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

Solution:

(i) $103 \times 107=(100+3)(100+7)$

$=(100)^{2}+(3+7) 100+(3)(7)$

[By using the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$, where

$x=100, a=3$, and $b=7]$

$=10000+1000+21$

$=11021$

(ii) $95 \times 96=(100-5)(100-4)$

$=(100)^{2}+(-5-4) 100+(-5)(-4)$

[By using the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$, where

$x=100, a=-5$, and $b=-4]$

$=10000-900+20$

$=9120$

(iii) $104 \times 96=(100+4)(100-4)$

$=(100)^{2}-(4)^{2}\left[(x+y)(x-y)=x^{2}-y^{2}\right]$

$=10000-16$

$=9984$