Two half reactions are
a. . `Mn^(2+)rarrMnO_(4)^(ө)`
Balancing the these two half reactions:
`2Mn^(2+)+4H_(2)OrarrMnO_(4)^(ө)+5e^(-)+8H^(o+)`
Adding the two equations, we have the balanced equation. `2Mn^(2+)+5S_(2)O_(8)^(2)+8H_(2)Orarr2MnO_(4)^(ө)+6H^(o+)+10HSO_(4)^(ө)`
b. `S_(2)O_(8)^(2-)rarrHSO_(4)^(ө)`
`S_(2)O_(8)^(2-)+2H^(o+)+2e^(-)rarr2HSO_(4)^(ө)`