# Cows eating Grass

A man has an acreage where grass grows continuously and evenly. If 20 cows eat all the grass in 96 days and 30 cows eat all of the grass in 60 days, then how long does it take 70 cows to eat all of the grass?

### 10 guesses to Cows eating Grass

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• Roolstar

is it 96 days or 90 days for 20 cows?

• carole

it should take 26 days

• Dude

Carole, you are very close but that is not the answer. And yes, it is 96 days.

• det

there is infinite solutions, since the product of the two given data pairs are different we can choose “any” function(s) to describe the cows eating process.
hyperbola with x=0 asymptote:

days(cows) = 2160*1/cows – 12

days(20)=96; days(30)=60;

days(70)=~18.86 -> 19 days

hyperbola with y=0 asymptote (probably you are looking for this function):

days(cowd) = 1600*1/(cows-10/3)

70-> 24 days

• Dude

24 days is correct!!!

Here’s how the Dude figured it:

Let’s say one cow can eat X pounds of grass per day. The field grows Y pounds of grass per day. At the start, the field already has N pounds of grass available. At the end, the field has 0 pounds of grass remaining. So we set up an equation for the amount of grass for the two cases, adding growing grass and subtracting eaten grass.
For 20 cows eating for 96 days: N + 96Y – 20x96X =0
For 30 cows eating for 60 days: N + 60Y – 30×60 X =0
Or: N + 96Y – 1920X = 0 and N + 60Y – 1800X = 0
When you subtract the two equations, you get:
36Y – 120X = 0, or 36Y = 120X or 3Y = 10X
In other words, in one day ten cows can eat as much as the field can grow in three days.
Now, substitute this back into either of the two original equations to find out how much grass was in the field to begin with.
N + 60Y – 1800X = N + 60Y Y = 0 and N = 480Y
Now we set up the equation for 70 cows eating and the field growing for T days, until the grass is gone.
N + TY – 70 TX = 0, or 480Y + TY – 21TY = 0 480Y, TY = 0
Hence, T – 24
So the 70 cows finish off the field in 24 days.

Great job, Det. You’re today’s winner!

• Sarah

96? Not very sure but could be right

• nicole

24 days

• BigSexy

24

• georgis

25/7

• MemeZero

Lets find the grass growth rate.
Condition 1 ==> 20 x 96 = 1920
Condition 2 ==> 30 x 60 = 1800
So rate of grass growing in 36 days (96-60) is 120 (1920-1800)
So daily grass growing rate = 120/36
Initial grass 1600 (1920 – (96 * 120/36) or 1800-(60 * 120/36))
Let say N is number of days
Then
Initial Grass + N * Rate = Number of Cows x N
1600 + (120/36)N = 70N
70N – (120/36)N = 1600
200N = 1600 x 3
N = 4800 / 200
=24 days