Each good worker can paint my new house alone in 12 hours.  Each bad worker can paint my house alone in 36 hours.  I need my house painted in 3 hours.  If I can only find 3 good workers, how many bad workers must I also find in order to have my house painted on time?
I want to know how many bad workers, let's call them x, I need to add to the 3 good workers to finish the job in 3 hours. I can use the idea of rate of work to figure this out. Each good worker can paint 1/12 of the house in an hour, and each bad worker can paint 1/36 of the house in an hour. So, the combined rate of work of the 3 good workers and the x bad workers is 3/12 + x/36. I want this rate to be equal to the rate of work that would finish the whole house in 3 hours, which is 1/3. So I can write an equation: 3/12 + x/36 = 1/3. I can simplify this equation by multiplying both sides by 36 to get rid of the fractions: 9 + x = 12. Subtracting 9 from both sides, I get: x = 3. So I need 3 bad workers in addition to the 3 good workers to paint my house in 3 hours. 