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# Distributive Property

## The distributive property dictates that when multiplying, the multiplication of real numbers distributes over two or more terms in parentheses. Learn more.

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Practice Distributive Property
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Distributive Property

Remember Kyle from the Expressions for the Product of a Number and a Sum Concept? Well, he wrote a numerical expression for the situation at the science museum, but he didn't evaluate it, which means that he doesn't have an answer to his teacher's question about cost.

Here is what Kyle wrote.

\begin{align*}22(8.95 + 2.00)\end{align*}

But there was more to the problem. Kyle also needed to figure out additional costs.

Kyle knows that there is a way to solve this with the Distributive Property, but he can’t remember exactly what to do.

In this Concept, you will learn how to use the Distributive Property to evaluate numerical expressions. Then we'll revisit this problem.

### Guidance

Previously we worked on how to write numerical expressions, and now you are going to learn how to evaluate those expressions.

What does the word “evaluate” mean?

When we evaluate an expression, we figure out the value of that expression or the quantity of the expression.

When we evaluate expressions that have a product and a sum, we use a property called the Distributive Property.

What is the Distributive Property?

The Distributive Property is a property that is a true statement about how to multiply a number with a sum. Multiply the number outside the parentheses with each number inside the parentheses. Then figure out the sum of those products.

In other words, we distribute the number outside the parentheses with both of the values inside the parentheses and find the sum of those numbers.

Let’s see how this works.

\begin{align*}4(3 + 2)\end{align*}

To use the Distributive Property, we take the four and multiply it by both of the numbers inside the parentheses. Then we find the sum of those products.

\begin{align*}& 4(3) + 4(2) \\ & 12 + 8 \\ & 20\end{align*}

Our answer is 20.

Here is another one.

\begin{align*}8(9 + 4)\end{align*}

Multiply the eight times both of the numbers inside the parentheses. Then find the sum of the products.

\begin{align*}& 8(9) + 8(4) \\ & 72 + 32 \\ & 104\end{align*}

Our answer is 104.

Now it is your turn. Evaluate these expressions using the Distributive Property.

#### Example A

\begin{align*}5(6 + 3)\end{align*}

Solution: 45

#### Example B

\begin{align*}2(8 + 1)\end{align*}

Solution: 18

#### Example C

\begin{align*}12(3 + 2)\end{align*}

Solution: 60

Now we can take the expression that Kyle wrote and use the Distributive Property to figure out the total amount of money needed for the trip.

\begin{align*}& 22(8.95 + 2) \\ & 22(8.95) + 22(2)\end{align*}

Next, we can multiply 22 by 8.95.

\begin{align*}& \quad \ \ \ \ 895 \\ & \underline{\times \quad \ \ \ 22 \;} \\ & \quad \ \ 1790 \\ & \underline{+ \ 1790 \; \; \;} \\ & \ \ \ 196.90 \ \text{this is the amount of all of the tickets}.\end{align*}

Next, we complete the second part of the problem.

2(22) = 44

It will cost the students an additional $44.00 to attend the Omni Theater. The good news is that there is enough money in the student account to help cover the additional costs. There are fifty dollars in the account and the class only needs$44.00 to help cover the costs.

The total amount of money needed is \$240.90.

Kyle gives his information to Mrs. Andersen and she is thrilled! Now the students are off to the Science Museum and the Omni Theater!

### Guided Practice

Here is one for you to try on your own.

Use the distributive property to evaluate this expression.

\begin{align*}4(9 + 2)\end{align*}

First, we can distribute the four and multiply it by each value in the parentheses. Then we can add.

\begin{align*}36 + 8 = 44\end{align*}

This is our answer.

### Video Review

This video presents the distributive property from whole numbers to more complicated algebraic expressions.

### Explore More

Directions: Evaluate each expression using the Distributive Property.

1. 4(3 + 6)

2. 5(2 + 8)

3. 9(12 + 11)

4. 7(8 + 9)

5. 8(7 + 6)

6. 5(12 + 8)

7. 7(9 + 4)

8. 11(2 + 9)

9. 12(12 + 4)

10. 12(9 + 8)

11. 10(9 + 7)

12. 13(2 + 3)

13. 14(8 + 6)

14. 14(9 + 4)

15. 15(5 + 7)

### Vocabulary Language: English

Evaluate

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Numerical expression

Numerical expression

A numerical expression is a group of numbers and operations used to represent a quantity.
Product

Product

The product is the result after two amounts have been multiplied.
Property

Property

A property is a rule that works for a given set of numbers.
Sum

Sum

The sum is the result after two or more amounts have been added together.

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