Shortcut Trick
Total work = LCM(30, 33, 36, 44) = 1980 units.
Efficiencies: A = 66, B = 60, C = 55, D = 45 units/day.
Work in 4-day cycle = 2 × (A + B + C) + 2 × (A + B + D) = 2 × 181 + 2 × 171 = 704 units.
Work in 8 days (2 cycles) = 1408 units. Remaining = 1980 − 1408 = 572 units.
In next 3 days, work done = 362 (by A, B, C in 2 days) + 171 (by A, B, D in 1 day) = 533 units.
Remaining 39 units takes 39 ÷ 171 = 13/57 days. Total time = 11 + 13/57 days.
∴ The correct answer is 1113⁄57 days.
Alternate Method
Given:
Time taken by A = 30 days
Time taken by B = 33 days
Time taken by C = 36 days
Time taken by D = 44 days
Let total work be the LCM of (30, 33, 36, 44).
| Person | Days | Total Work | Efficiency |
|---|
| A | 30 | 1980 | 66 |
| B | 33 | 60 | |
| C | 36 | 55 | |
| D | 44 | 45 | |
Calculations:
⇒ Combined efficiency of A, B and C = 66 + 60 + 55 = 181 units/day
⇒ Combined efficiency of A, B and D = 66 + 60 + 45 = 171 units/day
⇒ For the first 2 days, A, B and C work together. Work done = 181 × 2 = 362 units
⇒ For the next 2 days, A, B and D work together. Work done = 171 × 2 = 342 units
⇒ Total work done in 1 cycle of 4 days = 362 + 342 = 704 units
⇒ Work done in 2 complete cycles (8 days) = 704 × 2 = 1408 units
⇒ Remaining work after 8 days = 1980 − 1408 = 572 units
⇒ On days 9 and 10, A, B and C work again. Work completed = 362 units
⇒ Remaining work after 10 days = 572 − 362 = 210 units
⇒ On day 11, A, B and D work and complete 171 units.
⇒ Remaining work after 11 days = 210 − 171 = 39 units
⇒ On day 12, A, B and D continue. Time taken for remaining work = 39 ÷ 171 = 13/57 days
⇒ Total time taken = 8 + 2 + 1 + 13/57 = 1113⁄57 days
∴ The correct answer is 1113⁄57 days.
Additional Information
LCM Method for Time & Work
Assuming the total work to be the Least Common Multiple (LCM) of the given times ensures that each person's per-day work (efficiency) is an integer, significantly speeding up calculations.
Alternate Days Concept
When individuals or groups work on alternating days, the work forms a repeating cycle. Find the total work done in one complete cycle and determine the maximum number of full cycles that fit into the total work before dealing with the remainder step-by-step.
Efficiency and Total Work Rule
Total Work = Efficiency × Time. If multiple entities work simultaneously, their individual efficiencies are simply added up: Combined Efficiency = E1 + E2 + E3.