Identify the oxidation and reduction half-reactions. MnO4⁻ is reduced to MnO2, and Br⁻ is oxidized to BrO3⁻.
Balance the atoms other than oxygen and hydrogen in each half-reaction. Mn and Br are already balanced.
Balance the oxygen atoms by adding H2O molecules. For MnO4⁻ to MnO2, add 2 H2O to the right side. For Br⁻ to BrO3⁻, add 3 H2O to the left side.
Balance the hydrogen atoms by adding OH⁻ ions. For MnO4⁻ to MnO2, add 4 OH⁻ to the left side. For Br⁻ to BrO3⁻, add 6 OH⁻ to the right side.
Combine the balanced half-reactions, ensuring that the electrons cancel out, and verify that the final equation is balanced in terms of atoms and charge.