5 cows can eat 2 acres of grass in 10 days. 7 cows can eat 3 acres of grass in 30 days. The grass grows at a constant rate and each cow eats at a constant rate. The length of the grass before the cows begin grazing is constant. How many days will it take 16 cows to eat 7 acres of grass?
Solution
By the time the cows have eaten all the grass total grass consumed will equal total initial grass plus total grass regrowth. Next let’s define some terms. x=initial amount of grass in an acre.
y=amount of grass grown in one acre in one day.
Putting the given information in the form of equations: 50=2x+20y
210=3x+90y
To put the first equation in more simple English, 5 cows eating for 10 days results in a consumption of 50 units of grass. This is equal to the sum of 2x initial units of grass and 20y units of grass growth (10 days times 2 acres).
Next we must solve for x and y. Rewriting the above equations: 150=6x+60y (multiplying by 3)
420=6x+180y (multiplying by 2)
Subtracting the first equation from the second we get 270=120y, so y=9/4. Plugging this into either equation we get x=5/2.
The question to be answered is “How many days will it take 16 cows to eat 7 acres of grass?” Let’s let d be the number of days. So setting this up as an equation: 16d=7x + 7dy
16d=35/2 + 63d/4
d/4=35/2
d=70.
So it will take 70 days for 16 cows to eat 7 acres of grass.
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