Question 320236
if sally can paint a house in 4hrs and john can paint the same house in 6hrs, how long will it take for both of them to paint the house together

<pre><b>
The easy way to do it is this way, which you can do in your head:

4 and 6 both go into 12.  In 12 hours Sally can paint 3 houses and
John can paint 2 houses.  So that's 5 houses in 12 hours, or 1 house
in {{{12/5}}} hours or {{{2&2/5}}} hours, or 2 hours 24 minutes.

But your teacher doesn't want you to do it that easy way.  

Your teacher wants you to make this chart, get an equation 
and solve it:

          number        time        rate in
            of         required     houses
          houses         in          per  
          painted       hours        hour
--------|-----------|-----------|----------
Sally   |           |           |
--------|-----------|-----------|----------
John    |           |           |  
--------|-----------|-----------|----------
Both    |           |           | 
--------|-----------|-----------|---------- 


Let x be the number of hours required for both.
So fill in x in the middle on the bottom row,
for the time for both to paint one house working
together.

          number        time        rate in
            of         required     houses
          houses         in          per  
          painted       hours        hour
--------|-----------|-----------|----------
Sally   |           |           |
--------|-----------|-----------|----------
John    |           |           |  
--------|-----------|-----------|----------
Both    |           |     x     | 
--------|-----------|-----------|---------- 
  
Next fill in the times, 4 for Sally and 6 for John


          number        time        rate in
            of         required     houses
          houses         in          per  
          painted       hours        hour
--------|-----------|-----------|----------
Sally   |           |     4     |
--------|-----------|-----------|----------
John    |           |     6     |  
--------|-----------|-----------|----------
Both    |           |     x     | 
--------|-----------|-----------|---------- 

We are only talking about them painting 1 house so we
fill in 1 for the number of houses painted in all three
cases:

          number        time        rate in
            of         required     houses
          houses         in          per  
          painted       hours        hour
--------|-----------|-----------|----------
Sally   |     1     |     4     |
--------|-----------|-----------|----------
John    |     1     |     6     |  
--------|-----------|-----------|----------
Both    |     1     |     x     | 
--------|-----------|-----------|----------

Next we fill in the rates in houses/hour by
dividing the number of houses by the number
of hours

          number        time        rate in
            of         required     houses
          houses         in          per  
          painted       hours        hour
--------|-----------|-----------|----------
Sally   |     1     |     4     |    {{{1/4}}}
--------|-----------|-----------|----------
John    |     1     |     6     |    {{{1/6}}}
--------|-----------|-----------|----------
Both    |     1     |     x     |    {{{1/x}}}
--------|-----------|-----------|---------- 

Then we form the equation from:

       (Sally's rate) + (John's rate) = (rate for both together)
                          
                                 {{{1/4}}}{{{"+"}}}{{{1/6}}}{{{"="}}}{{{1/x}}} 
 Multiply through by LCD = 12x
                         {{{12x(1/4)}}}{{{"+"}}}{{{12x(1/6)}}}{{{"="}}}{{{12x(1/x)}}}
                               {{{3x}}}{{{"+"}}}{{{2x}}}{{{"="}}}{{{12}}}                              
                                   {{{5x}}}{{{"="}}}{{{12}}}

                                    {{{x}}}{{{"="}}}{{{12/5}}}
or {{{2&2/5}}} hours, or 2 hours 24 minutes 

Edwin</pre>