Associative Property: Definition, Explanation & Real-World Applications

Learn about the associative property, its mathematical principles, practical applications in physics, finance, computing, and everyday life. Discover how easily calculations may be made simpler by grouping numbers.

Consider organizing a set of numbers for fast mental calculations or breaking down difficult equations. The associative property is in play when you see that altering how you arrange the numbers doesn't affect the outcome! This basic principle makes computations simpler and more effective, whether you're solving physics equations, programming algorithms, or figuring out food budgets.

The following topics will be covered in this article:

• What is the associative property?
• How associative property is used in addition and multiplication?
• What distinguishes associative property from other properties of mathematics?
• Where in real life is associative property idea used?

What is the Associative Property?

As the name suggests, associative refers to grouping. The word "associate" is where the term "associative" originates. The association property can be used for addition and multiplication, two fundamental mathematical operations. Usually, this applies to more than two integers.

When three or more numbers are added or multiplied, the outcome (sum or the product) stays the same regardless of how the numbers are arranged, according to the associative property.

Associative Property of Addition

The associative property of addition states that no matter how the numbers are arranged, the sum of three or more of them stays the same. Assume that p, q, and r are three numbers. For these, the following formula will be used to express the associative property of addition:

Associative Property of Addition Formula:
(P + Q) + R = P + (Q + R)

Example:

(2 + 8) + 3 = 2 + (8 + 3) = 13 
10 + 3 = 13 is the result of solving the left-hand side. We now obtain 2 + 11 = 13 if we solve the right-hand side. As a result, even when the numbers are classified differently, the sum stays the same.

Associative Property of Multiplication

The product of three or more numbers stays the same regardless of how the numbers are arranged, according to the associative feature of multiplication. The following formula can be used to express the associative property of multiplication:

Associative Property of Multiplication Formula
(P × Q) × R = P × (Q × R)

Example:

(2 × 4) × 3 = 2 × (4 × 3) = 24 
8 × 3 = 24 is the result of solving the left-hand side. Now, 2 × 12 = 24 is the result of solving the right-hand side. As a result, it is evident that the product of the numbers is constant regardless of how the numbers are grouped.

Subtraction and Division

The associative property does not work with subtraction and division. This means if we try to apply the associative law to subtraction, it will not work.

Example:

• (8 - 4) - 3 is not equal to 8 - (4 - 3). If LHS is solved, we get, 2 - 3 = -1. If RHS is solved, we get, 8 - (1) = 7. Hence, associative property is not applicable to subtraction.
• (24 ÷ 4) ÷ 3 is not equal to 24 ÷ (4 ÷ 3). If LHS is solved, we get, 6 ÷ 3 = 2. Now, if RHS is solved, we get, 24 ÷ (4 ÷ 3) ≠ 2. Hence, associative property is not applicable for division.

Difference between Associative Property and Commutative Property

The commutative property states that no matter how two numbers are arranged, the outcome of multiplication or addition stays the same. Commutative property involves two numbers, while associative property involves more than two numbers. While the associative property deals with altering how numbers are grouped, the commutative property deals with altering the order of numbers.

Examples on Associative Property of Addition and Multiplication

Example 1: If (10 × 20) × 15 = 3000, then use associative property to find (15 × 10) × 20.
Solution: According to the associative property of multiplication, (10 × 20) × 15 = (15 × 10) × 20.
Given that (10 × 20) × 15 = 3000, (15 × 10) × 20 = 3000.

Example 2: Check whether the associative property of addition is implied in the following equation. 50 + (40 + 10) = (50 + 40) + 10
Solution: LHS = 50 + (40 + 10) = 50 + 50 = 100
RHS = (40 + 50) + 10 = 50 + 50 = 100 = LHS
Thus, the associative property of addition is implied in this equation.

Practice Questions

Test yourself with these problems:

1. Complete the following equation: 3 × (… × 4) = (3 × 2) × …
2. Fill in the blanks: 12 + 9 + … = 9 + 5 + …
3. If 4 × (6 × 5) = 120, find the product of (4 × 6) × 5 using the associative property.
4. Solve for y using the associative property formula: 3 + (y + 8) = (3 + 5) + 8.

Where Do We Use This in Real Life?

• Mental Math & Quick Calculations - Grouping numbers strategically while adding or multiplying makes mental math faster and more efficient.
• Computer Science & Programming - Algorithms use the associative property to optimize calculations and improve processing speed.
• Banking & Finance - Interest calculations and financial transactions rely on associative grouping for accurate computations.
• Physics & Engineering - The associative property helps simplify force equations, electrical circuits, and structural calculations.

The associative property is an essential mathematical rule that simplifies calculations across various fields, from computing to engineering. By understanding and applying this property, you can solve problems more efficiently.