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?
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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
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Let the completed job = 1 (all grass consumed)
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Write an equation for each statement:
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"A goat and a sheep can eat all the grass on a field in 90 days."
{{{90/g}}} + {{{90/s}}} = 1
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" A sheep and a cow can eat all the grass on the field in 60 days."
{{{60/s}}}+ {{{60/c}}} = 1
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"A cow and a goat can eat all the grass on the field in 45 days."
{{{45/c}}} + {{{45/g}}} = 1
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Let's use elimination method here, rearrange and multiply the last two equations
by 1.5 and -2
:
{{{90/g}}} + {{{90/s}}} + {{{(0)}}} = 1
{{{(0)}}} + {{{90/s}}} = {{{90/c}}} = 1.5; multiplied by 1.5
{{{-90/g}}} + {{{(0)}}} = {{{-90/c}}} = -2; multiplied by -2
--------------------------------------------- adding eliminates g & c, find s
0 + {{{180/s}}} + 0 = .5
.5s = 180
s = 360 days for the sheep alone to consume the grass
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Find g using: {{{90/g}}} + {{{90/s}}} = 1
{{{90/g}}} + {{{90/360}}} = 1
{{{90/g}}} + .25 = 1
{{{90/g}}} = 1 -.25
{{{90/g}}} = .75
.75g = 90
g = {{{90/.75}}}
g = 120 days for goat alone to devour all the grass
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Find c using: {{{45/c}}} + {{{45/g}}} = 1
{{{45/c}}} + {{{45/120}}} = 1
{{{45/c}}} + .375 = 1
{{{45/c}}} = 1 - .375
{{{45/c}}} + .625
.625c = 45
c = {{{45/.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?
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Let t  = time for all the animals to clear the field
{{{t/360}}} + {{{t/120}}} + {{{t/72}}} = 1
Multiply by 1800 to clear the denominators, resulat
5t + 15t + 25t = 1800
45t = 1800
t = {{{1800/45}}}
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.