Beautiful (high-level) problem... let me use it as a nice example to show some of the features of my method, ok?!
(i) Let us put the question stem into Math....
Given: m (1-x/100)(1-y/100) = n
Asked (therefore our "focus"): (I will use the fact that m is positive... where?)
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n/m = (1-x/100)(1-y/100) < 1 ?
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(1) Focusing in the question above, it is easy to see that sttm (1) is equivalent to:
y. (1-x/100) = x then (y/100).(1-x/100) = x/100 then (-y/100).(1-x/100) = -x/100 then the last one, a bit more tricky...
then (1-y/100).(1-x/100) = -x/100 + (1-x/100) , therefore sttm (1) tells us that (1-x/100)(1-y/100) = 1-x/50 correct ?!
(Please note that there is no room here for "random manipulations"... I am 100% focus on the question above!!!)
Therefore we must ask ourselves: is n/m = 1-x/50 less than 1? Answer: YES, because x/50 is positive (we assume x is not zero by the context) therefore 1 - positive is less than 1.
We are done for sttm(1), that DECIDES about the question asked, so it is sufficient.
(2) Let us go back to the focus and let us use the proportion y : x = 3 : 4 using my "k technique", I mean, y = 3k and x = 4k, where (in this context) k is certainly positive. For a reason that deals ONLY with easiness in calculations, we MAY and we will change y = 3k to y = 100.3k and x = 4k to x = 100.4k .... no generallity was lost, but the calculations will be much easier... observe:
n/m = (1-100.4k/100)(1-100.3k/100) = (1-4k)(1-3k) < 1 ?
Now it´s just a matter of studying an easy 2o degree inequality... trivial!
Let´s do it: (1-4k)(1-3k) < 1 is equivalent to 12 k^2 - 7k < 0 therefore k(12k-7) < 0 is equivalent to k < 7/12 ? (because I know k is positive)
Now we have to be careful... y = 300k and x = 400k must be between 0 and 100, correct?
If k is greater than 7/12, that means that 400k is greater than 400.7/12 but this is more than 100, therefore we guarantee that k is less than 7/12... therefore sttm (2) is also sufficient.
The answer is D!
Regards,
Fabio.
P.S.: although I may have done something wrong (I´m in a hurry, therefore I typed directly here...sorry), the "philosophy" I use is my method, I mean, we are not randomly "shooting" for numbers, for smart tricks, it is really applied elementary math, but with FOCUS, with care and with techniques. I really hope you see the difference...
