Question

How many different ways can you write a fraction that has a numerator of 2 as a sum of fractions

Answer 1

Infinitely many ways! 
Suppose you have the fraction 2/d.

Pick any pair of integers a and b where b ≠ 0.
Then 2b-ad is and integer, as is bd so that (2b - ad)/bd is a fraction.

Consider the fractions a/b and (2b - ad)/bd

Their sum is 
a/b + (2b-ad)/bd = ad/bd + (2b-ad)/bd = 2b/bd = 2/d - as required.

Since a and b were chosen arbitrarily, there are infinitely many possible answers to the question.