Let x represent the number of kilograms of the 1/2:1/2 mix, and y the number of kilograms of the 3/4:1/4 mix. (Here the ratio is nuts to raisins.)
The amount of profit can be represented by P=7x+9.5y.
There are some constraints -- namely the amount of raw materials available. The constraints can be represented by a system of inequalities:
`x>=0,y>=0` are the natural constraints.
`1/2 x+3/4y<=120` represents the amount of nuts available. Each kg of mix x uses 1/2 kg of nuts, while each kg mix y uses 3/4 kg.
`1/2x+1/4y<=90` represents the amount of raisins available.
Graphing the system of inequalities yields a feasible region (in this case a bounded polygon) where every point in the region satisfies all of the inequalities. By the corner point principle, there is a maximum and a minimum for the objective function (the profit function in this case) and they occur at the corner points of the feasible region.
The points are (0,0),(0,160),(180,0) and (150,60). These points represent, in order, making no product, only producing 160kg of product y, only producing 180kg of product x, and finally producing 150kg of x and 60kg of y.
Substituting the values into the objective function we get:
Thus your maximum profit occurs when you produce 150kg of x, and 60kg of y.
(Note that 150kg of x uses 75kg nuts and 75kg of raisins. 60kg of y uses 45kg of nuts and 15kg of raisins. Thus we use a total of 120kg of nuts and 90kg of raisins.)