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, h - 45 × (v1 + v2 - k) = 0, so v1 + v2 - k = 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, h - 60 × (v1 + v3 - k) = 0, so v1 + v3 - k = h/60 = 3 × h/180.
• When the cow grazes on the field alone, there is no grass left after 90 days. Therefore, h - 90 × (v1 - k) = 0, so v1 - k = 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, h - 90 × (v2 + v3 - k) = 0, so v2 + v3 - k = 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: h - t × (v1 + v2 + v3 - k) = 0, so t = h/(v1 + v2 + v3 - k) = h/(3 × h/180 + 2 × h/180 + 1 × h/180 - 1 × h/180) = 36. The three animals can graze together for 36 days.

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