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?

Let the amount of grass on the whole pasture at the given time be 1 unit.
Let the rate of grass eating of the goat be g.
Let the rate of grass eating of the cow be c.
Let the rate of grass eating of the sheep be s.
Let the rate of growing of the grass be x.

All the rates are in the above arbitrary whole pasture units per day.

45(c+g-x)=1
60(c+s-x)=1
90(c-x)=1
90(g+s-x)=1

This set solves to:

c=5/180; g=1/90; s=1/180; x=1/60

c+g+s-x = 1/36

So it will take 36 days of combined effort to denude the pasture.

This assumes that in no instance does any portion of the field become prematurely denuded to the extent that it no longer grows, as we assume that the rate x does not decrease as time goes by.

Since farmer Joe "occasionally lets them eat the grass", all the figures in the puzzle and answer must be dependent on the state of the field at some particular time.  If the animals arrived just one day later, all the times would be increased by a factor of 61/60; two days later, 62/60; etc., as there would be more than 1 unit of grass in the field at the start.  In general, it would depend on how long it was since the last "occasional" visit.

Posted by Charlie on 2004-10-12 13:53:10