Does the Commutative Property work for subtraction?

No. Subtraction is not commutative. 10 - 3 = 7, but 3 - 10 = -7. The order of the numbers matters for subtraction, so you cannot swap them like you can with addition.

Why is the Commutative Property useful?

It lets you rearrange a problem to make it easier to calculate or understand. For example, you might write 13 + 27 as 27 + 13 to count up from 27 instead of doing 13 + 27 in your head.

How is the Commutative Property different from the Associative Property?

The Commutative Property lets you change the order of numbers: a + b = b + a. The Associative Property lets you change the grouping with parentheses: (a + b) + c = a + (b + c). Both let you rearrange, but in different ways.

Can I use the Commutative Property in algebra?

Yes. The Commutative Property still works with variables. For example, x + 5 = 5 + x and 3y = y \cdot 3. This is useful for simplifying expressions and solving equations.

Commutative Property

The Commutative Property says you can switch the order of numbers when adding or multiplying without changing the result.

a + b = b + a \qquad ab = ba

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What each symbol means

What Commutative Property takes

Symbol	Meaning
$a$	A number that can be added to or multiplied by another number; if you mistake its value or sign, the reordered operation produces an incorrect answer.
$b$	The second number in the operation; if you misread its value as a different number or forget its sign, the Commutative Property will not yield an equivalent expression.

When to use it

When you need to rearrange numbers in a sum or product to make mental math easier or to match a different form.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Add two positive integers using Commutative Property

Problem

Show that

3 + 5

equals

5 + 3

$3 + 5 = 8$
Add the first number 3 and the second number 5.
$5 + 3 = 8$
Add the numbers in the switched order: 5 first, then 3.
$3 + 5 = 5 + 3$
Both orders give the same sum of 8, confirming the Commutative Property.

Answer: $3 + 5 = 5 + 3$

This example shows the Commutative Property for addition with simple positive whole numbers. No matter which order you add them, the sum stays the same.

2. Multiply fractions using Commutative Property

Problem

Verify that

\frac{1}{3} \cdot \frac{2}{5}

equals

\frac{2}{5} \cdot \frac{1}{3}

$\frac{1}{3} \cdot \frac{2}{5} = \frac{1 \cdot 2}{3 \cdot 5}$
Multiply the numerators and multiply the denominators.
$= \frac{2}{15}$
Simplify the fraction.
$\frac{2}{5} \cdot \frac{1}{3} = \frac{2 \cdot 1}{5 \cdot 3} = \frac{2}{15}$
Multiply in the switched order and get the same result.

Answer: $\frac{1}{3} \cdot \frac{2}{5} = \frac{2}{5} \cdot \frac{1}{3} = \frac{2}{15}$

Fractions also follow the Commutative Property for multiplication. This example shows that even with fractional numbers, switching the order does not change the product.

3. Basketball scoring with Commutative Property for addition

Problem

Team A scores 25 points in the first quarter and 18 points in the second quarter. Team B scores 18 points in the first quarter and 25 points in the second quarter. Use the Commutative Property to show both teams have the same total after two quarters.

$\text{Team A:} \; 25 + 18 = 43$
Add Team A's points from the first quarter and second quarter.
$\text{Team B:} \; 18 + 25 = 43$
Add Team B's points, which are the same numbers in reversed order.
$25 + 18 = 18 + 25 = 43$
The Commutative Property confirms both totals are equal.

Answer: $25 + 18 = 18 + 25 = 43$

This real-world example shows that the order in which you add up scores does not matter. Both teams have the same total points, demonstrating the Commutative Property for addition.

Common mistakes

Where Commutative Property usually goes wrong

The mistake	Why it is wrong	The fix
Writing $7 - 2 = 2 - 7$ because subtraction looks like addition and students think the Commutative Property applies to subtraction too.	The Commutative Property only applies to addition and multiplication, not subtraction, because subtraction depends on the order.	$7 - 2 = 5$ but $2 - 7 = -5$ ; they are not equal, so you cannot use the Commutative Property here.
Writing $20 \div 4 = 4 \div 20$ and assuming division is commutative like multiplication.	Division is not commutative; switching the order of the dividend and divisor produces a completely different result.	$20 \div 4 = 5$ but $4 \div 20 = 0.2$ , so the Commutative Property does not apply to division.
Rearranging $10 - 3 + 2$ to $10 + 3 - 2$ thinking the order of operations doesn't matter.	The Commutative Property applies within a single operation, not when mixing different operations like addition and subtraction.	$10 - 3 + 2 = 9$ but $10 + 3 - 2 = 11$ , so reordering across operations changes the answer.

Tips and when to use something else

Rearrange numbers in a sum or product using the Commutative Property to make the calculation mentally easier—for example, $13 + 27$ is easier to compute as $27 + 13$ by counting up from 27.
Remember: only addition and multiplication are commutative; for subtraction and division, use the Order of Operations instead.
Combine the Commutative Property with the Associative Property to rearrange and group terms flexibly in longer expressions like $2 + (3 + 5) = (2 + 3) + 5$ .
Always verify your rearrangement by computing both forms to catch any arithmetic mistakes.

Related formulas

Associative PropertyGrouping numbers differently in addition doesn't change the sum, so you can rearrange parentheses to make calculations easier or more convenient.Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.

Frequently asked questions

Does the Commutative Property work for subtraction?: No. Subtraction is not commutative. $10 - 3 = 7$ , but $3 - 10 = -7$ . The order of the numbers matters for subtraction, so you cannot swap them like you can with addition.
Why is the Commutative Property useful?: It lets you rearrange a problem to make it easier to calculate or understand. For example, you might write $13 + 27$ as $27 + 13$ to count up from 27 instead of doing $13 + 27$ in your head.
How is the Commutative Property different from the Associative Property?: The Commutative Property lets you change the order of numbers: $a + b = b + a$ . The Associative Property lets you change the grouping with parentheses: $(a + b) + c = a + (b + c)$ . Both let you rearrange, but in different ways.
Can I use the Commutative Property in algebra?: Yes. The Commutative Property still works with variables. For example, $x + 5 = 5 + x$ and $3y = y \cdot 3$ . This is useful for simplifying expressions and solving equations.

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Reviewed 2026-09-18