# Fractions

## Solve Equations with Fractions


### Determine Whether a Fraction is a Solution of an Equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.
The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, or a fraction.

### Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.
We used the Subtraction Property of Equality in . Now we’ll use the Addition Property of Equality.
The next example may not seem to have a fraction, but let’s see what happens when we solve it.

### Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation  We want to know what number divided by  gives  So to “undo” the division, we will need to multiply by  The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
Let’s use the Multiplication Property of Equality to solve the equation 

### Solve Equations with a Coefficient of 

Look at the equation  Does it look as if  is already isolated? But there is a negative sign in front of  so it is not isolated.
There are three different ways to isolate the variable in this type of equation. We will show all three ways in .

### Solve Equations with a Fraction Coefficient

When we have an equation with a fraction coefficient we can use the Multiplication Property of Equality to make the coefficient equal to 
For example, in the equation:
The coefficient of  is  To solve for  we need its coefficient to be  Since the product of a number and its reciprocal is  our strategy here will be to isolate  by multiplying by the reciprocal of  We will do this in .

### Translate Sentences to Equations and Solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Evaluate, Simplify, and Translate Expressions.

### Key Concepts

1. Determine whether a number is a solution to an equation.
2. Addition, Subtraction, and Division Properties of Equality
3. The Multiplication Property of Equality

### Section Exercises


### Practice Makes Perfect

Determine Whether a Fraction is a Solution of an Equation 
In the following exercises, determine whether each number is a solution of the given equation.
Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality 
In the following exercises, solve.
Solve Equations with Fractions Using the Multiplication Property of Equality 
In the following exercises, solve.
Mixed Practice 
In the following exercises, solve.
Translate Sentences to Equations and Solve 
In the following exercises, translate to an algebraic equation and solve.

### 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


### Visualize Fractions

In the following exercises, name the fraction of each figure that is shaded.
In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.
In the following exercises, convert the improper fraction to a mixed number.
In the following exercises, convert the mixed number to an improper fraction.
In the following exercises, locate the numbers on a number line.
In the following exercises, order each pair of numbers, using  or 

### Multiply and Divide Fractions

In the following exercises, simplify.
In the following exercises, multiply.
In the following exercises, find the reciprocal.
In the following exercises, divide.

### Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.
In the following exercises, translate the English phrase into an algebraic expression.
In the following exercises, simplify the complex fraction
In the following exercises, simplify.

### Add and Subtract Fractions with Common Denominators

In the following exercises, add.
In the following exercises, subtract.

### Add and Subtract Fractions with Different Denominators

In the following exercises, find the least common denominator.
In the following exercises, change to equivalent fractions using the given LCD.
In the following exercises, perform the indicated operations and simplify.
In the following exercises, evaluate.

### Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.

### Solve Equations with Fractions

In the following exercises, determine whether the each number is a solution of the given equation.
In the following exercises, solve the equation.
In the following exercises, translate and solve.

### Chapter Practice Test

Convert the improper fraction to a mixed number.
Convert the mixed number to an improper fraction.
Locate the numbers on a number line.
In the following exercises, simplify.
Evaluate.
In the following exercises, solve the equation.