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What is the Associative Property?

Associative property

In mathematics, the term associative property states that when an expression has three terms, they can be grouped in any way to solve that expression. The grouping of numbers will never change the result of their operation. The associative property is true for the cases of addition and multiplication. Neither the sum nor the product of the expression changes. 

Before moving further, let us learn about the term associate. Associate means to group or come together. When many people gather together for a common goal, it is known as an association. Mathematics also similarly uses the term. To associate in mathematics means to group or coagulate together.

Using associative property, you can group or associate the terms in any order you want. In this article, we shall prove this property and learn in-depth about what is associative property, associative property definition, associative property math, and solve adequate associative property examples. 

Understanding Associative Property

Fundamentally, the associative property is a law of numeric and algebraic mathematics that states that when three or more terms ( integers or variables) are summed or multiplied, the outcome always remains the same irrespective of how the terms are grouped. 

You might be wondering how grouping is done, right? Well, grouping is done with the help of parentheses or the round brackets ‘( ).’ Let us look at the expression below to understand how grouping is done:

  • + b + c is a simple expression without any grouping.
  • ( a + b ) + c is the same expression with terms a and b grouped.
  • + + ( b + c), the same expression with terms b and c group.
  • ( a + c ) + b, this is the same expression with terms a and c grouped.

You must now be clear on how grouping is done. You must now group any expression easily in any number of ways.

Let us take a numeric example, say, 2 + 7 + 4

You can group this expression as (2 + 7) + 4, 2 + (7 + 4), or (2 + 4) + 7.  

Associative Property Definition

The associative rule asserts that the sum or product of any three or more integers is unaffected by the order in which the numbers are grouped by parenthesis. It solely applies to addition and multiplication. In other words, the outcome is the same if the same numbers are grouped in various ways for summation and multiplication.

We have learned a lot about what is associative property, associative property definition, associative property math, and seen some good associative property examples. Now let us focus on individual associative properties. Next, we are going to study the associative property of addition, the associative property of addition example, the associative property of multiplication, and the associative property of multiplication example.

Associative Property of Addition

According to the associative property of addition, the result of the summation of three or more integers stays the same regardless of how the numbers are arranged. Let’s say we’ve got three numbers: K, L, and M. The following formula will be used to express the associative property of addition in these cases:

Associative Property of Addition Formula:

K + (L + M) = (K + L )+ M.

Let’s look at an example below to assist our understanding of the associative property of addition.

Example: 10 + 3 + 7 = 20 

Using the associative property of addition => (10 + 3) + 7 = 10 + (3 + 7) = 20. 

We obtain 13 + 7 = 20 if we solve the left-hand side. We obtain 10 + 10 = 20 if we solve the right-hand side. 

Conclusion: Even though the numbers are categorized differently, the total remains the same.

Associative Property of Multiplication

Similarly, the result of three or more integers, according to the associative property of multiplication, stays the same regardless of how the numbers are grouped. Let’s say we’ve got three numbers: K, L, and M. The following formula will be used to express the associative property of addition in these cases:

Associative Property of Multiplication Formula:

K x (L x M) = (K x L ) x M.

Let’s look at the example below to assist our understanding of the associative property of multiplication.

Example: 10 x 3 x 7 = 210 

Using the associative property of addition => (10 x 3) x 7 = 10 x (3 x 7) = 210. 

When solving the left-hand side of the expression, we get 30 x 7 = 210. When we solve the right-hand side, we get 10 x 21 = 210. 

Conclusion: The product remains unchanged even though the numbers are grouped differently.

You all must now be well acquainted with the associative property of addition, associative property of addition example, the associative property of multiplication, and associative property of multiplication example. It is now time to verify why these laws are universally true.

Verification of Associative Property

Hitherto, you have studied that the associative law is only true for addition and multiplication. Did you ponder why this law is not true for the other two mathematics operations, that is, subtraction and division? 

This section will verify and see how the associative property is valid only for addition and multiplication. We will apply this property to all the fundamental operations in mathematics and compare the results, one by one.

  • For Addition: We know that the associative law for addition is given as (K + L) + M = K + ( L + M). For instance, let’s take an expression, say, 11 + 4 + 6. If we sum the expression, we get the answer as 21.

Now let us use the associative property to solve this:

( 11 + 4 ) + 6 => 15 + 6 = 21 

11 + ( 4 + 6 ) => 11 + 10 = 21

In both ways, we are getting the same result which is 21. This is also equal to the initial result we calculated before using the property. Thus associative property holds for addition.

  • For Multiplication: We know that the associative law for multiplication is given as (K x L) x M = K x ( L x M). Let us take a new instance, say, 2 x 13 x 5 . If we multiply the expression, we get the answer as 130,

Let us now utilize the associative property to solve this expression:

( 2 x 13 ) x 5 => 26 x 5 = 130 

2 x ( 13 x 5 ) => 2 x 65 = 130

In both ways, we get the same result, which is 130, equal to the initial result we calculated before using the associative property. Therefore associative property holds for multiplication also.

  • For Subtraction: There is no defined expression for the associative law of subtraction; therefore, let us say that ( K – L ) – M = K – ( L – M ) will be the associative law of subtraction.

Example: Let our expression be 18 – 5 – 10 = 3

Let us now use associative property on the example:

(18 – 5) – 10 = 13 – 10 = 3       

18 – (5 – 10) = 18 – (-5) = 18 + 5 = 23

Clearly, by grouping the terms in different ways, we get other answers that make the associative property of subtraction null and void. 

  • For Division: Last, let us verify if the associative property for division stands true or not. There is no well-written formula for the associative law of division so let us suppose that ( K ÷ L ) ÷ M = K ÷ ( L ÷ M ) will be the associative law of division.

Example: Let our expression be 32 ÷ 4 ÷ 2 = 4

Let us now use associative property on the example:

( 32 ÷ 4 ) ÷ 2 = 8 ÷ 2 = 4 

32 ÷ ( 4 ÷ 2 ) = 32 ÷ 2 = 16

We get unequal solutions by grouping the items differently, indicating that the associative characteristic of division is not valid.

Associative Property Examples 

Example 1: If 7 × (3 × 9) = 189, then find the product of (7 × 3) × 9 using the associative property.

Solution:

Multiplication fulfils the associative condition, therefore (7 × 3) × 9 = 21 x 9 = 189.

Example 2: Find the value of x using the associative property: 6 + (x + 21) = (6 +2) + 21

Solution:

Multiplication fulfils the associative condition, therefore (6 + 2) + 21 = 6 + (x + 21). So, the value of x is 2.

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