As far as I can tell, if you want to say something like "Would you...?" or
"Why don't you..?", it's common to say something like "食べるのだろうか？". I looked it
up and as I far as I can tell it's short form+の to make it a noun, and then
だろう which is the volitional of だ... this is after a lot of Googling around and
I'm still kind of confused, since I haven't even heard the volitional
mentioned in my class yet. Please tell me if that's right.

If I understand Wikipedia correctly, you could also say "食べようか？". First off,
is that conjugating it right; second, is that something you would say;
finally, how is it different from -の+だろう?

Thanks for any help.
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2012-10-22T22:17:23.490
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volitional-form invitation
"Would ____?" -のだろう or -おう/-こう/etc
I've listed up some forms here for you.

```

    食べませんか？　- Won't you eat (with me) [as in an invitation]
    食べようか - Shall we eat
    食べるのだろうか　- I don't think I have ever heard this said 
    but I guess it would be equivalent to the the below.
    食べようかな - I wonder if I will eat
    
```
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2012-10-23T01:03:16.107
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There are many ways to make requests/suggestions in Japanese, but (some of)
the prototypical ones would be:

1 Requests (in order of increasing politeness/formality):

> 食べてもらえる？  
>  食べてもらえますか？  
>  食べていただけます(でしょう)か？

Of course you could just say 食べてください, which would be the polite way to express
an imperative.

2 Suggestions (in order of increasing politeness/formality):

> 食べたらどう？  
>  食べたらどうですか？  
>  食べたらいかがでしょうか？

but I would be careful about using these when it's really not a suggestion,
but a request that I'm making, since that could sound sarcastic and rude.
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2012-10-23T10:14:58.423
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> Is it common to say "食べるのだろうか？".

It sounds to me like:  
"I wonder if (someone) is going to eat (something)"/  
"Is (someone) going to eat (something)?" or  
"I wonder if (someone) regularly eats (something)"/  
"Does (someone) regularly eat (something)?"  
食べるんだろうか / 食べるのかな would sound more casual.  

> You could also say "食べようか？". First off, is that conjugating it right;
> second, is that something you would say; finally, how is it different from
> -の+だろう?

  1. Yes.  

  2. Yes, you would say this to mean "Let's eat, shall we?" or "Shall I eat?" depending on the context.   

  3. Hmm... I think だろう is like "I think ~~ will ~~" and よう is like "I'm going to~/intend to do~" or "Let's~".   

> "Would you...?" or "Why don't you..?"

For requests I would say:

> 食べてくれる？(casual)  
>  食べてくれますか？(polite)  
>  召し上がっていただけます？(politer) etc.  
>

For suggestions I would say:

> 食べたら？/ 食べれば？(very casual)  
>  食べない？(casual)  
>  食べません(か)？(polite)  
>  召し上がりません(か)？(politer) etc.  
>
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answer