Firstly we would determine the oxidation number of the elements to determine which is undergoing oxidation and which is undergoing reduction:
[tex]\begin{gathered} Mn\text{ }in\text{ }MnO_4^-:Mn+(-2\times4)=-1 \\ Mn\text{ }in\text{ }MnO_4^-:Mn-8=-1 \\ Mn\text{ }in\text{ }MnO_4^-:Mn=-1+8=+7 \\ \\ Mn\text{ }in\text{ }MnO_4^{2-}:Mn+(-2\times4)=-2 \\ Mn\text{ }in\text{ }MnO_4^{2-}:Mn=-2+8=+6 \end{gathered}[/tex]Mn oxidation number goes from +7 to +6 which means it is undergoing a reduction. A decrease in oxidation number means reduction.
[tex]\begin{gathered} S\text{ }in\text{ }HSO_3^-:1+(-2\times3)+S=-1 \\ S\text{ }in\text{ }HSO_3^-:1-6+S=-1 \\ S\text{ }in\text{ }HSO_3^-:S=-1+5=+4 \\ \\ S\text{ }in\text{ }SO_4^{2-}:S+(-2\times4)=-2 \\ S\text{ }in\text{ }SO_4^{2-}:S-8=-2 \\ S\text{ }in\text{ }SO_4^{2-}:S=+6 \end{gathered}[/tex]S oxidation goes from +4 to +6 meaning it is undergoing oxidation. There is an increase in oxidation number.
We will now balance the oxidation half reaction:
[tex]HSO_3^-+H_2O\rightarrow SO_4^{2-}+3H^++2e[/tex]The coefficient for the water molecule is 1 in the balnced half reaction.
