Balance the redox reaction by oxidation number method.
`MnO_(4)^(-)(aq)+Br^(-)(aq)toMnO_(2)(s)+BrO_(2)^(-)(aq)` (Basic medium)

Explanation

`MnO_(4)^(-) overset(1_(cancel(3))"units")(to)MnO_(2)to` Reduction
`Br^(-)overset(2_(cancel(6))"units")(to)Bro_(3)^(-)to`Oxidation
`x^(1y)` The reduction eqn by 2 units
and oxidation eqn by 1 unit
`{:(MnO_(4)^(-)" " to " "2MnO_(2)),(Br^(-)" "to" "BrO_(3)^(-)),(______________________________),(2MnO_(4)^(-)+Br^(-)to2MnO_(2)+BrO_(3)^(-)):}`
As the Reaction occurs in basic medium to equal ionic charges `20 H^(-)` are added on RHS. To balance hydrogen atom add `H_(2)O` on LHS.
`2MnO_(4)^(-)+Br^(-)+H_(2)Oto2MnO_(2)+BrO_(3)^(-)+2OH^(-)`