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?

(In reply to re: Not sure if this is right, but.... by Charlie)

Hey Charlie:

I agree with you.  My addition was wrong.  However, I have a new problem.  I got the same wrong answer another way!  And everything I have tried to do to check works out correctly.  How do I do this to myself?  Check it out:

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Hmmm, I think these are the various rates:

C = P*3/180
G = P/90
S = P/180
X = P/180

So the sum is still P*7/180, so my answer for how many days it will take is still 180/7 days.

I just solved it using matrix stuff. I started with the matrix

( c   g  s   x  = _ )
|45 45  0  -45 = P|
|60 0  60 -60 = P|
|90  0   0  -90 = P|
| 0  90 90 -90 = P|

When I reduced it I got:

|1 1  0 –1 = P/45 |
|0 1 -1  0 = P/180|
|0 0  1  0 = P/180|
|0 0  0  1 = P/180|

Using back substitution, starting with the last equation, I get the various rates.

I then plugged those rates back into the equations, and everything checked out.

I just did this to prove my other solution to myself. I still like my other solution better… it’s cleaner. Though it’s weird that in the other solution I didn’t need the 4th equation at all.

Posted by nikki on 2004-10-12 19:53:39