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?

If initial grass in pasture is 'A' rate of growth of grass is 'a' rate at which cow goat & sheep eat is c, g & s repectively then

1) A + 45 a = 45c + 45g

2) A + 60 a = 60c + 60s

3) A + 90 a = 90c

4) A + 90 a = 90g + 90s

From 3 & 4 we get

5) c=g+s

substituting it in 1 & 2 and from 1 & 4 we get

6) a=s

7) and A = 60(s+g)

substituting these two in 3 we get

8) c= s5/3+g2/3

from 5 and 8 we get

9) s=g/2

hence when cow goat & sheep are grazing if no of days is 'd" then

60(s+g) +ds= d(s+g)+ds+dg

d=60(s+g)/(s+2g) substituting from 9

we get d=36 

Posted by salil on 2006-02-15 04:07:49