The distributive property is a well-known property related to numbers and algebra in mathematics. As the name suggests, this property focuses on distributing or dividing a quantity through proper conditions. The distributive property or distributive law is only operated in the multiplication of numbers and algebra. This is why it is also called the distributive law of multiplication.

**Note: **Distributive property can never be applied in the addition or subtraction of numbers. Even if you apply, the result will be void or produce errors in the solution.

Before we dive deep into the distributive property of multiplication, let us have a quick look at other important properties in mathematics. They are listed below:

- Commutative Property: This property states that the numbers or terms can commute or move their places in the expression without altering the result. This is true for addition and multiplication. For instance, (1 + 4) = (4 + 1) and (2 * 4) = (4 * 2). Subtraction doesn’t follow this property, for example, (1 – 4) = -3 is not equal to (4 – 1) = 3.
- Associative Property: This property states that the numbers of terms in an expression can associate themselves or groups with each other without altering the result. This is true for addition and multiplication. For instance (1 + 4) + 3 = 1 + (4 + 3).

Let us now discuss what distributive property means and some distributive property examples.

**Distributive Property Definition**

Let us first understand a simple concept. If you have to distribute something, let’s say chocolate, with your friends, you divide the chocolate bar into pieces to ease the distribution, right! Mathematics follows the same concepts. When we have to simplify a hard problem, the distributive property helps to break down the expression into a sum or a difference of 2 numbers.

Mathematically the distributive property states that any expression provided in the form K × (L + M) can be easily resolved as K × (L + M) = KL + KM. This is known as the distributive law of multiplication’s application in addition. Likewise, the distributive law also stands true for expression containing subtraction. This is expressed as K × (L – M) = KL – KM.

As you all can witness, K is being distributed to both the terms in addition or subtraction. Here K is known as an operand, and the terms inside the expression are known as addends.

Let us learn some important terms we have learned so far:

- Operand: The term being distributed is known as the operand.
- Addends: The terms inside the bracket which are either added or subtracted are known as addends.
- Distributive property of addition: K × (L + M) = KL + KM
- Distributive property of subtraction: K × (L – M) = KL – KM

We can visualize now that the distributive property states that when the operand is multiplied to the sum or difference to the addends, it is equal to the sum or difference of the individual product of operand and addend terms.

**Distributive Property Formula**

The formula for a given value’s distributive property can be stated as

c * ( a + b ) = ca + cb

This concludes all the theoretical aspects of the distributive property of multiplication. Next, let us look at the distributive law of multiplication over addition and subtraction in-depth with proper instances.

**Distributive Property of Addition**

When multiplying a number (operand) by the summation of two integers (addend), we use the distributive property of addition. Multiplying three by the sum of 10 + 8 is a good example. 3 x (10 + 8) is the mathematical expression for this.

**Example: **The distributive principle of addition may solve the formula 3 x (10 + 8).

**Solution:** Using the distributive property, we multiply each addend by three using the distributive property before solving the formula 3 x (10 + 8). After that, we may add the products by dividing the number 3 between the two addends. This signifies that the addition will take place before the multiplication of 3 (18) and 3 x (10) + 3 x (8) = 30 + 24 = 54 is the result of the distribution property of addition.

**Distributive Property of Subtraction**

Similarly, when multiplying a number (operand) by the difference of two integers (addend), we use the distributive property of subtraction. Multiplying three by the difference of 10 – 8 is a good example of subtraction’s distributive property. The mathematical expression for this equation is 3 x (10 – 8).

**Example: **The distributive principle of subtraction may be used to solve the formula 3 x (10 – 8).

**Solution:** Using the distributive property, we multiply each addend by three before solving the formula 3 x (10 – 8). After that, we may subtract the products by dividing the number 3 between the two addends. This signifies that the subtraction will take place before the multiplication of 3 x (18) and 3 x (10) – 3 x (8) = 30 – 24 = 6 is the result of the distributive property of subtraction.

We have talked so much about the distributive property, but how does it stand true in mathematics? Is there a way to verify this property? Indeed there is verification. Continue reading the article to know why.

**Verification of Distributive Property**

Let’s look at how the distributive property works for various operations. We’ll use the distributive law to apply the two basic operations of addition and subtraction separately.

**Distributive Property of Addition**: We already know that addition’s distributive property is given as k × (l + m) = kl + lm. Now it is time to verify this property by taking an example.

**Example:** Let us take an expression, say, 10 x ( 3 + 6).

**Solution:** We will normally solve this expression by using the rules of BODMAS as standard.

In the first step, we will always solve the expressions inside the bracket. In this case (3 + 6 ) = 9. In the second step, we will multiply 10 with the number obtained, i.e. 9. This will give us the result as 10 x 9 = 90.

Now solve this using the distributive property of addition:

10 x ( 3 + 6 ) = (10 x 3) + (10 x 6)

= 30 + 60

= 90

As we can see both the methods yield the same result.

**Distributive Property of Subtraction**: Now, let us verify the same for the distributive property of subtraction. We all already know that the distributive property of addition is given as k × (l – m) = kl – lm. Now it is time to verify this property by taking an example.

**Example:** Let us take an expression, say, 10 x (6 – 3).

**Solution:** We will normally solve this expression by using the rules of BODMAS as standard.

In the first step, we will always solve the expressions inside the bracket. In this case ( 6 – 3 ) = 3. In the second step, we will multiply 10 with the number obtained, i.e. 3. This will give us the result as 10 x 3 = 30.

Now solve this using the distributive property of addition:

10 x ( 6 – 3 ) = (10 x 6) – (10 x 3)

= 60 – 30

= 30

As we can see both the methods yield the same result again.

Hence, we have verified that the property of both addition and subtraction distribution is true.

**Distributive Property of Division**

The distributive property of division is the same as the distributive law of multiplication, with only the multiplication sign changing to division along with the operation. The larger term is divided into smaller factors (addend), and the divisor acts as the operand. You will understand this better with the example given below.

**Example:** Using the distributive property of division, solve 36 ÷ 12.

**Solution:** 36 can be written as 24 + 12

Therefore we can write 36 ÷ 12 = (24 + 12) ÷ 12

Now, let us distribute 12 inside the bracket

⇒ (24 ÷ 12) + (12 ÷ 12)

⇒ 2 + 1

This gives us the answer as 3.

**Distributive Property Examples**

**Example 1:** Solve the expression 2 (11 + 7) using the distributive property.

**Solution:**

Using the distributive property formula,

k × (l + m) = (k × l) + (k × m)

= (2 × 11) + (2 × 7)

= 22 + 14

= 36

Therefore, the value of 2 (11 + 7) = 36

**Example 2:** Prove that 5 x (3 – 12) has a negative result using the distributive property of multiplication.

**Solution:**

Using the distributive property formula,

k × (l – m) = (k × l) – (k × m)

= (5 × 3) – (5 × 12)

= 15 – 60

= – 45

Therefore, the value of 5 x (3 – 12) = – 45, which is a negative integer.

Now you must be 100 percent confident in what distributive property means and how to solve problems concerning this property. If you are not completely sure and have missed any of the concepts in the article, you can revisit this page again for theory and solutions. Moreover, start preparing for your upcoming exam now and outshine others by learning and practicing.