BaCl2 and MnCl2 are both soluble.
MnSO4 is soluble.
Hg2Cl2(s), Hg2SO4(s), and BaSO4(s) are insoluble
Hg(s) and HgO(s) form where one would expect Hg2(OH)2
Ba(OH)2(s) is slightly soluble, a precipitate will depend on the ion concentration
Mn(OH)2(s) is insoluble
Hg2^2+ + 2Cl- --> Hg2Cl2(s)
Ba2+ + Cl- --> no reaction
Mn2+ + Cl- --> no reaction
If there is no precipitate when Cl- added, there is no Hg2^2+ present.
Ba2+ + SO4^2- --> BaSO4(s)
Mn2+ + SO4^2- --> no reaction
If there is no precipitate when SO4^2-, added, there is no Ba^2+ present.
Mn2+ + 2OH- --> Mn(OH)2(s)
Since a reaction occurred with OH-, then Mn2+ must be present.