The Distributive Property

Learning Objectives

Introduction

The distributive property of multiplication is a very useful property that lets you simplify expressions in which you are multiplying a number by a sum or difference. The property states that the product of a sum or difference, such as `6(5 - 2)`, is equal to the sum or difference of the products; in this case, `6(5)-6(2)`. 

Remember that there are several ways to write multiplication. `3text( x )6=3(6)=3*6`

`3 * (2 + 4) = 3 * 6 = 18`

Distributive Property of Multiplication over Addition

The distributive property of multiplication over additionThe product of a number and a sum is the same as the sum of the product of the number and each of the addends making up the sum. For example, `3(4 + 2) = 3(4) + 3(2)`. can be used when you multiply a number by a sum. For example, suppose you want to multiply `3` by the sum of `10 + 2`.

`3(10 + 2)=?` 

According to this property, you can add the numbers and then multiply by `3`.

`3(10 + 2) = 3(12) = 36` 

Or, you can first multiply each addend by the `3`. (This is called distributingTo rewrite the product of the number and a sum or difference using the distributive property. the `3`.) Then, you can add the products.

The image shows the expression 3 times open parenthesis 10 plus 2, closed parentheses, equals 3 times open parenthesis 10, closed parenthesis, plus 3 times open parenthesis 2, close parenthesis. Arrows point right from the first 3 on left side of the equal sign to the first 10 and 2 in parentheses.

The multiplication of `3(10)` and `3(2)` will each be done before you add.

`3(10) + 3(2) = 30+6 = 36` 

Note that the answer is the same as before.

You probably use this property without knowing that you are using it. When a group (let’s say `5` of you) orders food, and orders the same thing (let’s say you each order a hamburger for `$3` each and a soda for `$1` each), you can compute the bill (without tax) in two ways. You can figure out how much each of you needs to pay and multiply the sum times the number of you. So, you each pay `(3 + 1)` and then multiply times `5`. That’s `5(3 + 1) = 5(4) = 20`. Or, you can figure out how much the `5` hamburgers will cost and the `5` sodas and then find the total. That’s `5(3) + 5(1) = 15+5 = 20`. Either way, the answer is the same, `$20`.

The two methods are represented by the equations below. On the left side, we add `10` and `2`, and then multiply by `3`. The expression is rewritten using the distributive property on the right side, where we distribute the `3`, then multiply each by `3` and add the results. Notice that the result is the same in each case.

`3(10+2)=3(10)+3(2)`

`3(12)=30+6`

`36=36`

The same process works if the `3` is on the other side of the parentheses, as in the example below.

The image shows the expression open parenthesis 10 plus 2, closed parenthesis, times 3 equals open parenthesis 10, closed parenthesis, times 3 plus open parenthesis 2, closed parenthesis, times 3. Arrows point left from the 3 on the left side of the equal sign to the 10 and 2 in parentheses.

Example

Problem

 

 

Rewrite the expression `5(8 + 4)` using the distributive property of multiplication over addition. Then simplify the result.

The image shows the expression 5 times open parenthesis 8 plus 4, closed parentheses, equals 5 times open parenthesis 8, closed parenthesis, plus 5 times open parenthesis 4, close parenthesis. Arrows point right from the first 5 on left side of the equal sign to the first 8 and 4 in parentheses.

In the original expression, the `8` and the `4` are grouped in parentheses. Using arrows, you can see how the `5` is distributed to each addend. The `8` and `4` are each multiplied by `5`.

`40 + 20 = 60`

The resulting products are added together, resulting in a sum of `60`.

Answer    `5(8 + 4)=5(8) + 5(4)=60` 
     

Rewrite the expression `30(2 + 4)` using the distributive property of addition.

 

A) `30(2+4) + 30(2+4)`

 

B) `30(2) + 30(4)`

 

C) `30(6)`

 

D) `30(24)`

 

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Distributive Property of Multiplication over Subtraction

The distributive property of multiplication over subtractionThe product of a number and a difference is the same as the difference of the product of the number and each of the numbers being subtracted. For example, `8(10 - 2) = 8(10) - 8(2)`. is like the distributive property of multiplication over addition. You can subtract the numbers and then multiply, or you can multiply and then subtract as shown below. This is called “distributing the multiplier.”

The image shows the expression 5 times open parenthesis 6 minus 3, closed parentheses, equals 5 times open parenthesis 6, closed parenthesis, minus 5 times open parenthesis 3, close parenthesis. Arrows point right from the first 5 on left side of the equal sign to the first 6 and 3 in parentheses.

The same number works if the `5` is on the other side of the parentheses, as in the example below.

The image shows the expression open parenthesis 6 minus 3, closed parenthesis, times 5 equals open parenthesis 6, closed parenthesis, times 5 minus open parenthesis 3, closed parenthesis, times 5. Arrows point left from the first 5 on left side of the equal sign to the first 6 and 3 in parentheses.

In both cases, you can then simplify the distributed expression to arrive at your answer. The example below, in which `5` is the outside multiplier, demonstrates that this is true. The expression on the right, which is simplified using the distributive property, is shown to be equal to `15`, which is the resulting value on the left as well.

`5(6-3)=5(6)-5(3)`

`5(3)=30-15`

`15=15`

Example

Problem

 

 

Rewrite the expression `20(9 - 2)` using the distributive property of multiplication over subtraction. Then simplify.

The image shows the expression 20 times open parenthesis 9 minus 2, closed parentheses, equals 20 times open parenthesis 9, closed parenthesis, minus 20 times open parenthesis 2, close parenthesis. Arrows point right from the first 20 on left side of the equal sign to the first 9 and 2 in parentheses.

In the original expression, the `9` and the `2` are grouped in parentheses. Using arrows, you can see how the `20` is distributed to each number so that the `9` and `2` are both multiplied by `20` individually.

`180 - 40 = 140`

Here, the resulting product of `40` is subtracted from the product of `180`, resulting in an answer of `140`.

Answer     `20(9 - 2) = 20(9) - 20(2) = 140`
     

Rewrite the expression `10(15-6)` using the distributive property of subtraction.

 

A) `10(6)-10(15)`

 

B) `10(9)`

 

C) `10(6-15)`

 

D) `10(15)-10(6)`

 

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Summary

The distributive properties of addition and subtraction can be used to rewrite expressions for a variety of purposes. When you are multiplying a number by a sum, you can add and then multiply. You can also multiply each addend first and then add the products. This can be done with subtraction as well, multiplying each number in the difference before subtracting. In each case, you are distributing the outside multiplier to each number in the parentheses, so that multiplication occurs with each number before addition or subtraction occurs. The distributive property will be useful in future math courses, so understanding it now will help you build a solid math foundation.