Question

Simplify the following:
(i) $ 706 \times 694 $
(ii) $ 72 \times 68 $
(iii) $ 101 \times 99 $
(iv) $ 73 \times 67 $
 (v) $ 1010 \times 990 $
(vi) $ 91 \times 89 $

Answer 1

Hint: We are given 6 multiplications. So write one term of the multiplication in the form of $ \left( {a + b} \right) $ and another term of the multiplication in the form of $ \left( {a - b} \right) $ . So now the product of these two terms will be $ \left( {a + b} \right)\left( {a - b} \right) $ which is $ {a^2} - {b^2} $ . Use this info to simplify the given multiplications.

Complete step-by-step answer:
(i) $ 706 \times 694 $
706 can also be written as $ 700 + 6 $
694 can also be written as $ 700 – 6 $
Then $ 706 \times 694 $ will be $ \left( {700 + 6} \right)\left( {700 - 6} \right) $
It is in the form of $ \left( {a + b} \right)\left( {a - b} \right) $ which is equal to $ {a^2} - {b^2} $
 $ \Rightarrow \left( {700 + 6} \right)\left( {700 - 6} \right) = {700^2} - {6^2} = 490000 - 36 = 489964 $
 $ 706 \times 694 $ is equal to 489964.
So, the correct answer is “ 489964”.

(ii) $ 72 \times 68 $
72 can also be written as $ 70 + 2 $
68 can also be written as $ 70 - $
Then $ 72 \times 68 $ will be $ \left( {70 + 2} \right)\left( {70 - 2} \right) $
It is in the form of $ \left( {a + b} \right)\left( {a - b} \right) $ which is equal to $ {a^2} - {b^2} $
 $ \Rightarrow \left( {70 + 2} \right)\left( {70 - 2} \right) = {70^2} - {2^2} = 4900 - 4 = 4896 $
 $ 72 \times 68 $ is equal to 4896.
So, the correct answer is “ 4896.”.

(iii) $ 101 \times 99 $
101 can also be written as $ 100 + 1 $
99 can also be written as $ 100 – 1 $
Then $ 101 \times 99 $ will be $ \left( {100 + 1} \right)\left( {100 - 1} \right) $
It is in the form of $ \left( {a + b} \right)\left( {a - b} \right) $ which is equal to $ {a^2} - {b^2} $
 $ \Rightarrow \left( {100 + 1} \right)\left( {100 - 1} \right) = {100^2} - {1^2} = 10000 - 1 = 9999 $
 $ 101 \times 99 $ is equal to 9999.
So, the correct answer is “9999.”.

(iv) $ 73 \times 67 $
73 can also be written as $ 70 + 3 $
67 can also be written as $ 70 – 3 $
Then $ 73 \times 67 $ will be $ \left( {70 + 3} \right)\left( {70 - 3} \right) $
It is in the form of $ \left( {a + b} \right)\left( {a - b} \right) $ which is equal to $ {a^2} - {b^2} $
 $ \Rightarrow \left( {70 + 3} \right)\left( {70 - 3} \right) = {70^2} - {3^2} = 4900 - 9 = 4891 $
 $ 73 \times 67 $ is equal to 4891.
So, the correct answer is “4891”.

(v) $ 1010 \times 990 $
1010 can also be written as $ 1000 + 10 $
990 can also be written as $ 1000 – 10 $
Then $ 1010 \times 990 $ will be $ \left( {1000 + 10} \right)\left( {1000 - 10} \right) $
It is in the form of $ \left( {a + b} \right)\left( {a - b} \right) $ which is equal to $ {a^2} - {b^2} $
 $ \Rightarrow \left( {1000 + 10} \right)\left( {1000 - 10} \right) = {1000^2} - {10^2} = 4900 - 9 = 999900 $
 $ 1010 \times 990 $ is equal to 999900.
So, the correct answer is “999900.”.

(vi) $ 91 \times 89 $
91 can also be written as $ 90 + 1 $
89 can also be written as $ 90 – 1 $
Then $ 91 \times 89 $ will be $ \left( {90 + 1} \right)\left( {90 - 1} \right) $
It is in the form of $ \left( {a + b} \right)\left( {a - b} \right) $ which is equal to $ {a^2} - {b^2} $
 $ \Rightarrow \left( {90 + 1} \right)\left( {90 - 1} \right) = {90^2} - {1^2} = 8100 - 1 = 8099 $
 $ 91 \times 89 $ is equal to 8099.
So, the correct answer is “8099”.

Note: We can only apply $ {a^2} - {b^2} $ when the value of ‘a’ and ‘b’ of one term is the same as the value of ‘a’ and ‘b’ of another term. Here in every multiplication, the values of a and b of both the terms are the same, so we have applied the formula. For example, $ 75 \times 63 $ is given which can also be written as $ \left( {70 + 5} \right)\left( {70 - 7} \right) $ , as we can see the value of a of both the terms is same, but the value of b is different. Hence the formula cannot be applied to simplify it.