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Distributive Property
The distributive property is also known as the distributive law of multiplication over addition and subtraction. The name itself signifies that the operation includes dividing or distributing something. The distributive law is applicable to addition and subtraction. Let us learn more about the distributive property of multiplication along with some distributive property examples, how to use the distrivutive property on this page.
What is the Distributive Property?
The distributive property states that an expression which is given in form of A (B + C) can be solved as A × (B + C) = AB + AC. This distributive law is also applicable to subtraction and is expressed as, A (B  C) = AB  AC. This means operand A is distributed between the other two operands.
Distributive Property Definition
According to the distributive property definition, the distributive property allows us to take a factor and distribute it to each member (term) of the group of things that have been added or subtracted. Instead of multiplying the factor by the group as a whole, we can distribute it to be multiplied by each member (term) of the group individually.
Distributive Property Formula
The distributive property formula of a given value is expressed as,
Let us discuss the distributive property of multiplication over addition and subtraction in detail with examples.
Distributive Property of Multiplication Over Addition
The distributive property of multiplication over addition is applied when we need to multiply a number by the sum of two numbers. For example, let us multiply 7 by the sum of 20 + 3. Mathematically we can represent this as 7(20 + 3).
Example: Solve the expression 7(20 + 3) using the distributive property of multiplication over addition.
Solution: When we solve the expression 7(20 + 3) using the distributive property, we first multiply every addend by 7. This is known as distributing the number 7 amongst the two addends and then we can add the products. This means that the multiplication of 7(20) and 7(3) will be performed before the addition. This leads to 7(20) + 7(3) = 140 + 21 = 161.
Distributive Property of Multiplication Over Subtraction
The distributive property of multiplication over subtraction is similar to the distributive property of multiplication over addition except for the operation of addition and subtraction. Let us consider an example of the distributive property of multiplication over subtraction.
Example: Solve the expression 7(20  3) using the distributive property of multiplication over subtraction.
Solution: Using the distributive property of multiplication, we can solve the expression as follows: 7 × (20  3) = (7 × 20)  (7 × 3) = 140  21 = 119
Verification of Distributive Property
Let us try to justify how distributive property works for different operations. We will apply the distributive law individually on the two basic operations, i.e., addition and subtraction.
Distributive Property of Addition: The distributive property of multiplication over addition is expressed as A × (B + C) = AB + AC. Let us verify this property with the help of an example.
Example: Solve the expression 2(1 + 4) using the distributive law of multiplication over addition.
Solution: 2(1 + 4) = (2 × 1) + (2 × 4)
⇒ 2 + 8 = 10
Now, if we try to solve the expression using the law of BODMAS, we will solve it as follows. First, we will add the numbers given in brackets, and then we will multiply this sum with the number given outside the brackets. This means, 2(1 + 4) ⇒ 2 × 5 = 10. Therefore, both the methods result in the same answer.
Distributive Property of Subtraction: The distributive law of multiplication over subtraction is expressed as A × (B  C) = AB  AC. Let us verify this with the help of an example.
Example: Solve the expression 2(4  1) using the distributive law of multiplication over subtraction.
Solution: 2(4  1) = (2 × 4)  (2 × 1)
⇒ 8  2 = 6
Now, if we try to solve the expression using the order of operations, we will solve it as follows. First, we will subtract the numbers given in brackets, and then we will multiply this difference with the number given outside the brackets. This means 2(4  1) ⇒ 2 × 3 = 6. Since both the methods result in the same answer, this distributive law of subtraction is verified.
Distributive Property of Division
We can show the division of larger numbers using the distributive property by breaking the larger number into two or more smaller factors. Let us understand this with an example.
Example: Divide 24 ÷ 6 using the distributive property of division.
Solution: We can write 24 as 18 + 6
24 ÷ 6 = (18 + 6) ÷ 6
Now, let us distribute the division operation for each factor (18 and 6) in the bracket.
⇒ (18 ÷ 6) + (6 ÷ 6)
⇒ 3 + 1
Therefore, the answer is 4.
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 Distributive Property of Multiplication
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Distributive Property Examples

Example 1: Solve the expression 3(4 + 5) by using the distributive property.
Solution:
Using the distributive property formula,
a × (b + c) = (a × b) + (a × c)
We will multiply the outside term by both the terms inside the brackets.
3(4 + 5)
= (3 × 4) + (3 × 5)
= 12 + 15
= 27
Therefore, the value of 3(4 + 5) = 27

Example 2: Solve 6(7 + 9) by using the distributive property formula.
Solution:
The distributive property formula is expressed as,
a × (b + c) = (a × b) + (a × c)
Now, let us multiply the outside term by both the terms inside the brackets.
= 6(7 + 9)
= (6 × 7) + (6 × 9)
= 42 + 54
= 96
Therefore, the solution of 6(7 + 9) is 96.

Example 3: Solve using distributive property formula: 10(12 + 15)
Solution:
The distributive property formula is expressed as,
a × (b + c) = (a × b) + (a × c)
Let us multiply the outside term by both the terms inside the parenthesis,
10(12 + 15)
= (10 × 12) + (10 × 15)
= 120 + 150
= 270
Therefore, the value of 10(12 + 15) = 270
FAQs on Distributive Property
What is the Distributive Property in Math?
The distributive property is also known as the distributive law of multiplication. This distributive property of multiplication is applicable over addition and subtraction. The formula for the distributive property is expressed as, a × (b + c) = (a × b) + (a × c).
What is the Formula for Distributive Property?
The formula for the distributive property is expressed as, a × (b + c) = (a × b) + (a × c); where, a, b, and c are the operands. Here, the number outside the brackets is multiplied with each term inside the brackets and then the products are added.
How Does the Distributive Property Work?
When we use the distributive property formula, we multiply the outside term with the terms inside the brackets and then add the terms to get the solution. For example, let us solve 15(4 + 3). First, we will multiply 15 with 4, then multiply 15 with 3, and then add the products to get the answer. This means 15 × (4 + 3) = (15 × 4) + (15 × 3) = 60 + 45 = 105.
Use the Distributive Property Formula to Solve the Equation 2(m + 2) = 22.
Using the distributive property formula, a × (b + c) = (a × b) + (a × c), we will multiply the outside term by both the terms inside the brackets. This means 2(m + 2) = 22 ⇒ 2m + 4 = 22. Now, the value of 'm' can be calculated. That is, 2m = 22  4 which can be further solved as, m = 9.
What is the Distributive Property of Multiplication in Math?
The distributive property of multiplication is used when we need to multiply a number with the sum of two or more addends. The distributive property of multiplication is applicable to addition and subtraction of two or more numbers. It is used to solve expressions easily by distributing a number to the numbers given in brackets. For example, if we apply the distributive property of multiplication to solve the expression: 4(2 + 4), we would solve it in the following way: 4(2 + 4) = (4 × 2) + (4 × 4) = 8 + 16 = 24.
What is the Distributive Property for Rational Numbers?
The distributive property states, if p, q, and r are three rational numbers, then the relation between the three is given as, p × (q + r) = (p × q) + (p × r). For example, 1/3(1/2 + 1/5) = (1/3 × 1/2) + (1/3 × 1/5) = 7/30.
Where is the Distributive Property is Used?
The distributive property is used while adding, subtracting, multiplying, and dividing large numbers. By grouping the numbers we can create smaller parts irrespective of the order to solve the bigger equations. It makes calculations easier and faster.
How to Use the Distributive Property with Variables?
The distributive property is applied to variables in the same way as it is done for numbers. For example, let us find the value of 'x' in the equation 4(x  3) = 8 using the distributive property. We will first multiply 4 with x and then with 3. This means, 4(x  3) = 8 ⇒ 4x + 12 = 8. So, the value of x = 1.
How to Use Distributive Property with Fractions?
The distributive property is applicable to fractions in a similar way as it is used for numbers and variables. For example, let us solve the expression, 1/3(2/6 + 4/6) using the distributive property. We will first multiply 1/3 with 2/6 and then with 4/6. This means, 1/3(2/6 + 4/6) ⇒ (1/3 × 2/6) + (1/3 × 4/6) = 2/18 + 4/18 = 6/18 = 1/3.
What is the Distributive Property of Addition?
The distributive property of addition is another name for the distributive property of multiplication over addition. This is expressed as, a × (b + c) = (a × b) + (a × c).
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