Simplify using distributive property: - $\left( 126\times 33 \right)+\left( 126\times 67 \right)$?
(a) 1260
(b) 12600
(c) 126000
(d) 1000

Hint: Here, first we will understand the distributive property by considering three numbers a, b and c and see the process by which any expression is simplified using this property. Now, we will compare 126 with ‘a’, 33 with ‘b’ and 67 with ‘c’. We will take 126 common from both the terms and add 33 and 67 whose product we will finally take with 126 to get the answer.

Complete step-by-step solution:
Here, we have been provided with the expression $\left( 126\times 33 \right)+\left( 126\times 67 \right)$ and we are asked to simplify it using the distributive property. But first we need to know about the distributive property of numbers.
Now, let us assume that there are three numbers a, b and c. So, the distributive property states that if we take the sum of any two numbers, say b and c, and then multiply it with a then the result will be equivalent to the sum of product of a, b and a, c. It can be represented mathematically as: -
\[\Rightarrow a\times \left( b+c \right)=a\times b+a\times c\]
Similarly we have,
\[\begin{align}
  & \Rightarrow b\times \left( a+c \right)=a\times b+b\times c \\
 & \Rightarrow c\times \left( a+b \right)=a\times c+b\times c \\
\end{align}\]
Let us come to the question so we can compare 126 with ‘a’, 33 with ‘b’ and 67 with ‘c’, so we have,
$\begin{align}
  & \Rightarrow \left( 126\times 33 \right)+\left( 126\times 67 \right)=\left( a\times b \right)+\left( a\times c \right) \\
 & \Rightarrow \left( 126\times 33 \right)+\left( 126\times 67 \right)=a\times \left( b+c \right) \\
\end{align}$
So substituting the values we get,
$\begin{align}
  & \Rightarrow \left( 126\times 33 \right)+\left( 126\times 67 \right)=126\times \left( 33+67 \right) \\
 & \Rightarrow \left( 126\times 33 \right)+\left( 126\times 67 \right)=126\times 100 \\
 & \therefore \left( 126\times 33 \right)+\left( 126\times 67 \right)=12600 \\
\end{align}$
Hence, option (b) is the correct answer.

Note: One must remember three types of properties namely: - distributive property, commutative property and associative property which are performed on basic arithmetic operations like: - addition, subtraction, multiplication and division. In the above question we were performing the distributive property over addition. This property is followed because it will be difficult to multiply 126 with 33 and then with 67 on by one and therefore the property helps us to reduce the calculations.