SOLUTION: I am having trouble setting up an equation for the following: Together Bob and Dave can roof a house in 16 hours. By himself, Bob can roof the house in 4 hours less time than Dave

Question 529683: I am having trouble setting up an equation for the following: Together Bob and Dave can roof a house in 16 hours. By himself, Bob can roof the house in 4 hours less time than Dave can by himself. How long will it take Dave to roof the house by himself?
I am currently studying Quadratic equations and Radicals, so the answer to this might be in Radical form. I tried to assign x to Dave's time, and x-4 to Bob's time, and then average the sum of these. I then made this equal to 16, but I got 18 hours for Dave, and 14 for Bob, which is faster than the 2 of them working together (16)! Any help would be appreciated,
Thank you,
Kevin
Found 2 solutions by scott8148, solver91311:
Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
they each do some FRACTION of the job, based on their individual rates
___ the fractions sum to one, the WHOLE job

(16 / x) + [16 / (x - 4)] = 1

multiplying by the LCD will generate your quadratic
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Your assignment of variables was spot on, though you could just as easily used for Bob and for Dave. Either way it will all come out in the wash. Let's use your assignments, but first take a look at a general discussion of "Working Together" problems.

If A can do a job in x time periods, then A can do of the job in 1 time period. Likewise, if B can do the same job in y time periods, then B can do of the job in 1 time period.

So, working together, they can do

of the job in 1 time period.

Therefore, they can do the whole job in:

time periods.

For this particular problem we need to work backwards through the discussion above. We know that working together they get the job done in 16 time periods, so working together they must be able to accomplish of the job in one time period, i.e. 1 hour.

Using your variable assignments, we can say that Dave can do of the job in one time period, and Bob can do of the job in one time period, so working together they do:

of the job in one time period. But we have already established that they can do of the job in one time period. Hence:

Just solve for , then calculate

Note that this equation does, indeed, reduce to a quadratic equation. It will not factor so you will need to use the quadratic formula. Also note that one of your roots will be less than 4, meaning that once you subtract 4 to get Bob's time, you get a negative number. To avoid having to deal with the absurd notion that Bob can finish painting a house some number of hours before he starts, discard the smaller root as extraneous.

John

My calculator said it, I believe it, that settles it

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