After the comma we need the name of the person. In this sentence the dependent introductory clause (which is not underlined) is "her" which cannot describe a "toe." In order to fix this dangling modifier, the subject of the main clause needs to be changed to "the trapeze artist."

There are two major errors here: a modification error; and an idiomatic expression that must be in the correct form, "the idiom X is not nearly as well-known a Y as Z[is]."

As the phrase Although not… Charlemagne would be nonsensical as a description of the historians instead of Pepin the Short, the first clause must become an independent clause by inserting "he is." Moreover, the idiom must be corrected to "he is not nearly as well-known a monarch as … Charlemagne [is]."

This is a classic dangling modifier error. The dangling modifier should always be followed by the noun it modifies. In this case, the only answer choice which correctly resolves this issue also implements the passive voice. Although this might dissuade you, it is important to weigh your options and ask yourself what the nature of the problem is and which answer choice resolves that issue.

The adverb "sharply" is awkwardly placed in the sentence, creating confusion and ambiguity in the phrase. An adverb should be placed immediately next to the word it is describing, which in this case is "down." The only answer choice that  puts "sharply" correctly in the sentence is "corporate profits will go down sharply over the final quarter of the year."

The adverb "measly" is awkwardly placed in the middle of this sentence; it is confusing because its placement does not make it clear which noun the adverb is modifying. As a qualification of an amount, it makes the most sense to associate "measly" with the subject, "five dollars." "A measly five dollars was the largest amount the scouts were able to collect" is the only answer choice which places "measly" in the correct location.

The phrase "overly soggy with too much dressing" describes the salad that the young woman was eating, but is placed within the sentence in a way that it could describe the woman herself. The phrase needs to be set apart from the main sentence structure and more directly linked to the salad. As the only answer choice to make these changes, "the salad, which was overly soggy with too much dressing, during" is the correct answer choice.

The example sentence contains a misplaced modifier. The structure of the example sentence suggests that the guests are "in expensive crystal glasses" rather than the refreshments they are being served. The error can be corrected by placing the modifier as close as possible to the object with which it is meant to be associated, in this case by moving "the guests" to precede the mention of the refreshments, thus clarifying what is in the expensive crystal glasses.

The example sentence contains a misplaced modifier. "Only" in this context works best when applied to the encryption program's effects. The "only"  thing the software did was confuse the programmers. The modifier should be placed as closely to the action, idea, or thing it modifies. The best version of this sentence reads, "The addition of the encryption program only served to confuse the programmers."

The credited response shows with the greatest clarity what is being modified by "most commonly" by placing it within the verb phrase "is associated." The response also omits the unnecessary use of the pronoun "one," which makes the sentence clearer, especially later on when the pronoun "it" refers back to "nominalism" as the subject of the adjusted sentence.

The credited response is the only one in which every modifier is clearly paired with its object. In this instance, "Donatist crisis" is treated as a singular object modified by "bloody," rather than "Donatist" and "crisis" treated as equally important (and interchangeable) modifiers of "crisis," as "bloody" refers not to a crisis that happened to be "Donatist," but to a specific incident in history. "Belated" modifies "conclusion"; placing it after "conclusion" implies that the efforts delayed the conclusion, rather than that the belated conclusion was reached due to the efforts. While it is common to speak of "tireless efforts," properly speaking, it would be the officials who do not tire, rather than their inanimate efforts.