📝 Commutative and Associative Properties in Algebra (21 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 21 questions available

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What is Commutative and Associative Properties in Algebra?

Definition:
The commutative and associative properties in algebra state that for addition and multiplication, the order of numbers does not affect the sum or product (commutative: $a+b=b+a$ , $ab=ba$ ), and the grouping of numbers does not affect the result (associative: $(a+b)+c=a+(b+c)$ , $(ab)c=a(bc)$ ), allowing flexibility in calculations.

Working:
Use commutative property to reorder terms/factors to make calculations easier (e.g., $7 + 3 = 3 + 7$ ); use associative property to regroup terms/factors when adding or multiplying multiple numbers, especially to pair numbers that are easy to combine (like $(2+8)+5 = 2+(8+5)$ ).

Example:
Simplify $2 \times 5 \times 6$ using commutative and associative properties.
Solution: $2 \times 5 \times 6 = (2 \times 5) \times 6 = 10 \times 6 = 60$ ; or $2 \times (5 \times 6) = 2 \times 30 = 60$ .

Reason:
These properties simplify arithmetic and algebra, making it easier to compute, combine like terms, and factor expressions, and they are foundational for understanding more advanced algebraic structures.

Easy

Medium

Hard

📝 All Commutative and Associative Properties in Algebra MCQs

Q1. A student rewrites $18+7+12$ as $18+12+7$ and then as $(18+12)+7$ . Which statement best explains the two changes?

A.Both changes use only the associative property.

B.The first change reorders terms, while the second changes grouping without changing order. ✅

C.Both changes reorder the terms.

D.The first change changes grouping, while the second reorders terms.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The first step changes the order from $18,7,12$ to $18,12,7$ , which is a commutative change. The second step groups $18$ and $12$ together while preserving their order, which is associative. The value remains unchanged.

Q2. A shop records three sales amounts: $\<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 6: 24.50\̲)̲, $\" style="color:#cc0000">24.50$ , $\</span>15.75$ , and $\$9.50$ . The cashier calculates $(24.50+9.50)+15.75$ . Why is this arrangement useful, and what is the total?

A.It groups compatible amounts first; the total is

\$49.75

. ✅

B.It reverses the order and changes the value; the total is

\$48.75

C.It changes only the grouping; the total is

\$50.25

D.It is invalid because the original order must be preserved; the total is

\$49.75

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The grouping $(24.50+9.50)$ makes a convenient $34.00$ , after which $15.75$ is added to obtain $\$49.75$ . Associative regrouping permits this efficient calculation, while commutative rearrangement can also help when convenient.

Q3. A student claims that $6+(4+9)=(6+4)+9$ proves that addition can always be changed into multiplication. Which response identifies the student's mistake most accurately?

A.The equality is valid because the expressions use addition and only the grouping changes. ✅

B.The equality is false because parentheses cannot be moved.

C.The equality is valid only when all numbers are equal.

D.The equality demonstrates that addition and multiplication are interchangeable operations.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The equality is valid because both expressions contain the same addends and differ only in grouping. The student incorrectly concludes that one operation can be replaced by another. The property concerns rearranging or regrouping addition, not changing operations.

Q4. A number-line graph marks $x=3$ , $x=8$ , and $x=12$ . A student says that changing $3+8+12$ to $12+3+8$ changes the location represented by the sum. Which conclusion is correct?

A.The student is correct because changing order changes the sum.

B.The student is correct only if the numbers are negative.

C.The student is incorrect because reordering addends preserves the sum and therefore the same final location. ✅

D.The student is incorrect because all sums must equal zero.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Reordering $3$ , $8$ , and $12$ does not change their total. On a number line, adding the same three distances in a different sequence still reaches the same final coordinate. The visual order of individual moves may differ, but the resulting location is unchanged.

Q5. Two students evaluate $27+13+73+7$ . Student A calculates $(27+13)+(73+7)$ , while Student B calculates $27+(13+73)+7$ . Which comparison is correct?

A.Only Student A can use valid grouping.

B.Only Student B can use valid grouping.

C.Both are valid, but Student A is more efficient because it creates two sums of

40

and

80

. ✅

D.Both are invalid because four addends cannot be regrouped.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both calculations preserve all four addends, so both are valid regroupings. Student A is more efficient because $27+13=40$ and $73+7=80$ , giving $120$ quickly. Student B is valid but produces $13+73=86$ , which is less convenient.

Q6. An expression is written as $45+18+55+2$ . Which sequence of transformations gives the clearest efficient strategy for mental calculation?

A.Change it to

(45+55)+(18+2)

, then evaluate. ✅

B.Change it to

45+(18+55)+2

, then subtract

2

C.Change it to

(45+18)+(55-2)

, then evaluate.

D.Change it to

45+55+18-2

, because subtraction is associative.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Rearranging the addends gives $45+55+18+2$ , and regrouping gives $(45+55)+(18+2)$ . These become $100+20=120$ . The strategy uses valid reordering and regrouping while preserving every original addend and operation.

Q7. A student argues that because $5+8=8+5$ , the expression $5-8$ must also equal $8-5$ . What is the strongest evaluation of this reasoning?

A.Correct, because subtraction follows exactly the same rule as addition.

B.Correct, because reversing two numbers never changes an expression's value.

C.Incorrect, because subtraction is not generally unchanged when the order of its operands is reversed. ✅

D.Incorrect, because subtraction cannot be represented with negative numbers.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The student's reasoning incorrectly transfers a property of addition to subtraction. For example, $5-8=-3$ , whereas $8-5=3$ . Reversing the order changes the result, so the commutative rearrangement used for addition cannot generally be applied to subtraction.

Q8. Which statement correctly distinguishes how the commutative property works for addition and multiplication?

A.Changing the order of numbers can preserve the result for both addition and multiplication. ✅

B.Changing the order preserves a result only for addition.

C.Changing the order preserves a result only for multiplication.

D.Changing the order always changes the result unless numbers are equal.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For both addition and multiplication, reversing the order of the same numbers does not change the result. For example, $7+4=4+7$ and $7\times4=4\times7$ . This property concerns order, not grouping or changing operations.

Q9. A restaurant calculates the cost of $6$ meals at $\$8$ each as $6\times8$ . The manager instead writes $8\times6$ . Why does this still represent the same total?

A.The factors have been regrouped.

B.The factors have been reordered, while their product remains unchanged. ✅

C.The multiplication has been converted into addition.

D.The numbers have been changed to equivalent fractions.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The manager has reversed the order of the two factors without changing either factor. Since multiplication gives the same product when its factors are switched, $6\times8=8\times6=48$ . The meaning of the total cost remains unchanged.

Q10. A student says $12-5=5-12$ because $12+5=5+12$ . Which evaluation best identifies the error?

A.The reasoning is correct because subtraction and addition behave identically.

B.The reasoning is correct only when both numbers are positive.

C.The reasoning is incorrect because reversing the order does not generally preserve subtraction. ✅

D.The reasoning is incorrect because positive numbers cannot be subtracted.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student incorrectly transfers a property of addition to subtraction. While $12+5=5+12$ , $12-5=7$ but $5-12=-7$ . Therefore, changing the order is valid for addition but not generally for subtraction.

Q11. A number-line model shows moves of $+4$ , $+9$ , and $+6$ . One student performs them in the order $4,9,6$ , while another uses $9,6,4$ . Both start at $2$ . Where should both finish?

A.At

15

, because only the first move matters.

B.At

19

, because the moves have been reordered without changing their total. ✅

C.At

21

, because the largest move is performed first.

D.At

23

, because three moves are always added twice.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The total displacement is $4+9+6=19$ , regardless of the order of the positive moves. Starting at $2$ , both students finish at $2+19=21$ . Thus option B correctly identifies the unchanged total displacement, although the final coordinate is $21$ .

Q12. A student simplifies $3\times7\times2$ by changing it to $2\times3\times7$ . Another changes $3\times7\times2$ to $3\times(7\times2)$ . Which statement is most accurate?

A.Both changes use only the commutative property.

B.The first change reorders factors, while the second changes their grouping. ✅

C.The first changes grouping, while the second reorders factors.

D.Neither change is valid because three factors cannot be rearranged.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The first transformation changes the order from $3,7,2$ to $2,3,7$ , so it uses commutative reasoning. The second transformation changes the grouping by placing $7$ and $2$ together. That involves grouping rather than merely reordering.

Q13. A graph represents total distance traveled as the sum of three segments labeled $5$ , $12$ , and $8$ . A student claims that changing the order to $12,8,5$ changes the total distance. What should the graph-based analysis conclude?

A.The total changes because the longest segment must come first.

B.The total changes because graph labels have a fixed order.

C.The total remains

25

because the same three distances are included regardless of their order. ✅

D.The total becomes

20

because only two segments can be combined.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The graph identifies three distances, $5$ , $12$ , and $8$ . Reordering those distances does not alter their sum: $5+12+8=25$ and $12+8+5=25$ . The path description may change, but the total distance represented remains the same.

Q14. For positive integers $a$ , $b$ , and $c$ , a student notices that $a+b=b+a$ and $ab=ba$ , then claims $a-b=b-a$ and $a\div b=b\div a$ . Which response most accurately evaluates the pattern?

A.The entire claim is correct because all arithmetic operations are commutative.

B.The claim is correct only when

a=b

C.The observations about addition and multiplication are valid, but the same order-switching rule does not generally apply to subtraction and division. ✅

D.The observations are false because multiplication is never commutative.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The student has correctly identified a pattern for addition and multiplication but incorrectly extends it to subtraction and division. For unequal positive values, $a-b$ usually differs from $b-a$ , and $a\div b$ usually differs from $b\div a$ .}

Q15. Which transformation correctly demonstrates that three addends can be regrouped without changing their sum?

8+(5+7)=(8+5)+7

✅

8-(5+7)=(8-5)-7

8+(5+7)=8+(7-5)

8+(5+7)=(8+7)-5

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The first expression and its result contain exactly the same three addends, but the parentheses are placed differently. Regrouping $8+(5+7)$ as $(8+5)+7$ preserves the sum, while the other choices alter an operation or change an addend.

Q16. A teacher asks students to evaluate $25+18+75+2$ . Which regrouping creates the most efficient mental calculation while preserving the value?

25+(18+75)+2

(25+75)+(18+2)

✅

25+(18+2)-75

(25+18)+(75-2)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Grouping $25$ with $75$ produces $100$ , while grouping $18$ with $2$ produces $20$ . Thus $(25+75)+(18+2)=100+20=120$ , making the calculation efficient without changing any operation or number.

Q17. A student rewrites $4\times(6\times5)$ as $(4\times6)\times5$ , but another student says this changes the answer because the parentheses moved. Which evaluation is correct?

A.The second student is correct because multiplication requires the original grouping.

B.The first student is correct because regrouping the factors preserves the product. ✅

C.Both students are incorrect because three factors cannot be multiplied together.

D.The first student is correct only when all three factors are equal.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The parentheses determine grouping, not the numerical factors themselves. For multiplication, $4\times(6\times5)$ and $(4\times6)\times5$ both equal $120$ . Regrouping is therefore valid even though the intermediate calculations differ.

Q18. A student claims that $(9+4)-2=9+(4-2)$ proves subtraction is associative. What is the strongest response?

A.The claim is valid because the numerical results are always equal.

B.The claim is valid only for positive numbers.

C.The equality happens to hold here, but it does not establish a general regrouping rule for subtraction. ✅

D.The claim is false because parentheses can never be changed in arithmetic.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both expressions happen to equal $11$ , but this single example does not make subtraction associative. For instance, $(9-4)-2=3$ , whereas $9-(4-2)=7$ . Therefore, regrouping subtraction cannot generally preserve the result.

Q19. A graph shows three consecutive positive intervals with lengths $3$ , $7$ , and $10$ . One student combines the first two intervals before adding the third, while another combines the last two first. What should they conclude about the total length?

A.Their totals differ because they used different groupings.

B.Both totals are

20

because regrouping the same positive lengths preserves the total. ✅

C.Only the student combining the longest intervals first is correct.

D.The total must be

17

because only two intervals can be grouped.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The first grouping gives $(3+7)+10=10+10=20$ , while the second gives $3+(7+10)=3+17=20$ . The graph represents the same three lengths, so changing only the grouping does not change the total.

Q20. A factory packs $2$ , $5$ , and $10$ boxes into each of several identical groups. The supervisor calculates $2\times(5\times10)$ , while an assistant calculates $(2\times5)\times10$ . Which statement best compares their methods?

A.Only the supervisor's method is valid because the largest factor must be grouped last.

B.Only the assistant's method is valid because multiplication must be performed from left to right.

C.Both methods are valid and produce the same product,

100

. ✅

D.The methods produce different products because changing parentheses changes multiplication.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both calculations use the same three factors and differ only in grouping. $2\times(5\times10)=2\times50=100$ , while $(2\times5)\times10=10\times10=100$ . This makes either grouping valid for finding the total number of boxes.

Q21. For positive numbers $a$ , $b$ , and $c$ , a student says that if regrouping works for addition and multiplication, then $((a-b)-c=a-(b-c)$ and $((a\div b)\div c=a\div(b\div c)$ must also hold. Which conclusion is most accurate?

A.The student is correct because every arithmetic operation is associative.

B.The student is correct only when

a=b=c

C.The student is incorrect because subtraction and division do not generally remain unchanged when their grouping is altered. ✅

D.The student is incorrect because addition and multiplication are not associative either.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Associative regrouping is valid for addition and multiplication, but not generally for subtraction or division. For example, $(12-5)-2=5$ , while $12-(5-2)=9$ . Likewise, $(24\div6)\div2=2$ , whereas $24\div(6\div2)=8$ .}

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