Learn Hardcore Korean: bangbeobeul bakkulkkayo?

It’s a polite, inclusive proposal: “Shall we…?” It can also ask for the listener’s opinion or a guess: “Do you think…?” In this sentence it’s a suggestion: “Shall we change (it)?”

Who is the subject? Does it mean “Shall I” or “Shall we”?

Korean often omits the subject. With -(으)ㄹ까요?, the default reading is inclusive “we.” Context can shift it:

제가 방법을 바꿀까요? = “Shall I change the method?”
우리 방법을 바꿀까요? = “Shall we change the method?”

Why is it 을 after 방법 and not 를?

을/를 marks the direct object. Use 을 after a final consonant (받침) and 를 after a vowel. 방법 ends with ㅂ, so 방법을 is correct. If the noun ended in a vowel, you’d use 를 (e.g., 계획을/차를).

How is 바꾸다 changed to 바꿀까요? exactly?

Dictionary form: 바꾸다 (stem: 바꾸-)
Add the proposal/guess ending -ㄹ까요? (since the stem ends in a vowel)
Result: 바꾸 + ㄹ까요 → 바꿀까요? Compare: 가다 → 갈까요?, 먹다 → 먹을까요? (consonant-ending stem takes -을까요?)

What’s the difference between 바꾸다 and 바뀌다?

바꾸다 = transitive “to change something” (you do the changing). Example: 방법을 바꾸다.
바뀌다 = intransitive “to be changed/changed (by itself or by someone)” Example: 방법이 바뀌다. Your sentence uses the transitive verb.

How is -(으)ㄹ까요? different from -(으)ㄹ래요? or honorific options?

-(으)ㄹ까요?: inclusive suggestion or seeking opinion (“Shall we…?”).
-(으)ㄹ래요?: asks about the listener’s preference/willingness (“Would you like to…?”). Slightly more casual/intimate.
-(으)실래요?, -(으)시겠어요?: politer/honorific ways to ask the listener’s preference (“Would you care to…?”), often to seniors or customers.

How does 바꿀까요? compare with 바꿔요?, 바꾸자, 바꿉시다, 바꿀게요?

바꿀까요?: neutral-polite suggestion (“Shall we change it?”).
바꿔요?: “So we change it?” (checking/confirming a plan) or a softer suggestion; slightly more assertive than -ㄹ까요?.
바꾸자: plain casual “Let’s change it.”
바꿉시다: formal “Let’s change it.”
바꿀게요: “I’ll change it (then).” volunteering your own action.

How can I soften or nuance the suggestion?

Add softeners or “try”:

방법을 좀 바꿀까요? (a bit/please)
방법을 한(번) 바꿔 볼까요? (shall we try changing it once?)
같이 방법을 바꿔 볼까요? (shall we try changing it together?)
다른 방법으로 해 볼까요? (shall we try a different way?)

Can I drop the object particle in conversation?

Yes, in casual speech: 방법 바꿀까요? is natural. In careful or written Korean, keep 을.

What’s the nuance if I say 방법은 바꿀까요? instead?

은/는 topicalizes or contrasts. 방법은 바꿀까요? implies “As for the method (at least), shall we change it?”—perhaps contrasting with other things that will stay the same.

How do I pronounce and romanize it?

Pronunciation: 방법을 바꿀까요? ≈ [방버블 바꿀까요] (연음/assimilation makes 법을 sound like [버블]).
Revised Romanization: bangbeobeul bakkulkkayo?

Is the spelling/spacing 바꿀까요 or 바꿀 까요?

It’s one word: 바꿀까요. Endings attach directly to the verb. Also note 까요 uses the tense consonant ㄲ; 가요 here would be wrong.

Can -(으)ㄹ까요? also mean “Do you think…?” in other contexts?

Yes. With non-first-person subjects it often expresses conjecture:

비가 올까요? = “Do you think it’ll rain?”
For your sentence, the conjectural version would use the intransitive: 방법이 바뀔까요? = “Do you think the method will change?”

How would I politely ask a senior/customer “Would you like to change the method?”

Use a preference/honorific form:

방법을 바꾸시겠어요?
방법을 바꾸실래요? If you mean “Let’s (together),” prefer inclusive: 방법을 바꾸시죠 or 같이 바꿀까요?

Any close synonyms or alternate wordings?

방법 ≈ 방식 (method/manner), 수 (means/way).
Natural alternatives:
- 방식을 바꿀까요?
- 다른 방법으로 해 볼까요?
- 다른 수가 있을까요? (“Might there be another way?”)

How do people typically answer this question?

Agreeing: 네, 그럴까요? / 좋아요, 바꿔요. / 바꿉시다.
Disagreeing: 아니요, 그대로 하죠/합시다.
Hedging: 음… 조금만 바꿔 볼까요? / 다른 의견도 들어볼까요?