Consider the hypothetical eutectic phase diagram for metals A and $B,$ which is similar to that for the lead$-$tin $(Pb-Sn)$ system $($Figure given above$).$ Assume that $(1)$ and $$ phases exist at the A and B extremities of the phase diagram, respectively; $(2)$ the eutectic composition is $47wt\%B-53wt\%A$; and $(3)$ the composition of the $$ phase at the eutectic temperature is $92\mathrm{.}6wt\%B-7\mathrm{.}4wt\%A.$ Determine the composition of an alloy that will yield primary $$ and total $$ mass fractions of $0\mathrm{.}356$ and $0\mathrm{.}693,$ respectively.