Painter        Rate (house per hours)   Alternative variable rate

Abe                                 A

Bart                                B

Cain                                C

Abe&Bart     

Bart&Cain    

Abe&Cain     

Suppose Abe can paint a house by himself in A days.
Then Abe's painting rate is 

1 house in A days =    

Suppose Bart can paint a house by himself in B days.
Then Bart's painting rate is 

1 house in B days =  

Suppose Cain can paint a house by himself in C days.
Then his painting rate is 

1 house in C days =  

So the sum of Abe's and Bart's rates equals 1 house per 15 days.

That's 1 house in 15 days =  

So 

So the sum of Bart's and Cain's rates equals 1 house per 12 days.

That's 1 house in 12 days =  

So 
--

So the sum of Abe's's and Cain's rates equals 1 house per 20 days.

That's 1 house in 20 days =  

So 

So we have this system of three equations:

  
 
Subtracting the first two equations:

 

Adding that to the third equation:


-------------


Multiplying both sides of

 by 1/2 gives:





-------------------

Substituting in

 
  




Substituting



in

Suppose working together they can paint a house in D days.
Then their combined painting rate is 

1 house in D days = 

Then the sum of their painting rates will equal 1/D:



 









It will take them 10 days.

Edwin

Let "a", "b" and "c" be the rates of work of each of the boys Abe, Bart, and Cain respectively.

We are given that 

a + b = ,    (1)
a + c = ,    (2)
b + c = .    (3)

To solve the problem, let us add the equations (1), (2) and (3).
You will get 

2a + 2b + 2c =  =  =  =  = .

Hence, 

a + b + c = .

Thus we just found the combined rate-of-work of the three boys working togetther. It is  job per day.
It implies that it will take 10 days for three boys to complete the job working together.

Answer.  10 days.