Steam at 100 degree celcius is added to ice at 0 degree Celcius

a) find amount of ice melted and the final temperature when mass of steam is 10 g and mass of ice is 50g.

hv = heat of vaporization of steam
hf = heat of fusion of ice
cp = specific heat of water
M = original mass of steam = 10 g = 0.010 kg
Th = original temp of steam = 100 C
m = original mass of ice
Tc = original temp of ice = 0 C
T = final temp of melted ice and condensed steam

heat lost by steam in being condensed to water at 100 C = Qa = (M)(hv) = 10hv
heat lost by steam in being cooled as water to T° = Qb = M(cp)(100 - T)
heat gained by ice in being melted into water at 0 C = Qc = m(hf)
heat gained by water from ice being warmed to T° = Qd = m(cp)(T - 0)

Conservation of Heat:
Qa + Qb = Qc + Qd
M(hv) + M(cp)(Th - T) = m(hf) + m(cp)(T-Tc)
substitute KNOWNS from question:
0.01(hv) + (0.01)(cp)(100 - T) = m(hf) + m(cp)(T = 0)
altho "hv", "cp" and "hf" are also known - just look them up on Web or in textbook,
the TWO remaining unknowns are "m" and "T"
=> I never learned how to solve a single equation for 2 unknowns <=
so the only numeric solution is to assume that all of the ice did not melt, thus the FINAL temperature of the ice/water mix would be = T = 0°C => THEN there is only one(1) unknown and
U can, using the above equation, solve for m = mass of the ice <=