David and Michael together can finish a job in 4 days 19 hrs 12 min. If David works at two-thirds Michael's speed, how long does it take Michael alone to finish the same job?

### Concept & Strategy When dealing with mixed time units (days, hours, minutes), always convert the total time into a single standard unit (like days or fractions of a day). Then, use efficiency ratios to find individual times. $$ \text{Total Work} = \text{Total Time} \times \text{Total Efficiency} $$ ### Step-by-Step Solution 1. **Convert time into a single unit (Days):** 19 hrs 12 min = 19 hours + $\frac{12}{60}$ hours = 19 + 0.2 = 19.2 hours. Convert 19.2 hours into days: $\frac{19.2}{24} = \frac{192}{240} = \frac{4}{5} = 0.8$ days. Total time taken together = 4 days + 0.8 days = 4.8 days = $\frac{24}{5}$ days. 2. **Establish efficiency ratios:** David works at $\frac{2}{3}$ of Michael's speed. Let Michael's efficiency be 3 units/day. David's efficiency will be $\frac{2}{3} \times 3 = 2$ units/day. 3. **Calculate total work:** Combined efficiency = $3 + 2 = 5$ units/day. Total Work = Combined Efficiency $\times$ Total Time Total Work = $5 \times \frac{24}{5} = 24$ units. 4. **Calculate Michael's individual time:** Time taken by Michael alone = $\frac{\text{Total Work}}{\text{Michael's Efficiency}}$ Time taken by Michael = $\frac{24}{3} = 8$ days. ### Exam Strategy & Shortcut If David's speed is $\frac{2}{3}$ of Michael's, their speed ratio is 2:3. Combined speed is 5 parts. Michael alone is 3 parts. Because time and speed are inversely proportional, the time taken by Michael compared to their combined time will be the inverse of their efficiency ratio. Time ratio (Michael alone : Combined) = Combined Speed : Michael's Speed = 5 : 3. Combined time is 4.8 days. Michael's time = $\frac{5}{3} \times 4.8 = 5 \times 1.6 = 8$ days. ### Common Pitfall A frequent error occurs during the time conversion step. Dividing 12 minutes by 100 instead of 60, or miscalculating 19.2 hours into days, will completely derail the rest of the calculations. ### Final Answer Therefore, the correct answer is **8 days**.