Which one is bigger: $\;35{,}043 × 25{,}430\,$ or $\,35{,}430 × 25{,}043\;$?

Question

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?

Answer 1

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.)

Answer 2

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

Answer 3

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.

Answer 4

The idea of cross term can help you, if you can do it mentally:

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

Answer 5

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.

Answer 6

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 $x>0$.

Answer 7

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.

Answer 8

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.

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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.

Answer 13

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.

Answer 14

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.

Answer 15

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.

Answer 16

For the layman

Think of the question as big vs small; 35000 (big) vs 25000 (small), respectively.

When comparing column A to B, 35K is being multiplied by small in both situations However, in column A, it's being multipled ~400 times vs ~40 times in B. That's an entire order of magnitude larger (a.k.a extra digit).

Thus, you can intuit that A is bigger.

Note: Once they are both in the same order of magnitude, e.g. if B was >100, you'll probably want to use calculator.