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What is the Associative Property?

Rowena-Renee Xymines, Jennifer Beddoe, Robert Ferdinand
  • Author
    Rowena-Renee Xymines

    With a BSc (Hons) Mathematics from the Northern Caribbean University, I have served as a mathematics teacher, tutor, content creator, and instructional coach for over 9 years. This has seen me guiding both students and teachers of mathematics to mastery of the subject.

  • Instructor
    Jennifer Beddoe

    Jennifer has an MS in Chemistry and a BS in Biological Sciences.

  • Expert Contributor
    Robert Ferdinand

    Robert Ferdinand has taught university-level mathematics, statistics and computer science from freshmen to senior level. Robert has a PhD in Applied Mathematics.

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

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Principle Properties in Mathematical Operations

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

Distributive 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}

Commutative Property

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}

Associative Property

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.

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What is the Associative Property?

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}

Associative Property of Addition

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.

Example 1 - Numeric:

{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}

Example 2 - Worded:

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.

Associative Property of Multiplication

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}

Example 1 - Numeric:

{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}

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  • Activities
  • FAQs

Associative Property Practice Problems

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

Key Terms

  • 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)

Materials Needed

  • Paper
  • Pencil

Example

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.

Problems

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

  1. 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).
  2. Demonstrate the associate property of multiplication by multiplying the three numbers -4, -3 and -5 (in that order).
  3. Does the associate property hold for division? Try it out using the numbers, 8, 4 and 2 (in that order).

Solutions

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

What is an example of associative property of multiplication?

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)

What is associative property example?

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

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

What is associative and commutative property?

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.

What is associative property of addition in math?

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.

What is associative property formula?

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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