A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe mix and how many bags of

standard mix of Peanut$/$Raisin Delite to put up$.$ The deluxe mix has $\frac{2}{3}$ pound raisins and $\frac{1}{3}$ pound peanuts, and the standard mix has

$\frac{1}{2}$ pound raisins and $\frac{1}{2}$ pound peanuts per bag. The shop has $90$ pounds of raisins and $60$ pounds of peanuts to work with. Peanuts

cost $$\mathrm{.}60$ per pound and raisins cost $$1\mathrm{.}50$ per pound. The deluxe mix will sell for $$3\mathrm{.}40$ for a one$-$pound bag, and the standard mix will

sell for $$3\mathrm{.}05$ for a one$-$pound bag. The owner estimates that no more than $160$ bags of one type can be sold.

a$.$ If the goal is to maximize profits, how many bags of each type should be prepared?

b$.$ What is the expected profit?