Three grazing animals clearing a field together

A cow and a goat together would graze a field bare in $45$ days; the cow with a goose would take $60$ days; the goat with the goose would take $90$ days. Assuming constant eating rates and no regrowth, how many days would all three animals need together?

$40$ days. Adding the three pairwise rates counts each animal twice: $\tfrac1{45}+\tfrac1{60}+\tfrac1{90}=\tfrac{1}{20}$ fields per day. The combined rate is half of that, $\tfrac1{40}$, so the field lasts $40$ days.