SOLUTION: A machine can produce a certain number of pages twice as fast as another machine. Operating together these machines can produce this number of pages in 8 minutes. How long would

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Question 120244: A machine can produce a certain number of pages twice as fast as another
machine. Operating together these machines can produce this number of pages
in 8 minutes. How long would it take each machine, working alone, to produce
this number of pages?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
A machine can produce a certain number of pages twice as fast as another
machine. Operating together these machines can produce this number of pages
in 8 minutes. How long would it take each machine, working alone, to produce
this number of pages?

Make this chart:

              No. of Pages      Rate in pages/min       Time
Fast machine                                                              
Slow machine                                               
Both together                                             

Let N = the number of pages.
Let x = the number of minutes it takes the fast machine to
       produce N pages,

Fill those in:
 
              No. of Pages      Rate in pages/min       Time
Fast machine       N                                     x                
Slow machine       N                                        
Both together      N                                         

We read:
>>...Operating together these machines can produce this number 
of pages in 8 minutes...<<

So we fill in the 8:
              No. of Pages      Rate in pages/min       Time
Fast machine       N                                     x                
Slow machine       N                                      
Both together      N                                     8 

We read:
>>...A machine can produce a certain number of pages twice as 
fast as another machine...<<

So the slow machine takes twice as long as the fast machine, so
fill in twice x or 2x for the slow machine's time:

              No. of Pages      Rate in pages/min       Time
Fast machine       N                                     x                
Slow machine       N                                    2x
Both together      N                                     8 

Now we get the three rates in pages/minute by dividing the number
of pages by the number or minutes, that is, 

                        Rate = (No. of pages)/Time:

              No. of Pages      Rate in pages/min       Time
Fast machine       N                  N%2Fx                x                
Slow machine       N                 N%2F%282x%29              2x
Both together      N                  N%2F8                8 

So to get the equation we use this idea:

Rate of 1st + Rate of 2nd = Their rate together

N%2Fx + N%2F%282x%29 = N%2F8

Multiply thru by LCD = 8x written as %28%288x%29%2F1%29

%28%288x%29%2F1%29N%2Fx + %28%288x%29%2F1%29N%2F%282x%29 = %28%288x%29%2F1%29N%2F8

Cancel the x's in the first term, the 2 into the 8, 
leaving 4 in the second term, also cancel the x's in 
the second term. Cancel the 8's in the term on the
right:
               4
%28%288cross%28x%29%29%2F1%29N%2Fcross%28x%29 + %28%28cross%288%29cross%28x%29%29%2F1%29N%2F%28cross%282%29cross%28x%29%29 = %28%28cross%288%29x%29%2F1%29N%2Fcross%288%29

So we have

8N + 4N = xN

    12N = xN

Divide both sides by N

     12 = x

So the fast machine's time is x = 12 minutes. 
Then the slow machine's time is 2x = 24 minutes.

Edwin