Undergraduate students who are pre-service teachers need to make connections between the college mathematics they are learning and the pre-college mathematics they will be teaching. Spanning a broad range of undergraduate curricula, this article describes useful lesser-known connections, explorations, and original proofs involving fractions. In particular, we use standard tools of number theory to investigate when an integer results from "bc/(b + c)", an expression that arises from the not uncommon situation of adding fractions with a common numerator.