# Cows eat too much [closed]

Cows are always hungry...

• 70 Cows eat all the plants on a pasture in 24 days.
• 30 Cows clear the same pasture in 60 days.

How many cows are needed to clear the pasture by the end of 96th day?

## closed as off-topic by Lopsy, xnor, Len, Tryth, McMagisterJan 14 '15 at 1:19

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• This is a textbook math problem. I don't think it's actually a puzzle. – xnor Jan 13 '15 at 20:22
• IMO, this question belongs on math.SE. – Lopsy Jan 13 '15 at 20:48

20 cows.

The 70 cows over the 24 days eat 1680 cow-nom units.
The 30 cows over the 60 days eat 1800 cow-nom units.

Based on the growth from 24 days to 60 days, we can work out that at 96 days, the cows would need to eat 1920 cow-nom units. Divide that by the 96 days and we get $20 cows$.

• Can we assume that the growth is linear? – Ivo Beckers Jan 14 '15 at 9:39

Since 70 cows clear the pasture in 24 days, we know that there should be more than $70*23=1610$ and at most $70*24=1680$ cow-days of plants in the pasture.

Since 30 cows clear the pasture in 60 days, we know that there should be more than $30*59=1770$ and at most $30*60=1800$ cow-days of plants in the pasture.

Here we run into a problem - the two ranges don't overlap. Either there's an error in the question, or we have to deal with plants growing (they do that).

• yes, its normal for plants to grow – Rafe Jan 13 '15 at 20:38

First Try

>>> (70 * 24 + 30 * 60) / (2 * 96)
18.125

It looks like you need 18 cows for 96 days.

Second Try

>>> (30 * 60) / (70 * 24) / (60 - 24) * (96 - 60) * (30 * 60) / 96
20.08928571428571

After looking at what others had to say, it might be 20 cows instead.

In more detail, it the previous calculation can be read as follows:

>>> food_for_24_days = 24 * 70
>>> food_for_60_days = 60 * 30
>>> ratio_of_increase = food_for_60_days / food_for_24_days
>>> ratio_of_increase_per_day = ratio_of_increase / (60 - 24)
>>> increase_from_60_to_96_days = ratio_of_increase_per_day * (96 - 60)
>>> food_for_96_days = food_for_60_days * increase_from_60_to_96_days
>>> cows_needed_for_96_days = food_for_96_days / 96
>>> cows_needed_for_96_days
20.08928571428571
• By your answer it looks like you need 19. Unless you have 1/8 of a cow handy. – Paul Jan 13 '15 at 19:58
• @Paul Assume a spherical cow with half the diameter… – Gilles Jan 13 '15 at 21:46