Mahmud, I've just been fighting this topic, so I'll give you my thoughts on this.. I think that I've finally worked through all of the confusion.
The reason why Ex1 is so confusing is because 2 processes are being used here:
(1) Translating the question
(2) Negating
The two tasks are mutually exclusive processes. When translating the question, what we are really doing is figuring out the size of two sets e.g. performed better vs. not performed better. One needs to use simple addition to ensure that the sum of both of these sets equals the value of the superset (In the case of Ex 1, the superset is 100 test takers).
So when figuring out the size of the sets / translating the question “ Ex1: Most GMAT test takers who take the exam after 1 month of preparation perform better than they did in their first mock", we merely use simple addition:
Size of Set 1: Most performed better = 51-100.
Size of Set 2: 51+?=100. 49. 100+?=100. 0. Thus, 0-49 did not perform better.
This is ‘translating the question’.
However, when negating, it is necessary to remember two key things (1) the formula "Negation Space = Superset - answer choice space”; (2) Ranges or values for quantity words (All/Most/None etc.) occupy 'places' on the number-line.
Lets take an example “Most performed better". Thinking about this in terms of sets. Most performed= 51-100. The superset is 'all' which is "0-100".Using the formula: Negation Space [did not perform better] = All - Most ---> (0-100) - (51-100) —> We subtract the place values of 51 through 100 from the total number of place values of “all” which are zero through 100. We are left with place values zero to 50 (0-50). We only do this when Negating.
Negation space = 0-50 did not perform better.
Back to the question. We translated the question stem (51-100 performed better, 0-49 did not perform better). We worked out the negation space for the question stem: 0-50 did not perform better. We want to now TRANSLATE this space (as we did initially when translating the original question) —> back to simple addition:
Size of Set 1: 0 - 50 did not perform better
Size of Set 2: 0+?=100. 100. 50+?=100. 50. Thus, 50 - 100 did perform better.
So the complete picture for the negation space for the question stem is: 0-50 did not perform better or 50-100 did perform better.
It is this statement to which A to G are compared. Note: we are not negating the answer choices. We are merely ‘translating’ them (using addition) and matching them to our negated question stem.
Hope this extended explanation helps. In earnest - I know that I’m going to be re-reading this at some point in the future to remind myself of the process as it is so confusing!
e-Gmat - can you confirm that my reasoning above is correct?
Cheers,
RS