Problem 168 Solution

The problem again is, "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?"

By the time the cows have eaten all the grass total grass consumed will equal total initial grass plus total grass regrowth. Next lets 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 we get:

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 we get:

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 we get.

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.

Michael Shackleford, ASA - August 10, 2001