Seeing lots of good explanations here already! But also a little bit of disagreement.
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(A) in eliminating blood, contaminated with the virus, from the nation's blood supply
I don't see anything wrong here grammatically. Some test-takers like to skip ahead, eliminate anything with definite grammar errors, and then circle back to choices such as (A) that don't have any obvious mechanical errors -- and that's not a bad approach at all!
But whenever you get to (A), the problem is that the modifier tweaks the meaning of the sentence. It sounds like the test eliminates all blood from the nation's blood supply, and the modifier inside the commas ("contaminated with the virus") is presented as extra, non-essential information that just describes the blood. And that doesn't make sense: the test will be helpful in eliminating
only the contaminated blood, not all of the blood. (A) is gone.
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(B) in eliminating blood from the nation's blood supply that is contaminated with the virus
This one has a much clearer modifier error. The phrase "that is contaminated with the virus" modifies the noun it "touches": "the nation's blood supply." And that doesn't make sense, either, since we can't really say that the entire blood supply is contaminated. Eliminate (B).
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(C) in eliminating from the nation's blood supply blood that is contaminated with the virus
Wow, this sounds like hot garbage. I used to manage a team of writers, and if any of them had written this, I would have wondered if they were drunk or something. I don't think anybody would ever actually say this sentence in real life. It sounds terrible.
But your ear is not your friend on the GMAT! The important thing is that I don't see any grammatical errors or any meaning errors. "That" jumps out at me, but it's used well here: the blood is contaminated with the virus, not the entire supply. I guess we'll have to keep this piece of hot garbage -- it's not wrong. And we're always looking for four wrong answers, not one right answer. Since it has no clear mistakes, let's keep (C).
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(D) to eliminate blood that is contaminated with the virus from the nation's blood supply
(D) doesn't make logical sense, either. Strictly and literally, the sentence seems to be saying that the virus came from the nation's blood supply. That doesn't work. Eliminate (D).
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(E) to eliminate blood from the nation's blood supply that is contaminated with the virus
Same error as (B): the phrase beginning with "that" modifies the nation's blood supply, and that doesn't make sense. (E) is gone, too.
So (C) wins. In "real life", it's a lousy sentence, in my opinion. But the GMAT doesn't care: the other four answer choices have clear meaning errors, so we're stuck with crappy answer choice (C).