## Wednesday, May 12, 2010

### Maybe We Should Try Some Goats Instead

Grass grows in a field at some rate r, where r is the units of grass grown per day. It is known that if 10 sheep are turned out in the field, the grass will be gone in 20 days. On the other hand, if 15 sheep are turned out in the field, the grass will be gone in 10 days. If 25 sheep are turned out in the field, when will the grass be gone?

1. 5 days.

r=rate of grass grown per day
x=rate that one sheep eats grass per day
a=starting amount of grass

From the given info:
a + 20r = 10(20)x
a + 10r = 15(10)x

Subtracting those equations:
10r = 50x
r = 5x

Plug that back in the first equation to solve for a:
a + 20(5x) = 200x
a + 100x = 200x
a = 100x

Now for 25 sheep, using d as the number of days to clear the grass, plug in the values for a and r to solve for d:

a + dr = 25dx
100x + 5dx = 25dx
100x = 20dx
5 = d

2. 1/10 +1/20 =1/x

x = 20/3 = 6.67 days.

3. I believe Andy's answer of 5 days is right.

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