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When you say quarter of, you are supposed to say the next hour. Quarter after 4 is a quarter of 5.
That’s… I guess that makes sense but that’s really weird and ambiguous.
Quarter after four is 4:15.
Quarter of five is 4:45. Also quarter to five and quarter til five.
I’m seeing other comments that suggest I might be wrong. Especially in regards to other languages.
I would not have guessed that meaning of “of.” I think we should stick to “til” or “past” for clarity.