P, Q, and R can complete a work in 20, 25, and 15 days respectively. They all start working together. After 4 days, P and Q leave the work. After 2 more days, R also leaves, and P resumes working alone. After 3 days, P leaves and Q joins to complete the remaining work. What is the total time taken to finish the entire work?

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**Solution:**

1. Calculate the daily work rates of P, Q, and R:
   - P's rate = $\frac{1}{20}$ work/day
   - Q's rate = $\frac{1}{25}$ work/day
   - R's rate = $\frac{1}{15}$ work/day

2. For the first 4 days, all three work together:
   - Combined rate = $\frac{1}{20} + \frac{1}{25} + \frac{1}{15}$
   - Find common denominator: 300
   - $\frac{15}{300} + \frac{12}{300} + \frac{20}{300} = \frac{47}{300}$ work/day
   - Work done in 4 days = $4 	imes \frac{47}{300} = \frac{188}{300} = \frac{47}{75}$

3. Next 2 days, only R works:
   - Work done = $2 	imes \frac{1}{15} = \frac{2}{15}$

4. Total work done so far:
   - $\frac{47}{75} + \frac{2}{15} = \frac{47}{75} + \frac{10}{75} = \frac{57}{75} = \frac{19}{25}$

5. Then P works alone for 3 days:
   - Work done = $3 	imes \frac{1}{20} = \frac{3}{20}$

6. Total work done till now:
   - $\frac{19}{25} + \frac{3}{20} = \frac{76}{100} + \frac{15}{100} = \frac{91}{100}$

7. Remaining work = $1 - \frac{91}{100} = \frac{9}{100}$

8. Q completes the remaining work:
   - Time taken by Q = $\frac{	ext{Work}}{	ext{Rate}} = \frac{9/100}{1/25} = \frac{9}{100} 	imes 25 = \frac{225}{100} = 2.25$ days

9. Total time to complete the work:
   - $4 + 2 + 3 + 2.25 = 11.25$ days or $11 \frac{1}{4}$ days

**Answer:** $11 \frac{1}{4}$ days

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