Monday, June 27, 2011

GRE: Quantity Comparision of Multiplication of swapped numbers (12,964 * 31,526 vs 12,526 * 31,964)

Compare multiplication values of two of the numbers with some digits swapped type question is one of the possible questions in the GRE. Confused?

I mean comparision of something like 12,964 * 31,526 vs 12,526 * 31,964. Wait a minute, don't you think those values can be calculated easily with on screen calculator provided? No cause it will generate large number and on screen calculator provided in GRE can only calculate if the result does not exceeds 8 digits, but the result in this case would be of more digits; so no use of calculator; don't waste your time during GRE exam by trying to calculate such large number with on screen calculator.

So how do we solve such questions? Below is some derivation I did to give you my own theorem to do such calculations in no time. But don't be afraid; the derivations are not required for exams but just for understanding; you will learn how to in concluding part.

For derivation let us take two 5 digit numbers xymno and cdpqr and another pair is formed by swapping last three digits; although similar derivation can be used to reach same conclusion for numbers with more digits and swapping different number of last digits (swap last one digit or two or three etc) so question now is:
(xymno * cdpqr) vs (xypqr * cdmno) which quantity is larger?
or, (xy * 1000 + mno) * (cd * 1000 + pqr) vs (xy * 1000 + pqr) * (cd * 1000 + mno)
or, (xy * cd * 1000000 + xy * pqr * 1000 + cd * mno * 1000 + mno * pqr) vs (xy * cd * 1000000 + xy * mno * 1000 + cd * pqr * 1000 + mno * pqr)
or, (xy * pqr + cd * mno) vs (xy * mno + cd * pqr)

This is comparision of sum of cross multiplications (first two number from one and rest of number from other) rather than multiplication of all numbers.

And guess what, you don't even have to calculate those either; what you need to do is to see which of them involve multiplication of larger numbers together. Take an example for clearity: in 12,964 and 31,526 - 31 is larger number than 12 while 964 is larger than 526; now check which of combination in options involves multiplication of 31 (larger) and 964 (larger): when both large number does not comes in same number i.e. its 12,964 * 31,526 not 12,526 * 31,964, so first one is larger.

Need proof for that as well?
Ok here we go.

Take four numbers such that a >c and b >d
now lets see which one is larger (ab + cd) or (ad + bc), I said (ab + cd)

Lets find out the difference:
(ab + cd) - (ad + bc)
= a (b - d) - c (b-d)
= (a - c) (b - d)

Look closely, both (a - c) and (b - d) are positive (do I need to explain) and hence ab + cd is larger than ad + bc; sum of multiplication of numbers will be larger if it includes multiplication of larger number with another larger one and smaller number with another smaller one.

So in short if you are given two numbers 12,964 * 31,526 vs 12,526 * 31,964 then larger among 12 and 31 is 31 and larger among 964 and 526 is 964 and they should not come in same number i.e. the larger number is not the pair which has 31,964 but the other one.

Now wait a minute, what if either of the number is negative? Eg. 12,964 * (- 31,526) vs its counter part 12,526 * (-31,964)?

At last I am no GRE expert and hence I don't claim everything I derived to be cent percent correct but I can't see any fault here. And I have derived those things myself and please do not copy and paste the content to your site and claim it to be your work or at least give me credit by providing a link to my post/site.

I hope the post is useful; any comment, suggestion is welcomed.

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