# Rational Expressions and Functions

## Solve Applications with Rational Equations


### 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.”
Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved rational equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.
Notice in the last example that when we cleared the fractions by multiplying by the LCD, the result is the same as if we had cross-multiplied.
For any proportion,  we get the same result when we clear the fractions by multiplying by the LCD as when we cross-multiply.
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 each other and the units in the denominators must also match each other.

### 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 have the same ratio.
For example, the two triangles in  are similar. Each side of  is four 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.
On the map, Seattle, Portland, and Boise form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. The actual distance from Seattle to Boise is 400 miles.
We can use similar figures to find heights that we cannot directly measure.

### Solve Uniform Motion Applications

We have solved uniform motion problems using the formula  in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula  assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula 
We have also explained how flying with or against the wind affects the speed of a plane. We will revisit that idea in the next example.
In the next example, we will know the total time resulting from travelling different distances at different speeds.
Once again, we will use the uniform motion formula solved for the variable 

### Solve Work Applications

The weekly gossip magazine has a big story about the Princess’ baby and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours to do the job and Press #2 takes 12 hours to do the job. How long will it take the printer to get the magazine printed with both presses running together?
This is a typical ‘work’ application. There are three quantities involved here—the time it would take each of the two presses to do the job alone and the time it would take for them to do the job together.
If Press #1 can complete the job in 6 hours, in one hour it would complete  of the job.
If Press #2 can complete the job in 12 hours, in one hour it would complete  of the job.
We will let t be the number of hours it would take the presses to print the magazines with both presses running together. So in 1 hour working together they have completed  of the job.
We can model this with the word equation and then translate to a rational equation. To find the time it would take the presses to complete the job if they worked together, we solve for 
A chart will help us organize the information. We are looking for how many hours it would take to complete the job with both presses running together.
Keep in mind, it should take less time for two presses to complete a job working together than for either press to do it alone.

### Solve Direct Variation Problems

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.
Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation
Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.
In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.
We’ll list the steps here.
Now we’ll solve an application of direct variation.

### Solve Inverse Variation Problems

Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is 
The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is 
We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.

### Key Concepts

1. A proportion is an equation of the form  where  The proportion is read “a is to b as c is to d.”
2. Property of Similar Triangles
If  is similar to  then their corresponding angle measure are equal and their corresponding sides have the same ratio.



3. Direct Variation


4. Inverse Variation



### Practice Makes Perfect

Solve Proportions
In the following exercises, solve each proportion.
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises, the triangles are similar. Find the length of the indicated side.
In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.
In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.
In the following exercises, answer each question.
Solve Uniform Motion Applications
In the following exercises, solve the application problem provided.
Solve Work Applications
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
In the following exercises, write an inverse variation equation to solve the following problems.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?