> I'm looking for a thorough answer to this including concepts behind it and how to approach such a question.
>> Perfect! That´s how I teach my students in more than 10 years, and this is the only way I believe one should be really prepared to go over the scaled score of 48 in Math...
Let us start with the question stem!
Let D and N be the numbers (in the past) of Doctorates and Non-Doctorates (respectively), therefore we know that
D = 2k and N = 3k , where k is a positive integer. (That´s the nice way of dealing with proportions...)
Now let DH and NH be the number of Doct. and Non-Doct. hired in the 5-year period.
Our focus is then to the ratio: (D+DH)/(N+NH)
Let us start with statement (2), because it seems to me that it is the easier to begin with!
(2) Not enough:
Given that DH = 50, we look for (2k+50)/(3k+NH) and to guarantee that there is not a single value possible,
> TAKE NH = 0 and k = 1 to obtain the particular answer 52/3 = 14
> TAKE NH = 0 and k =2 to obtain the particular answer 54/6 = 9
This is enough to guarantee that statement (2) is insufficient to answer the question asked.
(1) Please consider DH = 2w and NH = w, where w is a positive integer (perhaps even w=0 could be considered...)
Then we are looking for the ratio (2k+2w)/(3k+w) and again it is easy to BIFURCATE (I created this word to mean "find a bifurcation, that is, two possible answers/"roads" satisfying all the conditions imposed at that moment"):
> Take k = w = 1 to obtain 4/4 = 1
> Take k = 2 and w = 1 to obtain 6/7
This is enough to guarantee that statement (1) is insufficient, too.
(1+2) From the fact that DH = 2w = 50, we know that w = 25 and we are looking for the ratio (2k+50)/(3k+25) but we are still with answers depending on the k variable! Observe this:
> Take k =1 to obtain 13/7
> Take k =2 to obtain 54/31 (different from 13/7)
From the fact that we were able to BIFURCATE (1+2), the answer is "E" without any doubt.
I hope you like that !
Obs.: this takes much less time that it seems by the length of my solution. It´s just a matter of practice this tool till you start to bifurcate and structurate without writing that much!
