The correct answer is **B) 7.75 days**. Let's break down the problem step-by-step to determine the total time taken. **Step 1: Calculate the total work and individual efficiencies.** To find a common unit for total work, we take the Least Common Multiple (LCM) of the number of days each person takes to complete the work. LCM(12, 15, 20) = 60 units. Now, we calculate the efficiency of each worker in units per day: * P's efficiency ($E_P$) = $\frac{\text{Total Work}}{\text{Days P takes}} = \frac{60}{12} = 5$ units/day. * Q's efficiency ($E_Q$) = $\frac{\text{Total Work}}{\text{Days Q takes}} = \frac{60}{15} = 4$ units/day. * R's efficiency ($E_R$) = $\frac{\text{Total Work}}{\text{Days R takes}} = \frac{60}{20} = 3$ units/day. **Step 2: Calculate work done in Stage 1.** Initially, P and Q work together for 4 days. * Combined efficiency of P and Q = $E_P + E_Q = 5 + 4 = 9$ units/day. * Work done in the first 4 days = $9 \times 4 = 36$ units. * Remaining work = $60 - 36 = 24$ units. **Step 3: Calculate work done in Stage 2 (R with reduced efficiency).** After 4 days, P leaves, and R joins. R's efficiency is 25% less for the first 3 days he works. * R's reduced efficiency ($E_{R_{reduced}}$) = $E_R \times (1 - 0.25) = 3 \times 0.75 = 2.25$ units/day. * The team now consists of Q and R (with reduced efficiency). * Combined efficiency of Q and (reduced) R = $E_Q + E_{R_{reduced}} = 4 + 2.25 = 6.25$ units/day. * They work for 3 days in this phase. * Work done in these 3 days = $6.25 \times 3 = 18.75$ units. * Remaining work = $24 - 18.75 = 5.25$ units. **Step 4: Calculate work done in Stage 3 (R with full efficiency).** After 3 days, R regains his full efficiency. The team still consists of Q and R. * Combined efficiency of Q and (full) R = $E_Q + E_R = 4 + 3 = 7$ units/day. * Time taken to complete the remaining $5.25$ units of work = $\frac{\text{Remaining Work}}{\text{Combined Efficiency}} = \frac{5.25}{7}$ days. * $\frac{5.25}{7} = \frac{525}{700} = \frac{21 \times 25}{28 \times 25} = \frac{21}{28} = \frac{3}{4} = 0.75$ days. **Step 5: Calculate the total time taken.** Total time = Time in Stage 1 + Time in Stage 2 + Time in Stage 3 Total time = $4 \text{ days} + 3 \text{ days} + 0.75 \text{ days} = 7.75$ days. Therefore, the entire task is completed in 7.75 days. **Why other options are incorrect:** * **A) 7.5 days:** This would imply either R's reduced efficiency phase was shorter or the remaining work was less, or the efficiencies were miscalculated. Our step-by-step calculation shows 7.75 days precisely. * **C) 8 days:** This would mean that the final stage took longer than 0.75 days, which is not consistent with the remaining work and full efficiencies. * **D) 8.25 days:** This is a higher value and would result from errors in calculation, potentially overestimating the time taken in one or more stages. The detailed calculation confirms that 7.75 days is the precise answer.