Solution to: Green Green Grass

Assumptions:

• The cow, the goat, and the goose eat grass at a constant rate (amount per day): v1 for the cow, v2 for the goat, and v3 for the goose.
• The grass grows at a constant rate k per day.
• The initial amount of grass is h.

The following information is given:

• When the cow and the goat graze in 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 in 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 in 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 in 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, we derive:

v1 = 3 × h / 180,
v2 = 2 × h / 180,
v3 = 1 × h / 180,
k = 1 × h / 180.

The time t that the three animals can graze together is given by: h - t × (v1 + v2 + v3 - k) = 0. Thus, t = h / (v1 + v2 + v3 - k) = h / (3 × h / 180 + 2 × h / 180 + 1 × h / 180 - 1 × h / 180) = 36.

Conclusion: The three animals can graze together for 36 days.

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