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?
<pre>
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(1,3,1,house,cleaned)/matrix(1,2,x,hours)}}}{{{""=""}}}{{{matrix(1,2,1/x,matrix(1,2,house,cleaned)/hour)}}} 
</pre>
>>the time that Jane takes in cleaning the house alone is 4 hours more than the time Maria takes in cleaning the same house.<<
<pre>
Therefore Jane's work rate is 1 house cleaned per x+4 hours or
 
{{{matrix(1,3,1,house,cleaned)/matrix(1,2,x+4,hours)}}}{{{""=""}}}{{{matrix(1,2,1/(x+4),matrix(1,2,house,cleaned)/hour)}}} 
</pre>
>>Jane and Maria clean the house in 8 hours if they work together.<<
<pre>

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

{{{matrix(1,3,1,house,cleaned)/matrix(1,2,8,hours)}}}{{{""=""}}}{{{matrix(1,2,1/8,matrix(1,2,house,cleaned)/hour)}}} 

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

       {{{1/x}}}{{{""+""}}}{{{1/(x+4)}}} {{{""=""}}} {{{1/8}}}

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

       {{{8(x+4)}}}{{{""+""}}}{{{8x}}} {{{""=""}}} {{{x(x+4)}}}

       {{{8x+32+8x}}} {{{""=""}}} {{{x^2+4x}}}
       
       {{{16x+32}}} {{{""=""}}} {{{x^2+4x}}}
  
       {{{"0"}}} {{{""=""}}} {{{x^2-12x-32}}}

       {{{x}}}{{{""=""}}}{{{(-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
 
       {{{x}}}{{{""=""}}}{{{(-(-12) +- sqrt((-12)^2-4*1*(-32) ))/(2*1) }}} 

       {{{x}}}{{{""=""}}}{{{(12 +- sqrt(144+128))/2 }}}

       {{{x}}}{{{""=""}}}{{{(12 +- sqrt(272))/2 }}}

       {{{x}}}{{{""=""}}}{{{(12 +- sqrt(16*17))/2 }}}

       {{{x}}}{{{""=""}}}{{{(12 +- 4sqrt(17))/2 }}}

       {{{x}}}{{{""=""}}}{{{(4(3 +- sqrt(17)))/2 }}}

       {{{x}}}{{{""=""}}}{{{2(3 +- sqrt(17)) }}}

       {{{x}}}{{{""=""}}}{{{14.24621125}}}

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</pre>