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I would choose option (B):

(B) With a population equal to that of Kansas in an area one three-hundredth its size, Singapore is the most densely populated nation in the world, averaging

This option maintains the clarity and correctness of the original sentence while using appropriate phrasing and parallel structure. It accurately states that Singapore has a population equal to that of Kansas in an area one three-hundredth its size, and it correctly identifies Singapore as the most densely populated nation in the world. The word "averaging" appropriately connects the statement about population density.
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The correct option is (B) "amount of research into the cause of acoustic neuromas has proven unsuccessful."

This option maintains subject-verb agreement by using the singular form "has" to agree with "amount." Additionally, it correctly states that the research into the cause of acoustic neuromas has been unsuccessful.

Option (A) is incorrect because it uses the plural verb form "have" with the singular subject "amount."

Option (C) is grammatically incorrect because it uses the plural verb form "have" with the singular subject "quantity."

Option (D) is incorrect because it rearranges the sentence structure and does not accurately convey the intended meaning.

Option (E) is incorrect because it places the phrase "has proven unsuccessful" too far from its intended subject "amount of research."
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Official Explanation
When a sentence has a comparison, make sure the sentence is comparing like terms. It's an error to say "a population equal to Kansas," because a population can only be compared to another population, not to a state. Due to this error, eliminate (A). An introductory modifying phrase is also a hint to check for modification errors. Here, there are no issues because the phrase beginning "With a population ..." properly modifies "Singapore," which comes next. However, keep an eye on the answer choices, which may introduce such an error. No clear grouping is apparent, so check out each choice individually.

In (B), "that of" stands in for "population," so this choice fixes the modification issue. It seems to be the correct answer, but check the others to be sure.

In (C), the introductory phrase that should modify "Singapore" now modifies "the densest population," so this choice can be eliminated. (D) turns the sentence into a compound sentence with two clauses connected by "and," but the second clause has no subject, so it's incorrect. (E) seems to imply not that Singapore has a population of similar size to that of Kansas but that it has Kansas's actual population. (B) is the correct answer.

TAKEAWAY: Comparisons have to be apples to apples, not apples to oranges. When you see a word indicating a comparison, such as "equal," make sure the things being compared are of the same type.
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