Question

# Using the following formula evaluate: $103\times 97$$(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$

Hint: We will consider variables a and b and we will take a+b=103 and a-b=97 and then solve the above question using the formula $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$.

According to the above question we have to evaluate the value of $103\times 97$.
Let us assume that the value of $103\times 97$ is $(a+b)(a-b)$.
Now we will compare 103 with a+b and 97 with a-b to obtain the values of a and b and then we will proceed to do multiplication using an algebraic formula.
i.e. a+b=103 let it be equation (1) and a-b=97 and let this be equation (2).
So, a+b=103….(1) and a-b=97….(2)
Now we will solve the equations (1) and (2) and then we will obtain the values of a and b.
Adding (1) and (2) $\Rightarrow$
We will get 2a=200, that means the value of a is 100, i.e. a=100.
We found the value of a. Now we have to find the value of b by substituting the value of a in either in equation(1) or in equation(2).
We will substitute the value of a in equation(1), we will get 100+b=103.
Then b=103-100 which is equals to 3.
So, b=3.
The value of a= 100 and the value of b=3.
Now we will express $103\times 97$in the form of $(a+b)(a-b)$.
$\Rightarrow 103\times 97=(100+3)(100-3)$, we substituted the values of a and b in $(a+b)(a-b)$.
We know that the formula, $(a+b)(a-b)={{a}^{2}}-{{b}^{2}}$, now we will express $(100+3)(100-3)$ as${{100}^{2}}-{{3}^{2}}$.
$\Rightarrow 103\times 97={{100}^{2}}-{{3}^{2}}$ and ${{100}^{2}}=10000$ , ${{3}^{2}}=9$
\begin{align} & \Rightarrow 10000-9 \\ & \Rightarrow 9991 \\ \end{align}
Hence, the value of $103\times 97=9991$.

Note: We can obtain the value of $103\times 97$ directly by multiplying it but this procedure requires time, rather if we use the algebraic formula to find the multiplication we can save the time. If this question is asked as an objective question then look after the last digit as it should be 1 because when the last digit of 103 is 3 and the last digit of 97 is 7 when we multiply both of them we will get 21 the last digit of 21 is 1. So, finally the last digit will be 1 when we multiply 103 with 97. So, that we can mark the correct answer fast.