Question 1003709
Working together, it takes Sam, Jenna, and Frank two hours to paint
one room. When Sam is working alone, she can paint one room in 6 
hours. When Jenna works alone, she can paint one room in four 
hours. 

When Sam is working alone, she can paint one room in 6 hours.
<pre>
Since Sam can paint 1 room in 6 hours, her painting rate is 
1 room per 6 hours or {{{matrix(1,2,1,room)/matrix(1,2,6,hr)}}} or
{{{matrix(1,2,1/6,room/hr)}}}
</pre>
When Jenna works alone, she can paint one room in four hours. 
<pre> 
Since Jenna can paint 1 room in 4 hours her painting rate is
1 room per 4 hours or {{{matrix(1,2,1,room)/matrix(1,2,4,hr)}}} or
{{{matrix(1,2,1/4,room/hr)}}}
</pre>
Determine how long it would take Frank to paint one room on his own.
<pre>
Suppose that when Frank works alone, he can paint one room in x hours.

Since Frank can paint 1 room in x hours, his painting rate is 
1 room per x hours or {{{matrix(1,2,1,room)/matrix(1,2,x,hr)}}} or
{{{matrix(1,2,1/x,room/hr)}}}
</pre>
Working together, it takes Sam, Jenna, and Frank two hours to paint 
one room.
<pre>
Since all three working together can paint 1 room in 2 hours, the sum 
of their painting rates is 1 room per 2 hours
or {{{matrix(1,2,1,room)/matrix(1,2,2,hr)}}} or {{{matrix(1,2,1/2,room/hr)}}}

So the equation comes from that fact, that the sum of their painting 
rates is {{{matrix(1,2,1/2,room/hr)}}}:

{{{matrix(1,2,1/6,room/hr)}}}{{{""+""}}}{{{matrix(1,2,1/4,room/hr)}}}{{{""+""}}}{{{matrix(1,2,1/x,room/hr)}}}{{{""=""}}}{{{matrix(1,2,1/2,room/hr)}}}

{{{1/6}}}{{{""+""}}}{{{1/4}}}{{{""+""}}}{{{1/x}}}{{{""=""}}}{{{1/2}}}

Multiply through by LCD 12x

{{{2x}}}{{{""+""}}}{{{3x}}}{{{""+""}}}{{{12}}}{{{""=""}}}{{{6x}}}

{{{5x}}}{{{""+""}}}{{{12}}}{{{""=""}}}{{{6x}}}

{{{12}}}{{{""=""}}}{{{x}}}

So it will take Frank 12 hours to paint a room by himself.

Edwin</pre>