'will stay' or 'will be staying'?
This is quite a difficult question to answer. First off, I should say that if you used either of these forms you would be understood without difficulty and they are both asking for the same information. The answer would be a period of time, three weeks, 10 minutes, a couple of years, for example.
But which is the most natural, which are you likely to hear? Well first let's look at the different forms. How long will you stay in London? This is what's commonly called the 'future simple'. How long will you be staying in London is the 'future continuous', also called the 'future progressive'.
To understand the difference, I think it might be useful to look at an example showing the differences between a present simple and continuous.
Let's compare these two - "I work in London" and "I'm working in London". Both of these are similar in that they identify the place where I work. One form is present simple, "I work in London", the other, "I'm working in London", is present continuous.
So which is correct? Well, the answer is both of them are correct, depending on the attitude of the person who is speaking. One of the general meanings of simple verb forms is that they describe things that are seen to be a fact, a statement of what is believed to be true and therefore permanent. Often there isn't actually any time connection with the present simple. This might sound strange, but think of a sentence like - "Fish live in water" - this is a statement of fact which is always true, past present and future, there is no real time connection.
In our example, "I work in London", this is just my stating a fact about me in the same way.
Now, "I'm working in London" is a little different. There's more information here. Present continuous verb forms are often used to describe things which the speaker believes to be temporary or in progress. They started before now and will end sometime after now. So this suggests perhaps that if I say "I'm working in London", I don't necessarily believe that to be a permanent thing, I imagine that sometime in the future I might work somewhere else.
So, in very simple terms you could say that often simple verb forms are for permanent things and continuous verb forms for temporary things.
Now, let's go back to the original question - "How long will you stay?" or "How long will you be staying?"
I think it would probably be unusual for a native speaker to say "How long will you stay?" This is the future simple. Simple forms often go with permanent ideas - but if you are asking someone this question, then you believe that they will not stay permanently, their stay will be temporary, they're going to leave at some point. So I don't think it quite matches. I don't think we'd use the future simple to ask a question about a temporary condition.
If someone is visiting you or your country, I think it'll be much more likely that we'd ask, 'How long will you be staying.'
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