Question 1155727
<pre>
Josemicelli was right.  I made a mistake before so I reposted it correctly in
my other pseudo-name AnlytcPhil:

Michael can do a job 2 hours faster than Dennis. Together they can complete the work in 5 hours. How long would it take Michael to do the job alone? 
(Round your answer to the nearest tenth of an hour.) 
<pre>
Instead of a "D=RT" problem, this is a "J=RT" problem. "Jobs" instead of "Distance"

We put in 1 job for all three situations:

                                  Jobs      =     Rate      *    Time
                                              [In jobs/hr]     [In hrs.]  
-------------------------------------------------------------------------
Michael when working alone          1                             
Dennis when working alone           1                             
Michael & Dennis working together   1</pre>Michael can do a job 2 hours faster than Dennis.<pre>So we put t for Dennis' time, and t-2 for Michael's time because Michael 
take 2 LESS hours than Dennis.

                                  Jobs      =     Rate      *    Time
                                              [In jobs/hr]     [In hrs.]  
-------------------------------------------------------------------------
Michael when working alone          1                            t-2 
Dennis when working alone           1                             t
Michael & Dennis working together   1

Then we fill in the rates using R=J/T like we fill in R=D/T in other problems.  

                                  Jobs      =     Rate      *    Time
                                          [In jobs/hr]     [In hrs.]  
-------------------------------------------------------------------------
Michael when working alone          1            1/(t-2)         t-2 
Dennis when working alone           1             1/t             t
Michael & Dennis working together   1</pre>Together they can complete the work in 5 hours.<pre>When they work together, their rate is the sum of their individual rates,
so we express this sum by putting + between them.  Then we fill in 5 for 
the number of hours working together.

                                  Jobs      =     Rate      *    Time
                                             [In jobs/hr]     [In hrs.]  
-------------------------------------------------------------------------
Michael when working alone          1            1/(t-2)         t-2 
Dennis when working alone           1             1/t             t
Michael & Dennis working together   1         1/(t-2) + 1/t       5

Then we use JOBS = RATE × TIME

                                    1  =     [1/(t-2) + 1/t]  ×   5

{{{1=(1/(t-2)+1/t)*5}}}

Distribute

{{{1=5/(t-2)+5/t}}}

Multiply through by LCD of t(t-2)

{{{t(t-2) = 5t + 5(t-2)}}}

{{{t^2-2t = 5t + 5t-10}}}

{{{t^2-2t=10t-10}}}

{{{t^2-12t+10=0}}}

{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

{{{t = (-(-12) +- sqrt( (-12)^2-4*1*10 ))/(2*1) }}} 

{{{t = (12 +- sqrt(144-40 ))/(2) }}}

{{{t = (12 +- sqrt(104))/2 }}}

{{{t = (12 +- sqrt(4*26))/2 }}} 

{{{t = (12 +- 2sqrt(26))/2 }}} 

{{{t=12/2 +- sqrt(26)}}}

{{{t=6+- sqrt(26)}}}

Two answers:

t = 11.1 hours and 0.9 hours

We must discard Dennis taking only 0.9 hours because Michael's time
to do it would then be negative.  So we discard t=2.

So Dennis' time to do it alone is 11.1 hours.  Therefore Michael can do the 
job in 2 hours less, so Michael can do the job in 9.1 hours.

Edwin</pre>