Basic operations on numbers include addition, subtraction, multiplication, and division. Some attributes are introduced to allow for faster algebraic computations by grouping numbers to reduce calculation time. The three main properties of the algebra of numbers are the associative property, distributive property, and commutative property. For students to gain a better understanding of the concept, we shall explore the Associative Property Formula here.

The associative property is the rule that relates to grouping, and the term “associative” derives from “associate” or “group.” We can add/multiply integers in an equation regardless of how they are grouped. Only addition and multiplication operations use the associative formula. Continue reading to know more.

Associative Property Definition

The associative property states that we can group integers in any order or combination when we add (or multiply). The term “associative” refers to a set of values (numbers) connected by operators that give the same result. 

For example: \((4+3)+7=14=4+(3+7)\)
\((4×3)×7=84=4×(3×7)\)

Associative Property Over Addition

While performing addition, the associative property of addition states that we can group the numbers in any order or combination to get the same result. The placement of parenthesis to group numbers has been described in the grouping. The property is only applicable when three or more integers are combined. Numbers can be made up of natural numbers, whole numbers, decimal numbers, and fractions.

After the grouping, the numbers within the parenthesis are added and evaluated first. Associative property over addition is also expanded to matrix algebra by rearranging the matrices.

Associative Property Over Addition Formula

The rule that involves number grouping is known as the associative property. The rule for the associative property of addition is
\((A+B)+C=A+(B+C)\)

For example: \((3+1)+2=3+(1+2)\)
\(⇒4+2=3+3\)
\(⇒6=6\)

Take a look at the image below to see how the total remains constant regardless of how the addends are arranged.

Examples of Associative Property Over Addition

(i) For natural numbers,

Consider \(A=1, B=5, C=13\)
Then \((A+B)+C=(1+5)+13=6+13=19\)
And \(A+(B+C)=1+(5+13)=1+18=19\)

Hence, \((A+B)+C=A+(B+C)\)

(ii) For whole numbers,

Consider \(A=0, B=5, C=3\)
Then \((A+B)+C=(0+5)+3=5+3=8\)
And \(A+(B+C)=0+(5+3)=0+8=8\)

Hence, \((A+B)+C=A+(B+C)\)

(iii) For whole integers,

Consider \(A=-2, B=5, C=-3\)
Then \((A+B)+C=(-2+5)+(-3)=3-3=0\)
And \(A+(B+C)=-2+(5+(-3))=-2+2=0\)

Hence, \((A+B)+C=A+(B+C)\)

(iv) For rational numbers,

Consider \(A = \frac{1}{2},B = \frac{2}{5},C = \frac{3}{4}\)
Then \((A + B) + C = \left( {\frac{1}{2} + \frac{2}{5}} \right) + \frac{3}{4} = \frac{9}{{10}} + \frac{3}{4} = \frac{{33}}{{20}}\)
And \(A + (B + C) = \frac{1}{2} + \left( {\frac{2}{5} + \frac{3}{4}} \right) = \frac{1}{2} + \frac{{23}}{{20}} = \frac{{33}}{{20}}\)

Hence, \((A + B) + C = A + (B + C)\)

Therefore, we can say the natural numbers, whole numbers, integers, and rational numbers are associative over addition.

Associative Property Over Multiplication

While performing multiplication, the associative property of multiplication states that we can group the numbers in any order or combination. The placement of parenthesis to group numbers has been described in the grouping. The property is only applicable when three or more integers are combined.

Numbers can be made up of natural numbers, whole numbers, decimal numbers, and fractions. After the grouping, the numbers within the parenthesis are multiplied and evaluated first.

Associative Property Over Multiplication Formula

The rule that involves number grouping is known as the Associative Property. The rule for the associative property of multiplication is
\((A×B)×C=A×(B×C)\)

For example: \((3×1)×2=3×(1×2)\)
\(⇒3×2=3×3\)
\(⇒6=6\)

Take a look at the image below to see how the total remains constant regardless of how the addends are arranged.

Examples of Associative Property Over Multiplication

(i) For natural numbers,

Consider \(A=1, B=5, C=13\)
Then \((A×B)×C=(1×5)×13=5×13=65\)
And \(A×(B×C)=(15×13)=1×65=65\)

Hence, \((A×B)×C=A×(B×C)\)

(ii) For whole numbers,

Consider \(A=0, B=5, C=3\)
Then \((A×B)×C=(0×5)×3=0×3=0\)
And \(A×(B×C)=0×(5×3)=0×15=15\)

Hence, \((A×B)×C=A×(B×C)\)

(iii) For integers,

Consider \(A=-2, B=5, C=-3\)
Then \((A×B)×C=(-2×5)×(-3)=(-10)×(-3)=30\)
And \(A×(B×C)=-2×(5×(-3))=-2×(-15)=30\)

Hence, \((A×B)×C=A×(B×C)\)

(iv) For rational numbers,

Consider \(A = \frac{1}{2},B = \frac{2}{5},C = \frac{3}{4}\)
Then \((A + B) + C = \left( {\frac{1}{2} + \frac{2}{5}} \right) + \frac{3}{4} = \frac{2}{{10}} + \frac{3}{4} = \frac{3}{{20}}\)
And \(A + (B + C) = \frac{1}{2} + \left( {\frac{2}{5} + \frac{3}{4}} \right) = \frac{1}{2} + \frac{3}{{10}} = \frac{3}{{20}}\)

Hence, \((A + B) + C = A + (B + C)\)

Therefore, we can say the natural numbers, whole numbers, integers, and rational numbers are associative over multiplication.

Associative Property Formula Over Subtraction

Subtraction, unlike addition, does not have the associative property.

Let us just take a look at an example.

\((10-5)-3=5-3=2\)
\(10-(5-3)=10-2=8\)

We get \(5\) by subtracting the first two integers, \(10\) minus \(5.\) When we deduct \(3\) from that, we get \(2.\) If we first subtract the last two numbers, then \(5\) minus \(3\) equals \(2.\) We get \(8\) when we subtract \(2\) from \(10.\) So, both the values are not equal.

In subtraction, changing the way the numbers are associated alters the answer. As a general result, subtraction fails the associative property.

Hence, for any three numbers \(A, B,\) and \(C,\) generally associative property for division is given as, \((A-B)-C≠A-(B-C).\)

Associative Property Formula Over Division

Division, unlike multiplication, does not have the associative property.

Let us just take a look at another example.

\((10 \div 5) \div 4 = 2 \div 4 = \frac{1}{2}\)
\(10 \div (5 \div 4) = 10 \div \frac{5}{4} = 8\)

We get \(2\) by dividing the first two integers, \(10\) divided by \(5.\) When we divide the result by \(4,\) we get \(\frac{1}{2}\) If we first divide the last two numbers, \(5\) divided by \(4\) equals \(\frac{5}{4}\) We get \(8\) when we divide \(10\) by \(\frac{5}{4}.\)

In division, changing the way the numbers are associated alters the answer. As a result, division fails the associative property.

Hence, for any three numbers \(A, B,\) and \(C\) associative property for division is given as, \(\left({A \div B} \right) \div C \ne A \div \left({B \div C} \right)\).

Commutative and Associative Property Formula

It’s essential to know the difference between the commutative and associate properties. The commutative property refers to the order in which certain mathematical operations are carried out. The equation \(a+b=b+a\) follows commutative property.

On the other hand, the associative property is concerned with the grouping of items in operation. The equation \((a+b)+c=a+(b+c)\) shows this.

Solved Examples – Associative Property Formula

Q.1. Prove that the numbers below follow the associative property of addition:
\(2,6\) and \(9\)
Ans: The rule for the associative property of addition is \((A+B)+C=A+(B+C)\)
Let \(A=2,B=6\) and \(C=9\)
Then \((A+B)+C=(2+6)+9=8+9=17\)
And \(A+(B+C)=2+(6+9)=2+15=17\)
So, \((2+6)+9=2+(6+9)\)
Hence, the given numbers obey the associative property of addition.

Q.2. Show that the numbers below follow the associative property of multiplication:
\(21,62\) and \(19\)
Ans: The rule for the associative property of addition is \((A×B)×C=A×(B×C)\)
Let \(A=21,B=62\) and \(C=19\)
Then \((A×B)×C=(21×62)×19=1302×19=24738\)
And \(A×(B×C)=21×(62×19)=21×1178=24738\)
Hence, \((A×B)×C=A×(B×C)\)
So, \((21×62)×19=21×(62×19)\)
Hence, the given numbers obey the associative property of multiplication.

Q.3. Find the value of \(p.\)
\(p + [15 + ( – 7)] = [4 + 15] + ( – 7)\)
Ans: Since addition satisfies the associative property formula.
So, \(p + [15 + ( – 7)] = [4 + 15] + ( – 7)\)
On comparing the Left-Hand Side and Right-Hand Side, we get
\(p=4\)
Hence, the value of \(p\) is equal to \(4.\)

Q.4. Show that \(\left( {\frac{1}{2}} \right) + \left[ {\left( {\frac{3}{4}} \right) + \left( {\frac{5}{6}} \right)} \right] = \left[ {\left( {\frac{1}{2}} \right) + \left( {\frac{3}{4}} \right)} \right] + \left( {\frac{5}{6}} \right)\) and \(\left({\frac{1}{2}} \right) \times \left[{\left({\frac{3}{4}} \right) \times \left({\frac{5}{6}} \right)} \right] = \left[{\left({\frac{1}{2}} \right) \times \left({\frac{3}{4}} \right)} \right] \times \left({\frac{5}{6}} \right)\)
Ans: The rule for the associative property of addition is
\((A + B) + C = A + (B + C)\)
Consider \(A = \frac{1}{2},B = \frac{3}{4},C = \frac{5}{6}\)
Then \((A + B) + C = \left( {\frac{1}{2} + \frac{3}{4}} \right) + \frac{5}{6} = \frac{5}{4} + \frac{5}{6} = \frac{{25}}{{12}}\)
And \(A + (B + C) = \frac{1}{2} + \left( {\frac{3}{4} + \frac{5}{6}} \right) = \frac{1}{2} + \frac{{19}}{{12}} = \frac{{25}}{{12}}\)
Hence, \((A + B) + C = A + (B + C)\)
The rule for the associative property of multiplication is
\((A \times B) \times C = A \times (B \times C)\)
Then \((A \times B) \times C = \left( {\frac{1}{2} \times \frac{3}{4}} \right) \times \frac{5}{6} = \frac{3}{8} \times \frac{5}{6} = \frac{5}{{16}}\)
And \(A \times (B \times C) = \frac{1}{2} \times \left( {\frac{3}{4} \times \frac{5}{6}} \right) = \frac{1}{2} \times \frac{5}{8} = \frac{5}{{16}}\)
Hence, \((A \times B) \times C = A \times (B \times C)\)
Therefore, it is proved that \(\left( {\frac{1}{2}} \right) + \left[ {\left( {\frac{3}{4}} \right) + \left( {\frac{5}{6}} \right)} \right] = \left[ {\left( {\frac{1}{2}} \right) + \left( {\frac{3}{4}} \right)} \right] + \left( {\frac{5}{6}} \right)\) and \(\left( {\frac{1}{2}} \right) \times \left[ {\left( {\frac{3}{4}} \right) \times \left( {\frac{5}{6}} \right)} \right] = \left[ {\left( {\frac{1}{2}} \right) \times \left( {\frac{3}{4}} \right)} \right] \times \left( {\frac{5}{6}} \right).\)

Q.5. Use the associative property of multiplication to make the problem \(32×0.25\) easy to solve mentally.
Ans: The associative property of multiplication states that we can group the numbers in any order or combination.
The rule for the associative property of multiplication is
\((A×B)×C=A×(B×C)\)
\(32×0.25=(8×4)×0.25\)
\(=8×4×0.25\) (By using the associative property)
\(=8×1\)
\(=8\)

Q.6. Fill the missing number and then find the sum:
\(7 + (10 + 7) = (7 + 10) + \_\_\_\_ = \_\_\_\_\)
Ans: According to the associative property of the addition formula
\((A + B) + C = A + (B + C)\)
\(7 + (10 + 7) = (7 + 10) + \_\_\_\_\)
On comparing the LHS and RHS of the above equation, we see that
7 will come in missing place, i.e. \(7 + (10 + 7) = (7 + 10) + 7\)
And the sum of the number is \(7 + (10 + 7) = (7 + 10) + 7 = 24.\)
Hence, the required sum is \(24.\)

Summary

The definition of associative property is given in this article. We learned and validated the associative property formula for addition, subtraction, multiplication, and division, along with examples of natural numbers, whole numbers, integers, and rational numbers.

We also noted that the associative property does not always apply to subtraction and division. Finally, we learned the difference between the commutative and associative property formulas.

Frequently Asked Questions

We have provided some frequently asked questions about Associative Property Formula here:

Q.1. What is associative property applicable to?
Ans: When three or more numbers are added (or multiplied), this property states that the sum (or product) is the same regardless of how the addends are grouped (or the multiplicands). Subtraction and division are not compatible with the associative attribute. When adding or multiplying many integers, the associative property proves useful.

Q.2. Give an example of the associative property of addition?
Ans: Consider \(A=-2, B=5, C=-3\)
Then \((A+B)+C=(-2+5)+(-3)=3-3=0\)
And \(A+(B+C)=-2+(5+(-3))=-2+2=0\)
Hence, \((A+B)+C=A+(B+C)\)

Q.3. How is the associative property of addition used in everyday life?
Ans: The associative property of addition can be used in a lot of circumstances. When we go shopping and purchase three items, we may sum up the cost of the items as follows:
(Price of item\(1+\)Price of item \(2\))\(+\)Price of item \(3\) (or) Price of item\(1+\)(Price of item \(2+\)Price of item \(3\)).
In any case, the overall cost remains the same.

Q.4. What is the associative property formula?
Ans: The associative property states that we can group integers in any order or combination when we add (or multiply). The term “associative” refers to a set of values (numbers) connected by operators that give the same result.
Let \(A,B\) and \(C\) be three numbers,
The rule for the associative property of addition is
\((A+B)+C=A+(B+C)\)
The rule for the associative property of multiplication is
\((A×B)×C=A×(B×C)\)
For example: \((4+3)+7=14=4+(3+7)
(4×3)×7=84=4×(3×7)\)

Q.5. What is the difference between associative property and distributive property?
Ans: The associative property states that we can group integers in any order or combination when we add (or multiply). The term “associative” refers to a set of values (numbers) connected by operators that give the same result. According to the distributive property, multiplying the sum of two or more addends by a number will multiply each addend individually by the number and then add the products together. The distributive property holds true for multiplication over addition and subtraction.

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