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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?

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You can also look at this way. For Quantity A, there are more multiples of the larger number (call this larger one x); whereas, for Quantity B, there are fewer multiples of the larger number (call this larger one y). And notice, y > x, but not by that much, and that difference won't count for much when you're multiplying x by 25,430, as compared to y by 25,043. Indeed, this can (and should) be made more rigorous, and as such has been done adequately in the answers. This was more for the intuition behind it. Though, you should avoid it altogether if you don't feel comfortable exercising... –  ThisIsNotAnId Jun 2 '13 at 5:45
    
intuition with numbers. –  ThisIsNotAnId Jun 2 '13 at 5:46
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Calculation would get you the answer far quicker than consulting MSE. :) –  Hurkyl Jun 2 '13 at 15:16
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@Hurkyl Training your intuition will allow getting the answer much faster than doing a longhand calculation when you don't have access to a calculator/computer. –  Dan Neely Jun 3 '13 at 14:53
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@Dan: Agreed, and that's what the OP is asking for which is why I only made a comment rather than an answer. But conversely, I see a lot of people spend a lot of effort avoiding straightforward solutions to their problems (sometimes to the point of not even looking for one, or even not even acknowledging one as a solution), so I think it's worth pointing out. –  Hurkyl Jun 3 '13 at 17:10
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15 Answers

up vote 103 down vote accepted

If we have two pairs of numbers, and both pairs add up to the same total, then the pair with the larger product will be the pair that's closer together. So the answer is the first pair.

(This picture might help: if you want a rectangle with a fixed perimeter to have the biggest possible area, you want it to be a square.)

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Intuitively, you want the $430$ to multiply the biggest thing it can, which is $35{,}000$, so the first.

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+1 i think this should be the accepted answer –  user31280 Jun 2 '13 at 14:38
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No calculation or equation needed at all.

When you have a fixed length to distribute over the 4 borders of a rectangle, and you want to get the maximum possible surface, you will get a square (equal length borders).

In other words, when adding a value to two multiplicants, you will get the biggest result when minimizing the difference of the multiplicants: The bigger one is the equation wher you add 0,43 to the smaller number and 0,043 to the bigger one.

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Quantity A is $(35+0.043)(25+0.430)$.

Quantity B is $(35+0.430)(25+0.043)$.

Imagine expanding $(a+x)(b+y)$, where $a$ and $b$ are "big."

The main term in the product is $ab$. The next in importance are the cross-terms $ay$ and $bx$.

Finally, the terms $xy$ are negligible. Actually, in our case they are not only negligible, they are the same in product A and product B.

We get a bigger product if the cross term is bigger. In quantity A, the number $0.430$ gets multiplied by the big guy, namely $35$. Thus A is bigger than B.

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+1 for being generalizable to when the two pairs don't have the same sum. –  Dan Neely Jun 3 '13 at 14:51
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The idea of cross term can help you, if you can do it mentally:

$$35{,}043 \times 25{,}430-35{,}430 \times 25{,}043$$ $$=35{,}043 \times 25{,}430-35{,}043 \times 25{,}043+35{,}043 \times 25{,}043-35{,}430 \times 25{,}043$$ $$=35{,}043 (25{,}430- 25{,}043)-25{,}043 \times (35{,}430- 35{,}043)$$ $$=35{,}043 (430- 43)-25{,}043 \times (430-43)$$ $$=10{,}000 \times (430-43)$$

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Hint: Compare $a\times b$ with $$(a+x)\times (b-x)=ab-ax+bx-x^2=ab-x(a-b)-x^2$$ keeping in mind that in your question, $a> b$ and $x>0$.

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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:

enter image description here

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.

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great work, it helps –  Complex Guy Jun 4 '13 at 17:34
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Another simple method particularly suited to a deliberately 'symmetric' problem like this: $35{,}043\times 25{,}430=(25{,}043\times25{,}430)+(10{,}000\times25{,}430)$ while $25{,}043\times35{,}430=(25{,}043\times25{,}430)+(10{,}000\times25{,}043)$, and written out this way it's clear that the former has to be larger.

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$\begin{eqnarray}{\bf Hint}\quad 35043 \times 25430 &-\,&\ \, 35430 \times 25043 \\ A\ (B\! +\! N)&-\,& (A\!+\!N)\ B\ =\ (A\!-\!B)\,N > 0\ \ \ {\rm by}\ \ \ A > B,\ N> 0\end{eqnarray}$

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A = 35,043×25,430 = 35,043 × 25,043 + 35,043 × 0,387

B = 35,430×25,043 = 35,043 × 25,043 + 25,043 x 0,387

so A is bigger

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from the first equation we get:

(35,000 + 43) * (25,000 + 430) =

= 35,000 * 25,000 + 430*35,000 + 43*25,000 + 43*430 ( let it be a)

from the second second equation we get:

(35,000 + 430)*(25,000 + 43) =

= 35,000 * 25,000 + 43 * 35,000 + 430 * 25,000 + 43 * 430 (let it be b)

suppose that a > b (1)

35,000 * 25,000 + 430*35,000 + 43*25,000 + 43*430 > 35,000 * 25,000 + 43 * 35,000 + 430 * 25,000 + 43 * 430 <=>

430 * 35,000 + 43 * 25,000 > 430 * 25,000 + 43 * 35,000 <=>

43 ( 350,000 + 25,000) > 43 ( 250,000 + 35,000) <=>

350,000 + 25,000 > 250,000 + 35,000 <=>

375,000 > 285,000

which is true, so from (1) a > b

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I would separate the numbers into 2 terms so that there is a common factor between A and B:

A = 35043×25430 = (35043)×(25043 + 387) = 35043×25043 + 35043×387
B = 35430×25043 = (35043 + 387)×(25043) = 35043×25043 + 25043×387

Comparing the last equality of A and B, it is clear that A is greater.

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If you have to go that far, however, your "intuition" (the whole point of this) is maybe not that good. –  Kaz Jun 2 '13 at 15:25
    
Why are you quoting "intuition"? That word is not even mentioned in the question. The question only states, "What is the best and quickest way to get the answer without using calculation?" I interpreted this to mean without doing the full multiplication and getting an exact number. I agree, an intuitive reasoning such as "for two rectangles having the same perimeter, the largest area is the most square," is probably better. This is just another approach. –  bcorso Jun 2 '13 at 18:33
    
@Kaz: Depending upon what precisely one is thinking, this can be essentially the same answer as the second highest rated answer. –  Hurkyl Jun 3 '13 at 17:07
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Closer numbers will yield the higher product when they got the same sum.

Prove lies in

(a-x)*(a+x) = a^2-x^2

Since the terms added results in 2*a, independent of x, closer together (smaller x) will always be larger.

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Quantities $A$ and $B$ can be written in the form $A = (35,000 + i 0,043)(25,000 + i 0,430)$ and $B = (35,000 + i 0,430)(25,000 + i 0,043)$, where i=1. Now application of complex arithmetic says that real parts are equal so we forget them. Imaginary parts are $35,000 \cdot 0,430 + 25,000 \cdot 0,043$ for $A$ and $35,000 \cdot 0,043 + 25,000 \cdot 0,430$ for $B$. Here the intuition says that big numbers have to be multiplied together, because it maximizes correlation. So the quantity $A$ is bigger.

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Imagine an extreme case of shopping where you have 3 Louis Vuitton bags costing £1,000,000 each. Which is cheaper, getting a discount of £1 per item or putting one bag back? This tells you that 3 x 999,999 is larger than 2 x 1,000,000. And that tells you whether increasing the little term or the big term has a greater effect.

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