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Job Completion Problems Math Review #3 --
Calculating the Calendar Time for a Team to Complete a Job 
When All Members Have the Same Productivity
by Crystal Sloan 

Home  --- Resources --- Job Completion Problems Math Review.
See also: Job Completion Math Review #1, #2

Job Completion Problem Math Review #3

From review #2:

  • 1.  Productivity is the units of work W to do, divided by units of time T it takes to do it: P = W/T  * (units of work/units of time).  Productivity is always expressed in units of work/units of time.  

  • 2. The number units of time T needed to complete a job is the number of units of work W divided by the worker's productivity P in the same units.

  • 3. Remember that the units of work and time used must match, or you have to convert them to match.

Moving on...

THE PROBLEM: Given the productivity value P units of work/unit of time for each of N identical workers doing a certain type of work, and the total amount W of work to be completed, determine C, the number of units of calendar time needed for those workers, working together, to complete all the work.  Assume that the work being done does not have any productivity loss due to workers having to coordinate their work, go to meetings, etc.

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

C units of calendar time needed = T units of individual working time needed / N workers

Since all workers in this example have the same productivity P, all we need to do is calculate the time it would take one worker to do all the work (as was detailed in Job Completion Math Review #2), then DIVIDE by the number of workers N.  Since all workers will be working at the same time, it takes LESS calendar time to finish the job than when just one worker has to do all the work.

  1. EXAMPLE: 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 found that Worker Joe's productivity was  2 (units of product A)/hour.  How long will it take Joe and two of his co-workers, both with productivity the same as Joe's, to complete 25 units of Product A? First we ask, as we did last week: 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.

    Now that we know how long it will take Joe to do the work all alone, we can divide 12.5 hours by 3 people to get the time it will take Joe's team of 3 to do the work: 

    Calendar time C (hours/(3-person team)) = 12.5 hours / 3 people on team = 4 1/6 (hours/3-person team)

    Note that in job completion problems, calendar time C is always measured in (hours/particular assignment of resources, such as a team). If the team size or average productivity changes, the calendar time to complete a task will change, too.  

    Tip: Do not be confused by people referring to calendar time as simple hours.        
  2. PITFALL TO AVOID: An amazing fact about arithmetic and job completion problems is that some problems are not intuitive.  This means that the average person's feeling about how to work the problem can be the opposite of what really should be done. 

    You cannot work these problems by "feelings." If you do, you will very often get the wrong answer.  

    When presented with two numbers X and Y, and the task of deciding what to do with them (divide X by Y, divide Y by X, or multiple X times Y), many people will find their "feeling" about the problem will tend to go against dividing a smaller number by a larger number, or dividing a small number by another small number, or dividing one number by another that is near it in value.  

    Most people feel more comfortable dividing a big number by a small number, or multiplying two smaller numbers, but mentally shy away from other combinations.  

    It is very important to not follow your feelings in deciding how to manipulate the numbers in the problem, but instead be careful to follow the formula, and afterwards check the result to make sure it makes sense.   

    For example, it will always be true that no matter how large or small the numbers involved, C units of calendar time needed = T units of individual working time needed / N workers. 

    Check your answer against the following fact:

    If you are using more than one person on your team, the calendar time C needed for the team to do the work will always be less than the time it takes one person to do the work. If you find your team taking longer to do the work than one person does, you have made a mistake, probably by multiplying instead of dividing, or by reversing the divisor and dividend in your division. 
  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 one 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.
    Step 5: Divide the time it would take one worker to do the work by the number of workers who will be working simultaneously on the job.  
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