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, then it takes 24 days to complete the job. The possible number of days that Nikita requires to complete the entire job, working alone, is

Create your PRT table (simplest way to solve it).

We know that both working together (A + N) can complete the work in 12 days.

$\begin{array}{l|c|c|c} \text{Person: } & A+N & A & N \\ \hline \text{Rate:} & & \\ \hline \text{Time:} &12 & \\ \hline \text{Total Work:} & & \\ \end{array}$

If Aprita does 40% of the work and then Nikita does 60%, they'll complete the work in 24 days.

Let's assume the total work to be a bigger number that is divisible by 24 too (for ease in calculation), say 240 units.

We know that Rate $\times$ Time $=$ Work Done.

$\begin{array}{l|c|c|c} \text{Person: } & A+N & A &N \\ \hline \text{Rate:} &20 & \\ \hline \text{Time:} &12 & \\ \hline \text{Total Work:} & 240 &0.4 \times 240 = 96&0.6 \times 240 = 144 \\ \end{array}$

Let's assume the time taken by $A$ and $N$ as $x$ and $y$ respectively.

$\begin{array}{l|c|c|c} \text{Person: } & A+N & A &N \\ \hline \text{Rate:} &20 & \\ \hline \text{Time:} &12 &x &y \\ \hline \text{Total Work:} & 240 &96&144 \\ \end{array}$

Using the formula, we can get the rate:

$\begin{array}{l|c|c|c} \text{Person: } & A+N & A &N \\ \hline \text{Rate:} &20 & \frac{96}{x}& \frac{144}{y} \\ \hline \text{Time:} &12 &x &y \\ \hline \text{Total Work:} & 240 &96&144 \\ \end{array}$

Now, we have two equations:

$x+y = 24$ (they took 24 days to complete the job)

$\frac{96}{x} + \frac{144}{y} = 20$ (because their rate while working together is 20)

Since the numbers are clean, we can do hit & trial (try $x=12, y=12$) and get the answer too.

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Let $y = 24-x$.

Taking the 2nd equation and solving using the quadratic formula:

$\begin{aligned} \frac{96}{x} + \frac{144}{24 - x} &= 20\\ 96(24 - x) + 144x &= 20x(24 - x)\\ 2304 + 48x &= 480x - 20x^2\\ 20x^2 - 432x + 2304 &= 0\\ 5x^2 - 108x + 576 &= 0\\ \Delta = 144,\quad x &= \frac{108 \pm 12}{10} = \{12,\,9.6\}\\ y = 24 - x &= \{12,\,14.4\} \end{aligned}$

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Hence, one of the answers is $x=12, y=12$.

$\begin{array}{l|c|c|c} \text{Person: } & A+N & A &N \\ \hline \text{Rate:} &20 & \\ \hline \text{Time:} &12 &x &y \\ \hline \text{Total Work:} & 240 &96&144 \\ \end{array}$

Let's plug the same:

$\begin{array}{l|c|c|c} \text{Person: } & A+N & A &N \\ \hline \text{Rate:} &20 & 8 & 12 \\ \hline \text{Time:} &12 &12 &12 \\ \hline \text{Total Work:} & 240 &96&144 \\ \end{array}$

So, Nikita's rate of work is $12$. If she had to do the full work (of 240 units), she'll take $\dfrac{240}{12} = 20$ days.

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You can also solve with $x= 9.6, y = 14.4$. You'll get a different answer (but also valid). IPMAT Indore 2025 gave marks to both the answers.