SOLUTION: 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 nea

Algebra ->  Rational-functions -> SOLUTION: 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 nea      Log On


   

Question 1155727: 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.)
Found 5 solutions by josmiceli, Edwin McCravy, AnlytcPhil, Alan3354, josgarithmetic:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +t+ = the time in hrs it takes Dennis to do the job
+t+-+2+ = the time in hrs it takes Michael to do the job
---------------------------
Dennis's rate of working:
[ 1 job ] / [ t hrs ]
Michael's rate of working:
[ 1 job ] / [ t - 2 hrs ]
---------------------------
Add their rates of working to get their rate working together
+1%2Ft+%2B+1%2F%28+t-2+%29+=+1%2F5+
Multiply both sides by +t%2A%28+t-2+%29%2A5+
+5%2A%28+t+-+2+%29+%2B+5t+=+t%2A%28+t-2+%29+
+5t+-+10+%2B+5t+=+t%5E2+-+2t+
+t%5E2+-+12t+%2B+10+=+0+
-----------------------------
Use quadratic formula
+t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+1+
+b+=+-12+
+c+=+10+
----------------
+t+=+%28-%28-12%29+%2B-+sqrt%28+%28-12%29%5E2-4%2A1%2A10+%29%29%2F%282%2A1%29+
+t+=+%28+12+%2B-+sqrt%28+144+-+40+%29%29%2F2+
+t+=+%28+12+%2B+10.198+%29%2F2+
+t+=+22.198+%2F+2+
+t+=+11.099+
and
+t+-+2+=+9.099+
------------------------
Rounded off, Michael's time to do the job alone is:
9.1 hrs
-----------
check the answer:
+1%2Ft+%2B+1%2F%28+t-2+%29+=+1%2F5+
+1%2F11.099+%2B+1%2F9.099+=+1%2F5+
+.0901+%2B+.1099+=+.2+
+.2009+=+.2+
Error due to rounding off, I believe
------------
Get a 2nd opinion also, if needed



Answer by Edwin McCravy(20054) About Me  (Show Source):
Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
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.) 

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
Michael can do a job 2 hours faster than Dennis.
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
Together they can complete the work in 5 hours.
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=%281%2F%28t-2%29%2B1%2Ft%29%2A5

Distribute

1=5%2F%28t-2%29%2B5%2Ft

Multiply through by LCD of t(t-2)

t%28t-2%29+=+5t+%2B+5%28t-2%29

t%5E2-2t+=+5t+%2B+5t-10

t%5E2-2t=10t-10

t%5E2-12t%2B10=0

t+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ 

t+=+%28-%28-12%29+%2B-+sqrt%28+%28-12%29%5E2-4%2A1%2A10+%29%29%2F%282%2A1%29+ 

t+=+%2812+%2B-+sqrt%28144-40+%29%29%2F%282%29+

t+=+%2812+%2B-+sqrt%28104%29%29%2F2+

t+=+%2812+%2B-+sqrt%284%2A26%29%29%2F2+ 

t+=+%2812+%2B-+2sqrt%2826%29%29%2F2+ 

t=12%2F2+%2B-+sqrt%2826%29

t=6%2B-+sqrt%2826%29

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

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
"... 2 hours faster …" is nonsensical.
… in 2 hours less time... makes sense.
========================
I can run 10 mi/hr.
I can run 2 hours faster than that.
----------------
Hours is a measure of time.
Fast is an indication of speed, not time.

Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
Question asks time for Michael by himself.

1%2Fx, Michael's rate
1%2F%28x%2B2%29, Dennis'es rate
1%2F5, combined rate for both

highlight_green%281%2Fx%2B1%2F%28x%2B2%29=1%2F5%29