Craftsmen install 500 sq ft of ceramic tile and 100 sq ft of vinyl tile in one day. An apprentice installs 100 sq ft of ceramic tile and 200 sq ft of vinyl tile in one day. The firm has a job that requires 2000 sq ft of ceramic tile and 1600 sq ft of vinyl tile. Tessa pays craftsmen $200 per day and apprentices $120 per day. How many Craftsman and apprentices should she employ to minimize costs?

To minimize costs, we need to determine the number of craftsmen and apprentices that Tessa should employ.

Let's assume the number of craftsmen is x and the number of apprentices is y.

The amount of ceramic tile installed by craftsmen in one day = 500 sq ft
The amount of ceramic tile installed by an apprentice in one day = 100 sq ft
The total amount of ceramic tile required = 2000 sq ft

Thus, the time (in days) craftsmen take to install all the ceramic tile = 2000 / (500 * x)
The time (in days) an apprentice takes to install all the ceramic tile = 2000 / (100 * y)

Similarly, the time (in days) craftsmen take to install all the vinyl tile = 1600 / (500 * x)
The time (in days) an apprentice takes to install all the vinyl tile = 1600 / (100 * y)

Since they are working together, the time taken will be the same, so we can equate them:

2000 / (500 * x) + 2000 / (100 * y) = 1600 / (500 * x) + 1600 / (100 * y)

Simplifying the equation further:

4 / (x * 5) + 20 / (y) = 8 / (x) + 16 / (y)

Now, let's consider the cost. The cost for craftsmen per day = $200 and the cost for an apprentice per day = $120.

The total cost for craftsmen per day = $200 * x
The total cost for apprentices per day = $120 * y

The total cost for the job will be the product of the cost per day and the number of days taken, so:

Total cost for craftsmen = $200 * x * (2000 / (500 * x) + 2000 / (100 * y))
Total cost for apprentices = $120 * y * (2000 / (500 * x) + 2000 / (100 * y))

The total cost for the job will be the sum of the costs for craftsmen and apprentices:

Total cost = Total cost for craftsmen + Total cost for apprentices

Therefore, to minimize costs, we need to find the values of x and y that minimize the total cost.

This problem can be solved using optimization techniques such as linear programming or numerical methods. You can set up the objective function and constraints, and then use a solver or numerical optimization method to find the optimal values for x and y.