Ronak and Riya can do a work in 12 and 18 days, respectively. They worked together for six days and then Ronak left. Find the time taken by Riya to complete the work.

The correct answer is 3 days.

We are given that Ronak can complete a work in 12 days and Riya can complete the same work in 18 days. They work together for 6 days, after which Ronak leaves, and Riya finishes the remaining work alone.

Step 1: Find the one-day work of each person.

Ronak's one-day work = \frac{1}{12}

Riya's one-day work = \frac{1}{18}

Step 2: Find their combined one-day work.

Combined one-day work of Ronak and Riya = \frac{1}{12}+\frac{1}{18}

Finding LCM of 12 and 18 → LCM = 36

\frac{1}{12}+\frac{1}{18}=\frac{3}{36}+\frac{2}{36}=\frac{5}{36}

Step 3: Find the work done together in 6 days.

Work done by both in 6 days = 6\times \frac{5}{36}=\frac{30}{36}=\frac{5}{6}

Step 4: Find the remaining work after Ronak leaves.

Remaining work = 1-\frac{5}{6}=\frac{1}{6}

Step 5: Find the time taken by Riya to finish the remaining work alone.

Riya's one-day work = \frac{1}{18}

Time taken by Riya to complete \frac{1}{6} of the work:

\text{Time}=\frac{\frac{1}{6}}{\frac{1}{18}}=\frac{1}{6}\times 18=3\text{days}

Conclusion: Riya takes 3 days to complete the remaining work after Ronak leaves.