Distributive Property

Multiplication is one of the four operational functions of mathematics, the other three are, Addition, Subtraction, and Division. And the students of Maths need to understand all the aspects of this operational function. Now, coming to the properties, there are three main types of properties of Number, Commutative, which can be achieved with the addition and multiplication operations, then there is Associative Property, which deals with the binary functions of the number, and then there is distributive property also known as distributive law of multiplication.

Understanding the Distributive Property

The Distributive property is used when you have to multiply the given number by the sum of the two numbers. In such cases what you generally do by following the rule of order of operations, is to first add the numbers, and then multiply them by the given number. For example, if you have to multiply the number 5 with the sum of 10 + 5, then you can represent it in this manner, 5(10 + 5), now according to the rule of order of operation, you first have to find the sum of 10 + 5, which becomes 15, and then you have to multiply 15 with the 5. The whole solution may look like,

5(10 + 5)

= 5(15)

= 75

But in the distributive law of multiplication, what you have to do is to distribute the sum first and then multiply the same with the given number, and then you have to add the results of both. Take the same example once again, which was 5(10 + 5), What you have to do now is first multiply 5 with the 10, which is 50, and then multiply the 5 with the 5, the answer is 25, now add 50 and 25 together, you have got 75. The whole solution looks like this:

5(10 + 5)

= 5(10) + 5(5)

= 50 + 25

= 75

The distributive property can be present in the equation form as: 

One thing you must keep in mind, that the distributive law of multiplication remains the same, even when the variables inside the sum bracket increase. That is to say, if the question is 5(10 + 5 + 5) then the rule still remains the same, it goes like this, 5(10) + 5(5) + 5(5), which becomes 50 + 25 + 25, and the total becomes 100.

Steps to Follow in the Distributive Property

It is always better to have a summarization of the steps, for an even better understanding of the distributive law of Multiplication, and hence following are the steps of the same.

• The very first step in the Distributive property is to multiply the number which is outside the bracket or the parenthesis, with the first number given inside the parenthesis.

• Now, multiply the number, which is given outside the parentheses with the second number given inside the parenthesis.

• Lastly, you have to do the addition of both the results.

Types of Distributive Property

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

Distributive property means dividing 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.

The 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 \[a \times b + a \times c = ab + ac.\]

Distributive Property Formula

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Let us understand this concept with distributive property examples

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

or

By distributive law

\[3( 2 + 4) = 3 \times 2 + 3 \times 4\]

                       = 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 are called distributive property with variables.

The expression can be monomial, binomial or polynomial.

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

• The 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)

= \[(x  \times 2x) + (x \times 8)\]

             = 2 x² + 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)

= \[x^{2} + 5xy + xy + 5y^{2}\]

= \[x^{2} + 6xy + 5y^{2}\]

The 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 to various operations such as 

• Distributive Property of Addition

• Distributive Property of Subtraction

• Distributive Property of Multiplication

• Distributive Property of Division

Distributive Property of Addition

We add 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

\[(p + q) \times r = (p \times q) + (p \times q)\]

Distributive Property of Subtraction

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

\[(p - q) \times r = (p \times q) - (p \times q)\]

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) 

= \[6 \times 2 + 6 \times 3\]

=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 division operations easier.

Let us understand this concept with more distributive property examples.

Solved Examples

1. 4(8x + 4)

Solution:

4(8x + 4)

=  \[(4 \times 8x) + ( 4 \times 4)\]

= 32x + 16

2. 9a(5a + 2b)

Solution:

         9a(5a + 2b)

         = \[(9a \times 5a) + (9a \times 2b)\]

         = \[45a^{2} + 18ab\]

Quiz Time

Applying distributive law solve

1. 7a(5a + 2)

2. \[3 \times (x - 2)\]