Arpita and Nikita, working together, can complete an assigned job in 12 days. If Arpita works initially to complete 40% of the job, and the remaining job is completed by Nikita alone, it takes 24 days to complete the job. The possible number of days that Nikita requires to complete the entire job, working alone, is

Let the total work done be 1 unit.

Now, let us assume the total number of days taken by Arpita is A, then her efficiency will be = $$\dfrac{1}{A}$$

And the total number of days taken by Nikita is N, then her efficiency will be = $$\dfrac{1}{N}$$

It is given that they complete the work together in 12 days. Thus => $$\dfrac{1}{A}+\dfrac{1}{N}=\dfrac{1}{12}\rightarrow1$$

Now, it is given that Arpita works initially to complete 40% of the job, which means she does 0.4 units of work. For 1 unit, she was taking A days; for 0.4 units of work, she will take 0.4 A days. 

Then, Nikita completes the remaining job alone. Thus, Nikita will do 0.6 units of work, which will take 0.6N days. It is given that in this way, they take 24 days to complete the job.

=> $$0.4A+0.6N=24\rightarrow2$$

=> $$2A+3N=120$$

=> $$A=\dfrac{\left(120-3N\right)}{2}\rightarrow3$$

From eq. 1, we can get => $$\dfrac{\left(A+N\right)}{AN}=\dfrac{1}{12}$$

=> $$12A+12N=AN$$

Substituting the value of A from eq. 3 -

=> $$12\left(\dfrac{120-3N}{2}\right)+12N=\left(\dfrac{120-3N}{2}\right)N$$

=> $$1440-36N+24N=120N-3N^2$$             (Dividing the entire equation by 3)

=> $$480-4N=40N-N^2$$

=> $$N^2-44N+480=0$$

=> $$N^2-24N-20N+480=0$$          (Splitting the middle term)

=> $$\left(N-24\right)\left(N-20\right)=0$$

=> $$N=20$$ or $$N=24$$ 

Thus, Nikita alone takes to complete that work is either 20 or 24 days.