Joe and Skyler can clean an entire building in 44 hours. Joe can clean the entire building by himself in 66 fewer hours than Skyler can. How long does it take Joe to clean the building by himself?
look at the solution both Damon and I gave here for a similar question
http://www.jiskha.com/display.cgi?id=1487716112
The only difference is that here you are given the combined rate.
To solve this problem, we can start by assigning variables to the unknown quantities.
Let's say x represents the number of hours it takes Skyler to clean the entire building by herself.
We also know the following information:
1. Joe and Skyler together can clean the entire building in 44 hours.
2. Joe can clean the entire building by himself in 66 fewer hours than Skyler can.
From the first statement, we can write the equation:
1/x + 1/(x+66) = 1/44
This equation comes from the idea that in one hour, Joe completes 1/x of the task, and Skyler completes 1/(x+66) of the task. Together, they complete 1/44 of the task in one hour.
Now, we can solve this equation to find the value of x, which represents the number of hours it takes Skyler to clean the building alone.
Multiplying both sides of the equation by 44x(x+66), we get:
44(x+66) + 44x = x(x+66)
Expanding and rearranging, we have:
44x + 2904 + 44x = x^2 + 66x
Combining like terms:
88x + 2904 = x^2 + 66x
Moving all terms to one side of the equation:
x^2 - 22x - 2904 = 0
This equation is quadratic, so we can solve it by factoring or using the quadratic formula. After solving, we get two possible values for x: 96 and -30.
Since we're looking for a positive number of hours, we can discard the -30. Therefore, x = 96, which represents the number of hours it takes Skyler to clean the building alone.
Finally, we can use this value to answer the question: How long does it take Joe to clean the building by himself?
Joe can clean the building in 96 - 66 = 30 hours.
So, it takes Joe 30 hours to clean the building by himself.