Let the time taken by Jack to complete the work be $x$ days. Then time taken by John is $2x.$

$\begin{array} {lccc} & \text{John} & \text{Jack} & \text{Jack + Jim} \\ \text{Time}: & 2x & x & \frac {2x}{3} \end{array}$

Let the time taken by Jim be $y$ days to finish the work.

Now, $\frac{2x}{3} = \frac{xy}{x+y} \quad \left[\because \text{Time} \propto \frac{1}{\text{Efficiency}} \right]$

$ \Rightarrow \frac{2}{3} = \frac{y}{x+y} $

$\Rightarrow 2(x+y) = 3y$

$\Rightarrow 2x + 2y = 3y$

$\Rightarrow \boxed {y = 2x}$

John takes three days more than, time is taken by all of them working together.

So, one day work by all of them $ = \frac{1}{2x} + \frac{1}{x} + \frac{1}{y} \quad [\because \text{Efficiency for all of them}]$

$ \qquad \qquad = \frac{1}{2x} + \frac{1}{x} + \frac{1}{2x} \quad [ \because y=2x]$

$ \qquad \qquad = \frac{1+2+1}{2x} = \frac{4}{2x} = \frac{2}{x} $

Time taken by all of them $ = \frac{x}{2}$

Now, $2x = \frac{x}{2} + 3$

$\Rightarrow 2x – \frac{x}{2} = 3$

$ \Rightarrow \frac{4x-x}{2} = 3$

$ \Rightarrow \frac{3x}{2} = 3$

$ \Rightarrow \boxed {x = 2\;\text{days}}$

$\therefore$ Time taken by Jim alone to finish the work $= y = 2x = 2 (2)= 4 \; \text{days}.$

Correct Answer $: 4$

$\textbf{PS:}$ If one person takes $x$ days, and the second person takes $y$ days, then total time is taken by both of them working together $ = \underbrace{\frac{1}{x} + \frac{1}{y}}_{\text{Efficiency}} = \frac{x + y}{xy} = \underbrace{\frac{xy}{x+y}}_{\text{Time}}. \quad [\because \text{Assume total work = 1 unit}]$