### Distributive Property Basics

All the numbers that are used in Mathematical calculations and have a specific value is called the real numbers. All the real numbers obey certain laws or have a few properties. These properties are shown when basic Mathematical operations are performed on these numbers. The most fundamental properties of numbers include closure property, commutative property, associative property, identity property, inverse property and distributive property. Distributive property is one of the most popular and very frequently used properties of numbers in Mathematical calculations. This section briefs about a few distributive property examples for better understanding.

### What is Distributive Property?

Distributive property is one of the fundamental properties of numbers and basic operations of Mathematics. This property refers to the distribution of multiplication over addition or subtraction. Hence it is most popularly known as the distributive property of multiplication over addition or subtraction. Distribution in English refers to the breaking down into parts. What is distributive property is math can be analysed as a property used to simplify complex algebraic expressions to arrive at the solution easily. By using the distributive property of multiplication over addition or subtraction, the expressions can be written as the sum or difference of the two numbers.

### How to use the Distributive Property of Multiplication Over Addition/Subtraction?

The distributive property of multiplication over addition allows the user to multiply a value of the sum by multiplying the addends individually. The individual products are then added to get the final answer. For instance, if ‘m’, ‘n’ and ‘p’ are the three numbers, then distributive property states that

m (n + p) = m n + m p

The answer of the product of a number and the addition of two other numbers can be obtained as the sum of the individual products obtained when each number inside the parentheses is multiplied by the number outside.

Any algebraic expression may contain the terms this form with several variables.

### Distributive Property of Multiplication Over Addition:

While multiplying a number or a term with the sum of the other two numbers or variables, the number or the term outside the parentheses is distributed over the individual addends within the parentheses. The individual products obtained by multiplying the outer term withe inner terms are added to arrive at the final answer.

If the variables ‘p’, ‘q’ and ‘r’ represent any real numbers or the terms in an algebraic expression, then distributive property of multiplication over addition states that:

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

### Distributive Property of Multiplication Over Subtraction:

Distributive property of multiplication over subtraction is the same as that incase of addition. However, the only change is that the difference of the individual products is found to arrive at the final answer. It does not make a great difference with the distributive property of multiplication over addition or subtraction. The only rule is that the term outside the bracket is distributed over the terms inside the bracket using multiplication operation and the basic operation indicated between the terms within the parentheses is performed between the individual products.

### Distributive Property Examples

Let us consider that there are three friends sitting on the same bench. Imagine that each individual student has 4 black pencils and 5 blue pencils. Just think about “How do we calculate the total number of pencils with them all together?”. This can be done in two ways.

Method 1:

Identify the total number of pencils with each individual person and then multiply it with the total number of friends. This method follows the order of operations.

No. of pencils with each individual = 4 + 5 = 9

Total no. of pencils = 9 x 3 = 27

Method 2:

We can first find the total number of blue pencils and the total number of black pencils. Then, to find the total number of pencils, the number of black pencils and blue pencils are added. Here, the number of individuals is distributed over the number of blue and black pencils.

Total number of pencils

= No. of friends x No. of black pencils with each + No. of friends x No. of blue pencils with each

= (3 x 4) + (3 x 5) = 12 + 15 = 27

### Fun Facts

The answer remains unaltered in case the distributive property example is used or the order of operations is followed.

Algebraic expressions can also be evaluated by following the PEMDAS rule which gives the order of operations. It is not mandatory to use the distributive property in all cases.

1. Give the List of Properties of Numbers.

The fundamental properties of numbers in Mathematics are:

Closure property

Commutative property

Associative property

Identity property

Inverse property

Distributive property

2. What is Distributive Property? Give Distributive Property Examples.

Distributive property is one of the fundamental properties of multiplication of numbers or expressions or variables. To distribute is to divide or to spread. According to distributive property, when a number or a variable or a term in an algebraic expression is multiplied with the sum of two numbers / variables / terms, the final product is equal to the sum of the individual products obtained by multiplying the term outside the parentheses with the individual addends represented within the parentheses.

Example: 5 (6m + 9q) = (5 x 6m) + (5 x 9q) = 30m + 54q