## Friday, September 01, 2006

### Sheep-days

Imagine a pasture that is just big enough to feed 11 sheep for a total of 8 days. It turns out that if we reduce the number of sheep to 10, they would be able to eat for 9 days.

Theoretically, how long could two sheep last?

1. about 45 days. if you assume there is a fixed amount of sustenance in the pasture, and the sheep eat at a constant rate, there there are approximately 90 sheep-days of food. two sheep would take about 45 days to consume that much. However, the numbers aren't exact (8*11=88, 9*10=90), so there is some degree of error here.

2. Possibly almost indefinitely. It depends on the climate, rainfall etc. Grass grows remember!

3. I get between 45 and 49 days, but what do I know? I don't eat grass.

4. more detail.

Say a sheep eats 1 sd of grass per day. If 11 sheep could eat for 8 days, there must be at least 88 sd. of grass. If there were 99 sd, then the sheep could eat for 9 days, so we are between 88 and 99, but not 99 sd.

Since 10 sheep can eat for 9 days, there must be at least 90 sd of grass. If there were 100 sd of grass, the sheep could eat for ten days, so we are less than 100 (but that doesn't help us)

So we are between 90 and 98 sd of grass (inclusive), meaning we could feed 2 sheep for at least 45 but no more than 49 days.

5. My answer, but not necessarily the Answer given all the solutions you provided:

If 11 sheep can last 8 days, it would seem the pasture provides 88 sheep days. But if 10 sheep can last 9 days for a total of 90 sheep-days, then the pasture is providing two sheep-days growth every day. And so, two sheep could, theoretically, last a life-time in this pasture.

This sort of reminds me of those physics problems where you ignore friction.

6. Let total units of pasture is X. As given in the question :
11 sheep can be fed for 8 days. So in one day 11 sheep eat X/8 units. Also given in one day 10 sheep eat X/9 units. We are figure out here that 1 sheep eats in a day total X/8-X/9 units i.e X/72 units. Therefore 2 sheep eat 2*X/72 i.e. X/36 units. Hence two sheep can be fed for total of 36 days.

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