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## Sunday, July 27, 2008

### Green Green Grass

Farmer Bob owns a piece of grassland and three animals: a cow, a goat, and a goose. Bob discovered the following:

* When the cow and the goat graze on the field together, there is no more grass after 45 days.
* When the cow and the goose graze on the field together, there is no more grass after 60 days.
* When the cow grazes on the field alone, there is no more grass after 90 days.
* When the goat and the goose graze on the field together, there is no more grass after 90 days also.

The Question: For how long can the three animals graze on the field together?

A Hint: The grass also still grows!

* 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.

There 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.