Thanks for the detailed explanations, masaegu san. :)

Hi masaegu san. I have one more thing to clarify which is what if my answer is "I know"? Am I right that I should say "知っています" instead of "わかります"?

No. You cannot do that. You have to repeat the same verb that the other person used in asking the question.

1. Do you know that girl's name?
2. What is the capital of China?
3. Can you tell me the answer to this math question?
4. Do you know what you want to do after college?

1.　The question will be asked with either verb. Use the same verb in your answer. This is basically about knowing or not knowing a stranger's name, which is why you won't sound indifferent even if you said 知りません.

2. If you know the answer, you just say 「北京です。」. If you don't, say 分かりません.

3. & 4. Only 分かる will be used in asking these, so that's the verb you use in your answer.

I get it now. Thanks masaegu san. By the way, do you mind translating number 3 and 4 into Japanese?

1. Do you know that girl's name?
　あの女の人の名前を知っていますか。

2. What is the capital of China?
　中国の首都は何ですか。

3. Can you tell me the answer to this math question?

4. Do you know what you want to do after college?

I will give it a try first, hopefully it doesn't sound too ridiculous

3.　この数学の答えがわかりますか。

4. 　大学のあとで何をしたいのがわかりますか。

３．この数学の問題の答えが分かりますか。

４．大学卒業後、何がしたいか分かっていますか。

One more thing I forgot to to ask which is, which of the two sentences below is appropriate when asking "Do you know the phone number of Mr. Tanaka?".

Personally I would usually say the first one but now I have second thoughts.


田中さんの電話番号をしっていますか。

田中さんの電話番号がわかりますか。

We use both. I'm pretty sure the second one is used more often, though, as it sounds more natural to me.

You really need to forget "know vs. understand" in English.

Yeah, that's true. And this is also why when I look at this sentence "この数学の問題の答えが分かりますか", the first thing that came out of my mind is that it also sounds like "Do you understand the answer of this question?".

By the way, can I use が instead of か in this sentence "４．大学卒業後、何がしたいか分かっていますか。"?

Is it true though, that when someone gives you some information you didn't know before, and obviously, they've not prompted you with a verb, you should use 知る not 分かる。
Like-
B: あの木の上にある物を見えますか。あれは「森の鳥」と 言うキノコです。おいしいです。
A: そうですか。知りませんでした。