Which one is bigger: $\;35{,}043 × 25{,}430\,$ or $\,35{,}430 × 25{,}043\;$? 
Which of the two quantities is greater?
Quantity A: $\;\;35{,}043 × 25{,}430$
Quantity B: $\;\;35{,}430 × 25{,}043$

What is the best and quickest way to  get the answer without using calculation, I mean using bird's eye view?

If we look at multiplying like amplification, then $35000$ is a bigger "gain stage" than $25000$. Whatever we add to one side is multiplied by the gain of the opposite side. So, relative to $35000\times 25000$, $35001\times 25000$ increases the output by $25000$: each extra $1$ we add to the $35000$ side is amplified to add an extra $25000$ to the result.  But an extra $1$ added on the other side, namely $35000\times 25001$, increases the output by $35000$.
Thus having the bigger "extra" amount opposite to the bigger number results in a larger increase.
If we add $430$ to $35000$, and $43$ to $25000$, the result will increase approximately by $25000\times 430 + 43\times 35000$.  This is a smaller gain than the opposite $25000\times 43 + 430\times 35000$.
Of course, these approximations ignore that fact that the gains have increased in both "amplifiers", but we are working with the original gains. But the gains have increased by only an insignificant amount. In fact the only difference between our approximation and the true increase is the missing small factor $430 \times 43$, which is only $18490$. That is less than the output increase from just adding a $1$ to either $25000$ or $35000$, therefore negligible when we are dealing with adding $43$ or $430$.
To see why, it helps to look at this picture:

When we increase the inner rectangular area $A\times B$ to the outer one $(A+a)\times(B+a)$, most of the increase comes from the long, thin horizontal and vertical strips $A\times b$ and $B\times a$. (Let us call those the flanks). Only a small contribution comes from the small square in the upper left $a\times b$. (Let us call that the tip).  Of course this is only true when $a$ and $b$ are small compared to $A$ and $B$, but neglecting $a\times b$ simplifies our reasoning in situations when this is the case.
Furthermore, if we are reasoning about a dilemma about involving swapping the $a$ and $b$, to obtain the larger product, we can always ignore the tip and look at just the flanks, even if $a$ and $b$ are large relative to $A$ and $B$. because the tip does not change between the two choices: it is $a\times b$ both ways.   We want the longer flank to be the thicker of the two and the thickness of the longer flank is determined by what we add to the shorter side.
For instance, quick! We have a $10'\times 12'$ room. What's bigger, adding $4'$ to the length and $3'$ to the width, or vice versa?  Of course, adding $4'$ to the width. Why? Because this will create a $4'$ wide flank along the length, and a $3'$ wide flank along the width. The $4'\times 3'$ tip is the same both ways, so we can neglect it.
This does not change even if we change the extras $4'$ and $3'$ to $40'$ and $30'$! A $40'$ wide extra "flank" along the $12'$ side plus a $30'$ flank along the $10'$ side are bigger than vice versa.
