That answer above 1 hour 40 minutes is incorrect, although the 4/3 hours
is correct.
Max can finish repairing a machine in 4 hours.
His helper can do the same job in 8 hours. Max began the job alone
and work for 2 hours.His helper joined until the was completed.
How long did it take them to finish the job?
Let the answer be x. Make this chart and put x in the middle column
on the 4th row,
fraction
of
jobs time rate in
done in hrs jobs/hr
Max alone doing 1 job
Helper alone doing 1 job
Max alone for 2 hours
Both together for x hours x
Max can finish repairing a machine in 4 hours.
So he can do 1 job in 4 hours so we fill in the first two blanks
in the first row:
fraction
of
jobs time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4
Helper alone doing 1 job
Max alone for 2 hours
Both together for x hours x
His helper can do the same job in 8 hours.
So the helper can do 1 job in 8 hours so we fill in the first two blanks
in the second row:
fraction
of
jobs time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4
Helper alone doing 1 job 1 8
Max alone for 2 hours
Both together for x hours x
Next we fill in the rates in the first two rows by dividing the
jobs done by the time in hours.
fraction
of
jobs time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4 1/4
Helper alone doing 1 job 1 8 1/8
Max alone for 2 hours
Both together for x hours x
Max began the job alone and work for 2 hours.
Max's rate for those 2 hours is the same as when doing 1 job,
namely 1/4, so we put 2 for the hours in the 3rd row and 1/4 for the rate
fraction
of
jobs time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4 1/4
Helper alone doing 1 job 1 8 1/8
Max alone for 2 hours 2 1/4
Both together for x hours x
We fill in the number of jobs done by multiplying the rate by
the time (1/4)(2) = 1/2, which is half a job, so we fill 1/2
in the 1st column on the 3rd row:
fraction
of
job time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4 1/4
Helper alone doing 1 job 1 8 1/8
Max alone for 2 hours 1/2 2 1/4
Both together for x hours x
His helper joined until the was completed.
Their combined rate is the sum of their rates, 1/4+1/8 = 2/8+1/8 = 3/8
So we fill that in the last column on the 4th row:
fraction
of
job time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4 1/4
Helper alone doing 1 job 1 8 1/8
Max alone for 2 hours 1/2 2 1/4
Both together for x hours x 3/8
We fill in the number of jobs done in the x hours by multiplying the
rate by the time (3/8)x, so we fill 1/2 in the 1st column on the 4th row:
fraction
of
jobs time rate in
done in hrs jobs/hr
Max alone doing 1 job 1 4 1/4
Helper alone doing 1 job 1 8 1/8
Max alone for 2 hours 1/2 2 1/4
Both together for x hours (3/8)x x 3/8
How long did it take them to finish the job?
The sum of the fractions of a job done in the 2 hours plus the fraction
of the job done in the x hours must total up to 1 job, so
1/2 + (3/8)x = 1
Multiply through by LCD = 8
4 + 3x = 8
3x = 4
x = 4/3 = 1 1/3 hours = 1 hour 20 minutes
Edwin
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