I hope this is useful.Now that I have your attention,
Q1. "We will ship the product within 2 weeks of the receipt of the payment."
Can you say "2 weeks from" instead of "2 weeks of"? A quick Googling shows many (but fewer) people use 'from' but please confirm whether this is strictly correct. I feel 'within' and 'from' are kind of redundant and shouldn't be used together.
The two sentences are each correct but mean something different. The first means that it will ship sometime within those two weeks. It could be a day or a week or two weeks. The second one implies that it will be a full two weeks after the receipt of payment before it ships.
Q2. "The size of the material was measured by using a special device."
Can you just say "...was measured using a special device", that is, without "by"? Again, the omission of the 'by' is quite commonly seen, but I feel this 'by' is indeed necessary. Here is my reasoning.
"John was killed by using a knife." --->Means the murderer used a knife to kill John.
"John was killed using a knife." --->Means John was using a knife when he was murdered.
You do not need to use "by" in the first sentence. The murder sentences are quite a different case, however. Here it would be strange to use the passive voice. Far better would be to say: "The murderer killed John with a knife." ("using" is implied.) If you do for some reason want to employ the passive, you might say:
"John was killed by a [murderer wielding a] knife."
I agree that the last sentence "John was killed using a knife" is highly ambiguous. He might not have been murdered at all. (Perhaps it was lightning that killed him.) In fact, it sounds like the beginning of a fine mystery novel!
Q3. "Simultaneously to the merger of the company, John was appointed the chief of the marketing division."
"Simultaneous to the merger of the company, John was appointed the chief of the marketing division.
Which sentene is grammatically correct? Both? I feel the first one may be grammatically sounder, but the second one actually sounds as natural for some reason. Plus, I have sometimes seen/heard people use adjectives like adverbs to begin a sentence. The most notable example is "Most important, this decision will affect...", which has been bothering me, actually. Shouldn't this be 'Most importantly,"?
Another (unconvincing) theory of mine is, the second sentence is grammatically incorrect, but the following sentence is correct; "Simultaneous to the merger of the company, John's appointment to the chief of the marketing division became official." The reasoning here is, 'merger' and 'appointment' are both nouns and they can be both modified by the adverb "simultaneous".
I agree that "Simultaneous" sounds better here, and I would bet that most people--including grammar teachers--would say this. As for which is grammatically more correct, I really can't say.
The whole sentence seems a bit awkward to me, however. I would reverse the order of the clauses, possibly replacing the word "simultaneous" with "just as":
John's appointment to the chief of the marketing division became official just as the merger of the company was completed.
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