# Linear Programming Word Problem With 3 Variables

A company makes three types of candy and packages them in three assortments. Assortment I contains 4 sour​, 4 lemon​, and 12 lime ​candies, and sells for ​$9.40. Assortment II contains 12 sour​, 4 lemon​, and 4 lime ​candies, and sells for ​$7.60. Assortment III contains 8 sour​, 8 lemon​, and 8 lime ​candies, and sells for $11.00. Manufacturing costs per piece of candy are ​$0.20 for soursour​, ​$0.25 for lemon, and$ 0.30 for lime. They can make 5,200 sour​, 3,800 lemon​, and 6,000 lime candies weekly. How many boxes of each type should the company produce each week in order to maximize its​ profit? What is the maximum​ profit? I'm struggling with getting these into an equation(s) that I can solve

ass1 [  4s + 4le + 12li =  9.4]
ass2 [ 12s + 4le +  4li =  7.6]
ass3 [  8s + 8le +  8li = 11.0]

s=  0.2
l=  0.25
li= 0.3


I'd guess you would want to put these into a matrix form and then reduce it,

 4  4  12 | 9.4
12  4   4 | 7.6
8  8   8 | 11.0


Let $n_1$ be the number of assortment of type I. Let $n_2$ be the number of assortment of type II. Let $n_3$ be the number of assortment of type III.We have the following constraints for sour, lemons and limes respectively \begin{eqnarray*} 4n_1+12n_2+8n_3 \leq 5200 \\ 4n_1+4n_2+8n_3 \leq 3800 \\ 12n_1+4n_2+8n_3 \leq 6000 \\ \end{eqnarray*} Let $C$ be the cost to make these \begin{eqnarray*} C=0.2(4n_1+12n_2+8n_3)+0.25(4n_1+4n_2+8n_3)+0.3(12n_1+4n_2+8n_3 )= ... \end{eqnarray*} Let $P$ be the price they will be sold at \begin{eqnarray*} P=9.4 n_1 +7.6 n_2 +11 n_3 \end{eqnarray*} Then the profit is $P-C$ and the linear problem is to maximise $P-C$ subject to the the sour,lemon & lime constraints.