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 $c''$?

Solution
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:

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

Given that each of $M$, $m$ and $Q$ are not zero, by Matrix is Non-Invertible iff Product with Non-Zero Vector is Zero, $A$ is non-invertible.

It follows that $\det A = 0$.

After algebra, we have:
 * $b c c' \paren {a b' - b a'} + c b \paren {b c' a' - b' c a} + c a'' b b' \paren {c - c'} = 0$