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Distributive Property

The distributive property is one of the most frequently used properties in basic Mathematics. In general, it refers to the distributive property of multiplication over addition or subtraction. It is also known as the distributive law of multiplication. When we distribute something, we are dividing it into parts. In math, the distributive property helps to simplify difficult problems. This is because it breaks down expressions into the sum or difference of two numbers. In this topic, we will learn about the distributive property and its examples. Let us begin it!

Distributive Property

What is the distributive property?

The distributive property will allow multiplying a sum value by multiplying each addend separately. And, then add the products. Multiplying the number immediately outside the parentheses with those given inside values. And, then adding the products together.

When an algebraic expression is having parentheses with variables. Here variable is a quantity that may change within the given context of a mathematical problem. It is usually represented by a single letter.

Two main laws under this property:

  1. Distributive property of multiplication over addition:

It is a fact that whether we use the distributive property or follow the order of given operations, we will arrive at the same answer. This is the main concept. So, we can see it as given below:

Using the distributive law, we will Multiply, or distribute, the outer term to the inner terms. Then combine like terms, and solve the equation.

This property states that:

m × (n + o) = (m × n) + (m ×o)

Here variables m, n and o are the real numbers.

Let’s use a real-life example to help make this clearer. Imagine one student and her two friends each have seven mangos and four bananas. How many pieces of fruit do all three students will have in total?

In their lunch bags i.e. as the parentheses, each student will have seven mangos and four bananas. To know the number of pieces of fruit in total, students will need to multiply the whole thing by 3.

When we break it down, we are multiplying 7seven mangos and four bananas by 3 students. So, we end up with 21 mangos and 12 bananas for a total of 33 pieces of fruit.

  1. Distributive property of multiplication over subtraction:

Similar to the above operation with addition, performing the distributive property with subtraction follows the same rules. But here exception is that we are finding the difference instead of the sum. Although, it doesn’t matter if the operation is addition or subtraction, keep whichever one is in the parentheses.

This property states that:

\(m \times (n- o) = (m \times n) – (m \times o)\)

Where m, n and o are the real numbers.

 Solved examples for You

Example-1: Prove the distributive property for the following expression:

\(3 \times (4+8) = (3 \times 4) + (3 \times 8)\)

Solution:

LHS: 3 × (4+8)

=3 × (12)

=36

RHS: (3×4) +(3×8)

=(12)+(24)

=36

Since  LHS = RHS

Hence Proved.

Example-2: Prove the distributive property for the following expression:

\(5 \times (7-3) = (5 \times 7) – (5 \times 3)\)

Solution:

LHS: 5 × (7-3)

=20

RHS: \((5 \times 7) -(5 \times 3)\)

=(35)-(15)

=20

Since  LHS = RHS

Hence Proved.

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