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The amount Neeta and Geeta together earn in a day equals what Sita alone earns in $6 \; \text{days}.$The amount Sita and Neeta together earn in a day equals what Geeta alone earns in $2 \; \text{days}.$ The ratio of the daily earnings of the one who earns the most to that of the one who earns the least is

- $7 : 3$
- $3 : 2$
- $11 : 3$
- $11 : 7$

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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 $:\text{C}$

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 $:\text{C}$