How long will it take for 3 of them to complete a job ?

Let the work of the job be $W$ . And the rate of doing work, the amount of $W$ done per hour, by $X$ , $Y$ , and $Z$ be $x$ , $y$ , and $z$ respectively. Then we have:

$\begin{cases} 2 x + 2 y = W & \implies x + y = \dfrac W2 & ...(1) \\ 3x + 3z = W & \implies x + z = \dfrac W3 & ...(2) \\ 4y + 4 z = W & \implies y + z = \dfrac W4 & ...(3) \end{cases}$

$\begin{array} {rll} (1) - (2): & y - z = \dfrac W2 - \dfrac W3 = \dfrac W6 &...(4) \\ (3)+(4): & 2y = \dfrac W4 + \dfrac W6 = \dfrac 5{12}W \\ & \implies y = \dfrac 5{24} W \\ (1): & \implies x = \dfrac W2 - \dfrac 5{24} W = \dfrac 7{24}W \\ (3): & \implies z = \dfrac W4 - \dfrac 5{24} W = \dfrac 1{24}W \end{array}$

Then we have

$\begin{aligned} x + y + z & = \frac 7{24}W + \frac 5{24}W + \frac 1{24}W = \frac {13}{24}W \\ \implies \red{\frac {24}{13}}(x+y+z) & = W \end{aligned}$

Therefore it will take $\dfrac {24}{13}\ \rm hours \approx \boxed{\text{1 hour 50.77 minutes}}.$

If it takes x, y and z hours for the three persons to complete the job if they worked alone. We arrive at the following equations based on the details provided.

1/x + 1/y = 1/2 -- (1)

1/y + 1/z = 1/3----(2)

1/x + 1/z = 1/4----(3)

Adding (1), (2) and (3) yields 2(1/x + 1/y + 1/z) = 1/2 + 1/3 + 1/4 = 13/12

Simplifying we get the value of (1/x + 1/y + 1/z) = 13/24 = 1/T where T is the time taken for all three of them to complete a job if they worked together.

Hence T = 24/13 or 1 hour and 50.77 Minutes.

If X and Y can do it in 2 hrs, X,Y, & Z working together must complete the work in less than 2 hrs! Perhaps the writer should adjust some of the multiple choice answers...hint hint