Question $2-1($Thermal Equilibrium$)$

$($a$)$ A small cup holding $0\mathrm{.}100$ kg of room temperature water at $20\mathrm{.}0C(293K)$ is dumped into a larger cup holding $0\mathrm{.}500$ kg of very hot water at $97\mathrm{.}0C(370K).$ Assuming the large cup is perfectly insulated from the surrounding room, calculate the final temperature of the combined $0\mathrm{.}600$ kg of water once it reaches thermal equilibrium. You may approximate the specific heat of water to be $4\mathrm{.}00k\frac{J}{kgK}.$

$($b$)$ An extremely cold ice cube with a mass $m_{\text{i}\text{c}\text{e}}=1\mathrm{.}50kg$ is taken from a freezer at $-150C(123K)$ and placed into the cup holding the combined $0\mathrm{.}600$ kg of water that is still at the same equilibrium temperature you calculated in part $($a$).$ Determine the final state of the system $($will all the ice melt? Will all the water freeze? Somewhere in between?$)$ and the final equilibrium temperature of the system. You may approximate the specific heat of ice as $2\mathrm{.}00k\frac{J}{kgK}$ and the latent heat of fusion for water as $300\mathrm{.}0k\frac{J}{k}g.$