The associative property of addition is the property of numbers which states that the way in which three or more numbers are grouped does not change the sum of these numbers. This means that the sum of three or more numbers remains the same irrespective of the way in which they are grouped. Let us learn more about the associative property of addition along with some examples related to the associative law of addition in this article.
| 1. | What is the Associative Property of Addition? |
| 2. | Associative Property of Addition Formula |
| 3. | Associative Property of Addition and Multiplication |
| 4. | FAQs on Associative Property of Addition |
What is the Associative Property of Addition?
The associative property of addition is a rule which states that while adding three or more numbers, we can group them in any combination, and the sum that we get remains the same irrespective of the manner in which they are grouped. In this case, grouping refers to the placement of brackets. For example, the figure given below shows that the sum of the numbers does not change regardless of how the addends are grouped.
Associative Property of Addition Formula
The formula for the associative property of addition shows that grouping of numbers in a different way does not affect the sum. The brackets that group the numbers help to make the process of addition simpler. Observe the following formula for the associative property of addition.
Let us take an example to understand and prove the formula. Let us group 13 + 7 + 3 in three ways.
- Step 1: We can group the set of numbers as (13 + 7) + 3, 13 + (7 + 3), and (13 + 3) + 7.
- Step 2: Add the first set of numbers, that is, (13 + 7) + 3. This can be further solved as 20 + 3 = 23.
- Step 3: Add the second set, i.e., 13 + (7 + 3) = 13 + 10 = 23.
- Step 4: Now, solve the third set, i.e., (13 + 3) + 7 = 16 + 7 = 23.
- Step 5: The sum of all three expressions is 23. This shows that no matter how we group the numbers with the help of brackets, the sum remains the same.
Associative Property of Addition and Multiplication
The associative property is applicable to addition and multiplication, but it does not exist in subtraction and division. We know that the associative property of addition says that the grouping of numbers does not change the sum of a given set of numbers. This means, (7 + 4) + 2 = 7 + (4 + 2) = 13. Similarly, the associative property of multiplication says that the grouping of numbers does not change the product of the given set of numbers. This formula is expressed as (a × b) × c = a × (b × c). For example, (2 × 3) × 4 = 2 × (3 × 4) = 24.
Important Notes:
- The associative property is applicable only to 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.
☛ Related Topics
- Properties of Addition
- Commutative Property of Addition
- Zero Property of Multiplication
- Multiplicative Identity Property
- Distributive Property
- Commutative Property
- Additive Identity vs Multiplicative Identity
- Distributive Property
- Associative Property of Addition Worksheets
FAQs on Associative Property of Addition
What is the Associative Property of Addition?
The associative property of addition says that no matter how a set of three or more numbers are grouped together, the sum remains the same. The grouping of numbers is done with the help of brackets. The formula for this property is expressed as, a + (b + c) = (a + b) + c = (a + c) + b. For example, if we group the numbers 3 + 4 + 5 as, 3 + (4 + 5) or (3 + 4) + 5, the sum that we get from both the sets is 12.
What is an Example of the Associative Law of Addition?
The associative property of addition states that the grouping of numbers does not change their sum. For example, (75 + 81) + 34 = 156 + 34 = 190; and 75 + (81 + 34) = 75 + 115 = 190. The sum of both the sides is 190.
What is the Benefit of Using the Associative Law of Addition?
The benefit of the associative law of addition is that it helps to form smaller components and this makes the calculation of addition simpler. The grouping of numbers with the help of brackets eases the process of simplifying an expression.
How to Verify the Associative Property of Addition?
The associative law of addition can be easily verified by adding the given set of numbers. For example, let us group 6 + 7 + 8 in two ways.
- Step 1: We can group the given set of numbers as (6 + 7) + 8 and 6 + (7 + 8).
- Step 2: Now, let us add the first set of numbers, i.e., (6 + 7) + 8. This results in 13 + 8 = 21.
- Step 3: Now, let us add the second set, i.e., 6 + (7 + 8) = 6 + 15 = 21.
- Step 4: The sum of both the expressions is 21. This proves the associative property of addition which shows that no matter how we group the numbers with the help of brackets, the sum remains the same.
Does the Associative Property of Addition Always Involve 3 or more Numbers?
Yes, the associative property of addition always involves 3 or more numbers because the property rule states that changing the grouping of addends does not change the sum and in the case of only two numbers we cannot make groups.
What is the Formula for the Associative Property of Addition?
The formula for the associative property of addition states that the sum of three or more numbers remains the same no matter how the numbers are grouped. It is expressed as, a + (b + c) = (a + b) + c = (a + c) + b.
What is the Difference Between the Commutative and Associative Property of Addition?
The following points show the difference between the commutative and the associative property of addition:
- The commutative property of addition states that changing the order of the addends does not change the sum. For example, 4 + 6 = 6 + 4 = 10. The associative property of addition states that the grouping of numbers does not change the sum. For example, 8 + (2 + 3) = (8 + 2) + 3 = 13.
- The commutative property of addition can be applied to two numbers, but the associative property is applicable to 3 or more numbers.
- In the commutative property of addition, the order of the addends does not matter, while in the associative property of addition, the grouping of the addends does not matter.
How is the Associative Property of Addition Used in Everyday Life?
There are many places where we can apply the associative property of addition. For example, if we spend $3 on a cupcake, $6 on ice cream, and $2 on candy, we can add up the cost of the items in any order as, 3 + (6 + 2), or, (3 + 6) + 2. Both the expressions result in the same sum, that is, 11. This shows the associative property of addition which says that no matter how we group 3 or more numbers, the sum remains the same.
