Shortcut Trick
Total Work = LCM(6, 10, 5) = 30 units.
Efficiency of (A + B) = 5, (B + C) = 3, and (A + B + C) = 6.
Efficiency of A = (A + B + C) − (B + C) = 6 − 3 = 3 units/day.
Efficiency of C = (A + B + C) − (A + B) = 6 − 5 = 1 unit/day.
Time for (A + C) = Total Work ÷ Efficiency(A + C) = 30 ÷ (3 + 1) = 7.5 days.
∴ The correct answer is 7½ days.
Alternate Method
Given:
Time taken by A and B together = 6 days
Time taken by B and C together = 10 days
Time taken by A, B, and C together = 5 days
Formula Used:
Total Work = Efficiency × Time
| Person | Days | Total Work | Efficiency |
|---|
| A + B | 6 | 30 | 5 |
| B + C | 10 | 3 | |
| A + B + C | 5 | 6 | |
Calculations:
⇒ Efficiency of A = Efficiency of (A + B + C) − Efficiency of (B + C)
⇒ Efficiency of A = 6 − 3 = 3 units/day
⇒ Efficiency of C = Efficiency of (A + B + C) − Efficiency of (A + B)
⇒ Efficiency of C = 6 − 5 = 1 unit/day
⇒ Combined Efficiency of (A + C) = 3 + 1 = 4 units/day
⇒ Time taken by (A + C) = Total Work ÷ Combined Efficiency
⇒ Time = 30 ÷ 4 = 15/2 = 7.5 days
∴ The correct answer is 7½ days.
Additional Information
Pipes & Cisterns
Conceptually identical to Time & Work. The only difference is an emptying pipe (leak) acts as negative efficiency, while a filling pipe acts as positive efficiency.
Alternate Day Work
For people working on alternate days, calculate the combined work done in one complete cycle to approach the total work, then find the remaining single days.
Work & Wages
Wages are directly distributed in proportion to the amount of total work completed by each individual, not by the time spent working.