1. **State the problem:** We have two tasks, A and B. Task A alone takes 4 days, and Task B alone takes 6 days. When done together, productivity is reduced by 20% due to coordination issues. We want to find how long both tasks will take when done together. 2. **Formula and rules:** The work rate is the reciprocal of the time taken. If $T_A$ is the time for task A alone, and $T_B$ for task B alone, then their rates are $\frac{1}{T_A}$ and $\frac{1}{T_B}$ respectively. When working together without any loss, combined rate is: $$\text{Rate}_{combined} = \frac{1}{T_A} + \frac{1}{T_B}$$ With a 20% productivity reduction, effective combined rate is: $$\text{Rate}_{effective} = 0.8 \times \text{Rate}_{combined}$$ The time taken together is the reciprocal of the effective rate: $$T = \frac{1}{\text{Rate}_{effective}}$$ 3. **Calculate individual rates:** $$\frac{1}{T_A} = \frac{1}{4} = 0.25$$ $$\frac{1}{T_B} = \frac{1}{6} \approx 0.1667$$ 4. **Calculate combined rate without reduction:** $$0.25 + 0.1667 = 0.4167$$ 5. **Apply 20% reduction:** $$\text{Rate}_{effective} = 0.8 \times 0.4167 = 0.3333$$ 6. **Calculate total time together:** $$T = \frac{1}{0.3333} = 3$$ **Final answer:** Both tasks done together with 20% productivity loss will take **3 days**.