Attached is the Ca$-$Mg phase diagram; the solubility of Ca in Mg is $0\mathrm{.}3$ wt$\%$ and that of Mg in Ca is $0\mathrm{.}2$ wt$\%$ at the applicable eutectic temperatures, respectively.

$($a$)$ Determine the composition of the alloys that will have $15\%$ primary Ca$3$Mg$4(44\mathrm{.}73$wt$\%$Mg$)$ after equilibrium solidification.

$($b$)$ Consider a Ca$-$Mg alloy containing $83$ wt$\%$ Ca$.$ What is the primary phase? What is the eutectic temperature? What is the fraction and composition of the eutectic for this alloy?