Wow for the Cow!

Suppose that the field originally has $G$ units of grass on it, and that the growth rate of the grass is $g$ units per day. In addition, let each cow eat $c$ units of grass per day.

Now, if $6$ cows are at the field for $3$ days, then, $G + 3g - 18c = 0$ Similarly, if $3$ cows are at the field for $7$ days, then $G + 7g - 21c = 0$ .

Solving the values of $g$ and $c$ in terms of $G$ gives $\frac{G}{21}$ and $\frac{4G}{63}$ , respectively.

Hence, if one cow takes $x$ days, then $G + xg - xc = 0$

$G = x(c - g)$

$x = \frac{G}{c-g}$

i.e., $x = 63 days$