Question

Evaluate the following products without multiplying directly

(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

Answer 1

$\mathbb{ANSWER}$ :

$\bf{(i) 103 \times 107 }$
$\boxed{\underline {Method 1}}$

(100 + 3) (100 + 7)
Now, by using identity
(x + a) (x + b) = x² + (a+b)*x + ab
So,
x = 100 , a = 3 , b = 7

= (100)² + (3+7)*100 + (3*7)
= 10000 + 1000 + 21
= 11021
.
$\boxed{\underline {Method 2}}$

(110 - 7) (110 - 3)
by using identity
(x + a) (x + b) = x² + (a+b)*x + ab
So,
x = 100 , a = (-7) , b = (-3)

= (110)² + { (-7) + (-3) }*110 + {(-7)*(-3)}
= 12100 + (-10)*110 + 21
= 21200 - 1100 + 21
= 11021

.
➖➖➖➖➖➖➖➖➖➖
.

$\bf{(ii) 95 \times 96}$
$\boxed{\underline {Method 1}}$

(90 + 5) (90 + 6)
by using identity
(x + a) (x + b) = x² + (a+b)*x + ab
So,
x = 90 , a = 5 , b = 6

= (90)² + (5+6)*90 + (5*6)
= 8100 + 990 + 30
= 9120
.

$\boxed{\underline {Method 2}}$

(100 - 5) (100 - 4)
by using identity
(x + a) (x + b) = x² + (a+b)*x + ab
So,
x = 100 , a = (-5) , b = (-4)

= (100)² + { (-5) + (-4) }*100 + 20
= 10000 + (-9)*100 + 20
= 10000 - 9000 + 20
= 10020 - 900
= 9120

.
➖➖➖➖➖➖➖➖➖➖
.

$\bf{(iii) 104 \times 96}$
$\boxed{\underline {Method 1}}$

(100 + 4) (100 - 4)
by using identity
(x + a) (x + b) = x² + (a+b)*x + ab
So,
x = 100 , a = 4 , b = (-4)

= (100)² + { 4 + (-4) }*100 + 4*(-4)
= 10000 + (4 - 4)*100 - 16
= 10000 + 0*100 - 16
= 10000 - 16
= 9984
.

$\boxed{\underline {Method 2}}$

(90 + 14) (90 + 6)
by using identity
(x + a) (x + b) = x² + (a+b)*x + ab
So,
x = 90 , a = 14 , b = 6

= (90)² + (14 + 6)*90 + (14*6)
= 8100 + 20*90 + 84
= 8100 + 1800 + 84
= 9984

Answer 2

Here is your solution

Evaluate the following products without multiplying directly

(i)103×107 

(100 + 3) (100 + 7)

Now, by using identity :-

(x + a) (x + b) = x² + (a+b)*x + ab

x = 100 , a = 3 , b = 7

Now put value of x , a & b

=> (100)² + (3+7)*100 + (3*7)

=> 10000 + 1000 + 21

=> 11021

(ii)95×96 

(90 + 5) (90 + 6)

by using identity :-

(x + a) (x + b) = x² + (a+b)*x + ab

x = 90 , a = 5 , b = 6

Now put value of x , a & b

=> (90)² + (5+6)*90 + (5*6)

=> 8100 + 990 + 30

=> 9120

(iii)104×96 

(100 + 4) (100 - 4)

by using identity :-

(x + a) (x + b) = x² + (a+b)*x + ab

x = 100 , a = 4 , b = (-4)

now put value of x , a & b

=> (100)² + { 4 + (-4) }*100 + 4*(-4)

=> 10000 + (4 - 4)*100 - 16

=> 10000 + 0*100 - 16

=> 10000 - 16

=> 9984

hope it helps you