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?

The problem does not say that in all given animal combinations the level (or volume?) of grass on zero date is same. Also, the rate of growth of grass is constant. Without such assumptions there cnnot be any solution.

Tristan's 'Dimentional analysis' - (actualy plain algebra) does give the solution with the assumptions implied.

An alternative algebra equations - I think easier to understand would be:

Lets set variables: x, c, g, s
It takes x days for grass to grow to volume on zero date
Cow can eat the volume on zero date (assuming no growth) in c days
Goat can eat the volume on zero date (assuming no growth) in g days
Sheep can eat the volume on zero date (assuming no growth) in s days

Now from the given eating rates we can set up equations:
1/c +1/g = 1/45 + 1/x -- (1)
1/c + 1/s = 1/60 + 1/x -- (2)
1/c = 1/90 + 1/x -- (3)
1/g + 1/s = 1/90 + 1/x -- (4)

Taking 1/c value in (3) to (1) and (2) and so on you will get 1/c + 1/g + 1/s = 1/36 + 1/ x which means in 36 days the pasture will be bare.

Posted by ravi on 2004-10-15 04:15:11