Two tanks $A$ and $B$ contain water at $30,^{\circ}C$ and $80,^{\circ}C$ respectively. Calculate the amount of water that must be taken from each tank to prepare $40,kg$ of water at $50,^{\circ}C$.

$10\; g$ of ice cubes at $0\;^{\circ} C$ are released in a tumbler (water equivalent $55\; g$) at $40\;^{\circ} C$. Assuming that negligible heat is taken from the surroundings,the temperature (in $^{\circ} C$) of water in the tumbler becomes nearly $(L_f = 80\; cal/g)$.

The temperature of $100 \, g$ of water is to be raised from $24^{\circ} C$ to $90^{\circ} C$ by adding steam at $100^{\circ} C$ to it. The mass of the steam required for this purpose is ........... $g$.

$50\, g$ of ice at $0^{\circ}C$ is mixed with $50\, g$ of water at $80^{\circ}C$. The final temperature of the mixture will be........ $^{\circ}C$.

$A$ metal ball of mass $0.1 \, kg$ is heated up to $500 \, ^\circ C$ and dropped into a vessel of heat capacity $800 \, J K^{-1}$ containing $0.5 \, kg$ of water. The initial temperature of the water and the vessel is $30 \, ^\circ C$. The approximate percentage increment in the temperature of the water is ........ $\%$. [Specific heat capacities of water and metal are $4200 \, J kg^{-1} K^{-1}$ and $400 \, J kg^{-1} K^{-1}$ respectively.]

$A$ small quantity,mass $m$,of water at a temperature $\theta$ (in $^oC$) is poured onto a large mass $M$ of ice which is at its melting point. If $c$ is the specific heat of water and $L$ is the latent heat of fusion of ice,then the mass of ice melted is given by