How to ask to whom an item belongs to

Question

The given answer to the question I'm supposed to ask in my workbook is:

それは　たけしさんの　ぼうし　です。

But I can't figure out if my question should be:

このぼうしは　だれの　ですか。

OR

だれの　このぼうし　ですか。

Thank you!

Answer 1

I can't figure out if my question should be このぼうしは　だれの　ですか。OR だれの　このぼうし　ですか。

The first one is correct, because...

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The genitive case particle の works like the English 's or of, as in:

わたしのぼうし my hat
あなたのぼうし your hat
たけしさんのぼうし Takeshi-san's hat
だれのぼうし whose hat
わたしのこのぼうし this hat of mine (lit. my this hat)

の can also be used for (someone)'s (stuff), as in:

わたしの mine
あなたの yours
たけしさんの Takeshi-san's (stuff)
だれの whose (stuff)

So...

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それはたけしさんのぼうしです。 That is Takeshi-san's hat.

To turn this into a question asking だれの(whose), you'd replace たけしさん with だれ, add the interrogative particle か at the end (and replace それ with これ), as in:

これはだれのぼうしですか。 Whose hat is this?

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そのぼうしはたけしさんのです。 That hat is Takeshi-san's.

To turn this into a question, you'd do the same: たけしさん→だれ, です→ですか (and replace その with この this time):

このぼうしはだれのですか。 Whose is this hat?

Answer 2

これは　だれの　ぼうし　ですか。
Whose hat is this?

それは　たけしさんの　ぼうし　です。
That is Mr. Takeshi's hat.