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?
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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:
</pre></b></font>
>>...How long would it take Carl to do the work alone?...<<
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So we put x for Carl's time:

                NUMBER OF TASKS       RATE       TIME
Bart                  1                           
Carl                  1                           x                
Both together         1               
</pre></b></font>
>>...Bart can complete his science task twice as quickly as Carl can...<<
<pre><font size = 4 color = "indigo"><b>
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                         
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>>...When they work together, the task takes 3 hours...<<
<pre><font size = 4 color = "indigo"><b>
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 = "NUMBER_OF_TASKS"/TIME}}}

Bart's rate = {{{1}}}{{{"÷"}}}{{{x/2}}}{{{""=1""}}}{{{"*"}}}{{{2/x=2/x}}}

Carl's rate = {{{1/x}}}

"Both together" rate = {{{1/3}}}

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/x + 1/x = 1/3}}}

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

         {{{(3x)(2/x) + (3x)(1/x) = (3x)(1/3)}}}

         {{{(3cross(x))(2/cross(x)) + (3cross(x))(1/cross(x)) = (cross(3)x)(1/cross(3))}}}

         {{{6 + 3 = x}}}

         {{{9=x}}}

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

Edwin</pre>