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

Last updated date: 26th Feb 2024
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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.

(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.