This exercise is a special case of a problem discussed in Isaac Newton’s Arithme

Question

This exercise is a special case of a problem discussed in Isaac Newton’s Arithmetica
Universalis, written in 1707 (http://cudl.lib.cam.ac.uk/ view/MS-ADD-03993/85):

3 cows graze 1 field bare(empty) in 2 days and
7 cows graze 4 fields bare in 4 days.

How long will it take 2 cows to graze 3 fields bare?

Hint: You may assume that each field initially provides the same amount, x, of grass;
that grass in each field continues to grow at a constant daily rate, y; and that all
cows eat the same amount of grass, z, each day. Quantities x, y, z are measured by
weight.

Explanation / Answer

Rearranging the data, and using the given notation,
6z = x + 2y ............ (i)
28z = 4x + 16y ..... (ii)

[(ii) - (i) * 4]

4z = 8y

z = 2y

Putting in i we get

12y = x + 2y

x = 10y

Let number of days required is 'a'

So,

2az = 3x + 3a

Putting value of z and x

4ay = 30y + 3a

4ay - 3a = 30y

a (4y - 3) = 30y

a = 30y/(4y-3)

So, this problem can have multiple solutions.

We can check with y = 1, a will be 30. But several solutions are possible.

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