Kathleen is getting in shape for the “Relay for Life” walk. She tells her brother, Mack, that on Monday she walked 4 miles and on Tuesday one-third as many miles as she walked on Wednesday, for a total of 24 miles. How can Mack create an equation to figure out how far his sister walked on each day?

In this concept, you will learn to solve multi-step equations involving fractions.

### Multi Step Equations with Fractions

Before you begin to solve equations involving fractions, you may have to review the rules for operations with fractions. Here are a few tips on how to perform operations with fractions:

- To add or subtract fractions they must have the same denominator.
- To multiply fractions you multiply the numerators and write the product over the product of the denominators.
- To divide fractions you multiply the first fraction by the reciprocal of the fraction after the division sign.

Let’s look at a problem with fractions before you begin solving equations involving fractions.

First, perform the addition and subtraction in the first set of parenthesis. The least common denominator is 12.

Next, multiply each fraction by multiplying the numerators and the denominators.

Next, express the fractions as a single fraction.

Then, do the addition and subtraction in the numerator.

This answer is .

Next, perform the division in the second set of parenthesis.

First, express the division as the first fraction multiplied by the reciprocal of the second fraction.

Then, multiply the numerators and multiply the denominators.

This answer is .

Next, multiply the answers from each set of parenthesis.

Remember, multiply the numerators and multiply the denominators.

Then, simplify the answer.

The answer is .

Let’s look at solving an equation involving fractions.

Solve the following equation for ‘’.

First, isolate the variable ‘’ by adding to both sides of the equation.

Next, simplify both sides of the equation.

Next, simplify the left side of the equation by performing the subtraction.

Next, multiply both sides of the equation by 12.

Then, divide both sides of the equation by 6.

The answer is .

Some equations involving fractions will require you to apply the distributive property to clear parenthesis as you solve the equation.

Solve the following equation:

First, clear the parenthesis by applying the distributive property.

Next, isolate the variable by subtracting from both sides of the equation.

Next, simplify each side of the equation.

Next, multiply both sides of the equation by 15.

Then, divide both sides of the equation by 10.

The answer is .

### Examples

#### Example 1

Earlier, you were given a problem about Kathleen who was walking for “Relay for Life?”

Her brother, Mack, wants to write an equation to figure out how many miles she walked each day.

First, name the variable. Let ‘’ represent the number of miles Kathleen walked on Wednesday.

Next, represent “*On Tuesday Kathleen walked one-third as many miles as… on Wednesday*.”

Next, represent the information given in the problem using a verbal model.

Verbal Model:

Next, write an equation to represent the verbal model.

Next, solve the equation for the variable.

First, simplify the left side of the equation.

Next, isolate the variable by subtracting 4 from both sides of the equation.

Next, simplify both sides of the equation.

Next, divide both sides of the equation by to solve for ‘’.

The answer is .

Kathleen walked 15 miles on Wednesday.

Then, use the answer to figure the number of miles Kathleen walked on Tuesday.

The answer is 5.

Kathleen walked 5 miles on Tuesday.

#### Example 2

Solve the following equation for the variable:

First, add the fractions on the left side of the equation. Remember the fractions must have a common denominator.

Next, isolate the variable by subtracting from both sides of the equation.

Next, simplify both sides of the equation.

Then, divide both sides of the equation by -1 to solve for ‘’. When you solve an equation you are finding the value of .

The answer is .

#### Example 3

Solve the following equation for the variable:

First, subtract the fractions on the left side of the equation. Remember the fractions must have a common denominator.

Next, express the mixed number as an improper fraction.

Next isolate the variable by subtracting from both sides of the equation.

Next, simplify both sides of the equation.

Then, express the answer in simplest form.

The answer is .

#### Example 4

Solve the following equation for the variable:

First, apply the distributive property to clear the parenthesis.

Next, isolate the variable by subtracting from both sides of the equation.

Next, simplify both sides of the equation.

Next, multiply both sides of the equation by 5.

Then, divide both sides of the equation by 2 to solve for ‘’.

The answer is .

#### Example 5

Solve the following equation for the variable.

First, isolate the variable by adding to both sides of the equation.

Next, simplify both sides of the equation.

Next, divide both sides of the equation by to solve for ‘’.

Then simplify the answer.

The answer is .

### Review

Solve the following equations:

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.10.