Priya can do a certain piece of work in 26 days. Priya and Varsha can together do the same work in 13 days, and Priya, Varsha and Nisha can do the same work together in 10 days. In how many days can Priya and Nisha do the same work together?

The correct option is 65/4 (or 16¹/₄ days).

Let's solve the problem step-by-step using work rates.

Step 1: Determine individual work rates.

Let total work be equal to 1 unit.

Priya can complete the work in 26 days.
Therefore, Priya's 1-day work rate = $\frac{1}{26}$

Priya and Varsha together can complete the work in 13 days.
Therefore, (Priya + Varsha)'s 1-day work rate = $\frac{1}{13}$

Priya, Varsha, and Nisha together can complete the work in 10 days.
Therefore, (Priya + Varsha + Nisha)'s 1-day work rate = $\frac{1}{10}$

Step 2: Find Nisha's 1-day work rate.

Nisha's 1-day work rate = (Priya + Varsha + Nisha)'s rate - (Priya + Varsha)'s rate

$\text{Nisha's rate}=\frac{1}{10}-\frac{1}{13}$

Taking LCM of 10 and 13, which is 130:

$\text{Nisha's rate}=\frac{13-10}{130}=\frac{3}{130}$

Step 3: Find the combined 1-day work rate of Priya and Nisha.

(Priya + Nisha)'s 1-day work rate = Priya's rate + Nisha's rate

$\text{Combined rate}=\frac{1}{26}+\frac{3}{130}$

Taking LCM of 26 and 130, which is 130 (since 26 × 5 = 130):

$\text{Combined rate}=\frac{5+3}{130}=\frac{8}{130}=\frac{4}{65}$

Step 4: Calculate the time taken by Priya and Nisha working together.

Time required = $\frac{1}{\text{Combined rate}}$

$\text{Time taken}=\frac{65}{4}\text{days}$

Thus, Priya and Nisha can do the work together in 65/4 days.