$40$ persons can take $30$ days to complete a work. How long will it take if $20$ more people join them?

Hint: The given questions revolve around the concepts of proportions. There are mainly two types of relations between any two given variables: direct relation or direct proportion and inverse relation or inverse proportion. The given question is an illustration where a certain number of people take a specific number of days to complete a work and we have to calculate the number of days they will take if some more people join in. We first find out the type of relation or proportion in the number of days and the number of workers and then form a table of values for the same.

Complete step-by-step solution:
In the given problem,
Number of people initially is $40$.
So, the number of days taken to complete a work is $30$ days.
Now, we are given that $20$ more people join them and we have to calculate the number of days they will take in the new case.
So, let the number of days taken in the new case be x days.
Now, we know that if the number of workers increases, they will take less number of days to complete the same work and vice versa.
Hence, we know that the number of workers and the number of days they take to complete the work are indirectly proportional to each other.
Now,
So, making a table for the number of workers and the time taken by them to complete the work, we get,


Now, calculating the value of x following the inverse relation between the number of workers and the time taken by them to complete the task, we get,
$ \Rightarrow x = \dfrac{{40 \times 30}}{{60}}$
Carrying out the calculations in numerator, we get,
$ \Rightarrow x = \dfrac{{1200}}{{60}}$
Cancelling the common factors in numerator and denominator, we get,
$ \Rightarrow x = 20$
So, the time taken by workers if $20$ more people join in to complete the work is $20$ days.

Note: In any inverse proportion relation, the product of the two quantities remains constant. So, here, the product of number of workers and time duration remains constant in both the cases.