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The distributive property can be defined as “multiplication of the sum of two or more variables by a number results in the same product as multiplication of the variables by the number individually and adding the sum of all the products of the variables.” The distributive property formula is used in division functions as well to break down factors and solve the equations. It is also known as the distributive law of multiplication over addition and subtraction.
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Key Terms: Addition, Subtraction, Multiplication, Division, Variable, Real Numbers, Equation
What is Distributive Property?
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The distributive property states that in a multiplication problem when one factor is rewritten as the sum of two numbers, the product does not change. The word “distribute” means to divide something or give a share or part of something. The distributive property is one of the most basic and most common functions which is used in the field of Mathematics, as it helps to simplify very complex equations.
Distributive Property Formula
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As explained above, the distributive property formula helps to simplify complex equations by breaking them into simpler equations. The formula used are:
a (b + c) = (a x b) + (a x c)
This is the distributive property of multiplication over addition. The distributive property formula can also be used in the form of
a (b - c) = (a x b) - (a x c)
This is the distributive property of multiplication over subtraction.
Distributive Property of Multiplication Over Addition
The distributive property of multiplication over addition states that:
a (b + c) = (a x b) + (a x c)
where, a, b and c are real numbers.
To understand things better, let us look at an example where we multiply two real numbers 5 and 7 to yield the result 35, i.e, 5 x 7 = 35
Breaking down 5 into (3+2) and using distributive property, we can rewrite 5 x 7 as,
(3+2) x 7
= (3 x 7) + (2 x 7)
= 21 + 14
= 35
Here, 7 gets ‘distributed to’ 3 and 2.
While solving such equations we can use this distributive property to simplify bigger and harder numbers; such as 19 x 6 = 114
The above can be simplified using distributive property where 19 is rewritten as (10 + 9) = 19
Thus, the equation (19 x 6) can be written as:
(10 + 9) x 6
= (10 x 6) + (9 x 6)
= 60 + 54
= 114
Verification of Distributive Property Formula in Addition
Let us verify the distributive property of multiplication over addition with the help of an example mentioned below:
Example: Zoya, Pia and Kathy planned to have a small party for the whole class during their lunch break to celebrate the re-opening of schools after the COVID-19 pandemic. They decided to carry 13 packets of chips, 9 packs of chocolates, and 4 bottles of soft drinks each. How many food items do they have in total?
Solution: Each student has 13 packets of chips, 9 packs of chocolates, and 4 bottles of soft drinks. One student is having (13 + 9 + 4 = 26) food items.
Therefore, to get the total number of food items they have, we need to multiply 26 by 3.
Thus, we arrive at the answer which is: (26 x 3 = 78)
When we break it down as per the distributive property of multiplication over addition, we get:
13 packets of chips by 3, 9 packs of chocolates by 3, and 4 bottles of soft drinks by 3.
So, we have 39 packets of chips, 27 packs of chocolates, and 12 bottles of soft drinks, which adds up to (39 + 27 + 12 = 78) food items.
Hence, the property is verified.
Distributive Property of Multiplication over Subtraction
The distributive property of multiplication over subtraction formula states that:
a (b - c) = (a x b) - (a x c)
where, a, b and c are real numbers.
The distributive property of multiplication over subtraction essentially performs the same function as the distributive property of multiplication over addition but instead of finding the sum, we find the difference.
To understand things better, let us look at an example,
5 (25 - 20) = ?
5 (25 - 20) = 5 x 5 = 25 (LHS)
By using distributive property of multiplication over subtraction:
5 (25 - 20)
= (5 x 25) - (5 x 20)
= 125 - 100 = 25 (RHS)
Since, LHS = RHS, hence, proved.
Here, 5 gets ‘distributed to’ both 25 and 20, and they are individually multiplied for us to arrive at the same result in both LHS and RHS.
Verification of Distributive Property Formula in Subtraction
To verify the distributive property of multiplication over subtraction, let us consider an example as mentioned below:
Example: We take three numbers 7,5 and 3 and perform both multiplication and subtraction operations between them as, 7 (5 - 3) = ?
Solution: Here, 7 (5 - 3) = 7 x 2 = 14 (LHS)
By using distributive property of multiplication over subtraction:
7 (5 - 3) = (7 x 5) - (7 x 3) = 35 - 21 = 14 (RHS)
Since, LHS = RHS, hence, proved.
Distributive Property of Division
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The distributive property of division states that when dividing a sum of two variables (or numbers) by another variable (or number), it gives the same result as dividing each variable individually and the sum of the resulting quotients.
The distributive property formula can be used in the division of larger numbers by breaking the bigger number into smaller factors of its own.
For example, let us solve 69 ÷ 3.
We know that, 69 ÷ 3 = 21 (LHS)
Using the distributive property of division,
69 can be re-written as 60 + 9
69 ÷ 3 = (60 + 9) ÷ 3
We can now ‘distribute’ or ‘divide’ each factor (60 + 9) individually in the parentheses.
= (60 ÷ 3) + (9 ÷ 3)
= 20 + 3
= 23 (RHS)
Since, LHS = RHS, hence, proved.
Things to Remember
- The Distributive Property of Multiplication over Addition states that a (b + c) = (a x b) + (a x c), where, a, b and c are real numbers.
- The Distributive Property of Multiplication over Subtraction states that a (b - c) = (a x b) - (a x c), where, a, b and c are real numbers.
- When we use distributivity, we split a product as a sum or difference of two products.
- The distributive property formula of multiplication can be applied over addition and subtraction.
- The distributive property formula can be applied in the division function as well.
- The distributive property formula is used to simplify complex equations.
- When we use the formula of distributive property we will get the same answer easily because the formula helps us to simplify complex equations involving variables, fractions, integers etc.
Sample Questions
Ques. Solve: 420 x 69 using Distributive Property of Multiplication over Addition. (3 marks)
Ans. We can write 69 as (60+9)
Therefore,
= 420 x (60+9)
= (420 x 60) + (824 x 9)
= (420 x 6 x 10) + (420 x 9)
= (25200 + 3780)
= 28980
Ques. Solve: 2222 x 107 using Distributive Property. (3 marks)
Ans. We can write 107 as (100+7)
Therefore,
= 2222 x (100+7)
= (2222 x 100) + (2222 x 7)
= (2222 x 10 x 10) + (2222 x 7)
= (222200 + 15554)
= 237754
Ques. Solve: (- 3) (2x + 5) using Distributive Property. (2 marks)
Ans. Given,
(- 3) (2x + 5) = [( -3) (2x)] + [( -3) (5)]
= ( -6x) + ( -15)
= -6x - 15
Ques. Solve: (x - 3) = (x/3 + 6) using Distributive Property. (2 marks)
Ans. Given, (x - 3) = (x/3 + 6)
(x- 3) {(2x + 1) ÷ 6} (Using LCM)
⇒ ( 6x - 18) = 2x + 1
⇒ ( 6x - 2x) = 1 + 18
⇒ 4x = 19
⇒ x = 19/4
⇒ x = 4 ¾
Ques. Solve: (11x + 4)2 using Distributive Property. (2 marks)
Ans. Given, (11x + 4)2
(11x + 4) ( 11x + 4)
= 121x 2 + 44x + 44x + 16
= 121x 2 + 88x + 16
Ques. Solve: {(9 x 16) (4 x 12)} + {(9 x 16) ( - 3 x 9)} using Distributive Property. (2 marks)
Ans. Given, {(9 x 16) (4 x 12)} + {(9 x 16) ( - 3 x 9)}
= (9 x 16) {(4 x 12)} + (- 3 x 9)}
= (9 x 16) {(12 -12) 36} (Using LCM)
= 0
Solving via distributive property of multiplication,
(9 x 16) x (4 x 12) = (3 x 16)
(9 x 16) x ( - 3 x 9) = (- 3 x 16)
(3 x 16) + (- 3 x 16) = (3 - 3) x 16 = 0
Since, LHS = RHS, hence, proved.
Ques. Divide 24 ÷ 6 using the distributive property of division. (2 marks)
Ans. We can write 24 as 18 + 6
24 ÷ 6 = (18 + 6) ÷ 6
Now, let us distribute the division operation for each factor (18 and 6) in the bracket.
⇒ (18 ÷ 6) + (6 ÷ 6)
⇒ 3 + 1
Therefore, the answer is 4.
Ques. Solve the expression 7(20 – 3) using the distributive property of multiplication over subtraction. (2 marks)
Ans. Using the distributive property of multiplication, we can solve the expression as follows: 7 × (20 – 3)
= (7 × 20) – (7 × 3)
= 140 – 21
= 119
Ques. Prove the distributive property for the following expression: 3 × (4 + 8) = (3 × 4) + (3 × 8). (3 marks)
Ans. Given, 3 × (4 + 8) = (3 × 4) + (3 × 8)
LHS: 3 × (4 + 8)
= 3 × (12)
= 36
RHS: (3 × 4) + (3 × 8)
= (12) + (24)
= 36
Since LHS = RHS
Hence Proved.
Ques. Prove the distributive property for the following expression: 5 × (7 − 3) = (5 × 7) – (5 × 3). (2 marks)
Ans. Given, 5 × (7 − 3) = (5 × 7) – (5 × 3)
LHS: 5 × (7 - 3)
= 20
RHS: (5 × 7) − (5 × 3)
= (35) - (15)
= 20
Since LHS = RHS
Hence Proved.
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