Wikipedia:Reference desk/Archives/Mathematics/2016 January 16

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January 16

Time and Work question anomaly

A question related to Time and Work came across me..... and I found that two different answers come out using two different methods- 1. Theorem 2. Logic

Q. A work could be completed by certain workers in 22 days. However, due to 3 workers being absent, it was completed in 24 days. Find out the original number of workers.

A. Theorem- Number of workers × Time Taken (Number of days)= Work Done

  So, it goes as-
  x=Original number of workers
  22x=24(x-3)
  22x=24x-72
 ∴x=36
  

A. Logic-

  x=Original number of workers
  If 24 days of work is caused by x-3 workers and 22 days of work is done by x workers, then 2 days of work is caused by absence of 3 workers
 ∴3(workers)=2(days)
  1.5(workers)=1(day)
  So original number of workers is (22×1.5)33.
  
  Theorem-36
  Logic-33

Please help me sort out this problem and inform me where I am wrong. --Varun T.^Talk 14:42, 16 January 2016 (UTC)

The second argument does not take into account that the missing 3 worker's contribution reduces the amount of work needed from the remaining workers. An way to check is to compute the number of workers days needed. In the first solution you get 22×36 = 24×33 worker days, but for the second solution you get 22×33 ≠ 24×30. --RDBury (talk) 17:44, 16 January 2016 (UTC)

So, if the original poster were to re-write the second argument (logic), what step should he re-write? And what would be the correct revision to that step? Thanks. Joseph A. Spadaro (talk) 20:05, 16 January 2016 (UTC)

The problem was in generalizing "the loss of 3 workers makes the project take an additional 2 days". While this was true when going from 36 workers to 33, this is not true in general. For example, when you lose the last 3 workers, and go down to no workers, will that just make the project take 2 more days ? Obviously not, since the project will take an infinite amount of time to complete with nobody working on it. StuRat (talk) 03:40, 17 January 2016 (UTC)

See also The Mythical Man-Month. But for problem purposes it's clear we are supposed to ignore these real-world complexities. --76.69.45.64 (talk) 08:00, 17 January 2016 (UTC)

You logic is wrong. It goes.. "The absence of 3 workers" results in "2 additional day of work". Thus "The absence of 1.5 workers" results in "1 additional day of work". Therefore "22 additional days of works" is a result of "The absence of 22*1.5 or 33 workers". WRONG! You confused the absence of Q workers with the number of workers. 175.45.116.66 (talk) 23:00, 17 January 2016 (UTC)

Another way to describe the logic error is that it assumes a linear relationship between number of workers and time to complete the project. However, this relationship is nonlinear. Let's look at a simplified case of a project which requires 12 man-days to complete. Here's a noticeably non-linear chart showing how long it should take with different numbers of workers:

Days
  ^
12| *
11|
10|
 9|
 8|
 7|
 6|   *
 5|
 4|     * 
 3|       *
 2|           *
 1|                       *
 0+-------------------------> Workers
  0 1 2 3 4 5 6 7 8 9 1 1 1
                      0 1 2

StuRat (talk) 00:11, 20 January 2016 (UTC)