A calorimeter contains 400gm of water at a temperature of $5^{0}C$ .Then 200gm of water at a temperature of $+{10}^{0}C$ and 400gm of ice at a temperature of $-{60}^{0}C$ are added. Specific heat capacity of water 1000 cal/kg K. Specific latent heat of fusion of ice $=80\times 1000 \mathrm{cal}/\mathrm{kg}$. Relative specific heat of $\mathrm{ice}=0.5$ . Ratio between masses of ice and water in equilibrium state is______ 

Equilibrium temperature of 200g water mixed with 400g water is let say $\theta$.

$200\times 1\times \left(10-\mathrm{\theta }\right)=400\times 1\times \left(\mathrm{\theta }-5\right)$

$\Rightarrow \mathrm{\theta }=\frac{20}{3}{}^{\mathrm{o}}\mathrm{C}$

Heat released by water to come at $0 {}^{\mathrm{o}}\mathrm{C}$.

${\mathrm{H}}_1=\left(600\times 1\times \frac{20}{3}\right)=4000 \mathrm{cal}$

Heat gained by ice to come at $0 {}^{\mathrm{o}}\mathrm{C}$.

${\mathrm{H}}_2=400\times 0.5\times 60=12000 \mathrm{cal}$

Some mass of water will get freeze by releasing remaining 8000 cal.

$\mathrm{Q}=\mathrm{mL}$

$\mathrm{m}=\frac{8000}{80}=100\mathrm{g}$

Presently, mass of water is 500g and mass of ice is also 500g.