One of the core classes at Hogwarts is Potions. Three potions taught in this class are the antidote to common poisons, the forgetfulness potion and the Swelling Solution. In order to brew one antidote to common potions you need one gram of unicorn Horn, one bezoar and two mistletoe berries. To brew a forgetfulness potion you need three grams of unicorn horn, two bezoars and three mistletoe berries. To brew a swelling solution you need three grams of unicorn Horn, one bezoar and two mistletoe berries. The currency at Hogwarts are called Galleons. Harry can sell an antidote to common poisons for one Galleon, a forgetfulness potion for 4 Galleons and a Swelling Solution for 3 Galleons. Harry has 100 grams of unicorn horn, 20 bezoars and 40 mistletoe berries. Which potions should he brew to maximize his profit? Formulate the problem as a linear programming model. Explain the model and comment on the fact that variables can take fractional values. Solve the LP using the Simplex method, step by step. Remember to explain your calculations. What is the solution to the problem? How many Galleons does Harry earn?