Distributive Property of Multiplication with Fractions and Variables – Free Exercises
Instructions: Below you will find free exercises on the distributive property of multiplication with fractions and variables. The answers and explanations are given in the next section. If you want to see the examples first, please scroll down.
1) 6(4x + 7y + 5) = ?
2) Perform the operation: 5(2x – 8y + 4)
3) Simplify: 4(3x + 2x + 4y + 5y + 3 + 7)
4) –7(3x – 6y + 4) = ?
5) Simplify the equation that follows:
Answers to the Distributive Property with Fractions and Variables Exercise
Problem 1:
Perform the distributive property of multiplication with the variable by multiplying each item inside the parentheses by 6 as shown below:
6(4x + 7y + 5) =
(6 × 4x) + (6 × 7y) + (6 ×5) =
24x + 42y + 30
Problem 2:
Multiply each item inside the parentheses by 5 as shown below:
5(2x – 8y + 4) =
(5 × 2x) – (5 × 8y) + (5 × 4) =
10x – 40y + 20
Problem 3:
Add the like terms together as shown below:
4(3x + 2x + 4y + 5y + 3 + 7) =
4(5x + 9y + 10)
Then carry out the distributive property by multiplying the terms, variables or integers inside the parentheses by 4.
4(5x + 9y + 10) =
(4 × 5x) + (4 × 9y) + (4 × 10) =
20x + 36y + 40
Problem 4:
Be careful with the negative sign when you multiply in this problem.
–7(3x – 6y + 4) =
(–7 × 3x) + (–7 × –6y) + (– 7 × 4) =
–21x + 42y – 28
Problem 5:
Here we have a problem on the distributive property with fractions.
Distributive Property of Multiplication – Step by Step
The distributive property of multiplication with variables and fractions is performed when the variable or number in front of a set of parentheses is multiplied by each item inside the parentheses.
The “items” within this type of algebra problem are called terms.
A term consists of a variable, number, or variable and number combination.
All of the following are examples of terms:
a
4
xy
5b
Terms also include decimals and fractions. Terms can be positive as well as negative.
Algebraic Distribution – Example
You might see a question like the following one on your entrance exam:
Distribute the number 3 over the terms 5x + 2y + 3
Remember that the word “distribute” means multiply.
STEP 1: Set up the problem in equation form.
3(5x + 2y + 3)
STEP 2: Multiply each term inside the parentheses by the term in front of the parentheses.
(3 × 5x) + (3 × 2y) + (3 × 3)
STEP 3: Perform the multiplication to solve.
(3 × 5x) + (3 × 2y) + (3 × 3) =
15x + 6y +9
Tip: Add Before Distributing
Check to see if any of the terms inside the parentheses can be added together before you do the multiplication.
You can add numbers together and variables together.
For example: 2(2x + 3x + 2 + 5) = ?
2(2x + 3x + 2 + 5) =
2(5x + 7) =
10x + 14
Distribution of Negatives
If the term in front of the parentheses is negative, each term inside the parentheses has to change its sign.
This means that positive terms inside the parentheses become negative, and negative terms become positive.
Look at this example:
−5(6x − 3y + 4) =
(−5 × 6x) + (−5 × −3y) + (−5 × 4) =
−30x + 15y − 20
Instructions on Distribution Problems
As shown above, the instructions to your problem may tell you to distribute a number over the terms.
You may also see instructions that are worded slightly differently, like the following examples.
Simplify:
Perform the operation:
3(2x + 3y)= ?
In the last example, the question mark indicates that you need to perform algebraic distribution.
Distribution – More Help
If you have had difficulties solving the above problem, please visit our pages on fractions and finding the lowest common denominator.
We will look at distribution again in our post on exponent properties.