Let the amount earned in a day by Neeta, Geeta and Sita be $x,y, \text{and}\;z$ rupees.
Now, $x+y=6z\quad \longrightarrow(1)$
And, $z+x=2y\quad \longrightarrow(2)$
In equation $(2),$ multiply both sides by $6.$
$\Rightarrow 6z + 6x= 12y$
$\Rightarrow x + y + 6x = 12y \quad [\because \text{From equation}\; (1)]$
$\Rightarrow \boxed{7x = 11y} $
$\Rightarrow \boxed{x = \frac{11}{7}y}$
Put the value of $x$ in equation $(1).$
$\Rightarrow \frac{11}{7}y + y = 6z$
$\Rightarrow \frac{11y+7y}{7} = 6z$
$\Rightarrow \frac{18y}{7} = 6z$
$\Rightarrow \frac{3y}{7} = z$
$\Rightarrow \boxed{z = \frac{3}{7}y}$
Let, $y=7k$
Then, $x=11k$
$\Rightarrow z=3k$
$\therefore$ The rate of the daily earning of the one who earns the most to that of the one who earns the least $=x:z$
$\qquad \qquad = 11k:3k = 11:3.$
Correct Answer $:\text{C}$