Isn't it true that if you divide profit by shares outstanding you get your earnings per shares

I believe so

Since earnings is the same as profit, yes. But you also have to account for any preferred dividends coming off first, if there are any. But the division works. Something PER something is always a division, in that order, regardless of what it is. (But schools seem to be teaching plug n chug these days and not getting the concept behind it across.)