A factory for the construction of quality furniture has 2 divisions: a machine shop where the parts of the furniture are fabricated, and an assembly and finishing division where the parts are put together into the finished product. Suppose that there are 12 employees in the machine shop and 20 in the assembly and finishing division and that each employee works an 8-hour day. Suppose further that the factory produces only two products: chairs and tables. A chair requires 384/17 hours of machine shop time and 480/17 hours of assembly and finishing time. A table requires 240/17 hours of machine shop time and 640/17 hours of assembly and finishing time. Assuming that there is an unlimited demand for these products and that the manufacturer wishes to keep all employees busy, how many chairs and how many tables can this factory produce each day?
Let t and c represent the number of tables and chairs produced each day, respectively. We want the total number of production hours to match the number available. .. (240/17)t + (384/17)c = 12*8 ... machine shop hours .. (640/17)t + (480/17)c = 20*8 ... finishing hours We can eliminate fractions by multiplying the first equation by 17/48 and the second equation by 17/160. .. 5t + 8c = 34 .. 4t + 3c = 17 These equations can be solved by any of the usual methods. Perhaps Cramer's rule is a useful method here. .. t = (34*3 - 17*8)/(5*3 - 4*8) = 17(6-8)/(-17) = 2 .. c = (5*17 - 4*34)/-17 = 17(5-8)/-17 = 3 Producing 2 tables and 3 chairs each day will keep the factory fully occupied.
