Question

Evaluate the following products without multiplying directly:
(i) $103 \times 107$
(ii) $95 \times 96$
(iii) $104 \times 96$

Answer 1

According to the given question, we will convert the numbers in the equations in the form of $\left( {100 - x} \right)$ or $\left( {100 + x} \right)$ .Then we will use different algebraic identities to get the desired result.
Formula used:
Here, we use the algebraic identities that is $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ and $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$ .

Complete step by step solution:
(i) $103 \times 107$
Firstly we will rewrite both numbers in the form of $\left( {100 - x} \right)$ or $\left( {100 + x} \right)$ depending on what is the best approach for that number.
$\Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right)$
Here, we will use the algebraic identity $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$
Putting $a = 3$ , $b = 7$ and $x = 100$ .
On substituting the values in the identity we get,
$\Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right) = {\left( {100} \right)^2} + \left( {3 + 7} \right)100 + 3 \times 7$
After opening the squares and simplifying the above equation of right hand side we get,
$\Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right) = 10000 + 1000 + 21$
On adding the right hand side we get,
$\Rightarrow \left( {100 + 3} \right)\left( {100 + 7} \right) = 11021$
Hence, $103 \times 107 = 11021$
(ii) $95 \times 96$
Firstly we will rewrite both numbers in the form of $\left( {100 - x} \right)$ or $\left( {100 + x} \right)$ depending on what is the best approach for that number.
$\Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right)$
Here, we will use the algebraic identity $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$
Putting $a = - 5$ , $b = - 4$ and $x = 100$ .
On substituting the values in the identity we get,

\right)100 + \left( { - 5 \times - 4} \right)$$ After opening the squares and simplifying the above equation of right hand side we get, $$ \Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right) = 10000 - 900 + 20$$ After solving the right hand side we get, $$ \Rightarrow \left( {100 - 5} \right)\left( {100 - 4} \right) = 9120$$ Hence, $$95 \times 96 = 9120$$ (iii) $$104 \times 96$$ Firstly we will rewrite both numbers in the form of $$\left( {100 - x} \right)$$ or $$\left( {100 + x} \right)$$ depending on what is the best approach for that number. $$ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right)$$ Here, we will use the algebraic identity $$\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$$ Putting $$a = 100$$ and $$b = 4$$ . On substituting the values in the identity we get, $$ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right) = {\left( {100} \right)^2} - {\left( 4 \right)^2}$$ Now we will open the squares and simplify on right hand side we get, $$ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right) = 10000 - 16$$ After solving the right hand side we get, $$ \Rightarrow \left( {100 + 4} \right)\left( {100 - 4} \right) = 9984$$ **Hence, $$104 \times 96 = 9984$$** **Note:** To solve these types of questions, you must remember the algebraic identities and convert the equations according to the requirement of the identity. So, carefully observe the value of a and b while substituting in the formula.