Question

The value of \[106 \times 106 - 94 \times 94\] is
a) \[2400\]
b) \[2000\]
c) \[1904\]
d) \[1906\]
e) \[None\,of\,these\]

Answer 1

Hint: We are given to find the difference of products' big numbers. Solving it normally by first multiplying it and then subtracting it will take time. When we look at it carefully, we find that it is of the form \[{a^2} - {b^2}\]. We know how to solve such forms. Using this identity we can also solve the above given problem.
Formula used: For any two numbers \[a\] and \[b\], given as \[{a^2} - {b^2}\], be can solve them as,
\[{a^2} - {b^2} = (a + b)(a - b)\]

Complete step-by-step solution:
We are given the problem \[106 \times 106 - 94 \times 94\],
To solve such problems we can always solve them by usually multiplying it first and then subtracting the product. But since the numbers given are quite big, they may take time to solve. So we can also solve it by the below given method.
We see that the question is of the form \[{a^2} - {b^2}\]. Where \[a = 106\] and \[b = 94\]. We know there is an identity that \[{a^2} - {b^2} = (a + b)(a - b)\].
Using this identity we move ahead as
\[
  106 \times 106 - 94 \times 94 = {106^2} - {94^2} \\
   \Rightarrow 106 \times 106 - 94 \times 94 = (106 + 94)(106 - 94) \\
   \Rightarrow 106 \times 106 - 94 \times 94 = 200 \times 12 \\
   \Rightarrow 106 \times 106 - 94 \times 94 = 2400 \]
Thus the result of the problem is \[2400\] .
Hence the answer to the given question is a).

Note: While solving such questions which involve the multiplication or division of large numbers, we should always try to carefully see if we can use any identity or formula which can be used to simplify them. This can save us a lot of time and energy. It can also be used in equations and expressions and can also help us to simplify the complex equations.