Grazing Cows

Problem

Let:

$a$ cows graze $b$ fields in $c$ days

$a'$ cows graze $b'$ fields in $c'$ days

$a$ cows graze $b$ fields in $c$ days.

What is the relationship between the $9$ magnitudes $a$ to $c$?

Suppose that:

each field initially contains the same quantity of grass $M$

the daily growth in each field is the same, $m$

each cow consumes the same amount of grass per day, $Q$.

Then:

$\ds b M + c b m - c a Q$

$=$

$\ds 0$

$\ds b' M + c' b' m - c' a' Q$

$=$

$\ds 0$

$\ds b M + c b m - c a Q$

$=$

$\ds 0$

Consider the matrix:

$A = \begin{pmatrix} b & b c & c a \\ b' & b' c' & c' a' \\ b & b c & c a \end{pmatrix}$

We have:

$A \tuple {M, m, -Q}^T = \mathbf 0$

It follows that $\det A = 0$.

After algebra, we have:

$b c c' \paren {a b' - b a'} + c b \paren {b c' a' - b' c a} + c a b b' \paren {c - c'} = 0$

$\blacksquare$