```Question 472710
Jun can finish an accounting work in 8 hours, while Leo can do the same in 6 hours. After 2 hours of working together, Jun left Leo for lunch and Leo finished the job. Find the amount of time for Leo to finish the job.
<pre>
Make this chart:

number of     Time
jobs or       spent    Rate
fraction of    in       in
job done      hours   jobs/hr
Jun working alone for 8 hrs
Leo working alone for 6 hrs
Jun & Leo working together for 2 hrs
Leo working alone finishing the job

Let x be the answer, so put x for the time Leo spent finishing the job.

Jun can do 1 complete job in 8 hours, so put 1 for the number of jobs on
the top line and 8 for his time spent.

Leo can do 1 complete job in 6 hours, so put 1 for the number of jobs on
the second line and 6 for his time spent.  Put 2 on the third line for
the time they spent working together:

number of     Time
jobs or       spent    Rate
fraction of    in       in
job done      hours   jobs/hr
Jun working alone for 8 hrs                   1           8
Leo working alone for 6 hrs                   1           6
Jun & Leo working together for 2 hrs                      2
Leo working alone finishing the job                       x

Fill in the rates in jobs/hr for the top two lines by dividing the
number of jobs by the number of hours.

number of     Time
jobs or       spent    Rate
fraction of    in       in
job done      hours   jobs/hr
Jun working alone for 8 hrs                   1           8       1/8
Leo working alone for 6 hrs                   1           6       1/6
Jun & Leo working together for 2 hrs                      2
Leo working alone finishing the job                       x

Next we can fill in 1/6 also for Leo's rate working alone to finish the job

number of     Time
jobs or       spent    Rate
fraction of    in       in
job done      hours   jobs/hr
Jun working alone for 8 hrs                   1           8       1/8
Leo working alone for 6 hrs                   1           6       1/6
Jun & Leo working together for 2 hrs                      2
Leo working alone finishing the job                       x       1/6

To fill in the rate for Jun & Leo working together we add their rates:

{{{1/8}}} + {{{1/6}}}
{{{3/24}}} + {{{4/24}}}
{{{7/24}}}

So we put 7/24 for their combined rate on the third line:

number of     Time
jobs or       spent    Rate
fraction of    in       in
job done      hours   jobs/hr
Jun working alone for 8 hrs                   1           8       1/8
Leo working alone for 6 hrs                   1           6       1/6
Jun & Leo working together for 2 hrs                      2      7/24
Leo working alone finishing the job                       x       1/6

Now we can fill in the fractions of the job done on the bottom two lines
by multiplying the rates in jobs/hr by the time in hours:

{{{7/24}}}·2 = {{{7/12}}}  and {{{1/6}}}·x = {{{x/6}}}

number of     Time
jobs or       spent    Rate
fraction of    in       in
job done      hours   jobs/hr
Jun working alone for 8 hrs                   1           8       1/8
Leo working alone for 6 hrs                   1           6       1/6
Jun & Leo working together for 2 hrs       7/12           2      7/24
Leo working alone finishing the job         x/6           x       1/6

The equation comes from:

{{{(matrix(5,1,

fraction,
"of_job",
working,
together,
"2_hrs")) +

(matrix(5,1,

fraction,
"of_job",
"Leo_did",
alone_to,
finish))=

(matrix(3,1,
one,
complete,
job)) }}}

{{{7/12 + x/6 = 1}}}

Multiply through by 12:

{{{7 + 2x = 12}}}

{{{2x = 5}}}

{{{x = 5/2 = 2&1/2}}} hours.

Edwin</pre>
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