I was going through an algebra book to brush up in anticipation of possibly going back to school (although I took Calculus in high school, it's been 18 years...). In an end-of-chapter quiz I came up with a solution to a problem as:

A

- -1 (A over P minus one)

P

But the the answer key for the quiz gave the answer as:

A-P

--- (A minus P over P)

P

Near as I can tell, those are equivalent. Am I right or wrong? If I'm right, is there a reason one is preferred over another?

...but we don't have A&P stores in North Texas...

Seriously, though, I think that A over P minus 1 is the preferred method, but I cannot give you a specific reason as to why. This is just the style of algebra that I learned.

That, and the first method is faster, easier, and reduces steps to resolve solution. In algebra, always take the fastest formula.

But remember... I attended public school in Texas, so I may not have any clue whatsoever...

Your solution is to be logically preferred, because it takes the relation nearer to being 'solved'. However, in a specific context where P has some extrinsic significance, the other solution may be used to emphasize the constant, as opposed to the variable, contribution of P to the relation, even though this resolves as unity.

Think of the two in plain language rather than in mathematics. The (unmentioned) quantity Q is equal to 1) the difference between A and P divided by P, or 2) one less than the quotient of A over P.

de