Job Completion Problem Math Review #2 --
--- Resources --- Job Completion Problems
Calculating the Time for One Person to Complete a Job
from Known Productivity
See also: Job Completion
Math Review #1, #3
Job Completion Problem Math Review #2
From Job Complete Problem Math Review #1:
= amount of work done per unit time
= total work done / total time worked
remember to use units in all steps of your calculations, and make sure
they match on both sides of your equations.
THE PROBLEM: Given
the productivity value P units of work/unit of time for a worker doing a
certain type of work, and the total amount W of work to be completed,
determine T, the number of units of time needed for that worker to
complete all the work.
units of time needed = (W units of work) / (P units of work/unit of time)
that the units portion of both sides of the equation simplifies to
"units of time." The "(units of work)/(units of
work)" on the right side of the equation "cancel out"
(simplify to 1, that is, a pure number with no units), leaving only
1/1/(units of time) = "units of time" on the right-hand side.
- EXAMPLE 1:
required to complete a job is the amount of work to do, divided by the
worker's productivity for that job. Last week, we saw that
every 3 hours, Worker Joe builds another 6 units of Product A, and
calculated that Joe's productivity was 2 units of Product A in one
hour = 2 (units of A)/hour. Now that we know Worker Joe's
productivity for Product A, we can ask: how long will it take Worker
Joe to complete 25 units of Product A? To calculate the time
needed, divide the total amount of work needed (25 units of Product A)
by Joe's productivity:
T hours = 25 (units of A)/(2 (units of A)/hour) )
Simplifying by bringing the numbers together with the numbers, and the
units with the units for easier reading:
T hours = (25/2) (units of A) / ((units of A)/hour)
Remember that dividing by a fraction is equivalent to multiplying by
its reciprocal, as long as the dividends are nonzero. Here we
are dividing by "(units of A)/hour", so we can simplify by
multiplying by "hours/(units of A)" instead:
T hours = 12.5 (units of A) (hours/(units of A) ) = 12.5 hours (units
of A)/(units of A) = 12.5 hours * 1 = 12.5 hours
Hence it should take Worker Joe 12.5 hours to produce 25 units of
2: There are 25 boxes of apples in one "bin." If
last week Worker Joe picked 75 boxes of apples in 8 hours, then his
productivity P is 75 boxes/8 hours = 75/8 boxes/hour = 9.375
boxes/hour. This week you need to have him pick 7 bins to fill
an order from a cider maker. How many 10-hour days will it take
him to pick 7 bins? (Yes, apple pickers often work 10-hour days.)
We already know the productivity, as boxes/hour. Next, we need
to make the units of the old work ("boxes") match those of
the new work ("bins"). It does not matter which way we
convert, as long as we make the units match. Since we already know
Worker Joe's productivity in boxes/hour, let's choose to convert the
units for the new amount of work (7 bins) to the same units we used in
the calculation of Worker Joe's productivity: New amount of work = 7
bins * 25 boxes/bin = 175 boxes.
Once we have the productivity and have made the units of work match,
we can calculate the total time needed for Worker Joe to do the new
amount of work: T = W/P, so T hours = 175 boxes/ (9.375 boxes/hour) =
18 2/3 hours.
We need the time in 10-hour days, not in hours, so convert the time to
complete the 25 bins from 18 2/3 hours (= 18.67 hours) by dividing by
10 hours/day to yield an answer of 1.867 days of 10 hours each.
If we needed the time in 8-hour days instead, we would divide 18 2/3
hours by 8 hours/day to get 2 1/3 days.
- GENERAL SOLUTION:
you know how long it takes a worker to do one amount of work, one way
of figuring out how long he or she takes to do a different amount of
the same work is to follow these steps:
Step 1: First calculate the worker's Productivity P:
P = known amount of work done / known time worked.
Step 2: If the units of work in the productivity value and
given with the new amount of work differ: Next convert the new amount
of work to be done to the same units as that used in the productivity
Step 3: Next calculate the time T for the worker to do the new
amount of the same work:
T = new amount of work to do / P
The resulting time to complete the new work will be given in the same
units as the time units part of the productivity units.
Step 4: If you need the time to complete the new work in
different units, convert it to those units.
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