A cow and a goat can eat the entire contents of a pasture, including an abandoned 1958 Edsel, in 40 days. The cow and a goose can do it in 90 days, and the goose and the goat take 60 days. How long should it take all three of them chomping away together?

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The trick is to find the daily rate for each animal, starting with the daily rates for pairs. Let c be the cow’s daily rate, g the goose’s, and G the goat’s daily rate. So:
c + G = 1/40
c + g = 1/90
g + G = 1/60

Add all 3 equations: 2c + 2g + 2G = 1/40 + 1/90 + 1/60
= (9 + 4 + 6)/360
= 19/360 in 2 days if all 3 are chomping together.

Thus: 19/720 gets eaten in 1 day, so they’ll need 720/19 ≈ 37.89 ≈ 38 days to clear the whole pasture. Apparently the goose doesn’t help a whole lot.

(Note: This problem does not take into account the fact that the grass in the pasture is growing while they’re there.)

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