Melting of ice with water thermodynamics problem

[Satis24]

1. 0.500kg of ice at -5.00°C is put into a vessel containing 1.80kg of water at 17.00°C and mixed together, the result being a mixture of ice and water at 0.00°C. Calculate that final masses of ice and water, taking the water equivalent of the vessel as 0.148kg, the specific heat of ice as 2.04kJ/kg*K, and the latent heat of fusion of ice as 335kJ/kg
2. Q = mcΔT and Q = mLf
3. I started by finding Q = mLf of the ice. Q = (0.500kg)*(335kJ/kg*K) = 167.5kJ However I wasn't sure exactly sure where to go from here. I know that the mass of the vessel has to be added into the mass of ice at come point, but I'm really unsure of where to go from here.

 

[haruspex]

I started by finding Q = mLf of the ice. Q = (0.500kg)*(335kJ/kg*K) = 167.5kJ

But the question implies not all the ice melts, so the above number is too high.
In general, there are three possible end states:
- the ice warms but none melts
- some but not all of the ice melts
- all of the ice melts
You're told all ends up at 0C, so you can safely assume the middle case.
Introduce a variable unknown for the mass of ice that does melt. Using that, write an equation relating the heat lost by the water and the vessel to that taken up by the ice. Can you calculate those two quantities?

 

[Satis24]

Yes I should be able to figure that out, thank you!