Brain Teasers
Madadian Farm Animals
Old Farmer McDoughnut owns a piece of grassland in the heart of Madadia and has three animals: a Madadian Burger Cow, a Spare Kebab Goat, and a Scotch Egg laying Goose. McDoughnut 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.
For how long can the three animals graze on the field together?
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.
For how long can the three animals graze on the field together?
Answer
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-45x(v1+v2-k) = 0, so v1+v2-k = h/45 = 4xh/180.
When the cow and the goose graze on the field together, there is no grass left after 60 days. Therefore, h-60x(v1+v3-k) = 0, so v1+v3-k = h/60 = 3xh/180.
When the cow grazes on the field alone, there is no grass left after 90 days. Therefore, h-90x(v1-k) = 0, so v1-k = h/90 = 2xh/180.
When the goat and the goose graze on the field together, there is also no grass left after 90 days. Therefore, h-90x(v2+v3-k) = 0, so v2+v3-k = h/90 = 2xh/180.
From this follows:
v1 = 3 x h/180,
v2 = 2 x h/180,
v3 = 1 x h/180,
k = 1 x h/180.
Then holds for the time t that the three animals can graze together: h-tx(v1+v2+v3-k) = 0, so t = h/(v1+v2+v3-k) = h/(3xh/180+2xh/180+1xh/180-1xh/180) = 36. The three animals can graze together for 36 days.
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