Solution to:
Green Green Grass
Some assumptions:
 The cow, the goat, and the goose eat grass with a constant speed
(amount per day): v1 for the cow, v2 for the goat, v3 for the goose.
 The grass grows with a constant amount per day (k).
 The amount of grass at the beginning is h.
The following is given:
 When the cow and the goat graze on the field together, there is no grass left after 45 days.
Therefore, h45×(v1+v2k) = 0, so v1+v2k = h/45 = 4×h/180.
 When the cow and the goose graze on the field together, there is no grass left after 60 days.
Therefore, h60×(v1+v3k) = 0, so v1+v3k = h/60 = 3×h/180.
 When the cow grazes on the field alone, there is no grass left after 90 days.
Therefore, h90×(v1k) = 0, so v1k = h/90 = 2×h/180.
 When the goat and the goose graze on the field together, there is also no grass left after 90 days.
Therefore, h90×(v2+v3k) = 0, so v2+v3k = h/90 = 2×h/180.
From this follows:
v1 = 3 × h/180,
v2 = 2 × h/180,
v3 = 1 × h/180,
k = 1 × h/180.
Then holds for the time t that the three animals can graze together:
ht×(v1+v2+v3k) = 0, so t = h/(v1+v2+v3k) = h/(3×h/180+2×h/180+1×h/1801×h/180) = 36.
The three animals can graze together for 36 days.
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