Triangles.
To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.
The next example shows how similar triangles are used with maps.
We can use similar figures to find heights that we cannot directly measure.

### Key Concepts

1. Property of Similar Triangles


2. Problem Solving Strategy for Geometry Applications



### Practice Makes Perfect

Solve Proportions
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises,  is similar to . Find the length of the indicated side.
In the following exercises,  is similar to .
In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.
In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?