Question

The value of $\dfrac{{{{(469 + 174)}^2} - {{(469 - 174)}^2}}}{{469 \times 174}}$ is
A. 2
B. 4
C. 689
D. 1023

Answer 1

Hint: We will first consider the expression and let ${\text{A}} = (469 + 174)$ and ${\text{B}} = (469 - 174)$. Now, we will use the identity ${{\text{A}}^2} - {{\text{B}}^2} = {\text{(A + B)(A}} - {\text{B)}}$ and substitute the values accordingly. Next, while applying the identity put the values of \[{\text{A}}\] and \[{\text{B}}\] carefully. Further solve the expression and get the value.

Complete step by step solution:
We will first consider the equation given $\dfrac{{{{(469 + 174)}^2} - {{(469 - 174)}^2}}}{{469 \times 174}}$.
Now, we will let ${\text{A}} = (469 + 174)$ and ${\text{B}} = (469 - 174)$.
Next, we will use the formula ${{\text{A}}^2} - {{\text{B}}^2} = {\text{(A + B)(A}} - {\text{B)}}$
Let us take the given expression $\dfrac{{{{(469 + 174)}^2} - {{(469 - 174)}^2}}}{{469 \times 174}}$ as equation $\left( 1 \right)$
Now, put the values in equation (1) as ${\text{A}} = (469 + 174)$ and ${\text{B}} = (469 - 174)$.
Use formula of ${{\text{A}}^2} - {{\text{B}}^2}$in equation $\left( 1 \right)$
Thus, we get,
$
   \Rightarrow \dfrac{{\left( {\left( {469 + 174} \right) + \left( {469 - 174} \right)} \right)\left( {\left( {469 + 174} \right) - \left( {469 - 174} \right)} \right)}}{{469 \times 174}} \\
   \Rightarrow \dfrac{{(2 \times 469)(2 \times 174)}}{{469 \times 174}} \\
   \Rightarrow 2 \times 2 \\
   \Rightarrow 4 \\
$
Hence, we get the value of the given expression as 4.
Thus, option B is the correct answer.

Note: Don’t simplify the question initially because it can lead the question in a very complex calculation. Use the identity ${{\text{A}}^2} - {{\text{B}}^2} = {\text{(A + B)(A}} - {\text{B)}}$ to simplify the expression and get the value. While substituting the values of \[{\text{A}}\] and \[{\text{B}}\] in the identity use the brackets for separation of numbers. Do not multiply the numerator else cut with the values in the denominator to simplify the calculations.