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Associative Property – Meaning & Examples

associative property

 What comes to mind when you hear the word associative? Associate means to connect or to group. In the same way, the associative property allows us to group terms that are joined by addition or multiplication in various ways. Parentheses are used to group the terms, and they establish the order of operations. Work inside the parentheses is always done first.

What is Associative Property?

The associative property states that the sum or the product of any three or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.

Mathematically it can be stated as

If $A$, $B$, and $C$ are any three numbers then, 

  • For addition: $\left(A + B \right) + C = A + \left(B + C \right)$
  • For multiplication: $\left(A \times B \right) \times C = A \times \left(B \times C \right)$

Note 

  • The associative law is applicable to addition and multiplication 
  • The associative law is not applicable to subtraction and division
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Associative Property of Addition

The associative property of addition states that $\left(A + B \right) + C = A + \left(B + C \right)$.

Let’s understand the associative property of addition by this example.

Consider three numbers $A = 5$, $B = 7$ and $C = 3$.

Left Hand Side of the statement becomes $\left(5 + 7 \right) + 3 = 12 + 3 = 15$.

And, the Right Hand Side of the statement is $5 + \left(7 + 3 \right) = 5 + 10 = 15$.

The result of both is the same ($=15$).

Consider one more example by taking $A = 18$, $B = 23$ and $C = 32$.

Left Hand Side of the statement becomes $\left(18 + 23 \right) + 32 = 41 + 32 = 73$.

And, the Right Hand Side of the statement is $18 + \left(23 + 32 \right) = 18 + 55 = 73$.

In this case, also the result of both is the same ($=73$).

So, based on the associative property of addition we can say that the sum of three or more numbers always remains the same whatever way we group the numbers to add them.

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Associative Property of Multiplication

The associative property of addition states that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$.

Let’s understand the associative property of multiplication by this example.

Consider three numbers $A = 3$, $B = 2$ and $C = 5$.

Left Hand Side of the statement becomes $\left(3 \times 2 \right) \times 5 = 6 \times 5 = 30$.

And, the Right Hand Side of the statement is $3 \times \left(2 \times 5 \right) = 3 \times 10 = 30$.

The result of both is the same ($=30$).

Consider one more example by taking $A = 12$, $B = 15$ and $C = 40$.

Left Hand Side of the statement becomes $\left(12 \times 15 \right) \times 40 = 180 \times 40 = 7200$.

And, the Right Hand Side of the statement is $12 \times \left(15 \times 40 \right) = 12 \times 600 = 7200$.

In this case, also the result of both is the same ($=7200$).

So, based on the associative property of multiply we can say that the product of three or more numbers always remains the same whatever way we group the numbers to multiply them.

Is Associative Property Applicable to Numbers of All Categories?

The associative property in both the two forms – associative property of addition and associative property of multiplication works well with any real number

Note: A set of real numbers $R$ is a superset of sets of natural numbers, a set of whole numbers, a set of integers, a set of rational numbers, and a set of irrational numbers.

Natural Numbers and Whole Numbers

Consider any three whole numbers $45$, $67$ and $13$.

Addition

We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$

$\left(45 + 67 \right) + 13 = 45 + \left(67 + 13 \right)$

$=>112 + 13 = 45 + 80 => 125 = 125$.

Also, $\left(0 + 12 \right) + 19 = 0 + \left(12 + 19 \right)$

$=>12 + 19 = 0 + 31 => 31 = 31$.

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Multiplication

We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$

$\left(45 \times 67 \right) \times 13 = 45 \times \left(67 \times 13 \right)$

$=>3015 \times 13 = 45 \times 871 => 39195 = 39195$.

Also, $\left(0 \times 12 \right) \times 19 = 0 \times \left(12 \times 19 \right)$

$=>0 \times 19 = 0 \times 228 => 0 = 0$.

Integers (Signed Numbers)

Consider any three integers $-11$, $B=+16$ and $C=-14$.

Addition

We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$

$\left(-11 + 16 \right) + \left(-14 \right) = -11 + \left(16 + \left(-14 \right) \right)$

$=> 5 + \left(-14 \right) = -11 + \left(16 – 14 \right)$

$=> 5 – 14 = -11 + 2$ $=>-9 = -9$

Multiplication

We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$

$\left(-11 \times 16 \right) \times \left(-14 \right) = -11 \times \left(16 \times \left(-14 \right) \right)$

$=> -176 \times \left(-14 \right) = -11 \times \left(-224 \right)$

$=> 2464 =2464$

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Decimal Numbers

Consider any three decimal numbers $A = 2.5$, $B = 5.6$ and $C = 0.8$.

Addition

We want to verify that We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$

$\left(2.5 + 5.6 \right) + 0.8 = 2.5 + \left(5.6 + 0.8 \right)$

$=> 8.1 + 0.8 = 2.5 + 6.4 => 8.9 = 8.9$

Multiplication

We want to verify that We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$

$\left(2.5 \times 5.6 \right) \times 0.8 = 2.5 \times \left(5.6 \times 0.8 \right)$

$=>14 \times 0.8 = 2.5 \times 4.48 => 11.2 = 11.2$

Fractions

Consider any three fractions $A = \frac {1}{2}$, $B = \frac {2}{3}$ and $C = \frac {3}{4}$.

Addition

We want to verify that We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$

$\left(\frac {1}{2} + \frac {2}{3} \right) + \frac {3}{4} =  \frac {1}{2} + \left(\frac {2}{3} + \frac {3}{4} \right)$

$=> \frac{23}{12} = \frac{23}{12}$

Multiplication

We want to verify that We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$

$\left(\frac {1}{2} \times \frac {2}{3} \right) \times \frac {3}{4} =  \frac {1}{2} \times \left(\frac {2}{3} \times \frac {3}{4} \right)$

$=> \frac {1}{3} \times \frac {3}{4} =  \frac {1}{2} \times \frac {1}{2}$

$=> \frac {1}{4}=  \frac {1}{4}$

Irrational Numbers

Consider any three fractions $A = 2\sqrt{3}$, $B = 3\sqrt{2}$ and $C = 5\sqrt{2}$.

Addition

We want to verify that $\left(A + B \right) + C = A + \left(B + C \right)$

$\left(2\sqrt{3} + 3\sqrt{2} \right) + 5\sqrt{2} = 2\sqrt{3} + \left(3\sqrt{2} + 5\sqrt{2} \right)$

$=>2\sqrt{3} + 3\sqrt{2} + 5\sqrt{2} = 2\sqrt{3} + 8\sqrt{2}$

$=>2\sqrt{3} + 8\sqrt{2} = 2\sqrt{3} + 8\sqrt{2}$

Multiplication

We want to verify that $\left(A \times B \right) \times C = A \times \left(B \times C \right)$

$\left(2\sqrt{3} \times 3\sqrt{2} \right) \times 5\sqrt{2} = 2\sqrt{3} \times \left(3\sqrt{2} \times 5\sqrt{2} \right)$

$=>6\sqrt{6} \times 5\sqrt{2} = 2\sqrt{3} \times 30$ $=>30\sqrt{12} = 60\sqrt{3}$

$=>30\sqrt{4 \times 3} = 60\sqrt{3}$ $=>30\times 2 \sqrt{3} = 60\sqrt{3}$ $=>60\sqrt{3} = 60\sqrt{3}$

Does Associative Property Hold For Subtraction and Division?

Let’s verify whether associative law holds for subtraction and division also, i.e.,

  • $\left(A – B \right) – C = A – \left(B – C \right)$
  • $\left(A \div B \right) \div C = A \div \left(B \div C \right)$

Again consider any three numbers $A = 5$, $B = -2$ and $C = 0.5$

Subtraction

LHS = $\left(5 – \left(-2 \right) \right) – 0.5 = 5 + 2 – 0.5 = 6.5$

RHS = $5 – \left(-2 – 0.5 \right) = 5 – \left(-2.5 \right) = 5 + 2.5 = 7.5$

Since, LHS $\ne$ RHS, therefore, the associative property does not hold for subtraction.

Division

LHS = $\left(5 \div \left(-2 \right) \right) \div 0.5 = -\frac {5}{2} \div 0.5 = -\frac {5}{2} \times \frac{10}{5} = -5$

RHS = $5 \div \left(-2\div 0.5 \right) = -5 \div \frac {2}{0.5} = -5 \times \frac {0.5}{2} = -5 \times \frac {5}{20} = -\frac {5}{4}$

 Since, LHS $\ne$ RHS, therefore, the associative property does not hold for division.

Conclusion

The associative property states that the result remains the same irrespective of the order in which the arithmetic operation is performed. It is applicable to addition and multiplication only and not for subtraction and division.

Practice Problems

Verify Associative Property of Addition and Multiplication for the following set of numbers

  • $A = 5$, $B = -7$, $C = 2$
  • $A = \frac {2}{5}$, $B = -7$, $C = \frac{1}{3}$
  • $A = 0.95$, $B = 1.5$, $C = -8.9$

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FAQs

What is the associative property? Give an example.

The associative property states that the sum or the product of any three or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.

For example, in the case of addition $12 + \left(14 + 18 \right) = \left(12 + 14 \right) + 18$. (The sum in both the cases is $44$ and in the case of multiplication $7 \times \left(5 \times 4 \right) = \left(7 \times 5\right) \times 4 = 140$.

What is the associative property formula?

There are two variations of associative property in math:
Associative property of addition: It states that for any three numbers $A$, $B$, and $C$, $A + \left(B + C \right) = \left(A + B \right) + C$.
Associative property of multiplication: It states that for any three numbers $A$, $B$, and $C$, $A \times \left(B \times C \right) = \left(A \times B \right) \times C$.

What is the difference between associative and commutative property?

The associative property deals with the grouping of numbers whereas the commutative property deals with the ordering of numbers while performing addition or subtraction.

The associative property states that in the case of addition and multiplication the grouping of the numbers does not matter i.e., for any three numbers $A$, $B$, and $C$, 
Addition: $A + \left(B + C \right) = \left(A + B \right) + C$ 
Multiplication: $A \times \left(B \times C \right) = \left(A \times B \right) \times C$ 

The associative property states that in the case of addition and multiplication the ordering of the numbers does not matter i.e., for any two numbers $A$, and $B$, 
Addition: $A + B = B + A$.
Multiplication: $A \times B = B \times A$.

To which operations Associative property is applicable?

The associative property is applicable to the operations of addition and multiplication. It does not hold for subtraction and division.

Is the associative property applicable to division and subtraction?

No, the associative property does not apply to subtraction and division.

Is multiplication always associative?

The operation multiplication always follows associate property for all the categories of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Even the operation multiplication always follows associate property for all the categories of numbers such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

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