Questions about example sentences with, and the definition and usage of "Quater"

A: it means it’s 11.45. There is quarter of an hour until it’s 12 o’clock.

A: @okinawangirl: yep. Repeating what Antiggi has said, "quarter to.." means "15 minutes before" or "15 minutes until." "Quarter after" means "15 minutes after." 😊

A: I'm not familiar with the word "quater." I looked it up and apparently it's used in pharmaceutical prescriptions. I've never used it. "Search by twos" means that two people pair up and go search for something or someone.

A: × It‘s a quater to twenty-two o'clock now.
✓ It‘s a quater to 10 PM now.

× Although I am pursuing my postgraduate degree, my ability of English writing is so poor that I can't bear it more.
✓ Although I am pursuing my postgraduate degree, my English writing abilities are so poor that I can't bear it anymore.

× I want to improve my capability in writing as soon as possible due to the English examination of vital importance to be took about half a month later.
✓ I want to improve my capability in writing as soon as possible, because of a vitally important English examination I have to take in about half a month.

× From now on, I will make a diary in English every day to record my improvement on writing.
✓ From now on, I will make a diary entry in English every day to record my improvement in writing.

I understood everything you meant, and your vocabulary is very impressive! Sometimes your writing sounds a bit stuffy, so I made a few corrections to help the flow.
There are other ways to make it sound more natural, and different people might have different corrections. Reading more books in English will help you write more naturally.
Good luck on your exam!

A: “A quarter of the apples have/has been lost.”

A: 9:45 Quarter to 10
10:15 - Quarter past 10.