Question

Evaluate the following products without actual multiplication:
\[50\dfrac{1}{2} \times 49\dfrac{1}{2}\]?

Answer 1

Hint: Here in the above question, we are provided with a expression in which the term given are in mixed fraction, and needed to be multiplied without doing actual multiplication, here we need to use the expansion of individual term and then solve further, for which we need to write the numbers by rearranging them.
Formulae Used: For solving mixed fraction we use:
\[ \Rightarrow a\dfrac{b}{c} = \dfrac{{\left( {a \times c} \right) + b}}{c}\]

Complete step by step answer:
Here in the given question we first need to solve for the mixed fraction, and then by rearranging the numbers we can solve further in order to solve for the expression given, on solving we get:
\[
   \Rightarrow 50\dfrac{1}{2} \times 49\dfrac{1}{2} \\
   \Rightarrow \dfrac{{100 + 1}}{2} \times \dfrac{{98 + 1}}{2} \\
   \Rightarrow \dfrac{{101}}{2} \times \dfrac{{99}}{2} \\
   \Rightarrow \left( {\dfrac{{100}}{2} + \dfrac{1}{2}} \right)\left( {\dfrac{{100}}{2} - \dfrac{1}{2}} \right) \\
   \Rightarrow \dfrac{{100}}{2}\left( {\dfrac{{100}}{2} - \dfrac{1}{2}} \right) + \dfrac{1}{2}\left( {\dfrac{{100}}{2} - \dfrac{1}{2}} \right) \\
   \Rightarrow {\left( {\dfrac{{100}}{2}} \right)^2} - \dfrac{{100}}{2} \times \dfrac{1}{2} + \dfrac{1}{2} \times \dfrac{{100}}{2} - {\left( {\dfrac{1}{2}} \right)^2} \\
   \Rightarrow \dfrac{{10000}}{4} - \dfrac{1}{4} \\
   \Rightarrow \dfrac{{10000 - 1}}{4} = \dfrac{{9999}}{4} \\
 \]
Here we get the solution for the given expression, without actually multiplying the given terms.

Note: Here the given question is to solve for the multiplication of two big numbers, where we are restricted to multiply directly, hence we break the number such that on solving brackets some of the terms get eliminated and the rest can be solved easily.