Question

If (x+1) men will do the work in (x+1) days, then find the number of days in which (x+2) men can do the same work?

Answer 1

Hint- Here, we will proceed by finding the number of man-days in both the cases mentioned in the problem statement and then using the concept that the total number of man-days will always remain constant.

Complete step-by-step solution -

Given, a specific amount of work is completed by (x+1) men in (x+1) days.
Here, we have to find the number of days taken to complete the same amount of work when (x+2) men are doing the work.
Let this number of days taken to complete the same amount of work when (x+2) men are doing the work be y days.
As we know that the number of men multiplied by the number of days that they take to complete the work will give the number of man-days required to do the work. The total number of man-days required to complete a specific task is always constant.
For the first case,
Number of men = (x+1) men
Number of days = (x+1) days
Total number of man-days = (Number of men)$ \times $(Number of days)
$ \Rightarrow $Total number of man-days = (x+1)$ \times $(x+1) = ${\left( {x + 1} \right)^2}$
For the second case,
Number of men = (x+2) men
Number of days = y days
Total number of man-days = (Number of men)$ \times $(Number of days)
$ \Rightarrow $Total number of man-days = (x+2)$ \times $(y) = y(x+2)
  Since, the total number of man-days needs to be constant (i.e., equal for both the above discussed cases)
So, \[{\left( {x + 1} \right)^2} = y\left( {x + 2} \right)\]
\[ \Rightarrow y = \dfrac{{{{\left( {x + 1} \right)}^2}}}{{\left( {x + 2} \right)}}\]
Therefore, the number of days in which (x+2) men can do the same work is equal to \[\dfrac{{{{\left( {x + 1} \right)}^2}}}{{\left( {x + 2} \right)}}\] days.

Note- This man-days concept states that for same amount of work to be done, the number of men and number of days are inversely proportional to each other i.e., if the number of men increases then the number of days required to complete the same work decreases and if the number of men decreases then the number of days required to complete the same work increases.