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. This formula is also known as the distributive property of addition over multiplication. Let us understand the distributive property using solved examples.
What Is Distributive Property?
The distributive property states that any expression with three numbers A, B, and C, given in form A (B + C) then it is resolved as A × (B + C) = AB + AC or A (B – C) = AB – AC. This means operand A is distributed among the other two operands. This property is also known as the distributivity of multiplication over addition or subtraction.
The distributive property formula of a given value can thus be expressed as,
Let us discuss in detail the distributive property of addition over multiplication with examples.
Distributive Property of Multiplication Over Addition
When we are required to multiply a respective number by a sum of two numbers in such a case the distributive property of multiplication over addition is applied. For example, let us multiply 7 by the sum of 20 + 3. Mathematically we can represent this as 7(20+3). Going by the rules of order of operations we first solve the sum within the parentheses and then multiply the value by 7.
7(20 + 3) = 7(23) = 161. If we solve the expression using the distributive property, we can 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.
The multiplication of 7(20) and 7(3) will be performed before the addition.
7(20) + 7(3) = 140 + 21 = 161
We can observe that the obtained result in both cases is the same before as well after.
Distributive Property of Multiplication Over Subtraction
In the above section, we covered the distributive property of multiplication over addition. In this section, we will talk about subtraction. There will be no such difference in the process other than a sign. Now, let us consider an example of the distributive property of multiplication over subtraction. Suppose we have to multiply 7 with a difference of 20 and 3, i.e. 7(20 – 3).
Let us use the two different approaches to solve the same.
Method 1: 7 × (20 – 3) = 7 × 17 = 119
Method 2: 7 × (20 – 3) = (7 × 20) – (7 × 3) = 140 – 21 = 119
In both the methods the final result is the same.
Verification of Distributive Property
Let us try to justify how distributive property works for different operations. We will apply the distributive property law individually on the three basic operations, i.e., addition, subtraction, and division.
Distributive Property of Addition: The general distributive property law for addition is expressed as A × (B + C) = AB + AC. Let us try to fix some numbers in the property to verify the same. For example,
⇒ 2(1 + 4) = 2×1 + 2× 8
⇒ 10 = 10.
LHS = RHS.
Distributive Property of Subtraction: The general distributive property law for subtraction is expressed as A × (B  C) = AB  AC. Let us try to fix some numbers in the property to verify the same. For example,
⇒ 2(4  1) = 2×4  2×1
⇒ 6 = 6.
LHS = RHS.
Distributive Property of Division: We can show the division of larger numbers using the distributive property simply by breaking the respective larger number into two or more smaller factors. Let us consider an example for the same. Divide 24 ÷ 6.
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;
⇒ 24 ÷ 6 = (18÷6) + (6÷6)
⇒ 4 = 3 + 1
Therefore, 4 = 4
LHS = RHS
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Let us take a look at a few examples to better understand the distributive property.
Examples of Distributive Property

Example 1: Solve 3(4 + 5) by distributive property.
Solution:
Using the distributive property formula,
a × (b + c) = a × b + a × c
Multiply the outside term by both the terms inside the parenthesis, we get,
= 3 × 4 + 3 × 5
= 12 + 15
= 27
Therefore, the value of 3(4 +5) = 27.

Example 2: Solve (3+2)(4+5) by distributive property formula.
Solution:
Using distributive property formula,
(a+b)(c+d) = ac + ad + bc + bd
Multiply the outside term by both the terms inside the parenthesis, we get,
= (3 + 2) × 4 + (3 + 2) × 5
= 3 × 4 + 2 × 4 + 3 × 5 + 2 × 5
= 12 + 8 + 15 + 10
= 45
Therefore, the solution of (3 + 2)(4 + 5) is 45.

Example 3: Solve 10(12 + 15) by distributive property formula.
Solution:
Using the distributive property formula,
a × (b + c) = a × b + a × c
Multiply the outside term by both the terms inside the parenthesis, we get,
= 10 × 12 + 10 × 15
= 120 + 150
= 270
Therefore, the value of 10(12 + 15) = 270
FAQs on Distributive Property
What is Meant by Distributive Property in Math?
The distributive property formula is also known as the distributive law of multiplication. The name itself signifies that the operation includes dividing or distributing something. This formula is also known as the distributive property of addition over multiplication. The formula to calculate the distributive property of an equation is a × (b + c) = a × b + a × c or (a + b)(c + d) = ac + ad + bc + bd
What is the Formula to Find the Distributive Property of an Equation?
The formula for distributive property can be written as,
a × (b + c) = a × b + a × c
or,
(a + b)(c + d) = ac + ad + bc + bd
where, a, b, c, and d are operands
How to Calculate the Equation Using the Distributive Property?
The distributive property formulas are a (b + c) = ab +ac and (a + b)(c + d) = ac + ad + bc + bd.
Expand the terms, remove the brackets by multiplying the outside terms with the terms inside, add the like terms and simplify.
For example, 5(y + 2) + y
Applying distributive property, 5y + 10 + y = 6y + 10
Using the Distributive Property Formula, Find the Value of (14+20)(10+12)
Using distributive property formula, (a+b)(c+d) = ac + ad + bc + bd
Multiply the outside term by both the terms inside the parenthesis, we get,
= (14 + 20) × 10 + (14 + 20) × 12
= 14 × 10 + 20 × 10 + 14 × 12 + 20 × 12
= 140 + 200 + 168 + 240
= 748
Therefore, the solution of (14+20)(10+12) is 748.
What is the Distributive Property of Multiplication in Math?
In Math, the distributive property of multiplication is described as when we multiply a number with the sum of two or more addends or minuends, we get a result that is equal to the result that is obtained when we multiply each addend or minuend separately by the number. The distributive property of multiplication applies to the sum and the difference 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. This property is also known as the distributivity of multiplication over addition. This property is also known as the distributivity of multiplication over subtraction p × (q  r) = (p × q)  (p × r). 1/3(1/2  1/5) = 1/3 × 1/2  1/3 × 1/5 = 1/10.
How the Distributive Property is useful?
The distributive property essential property in math 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.
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