Fe2+ + MnO4 -> Fe3+ + Mn2+

NO3- + S2-+ -> S+ NO

NH3+NO2 -> N2+H2O

Assign oxidation numbers to each element
Write half reactions for each
Balance these half reactions

For the given reactions:

1) Fe2+ + MnO4- -> Fe3+ + Mn2+

Oxidation numbers:
Fe2+ --> +2
MnO4- --> +7

Half-reactions:
Oxidation half-reaction: Fe2+ --> Fe3+
Reduction half-reaction: MnO4- --> Mn2+

2) NO3- + S2- -> S + NO

Oxidation numbers:
NO3- --> -1
S2- --> -2

Half-reactions:
Oxidation half-reaction: S2- --> S
Reduction half-reaction: NO3- --> NO

3) NH3 + NO2- -> N2 + H2O

Oxidation numbers:
NH3 --> -3
NO2- --> +3

Half-reactions:
Oxidation half-reaction: NH3 --> N2
Reduction half-reaction: NO2- --> H2O

Balancing of the half-reactions:

1) Fe2+ --> Fe3+ (oxidation)

Fe2+ -> Fe3+
To balance the charges, we need to add one electron on the left side:
Fe2+ + 1e- -> Fe3+

2) MnO4- --> Mn2+ (reduction)

MnO4- -> Mn2+
To balance the charges, we need to add five electrons on the left side:
MnO4- + 5e- -> Mn2+

3) S2- --> S (oxidation)

S2- -> S
To balance the charges, we need to add two electrons on the right side:
S2- -> S + 2e-

4) NO3- --> NO (reduction)

NO3- -> NO
To balance the charges, we need to add three electrons on the left side:
NO3- + 3e- -> NO

5) NH3 --> N2 (oxidation)

NH3 -> N2
To balance the charges, we need to add three electrons on the right side:
NH3 -> N2 + 3e-

6) NO2- --> H2O (reduction)

NO2- -> H2O
To balance the charges, we need to add two electrons on the left side:
NO2- + 2e- -> H2O