Question

determine the temperature that result when 1.0 kg of ice at 0 degree Celsius is mixed with 9 kg water at 50 degree Celsius.

Answer 1

you should use
Loss of energy = Gain in energy
here ice will lose energy while the water will gain the same amount of energy. here's the link where you can find these formulas and a fine explanation of this topic. you can read it all or just skip to 'Relating the Quantity of Heat to the Temperature Change' part which is of the concern here.
feel free if you get stuck somewhere.

Answer 2

oops, the link
http://www.physicsclassroom.com/class/thermalP/u18l2b.cfm