Associative Property of Addition
The associative property of addition is the property of numbers that states the sum of three or more numbers will not change however the numbers are grouped while adding. Here 'grouped' means 'the way we used parenthesis'. In other words, if we add three or more numbers we arrive at the same answer irrespective of how the numbers are grouped or parenthesized in any order. The associative property is one of the 3 important properties of addition. All the numbers in the number system are bound to follow the associative property of addition.
What is the Associative Property of Addition?
The associative property of addition is a law that states that when we add, we can group the numbers in any order or combination. The word Associative means to connect with something or in other words, a group of quantities (numbers) connected by operators gives the same result. The addition follows the associative property and it says that regardless of how numbers are parenthesized the final sum of three or more numbers will be the same. The grouping has described the placement of parentheses to group numbers. The property is only applicable in the association of three or more numbers. Natural numbers, whole numbers, decimal numbers, fractions can be used as numbers. Once the grouping is done, the numbers within the parentheses are added first and evaluated.
For example, look at the image given below and observe how the sum does NOT change regardless of how addends are grouped.
Associative Property of Addition Formula
The Associative Property is the rule that includes grouping of numbers. For associative property of addition, the rule is "a + (b + c) = (a + b) + c", for example, 2 + (3 + 4) = (2 + 3) + 4 ⇒2+7 = 5+4. Both are evaluated as 9.
Look at the image given below which shows the formula of associative property of addition where a, b, c are the numbers.
Let’s take an example to prove the formula of the associative property of addition.
The set of equation is (13 + 7) + 3 = 13 + (7 + 3)
Following are the steps to prove the formula.
 Step 1: Write the lefthand side of equation i.e. (13 + 7) + 3 = ?
 Step 2: Add the numbers in parentheses to obtain one number i.e. 20 + 3
 Step 3: Add this single number to the leftout numbers and obtain the answer for it.i.e.20 + 3 = 23
 Step 4: Now, write the righthand side of the equation i.e. 13 + (7 + 3) = ?
 Step 5: Add the numbers in parentheses i.e. 13 + 10
 Step 6: Add this single number to the leftout numbers and obtain the answer for it i.e. 3 + 10 = 23
Hence, we can observe that the value obtained from the lefthand side equation and the righthand side equation is the same.
Important Notes
 Associative properties only exist in addition and multiplication.
 Associative properties are in line with the ability to associate or group numbers, which is not possible in the case of subtraction and division.
 The associative property is among the list of properties in mathematics that are helpful in the manipulation of mathematical equations and their solutions.
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Associative Property of Addition Examples

Example 1: Can you help Paul determine whether the given equation follows the associative property of addition?
( 25 + 2 ) + 8 = 25 + ( 2 + 8)
Solution: Following are the steps Paul would follow to determine whether the given equation follows the associative property of addition or not:Step 1: Write the lefthand side of the equation i.e. (25 + 2) + 8 = ?
Step 2: Add the numbers in parentheses to obtain one number i.e. 25 + 2 = 27
Step 3: Add this single number to the leftout numbers and obtain the answer for it i.e. 27 + 8 = 35
Step 4: Now, write the righthand side of the equation i.e.25 + (2 + 8) = ?
Step 5: Add the number in parentheses i.e. 25 + 10
Step 6: Add this single number to the leftout numbers and obtain the answer i.e. 25 + 10 = 35
Thus, the value obtained from the lefthand side equation and the righthand side equation is the same. So the equation follows the associative property.

Example 2: Fill the missing number and then find the sum:
7 + (10 + 7) = (7 + 10) + ___ = ___
Solution: According to the associative property of addition formula "a + (b + c) = (a + b) + c".
Therefore, 7 will come in missing place i.e. 7 + (10 + 7) = (7 + 10) + 7 and the sum is 24. 
Example 3: Choose the correct option for the number on the blank.
6 + (7 + 10) = (6 + ___) + 10
a)10
b) 7
c) 6
Solution: Option b, "7" is correct as the sum of both sides are equal when we place 7 in the missing place. The sum of both the left and the right sides is 23.
FAQs on Associative Property of Addition
Given an Example of the Associative Property of Addition.
The associative property of addition states that the grouping of numbers doesn't change the sum.
For example, (75 + 81 ) + 34 = 156 + 34 = 190 and 75 + ( 81 + 34) = 75 + 115 = 190. The sum of both sides is same i.e. 190.
Is The Associative Property of Addition Always Involves 3 or More Numbers?
Yes, the associative property of addition always involves 3 or more numbers because the property rule says changing the grouping of addends does not change the sum and in the case of only two numbers we cannot make groups.
How The Associative Property of Addition is useful?
The associative property of addition is helpful while adding numbers. By grouping the numbers we can create smaller parts to solve the equations. We can create number bonds in our mind that makes calculations easier and faster.
What is the Formula of Associative Property of Addition?
The word "associative" comes from "associate" which means to connect or to make a group. The Associative Property of addition formula states that the sum of three or more numbers will be the same no matter how numbers are grouped. Therefore the formula is "a + (b + c) = (a + b) + c"; where a, b, c are the numbers, for example, 8 + (2 + 4) = (8 + 2) + 4.
What is the Difference Between Commutative and Associative Properties of Addition?
Commutative property of addition  Associative property of addition 
The commutative property of addition states that you can move the numbers around and the final sum will remain the same  The associative property of addition states that you can group the addends in different ways and the result of the sum of equations will be the same. 
The commutative property of addition is applicable for two numbers added in any order.  Associative property involves more than 2 numbers grouped in any order. 
In the commutative property of addition, the order of the addends doesn't matter.  In the associative property of addition, grouping the addends doesn't matter. 
How is the Associative Property of Addition Used in Everyday Life?
There are many places we can apply the associative property of addition. When we go shopping, purchase 3 items and move to the billing counter we can add up the cost of the items as:
(cost of item 1 + cost of item 2) + cost of item 3 (or) cost of item 1 + (cost of item 2 + cost of item 3). The total cost remains the same either way.