Commutative and Associative Properties

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

Use the Commutative and Associative Properties

Think about adding two numbers, such as $5$ and $3 .$

\begin{matrix} 5 + 3 & 3 + 5 \\ 8 & 8 \end{matrix}

The results are the same. $5 + 3 = 3 + 5$

Notice, the order in which we add does not matter. The same is true when multiplying $5$ and $3 .$

\begin{matrix} 5 \cdot 3 & 3 \cdot 5 \\ 15 & 15 \end{matrix}

Again, the results are the same! $5 \cdot 3 = 3 \cdot 5 .$ The order in which we multiply does not matter.

These examples illustrate the commutative properties of addition and multiplication.

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

Use the commutative properties to rewrite the following expressions:

ⓐ $−1 + 3 =_____$
ⓑ $4 \cdot 9 =_____$

Solution

This table demonstrates the commutative property of addition using a numerical example.
ⓐ
	$−1 + 3 =_____$
Use the commutative property of addition to change the order.	$−1 + 3 = 3 + (−1)$

This table demonstrates the commutative property of multiplication with an example, showing how the order of factors can be changed.
ⓑ
	$4 \cdot 9 =_____$
Use the commutative property of multiplication to change the order.	$4 \cdot 9 = 9 \cdot 4$

What about subtraction? Does order matter when we subtract numbers? Does $7 - 3$ give the same result as $3 - 7 ?$

\begin{matrix} 7 - 3 & 3 - 7 \\ 4 & - 4 \\ 4 \neq −4 \end{matrix}

The results are not the same. 7 - 3 \neq 3 - 7

Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

\begin{matrix} 12 \div 4 & 4 \div 12 \\ \frac{12}{4} & \frac{4}{12} \\ 3 & \frac{1}{3} \\ 3 \neq \frac{1}{3} \end{matrix}

The results are not the same. So 12 \div 4 \neq 4 \div 12

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

7 + 8 + 2

How would you do it and what would your answer be?

Some people would think $7 + 8 is 15$ and then $15 + 2 is 17 .$ Others might start with $8 + 2 makes 10$ and then $7 + 10 makes 17 .$

Both ways give the same result, as shown in Figure 1. (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below.

When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.

The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:

5 \cdot \frac{1}{3} \cdot 3

Changing the grouping of the numbers gives the same result, as shown in Figure 2.

The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below.

When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.

If we multiply three numbers, changing the grouping does not affect the product.

You probably know this, but the terminology may be new to you. These examples illustrate the Associative Properties.

Use the associative properties to rewrite the following:

ⓐ $(3 + 0.6) + 0.4 =__________$
ⓑ $(−4 \cdot \frac{2}{5}) \cdot 15 =__________$

Solution

Illustrates the associative property of addition with an example, showing how changing number grouping doesn't alter the sum.
ⓐ
	$(3 + 0.6) + 0.4 =__________$
Change the grouping.	$(3 + 0.6) + 0.4 = 3 + (0.6 + 0.4)$

Notice that $0.6 + 0.4$ is $1,$ so the addition will be easier if we group as shown on the right.

Illustrates applying an instruction to change the grouping in a mathematical expression, demonstrating problem-solving steps.
ⓑ
	$(−4 \cdot \frac{2}{5}) \cdot 15 =__________$
Change the grouping.	$(−4 \cdot \frac{2}{5}) \cdot 15 = −4 \cdot (\frac{2}{5} \cdot 15)$

Notice that $\frac{2}{5} \cdot 15$ is $6 .$ The multiplication will be easier if we group as shown on the right.

Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

Use the Associative Property of Multiplication to simplify: $6 (3 x) .$

Solution

Steps demonstrating the simplification of the expression 6(3x) to 18x using the associative property of multiplication.
	$6 (3 x)$
Change the grouping.	$(6 \cdot 3) x$
Multiply in the parentheses.	$18 x$

Notice that we can multiply $6 \cdot 3,$ but we could not multiply $3 \cdot x$ without having a value for $x .$

Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

Evaluate each expression when $x = \frac{7}{8} .$

ⓐ $x + 0.37 + (- x)$
ⓑ $x + (- x) + 0.37$

Solution

ⓐ

Substitute $\frac{7}{8}$ for $x$ .
Convert fractions to decimals.
Add left to right.
Subtract.

ⓑ

Substitute $\frac{7}{8}$ for x.	$A mathematical expression showing the sum of 7/8, its additive inverse -7/8, and the decimal 0.37. The fractions are displayed in red text, while the operations and the decimal are in black.$
Add opposites first.

What was the difference between part ⓐ and part ⓑ ? Only the order changed. By the Commutative Property of Addition, $x + 0.37 + (- x) = x + (- x) + 0.37 .$ But wasn’t part ⓑ much easier?

Let’s do one more, this time with multiplication.

Evaluate each expression when $n = 17 .$

ⓐ $\frac{4}{3} (\frac{3}{4} n)$
ⓑ $(\frac{4}{3} \cdot \frac{3}{4}) n$

Solution

ⓐ
	$A mathematical expression shows the fraction 4/3 multiplied by the quantity (3/4 multiplied by n), enclosed in parentheses.$
Substitute 17 for n.	$A mathematical expression showing the fraction 4/3 multiplied by a parenthesized term (3/4 multiplied by the number 17, which is highlighted in red).$
Multiply in the parentheses first.	$A mathematical expression showing the fraction 4/3 multiplied by the fraction 51/4, enclosed in parentheses, against a white background.$
Multiply again.

ⓑ
	$The image shows a mathematical expression: an open parenthesis, the fraction 4/3, a multiplication dot, the fraction 3/4, a close parenthesis, and the variable 'n'. The expression is (4/3 * 3/4)n.$
Substitute 17 for n.	$A mathematical expression showing the product of two fractions, (4/3) and (3/4), inside parentheses, which is then multiplied by 17, with 17 highlighted in red.$
Multiply. The product of reciprocals is 1.
Multiply again.

What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, $\frac{4}{3} (\frac{3}{4} n) = (\frac{4}{3} \cdot \frac{3}{4}) n .$ By carefully choosing how to group the factors, we can make the work easier.

Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in Example 4 part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is $0 .$ Likewise, part ⓑ in Example 5 was easier, with the reciprocals grouped together, because their product is $1 .$ In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

Simplify: $−84 n + (−73 n) + 84 n .$

Solution

Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.

Steps to simplify the expression -84n + (-73n) + 84n, demonstrating re-ordering and addition of terms.
	$−84 n + (−73 n) + 84 n$
Re-order the terms.	$−84 n + 84 n + (−73 n)$
Add left to right.	$0 + (−73 n)$
Add.	$−73 n$

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is $1 .$

Simplify: $\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7} .$

Solution

Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.

Step-by-step simplification of a fractional multiplication problem, demonstrating how to reorder terms to simplify the expression.
	$\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7}$
Re-order the terms.	$\frac{7}{15} \cdot \frac{15}{7} \cdot \frac{8}{23}$
Multiply left to right.	$1 \cdot \frac{8}{23}$
Multiply.	$\frac{8}{23}$

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

Simplify: $(\frac{5}{13} + \frac{3}{4}) + \frac{1}{4} .$

Solution

Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.

Step-by-step simplification of a fractional mathematical expression.
	$(\frac{5}{13} + \frac{3}{4}) + \frac{1}{4}$
Group the terms with a common denominator.	$\frac{5}{13} + (\frac{3}{4} + \frac{1}{4})$
Add in the parentheses first.	$\frac{5}{13} + (\frac{4}{4})$
Simplify the fraction.	$\frac{5}{13} + 1$
Add.	$1 \frac{5}{13}$
Convert to an improper fraction.	$\frac{18}{13}$

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

Simplify: $(6.47 q + 9.99 q) + 1.01 q .$

Solution

Notice that the sum of the second and third coefficients is a whole number.

This table demonstrates the step-by-step simplification of an algebraic expression, illustrating the associative property of addition.
	$(6.47 q + 9.99 q) + 1.01 q$
Change the grouping.	$6.47 q + (9.99 q + 1.01 q)$
Add in the parentheses first.	$6.47 q + (11.00 q)$
Add.	$17.47 q$

Many people have good number sense when they deal with money. Think about adding $99$ cents and $1$ cent. Do you see how this applies to adding $9.99 + 1.01 ?$

No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.

Simplify the expression: $[1.67 (8)] (0.25) .$

Solution

Notice that multiplying $(8) (0.25)$ is easier than multiplying $1.67 (8)$ because it gives a whole number. (Think about having $8$ quarters—that makes $$2.)$

Step-by-step simplification of a mathematical expression, demonstrating regrouping and multiplication.
	$[1.67 (8)] (0.25)$
Regroup.	$1.67 [(8) (0.25)]$
Multiply in the brackets first.	$1.67 [2]$
Multiply.	$3.34$

When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.

Simplify: $6 (9 x) .$

Solution

Demonstrates simplifying 6(9x) to 54x using the associative property of multiplication, showing each step.
	$6 (9 x)$
Use the associative property of multiplication to re-group.	$(6 \cdot 9) x$
Multiply in the parentheses.	$54 x$

In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression $3 x + 7 + 4 x + 5$ by rewriting it as $3 x + 4 x + 7 + 5$ and then simplified it to $7 x + 12 .$ We were using the Commutative Property of Addition.

Simplify: $18 p + 6 q + (−15 p) + 5 q .$

Solution

Use the Commutative Property of Addition to re-order so that like terms are together.

This table demonstrates the step-by-step simplification of an algebraic expression by re-ordering and combining like terms to reach a final simplified form.
	$18 p + 6 q + (−15 p) + 5 q$
Re-order terms.	$18 p + (−15 p) + 6 q + 5 q$
Combine like terms.	$3 p + 11 q$

Key Concepts

Commutative Properties
- Commutative Property of Addition:
  - If $a, b$ are real numbers, then $a + b = b + a$
- Commutative Property of Multiplication:
  - If $a, b$ are real numbers, then $a \cdot b = b \cdot a$
Associative Properties
- Associative Property of Addition:
  - If $a, b, c$ are real numbers then $(a + b) + c = a + (b + c)$
- Associative Property of Multiplication:
  - If $a, b, c$ are real numbers then $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

Practice Makes Perfect

Use the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

$8 + 9 =___$

$7 + 6 =___$

Solution

7 + 6 = 6 + 7

$8 (−12) =___$

$7 (−13) =___$

Solution

7(−13) = (−13)7

$(−19) (−14) =___$

$(−12) (−18) =___$

Solution

(−12)(−18) = (−18)(−12)

$−11 + 8 =___$

$−15 + 7 =___$

Solution

−15 + 7 = 7 + (−15)

$x + 4 =___$

$y + 1 =___$

Solution

y + 1 = 1 + y

$−2 a =___$

$−3 m =___$

Solution

−3m = m(−3)

In the following exercises, use the associative properties to rewrite the given expression.

$(11 + 9) + 14 =___$

$(21 + 14) + 9 =___$

Solution

(21 + 14) + 9 = 21 + (14 + 9)

$(12 \cdot 5) \cdot 7 =___$

$(14 \cdot 6) \cdot 9 =___$

Solution

(14 · 6) · 9 = 14(6 · 9)

$(−7 + 9) + 8 =___$

$(−2 + 6) + 7 =___$

Solution

(−2 + 6) + 7 = −2 + (6 + 7)

$(16 \cdot \frac{4}{5}) \cdot 15 =___$

$(13 \cdot \frac{2}{3}) \cdot 18 =___$

Solution

$(13 \cdot \frac{2}{3}) \cdot 18 = 13 (\frac{2}{3} \cdot 18)$

$3 (4 x) =___$

$4 (7 x) =___$

Solution

4(7x) = (4 · 7)x

$(12 + x) + 28 =___$

$(17 + y) + 33 =___$

Solution

(17 + y) + 33 = 17 + (y + 33)

Evaluate Expressions using the Commutative and Associative Properties

In the following exercises, evaluate each expression for the given value.

y = \frac{5}{8},

evaluate:

ⓐ $y + 0.49 + (- y)$
ⓑ $y + (- y) + 0.49$

z = \frac{7}{8},

evaluate:

ⓐ $z + 0.97 + (- z)$
ⓑ $z + (- z) + 0.97$

Solution

ⓐ 0.97
ⓑ 0.97

c = - \frac{11}{4},

evaluate:

ⓐ $c + 3.125 + (- c)$
ⓑ $c + (- c) + 3.125$

d = - \frac{9}{4},

evaluate:

ⓐ $d + 2.375 + (- d)$
ⓑ $d + (- d) + 2.375$

Solution

ⓐ 2.375
ⓑ 2.375

j = 11,

evaluate:

ⓐ $\frac{5}{6} (\frac{6}{5} j)$
ⓑ $(\frac{5}{6} \cdot \frac{6}{5}) j$

k = 21,

evaluate:

ⓐ $\frac{4}{13} (\frac{13}{4} k)$
ⓑ $(\frac{4}{13} \cdot \frac{13}{4}) k$

Solution

ⓐ 21
ⓑ 21

m = −25,

evaluate:

ⓐ $- \frac{3}{7} (\frac{7}{3} m)$
ⓑ $(- \frac{3}{7} \cdot \frac{7}{3}) m$

n = −8,

evaluate:

ⓐ $- \frac{5}{21} (\frac{21}{5} n)$
ⓑ $(- \frac{5}{21} \cdot \frac{21}{5}) n$

Solution

ⓐ 8
ⓑ 8

Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

$−45 a + 15 + 45 a$

$9 y + 23 + (−9 y)$

Solution

$\frac{1}{2} + \frac{7}{8} + (- \frac{1}{2})$

$\frac{2}{5} + \frac{5}{12} + (- \frac{2}{5})$

Solution

$\frac{5}{12}$

$\frac{3}{20} \cdot \frac{49}{11} \cdot \frac{20}{3}$

$\frac{13}{18} \cdot \frac{25}{7} \cdot \frac{18}{13}$

Solution

$\frac{25}{7}$

$\frac{7}{12} \cdot \frac{9}{17} \cdot \frac{24}{7}$

$\frac{3}{10} \cdot \frac{13}{23} \cdot \frac{50}{3}$

Solution

$\frac{65}{23}$

$−24 \cdot 7 \cdot \frac{3}{8}$

$−36 \cdot 11 \cdot \frac{4}{9}$

Solution

−176

$(\frac{5}{6} + \frac{8}{15}) + \frac{7}{15}$

$(\frac{1}{12} + \frac{4}{9}) + \frac{5}{9}$

Solution

$\frac{13}{12}$

$\frac{5}{13} + \frac{3}{4} + \frac{1}{4}$

$\frac{8}{15} + \frac{5}{7} + \frac{2}{7}$

Solution

$\frac{23}{15}$

$(4.33 p + 1.09 p) + 3.91 p$

$(5.89 d + 2.75 d) + 1.25 d$

Solution

9.89d

$17 (0.25) (4)$

$36 (0.2) (5)$

Solution

$[2.48 (12)] (0.5)$

$[9.731 (4)] (0.75)$

Solution

29.193

$7 (4 a)$

$9 (8 w)$

Solution

72w

$−15 (5 m)$

$−23 (2 n)$

Solution

−46n

$12 (\frac{5}{6} p)$

$20 (\frac{3}{5} q)$

Solution

12q

$14 x + 19 y + 25 x + 3 y$

$15 u + 11 v + 27 u + 19 v$

Solution

42u + 30v

$43 m + (−12 n) + (−16 m) + (−9 n)$

$−22 p + 17 q + (−35 p) + (−27 q)$

Solution

−57p + (−10q)

$\frac{3}{8} g + \frac{1}{12} h + \frac{7}{8} g + \frac{5}{12} h$

$\frac{5}{6} a + \frac{3}{10} b + \frac{1}{6} a + \frac{9}{10} b$

Solution

$a + \frac{6}{5} b$

$6.8 p + 9.14 q + (−4.37 p) + (−0.88 q)$

$9.6 m + 7.22 n + (−2.19 m) + (−0.65 n)$

Solution

7.41m + 6.57n

Everyday Math

Stamps Allie and Loren need to buy stamps. Allie needs four $$0.49$ stamps and nine $$0.02$ stamps. Loren needs eight $$0.49$ stamps and three $$0.02$ stamps.

ⓐ How much will Allie’s stamps cost?
ⓑ How much will Loren’s stamps cost?
ⓓ How many $$0.49$ stamps do the girls need altogether? How much will they cost?
ⓔ How many $$0.02$ stamps do the girls need altogether? How much will they cost?

Counting Cash Grant is totaling up the cash from a fundraising dinner. In one envelope, he has twenty-three $$5$ bills, eighteen $$10$ bills, and thirty-four $$20$ bills. In another envelope, he has fourteen $$5$ bills, nine $$10$ bills, and twenty-seven $$20$ bills.

ⓐ How much money is in the first envelope?
ⓑ How much money is in the second envelope?
ⓓ What is the value of all the $$5$ bills?
ⓔ What is the value of all $$10$ bills?
ⓕ What is the value of all $$20$ bills?

Solution

ⓐ $975
ⓑ $700
ⓒ $1675
ⓓ $185
ⓔ $270
ⓕ $1220

Writing Exercises

In your own words, state the Commutative Property of Addition and explain why it is useful.

In your own words, state the Associative Property of Multiplication and explain why it is useful.

Solution

Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment chart for students to rate their understanding of Commutative and Associative Properties, with options: Confidently, With some help, or No-I don't get it!

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?