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Recently I had a haircut in one of the local barber shops, during which I asked the owner “How long has the shop been open?” However, I feel something wrong in that sentence, but I couldn’t put my finger on it. First of all, is it grammatically correct? Second, Is there some ambiguity in that sentence? And how will you change some of the wording of that question to make it sounds better?

Another thing I want to ask about is that how shall the owner answer this question if his shop will be 25 years old in Dec 15th. Is it correct for him to say that “It will be 25 years this December 15th” or “25 years come next December?” Is there a better way to put it?

The ambiguity lies in the two states of openness for the shop: one refers to when it was opened for the first time and the other to when it opened for business on that particular day. The responses for the former could be a month and year (e.g., "since March 1993"), while for the latter they would be an hour, (e.g., "since 8:00 this morning").

So if you want to prevent ambiguity you can include a reference to either.

For today, simply:

How long has the shop been open today?

For the entire lifespan:

How long has the shop been in business?

And yes, it's fine to say "It will be 25 years this December 15th," and so is "25 years come next December," although if you mean the December of the year you are currently in it's fine to shorten that to "25 years come December."