SOLUTION: A goat and a sheep can eat all the grass on a field in 90 days. A sheep and a cow can eat all the grass on the field in 60 days. A cow and a goat can eat all the grass on the field

Algebra ->  Human-and-algebraic-language -> SOLUTION: A goat and a sheep can eat all the grass on a field in 90 days. A sheep and a cow can eat all the grass on the field in 60 days. A cow and a goat can eat all the grass on the field      Log On


   

Question 339663: A goat and a sheep can eat all the grass on a field in 90 days. A sheep and a cow can eat all the grass on the field in 60 days. A cow and a goat can eat all the grass on the field in 45 days. So...if I release all the animals on the field, how long will it take them to eat all the grass?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A goat and a sheep can eat all the grass on a field in 90 days.
A sheep and a cow can eat all the grass on the field in 60 days.
A cow and a goat can eat all the grass on the field in 45 days.
So...if I release all the animals on the field, how long will it take them to eat all the grass?
:
Let g = time for goat alone to consume all the grass
Let s = time for sheep to do the same
Let c = time for cow to do it
:
Let the completed job = 1 (all grass consumed)
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Write an equation for each statement:
:
"A goat and a sheep can eat all the grass on a field in 90 days."
90%2Fg + 90%2Fs = 1
:
" A sheep and a cow can eat all the grass on the field in 60 days."
60%2Fs+ 60%2Fc = 1
:
"A cow and a goat can eat all the grass on the field in 45 days."
45%2Fc + 45%2Fg = 1
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Let's use elimination method here, rearrange and multiply the last two equations
by 1.5 and -2
:
90%2Fg + 90%2Fs + %280%29 = 1
%280%29 + 90%2Fs = 90%2Fc = 1.5; multiplied by 1.5
-90%2Fg + %280%29 = -90%2Fc = -2; multiplied by -2
--------------------------------------------- adding eliminates g & c, find s
0 + 180%2Fs + 0 = .5
.5s = 180
s = 360 days for the sheep alone to consume the grass
:
Find g using: 90%2Fg + 90%2Fs = 1
90%2Fg + 90%2F360 = 1
90%2Fg + .25 = 1
90%2Fg = 1 -.25
90%2Fg = .75
.75g = 90
g = 90%2F.75
g = 120 days for goat alone to devour all the grass
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Find c using: 45%2Fc + 45%2Fg = 1
45%2Fc + 45%2F120 = 1
45%2Fc + .375 = 1
45%2Fc = 1 - .375
45%2Fc + .625
.625c = 45
c = 45%2F.625
c = 72 days for the cows to eat up the grass
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So...if I release all the animals on the field, how long will it take them to eat all the grass?
:
Let t = time for all the animals to clear the field
t%2F360 + t%2F120 + t%2F72 = 1
Multiply by 1800 to clear the denominators, resulat
5t + 15t + 25t = 1800
45t = 1800
t = 1800%2F45
t = 40 days for all the animals to eat all the grass.
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I'm going to send this off without a check, looks like a tornado is in the area here, we will probably lose power.