Which operations are associative in arithmetic?

Which operations are associative in arithmetic?

• A. Subtraction and division are associative
• B. Addition and subtraction are associative
• C. Subtraction and multiplication are associative
• D. Addition and multiplication are associative

Answer: (D)

The key idea is the associative property: when you combine several numbers with an operation, the grouping of those numbers doesn’t affect the result. For addition, grouping doesn’t matter because (a + b) + c equals a + (b + c). For example, (2 + 3) + 4 = 9, and 2 + (3 + 4) = 9. The same idea holds for multiplication: (a × b) × c equals a × (b × c). For instance, (2 × 3) × 4 = 24, and 2 × (3 × 4) = 24. Subtraction and division, however, don’t follow this property: (5 − 2) − 1 = 2 while 5 − (2 − 1) = 4, and (8 ÷ 2) ÷ 2 = 2 but 8 ÷ (2 ÷ 2) = 8. So the operations that are associative are addition and multiplication.

The key idea is the associative property: when you combine several numbers with an operation, the grouping of those numbers doesn’t affect the result. For addition, grouping doesn’t matter because (a + b) + c equals a + (b + c). For example, (2 + 3) + 4 = 9, and 2 + (3 + 4) = 9. The same idea holds for multiplication: (a × b) × c equals a × (b × c). For instance, (2 × 3) × 4 = 24, and 2 × (3 × 4) = 24. Subtraction and division, however, don’t follow this property: (5 − 2) − 1 = 2 while 5 − (2 − 1) = 4, and (8 ÷ 2) ÷ 2 = 2 but 8 ÷ (2 ÷ 2) = 8. So the operations that are associative are addition and multiplication.