Question

Evaluate: $203 \times 197$
A) $39991$
B) $39891$
C) $39981$
D) $38981$

Answer 1

Hint:
Observe the numbers. We can write these numbers as sum and difference using the number $200$. That is, $200 + 3$ and $200 - 3$. Then we can apply the result of $(a + b)(a - b)$. Simplifying we get the answer.

Useful formula:
For any two numbers $a,b$ we have,
$(a + b)(a - b) = {a^2} - {b^2}$

Complete step by step solution:
We are asked to find the product, $203 \times 197$.
We can see the speciality or connection of these numbers
We have,
$203 = 200 + 3$ and $197 = 200 - 3$
This can be substituted instead.
This gives $203 \times 197 = (200 + 3)(200 - 3)$ _______ (i)
Here we can use another result to see the product.
We have, for any $a,b$,
$(a + b)(a - b) = {a^2} - {b^2}$
So, let $a = 200,b = 3$
We get, $(200 + 3)(200 - 3) = {200^2} - {3^2}$
Substituting this in (i) we have,
$203 \times 197 = {200^2} - {3^2}$
We have, ${200^2} = 40000$ and ${3^2} = 9$
Substituting we get,
$203 \times 197 = 40000 - 9$
$203 \times 197 = 39991$

Therefore the answer is option A.

Note:
To find the product of two numbers, we can use another method.
$203 \times 197$ can be written as $203 \times 100 + 203 \times 90 + 203 \times 7$
So we get,
$203 \times 197 = 20300 + 18270 + 1421$
$ \Rightarrow 203 \times 197 = 39991$
This is difficult to calculate. But for random numbers we can only use this method.
But in our problem the numbers are in the form of \[(a + b)(a - b)\].
So the calculation was easy using the result.
Here we had one more advantage. The numbers $200$ and $3$ were easy to find squares also. If not so, this method would have been complicated.