Arun, Varun and Tarun, if working alone, can complete a task in 24, 21, and 15 days, respectively. They charge Rs 2160, Rs 2400, and Rs 2160 per day, respectively, even if they are employed for a partial day. On any given day, any of the workers may or may not be employed to work. If the task needs to be completed in 10 days or less, then the minimum possible amount, in rupees, required to be paid for the entire task is

The correct answer is 38400.

To find the minimum possible amount to be paid for the entire task, we first need to determine the cost-efficiency of each worker. We can do this by calculating the total cost if each worker were to complete the task entirely alone.

The total theoretical cost for each worker completing the whole task is:

Arun:
$24\times 2160=51840$
rupees

Varun:
$21\times 2400=50400$
rupees

Tarun:
$15\times 2160=32400$
rupees

From these calculations, we can see that Tarun is the most cost-efficient worker for the task, followed by Varun, while Arun is the least efficient (most expensive) per unit of work completed. To minimize the overall cost, we must maximize the amount of work done by the most cost-efficient worker.

Since the task needs to be completed in 10 days or less, we should employ the most efficient worker, Tarun, for the maximum possible duration, which is 10 days. The fraction of the work Tarun completes in these 10 days is:

$\frac{10}{15}=\frac{2}{3}$

The remaining fraction of the task that still needs to be completed is:

$1-\frac{2}{3}=\frac{1}{3}$

To complete this remaining work at the lowest possible cost, we should use the next most efficient worker, Varun. We calculate how many days Varun would need to finish this remaining 1/3 of the task:

$21\times \frac{1}{3}=7$

Varun will need 7 days to complete the remaining work. Since 7 days is less than the 10-day time limit, this distribution of labor is feasible within the given constraints.

Now, we can calculate the total minimum cost by adding the amounts paid to Tarun and Varun for their respective days of work.

Amount paid to Tarun for 10 days:
$10\times 2160=21600$
rupees

Amount paid to Varun for 7 days:
$7\times 2400=16800$
rupees

Total Minimum Cost:
$21600+16800=38400$
rupees