Sentence Correction

1. Gas prices in 2012 are higher than in 1997.
2. Gas prices are higher in 2012 than in 1997.
Answer: Both are correct. The comparative idiom "higher than" requires parallel construction between the two items being compared. The prepositional phrase "in 2012" is parallel to "in 1997". The placement of "in 2012" before or after "are higher" is not of material importance here, and both are idiomatic. This is the type of distinction that the GMAT might make between answer choices to trick you into think that this is the issue at hand, when it is not. You would never have to choose between 1 and 2 on the GMAT.

Yes, it might seem more thoroughly parallel to write 1 as "Gas prices in 2012 are higher than gas prices in 1997," but nothing essential is being left out because what's being compared is the situation "in 2012" and the situation "in 1997". BOTH sentences can be thought of as "ellipses", a term referring to "the omission from speech or writing of a word or words that are superfluous or able to be understood from contextual clues." Written out in full, the conjunction "than" in both examples would be followed by "[than] were gas prices in 1997." For example, in full example 1 would be "Gas prices in 2012 are higher than [were gas prices] in 1997." However, all that is necessary for comprehension is "in 1997" after "than".

3. There are more books in box A than in box B.
4. There are more books in box A than books in box B.
Answer: Both are correct, but 4 is less concise so 3 would be preferred. Again, as in examples 1 and 2, both sentences are ellipses for "There are more books in Box A than there are books in Box B." The difference between examples 3 and 4 is how many words are omitted in the ellipsis. The sentence is perfectly comprehensible as written in 3, but 4 is merely less concise, not grammatically incorrect.

5. There are more books in box A than those in box B.
Answer: INCORRECT. The demonstrative pronoun "those" would have to refer to a preceding plural noun, which can only be "books in box A". However, the books in box A are not the same as the books in box B. The demonstrative pronoun "those" cannot apply to both the books in box A and the books in box B, so its use here is incorrect and leads to a breakdown of logic in the sentence.

6. There are more books in box A than them in box B.
Answer: INCORRECT. The short answer is that this is unidiomatic. A construction with a pronoun followed by a prepositional phrase is going to use one of the demonstrative pronouns "that" or "those". Here's an example: "The quantitative logic questions on the GMAT are called 'Data Sufficiency,' while those [not they] on the GRE are called 'Quantitative Comparison.'"

If you need another reason, "them" is the wrong case. Remember, this is an ellipsis for "There are more books in box A than [there are books] in box B." Because "are" is a linking verb, a pronoun replacing "books" would have to be in the nominative case-i.e., "they" instead of "them", but "they" would still be unidiomatic as discussed in the preceding paragraph.

The third reason, if you needed one, is that--even if "them" were idiomatic and the correct case--it would make no sense for the same reason that "those" makes no sense in example 5: the pronoun would have to refer to the "books in box A" and couldn't be applied as well to the books in box B.

I also have some other questions.
3. There are more books in box A than in box B. What does the prepositional phrase "in box A" modify?
Answer: It modifies the verb "are" by indicating WHERE "the books" are located.

5. There are more books in box A than those in box B. What does the pronoun "those" refer to?
Answer: I explained this above. Grammatically, "those" would have to refer to the "books in box A", and therefore could not also apply to the books in box B. This leads to a breakdown in logic.

6. There are more books in box A than them in box B. What does the pronoun "them" refer to?
It doesn't matter because "them" here is both unidiomatic (the demonstrative pronoun "those" would be correct if it made any sense) and the wrong case (you would need the nominative case "they"). If these objections were somehow removed, it wouldn't make sense for the same reason that "those" doesn't make sense in example 5.

I know that this was a lot of detail, but the nuances among your examples called for in-depth explanations. Please feel free to respond or send a PM if you would like further clarification. Thanks for this provocative set of questions!