Problema Solution
Amber found that her cow and goat would eat all the grass in a certain field in 45 days, that her cow and goose would eat it in 60 days, but that it would take her goat and goose 90 days to eat it all. If she turns the cow, the goat, and the goose into the field together, how long will it take them to eat all the grass? Assume that the grass is not growing each day.
Answer provided by our tutors
let
c = the number of days the cow needs to eat all the grass alone thus it's rate is 1/c field of grass per day
g = the number of days the goat needs to eat all the grass alone thus it's rate is 1/g field of grass per day
d = the number of days the goose needs to eat all the grass alone thus it's rate is 1/g field of grass per day
Amber found that her cow and goat would eat all the grass in a certain field in 45 days,
1/c + 1/g = 1/45
her cow and goose would eat it in 60 days
1/c + 1/d = 1/60
it would take her goat and goose 90 days to eat it all
1/g + 1/d = 1/90
let x = the number of days the cow, the goat, and the goose need to eat all the grass together, thus their together rate is 1/x field of grass per day.
Also we have that the sum of each of the rates is equal to the together rate:
1/c + 1/g + 1/d = 1/x
by adding the equations:
1/c + 1/g = 1/45
1/c + 1/d = 1/60
1/g + 1/d = 1/90
we get
2(1/c + 1/g + 1/d) = 1/45 + 1/60 + 1/90
plug 1/c + 1/g + 1/d = 1/x into the last equation
2(1/x) = 1/45 + 1/60 + 1/90
by solving we find
x = 40 days
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it will take them 40 days to eat all the grass together.