On a certain 50-question test, each correct answer is worth

Hi langus91, you've got the logic down perfectly. Statement 1 indicates there could have been 10 correct answers in the first half and 0 in the second half, or 11-1, or 12-2, etc. So the minimum score the student could get while satisfying this equation is 20/100. Assuming more and more answers are answered correctly, we end up at 25-15 (that is, all questions correct on the first half and 60% correct on the second half, which leads to 80/100 total score.

As you indicated in your spoiler, you were unsure about a 30-20 split, which was addressed by JimGMATPrepster. However, even if you're unsure of this on test day it won't stop you from getting the correct answer as 20-10 or above will lead to a passing grade on the exam. Even putting the two conditions together, Ethel could have correctly answered questions in any of the following proportions:
18-8, (total 26/50, more than half)
19-9,
20-10,
21-11,
22-12,
23-13,
24-14,
25-15 (total 40/50, maximum score of 80/100)

The first two are failing grades, while the final five are passing grades. Hence why the answer is E. The GMAT can be a very unforgiving exam, but you can still get the right answer even if you make little conceptual mistakes like this one.

Hope this helps!
-Ron