SOLUTION: Bart can complete his science task twice as quickly as Carl can. When they Work together, the task takes 3 hours. How long would it take Carl to do the work alone?

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Bart can complete his science task twice as quickly as Carl can. When they Work together, the task takes 3 hours. How long would it take Carl to do the work alone?      Log On


   

Question 264387: Bart can complete his science task twice as quickly as Carl can. When they Work together, the task takes 3 hours. How long would it take Carl to do the work alone?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Bart can complete his science task twice as quickly as Carl can. When they Work together, the task takes 3 hours. How long would it take Carl to do the work alone?

Make this chart

                NUMBER OF TASKS       RATE       TIME
Bart                 
Carl                               
Both together        

In all three cases, Bart alone, Carl alone, and both together,
they will all do just 1 task so we put 1 for the NUMBER OF TASKS
in each of the three cases 



                NUMBER OF TASKS       RATE       TIME
Bart                  1               
Carl                  1                               
Both together         1              


The question asks:

>>...How long would it take Carl to do the work alone?...<<

So we put x for Carl's time:

                NUMBER OF TASKS       RATE       TIME
Bart                  1                           
Carl                  1                           x                
Both together         1

>>...Bart can complete his science task twice as quickly as Carl can...<<

That means Bart requires only HALF as much time, so we divide
Carl's by 2 to get Bart's time.  So we put x/2 for Bart's time.


                NUMBER OF TASKS       RATE       TIME
Bart                  1                          x/2
Carl                  1                           x                
Both together         1

>>...When they work together, the task takes 3 hours...<<

So we put 3 for the time for "both together"":


                NUMBER OF TASKS       RATE       TIME
Bart                  1                          x/2
Carl                  1                           x                
Both together         1                           3

Now we figure the three rates in tasks per hour.

RATE+=+%22NUMBER_OF_TASKS%22%2FTIME

Bart's rate = 1%22%F7%22x%2F2%22%22=1%22%22%22%2A%222%2Fx=2%2Fx

Carl's rate = 1%2Fx

"Both together" rate = 1%2F3

So fill those in:

                NUMBER OF TASKS       RATE       TIME
Bart                  1               2/x        x/2
Carl                  1               1/x         x                
Both together         1               1/3         3

Now we get the equation from the rates:

Bart's RATE + Carl's RATE = the RATE for "both together":

         2%2Fx+%2B+1%2Fx+=+1%2F3

Get a common denominator of 3x  and multiply
every term by 3x

         %283x%29%282%2Fx%29+%2B+%283x%29%281%2Fx%29+=+%283x%29%281%2F3%29

         

         6+%2B+3+=+x

         9=x

So it would take Carl 9 hours to complete the task.

Edwin