GMAT Prep Test1 simultaneously (A+B), (A+C) , and (B+C)

[rajdeep_duttaroy]

folks, this problem appeared in the GMAT Prep Test1. Unfortunately I forgot to copy the answer options that were provided. The problem stem reads...

A and B working simultaneously can paint a Tank in 5/6 hrs. A and C working simultaneously complete the same task in 3/2 hrs, while C and B working simultaneously complete it in 2 hrs. If A,B,and C were to work together how long would it take them to finish painiting the tank?

Can some one help me fathom the approach thats needed for this problem. Precisely, my confusion is with handling the combined workers.

[Ben Ku]

The first step is translating the information given and setting up the equations correctly.

Let Ra = rate of A painting the tank alone
Rb = rate of B painting the tank alone
Rc = rate of C painting the tank alone

We can use the work formula to express the information in the problem.
rate * time = work

When different people work together, we can combine/add their rates.
(1) A and B working together: (Ra + Rb)(5/6) = 1
(2) A and C working together: (Ra + Rc)(3/2) = 1
(3) B and C working together: (Rb + Rc)(2)= 1

Now the question is asking for t in the expression: (Ra + Rb + Rc)(t) = 1
To find out t, we need to find Ra + Rb + Rc.

Let's use the equations from the data and add them together:
(1) can be manipulated to be (Ra + Rb) = 6/5
(2) can be manipulated to be (Ra + Rc) = 2/3
(3) can be manipulated to be (Rb + Rc) = 1/2

When we add the three equations together, we get 2Ra + 2Rb + 2Rc = 71/30.
If Ra + Rb + Rc = 71/60, then t = 60/71.

I hope that's clear.