Let B be a subgroup of the arbitrary abelian group C. A simultaneous decomposition of B and C is a pair of decompositions \(B=\oplus_{i\in I}B_ i\) and \(C=\oplus_{i\in I}C_ i\) such that \(B_ i=B\cap C_ i\) for each i. The question of simultaneous decompositions is attacked in the case that C is a (mixed) direct sum of cyclic groups and B is a pure subgroup. The main tool is Theorem 1: Let C and C' be coproducts of cyclic groups with pure subgroups B and B' respectively. There exists an isomorphism \(\pi\) : \(C\to C'\) that maps B onto B' if and only if \(B\cong B'\) and C/B\(\cong C'/B'\). Moreover, if the isomorphism \(\phi\) : C/B\(\cong C'/B'\) is given then \(\pi\) can be chosen so that it induces \(\phi\). As a sample for the results obtained we state Theorem 4: Let B be a pure subgroup of the coproduct of cyclic groups C. Then B and C have a common summand isomorphic to a given coproduct of cyclic groups K if and only if (i) torsionfree rank (K)\(\leq\) torsion-free rank (B), (ii) for each prime p, \(f_ n(K_ p)\leq f_ n(B_ p)\), and (iii) if \(B_ p\) is unbounded, \(f_ n(K_ p)