Question 594695
<pre>
Here is a slightly different approach, though essentially it is the same as
that of the other tutor.
</pre>
A carpenter can complete a certain job in 5 hours. After working on the job for 2 hours, an assistant helped finish the job. Together they completed the job in 1 hour. How long might it take the assistant, working alone, to complete a job similar to this one? 
<pre>
Make this chart:

                                  Number of    
                                  jobs or        Time in      Rate
                                  fraction        hours        in 
                                of job done       worked    jobs/hour
Carpenter alone for 1 job
Carpenter alone for 2 hours
Carpenter and assistant for 1 hour
Assistant alone for 1 job

Let x equal the number of hours it would take the assistant to do
1 complete job.  So we fill in 1 for the number of jobs and x for
the number of hours:

                                    Number of    
                                    jobs or        Time in      Rate
                                    fraction        hours        in 
                                  of job done       worked    jobs/hour
Carpenter alone for 1 job
Carpenter alone for 2 hours
Carpenter and assistant for 1 hour
Assistant alone for 1 job              1              x
</pre>
>>...A carpenter can complete a certain job in 5 hours...<<
<pre>
So we fill in 1 job and 5 hours on the first line:

                                    Number of    
                                    jobs or        Time in      Rate
                                    fraction        hours        in 
                                 of job done       worked    jobs/hour
Carpenter alone for 1 job              1              5
Carpenter alone for 2 hours
Carpenter and assistant for 1 hour
Assistant alone for 1 job              1              x

Next we form the rates in jobs/hour by dividing the number of jobs by the
number of hours:

                                    Number of    
                                    jobs or        Time in      Rate
                                    fraction        hours        in 
                                  of job done      worked    jobs/hour
Carpenter alone for 1 job              1              5          1/5
Carpenter alone for 2 hours
Carpenter and assistant for 1 hour
Assistant alone for 1 job              1              x          1/x
</pre>
>>...(The carpenter) After working on the job for 2 hours,...<<
<pre>
Fill in the carpenter's rate as 1/5 and his time this time as 2 hours.

                                    jobs or        Time in      Rate
                                    fraction        hours        in 
                                   of job done          worked    jobs/hour
Carpenter alone for 1 job              1              5          1/5
Carpenter alone for 2 hours                           2          1/5
Carpenter and assistant for 1 hour
Assistant alone for 1 job              1              x          1/x


Next we use the fact that their rate together is the sum of their rates
(like in those math problems about a boat in a stream where you add
the rate of the stream to the rate of the boat). So we determine their
rate together by adding 1/5 and 1/x, and fill in 1 for the number of hours 

                                    Number of    
                                    jobs or        Time in      Rate
                                    fraction        hours        in 
                                 of job done       worked    jobs/hour
Carpenter alone for 1 job              1              5          1/5
Carpenter alone for 2 hours                           2          1/5
Carpenter and assistant for 1 hour                    1        1/5+1/x
Assistant alone for 1 job              1              x          1/x

Now we get the fractions of a job for the middle two lines by multiplying
the rate in jobs/hour by hours worked

                                    Number of    
                                    jobs or        Time in      Rate
                                    fraction        hours        in 
                                 of job done       worked    jobs/hour
Carpenter alone for 1 job              1              5          1/5
Carpenter alone for 2 hours           2/5             2          1/5
Carpenter and assistant for 1 hour  1(1/5+1/x)        1        1/5+1/x
Assistant alone for 1 job              1              x          1/x

The equation comes from:

              
{{{(matrix(11,1,Fraction,of,a,job, done, by,carpenter,alone,for,2,hours))}}} + {{{(matrix(13,1,Fraction,of,a,job, done,by,carpenter,and,assistant,alone,for,1,hour))}}} = {{{(matrix(3,1,

1,complete,job))}}}

2/5 + (1/5 + 1/x) = 1

Solve that and get 5/2 or 2.5
 
Edwin</pre>