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?
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.)