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Distributive Property

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Distributive Property in Maths as the name suggests refers to the distribution of the number over the other operations. It is usually called the Distributive Law of Multiplication over Addition and Subtraction as the number which is to be distributed over addition or subtraction is in multiple of the sum or difference of the numbers.

Distributive Property Formula

In this article, we will learn about the distributive property in detail.

Distributive Property Definition

Distributive Property states that when a number is multiplied by the sum or difference of two numbers then it is equal to the sum or difference of the product of the first number with the other two numbers individually.

Distributive Property Formula

Distributive Property is often expressed in the following formula:

a × (b + c) = (a × b) + (a × c)

In this formula:

  • a is a constant or variable that is multiplied by the sum or difference of b and c.
  • b and c can be constants, variables, or expressions.

As of now, we know that Distributive Property Formula is defined over two operations i.e. addition and subtraction. Let’s Learn them in detail.

Distributive Property of Multiplication over Addition

Distributive Property Formula of Multiplication over Addition, as the name suggests, is defined over the operation of addition. This Formula is used when a number is multiplied by the sum of two numbers. The Distributive Formula of Multiplication over Addition is given as P(Q + R) = PQ + PR. This can be better understood from an example.

Let’s understand this with the help of an example.

Example: Solve 15(6 + 5) using Distributive Property.

We have, 15(6 + 5)

Using the Distributive Law of Multiplication over Addition

15(6 + 5) = 15⨯6 + 15⨯5 = 90 + 75 = 165

Distributive Property of Multiplication over Subtraction

Distributive Law of Multiplication over Subtraction, as the name suggests, is defined over the operation of subtraction. In this case, if a number is multiplied to the difference of two numbers then it is equal to the difference of product of the numbers. The distributive Property of Multiplication over Subtraction is expressed as P(Q – R) = PQ – PR.

Let’s understand this with the help of an example.

Example: Solve 6(30 – 4) using Distributive Property.

We have, 6(30 – 4)

Using the Distributive Property of Multiplication over Subtraction, we have

6(30 – 4) = 6⨯30 – 6⨯4 = 180 – 24 = 156

Verification of Distributive Property

To verify the Distributive Property we need to check if the values on both Left Hand Side and Right Hand Side of the expression. Let’s see the verification of the Distributive Property of Multiplication over Addition and the Distributive Property of Multiplication over Subtraction one by one.

Verification of Distributive Property of Multiplication over Addition

Let’s verify if, 15(6 + 5) = 15 ⨯ 6 + 15 ⨯ 5.

We have LHS = 15(6 + 5)

using BODMAS Rule we will first solve the bracket.

⇒ 15(6 + 5) = 15 ⨯ 11 = 165

Hence, we have LHS = 165

We have RHS = 15 ⨯ 6 + 15 ⨯ 5

⇒ 90 + 75 = 165

Hence, RHS = 165

Thus we see that we have LHS = RHS. Hence, 15(6 + 5) = 15 ⨯ 6 + 15 ⨯ 5. Thus Distributive Property of Multiplication over Addition is verified.

Verification of Distributive Property of Multiplication over Subtraction

Distributive Property of Multiplication over Subtraction is given as P(Q – R) = PQ – PR. Let’s check if it’s true or not using an example.

Let’s verify if 6(30 – 9) = 6⨯30 – 6⨯9.

We have LHS = 6(30 – 9)

Solving bracket first using BODMAS Rule,

⇒ 6 ⨯ 21 = 126

We have RHS = 6⨯30 – 6⨯9

⇒ 180 – 54 = 126

Hence, we have LHS = RHS, it means 6(30 – 9) = 6⨯30 – 6⨯9. Thus Distributive Law of Multiplication over Subtraction is verified.

Distributive Property of Division

Distributive Property of Division is based on the same pattern as the Distributive Property of Multiplication just with a minor difference that the sum or difference inside the bracket is now divided by a number instead of multiplication.

The Distributive Property of Division can be expressed as :

(Q + R) ÷ P = Q ÷ P + R ÷ P for the Distributive Property of Division over Addition

(Q – R) ÷ P = Q ÷ P – R ÷ P for the Distributive Property of Division over Subtraction.

The same expression can be written in terms of multiplication as

(Q + R)1/P and (Q – R)1/P.

Example: Divide 76 ÷ 4 using the distributive property of division.

Given expression: 76 ÷ 4

We can write 76 as 64 + 12

So, 76 ÷ 4 = (64 + 12) ÷ 4

Now, let us distribute the division operation for each factor (64 and 12) in the bracket.

= (64 ÷ 4) + (12 ÷ 4)

= 16 + 3 = 19

Therefore, the answer is 19.

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Distributive Property Examples

Example 1: Solve equation 5 (y + 8) = 120 Using Distributive Property.

Given, 5 (y + 8) = 120

We have a set of two parentheses inside the bracket. So, distribute 5.

5 × y + 5 × 8 = 120

⇒ 5y + 40 = 120

⇒ 5y = 120 – 40 = 80

⇒ y = 80/5 = 16

Hence, y = 16.

Example 2: Solve 3x + 4(x – 6) + 17 = 28 Using Distributive Property.

Given, 3x + 4(x – 6) + 17 = 28

We have a set of two parentheses inside the bracket. So, distribute 4.

3x + 4 × x – 4 × 6 + 17 = 28

⇒ 3x + 4x – 24 + 17 = 28

⇒ 7x – 7 = 28

⇒ 7x = 28 + 7 = 35

⇒ x = 35/7 = 5

Hence, x = 5.

Example 3: Solve the equation (a + 3b) (2a + b) Using Distributive Property.

Given, (a + 3b) (2a + b)

From the distributive property, we have

(p + q) × r=(p × r)+(q × r)      

So, (a + 3b) (2a + b) =  a × (2a + b) + 3b × (2a + b)

⇒ (a + 3b) (2a + b) = 2a2 + ab + 6ab + 3b2

⇒ (a + 3b) (2a + b) = 2a2 + 7ab + 3b2

Thus, (a + 3b) (2a + b) = 2a2 + 7ab + 3b2.

Practice Questions on Distributive Property

Q1: Solve using Distributive Property 99 ⨯ 23.

Q2: Solve 24 ⨯ 96 + 24 ⨯ 4 with the help of Distributive Property.

Q3: Verify using Distributive Property: Is 39 ⨯ 101 = 39 (100 + 1)?

Q4: Solve 36 ⨯ 204 – 26 ⨯ 4 using Distributive Property.

Q5: Solve 19 ⨯ 47 + 19 ⨯ 3.

Distributive Property-FAQs

What is Distributive Property in Maths?

In mathematics, the distributive property is a fundamental property that applies to operations like addition and multiplication. It allows you to perform operations on terms within parentheses or brackets and is a key concept in algebra.

What is Distributive Property of Multiplication?

Distributive Property of Multiplication is a method of distributing numbers over addition and subtraction to the number present inside the bracket. It is expressed as P(Q ± R) = PQ ± PR.

What is Distributive Property Equation?

The distributive property expression is written as P(Q ± R) = PQ ± PR

What is Distributive Property Example?

The Distributive Property Example is, 6(6 + 10) = 36 + 160 = 196

How to Solve Distributive Property?

Distributive Property can be solved by the following method.

Let’s say we have to solve 7(6 + 10)

then we will open the bracket and multiply 7 by 6 and 7 by 10 and then multiply i.e. 7 ⨯ 6 + 7 ⨯ 10 = 42 + 70 = 112.

What is Distributive Property of Rational Numbers?

Distributive property applies to rational numbers just as it does to integers or real numbers. The distributive property of rational numbers states that for any rational numbers a, b, and c:

Distributive Property of Rational Numbers: a(b + c) = (ab) + (ac)

What is Distributive Property of Addition?

Distributive Property of Addition is just the same as Distributive Property of Multiplication over addition.

Why do we use Distributive Property?

Distributive Property is used to solve mathematical problems and simplify various algebraical expressions.



Last Updated : 08 Feb, 2024
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