Question 438028
Sally can clean the house in 6 hours her father can clean the house in 4 hours and her younger brother dennis can completely mess up the house in 8 hours if sally and her father clean and Dennis plays how long will it take to clean the house
<pre><font face = "DFKai-SB" color = "indigo" size = 4><b>
Make this chart:


        no. of  houses cleaned  |  time |  rate in houses/hour   
--------------------------------------------------------------
Sally                           |       |
Father                          |       |
Dennis                          |       | 
--------------------------------------------------------------
All 3                           |       |

Let x be the time for all three doing their thing, whether cleaning 
or playing, to get 1 house clean.  So we put x for the time for all 3:

        no. of  houses cleaned  |  time |  rate in houses/hour   
--------------------------------------------------------------
Sally                           |       |
Father                          |       |
Dennis                          |       | 
--------------------------------------------------------------
All 3                           |   x   |

Now fill in the 3 times: 


        no. of  houses cleaned  |  time |  rate in houses/hour   
--------------------------------------------------------------
Sally                           |   6   |
Father                          |   4   |
Dennis                          |   8   | 
--------------------------------------------------------------
All 3                           |   x   |

Sally can clean 1 house in 6 hours, so we put 1 for the
no. of houses Sally can clean.
Her father can clean 1 house in 4 hours, so we put 1 for the
no. of houses her father can clean.
Dennis can mess up 1 house in 8 hours, so we put -1 for the
no. of houses Dennis can clean, because to mess up 1 house is
mathematically the same as "cleaning -1 house"
Since all 3 will be cleaning 1 house, put 1 for the no. of
houses cleaned by all 3

        no. of  houses cleaned  |  time |  rate in houses/hour   
--------------------------------------------------------------
Sally              1            |   6   |
Father             1            |   4   |
Dennis            -1            |   8   | 
--------------------------------------------------------------
All 3              1            |   x   |

Fill in the rates in houses/hour by dividing the no. of
houses by the number of hours:

        no. of  houses cleaned  |  time |  rate in houses/hour   
--------------------------------------------------------------
Sally              1            |   6   |        1/6
Father             1            |   4   |        1/4
Dennis            -1            |   8   |       -1/8
--------------------------------------------------------------
All 3              1            |   x   |        1/x

Now the equation come from

Sally's rate + Father's rate + Dennis' rate = Rate for all 3

                   1/6 + 1/4 - 1/8 = 1/x

Clear of fractions by multiplying through by LCD of 24x

                     4x + 6x - 3x = 24
                               7x = 24
                                x = 24/7 = 3 3/7 hours

Edwin</pre>