Here is a sentence that I encountered in a textbook.
けがが治るまでに３週間かかった。
As far as I understand,「XまでYする」means that Y continues until time X, whereas「XまでにYする」means that Y takes/will take place before time X. In the sentence above, the 3 weeks lasted until the injury healed, so why is までに used?
「までに」は動作がその時点までに完了するときに使います
Ex
宿題を明日までに終わらせる。
君が帰るまでに掃除をしておく。
それに対して 「まで」は動作、状態が継続しているときに使います。
EX
祭りは明日までです。
寝ないで朝まで仕事をする。
「までに」→by（完了）
「まで」→until（継続）
という使い分けをするわけです。
As far as I understand,「XまでYする」means that Y continues until time X, whereas「XまでにYする」means that Y takes/will take place before time X.と書いてある通り、OPもすでに知っていることですよね。質問は、In the sentence above (けがが治るまでに３週間かかった), the 3 weeks lasted until the injury healed, so why is までに used? ですので、そこの説明が要るのではないでしょうか。
This is a very interesting question. Here is my analysis.
When the verb かかる is used to talk about an amount of time spent (or to be spent) before some event happened (or happens), as in your example, it is used with [event]-までに.
けがが治るまでに３週間かかった。
In this case, the verb itself doesn’t refer to a continuous action or state. It simply means “take” or “require”, which in Japanese is apparently not considered something that, well, takes time. In your example, that (“taking” of three weeks) happened before the healing of the injury. Hence までに.
However, the same verb can also be used with まで as in the following example.
けがを治すのに４月までかかった。
Here, the verb is used to talk about a continuous effort, and まで is used not with an event but with a specific point of time. The effort continued till that point of time. Hence まで.
