# Percents

## Solve Proportions and their Applications


### Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates 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  is to  as  is to 
If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion  we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
Look at the proportions  and  From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?
To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross product because of the cross formed. If, and only if, the given proportion is true, that is, the two sides are equal, then the cross products of a proportion will be equal.
Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are both equal, we have a proportion.

### Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.
When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions. 
We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

### Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

### Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.
For example,  and we can simplify  Since the equation  shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:
If we restate the problem in the words of a proportion, it may be easier to set up the proportion:
We could also say:
First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

### Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.
In the next example, the percent is more than  which is more than one whole. So the unknown number will be more than the base.
Percents with decimals and money are also used in proportions.

### Key Concepts

1. Proportion

2. Cross Products of a Proportion

3. Percent Proportion


### Section Exercises


### Practice Makes Perfect

Use the Definition of Proportion 
In the following exercises, write each sentence as a proportion.
In the following exercises, determine whether each equation is a proportion.
Solve Proportions 
In the following exercises, solve each proportion.
Solve Applications Using Proportions 
In the following exercises, solve the proportion problem.
Write Percent Equations as Proportions 
In the following exercises, translate to a proportion.
Translate and Solve Percent Proportions 
In the following exercises, translate and solve using proportions.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


### Chapter Review Exercises


### Understand Percent

In the following exercises, write each percent as a ratio.
In the following exercises, write as a ratio and then as a percent.
In the following exercises, convert each percent to a fraction.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each percent to ⓐ a simplified fraction and ⓑ a decimal.
In the following exercises, convert each decimal to a percent.
In the following exercises, convert each fraction to a percent.
In the following exercises, translate and solve.

### Solve General Applications of Percents

In the following exercises, solve.
In the following exercises, solve.

### Solve Sales Tax, Commission, and Discount Applications

In the following exercises, find ⓐ the sales tax ⓑ the total cost.
In the following exercises, find the sales tax rate.
In the following exercises, find the commission.
In the following exercises, find the rate of commission.
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)
In the following exercises, find 

### Solve Simple Interest Applications

In the following exercises, solve the simple interest problem.

### Solve Proportions and their Applications

In the following exercises, write each sentence as a proportion.
In the following exercises, determine whether each equation is a proportion.
In the following exercises, solve each proportion.
In the following exercises, solve the proportion problem.
In the following exercises, translate to a proportion.
In the following exercises, translate and solve using proportions.
In the following exercises, convert each percent to ⓐ a decimal ⓑ a simplified fraction.
In the following exercises, convert each fraction to a percent. (Round to  decimal places if needed.)
In the following exercises, solve the percent problem.