Question 439251
Let x=amount of time it takes the apprentice to paint the house
Then the apprentice paints at the rate of 1/x house per day
x+3=amount of time needed by the day laborer to paint the house
Then the day laborer works at the rate of 1/(x+3) house per day
x-3=amount of time needed by the master painter to paint the house
Then the master painter works at the rate of 1/(x-3) house per day
In 7 days the master painter can paint 7/(x-3) of the house
In 6 days the apprentice can paint 6/x house
In 6 days the day laborer can paint 6/(x+3) house 
Together the day laborer and the apprentice can paint (6/x + 6/(x+3)) house

Now we are told that 7/(x-3)=6/x + 6/(x+3) multiply each term by x(x-3)(x+3)
7x(x+3)=6(x-3)(x+3)+6x(x-3) expand
7x^2+21x=6x^2-54 + 6x^2-18x subtract 7x^2 and also 21x from both sides
0=6x^2-7x^2-21x-54+6x^2-18x or
5x^2-39x-54=0 ---quadratic in standard form
When we solve this quadratic we get:
x=(39+-51)/10
x=-1.2----no good--- can't have negative times
and
x=9----so it takes the apprentice 9 hours to paint the house working alone
It takes the day laborer x+3=9+3=12 hours
And it takes the master painter x-3=9-3=6 hours

CK
In 7 hours the master painter can paint 7/6 house 
In 6 hours the apprentice can paint 6/9 house
And in 6 hours the day laborer can paint 6/12 house
Does 6/9  + 6/12 = 7/6 ??
24/36 + 18/36 = 42/36
42/36=42/36
7/6=7/6


Hope this helps----ptaylor