# Lesson on Associative Property of Multiplication

The associative property of multiplication is one of the fundamental properties of arithmetic. This property applies to real numbers, integers, fractions, and other types of numbers. The associative property states that the way in which numbers are grouped in a multiplication expression does not change the product.

Mathematically, the associative property of multiplication can be expressed as follows:

(a · b) · c = a · (b · c)

In simpler terms, this means that when you are multiplying three numbers, it doesn’t matter how you group them. You can first multiply the first two numbers and then multiply the result by the third number, or you can first multiply the last two numbers and then multiply the result by the first number. The product will be the same in both cases.

For example:

(2 · 4) · 5 = 8 · 5 = 40

2 · (4 · 5) = 2 · 20 = 40

In both cases, the result is 40, demonstrating the associative property of multiplication. This property is a basic rule that influences arithmetic operations and is often taken for granted in mathematical computations.

The associative property of multiplication applies anytime you are multiplying three or more numbers together. Let’s look at another example, with four numbers.

(2 · 3) · (4 · 5) = 6 · 20 = 120

(2 · 5) · (3 · 4) = 10 · 12 = 120

In simple terms, it does not matter the way in which we group multiplication, we will still get the same answer. The associative property of multiplication is the way in which we group the numbers to multiply, and can often be very helpful when doing mental math. When doing mental math with multiplication, we can use the associative property of multiplication to group together numbers that more easily multiply in our heads. For example, if we had to multiply the following numbers:

(5 · 7) · (4 · 2) = 35 · 8 = 280

Doing, 35 · 8, in your head would be difficult for most, especially at a younger age. However, if we use the associative property of multiplication, we can regroup the numbers as shown below, which are easier to multiply in our head:

(5 · 2) · (7 · 4) = 10 · 28 = 280

Multiplying numbers by ten is fairly simple for most people, and therefore using the associative property of multiplication, makes this a much easier multiplication problem to do in our heads.

Examples

Example 1: Use the associative property of multiplication to multiply the following expression easier using mental math.

(4 · 6) · 5

Solution:

• (4 · 5) · 6
• 20 · 6
• 120

Example 2: Use the associative property of multiplication to fill in the blank in the problem below.

(3 · 4) · (5 · 6) = (3 · 5) · ( __ · 6)

Solution:

Since the numbers 4 and 5 from the left have been grouped together, we know that the missing number is 3. The 3 should be grouped with the 6, making the problem:

(3 · 4) · (5 · 6) = (3 · 5) · ( 3 · 6)

12 · 30 = 20 · 18

360 = 360

Example 3: Use the associative property of multiplication to fill in the blanks below. Then find the product.

(3 · __ ) · 5 = (2 · 5) · __

Solution:

Looking at both sides of the equation, we can see that 5 is used on both sides of the equation, and the other two numbers are 2 and 3.

The next step is to determine where the numbers 2 and 3 go on each corresponding side of the equation. The equation should be as follows:

• (3 · 2) · 5 = (2 · 5) · 3
• 6 · 5 = 10 · 3
• 30 = 30

FAQs on Associative Property of Multiplication

1. Why is the Associative Property of Multiplication important in mathematics?

• The property simplifies the manipulation of expressions and helps establish a consistent framework for mathematical operations. The property also allows you to regroup numbers in a manner that makes the use of mental math much easier.

2. What is the Associative Property of Multiplication?

• The associative property of multiplication states that the multiplication of three or more numbers results in the same answer regardless of the order or grouping of the numbers.

3. What is the General Rule for the Associative Property of Multiplication?

• The general rule for the associative property of multiplication is written as (a · b) · c = a · (b · c). This means that the grouping of any three or more numbers does not affect the product.

4. Can the Associative Property be used with other mathematical operations?

• The associative property can also be used with addition, as it does not matter the order or grouping in which we add numbers, we will get the same answer. However, the associative property does not work with subtraction or division.
• One example to demonstrate why it does not work with subtraction is:

3 – 2 ≠ 2 – 3

1 ≠ – 1

– Similarly it does not work with division, and here is an example to demonstrate this:

4 ≠ 1/4

5. Does the Associative Property of Multiplication apply only to whole numbers?

• No, theassociative property of multiplication applies to all real numbers, such as decimals, fractions, negative numbers, and more.

6. Does the Associative Property of Multiplication only apply to three numbers?

• No, the property is applicable to multiplication of three or more numbers. For example,

(a · b · c · d) · e = a · (b · c · d · e)