Riya and Sangeeta can finish a work in 6 days and 8 days, respectively. Riya started the work alone and then after 3 days, Sangeeta joined Riya. They both finish the remaining work. How long did the total work last?

The correct option is $4\frac{5}{7}\text{ days}$.

Step 1: Calculate the total work and individual efficiencies.
Let the total work be equal to the Least Common Multiple (LCM) of the number of days taken by Riya and Sangeeta.
LCM of 6 days and 8 days = 24 units.
Therefore, Total Work = 24 units.

Efficiency of Riya = $\frac{24\text{ units}}{6\text{ days}}=4\text{ units/day}$

Efficiency of Sangeeta = $\frac{24\text{ units}}{8\text{ days}}=3\text{ units/day}$

Step 2: Calculate work done by Riya initially.
Riya worked alone for 3 days.
Work completed by Riya in 3 days = $3\times 4=12\text{ units}$

Step 3: Calculate remaining work.
Remaining work = Total Work - Work done by Riya
Remaining work = $24-12=12\text{ units}$

Step 4: Calculate time taken to finish the remaining work together.
Combined efficiency of Riya and Sangeeta = $4+3=7\text{ units/day}$

Time taken to complete remaining work = $\frac{12\text{ units}}{7\text{ units/day}}=1\frac{5}{7}\text{ days}$

Step 5: Calculate total duration of work.
Total time = Time Riya worked alone + Time worked together
Total time = $3\text{ days}+1\frac{5}{7}\text{ days}=4\frac{5}{7}\text{ days}$

Thus, the total work lasted for $4\frac{5}{7}\text{ days}$.