Questions: Associative Property of Multiplication

Question 1

A student solves 4 × 5 × 2 by computing (4 × 5) × 2 = 20 × 2 = 40. A second student computes (4 × 2) × 5 = 8 × 5 = 40. Who got the right answer?

• A. Only the first student — you must always multiply left to right
• B. Only the second student — you must use parentheses as written
• C. Both students — the associative property guarantees the same product regardless of which pair is multiplied first
• D. Neither — the correct answer depends on which parentheses are shown in the original problem

Answer: (C)

The associative property states that grouping (which pair you multiply first) does not affect the product. Both students get 40 because all three factors — 4, 5, and 2 — are still in the problem. The only difference is the order of computation, not the result. This is what makes the property useful: you are free to choose the grouping that is easiest to calculate.

Question 2

How does the associative property differ from the commutative property of multiplication?

• A. They are the same thing — both let you rearrange factors
• B. The commutative property swaps the order of two factors; the associative property changes which pair among three or more factors is multiplied first
• C. The commutative property applies to three factors; the associative applies only to two
• D. The associative property works for addition but not multiplication

Answer: (B)

Commutative: 4 × 7 = 7 × 4 — the order of two factors is swapped, same product. Associative: (2 × 3) × 4 = 2 × (3 × 4) — the factors stay the same but the grouping (which multiplication happens first) changes. Confusing these two is common. Together, both properties give complete freedom to rearrange and regroup any multiplication expression.

Question 3

Using the associative property, (5 × 2) × 7 = 5 × (2 × 7), and both expressions equal 70.

• T. True
• F. False

Answer: True

(5 × 2) × 7 = 10 × 7 = 70. And 5 × (2 × 7) = 5 × 14 = 70. The associative property guarantees these are equal. Notice that (5 × 2) × 7 is much easier to compute because 5 × 2 = 10 and multiplying by 10 is trivial — this is exactly the kind of strategic simplification the property enables.

Question 4

Changing which factors are inside parentheses changes the final product because the operations are performed in a different order.

• T. True
• F. False

Answer: False

In multiplication, changing the grouping never changes the product. The parentheses only affect which calculation you do first, not what values are being multiplied. Unlike subtraction or division (where order and grouping do matter), multiplication is both commutative and associative — all factors contribute equally to the product regardless of grouping order.

Question 5

A student needs to compute 5 × 9 × 2. Show how the associative property can make this easier, and explain why the answer stays the same.

Model answer: Regroup as (5 × 2) × 9 = 10 × 9 = 90. The answer stays the same because the associative property guarantees that changing which pair you multiply first doesn't change the product — all three factors are still multiplied together.

The 'natural' left-to-right approach gives 5 × 9 = 45, then 45 × 2 = 90 — correct but harder. Noticing that 5 × 2 = 10 and multiplying by 10 is trivial (just append a zero) makes the problem much easier. This is the practical payoff of the associative property: you can shop around for the easiest grouping before computing, and the answer is guaranteed to be the same.