About Distributive Property of Multiplication
What is the Distributive Property?
The distributive property is used to multiply a single value by two or more values enclosed in a set of parentheses. The distributive property states, ‘multiply each of the numbers inside the parentheses by the number outside the parentheses’, meaning one expression is ‘divided into’ or ‘distributed across’ two or more parts.
The distributive property is applied to addition and subtraction. In case the negative numbers do not affect the distributive property.
Distributive property over addition:
a \(\times\) ( b + c) = a \(\times\) b + a \(\times\) c
Distributive property over Subtraction:
a \(\times\) (b – c) = a \(\times\) b – a \(\times\) c
Verification:
a \(\times\) (b + c) = a \(\times\) b + a \(\times\) c [Distributive property over addition]
Consider, a = 5, b = 6 and c = 7
First take the left hand side of the equation and substitute the values.
a \(\times\) (b + c)
= 5 \(\times\) (6 + 7)
= 5 \(\times\) 13
= 65
Now substitute the values on the right hand side of the equation.
a \(\times\) b + a \(\times\) c
= 5 \(\times\) 6 + 5 \(\times\) 7
= 30 + 35
= 65
Hence it can be seen that the LHS = RHS.
So, using the values we have verified the distributive property.
Rapid Recall
Solved Examples
Example 1: Simplify the following expressions using the distributive property.
a) 9(4+8) b) -6 (8 – 2)
Solution:
a) 9(4+8)
According to the distributive property,
a (b + c) = ab + ac [Write the distributive property expression]
9(4 + 8) = 9 \(\times\) 4 + 9 \(\times\) 8 [Apply the distributive property]
= 36 + 72 [Simplify]
= 108
So, 9(4 + 8) = 108.
b) – 6(8 – 2)
According to the distributive property,
a (b – c) = ab – ac [Write the distributive property expression]
= (- 6) \(\times\) 8 – (- 6) \(\times\) 2 [Apply the distributive property]
= – 48 + 12 [Simplify]
= – 36
So, – 6(8 – 2) = – 36.
Example 2: Prove ‘ a (b+ c) = ab + ac’ for the expression 13(6706+570).
Solution:
Substitute the values in LHS
a (b + c)
= 13(670 + 570) [Apply PEMDAS]
= 13(1240) [Simplify]
= 16120
Substitute the values in RHS
13 \(\times\) 670 + 13 \(\times\) 570 [Apply the distributive property]
= 8710 + 7410 [Simplify]
= 16120
So, the LHS = RHS
Hence, the expression a (b + c) = ab + ac is verified.
Example 3: Edward was asked to take out 20 chocolates from a total of 60 and keep the rest with him. His friend Ruby gave him a batch of 500 chocolates. Edward’s mother then asked, how many chocolates would he need if he was to distribute the number of chocolates he has now to 10 people from Ruby’s batch. Depict the scenario in the form of an equation and find the number of chocolates remaining from Ruby’s batch.
Solution:
As per the scenario, Edward has to find out the solution to 60 minus 20. This number has to be multiplied by 10. The expression representing the scenario would look like the following.
10(60 – 20)
As per the distributive property,
a (b – c) = ab – ac [Write the distributive property expression]
= 10 \(\times\) 60 – 10 \(\times\) 20 [Apply the distributive property]
= 600 – 200 [Simplify]
= 400
So, Edward needed 400 chocolates to distribute to 10 people, and he would be left with 500 – 400 = 100 chocolates from Ruby’s batch.
The distributive property can only be applied to addition and subtraction.
To divide a complex expression into two or more expressions, we use the distributive property.
The distributive property is extensively used in concepts like algebra, where variables are part of the problem along with numbers.