Diophantine approximation has its origins in the work of Diophantus: finding integer solutions to equations. Over the years this has morphed into studying rational approximations of real numbers. In particular, Dirichlet's Theorem begins the study of finding good rational approximations that are not too complex. While Dirichlet's Theorem tells us that every irrational number can be approximated ``well,'' Hurwitz and others showed that we cannot approximate every irrational ``too well.'' This naturally leads to studying how large the set of ``well'' approximable numbers are, landing us in the realm of metric number theory. Here, Khintchine's Theorem provides a beautiful answer to this: the set has either zero or full measure and this can be determined by the convergence of divergence of a particular series. The recently proven Duffin--Schaeffer Conjecture, a long-standing open problem in the field, can be viewed as an analogue to Khintchine's Theorem with the added restriction of only allowing rationals in reduced form. Other analogues of Khintchine's Theorem with restrictions have been studied, such as: numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. We prove versions of Khintchine's Theorem where the rational numbers are sourced from a ball in some completion of Q (i.e. Euclidean or p-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarn'ik's Theorem.