Question

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

Answer 1

Since, we have an identity: $(x+a)(x+b)=x^2+(a+b)x+ab$.........(1)
(i) Given $103\times 107$
$=(100+3)(100+7)$
Substitute $x=100,a=3$ and $b=7$ in equation(1), we get
$(100+3)(100+7)={100}^2+(3+7)\left(100\right)+(3\times 7)$
$=10000+1000+21$
$=11021$

(ii) Given $95\times 96$
$=(100-5)(100-4)$
$=(100+(-5\left)\right)(100+(-4\left)\right)$

Substitute $x=100$, $a=-5$ and $b=-4$ in equation(1), we get

$\Rightarrow (100+(-5\left)\right)(100+(-4\left)\right)={\left(100\right)}^2+(-5-4)\left(100\right)+(-5)(-4)$

$=10000-900+20$
$=9120$

(iii) Given $104\times 96$

$=(100+4)(100-4)$

Here, we can use an identity: (a+b)(a−b)=$a^2$−$b^2$ .......(2)
Putting values in $a=100$ and $b=4$ in equation (2),$(100+4)(100-4)={\left(100\right)}^2$−$4^2$
$=10000-16$
$=9984$