Here is a slightly different approach, though essentially it is the same as
that of the other tutor.
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?
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
>>...A carpenter can complete a certain job in 5 hours...<<
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
>>...(The carpenter) After working on the job for 2 hours,...<<
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:
+ =
2/5 + (1/5 + 1/x) = 1
Solve that and get 5/2 or 2.5
Edwin