## Table of Contents

- Principle Properties in Mathematical Operations
- What is the Associative Property?
- Associative Property of Addition
- Associative Property of Multiplication
- Why Not Subtraction or Division
- Lesson Summary

What is the associative property? Learn the associative property definition and see specific associative property examples of addition, and multiplication.
Updated: 07/26/2021

- Principle Properties in Mathematical Operations
- What is the Associative Property?
- Associative Property of Addition
- Associative Property of Multiplication
- Why Not Subtraction or Division
- Lesson Summary

In mathematics, there are three basic principles that govern how numbers can be manipulated to solve equations. These properties are:

- The Distributive Property
- The Commutative Property
- The Associative Property

This property states that a number multiplying a sum or a difference can be distributed among the numbers being added or subtracted without affecting the result.

Algebraically:

{eq}a(b+c) = ab+ac {/eq}

OR

{eq}a(b-c)=ab-ac {/eq}

This property states that numbers in an addition or multiplication problem can be rearranged without affecting the result.

Algebraically:

{eq}a \times b\times c=b\times c\times a {/eq}

OR

{eq}a+b+c=b+c+a {/eq}

This property states that the order in which numbers are added or multiplied does not affect the outcome of the operation performed. To further explore what the associative property means, we'll go over it in more detail in this lesson, and review a few associative property examples as we go.

In mathematics, the **associative property** of addition (or multiplication) states that when adding (multiplying) three or more numbers, the sum (product) remains the same regardless of how the numbers are grouped to be added (multiplied).

This definition can also be stated algebraically.

**Associative Property of Addition **

{eq}(a + b) + c = a + (b+c) {/eq}

**Associative Property of Multiplication**

{eq}(a\times b)\times c=a\times (b\times c) {/eq}

When adding three or more numbers, the associative property of addition is a convenient way to simplify the operation without affecting the sum. This simplification can happen because the way in which the numbers are grouped will not affect the sum.

The rule is:

{eq}(a + b) + c = a + (b+c) {/eq}

Now that we've gone over the basics, one might be left wondering what an example of the associative property looks like. Let's go over a few problems to see this property in action.

{eq}16 + 12 + 13 {/eq}

This can be grouped as:

{eq}(16 + 12)+ 13 {/eq}

{eq}=28 + 13 {/eq}

{eq}=41 {/eq}

OR

{eq}16 + (12+ 13) {/eq}

{eq}=16 + 25 {/eq}

{eq}=41 {/eq}

The table below shows the tips earned by a worker. The worker wishes to determine the sum of tips earned in this week.

Day | Tip |
---|---|

Monday | 55 |

Tuesday | 22 |

Wednesday | 28 |

Thursday | 25 |

Friday | 30 |

To find the sum, the worker needs to perform the operation below.

{eq}55+22+28+25+30 {/eq}

At first glance it may seem tedious to add the numbers in the order they are currently presented. However, the worker can apply the associative property as shown below.

{eq}55+(22+28)+(25+30) {/eq}

In this way, the worker could possibly even perform the operation mentally!

{eq}55+(22+28)+(25+30) {/eq}

{eq}=(55+50)+55 {/eq}

{eq}=105+55 {/eq}

{eq}=160 {/eq}

The important thing to remember is that the sum of these five numbers will still be 160, regardless of how the numbers are grouped.

Another option for grouping is shown below.

{eq}(55+22)+28+(25+30) {/eq}

{eq}=77+(28+55) {/eq}

{eq}=77+83 {/eq}

{eq}=160 {/eq}

Even more ways of grouping can be explored to show that the sum remains unchanged.

The associative property also applies to multiplication. The product of a multiplication problem will remain unchanged regardless of now the numbers are grouped. The rule is:

{eq}(a\times b)\times c=a\times (b\times c) {/eq}

{eq}(2\times 5)\times 8 {/eq}

{eq}=10\times8 {/eq}

{eq}=80 {/eq}

OR

{eq}2\times (5\times 8) {/eq}

{eq}=2\times40 {/eq}

{eq}=80 {/eq}

Using the conventions below, the associative property can be applied to the process of determining how many hours are in a year.

24 hours = 1 day |
---|

7 days = 1 week |

4 weeks = 1 month |

12 months = 1 year |

To convert, the product below must be found.

{eq}24\times7\times4\times12 {/eq}

Below are two of the different ways the associative property may be applied to this problem to find the product.

Option 1:

{eq}(24\times7)\times(4\times12) {/eq}

{eq}=168\times48 {/eq}

{eq}=8,064 {/eq}

Option 2:

{eq}24\times(7\times4)\times12 {/eq}

{eq}=24\times(28\times12) {/eq}

{eq}=24\times336 {/eq}

{eq}=8,064 {/eq}

It has been shown how the associative property applies to addition and multiplication. This may raise the question of whether or not the property applies to the other two of the four basic arithmetic operations, subtraction and division.

To determine the answer, examine the numerical examples below.

This example demonstrates that there is no associative property of subtraction.

Though the associative property definition cannot be widely applied to subtraction in the general sense, it is possible to manipulate subtraction problems by rewriting them as the addition of negative numbers in order to use the associative property. Examine the example below.

{eq}6-3-1 {/eq} can be rewritten as:

{eq}6+(-3)+(-1) {/eq}

The associative property definition may then be applied as:

{eq}[6+(-3)]+(-1) = 6+[(-3)+(-1)] {/eq}

{eq}3+(-1) = 6+-4 {/eq}

{eq}2 = 2 {/eq}

To be sure, examine the contrast when the problem is not rewritten as an addition problem.

{eq}(6-3)-1\neq6-(3-1) {/eq}

{eq}3-1\neq6-2 {/eq}

{eq}2\neq4 {/eq}

This example demonstrates that the associative property does not apply to division.

From these examples, we can see that changing the order in which numbers are subtracted or divided changes the answer. This is why the associative property does not work in these cases.

A property common to addition and multiplication is the **associative property** which states that numbers can be grouped in any order without affecting the outcome of the operation. The **associative property of addition** is written algebraically as:

{eq}(a+b)+c=a+(b+c) {/eq}

The **associative property of multiplication** is written as:

{eq}(a \times b) \times c=a \times (b \times c) {/eq}

- Though the associative property does not apply to subtraction, subtraction problems can be manipulated in order to use the property by taking the sum of the negative numbers.
- The associative property does not apply to division.

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- Activities
- FAQs

In this activity, you will put what you have learned about the associative property into practice.

- Operations: The four basic operations are addition, subtraction, multiplication, and division.
- Associative property: Application of this property involves combining terms in expression based on order of appearance (only works for addition and multiplication)

- Paper
- Pencil

Demonstrate the associate property by adding and multiplying the numbers 2, 3 and 5.

**Associate Property of Addition**

When adding 2, 3 and 5, we perform the operation inside the parentheses first to get:

(2 + 3) + 5 = 5 + 5 = 10

Then:

2 + (3 + 5) = 2 + 8 = 10

Hence, from above:

(2 + 3) + 5 = 2 + (3 + 5)

We have shown the associative property is true for addition of these three numbers.

**Associate Property of Multiplication**

When multiplying 2, 3 and 5, we perform the operation inside the parentheses first to get:

(2 * 3) * 5 = 6 * 5 = 30

Then:

2 * (3 * 5) = 2 * 15 = 30

Hence, from above:

(2 * 3) * 5 = 2 * (3 * 5)

We have shown the associative property is true for multiplication of these three numbers.

Practice using the associative property using the procedure outlined above (show your work).

- Demonstrate the associate property of addition by adding the three numbers 4, -3 and 11 (in that order).
*Hint: you can add 4 and -3 as 4 + (-3).* - Demonstrate the associate property of multiplication by multiplying the three numbers -4, -3 and -5 (in that order).
- Does the associate property hold for division? Try it out using the numbers, 8, 4 and 2 (in that order).

- The sum should be 12 for both associations of addition.
- The product should be -60 for both associations of multiplication.
- No. Since (8/4)/2 = 2/2 = 1 and 8/(4/2) = 8/2 = 4.

The associative property of multiplication states that when three or more numbers are multiplied, the product will not be affected regardless of how they are grouped. For example: (12*3)*4 = 12*(3*4)

An example of the associative property of addition is as follows.

(10 + 5) + 12 = 10 + (5 +12) = 27.

The associative property states that the way in which three or more numbers being added or multiplied are grouped will not affect the outcome. That is, (x + y) + z = x + (y + z). The commutative property states that three or more numbers can be rearranged in any order when adding or multiplying without affecting the outcome. That is, a + b + c = c + a + b.

The associative property of addition states that when three are more numbers are added, the way in which they are grouped will not affect the sum. This property allows us to rearrange addition problems to make them easier for us to solve.

The associative property of addition: (a + b) + c = a + (b + c)

The associative property of multiplication: (a * b) * c = a * (b * c)

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