  The Duffin--Schaeffer Conjecture answers a question on how well one can
approximate irrationals by rational numbers in reduced form (an imposed
condition) where the accuracy of the approximation depends on the rational
number. It can be viewed as an analogue to Khintchine's Theorem with the added
restriction of only allowing rationals in reduced form. Other conditions such
as numerator or denominator a prime, a square-free integer, or an element of a
particular arithmetic progression, etc. have also been imposed and analogues of
Khintchine's Theorem studied. We prove versions of Khintchine's Theorem where
the rational numbers are sourced from a ball in some completion of $\mathbb{Q}$
(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.
