Sowmika and Rachana can complete a piece of work in $40\text{days}$ and $60\text{days}$ respectively. Sowmika starts the work alone, and after $4\text{days}$, Rachana joins her to complete the remaining work. In how many days will the entire work be completed?

Let the total work be the LCM of $40$ and $60$:
$$\text{Total Work}=120\text{units}$$

Now, calculate their individual daily efficiencies:

* Sowmika's efficiency $=\frac{120}{40}=3\text{units/day}$
* Rachana's efficiency $=\frac{120}{60}=2\text{units/day}$

Sowmika works alone for the first $4\text{days}$:

$$\text{Work done by Sowmika in 4 days}=4\times 3=12\text{units}$$

Calculate the remaining work:

$$\text{Remaining Work}=120-12=108\text{units}$$

Now Rachana joins her, so their combined daily efficiency becomes:

$$\text{Combined Efficiency}=3+2=5\text{units/day}$$

Time taken by both together to complete the remaining work:

$$\text{Time taken together}=\frac{108}{5}=21.6\text{days}$$

Calculate the total time taken for the entire work:

$$\text{Total Days}=4\text{days (alone)}+21.6\text{days (together)}=25.6\text{days}$$