Question : X can do a work in 3 days, Y can do three times the same work in 8 days, and Z can do five times the same work in 12 days. If they have to work together for 6 hours a day, then how much time will they take to complete the work?
Option 1: 4 hours
Option 2: 5 hours 20 minutes
Option 3: 4 hours 10 minutes
Option 4: 5 hours
Correct Answer: 5 hours 20 minutes
Solution : X can complete the work in 3 days. Y can do three times the same work in 8 days. Z does five times the same work in 12 days. Working hours per day = 6 hours Efficiency = $\frac{\text{Total work}}{\text{Total Time}}$ Let the total work be 1 unit. Time to complete 1 unit by X alone = 3 × 6 = 18 hours Time to complete the 3 units by Y alone = 8 × 6 = 48 hours Time to complete the 1 unit by Y alone = $\frac{48}{3}$ = 16 hours Time to complete the 5 units by Z alone = 12 × 6 = 72 hours Time to complete the 1 unit by Z alone = $\frac{72}{5}$hours Efficiency of X = $\frac{1}{18}$ units/hour Efficiency of Y = $\frac{1}{16}$ units/hour Efficiency of Z = $ \frac{5}{72}$ units/hour Combined efficiency of X, Y and Z $=\frac{1}{18} + \frac{1}{16}+ \frac{5}{72}=\frac{(8 + 9 + 10)}{144}=\frac{27}{144}$ units/hour Time to complete total work by X, Y and Z together = $\frac{1}{\frac{27}{144}}$= $\frac{144}{27}$ hours = $\frac{144}{27}×60$ = $320$ minutes = 5 hours 20 minutes Hence, the correct answer is 5 hours 20 minutes.
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