$(1)$ At $-31\mathrm{.}2C,$ pure propane and $n-$butane have a vapor pressure of $1200$ and $200$Torr, respectively.

$($a$)$ Calculate the mole fraction of propane in the liquid mixture that boils at $-31\mathrm{.}2C$ at a pressure of $760$Torr.

$($b$)$ Calculate the mole fraction of propane in the vapor that is in equilibrium with the liquid of part $($a$).$

$(2)$ In an ideal solution of A and B$,3\mathrm{.}50$mol are in the liquid phase and $4\mathrm{.}75$mol are in the gaseous phase. The overall composition of the system is $Z_A=0\mathrm{.}300$ and $x_A=0\mathrm{.}250.$ Calculate $y_A.$