Manish, Nakul and Pintoo alone can complete a certain work in 21days, 28 days and 15 days, respectively. Manish and Pintoo started the work together while Nakul joined them after 5 days and worked with them till the completion of the work. For how many days did Nakul work?

Correct Answer: 2 6/7 days (or $2\frac{6}{7}$ days)

To find out for how many days Nakul worked, we can break down the problem step-by-step:

Step 1: Determine the individual work rates of Manish, Nakul, and Pintoo.
Let the total work to be completed be represented as 1 unit.
- Manish can complete the work in 21 days, so Manish's 1-day work rate is:
$\frac{1}{21}$
- Nakul can complete the work in 28 days, so Nakul's 1-day work rate is:
$\frac{1}{28}$
- Pintoo can complete the work in 15 days, so Pintoo's 1-day work rate is:
$\frac{1}{15}$

Step 2: Calculate the work completed in the first 5 days.
Manish and Pintoo start the work together and work for 5 days before Nakul joins them.
The combined 1-day work rate of Manish and Pintoo is:

The work completed by Manish and Pintoo in 5 days is:
$5\times \frac{4}{35}=\frac{4}{7}$

Step 3: Calculate the remaining work.
After 5 days, the fraction of work left is:
$1-\frac{4}{7}=\frac{3}{7}$

Step 4: Calculate the combined work rate when Nakul joins.
Now, Manish, Pintoo, and Nakul work together to complete the remaining work. Their combined daily rate is:

Taking the least common multiple (LCM) of 35 and 28, which is 140:
$\frac{16+5}{140}=\frac{21}{140}=\frac{3}{20}$

Step 5: Determine the number of days Nakul worked.
Let Nakul work for D days with Manish and Pintoo to finish the remaining work.
$D=\frac{\text{Remaining Work}}{\text{Combined Rate}}$
$D=\frac{3/7}{3/20}=\frac{3}{7}\times \frac{20}{3}=\frac{20}{7}=2\frac{6}{7}\text{days}$

Therefore, Nakul worked for 2 6/7 days.