What is the chemical equation for a dry cell battery with electrolytes?

Cathode: NH4+(aq) + MnO2(s) + H2O(l) + e- → Mn(OH)3(s) + NH3(aq)

Anode: Zn(s) → Zn2+(aq) + 2e-

NH4Cl(aq) + MnO2(s) + H2O(l) + Zn(s) → Mn(OH)3(s) + NH3(aq) + Zn2Cl(aq)