Lesson Had there were more workers, the job would be completed sooner

Had there were more workers, the job would be completed sooner

Problem 1

Let "n" be the number of workers in the first case.
Then the number of workers in the hypothetical case is (n+20).

All the job is  n*50  man-days in the first case, and (n+20)*45 man-days in the second case.

So the equation is

50n = 45(n+20).

Simplify and solve:

50n = 45n + 900,

5n = 900,

n =  = 180.

It is your answer: 180 workers are there in the base case.

Problem 2

Let "a" be the rate of work of one worker per day.



Then the number of days for 5 workers to complete the job is  ;

     the number of days for 6 workers to complete the job is  .



Thus, we can write this time equation 

     -  = 10  days.    (1)


It implies

     -  = 10,

     = 10,

    a = .     (2)


    +--------------------------------------------------------------+
    |   So, we found that under given condition the rate of work   |
    |       is  1/300  of the job per day for each worker.         |
    +--------------------------------------------------------------+


Now we want to find the number n of additional workers (to 5 workers) to complete 
the job 32 days earlier.  For it, we write similar time equation to (1)

     -  = 30.


Substitute here a = 1/300, based on (2). You will get

     -  = 30.


To solve, simplify step by step.  You will get

    60 -  = 30,

    60 - 30 = ,

       30   = 

    5 + n =  

    5 + n = 10.

        n = 10 - 5 = 5.


  +----------------------------------------------------------------+
  |   Second part of the solution can be worded in different way.  |
  +----------------------------------------------------------------+


We just found that the rate of work of one worker is  1/300  of the job per day.


It means that the entire job is 300 man-days.


5 workers can complete this job in  300/5 = 60 days.


We want the job be complete in 60-30 = 30 days.

Hence,  300/30 = 10 workers are needed, i.e. 10 - 5 = 5 workers should be added.


ANSWER.  5 workers should be hired in addition to the original 5 workers to complete the job 32 days earlier.