Distributive Property of Multiplication

The distributive property is multiplying a number by sum equals to multiplying the number by each addend separately and then adding the products. It breaks down and simplifies expressions. To distribute anything is to divide or assign a portion of something. We apply this property to describe the way by which multiplication extends to addition and subtraction.

Distributive Property of Multiplication Formula

Using the distributive property of multiplication, we solve an expression of the form A(B + C) by expanding it as:
A(B + C) = AB + AC
This property also applies to subtraction:
A(B – C) = AB – AC
where A, B, and C are any real numbers.

Distributive Property of Multiplication Over Addition 

The distributive property of multiplication shows that when the sum is multiplied by another number, each term in the sum can be multiplied individually by that number, and the results can then be added together for the same outcome. Consider the example: 5(6 + 4) by expanding it:
5(6 + 4) = 5(6) + 5(4) = 30 + 20 = 50
Here, we distribute the numbers 5 to both 6 and 4 and then sum up their products separately.
Using BODMAS rule also gives the same result
5 (6 + 4) = 5 (10) = 50
As both methods give the same result, the distributive property is true.

Distributive property of multiplication over subtraction 

The distributive property of multiplication over subtraction states that you can multiply a number by each value separately and subtract the results. This is the same as multiplying the number by the difference between the two values. The formula we use for the distributive property of multiplication over subtraction is:
a(b – c) = ab – ac. 
For example:
Solve 7 (15 – 5) using both methods:
As per this property, the result on both LHS and RHS is the same.
On the left-hand side (LHS), we evaluate directly:
7 (15 – 5) = 7 (10) = 70
In RHS, we apply the distributive property:
7 × 15 – 7 × 5 = 105 – 35 = 70
Since both sides give the same result, we confirm that the property holds.
 

Understanding the distributive property enhances students' ability to simplify complex calculations. It applies not only to math but also to real life. Some real-life applications are:

• When shopping for multiple items, we apply the distributive property of multiplication to determine the total cost.
  For example: Purchasing 3 units of each of two items priced at $30 and $12
  3 (30 + 12) = 3 × 30 + 3 × 12 = 90 + 36 = 126, which indicates that the total cost is $126.

• The distributive property can be used to divide the task evenly among students.
  
   
• In construction, the distributive property is used to calculate areas when rooms are divided into sections. 
  For example: Dividing the area of a single part:
  10 (3 + 2) = 10 × 3 + 10 × 2 = 30 + 20 = 50.