What is the Distributive Property?
The distributive property states that when a number is multiplied by the sum or difference of two numbers, then, the first number can be distributed to both the numbers that are being added or subtracted. It is then multiplied by each of them separately. In the next step, we add or subtract the two products to obtain the same result as multiplying the first number by the sum or difference of two numbers.
The distributive law of multiplication over addition and subtraction is another name for the distributive property. By its very name, the operation implies that a value will be divided or distributed. Both addition and subtraction are subject to the distributive law.
According to the distributive property, an expression in the form a(b + c) can be re-written as a × (b + c) = ab + ac. This indicates that the other two operands share the operand ‘a’.
Formula for the Distributive Property over Addition
The formula for the distributive property over addition is written as,
a × (b + c) = a × b + a × c
Where, a, b and c are the operands or the numbers that are a part of the operation.
Formula for the Distributive Property over Subtraction
The formula for the distributive property over subtraction is written as,
a × (b – c) = a × b – a × c
With a, b and c as the values in this operation.
Solved Examples
Example 1: Solve the expression 14(2 + 4) using the distributive property of addition.
Solution:
The given expression is, 14(2 + 4)
Using the distributive property over addition,
a × (b + c) = a × b + a × c
14(2 + 4) = 14 × 2 + 14 × 4
14(2 + 4) = 28 + 56
14(2 + 4) = 84
Therefore the expression 14(2 + 4) when solved, is equal to 84.
Example 2: Use the distributive property of addition to rewrite the expression 9 (7 + 3) and find the solution.
Solution:
The expression in the question, 9 (7 + 3)
Using the distributive property over addition,
a × (b + c) = a × b + a × c
9 (7 + 3) = 9 × 7 + 9 × 3
9 (7 + 3) = 63 + 27
9 (7 + 3) = 90
Therefore the expression 9 (7+3) when solved is equal to 90.
Example 3: Solve the equation 7(x – 2) = 14, and find the value of ‘x’ using the distributive property.
Solution:
The equation in the question: 7(x – 2) = 14
Using the distributive property over subtraction,
7(x – 2) = 14
7 × x – 7 × 2 = 14
7x – 14 = 14
7x = 14 + 14 (Solve for x)
7x = 28
x = 4
Therefore the value of x is 4.
Yes, the distributive property applies to fractions and decimals as well.
The distributive property can be used to solve equations involving larger numbers. Larger numbers can be split into smaller numbers, making calculation easier, and then the distributive property can be applied to solve the equation.
The distributive property can be used with variables in the same way as with the numbers.
For example let us solve the equation 3(x + 1) = 21 using distributive property.
Here, 3(x + 1) = 21
3x + 3 = 21 Using distributive property
3x = 21 – 3 Solve for x
3x = 18 Simplify
x = 6 Simplify further
So we have found the value of the variable x using the distributive property.