# What is Associative Property, Definition, Types, and Examples

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The associative property is a numeric law of mathematics. It states that adding and multiplying three or more numbers produce the same results irrespective of how they are grouped.

You might be thinking about how to group the numbers, right?

Grouping of numbers is done with parenthesis (brackets). Let’s take an example.

Assume a + b + c are three different numbers. It is a simple expression.

a + (b + c) This is the same expression, but the only difference is that numbers are grouped with brackets.

You must be clear on how to group numbers. Now you can group numbers for different expressions easily.

In this article, we will learn about different terms, like the definition of associative properties, associative law, associative property of addition, associative property of multiplication, and some examples.

## Definition

It is defined as the addition and multiplication of more than two numbers that result in the same no matter how they are grouped.

Example

If you want to add three numbers, like 4, 5, and 6 altogether. You have to group these numbers in the addition procedure; it will be something like this

4+ (5 + 6) = (4 + 5) + 6

4+11 = 9+6

15=15

The same rule applies to multiplication;

If you want to multiply three numbers, like 4, 5, and 6 altogether. You have to group these numbers in a multiplication procedure; it will be something like this

4 x (5 x 6) = (4 x 5) x 6

4×30 = 20×6

120 =120

## Associative Law

When the exact numbers are grouped differently for addition and multiplication, they give the same resuls.

Associative law applies to addition and multiplication only.

A general expression for associative properties

A general expression for associative properties is given right below;

For the associative property of addition:

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

For the associative property of multiplication:

a x (b x c) = (a x b) x c

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According to the associative property of addition, when three numbers are added, the result remains the same regardless of how they are grouped.

Suppose we add three numbers: a, b, and c.

### Associative Property of Addition Formula

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

To understand this, we take an example.

Example

Suppose we want to add 2, 4, and 5.

First of all, group the numbers according to the general formula

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

(2+ 4) + 5 = 2 + (4 + 5)

6+5 = 2+9

11=11

L.H.S = R.H.S

You can see that the result in both cases is the same. However, the numbers are grouped in different ways.

Let’s take another example

Suppose we want to add 5, 6, and 8.

First of all, group the numbers according to the general formula

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

(5+ 6) + 8 = 5 + (6 + 8)

11+8 = 5+14

19=19

L.H.S = R.H.S

Thus, the result in both cases is the same.

## Associative Property of Multiplication

According to the associative property of multiplication,

When three numbers are multiplied, the result remains the same regardless of how they are grouped.

Suppose we have to multiply three numbers a, b, and c.

### Associative Property of multiplication Formula:

(A x B) x C = A x (B x C)

To understand this, we take an example.

Example

Suppose we want to multiply 2, 4, and 5.

First of all, group the numbers according to the general formula

(A x B) x C = A x (B x C)

(2x 4) x 5 = 2 x (4 x 5)

8×5 = 2×20

40=40

L.H.S = R.H.S

You can see that the result in both cases is the same. However, the numbers are grouped in different ways.

Let’s take another example

Suppose we want to add 5, 6, and 8.

First of all, group the numbers according to the general formula

(A x B) x C = A x (B x C)

(5x 6) x 8 = 5 x (6 x 8)

30 x 8 = 5 x 48

240=240

L.H.S = R.H.S

Thus, the result in both cases is the same.

## FAQ’s

### Which two operations satisfy the condition of the associative properties?

The two operations which fulfill the condition of the associative property are;
Multiplication
It means that the associative property is only applicable to addition and multiplication.

### Define the associative property in math with an example.

The associative property is the addition and multiplication of three or more numbers that result in the same irrespective of how brackets group them.
Example
If you want to add three numbers, like 2, 3, and 6, together, you have to group these numbers in the addition procedure; it will be something like this
2+ (3 + 6) = (2 + 3) + 6
2+9 = 5+6
11=11
The same rule applies to multiplication.

### Does the associative property apply to fractional numbers?

Yes, the associative property applies to fractional numbers. When we add more than two fractions in any order, the sum of the fractions remains the same.

### How can you differentiate Associative Property and Commutative Property?

Associative and commutative properties are applicable only for addition and multiplication operations.
The associative property states that adding and multiplying three or more numbers produces the same results irrespective of how they are grouped by brackets.
A + (b + c) = (a + b) + c
Associative property of multiplication
A x (b x c) = (A x b) x c
While Commutative property states;
The addition and multiplication of numbers result the same regardless of how the numbers are ordered.
a + b = b + a
a x b = b x a

### Why does associative law not apply to subtraction and division?

Associative law does not apply to subtraction and division. Let’s take an example to explain this;
In the case of subtraction
10 – (5 – 2) = (10 – 5) – 2
10- 3 = 5 – 2
7 ≠ 3
L.H.S ≠ R.H.S
Thus, the associative property does not apply to subtraction.
In the case of division:
(18 ÷ 3) ÷ 2 = 18 ÷ (4 ÷ 2)
6 ÷ 2 = 18 ÷ 2
3 ≠ 9
L.H.S ≠ R.H.S
Thus, the associative property does not apply to the division.
I hope all your confusion related to associative properties has been cleared. You can contact us at scientific Sarkar.com if you need help with any problem.