Associative Property Formula

July 16, 2021

We can add/multiply the numbers in an equation irrespective of the grouping of those numbers. This property is called the associative property formula. So, the associative formula exists in only addition and multiplication operations. Let us learn the associative property formula with a few solved examples.

What Is the Associative Property Formula?
Suppose we have three numbers: a, b, and c. We will show two important associative property formulas: associative property of addition and associative property of multiplication.

Associative Property Formula
Associative Property Formula for Addition: The sum of three or more numbers remains the same irrespective of the way numbers are grouped.

(A + B) + C = A + (B + C)

Associative Property Formula for Multiplication: The product of three or more numbers remains the same irrespective of the way numbers are grouped.

(A × B) × C = A × (B × C)

Verification of Associative Property Formula
Let us try to justify how and why the associative property formula is only valid for addition and multiplication operations. We will apply the property formula individually on the four basic operations.

For Addition: The general associative property formula is expressed as (A + B) + C = A + (B + C). Let us try to fix some numbers in the formula to verify the same. For example, (1 + 4) + 2 = 1 + (4 + 2) = 7. We say that addition is associative for the given set of numbers.

For Subtraction: The general associative property formula is expressed as (A – B) – C ≠ A – (B – C). Let us try to fix some numbers in the formula to verify the same. For example, (1 – 4) – 2 ≠ 1 – (4 – 2) i.e., -5 ≠ -1. We say that subtraction is not associative for the given set of numbers.

For Multiplication: For any set of three numbers (A, B, and C) associative property for multiplication is given as (A × B) × C = A × (B × C). For example, (1 × 4) × 2 = 1 × (4 × 2) = 8. Here we find that multiplication is associative for the given set of numbers.

For Division: For any three numbers (A, B, and C) associative property for division is given as A, B, and C, (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). For example, (9 ÷ 3) ÷ 2 ≠ 9 ÷ (3 ÷ 2) = 3/2 ≠ 6. You will find that expressions on both sides are not equal. So division is not associative for the given three numbers.

Examples on Associative Property Formula
Let us take a look at a few examples to better understand the formula of associative property.

Example 1: If 3 × (6 × 4) = 72, then find (3 × 6) × 4 using associative property formula

Solution:

Since multiplication satisfies the associative property formula, (3 × 6) × 4 = 3 × (6 × 4) = 72

Example 2: Solve for x using associative property formula: 2 + (x + 9) = (2 + 5) + 9

Solution:

Since addition satisfies the associative property formula, (2 + 5) + 9 = 2 + (x + 9) = (2 + x) + 9. So, the value of x is 5.

Example 3: If 2 × (3 × 5) = 30, then find (2 × 3) × 5 using associative property formula.

Solution:

The associative property formula for any given set of three numbers says that for any three numbers (A, B, and C) expression can be expressed as (A × B) × C = A × (B × C)
Given = 2 × (3 × 5) = 30
Using the associative property formula, we can evaluate (2 × 3) × 5.
To verify: (2 × 3) × 5 = 30 or not first, solve the terms inside parentheses.
= 6 × 5
= 30
Hence, 2 × (3 × 5) = (2 × 3) × 5 = 30.


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