commutative property of multiplication problem solving

Commutative Property of Multiplication

The commutative property of multiplication states that the product of two or more numbers remains the same irrespective of the order in which they are placed. For example, 3 × 4 = 4 × 3 = 12. Let us study more about the commutative property of multiplication in this article.

What is the Commutative Property of Multiplication?

According to the commutative law of multiplication , if two or more numbers are multiplied, we get the same result irrespective of the order of the numbers. Here, the order of the numbers refers to the way in which they are arranged in the given expression. Observe the following example to understand the concept of the commutative property of multiplication.

5 × 6 = 6 × 5 30 = 30

Here, we can observe that even when the order of the numbers is changed, the product remains the same. This means 5 × 6 = 30; and 6 × 5 = 30.

Commutative Property of Multiplication Formula

The commutative property formula for multiplication shows that the order of the numbers does not affect the product. The commutative property of multiplication applies to integers, fractions, and decimals .

The Commutative property multiplication formula is expressed as:

A × B = B × A

According to the commutative property of multiplication, the order in which we multiply the numbers does not change the final product.

This can be applied to two or more numbers and the order of the numbers can be shuffled and arranged in any way.

Example: 5 × 3 × 2 × 10 = 10 × 2 × 5 × 3 = 300. We can see that even after we shuffle the order of the numbers, the product remains the same.

Commutative Property of Multiplication and Addition

The commutative property is applicable to multiplication and addition .

• For Addition : The Commutative law for addition is expressed as A + B = B + A. For example, (7 + 4) = (4 + 7) = 11. This shows that even after we change the order of the numbers, 7 and 4, the sum remains the same.
• For Multiplication : The Commutative law of multiplication is expressed as A × B = B × A. For example, (7 × 4) = (4 × 7) = 28. Here, we can see that the product of the numbers remains the same even when the order of the numbers is changed.

It should be noted that the Commutative property of multiplication is not applicable to subtraction and division.

Tips on the Commutative Property of Multiplication:

Here are a few important points related to the Commutative property of multiplication.

• The commutative property of multiplication and addition is only applicable to addition and multiplication. It cannot be applied to division and subtraction .
• The commutative property of multiplication and addition can be applied to 2 or more numbers.

☛ Related Articles

• Associative Property of Multiplication
• Multiplicative Identity Property
• Distributive Property of Multiplication
• Zero Property of Multiplication
• Associative Property of Addition
• Distributive Property
• Additive Identity Property

Examples of Commutative Property of Multiplication

Example 1: Fill in the missing number using the commutative property of multiplication: 6 × 4 = __ × 6.

According to the commutative property of multiplication formula, A × B = B × A. So, let us substitute the given values in this formula and check. (6 × 4) = (4 × 6) = 24. Hence, the missing number is 4.

Example 2: Shimon's mother asked him whether p × q = q × p is an example of the commutative property of multiplication. Can you help Shimon to find out whether it is commutative or not?

We know that the commutative property for multiplication states that changing the order of the multiplicands does not change the value of the product. pq = qp So, we see that changing the order will not alter the product value. So this is an example of the commutative property.

Answer: p q = q p is an example of the commutative property of multiplication.

Example 3: Which of the expressions follows the commutative property of multiplication?

a.) 7 × 8 × 5 × 6

b.) 4 × (- 2)

a.) Let us find the product of the given expression. It comes to 7 × 8 × 5 × 6 = 1680.

Now, let us reverse the order of the numbers and find the product of the numbers. It comes to 6 × 5 × 8 × 7 = 1680.

Both the products are the same. Therefore, the given expression follows the commutative property of multiplication because it shows that even when we changed the order of the numbers the product remains the same.

b.) Let us find the product of the given expression, 4 × (- 2) = -8. Now, let us reverse the order of the numbers and check, (- 2) × 4 = -8. This shows that the given expression follows the commutative property of multiplication.

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Practice Questions on Commutative Property of Multiplication

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FAQs on Commutative Property of Multiplication

The commutative law of multiplication states that the product of two or more numbers remains the same, irrespective of the order of the operands. For multiplication , the commutative property formula is expressed as (A × B) = (B × A). The commutative property of multiplication applies to integers, fractions, and decimals.

How do you find the Commutative Property of Multiplication?

The commutative property of multiplication states that if 'a' and 'b' are two numbers, then a × b = b × a. If the product of the values on the Left-hand side (LHS) and the product of the values on the right-hand side (RHS) terms is equal, then it can be said that the given expression follows the commutative property of multiplication.

What is an Example of Commutative Property of Multiplication?

An example of the commutative property of multiplication can be seen as follows. We know that (A × B) = (B × A). Let us substitute the value of A = 8 and B = 9. On substituting these values in the formula we get 8 × 9 = 9 × 8 = 72. Hence it is proved that the product of both the numbers is the same even when we change the order of the numbers. This means, if we have expressions such as, 6 × 8, or 9 × 7 × 10, we know that the commutative property of multiplication will be applicable to it.

What is the Commutative Property of Multiplication for the Numbers 7 and 6?

Let us arrange the given numbers as per the general equation of commutative law that is (A × B) = (B × A). Here A = 7 and B = 6. After substituting the values in the formula, we get 7 × 6 = 6 × 7 = 42. Hence, 6 × 7 follows the commutative property of multiplication.

What is the Commutative Property of Multiplication for Rational Numbers?

The commutative property of multiplication for rational numbers can be expressed as (P × Q) = (Q × P). Here the values of P, Q are in form of a/b, where b ≠ 0.

What is the Commutative Property of Multiplication for Fractions?

The commutative property of multiplication for fractions can be expressed as (P × Q) = (Q × P). Let us substitute the values of P, Q in the form of a/b. For example, if, P = 7/8 and Q = 5/2. On substituting the values in (P × Q) = (Q × P) we get, (7/8 × 5/2) = (5/2 × 7/8) = 35/16. Hence, the commutative property of multiplication is applicable to fractions.

What is the Commutative Property of Multiplication for Integers?

The commutative property of multiplication for integers can be expressed as (P × Q) = (Q × P). For example, let us substitute the value of P = -3 and Q = -9. On substituting the values in the formula, we get (-3 × -9) = (-9 × -3) = 27. Hence, the commutative property of multiplication is applicable to integers.

What is the Difference between the Associative and Commutative Property of Multiplication?

The associative property of multiplication states that the product of the numbers remains the same even when the grouping of the numbers is changed. The associative property of multiplication is expressed as (A × B) × C = A × (B × C). The commutative property of multiplication states that the product of two or more numbers remains the same even if the order of the numbers is changed. The commutative property of multiplication is expressed as A × B × C = C × B × A.

Commutative property of multiplication

The commutative property of multiplication says that changing the order in which the factors are multiplied does not change the product. Generally:

a × b = b × a

1. 8 × 5 = 40

  5 × 8 = 40

2. 1.3 × 4 = 5.2

  4 × 1.3 = 5.2

In either case, changing the order of the factors does not change the result. If we add more factors, the same still holds true. Regardless of how many factors they are, or in what order we multiply them, the result will be the same.

3. 4 × 6 × 7 × 2 = 336

  7 × 6 × 4 × 2 = 336

There are more combinations than those two listed above, but their result will still be 336.

To visualize how the commutative property works, use the figure below.

The orange lines in the figure show how the groups are separated. Multiplying 4 × 2 is the same as adding 2 groups of 4. On the other hand, multiplying 2 × 4 is the same as adding 4 groups of 2. Although we are grouping both differently, we can clearly see that the result is the same.

Commutative Property of Multiplication — Definition & Examples

Commutative property of multiplication definition.

Commutative property of multiplication says that the order of factors in a multiplication sentence has no effect on the product. The commutative property of multiplication works on integers, fractions, decimals, exponents, and algebraic equations.

The commutative property of multiplication is one of the four main properties of multiplication. It is named after the ability of factors to commute, or move, in the number sentence without affecting the product.

The word “commutative” comes from a Latin root meaning “interchangeable”.

Switching the order of the multiplicand (the first factor) and the multiplier (the second factor) does not change the product.

What is 4 × 5 ? The answer is 20 .

What is 5 × 4 ? The answer is also 20 .

The order of the two factors, 4 and 5 , did not affect the product, 20 .

Commutative property is also true for addition.

Commutative property of multiplication formula

The  generic formula  for the commutative property of multiplication is:

Any number of factors can be rearranged to yield the same product:

1 × 2 × 3 = 6

3 × 1 × 2 =  6

2 × 3 × 1 = 6

2 × 1 × 3 = 6

Often, when demonstrating the commutative property of multiplication, the product is shown in the middle of the multiple arrangements of the equation.

Commutative property of multiplication examples

Let's see the commutative property of multiplication in action with some examples:

In the first picture we can think of the set of five rubber ducks as the multiplicand, spread across from left to right. Beneath it, vertically, we have the multiplier, 4 .

In the second picture we have one set of four rubber ducks arrayed left to right, the multiplicand. Then we have the multiplier, 5 , vertically.

Whether we take a set of five rubber duckies and multiply them four times, as on the left, or we take a set of four rubber duckies and multiply them five times, as on the right, we still end up with 20 rubber duckies.

The commutative property of multiplication works on basic multiplication equations and algebraic equations. Here was see how to use commutative property of multiplication various multiplication sentences:

6 × 7 = 42 = 7 × 6 6\times 7=42=7\times 6 6 × 7 = 42 = 7 × 6

1 , 234 × 0 = 0 = 0 × 1 , 234 \mathrm{1,234}\times 0=0=0\times \mathrm{1,234} 1 , 234 × 0 = 0 = 0 × 1 , 234

717 × 11 = 7 , 887 = 11 × 717 717\times 11=\mathrm{7,887}=11\times 717 717 × 11 = 7 , 887 = 11 × 717

6 2 × 3 2 = 324 = 3 2 × 6 2 {6}^{2}\times {3}^{2}=\mathbf{324}={3}^{2}\times {6}^{2} 6 2 × 3 2 = 324 = 3 2 × 6 2

2 3 × 4 3 = 256 = 4 3 × 2 3 {2}^{3}\times {4}^{3}=\mathbf{256}={4}^{3}\times {2}^{3} 2 3 × 4 3 = 256 = 4 3 × 2 3

3 4 × 7 8 = 21 32 = ( 7 8 ) ( 3 4 ) \frac{3}{4}\times \frac{7}{8}=\frac{\mathbf{21}}{\mathbf{32}}=\left(\frac{7}{8}\right)\left(\frac{3}{4}\right) 4 3 ​ × 8 7 ​ = 32 21 ​ = ( 8 7 ​ ) ( 4 3 ​ )

9 10 × 75 100 = 75 100 × 9 10 = 675 1000 = 27 40 \frac{9}{10}\times \frac{75}{100}=\frac{75}{100}\times \frac{9}{10}=\frac{\mathbf{675}}{\mathbf{1000}}\mathbf{=}\frac{\mathbf{27}}{\mathbf{40}} 10 9 ​ × 100 75 ​ = 100 75 ​ × 10 9 ​ = 1000 675 ​ = 40 27 ​ (simplified by dividing by 25 25 \frac{25}{25} 25 25 ​ )

0.1234 × 0.987 = 0.1217958 = 0.987 × 0.1234 0.1234\times 0.987=0.1217958=0.987\times 0.1234 0.1234 × 0.987 = 0.1217958 = 0.987 × 0.1234

411.52 × 0.3 = 123.456 = 0.3 × 411.52 411.52\times 0.3=123.456=0.3\times 411.52 411.52 × 0.3 = 123.456 = 0.3 × 411.52

4 x 2 ( 2 ) = 32 = 2 ( 4 x 2 ) 4{x}^{2}\left(2\right)=\mathbf{32}=2\left(4{x}^{2}\right) 4 x 2 ( 2 ) = 32 = 2 ( 4 x 2 )

To get our answer 32 , we first solved for x .

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Understanding multiplication

Commutative property

Here you will learn about the commutative property, including what it is, and how to use it to solve problems.

Students will first learn about the commutative property as part of operations and algebraic thinking in 3rd grade.

What is the commutative property?

The commutative property states that when you add or multiply numbers, you can change the order of the numbers and the answer will still be the same.

For example,

This is also true when multiplying numbers.

The commutative property can be used to create friendly numbers when solving.

Friendly numbers are numbers that are easy to add or multiply mentally – like multiples of 10.

The commutative property lets us change the order and create friendlier numbers.

10 + 25 is easier to solve mentally than 3 + 25 + 7 = 28 + 7.

The commutative property lets us regroup and create friendlier numbers.

10 \times 8 is easier to solve mentally than 2 \times 8 \times 5=16 \times 5.

The commutative property can also be referred to as the commutative property of addition and the commutative property of multiplication, or more generally as the commutative law.

Common Core State Standards

How does this relate to 3rd grade math?

• Grade 3 – Operations and Algebraic Thinking (3.OA.B.5) Apply properties of operations as strategies to multiply and divide. Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.) 3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.) Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property.)

[FREE] Properties of Equality Check for Understanding Quiz (Grade 3 to 6)

Use this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!

How to use the commutative property

In order to use the commutative property:

Check to see that the operation is addition or multiplication.

Change the order of the numbers and solve.

Commutative property examples

Example 1: simple commutative property with addition.

Give an example of the commutative property using 4 + 9.

All the numbers are being added, so the commutative property can be used.

2 Change the order of the numbers and solve.

Changing the order in the equation does not change the sum.

Example 2: simple commutative property with multiplication

Give an example of the commutative property using 10 \times 6.

All the numbers are being multiplied, so the commutative property can be used.

Changing the order in the equation does not change the product.

Example 3: commutative property – addition with friendly numbers

Use the commutative property to create a friendly number and solve 6 + 32 + 14.

Example 4: commutative property – multiplication with friendly numbers

Use the commutative property to create a friendly number and solve 3 \times 8 \times 3.

Notice that when multiplying, friendly numbers can also be single digit numbers. If you know your basic facts, it is easier to solve 9 \times 8 than solving 3 \times 8 \times 3=24 \times 3.

Example 5: commutative property – addition with friendly numbers

Use the commutative property to create a friendly number and solve 41 + 17 + 9.

Example 6: commutative property – multiplication with friendly numbers

Use the commutative property to create a friendly number and solve 3 \times 5 \times 4.

Notice that when multiplying, friendly numbers can also be numbers that are basic facts. If you have memorized the basic multiplication facts from 1-12, it is easier to solve 12 \times 5 than solving 3 \times 5 \times 4=15 \times 4.

Teaching tips for the commutative property

• Be intentional about choosing problems where the commutative property makes solving easier, since it is not always useful or necessary in all solving situations.
• Instead of just telling students the commutative property definition, draw attention to examples of the commutative property when they naturally occur  in daily math activities. Record the different examples you see in the classroom on an anchor chart. Over time, students will start recognizing and using the property on their own. Then, after there are sufficient examples, you can introduce students to the property name and definition by using their own examples.
• Include plenty of student discourse around this property so that students understand changing the order of numbers when adding or multiplying does not change the final result. This could include students sharing their thinking or critiquing the thinking of others.

Easy mistakes to make

• Using the commutative property for subtraction or division The commutative property only works when changing the order of the numbers doesn’t change the answer. This is not true for subtraction or division and they are considered non-commutative arithmetic operations. For example, 11-5 = 6 \; AND \; 5-11 = -6 Changing the order of the numbers, changes the answer.
• Thinking there is only one way to use the commutative property to solve with friendly numbers Sometimes there is more than one way to use the commutative property when solving. For example, \begin{aligned} & 6 \times 4 \times 5 \hspace{2.1cm} 6 \times 4 \times 5 \\ & =5 \times 4 \times 6 \hspace{1.7cm} =6 \times 5 \times 4 \\ & =20 \times 6 \hspace{2.05cm} =30 \times 4 \\ & =120 \hspace{2.4cm} =120 \end{aligned}
• Confusing the order of operations Equations are always solved moving from left to right. It is not necessary to formally introduce students to the order of operations, but they need to understand and read equations in this way. Otherwise the commutative property may not mean anything to them.

Related properties of equality lessons

This commutative property topic guide is part of our series on properties of equality . You may find it helpful to start with the main properties of equality topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other related topic guides in this series include:

• Properties of equality
• Order of operations
• Associative property
• Distributive property

Practice commutative property questions

1. Which of the following equations shows the commutative property?

The commutative property says that changing the order in the equation does not change the product.

2. Which of the following equations shows the commutative property?

The commutative property says that changing the order in the equation does not change the sum.

3. Which of the following equations shows how to solve 2 \times 9 \times 5 using the commutative property?

4. Which of the following equations shows how to solve 37 + 28 + 23 using the commutative property?

5. Which of the following equations shows how to solve 8 \times 4 \times 5 using the commutative property to create a friendly number?

The commutative property says that changing the order in the equation does not change the product. Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

6. Which of the following equations shows how to solve 16+18+22 using the commutative property to create a friendly number?

The commutative property says that changing the order in the equation does not change the sum. Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

Commutative property FAQs

No, you can solve the numbers as they appear in the equation, without changing the order. The commutative property just gives you flexibility to add or multiply in a different order.

Yes, the commutative property can be used with integers, rational numbers and any real number, as long as they are all being added or multiplied.

The associative property of addition states that you can change the grouping of numbers when adding (using parentheses) and the sum will still be the same. The order of operations changes, but not the written order of the numbers in the equation. The commutative property of addition says you can change the written order of the numbers when adding and the sum will still be the same.

It is one of the properties of numbers for mathematical operations. This property states that any number added to 0 will still result in the same number (0 + a = a) or any number multiplied by 1 will still result in the same number (1 \times a=a).

The next lessons are

• Addition and subtraction
• Multiplication and division
• Types of numbers

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What are the Properties of Multiplication?

Discover the power of multiplication and its fundamental properties, making multiplication operations easier to understand and solve.

Author Michelle Griczika

Published October 26, 2023

• Key takeaways
• Understanding the five properties of multiplication —Associative, Commutative, Identity, Distributive, and Zero—provides a solid foundation for tackling more complex mathematical concepts.
• These properties simplify calculations, making it easier to solve multiplication problems.
• Consistent practice reinforces understanding and the application of these properties.

Table of contents

• The 5 properties of multiplication
• Practice questions

Let’s make math simple. Some special rules, or “properties,” always apply when multiplying numbers. Understanding these multiplication properties makes calculations easier and faster. In this article, we will walk through the five properties of multiplication and practice applying them so you feel confident when tackling any multiplication problem.

The 5 Properties of Multiplication

Associative property of multiplication.

The associative property of multiplication is about how we group numbers in a multiplication problem. This property suggests that how we group numbers doesn’t influence the outcome, aka product. Take 2 x 3 x 4, for example. If you multiply 2 and 3 first and then multiply the result by 4, or multiply 3 and 4 first, then multiply the result by 2, the final product will be the same. As a result, the associative property can simplify complex multiplication problems by allowing us to conveniently group numbers.

Commutative property of multiplication

The commutative property of multiplication states that the order of numbers in a multiplication problem doesn’t change the result. So, 7 x 3 = 3 x 7. This property is beneficial when solving problems because it allows us to rearrange numbers to make calculations easier. For example, when solving a multiplication problem involving numerous numbers, you can rearrange the numbers to multiply familiar combinations. This helps simplify the overall calculation.

Identity property of multiplication

The identity property of multiplication states that a number retains its original value when multiplied by one. Any number multiplied by one will remain the same. For example, when we multiply 5 by 1, the outcome is still 5. This is true of every number. The multiplicative identity property is crucial because it reminds us that multiplying by 1 doesn’t change the value of a number. It comes in handy when dealing with more complex mathematical equations in alegebra where 1 is often used to preserve the identity of a variable.

Distributive property of multiplication

The distributive property of multiplication states that a number multiplied by the sum of two others, like 2 x (3 + 4), is equivalent to multiplying the number by each of the other two separately and then adding those results, as in (2 x 3) + (2 x 4). This property is especially beneficial for simplifying intricate problems into smaller, more solvable pieces. It’s also the basis for many strategies used in mental math, allowing us to multiply larger numbers quickly in our heads.

Zero property of multiplication

The zero property of multiplication states that any number multiplied by zero equals zero. For example, when we multiply 5 by 0 the outcome will always be 0. This property is beneficial when solving equations or simplifying expressions, allowing us to “eliminate” terms multiplied by zero. It’s a fundamental rule in math that provides a quick way to solve any multiplication problem where one of the factors is zero.

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Multiplication Property Practice Problems

Apply the associative property of multiplication to simplify these problems:

• (2 * 3) * 4 = ?
• 5 * (3 * 4) = ?
• (7 * 2) * 3 = ?

Use the commutative property of multiplication to rewrite and solve these problems:

• 7 * 6 = ? Can you rewrite and solve this?
• 8 * 3 = ? What is it when rewritten?
• 2 * 9 = ? How about this one when rearranged?

Use the distributive property of multiplication to simplify these problems:

• 3 * (4 + 2) = ?
• 5 * (6 + 3) = ?
• 2 * (5 + 7) = ?

Use the identity property of multiplication to solve these problems:

Apply the zero property of multiplication to solve these problems:

FAQs About the Properties of Multiplication

The five properties of multiplication are the Associative Property of Multiplication, Commutative Property of Multiplication, Identity Property of Multiplication, Distributive Property of Multiplication, and Zero Property of Multiplication.

Applying the properties of multiplication is all about recognizing the conditions that allow each property to be used. For instance, if you see a problem that requires you to multiply a number by the sum of two other numbers, such as 2 * (3 + 4), you might recognize it as the distributive property.

The commutative property of multiplication means that the order of numbers in a multiplication problem doesn’t change the result. For instance, 5 x 3 = 3 x 5. Many consider it similar to the multiplication property of equality.

In summary...

Remember, practicing consistently is vital to mastering these properties. For additional practice, you can check out this math help app . It’s an excellent tool for extra help and provides more exercises.

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Michelle Griczika

Michelle Griczika is a seasoned educator and experienced freelance writer. Her years teaching first and fifth grades coupled with her double certification in elementary and early childhood education lend depth to her understanding of diverse learning stages. Michelle enjoys running in her free time and undertaking home projects.

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Associative Property of Multiplication

• (2 * 3) * 4 = 24
• 5 * (3 * 4) = 60
• (7 * 2) * 3 = 42

Commutative Property of Multiplication

• 7 * 6 = 42; 6 * 7 = 42
• 8 * 3 = 24; 3 * 8 = 24
• 2 * 9 = 36; 9 * 2 = 18

Distributive Property of Multiplication

• 3 * (4 + 2) = 18
• 5 * (6 + 3) = 45
• 2 * (5 + 7) = 24

Identity Property of Multiplication

Zero Property of Multiplication

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• Math Article

Commutative Property

In mathematics, commutative property or commutative law explains that order of terms doesn’t matter while performing arithmetic operations . 

Commutative property is applicable only for addition and multiplication processes. Thus, it means we can change the position or swap the numbers when adding or multiplying any two numbers. This is one of the major properties of integers .

For example: 1+2 = 2+1 and 2 x 3 = 3 x 2. 

What is Commutative Property?

As we already discussed in the introduction, as per the commutative property or commutative law, when two numbers are added or multiplied together, then a change in their positions does not change the result. 

• 2+3 = 3+2 = 5
• 2 x 3 = 3 x 2 = 6
• 5 + 10 = 10 + 5 = 15
• 5 x 10 = 10 x 5 = 50

So, there can be two categories of operations that obeys commutative property:

• Commutative property of addition
• Commutative property of multiplication

Although the official use of commutative property began at the end of the 18th century, it was known even in the ancient era.

The word, Commutative, originated from the French word ‘commute or commuter’ means to switch or move around, combined with the suffix ‘-ative’ means ‘tend to’. Therefore, the literal meaning of the word is tending to switch or move around. It states that if we swipe the positions of the integers, the result will remain the same.

Commutative Property of Addition

According to the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

Let A and B be the two integers, then;

Examples of Commutative Property of Addition

• 1 + 2 = 2 + 1 = 3
• 3 + 8 = 8 + 3 = 11
• 12 + 5 = 5 + 12 = 17

Commutative Property of Multiplication

As per the commutative property of multiplication, when we multiply two integers, the answer we get after multiplication will remain the same, even if the position of the integers are interchanged.

Examples of Commutative Property of Multiplication

• 1 × 2 = 2 × 1 = 2
• 3 × 8 = 8 × 3 = 24
• 12 × 5 = 5 × 12 = 60

Important Facts Of Commutative Property

• Commutative property is only applicable for two arithmetic operations: Addition and Multiplication
• Changing the order of operands, does not change the result 
• Commutative property of addition: A + B = B + A
• Commutative property of multiplication: A.B = B.A

Other Properties

The other major properties of addition and multiplication are:

• Associative Property
• Distributive Property

Now, observe the other properties as well here:

Associative Property of Addition and Multiplication

According to the associative law, regardless of how the numbers are grouped, you can add or multiply them together, the answer will be the same. In other words, the placement of parentheses does not matter when it comes to adding or multiplying.

• 1 + (2+3) = (1+2) + 3 → 6
• 3 x (4 x 2) = (3 x 4) x 2 → 24

Distributive Property of Multiplication

The distributive property of Multiplication states that multiplying a sum by a number is the same as multiplying each addend by the value and adding the products then.

According to the Distributive Property, if a, b, c are real numbers, then:

a x (b + c) = (a x b) + (a x c)

• 2 x (5 + 8) = (2 x 5) + (2 x 8)
• 2 x (13) = 10 + 16

There are certain other properties such as Identity property, closure property which are introduced for integers.

Non-Commutative Property

Some operations are non-commutative. By non-commutative, we mean the switching of the order will give different results. The mathematical operations, subtraction and division are the two non-commutative operations. Unlike addition, in subtraction switching of orders of terms results in different answers.

Example: 4 – 3 = 1 but 3 – 4 = -1  which are two different integers.

Also, the division does not follow the commutative law. That is,

2 ÷ 6  = 1/3

Hence, 6 ÷ 2 ≠  2 ÷ 6

Solved Examples on Commutative Property

Example 1: Which of the following obeys commutative law?

• 36 – 6

Solution: Options 1, 2 and 5 follow the commutative law

Explanation:

• 3 × 12 = 36 and

       12 x 3 = 36

=> 3 x 12 = 12 x 3 (commutative)

• 4 + 20 = 24 and

     20 + 4 = 24

     => 4 + 20 = 20 + 4 (commutative)

• 36 ÷ 6 = 6 and

     6 ÷ 36 = 0.167

=> 36 ÷ 6 ≠ 6 ÷ 36  (non commutative)

• 36 − 6 = 30 and 

      6 – 36 = – 30

=> 36 – 6 ≠ 6 – 36  (non commutative)

• −3 × 4 = -12 and

       4 x -3 = -12

=> −3 × 4 = 4 x -3  (commutative)

Q.2: Prove that a+ b = b+a if a = 10 and b = 9.

Sol: Given that, a = 10 and b = 9 

LHS = a+b = 10 + 9 = 19   ……(1)

RHS = b + a = 9 + 10 = 19 ……(2)

By equation 1 and 2, as per commutative property of addition, we get;

Hence, proved.

Q.3: Prove that A.B = B.A, if A = 4 and B = 3.

Sol: Given, A = 4 and B = 3.

A.B = 4.3 = 12  ….. (1)

B.A = 3.4 = 12  …..(2)

By eq.(1) and (2), as per the commutative property of multiplication, we get;

Practice Questions

Find which of the following is the commutative property of addition and multiplication.

• 3 + 4 = 4 + 3
• 10 x 7 = 7 x 10
• 8 x 9 = 9 x 8
• 6 + 4 = 4 + 6

Frequently Asked Questions – FAQs

What is commutative property give examples., what is commutative property of addition, what is the commutative property of multiplication, what are the major four properties in maths, what is the difference between commutative and associative property.

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Commutative property

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The order of multiplication does not matter

The commutative property of multiplication tells us that when multiplying numbers, the order of multiplication does not matter ( 3 x 4 = 4  x 3 ).

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Applying the Commutative Property of Addition and Multiplication in a Problem

Did you know that the commutative property can help us solve an operation faster?  Today we will look at the commutative property of addition and multiplication. The commutative property tells us that the result of an addition or multiplication is always the same, regardless of the order of the elements with which it operates. Let’s see it in more detail:

The Commutative Property of Addition

The commutative property of addition reads as follows: The order of the addends does not change the result That is, when you have to solve a problem, no matter the order in which you put the addends, you will always get the same result. Why does this happen? Let’s see an example with a problem:

Sara had 4 red apples in her shopping cart. She met Ruth, who gave her 2 green apples. How many apples does Sara have at the end?

To solve this problem, we must add the two types of apples to know how many there are in total. We can add them in two ways: If we add 4 red apples plus 2 green apples, we get 6 apples in total. Similarly, if we add 2 green apples plus 4 red apples, we also get 6 apples in total.

The Commutative Property of Multiplication

The commutative property of multiplication states:

The order of the factors does not change the product That is, when we have to solve a multiplication problem, we can arrange the factors in any way we want and always get the same product. Let’s see an example with a problem:

Mark is a baker and today he has received a cake order for a party. They have told him that the party will have 4 tables, and on each of the tables, they want to put 2 cakes. How many cakes will Mark have to make?

To solve this problem we must multiply. We can do this in two ways:

Do you want to learn more? Click below to learn more about the different properties of multiplication:

• Distributive Property of multiplication
• Associative Property of multiplication
• Properties of Multiplication

What did you think of this post? Do you understand the commutative property better now? If you liked it, share it with your friends so that they can learn too.

Learn More:

• Pedagogical Justification of the Commutative Property
• Review the Different Properties of Multiplication
• Learn the Different Properties of Multiplication
• Multiplication Tables and Tricks to Learn Them
• Distributive Property of Multiplication with Examples
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Commutative Property – Definition, Examples, FAQs

Commutative property.

• Commutative Property of Addition
• Commutative Property of Multiplication

Solved Examples on Commutative Property

Practice problems on commutative property, frequently asked questions on commutative property.

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division. Let’s see.

The above examples clearly show that the commutative property holds true for addition and multiplication but not for subtraction and division . So, if we swap the position of numbers in subtraction or division statements, it changes the entire problem. 

So, mathematically commutative property for addition and multiplication looks like this:

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Commutative Property of Addition:

a + b = b + a; where a and b are any 2 whole numbers

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Commutative Property of Multiplication:

a × b = b × a; where a and b are any 2 non zero whole numbers

Use Cases of Commutative Property

• Myra has 6 apples and 2 peaches. Kim has 2 apples and 6 peaches. Who has more fruits?

Even if both have different numbers of apples and peaches, they have an equal number of fruits, because 2 + 6 = 6 + 2.

• Sara buys 3 packs of buns. Each pack has 4 buns. Mila buys 4 packs of buns and each pack has 3 buns. Who bought more buns?

Even if both have different numbers of bun packs with each having a different number of buns in them, they both bought an equal number of buns, because 3 × 4 = 4 × 3.

Example 1: Fill in the missing numbers using the commutative property.

• _________ + 27 = 27 + 11
• 45 + 89 = 89 + _________
• 84 × ______ = 77 × 84
• 118 × 36 = ________ × 118
• 11; by commutative property of addition
• 45; by commutative property of addition
• 77; by commutative property of multiplication
• 36; by commutative property of multiplication

Example 2: Use 14 × 15 = 210, to find 15 × 14.

Solution: 

As per commutative property of multiplication, 15 × 14 = 14 × 15. 

Since, 14 × 15 = 210, so, 15 × 14 also equals 210.

Example 3: Use 827 + 389 = 1,216 to find 389 + 827. 

As per commutative property of addition , 827 + 389 = 389 + 827. 

Since, 827 + 389 = 1,216, so, 389 + 827 also equals 1,216.

Example 4: Use the commutative property of addition to write the equation, 3 + 5 + 9 = 17, in a different sequence of the addends.

3 + 9 + 5 = 17 (because 5 + 9 = 9 + 5)

5 + 3 + 9 = 17 (because 3 + 5 = 5 + 3)

5 + 9 + 3 = 17 (because 3 + 9 = 9 + 3)

Similarly, we can rearrange the addends and write:

9 + 3 + 5 = 17

9 + 5 + 3 = 17

Example 4: Ben bought 3 packets of 6 pens each. Mia bought 6 packets of 3 pens each. Did they buy an equal number of pens or not?

Ben bought 3 packets of 6 pens each.

So, the total number of pens that Ben bought = 3 × 6

Mia bought 6 packets of 3 pens each.

So, the total number of pens that Ben bought = 6 × 3

By the commutative property of multiplication, 3 × 6 = 6 × 3. 

So, both Ben and Mia bought an equal number of pens.

Example 5: Lisa has 78 red and 6 blue marbles. Beth has 6 packets of 78 marbles each. Do they have an equal number of marbles?

Since Lisa has 78 red and 6 blue marbles.

So, the total number of marbles with Lisa = 78 + 6

Beth has 6 packets of 78 marbles each.

So, the total number of marbles with Beth = 6 × 78

Clearly, adding and multiplying two numbers gives different results. (Except 2 + 2 and 2 × 2.

That is, 78 + 6 ≠ 6 × 78

So, Lisa and Beth don’t have an equal number of marbles.

Attend this Quiz & Test your knowledge.

Which of the following represents the commutative property of addition?

Which of the following represents the commutative property of multiplication, which of the following expressions will follow the commutative property, choose the set of numbers to make the statement true. 5 + _____ = 4 + ______.

Can you apply the commutative property of addition/multiplication to 3 numbers?

Yes. By definition, commutative property is applied on 2 numbers, but the result remains the same for 3 numbers as well. This is because we can apply this property on two numbers out of 3 in various combinations.

Which operations do not follow commutative property?

Commutative property cannot be applied to subtraction and division.

What is the associative property of addition (or multiplication)?

This property states that when three or more numbers are added (or multiplied), the sum (or the product) is the same regardless of the grouping of the addends (or the multiplicands). That is, 

(a + b) + c = a + (b + c) (a × b) × c = a × (b × c) where a, b, and c are whole numbers.

For which all operations does the associative property hold true?

The Associative property holds true for addition and multiplication.

What is the distributive property of multiplication?

By the distributive property of multiplication over addition, we mean that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. That is,

a × (b + c) = (a × b) + (a × c) where a, b, and c are whole numbers.

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• Associative Property – Definition, Examples, FAQs, Practice Problems
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• Distance Between Two Points
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Associative, Distributive and Commutative Properties

All together on 1 page!

To learn more about any of the properties below, visit that property's individual page.

Properties and Operations

Let's look at how (and if) these properties work with addition , multiplication , subtraction and division.

Multiplication

Subtraction

Practice Problems

Which of the following statements illustrate the distributive, associate and the commutative property?

Directions: Click on each answer button to see what property goes with the statement on the left .

All three of these properties can also be applied to Algebraic Expressions.

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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Commutative, Associative and Distributive Laws

Wow! What a mouthful of words! But the ideas are simple.

Commutative Laws

The "Commutative Laws" say we can swap numbers over and still get the same answer ...

... when we add :

a + b  =  b + a

... or when we multiply :

a × b  =  b × a

Percentages too!

Because a × b  =  b × a it is also true that:

a% of b  =  b% of a

Example: what is 8% of 50 ?

8% of 50 = 50% of 8   = 4

Why "commutative " ... ?

Because the numbers can travel back and forth like a commuter .

Associative Laws

The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first) ...

(a + b) + c  =  a + (b + c)

(a × b) × c  =  a × (b × c)

Sometimes it is easier to add or multiply in a different order:

What is 19 + 36 + 4?

19 + 36 + 4 =  19 + (36 + 4)   =  19 + 40 = 59

Or to rearrange a little:

What is 2 × 16 × 5?

2 × 16 × 5  =  (2 × 5) × 16   =  10 × 16 = 160

Distributive Law

The "Distributive Law" is the BEST one of all, but needs careful attention.

This is what it lets us do:

3 lots of (2+4) is the same as 3 lots of 2 plus 3 lots of 4

So, the 3× can be "distributed" across the 2+4 , into 3×2 and 3×4

And we write it like this:

a × (b + c)  =  a × b  +  a × c

Try the calculations yourself:

• 3 × ( 2 + 4 )  =  3 × 6  =  18
• 3×2 + 3×4  =  6 + 12  =  18

Either way gets the same answer.

We get the same answer when we:

• multiply a number by a group of numbers added together , or
• do each multiply separately then add them

Sometimes it is easier to break up a difficult multiplication:

Example: What is 6 × 204 ?

6 × 204  =  6×200 + 6×4   =  1,200 + 24   =  1,224

Or to combine:

Example: What is 16 × 6 + 16 × 4?

16 × 6 + 16 × 4  =  16 × (6+4)   = 16 × 10   =  160

We can use it in subtraction too:

Example: 26×3 - 24×3

26×3 − 24×3 = (26 − 24) × 3   =  2 × 3   =  6

We could use it for a long list of additions, too:

Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7

6 ×7 + 2 ×7 + 3 ×7 + 5 ×7 + 4 ×7 = (6+2+3+5+4) × 7 = 20 × 7 = 140

And those are the Laws . . .

                  . . . but don't go too far.

The Commutative Law does not work for subtraction or division:

• 12 / 3 = 4 , but

 The Associative Law does not work for subtraction or division:

• (9 – 4) – 3 = 5 – 3 = 2 , but
• 9 – (4 – 3) = 9 – 1 = 8

 The Distributive Law does not work for division:

• 24 / (4 + 8) = 24 / 12 = 2 , but
• 24 / 4 + 24 / 8 = 6 + 3 = 9