Associative Property of Whole Numbers With Examples

The Associative Property is a fundamental concept in mathematics, especially useful in simplifying calculations and understanding the structure of arithmetic operations. This property applies to addition and multiplication but not to subtraction or division.

Let’s break it down with examples for better clarity.

Associative Property of Addition

The associative property of addition states that the way in which numbers are grouped does not change their sum. In other words, when adding three or more numbers, you can regroup them in any way, and the sum will be the same.

Mathematical Expression:

For any whole numbers a, b, and c:

(a + b) + c = a + (b + c)

Example 1:

Let’s take the numbers 2, 3, and 4.

Group them in different ways and add:

(2 + 3) + 4 = 5 + 4 = 9

2 + (3 + 4) = 2 + 7 = 9

As you can see, both groupings result in the same sum, 9.

Example 2:

Consider the numbers 1, 5, and 7.

(1 + 5) + 7 = 6 + 7 = 13

1 + (5 + 7) = 1 + 12 = 13

Again, both groupings give the same sum, 13.

Associative Property of Multiplication

The associative property of multiplication states that the way in which numbers are grouped does not change their product. In other words, when multiplying three or more numbers, you can regroup them in any way, and the product will be the same.

Mathematical Expression:

For any whole numbers a, b, and c:

(a × b) × c = a × (b × c)

Example 1:

Let’s take the numbers 2, 3, and 4.

Group them in different ways and multiply:

(2 × 3) × 4 = 6 × 4 = 24

2 × (3 × 4) = 2 ×1 2 = 24

Both groupings result in the same product, 24.

Example 2:

Consider the numbers 1, 5, and 7.

(1 × 5) × 7 = 5 × 7 = 35

1 × (5 × 7) = 1 × 35 = 35

Both groupings give the same product, 35.

Key Points to Remember

1. Addition and Multiplication Only: The associative property applies only to addition and multiplication, not to subtraction or division.
2. For example, (8 − 3) − 2 ≠ 8 − (3 − 2).
3. Simplification: This property helps in simplifying complex arithmetic calculations, making it easier to group numbers in a way that simplifies the computation process.
4. Parentheses Indicate Grouping: In expressions, parentheses indicate which numbers are grouped together and should be computed first.

Real-Life Application

Consider splitting a bill among friends at a restaurant. If the bill is $60, and there are three friends contributing different amounts:

• Friend A pays $20
• Friend B pays $20
• Friend C pays $20

Using the associative property, you can group the contributions in any order:

(20 + 20) + 20 = 40 + 20 = 60

20 + (20 + 20) = 20 + 40 = 60

No matter how you group the payments, the total amount collected remains the same.

Understanding the associative property helps build a strong foundation in arithmetic and algebra, making it easier to tackle more complex mathematical problems in the future.