When trying to show possession in regard to a family owned business, would it be proper to display "Harpers' Bar" or "Harper's Bar". What is the difference between the two?
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The apostrophe is used for two things only:

To signify two words joined: What is = What's

To signify possesion, here's the confussion:

Your bar would be "Harper's Bar" because there's only one "Harper".

If you were all nuns it would be "Nuns' Bar" - Many nuns - the apostrophe is used after the "s"

or

"At the zoo, the children were most interested in seeing the lions' den." - Many lions
Just to add to Jason13_32's reply, there is another use for apostrophe that MLA recommends. Although at present, an apostrophe is no longer needed for plural numbers, letters, abbreviations, and words mentioned as words, it makes sense to put one on some, if not on all, of them.

Take this for example:

A's is not as vague as As.
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This is what I know:

If the family name is Harper, then the correct way to write it would be Harper's.
If the family name is Harpers, then Harpers's would be the correct one.
If the noun is singular and ends in s, then add 's.
If the noun is plural and ends in s, then add ' only.
I would like to clear the last of the two rules I posted just above this post.

...
If the noun is plural and ends in s, then add ' only.
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what about James? James's or James'?
It's James's, Maj.
You are probably right; but what about Cervantes' Don Quixote or Socrates' ideas?
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