Associative Property of Addition – Definition with Examples

Definition:

To “associate” means to connect or join with something.

According to the associative property of addition,the sum of three or more numbers remains the same regardless of how the numbers are grouped.

Here’s an example of how the sum does NOT change irrespective of how the addends are grouped.

As seen in the example above, grouping is defined with parentheses. Whether we group 5 and 3 or 3 and 4 within the parentheses, the final sum is 12. You can verify the final result by checking the colored blocks that stay the same in both the cases.

General Case:

For any three numbers a,b and c

a + (b + c) = (a + b) + c 

i.e. when you add, you can group the numbers in any combination. 

Let us take another example to understand and prove the formula. 

Let us group 14 + 7 + 5 in two ways.

• Step 1: We can group the set of numbers in two different ways as (14 + 7) + 5 or as  14 + (7 + 5).  
• Step 2: Add the first set of numbers, that is, (14 + 7) + 5. This can be further solved as 21 + 5 = 26.
• Step 3: Add the second set, i.e, 14 + (7 + 5) = 14 + 12 = 26.
• Step 4: The sum of both the expressions is 26. 

This shows that the sum remains the same irrespective of how we group the numbers with the help of brackets.

The minimum numbers we require for associative property of addition are 3. However, the associative property of addition holds true for more than three numbers too.

The associative property along with other properties in Mathematics are useful in manipulating equations and their solutions. 

Closely Related Concepts:

• The associative property holds for multiplication as well i.e. for any three numbers a, b and c, a $\times$ (b $\times$ c) = (a $\times$ b) $\times$ c

Let a = 2, b = 3, c = 4

a $\times$ (b $\times$ c) = 2 $\times$ (3 $\times$ 4) = 2 $\times$ 12 = 24

(a $\times$ b) $\times$ c = (2 $\times$ 3) $\times$ 4 = 6 $\times$ 4 = 24

Hence, a $\times$ (b $\times$ c) = (a $\times$ b) $\times$ c 

• The associative property does not hold for subtraction. Let us look at an example

Let a = 2, b = 3, c = 4

a- (b – c) = 2 – (3 – 4) = 3

(a – b) -c = (2 – 3) – 4 = -53

Hence, associative property does not hold for subtraction.

• The associative property does not hold for division. Let us look at an example

a $\div$ (b $\div$ c) = 2 $\div$ (3 $\div$ 4) = 2.67

(a $\div$ b) $\div$ c = (2 $\div$ 3) $\div$ 4 = 5.97 $\neq$ 2.67

Hence, associative property does not hold for division. 

Solved Examples

1. Is (5 + 10) + 4 the same as 5 + (10 + 4)?

Answer: Yes. Let’s solve and check:

(5 + 10) + 4 = 15 + 4 = 19

And, 5 + (10 + 4) = 5 + 14 = 19

If we group these three numbers differently, we get the same answers.

2. Fill in the missing numbers:

21 + (45 + 36) = (21 + 45) + _ = _

Answer: Using the associative property of addition,

21 + (45 + 36) = (21 + 45) + 36 = 102

3. Solve for x using the associative property formula: (2 + 3) + x = 2 + (3 + 6)

Answer: Given,  (2 + 3) + x = 2 + (3 + 6)

According to the associative property of addition, LHS = RHS,

Hence, 5 + x = 2 + 9

Or 5 + x = 11

Or x = 6

Practice Problems

Conclusion

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