Correct Answer: Option B\n\nLet's break down the problem step-by-step:\n\n**Step 1: Determine the individual efficiencies of P, Q, and R.**\nLet the efficiency of R be $x$ units per day. (This means R can complete $x$ units of work in one day).\n\nGiven that Q is twice as efficient as R:\nEfficiency of Q ($E_Q$) = $2 \times x = 2x$ units per day.\n\nGiven that P is $50\%$ more efficient than Q:\nEfficiency of P ($E_P$) = Efficiency of Q + $50\%$ of Efficiency of Q\n$E_P = E_Q + 0.5 \times E_Q = 1.5 \times E_Q$\n$E_P = 1.5 \times (2x) = 3x$ units per day.\n\nFor simplicity, we can assume $x=1$ unit/day.\nSo, $E_R = 1$ unit/day.\n$E_Q = 2$ units/day.\n$E_P = 3$ units/day.\n\n**Step 2: Calculate the total work based on the last phase of the project.**\nIn the final stage, R completes the remaining $10\%$ of the project in 4 days.\nWork done by R in 4 days = $E_R \times \text{Number of days} = 1 \text{ unit/day} \times 4 \text{ days} = 4$ units.\n\nSince these 4 units represent $10\%$ (or $1/10^{\text{th}}$) of the total work, we can find the total work ($W$):\n$10\%$ of $W = 4$ units\n$0.10 \times W = 4$\n$W = \frac{4}{0.10} = 40$ units.\n\nSo, the total project requires 40 units of work.\n\n**Step 3: Set up an equation for the total work based on all phases of the project.**\nThe project was completed in three phases:\n\n* **Phase 1:** P and Q work together for '$d$' days.\n Combined efficiency of P and Q ($E_{P+Q}$) = $E_P + E_Q = 3 + 2 = 5$ units/day.\n Work done in Phase 1 = $E_{P+Q} \times d = 5d$ units.\n\n* **Phase 2:** P leaves, and Q and R continue for $(d+2)$ days.\n Combined efficiency of Q and R ($E_{Q+R}$) = $E_Q + E_R = 2 + 1 = 3$ units/day.\n Work done in Phase 2 = $E_{Q+R} \times (d+2) = 3(d+2)$ units.\n\n* **Phase 3:** Q leaves, and R completes the remaining work in 4 days.\n Work done in Phase 3 = 4 units (as calculated in Step 2).\n\nThe sum of work done in all phases must equal the total work:\n$5d + 3(d+2) + 4 = 40$\n\n**Step 4: Solve the equation to find the value of 'd'.**\n$5d + 3d + 6 + 4 = 40$\n$8d + 10 = 40$\n$8d = 40 - 10$\n$8d = 30$\n$d = \frac{30}{8} = \frac{15}{4}$ days.\n\n**Step 5: Calculate the combined efficiency of P and R.**\nIf P and R were to work together, their combined efficiency ($E_{P+R}$) would be:\n$E_{P+R} = E_P + E_R = 3 + 1 = 4$ units/day.\n\n**Step 6: Calculate the time required for P and R to complete the entire project together.**\nTime = $\frac{\text{Total Work}}{\text{Combined Efficiency of P and R}}$\nTime = $\frac{40 \text{ units}}{4 \text{ units/day}} = 10$ days.\n\nThus, P and R working together would take 10 days to complete the entire project.