I'd suggest "learned" for the first sentence instead of all those other choices:
After I learned that two of my friends would not be able to {make it to / come to / attend [CHOOSE ONE]} the party, I decided to call it off.
And for the second sentence, I'd suggest this:
Every time I {notice / see [CHOOSE ONE] that the water level in the can is below X level, I call the store and they {replace it / fill it up [CHOOSE ONE]}.
I {come to know/find out} are verbose.
The prepositions of and about add some indirection to the sentence. For example:
I know him
I know about him
I don't know about him
I know of him
The first means that you know him personally: the second means that you have information or experience of him. The third sentence could either mean that you don't have information about him, or that you do have information and it's not very favourable. The final sentence means that you have heard about him (you know that he exists) but you don't know him personally and don't have any information about him.
Looking at the sentences 1 and 2:
1) I don't know about how you took care of her.
2) I don't know how you took care of her.
The first sentence suggests that you don't have any information about how you took care of her: this might be used to indicate that the speaker doesn't know whether you did the job well or not.
In the second sentence, how is ambiguous: it could relate to what care was given (what specifically was done), or to how it was possible for you to give care to her (what obstacles were overcome).
The latter might be the case if both speaker and listener know that the person whom the listener took care of has in the past been unkind to the listener. For example, if she is the listener's mother, and she had an argument with the listener five years ago and hasn't spoken to her since.
For sentences 3 and 4,
good is often used as an adverb in American English, but in Brtish English, we think that
good is an adjective and
well is the corresponding adverb. For avoidance of doubt (as the lawyers say), I have replaced
good by
well. The preposition
in makes sense if
party is a political party. If you want to talk about a social event, you would have to use the preposition
at.
3) I am thinking of how well he will perform in the party.
4) I am thinking how well he will perform in the party.
Sentence 3 indicates that the speaker is in the process of assembling information about how well he will perform, and has not yet made their mind up.
In sentence 4, the speaker is stating an opinion, and almost-fact, about
how well he will perform.
Best Answer
"Know" is a state of information being stored and recallable in your brain. It is present tense, meaning that you can recall and use the information now if you need to. "Appear to know" or "don't appear to know" is something that can be assessed through a person's responses to a question or situation. Their knowledge, or apparent lack of it, can be obvious through their ability to correctly answer questions or perform a task.
"Knowing" is the continuous form of the word, it describes a more active process. The information is actively in your mind, not just accessible. That's the case when you are actually using the information, but someone else would have no way to distinguish whether it is in your conscious mind because you just recalled it, or it is stuck there like a song you can't get out of your head. There is no way for someone else to distinguish what is actively in your mind on a continuing basis.
I suppose in a unique situation like someone complaining that they can't get a thought out of their head, you could respond that they appear to be "knowing" that information, although that wouldn't be idiomatic. If they weren't providing that context, there would be no basis for you to conclude that they "don't appear to be knowing..."