Break the equation into each of its two parts:
MnO4^- ---> MnO2
NH3 ---> NO3^-
Balance oxygens by adding H2O
Next, balance the hydrogens by adding H^+
MnO4^- + 4H^+ + 3e^- ---> MnO2 + 2H2O
NH3 + 3H2O ---> NO3^- + 9H^+ + 8e^-
Multiply the first equation by 8, and the second by 3;
8MnO4^- + 32H^+ + 24e^- ---> 8MnO2 + 16H2O
3NH3 + 9H2O ---> 3NO3^- + 27H^+ + 24e^-
Adding these two will give
8MnO4^- + 3 NH3 + 5H^+ ---> 8MnO2 + 3NO3^- + 7H2O
Now, add 5OH^- (since its acidic) to both sides, cancelling out the H2O
8MnO4^- + 3 NH3 + 5H2O ---> 8MnO2 + 3NO3^- + 7H2O + 5OH^-
8MnO4^- + 3 NH3 ---> 8MnO2 + 3NO3^- + 2H2O + 5OH^-
8KMNO4 + 3NH3 ---> 3KNO3 + 8MnO2 + 5KOH + 2H2O
I hope this helps..