 # Distributive Property

## Distributive Property Formula

There are three properties of numbers most widely used. They are commutative , associative and distributive property. Distributive property is also known as distributive law of multiplication. Distributive property is most frequently used property in mathematics. Distribute means the name itself implies that to divide something.

Distributive property means to divide the given operations on the numbers, so that the equation becomes easier to solve. So let us study distributive property definition, distributive property formula, distributive property example and distributive property with variables

### Distributive Property Definition

The distributive property applies to the multiplication of a number with the sum or difference of two numbers i.e., the distributive property holds true for multiplication over addition and subtraction.

Distributive property definition simply states that “multiplication distributed over addition.”

For instance take the equation

• a( b + c)

• when we apply distributive property we have to multiply a with both b and c and then add

• i.e a x b + a x c = ab + ac.

Distributive Property Formula

Let us understand this concept with distributive property examples

For example 3( 2 + 4) = 3 (6) = 18

Or

By distributive law

3( 2 + 4) = 3 x 2 + 3 x 4

= 6 + 12

= 18

Here we are distributing the process of multiplying 3 evenly between 2 and 4. We observe that whether we follow the order of the operation or distributive law the result is the same.

### Distributive Property with Variables

The distributive property law can also be used when multiplying or dividing algebraic expressions that include real numbers and variables that is called distributive property with variables.

The expression can be monomial, binomial or a polynomial.

You can multiply a polynomial by a monomial by using distributive law in the following ways

• First step is to multiply the outside term by the first term in parenthesis.

• Next multiply the outside term by the second term in parenthesis with the given operation in between.

• Now carry out the given operation.

For example consider  x (2x + 8) x (2x + 8)

= (x  x 2x) + (x x 8)

= 2x2 + 8x

You can use the distributive property law to find the product of binomials too.

For example consider (x + y)(x + 5y)

(x + y)(x + 5y)

=x( x + 5y) + y (x + 5y)

= x2 + 5xy + xy + 5y2

= x2 + 6xy + 5y2

Distributive property also allows us to simplify the algebraic equation and find the values of unknown variables.

Using distributive property finding the values of unknown variables

1. Multiplying the outer terms with the inner terms of parentheses.

2. Then combining the like terms.

3. Arrange terms so that variables and constants are opposite to the equal to sign.

4. Solve the equation and simplify, to get the unknown values.

For example : 4( x + 3) = 20

4(x) + 4(3) = 20

4x + 12 = 20

4x = 20 - 12

4x = 8

x = 8/4

x = 2

In Mathematics distributive property is applied on various operations such as

• Distributive Property of Subtraction

• Distributive Property of Multiplication

• Distributive Property of Division

We add to two or more numbers  to get their total. The distributive property tells us that the sum of two numbers multiplied by the third number is equal to the sum of each addition multiplied by the third number. Distributive property of addition is represented as

(p + q) x r = (p x q) + (p x q)

### Distributive Property of Subtraction

Distributive property of subtraction is the same as distributive property of addition. And is represented as:

(p - q) x r = (p x q) - (p x q)

### Distributive Property of Multiplication

When we want to multiply any number with the sum of a number we usually first add the numbers and then multiply it by the number.

For instance 6( 2+ 3) = 6(5) = 30

But by the law of distributive property of multiplication we first multiply  the number by every addend and then perform the addition on the products.

As, 6 ( 2 +  3)

= 6 x 2 + 6 x 3

=12 + 18

= 30

We observe that the results for the operations are the same.

### Distributive Property of Division

Dividing large numbers is a bit time consuming, hence distributive property allows us to make it easier by breaking those numbers into smaller factors and then distributing the divide operation between them.

For example, divide 96 8

Solution: we can write 96 = 80 + 16

(80 + 16) 8

Now distributing division operation for each value in the bracket we get,

( 80 ÷ 8) + (16 8)

=10 + 2

= 12

In this way we can make the divide operations more easier.

Let us understand this concept with more distributive property examples.

### Solved Examples

1.  4(8x + 4)

Solution:

4(8x + 4)

=  (4 x 8x) + ( 4 x 4)

= 32x + 16

1. 9a(5a + 2b)

Solution:

9a(5a + 2b)

=(9a x 5a) + (9a x 2b)

= 45a2 + 18ab

### Quiz Time

Applying distributive law solve

1. 7a(5a + 2)

2. 3x(x - 2)

1. What does Distributive Property Mean in Math?

Answer: Distribute, the name itself implies that to divide something.

Distributive property means to divide the given operations on the numbers, so that the equation becomes easier to solve.

Distributive property definition simply states that “multiplication distributed over addition.”

For instance take the equation

a( b + c)

when we apply distributive property we have to multiply a with both b and c and then add

i.e a x b + a x c = ab + ac.

2. What is the Distributive Property of Division?

Answer: Distributive property of division breaks numbers into smaller factors and then distributes the divide operation between them.

For example, divide 99 9

Solution: we can write 99 = 90 + 9

(90 + 9) 9

Now distributing division operation for each value in the bracket we get,

( 90 ÷ 9) + (9 9)

= 10 + 1

= 11

3. What is Associative Property?

Answer: The word associative comes from the word ‘associates’. Associative property refers to grouping. Associative property rule can be applied for addition and multiplication. If associative property for addition and multiplication operation is carried out regardless of the order of how they are grouped. In any order of grouping the result remains constant.

• Associative property for addition states that if

(p + q) + r = p + ( q + r)

For example: (2 +  5) + 4 = 2 + (5 + 4) the answer for both the possibilities will be 11.

• Associative property for multiplication states that if

(p x q) x r = p x ( q x r)

For example ( 2 x 3) x 5 = 2 x ( 3 x5) the answer for both the possibilities  will be 30.

Thus we can apply the associative rule for addition and multiplication but it does not hold true for subtraction and division.

4. What is Commutative Property?

Answer: Commutative property states that order of term does not matter when adding or multiplying the numbers. The result remains constant if the order changes. But this law does not hold true for subtraction and division.

For example : a + b = b + a

a x b = b x a

But ,

a - b b - a

ab b a