Math Problem: Farmer and his Animals

A farmer found that his cow and goat would eat all the grass in a field in 45 days, that his cow and goose would eat it in 60 days, but that it would take his goat and goose 90 days to eat it all.

If the farmer places the cow, the goat, and the goose into the field together, how long will it take them to eat all the grass? Assume the grass is not growing each day.

Can you explain and show steps as to how you solved it. Thanks

cow+goat=45
cow+goose=60
goat+goose=90
just from looking you can deduce 2 things:
-the cow eats the most then the goat
-the number of days will most certainly be less than 45
-the number will be more than 30 since the goose eats the least and therefore would make them take less than 1/3 of the time it currently takes.
now make the problem make sense by solving for one of the animals
cow + goat = 1/45
cow + goose= 1/60
goat + goose= 1/90
lets solve for goat:
cow +goat = 2goose + 2 goat
goat= cow- 2goose
cow +goat = 1/45
cow + cow -2 goose= 1/45
2 cow -2 goose = 1/45
cow- goose = 1/90
cow + goose = 1/60
3cow - 3 goose = 2 cow + 2 goose
1 cow = 5 goose
therefore cow eats 5 times more than goose
therefore in 60 days cow eats 83% of the food
or the cow eats(25/18)% of the food each day
the goose eats (5/18)% of the food each day
plug in for the goat:
(5/18) %* 90= 25 % or the amount the goose eats in the 90 days
that mean the goat eats the remainder 75% ---> (15/18 or 5/6)% per day
verify this by solving for the goat through the cow:
(25/18)% * 45 = 62.5% from the cow and 37.5 % from the goat. Therefor the goat eats 15/18% per day and It fits
add up all the percentages per day:
(5+25+15)/18= 45/18 =2.5% per day
therefore it takes 40 days for all the food to be consumed.

Cow * Goat / (Cow + Goat) = 45
Cow * Goose / (Cow + Goose) = 60
Goat * Goose / (Goat + Goose) = 90

(Cow, Goose, Goat) = (72, 360, 120)