Manvi and Darshan together can complete a project work in 40 days. They worked together for 25 days and then Darshan left. After another 24 days, Manvi completed the remaining project work. In how many days can Manvi alone complete the project?

Correct option is D

Given:
Manvi and Darshan together can complete the project in 40 days.
They worked together for 25 days, and then Darshan left.
After Darshan left, Manvi completed the remaining work in 24 days.

Formula Used:

$$\text{Work done per day} = \frac{1}{\text{Total time taken together}}$$

$$\text{Work done} = \text{Work done per day} \times \text{Number of days}$$​​

Solution:
Let, Total work = 1 unit
Work done by Manvi and Darshan together in 1 day = $$\frac{1}{40}$$​  units
Work done by Manvi and Darshan in 25 days:

​$$\text{Work done} = 25 \times \frac{1}{40} = \frac{25}{40} = \frac{5}{8} \text{ units}$$

​$$\text{Remaining work} = 1 - \frac{5}{8} = \frac{3}{8} \text{ units}$$​

Let Manvi's work rate be $$\frac{1}{M}$$​ units per day, where M  is the number of days Manvi takes to complete the work alone.

Work done by Manvi in 24 days:

​$$24 \times \frac{1}{M} = \frac{24}{M} \text{ units}$$

So,   $$\frac{24}{M} = \frac{3}{8}$$​

​$$M = \frac{24 \times 8}{3} = 64 \text{ days}$$

Option (D) is right.