A can do a piece of work in 90 days, B in 40 days and C in 12 days. They work for a day each in turn i.e., first day A does it alone, B does it the second day and C the third day. After that A does it for another day, and so on. After finishing the work they get ₹ 240. If the wages are divided in proportion to the work done by them, find what each will get. (M.A.T., 2006)

### Concept & Proportional Distribution for Irregular Work When workers take turns and the total work does not conclude at the exact end of a cycle, you must calculate the exact number of units each person completed individually to determine their precise share of the final payment. ### Step-by-Step Solution * **Determine Total Work and Efficiencies:** * LCM of 90, 40, and 12 is 360. Let Total Work = 360 units. * A's efficiency = $360 \div 90 = 4 \text{ units/day}$. * B's efficiency = $360 \div 40 = 9 \text{ units/day}$. * C's efficiency = $360 \div 12 = 30 \text{ units/day}$. * **Analyze the 3-Day Cycle:** * Day 1: A does 4 units. * Day 2: B does 9 units. * Day 3: C does 30 units. * Total work per cycle = $4 + 9 + 30 = 43 \text{ units}$. * **Calculate Cycles and Remaining Work:** * Number of full cycles = $360 \div 43 = 8 \text{ full cycles} \text{ (with a remainder)}$. * Work completed in 8 cycles = $8 \times 43 = 344 \text{ units}$. * Remaining work = $360 - 344 = 16 \text{ units}$. * **Allocate the Remaining Work:** * Next is A's turn (Day 25): A completes 4 units. Remaining = $16 - 4 = 12 \text{ units}$. * Next is B's turn (Day 26): B completes 9 units. Remaining = $12 - 9 = 3 \text{ units}$. * Next is C's turn (Day 27): C completes the last 3 units. * **Calculate Total Work Done by Each:** * A's total work = $(8 \text{ cycles} \times 4) + 4 = 32 + 4 = 36 \text{ units}$. * B's total work = $(8 \text{ cycles} \times 9) + 9 = 72 + 9 = 81 \text{ units}$. * C's total work = $(8 \text{ cycles} \times 30) + 3 = 240 + 3 = 243 \text{ units}$. * **Distribute the Wages (₹ 240):** * The ratio of work done is A : B : C = $36 : 81 : 243$. * Simplify by dividing by 9: $4 : 9 : 27$. * Sum of ratio parts = $4 + 9 + 27 = 40 \text{ parts}$. * Value of 1 part = $240 \div 40 = \text{₹ } 6$. * A's share = $4 \times 6 = 24$. * B's share = $9 \times 6 = 54$. * C's share = $27 \times 6 = 162$. ### Exam Strategy & Shortcut Track the total work accurately. Total work = 360. Efficiencies are 4, 9, 30. A cycle gives 43 units. 8 cycles give 344 units. The leftover 16 units go to A (4), then B (9), leaving exactly 3 for C. Work done is $36, 81, 243$. The total payment is 240, which is exactly $\frac{2}{3}$ of the total work units (360). Just multiply each person's work units by $\frac{2}{3}$: $36 \times \frac{2}{3} = 24$, $81 \times \frac{2}{3} = 54$, $243 \times \frac{2}{3} = 162$. ### Common Pitfall A major pitfall is dividing the total wages strictly by the efficiency ratio (4:9:30) without realizing the work cycle was broken at the end. Because the final cycle is incomplete, the exact work ratio is slightly different from the basic efficiency ratio. ### Final Answer Therefore, the correct answer is **A ₹ 24, B ₹ 54 and C ₹ 162**.