What would be the final temperature of the mixture when five grams of ice at $-{{10}^{0}}C$ are mixed with twenty grams of water at ${{30}^{0}}C$. Specific heat of ice and latent heat of water are $0.5,80ca{{\lg }^{-1}}$ respectively.

Hint: The total heat lost by the water will be equal to the sum of heat energy needed to change the temperature of ice from given to zero degrees and latent heat to change the state of matter and heat absorbed by water which is melted ice.
Formulas used:
$ms\Delta T={{m}_{1}}{{s}_{1}}\Delta T+{{m}_{1}}L+{{m}_{1}}{{s}_{1}}{{(\Delta T)}_{2}}$

Complete answer:
Let us write down all the given values,
mass of ice, mass of water, temperature of ice, temperature of water, specific heat of ice, latent heat of water and final temperature as ${{m}_{1}}=5g,m=20g,{{T}_{2}}=-{{10}^{0}}C,{{T}_{1}}={{30}^{0}}C,{{s}_{1}}=0.5ca{{\lg }^{-1}}^{0}C,L=80ca{{\lg }^{-1}},T$
Now, the total heat lost by water is equal to the sum of the energy needed to change ice to zero degrees plus the latent heat to change the state of ice plus the heat absorbed to get into final temperature.
That is,
$\begin{align}
  & ms\Delta T={{m}_{1}}{{s}_{1}}\Delta T+{{m}_{1}}L+{{m}_{1}}{{s}_{1}}{{(\Delta T)}_{2}} \\
 & 20(1)(30-T)=5(0.5)(0+10)+5(80)+5(1)(T-0) \\
 & 600-20T=25+400+5T \\
 & T={{7}^{0}}C \\
\end{align}$
Therefore, the final temperature of the mixture will be ${{7}^{0}}C$.

Additional information:
Specific heat is the energy required to change the temperature of one gram of any material by just one degree. The units for specific heat are energy per unit mass per degree. Latent heat is the net energy required or absorbed to change the state of a matter that could be solid, liquid or gas of a unit mass of any material without changing the temperature of the material. The latent heat associated with melting a solid or freezing a liquid is called latent heat of fusion and with that of vapour is latent heat of vaporization.

Note:
There will not be any change in temperature of the material when latent heat is released or absorbed. Also, specific heat of any material is independent of the temperature of the material. The heat capacity of water is abnormal because of the hydrogen bonding between the molecules.