When I first read through this problem I thought it was so easy. I completed it so quickly and wondered even why we had been assigned this. Then I realized that I didn’t read the whole thing and that I needed to have 100 animals. So then I went about solving the ACTUAL problem. I started off the way I start off any problem: write down what I know. Doing this led to two equations and three variables, which I knew I couldn’t solve. There had to be something else that I could extract from the problem. Combing through it again, this time much more frustrated, I decided to try and take the constraint on having at least one of each animal and turn that into something that I could use. All that resulted in, though, was my changing my original equations so that at least one of each animal would be accounted for. The only way I could think to solve this was to start substituting guesses in for how many sheep there were, thus reducing my two equations to two variables and therefore making them solvable. I started off by guessing there were 51 sheep. I knew that I needed an odd number of sheep because I had already accounted for one and I needed to bring the cost to $100 even. This gave me a weird answer that didn’t work within the constraints of the situation so I moved on and tried 85 sheep. This didn’t work either. I had the feeling that there was something I was missing, so I took a break and went back to the problem a few hours later. That break was exactly what I needed. After looking at the algebra that I had used to solve my previous attempts I saw that I could generalize this situation to help me make better guesses for the number of sheep. My previous solutions kept giving me negative answers, so I needed to rework my guesses so that I could produce positive answers. The cost of my pigs, for example, had to be less than the remaining total after I purchased all my sheep (and one of each animal). So that meant that 3(97-s)<86.5-0.5s. If I solved this then I could find a restriction on s, thus narrowing down the possibilities for me to begin my guessing. This led me to the fact that I had to have more than 81.8 sheep, so I began the remainder of my tests there. After some time I found my solution. It was annoying to solve this through an educated guess and check method, and I wish I could have remembered a more efficient way to do so, but sometimes math is all about the long, tedious, number crunching!
Cow, Pig, and Sheep Problem