Farmer Joe owns a cow, a goat, and a sheep. The animals each eat grass at a constant rate, and the grass grows at a constant rate. And Farmer Joe occasionally lets them eat the grass on a small pasture of his.

• If the cow and the goat graze together, the pasture is bare after 45 days.
• If the cow and the sheep graze together, the pasture is bare after 60 days.
• If the cow grazes alone, the pasture is bare after 90 days.
• If the goat and the sheep graze together, the pasture is bare after 90 days, also.

How long will it take for the pasture to be bare if all three animals graze together?

Sorry, I’m a little confused as to how to approach this problem. Could someone (preferably SilverKnight since he wrote the problem) help me understand which thoughts are correct interpretations of the puzzle, and which are not? Thanks in advance!

At first my thought process was this: Ok, call the consumption rates of the animals c, g, and s, in grass units per day. Then call the grass’s rate of growth r, also in grass units per day. Now look at the first statement. In order for the cow and the goat to eat all the grass in 45 days, that means that c*45 + g*45 = r*45. Or c+g=r. Following that logic, I’ll get c+s=r. But that doesn’t make sense. That makes it sound like the goat and the sheep eat at the same rate, but clearly they eat at different rates since when they each eat with a cow they clear the field in different amounts of time. Looking more closely, c+g=r means that the cow and goat eat at the same rate as the grass growth. But that means that if there was any grass to start out with, then all the grass could never get all eaten.

So then I had a new thought process. There must be some grass to start out with. As the animals are eating the initial amount of grass, the grass continues to grow. Since the field always ends up bare in the above situations, that means that the animal’s rate of consumption must be GREATER than the grasses rate of growth. Otherwise they would never be able to eat what was already there AND what was grown.

Following in this manner, I’ll call the initial amount of grass in the pasture P. I will assume that in each of the above situations, P is the same (otherwise, how can I know?). So then I get the following equations

45c + 45g = P + 45r
60c + 60s = P + 60r
90c = P + 90r
90g + 90s = P + 90r

Yes, I can find some relationships this way, but I can’t get rid of my big problem… I have 4 equations but 5 unknowns.

Maybe I don’t need to know each animal’s exact rate (or the exact rate of the grass). I just need to know some relationships. But I still don’t feel like I’ve found the right track yet.

Any thoughts / confirmations? Thanks!

Posted by nikki on 2004-10-12 13:24:13