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Job Completion Problem Math Review #2 --
Calculating the Time for One Person to Complete a Job 
from Known Productivity
by Crystal Sloan 

Home  --- Resources --- Job Completion Problems Math Review.
See also: Job Completion Math Review #1, #3

Job Completion Problem Math Review #2

From Job Complete Problem Math Review #1:

  • Productivity = amount of work done per unit time

  • Productivity = total work done / total time worked

  • Always remember to use units in all steps of your calculations, and make sure they match on both sides of your equations.

Moving on:

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.

T units of time needed = (W units of work) / (P units of work/unit of time)

Notice 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.

  1. EXAMPLE 1: The time 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 Product A.
  2. EXAMPLE 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.
  3. GENERAL SOLUTION If 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 figure. 

    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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