The distinction is not between old and new.
Generally speaking, "a" refers to a nonspecific person or thing while "the" refers to a specific person or thing.
A...Let's take a cab to the stadium. (there are many cabs but only one stadium that we want to go to)
B...Good idea, let's take the empty cab waiting over there. (B is referring to the specific cab waiting over there)
In your question, I would choose "the" to refer to the fare that you will split. (actually this is not the best example to illustrate the difference and some people may choose "a", but I wouldn't)
Student or Learner