P, Q and R can do a work in 20, 25 & 15 days respectively. P, Q & R started work together and after 4 days P & Q leave the work and after 2 days R leaves the work and P joins again. After 3 days P leaves and Q joins and he completes the remaining work. Find the total time to complete the whole work.

The correct option is $11\frac{1}{4}$ days.

Let us calculate the total work and the work efficiencies of P, Q, and R step-by-step.

Step 1: Determine the total work and individual work rates (efficiencies)
The time taken by P, Q, and R to complete the work individually is:
• P = 20 days
• Q = 25 days
• R = 15 days

Let the total work be the Least Common Multiple (LCM) of 20, 25, and 15.
LCM(20, 25, 15) = 300 units.

Now, calculate the units of work done per day (efficiency) by each person:
• Efficiency of P = $\frac{300}{20}=15$ units/day
• Efficiency of Q = $\frac{300}{25}=12$ units/day
• Efficiency of R = $\frac{300}{15}=20$ units/day

Step 2: Calculate work done in each phase

Phase 1: First 4 days (P, Q, and R work together)
Combined daily work rate of (P + Q + R) = 15 + 12 + 20 = 47 units/day.
Work done in 4 days = $4\times 47=188$ units.

Phase 2: Next 2 days (P & Q leave, so only R works)
Work done by R in 2 days = $2\times 20=40$ units.

Phase 3: Next 3 days (R leaves, P joins again, so only P works)
Work done by P in 3 days = $3\times 15=45$ units.

Step 3: Calculate remaining work and time required by Q to complete it
Total work completed so far = 188 + 40 + 45 = 273 units.
Remaining work = 300 - 273 = 27 units.

Now, P leaves and Q joins to complete the remaining work.
Time taken by Q to complete the remaining 27 units = $\frac{27}{12}=\frac{9}{4}=2\frac{1}{4}$ days.

Step 4: Calculate total time taken to complete the whole work
Total time = 4 days + 2 days + 3 days + $2\frac{1}{4}$ days

Total time = $9+2\frac{1}{4}=11\frac{1}{4}$ days.

Thus, the total time required to complete the whole work is $11\frac{1}{4}$ days.