(a) In order to treat permanent hard water, slaked lime is used, magnesium hydroxide and calcium carbonate precipitates out and can be easily filtered.
MgSO4 + Na2CO3 + Ca(OH)2 ⟶ Mg(OH)2 ↓ + CaCO3 ↓ + Na2SO4
(b) Washing soda is added to hard water, which results in settling down of insoluble carbonates which can be removed by filtration.
Washing sodaNa2CO3+Calcium sulphateCaSO4⟶Calcium carbonateCaCO3↓+Na2SO4
Washing sodaNa2CO3+Calcium chlorideCaCl2⟶Calcium carbonateCaCO3↓+2NaCl