SOLUTION: Jane and Maria clean the house in 8 hours if they work together. the time that Jane takes in cleaning the house alone is 4 hours more than the time Maria takes in cleaning the same

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Question 885861: Jane and Maria clean the house in 8 hours if they work together. the time that Jane takes in cleaning the house alone is 4 hours more than the time Maria takes in cleaning the same house. How long does it take Jane to clean the house alone? How about Maria?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Jane and Maria clean the house in 8 hours if they work together. the time that Jane takes in cleaning the house alone is 4 hours more than the time Maria takes in cleaning the same house. How long does it take Jane to clean the house alone? How about Maria?
Let x = the number of hours it takes Maria to clean the house.

Therefore Maria's work rate is 1 house cleaned per x hours or 

matrix%281%2C3%2C1%2Chouse%2Ccleaned%29%2Fmatrix%281%2C2%2Cx%2Chours%29%22%22=%22%22matrix%281%2C2%2C1%2Fx%2Cmatrix%281%2C2%2Chouse%2Ccleaned%29%2Fhour%29

>>the time that Jane takes in cleaning the house alone is 4 hours more than the time Maria takes in cleaning the same house.<<
Therefore Jane's work rate is 1 house cleaned per x+4 hours or
 
matrix%281%2C3%2C1%2Chouse%2Ccleaned%29%2Fmatrix%281%2C2%2Cx%2B4%2Chours%29%22%22=%22%22matrix%281%2C2%2C1%2F%28x%2B4%29%2Cmatrix%281%2C2%2Chouse%2Ccleaned%29%2Fhour%29

>>Jane and Maria clean the house in 8 hours if they work together.<<

Therefore their combined work rate is 1 house cleaned per 8 hours or

matrix%281%2C3%2C1%2Chouse%2Ccleaned%29%2Fmatrix%281%2C2%2C8%2Chours%29%22%22=%22%22matrix%281%2C2%2C1%2F8%2Cmatrix%281%2C2%2Chouse%2Ccleaned%29%2Fhour%29 

The equation comes from the sum of their individual work rates equals 
their combined work rate:

       1%2Fx%22%22%2B%22%221%2F%28x%2B4%29 %22%22=%22%22 1%2F8

Multiply through by LCD of 8x(x+4)

       8%28x%2B4%29%22%22%2B%22%228x %22%22=%22%22 x%28x%2B4%29

       8x%2B32%2B8x %22%22=%22%22 x%5E2%2B4x
       
       16x%2B32 %22%22=%22%22 x%5E2%2B4x
  
       %220%22 %22%22=%22%22 x%5E2-12x-32

       x%22%22=%22%22%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
 
       x%22%22=%22%22%28-%28-12%29+%2B-+sqrt%28%28-12%29%5E2-4%2A1%2A%28-32%29+%29%29%2F%282%2A1%29+ 

       x%22%22=%22%22%2812+%2B-+sqrt%28144%2B128%29%29%2F2+

       x%22%22=%22%22%2812+%2B-+sqrt%28272%29%29%2F2+

       x%22%22=%22%22%2812+%2B-+sqrt%2816%2A17%29%29%2F2+

       x%22%22=%22%22%2812+%2B-+4sqrt%2817%29%29%2F2+

       x%22%22=%22%22%284%283+%2B-+sqrt%2817%29%29%29%2F2+

       x%22%22=%22%222%283+%2B-+sqrt%2817%29%29+

       x%22%22=%22%2214.24621125

So it takes Maria about 14.25 hours or about 14 hours and 15 minutes
to clean the house by herself.

It takes Jane 4 hours longer or about 18 hours and 15 minutes to clean
the house by herself.

Edwin