Here is a problem that has been making the rounds of offices and stores. It has

Question

Here is a problem that has been making the rounds of offices and stores. It has most employees stumped. Use our skill to solve it.

A farmer has $100 to buy 100 chickens. Roosters cost $5 each, hens $3 each, and baby chicks 5 cents each. How many of each does the farmer buy if he must buy at least one of each and pay exactly $100 for exactly 100 chickens?

Do the following:
a) define all variables, and set up a system of equations that describes the example.
b) using any algebraic method, solve the problem.
c) using information obtained in #4 b above, determine an optimum viable solution.

Explanation / Answer

First off we can write out two equations Let us denote x=# of roosters y=# of hens z=# of baby chicks 5x+3y+0.05z=100 x+y+z=100 Now we can put this into matrix form Divide row1 by 5 1 3/5 1/100 20 1 1 1 100 Add (-1 * row1) to row2 1 3/5 1/100 20 0 2/5 99/100 80 Divide row2 by 2/5 1 3/5 1/100 20 0 1 99/40 200 Add (-3/5 * row2) to row1 1 0 -59/40 -100 0 1 99/40 200 Now we need to guess a solution for the first equation: x-59/40z=-100 We guess z=80 because we need to make sure x has a positive value. Therefore x=18 We then plug in these values into the second equation and get y=2

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