Distributive Property of Natural Numbers With Examples

The Distributive Property is a fundamental property of numbers that describes how multiplication interacts with addition and subtraction.

Specifically, it states that multiplying a number by a sum (or difference) is the same as doing each multiplication separately and then adding (or subtracting) the results.

Here’s the formal definition for natural numbers:

For any natural numbers a, b, and c:

a × (b + c) = (a × b) + (a × c)

Similarly, for subtraction:

a × (b − c) = (a × b) − (a × c)

Let’s break this down with some detailed examples.

Example 1: Distributive Property with Addition

Consider the expression 3 × (4 + 5).

Step 1: Calculate the value inside the parentheses first:

4 + 5 = 9

Step 2: Multiply the result by 3:

3 × 9 = 27

Now, using the distributive property:

Step 1: Distribute the multiplication over the addition:

3 × (4 + 5) = (3 × 4) + (3 × 5)

Step 2: Perform the individual multiplications:

3 × 4 = 12

3 × 5 = 15

Step 3: Add the results:

12 + 15 = 27

Both methods give the same result, illustrating the distributive property.

Example 2: Distributive Property with Subtraction

Consider the expression 6 × (8 − 3).

Step 1: Calculate the value inside the parentheses first:

8 − 3 = 5

Step 2: Multiply the result by 6:

6 × 5 = 30

Now, using the distributive property:

Step 1: Distribute the multiplication over the subtraction:

6 × (8 − 3) = (6 × 8) − (6 × 3)

Step 2: Perform the individual multiplications:

6 × 8 = 48

6 × 3 = 18

Step 3: Subtract the results:

48 − 18 = 30

Again, both methods give the same result, confirming the distributive property.

Visual Explanation

To help visualize this, imagine you have groups of objects.

Example with Addition

You have 3 groups of (4 + 5) objects:

• If you add the objects in each group first, you get 9 objects per group.
• 3 groups of 9 objects are 27 objects in total.

Alternatively:

• You can split each group into 4 objects and 5 objects.
• 3 groups of 4 objects give 12 objects.
• 3 groups of 5 objects give 15 objects.
• Adding these together, you again get 27 objects.

Example with Subtraction

You have 6 groups of (8 – 3) objects:

• If you subtract inside each group first, you get 5 objects per group.
• 6 groups of 5 objects are 30 objects in total.

Alternatively:

• You can consider each group to start with 8 objects and then remove 3 objects from each group.
• 6 groups of 8 objects give 48 objects.
• Removing 6 groups of 3 objects gives 18 objects removed.
• Subtracting, you get 30 objects.

The distributive property is a powerful tool in mathematics because it simplifies calculations and helps to understand how operations interact with each other. It’s especially useful in algebra when dealing with variables and more complex expressions.