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Associative property is a mathematical concept that is used in calculating arithmetic operations. Associative property states that three or more than 3 numbers that are being calculated can be grouped in different ways using the same binary operands and the resulting value will remain the same. The associative property is applicable only for addition and multiplication. There are two forms of associative formulae-addition and multiplication. This property of arithmetic operations is important and applicable in several mathematical calculations and is, therefore, a concept of high significance. Here, we will discuss more about the associative property along with some important questions.
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Key Takeaways: Associative Property, associative property of addition, associative property of multiplication, associative property formula, examples.
Associative Property Formula
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The associative property is an important mathematical function. Mathematics, or more specifically arithmetics has several operations including addition, subtraction, multiplication, division, and others. These functions or operations are performed using certain laws that govern the fundamental concepts of mathematical calculations. The associative property is one such fundamental rule established to be performed on operations such as addition and multiplication. It states that if three or more values are being added or multiplied the grouping of values does not affect the sum or product of the operation.
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Associative Property of Addition Formula
The associative property of addition states if three or more values are added then if two numbers are added first using brackets and then the third one is added, the sum is equal to that of the operation if the same three values are added but in a different order.
The associative property of the addition formula is written as -
a + (b + c) = (a + b) + c
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Associative Property of Multiplication Formula
The associative property of multiplication states if three or more values are multiplied then the two numbers that are multiplied first using brackets and then the third one is multiplied to the product of the first two, the product obtained is equal to that of the operation if the same three values are multiplied but in a different order.
The associative property of the multiplication formula is written as -
a x (b x c) = (a x b) x c
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Associative Property Uses
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The associative property can be useful in a lot of mathematical calculations.
- Associative properties can be used in manipulating mathematical equations to acquire the required answer.
- The associative property of integers proves useful in grouping addition or multiplication operations and leads to less confusion. However, in the case of subtraction and division, the associative property is not applicable.
- The associative property is only applicable for addition and multiplication operands.
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Things to Remember
- The associative property states that three or more than three numbers that are being calculated can be grouped or associated in different ways operating on the same binary operands and the resulting value will remain the same.
- The associative property is only applicable for addition and multiplication operands.
- In case of subtraction and division, associative property is not applicable.
- The associative property of multiplication formula is - a x (b x c) = (a x b) x c
- The associative property of addition formula is written as - a + (b + c) = (a + b) + c
- The associative property is a fundamental rule established to be performed on operations such as addition and multiplication to simplify their use.
- The associative property of addition states if three or more values are added then if two numbers are added first using brackets and then the third one is added, the sum is equal to that of the operation if the same three values are added but in a different order.
- The associative property of multiplication states if three or more values are multiplied then the two numbers that are multiplied first using brackets and then the third one is multiplied to the product of the first two, the product obtained is equal to that of the operation if the same three values are multiplied but in a different order.
Sample Questions
Ques. How is the Associative Property Different From the Commutative Property? [4 marks]
Ans. The associative property is a formula that says that the sum or the product of three or more than three numbers does not change if different numbers are grouped and calculated from the equation. The associative property is applicable to both addition and multiplication.
The associative property of addition is expressed as,
(a + b) + c = a + (b + c) and
the associative property of addition is expressed as,
(a x b) x c = a x (b x c).
The commutative property states that the order in which the values are calculated using binary operands does not change the end result of the given arithmetic equation. The commutative property is also applicable to only addition and multiplication. The commutative property is expressed as,
a x b = b × a
and
a + b = b + a.
Ques. Give an example of the Associative Property of Multiplication. [3 marks]
Ans. The associative property of multiplication can be explained with the help of an example.
Let us consider any three numbers
4 x 6 x 10
Now we group the numbers and then solve the operation,
(4 x 6) x 10
= 24 x 10
= 240.
Now let us group two separate values,
4 x (6 x 10)
= 4 x 60
= 240.
Thus, grouping the values in two separate ways still produce the same end result which proves the associative property of multiplication.
Ques. Which two operations satisfy the condition of the Associative Property?[1 mark]
Ans. The two arithmetic operations that satisfy the conditions of the associative property are addition and multiplication.
Ques. Give an example of the associative property of addition. [3 marks]
Ans. The associative property of addition can be explained with the help of an example.
Let us consider any three numbers
3 + 5 + 10
Now we group the numbers and then solve the operation,
(3 + 5) + 10
= 8 + 10
= 18.
Now let us group two separate values,
3 + (5 + 10)
= 3 + 15
= 18.
Thus, on grouping the values in two separate ways carrying out the operation still produces the same end result which proves the associative property of addition.
Ques. [i] Select the equations which stand true for the associative property.
(a) (a - b - c) - d = a - (b - c - d)
(b) (a + b + c) + d = a + (b + c + d)
(c) (a ÷ b) ÷ c = a ÷ (b ÷ c)
(d) (a x b) x c = a x (b x c)
[ii] The associative property of multiplication states that the product of three or more numbers remains the same irrespective of the way the numbers are grouped.
True
False [2 marks]
Ans. (i) (a + b + c) + d = a + (b + c + d)
(ii) True
Ques. If 2 × (3 × 5) = 30, find the product of (2 × 3) × 5 using the associative property. [2 marks]
Ans. The associative property formula is (a x b) x c = a x (b x c)
Given,
2 × (3 × 5) = 30
Using the associative property formula, we can also find the value of (2 × 3) × 5.
(2 × 3) × 5 = 30
= 6 × 5
= 30
Therefore, 2 × (3 × 5) = (2 × 3) × 5 = 30.
Ques. If 3 x (6 x 4) = 72, then find the product of (3 x 6) x 4 using the associative property. [1 mark]
Ans. Using the associative property of multiplication formula,
(3 x 6) x 4
= 3 x (6 x 4)
= 3 x 24
= 72
Ques. Find the value of x using the associative property formula in this equation: 2 + (x + 9) = (2 + 5) + 9. [1 mark]
Ans. By using the associative property of addition formula,
(2 + 5) + 9 = 2 + (x + 9) = (2 + x) + 9.
So, the value of x is 5.
Ques. (i) Solve 12 + 15 + 7 using Associative Property of Addition.
(ii) Solve 3 x (2 x 6) using Associative Property of Multiplication. [3 marks]
Ans. (i) The associative property of addition is (12 + 15) + 7
= (12 + 15) + 7
= 27 + 7
= 34
Or
12 + (15+ 7)
= 12 + 22
= 34
(ii) 3 x (2 x 6)
= (3 x 2) x 6
= 6 x 6
= 36
Ques. Complete the equations to solve 3 x 5 x 2 in two different ways. [2 marks]
Ans. This equation can be solved in two different ways using the associative formula-
Way I: 3 x 5 x 2 = 3 x (5 x 2)
= 3 x 10
= 30
Way II: 3 x 5 x 2 = (3 x 5) x 2
= 15 x 2
= 30
Both the values are equal thus, 3 x 5 x 2 = 3 x (5 x 2) = (3 x 5) x 2, which is the associative property.
Ques. What is the difference between associative property and distributive property? [4 marks]
Ans. The associative property states that we can add or multiply integers by grouping them in any particular order and the sum or product will be equal in both cases. The associative property has two forms expressed as-
(a + b) + c = a + (b + c) and,
(a x b) x c = a x (b x c).
The distributive property states that multiplying the sum of two or more numbers will be equal if we multiply each number with the numbers individually and then add the products together. The distributive property is applicable for multiplication over addition and subtraction. The distributive property can be expressed as-
a x (b + c) = ab + ac and,
a x (b – c) = ab – ac.
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