i'm buying a gas heater that costs $700 plus $250 per year for operating costs.
but i'll save 40% on the operating costs (so i'll save $100.).
how long until the heater pays for itself?
Do you mean $100 per year is what you save by saving 40% on operating costs prior to buying the gas heater?? If so, then your prior operating costs are also $250. So buying the heater saves you nothing and so it never pays for itself.
Or do you mean that after buying the gas heater, you're now saving $100 per year on the operating costs you previously had. In which case it takes 7 years (7 x 100 = 700) for the gas heater to pay for itself. (Except in this day and age of built-in-obsolescence, the heater will probably go kaput well before 7 years have elapsed. The warranty will probably only be for 12 months and all ....)