Lesson 56 of 100 | Quantitative Aptitude / Work and Time / कार्य और समय
Work and wages
Learning outcome
Distribute a fixed payment according to actual work, accounting for unequal efficiency, unequal attendance and a partial final day.
Concepts and assumptions
Assume an agreed payment pool for a completed job, divided strictly in proportion to productive contribution. No separate salary, bonus or reimbursement applies. Assume comparable-quality work, constant rates, additive outputs and equal daily working hours.
A worker’s contribution is rate × actual working time. Therefore:
Individual payment = total payment × individual contribution ÷ total contribution.
This follows because each equal unit of completed work earns an equal share of the pool. For a job counted as 1, the contributions sum to 1.
If only relative efficiencies are known, multiply each efficiency number by actual days worked. The unknown common rate factor cancels when these weights are divided by their sum. Days alone determine the ratio only when efficiencies are equal.
Count a partial final day proportionately. If daily working hours differ, use actual hours with hourly efficiency instead of simply counting days.
Worked examples
Example 1 — Equal efficiency. A and B complete a job, working 5 and 8 days respectively at equal efficiency. Divide ₹15,600.
Equal rates cancel, so contribution ratio A:B = 5:8. Total parts = 13; one part = 15,600/13 = ₹1,200. A receives 5 × 1,200 = ₹6,000. B receives 8 × 1,200 = ₹9,600.
Example 2 — Different rates and durations. A:B efficiency is 3:2. A works 4 days and B 9 days to complete a job. Divide ₹25,000.
Contribution weights = 3 × 4 : 2 × 9 = 12:18 = 2:3. A receives 25,000 × 2/5 = ₹10,000. B receives 25,000 × 3/5 = ₹15,000. B’s longer participation outweighs B’s lower efficiency.
Example 3 — A departure and a partial day. A, B and C alone need 12, 18 and 36 days. All start together. B leaves after 3 days; A and C finish. Divide ₹36,000.
Choose 36 work units: daily outputs are A = 3, B = 2, C = 1. First-stage work = (3 + 2 + 1) × 3 = 18 units. Remaining 18 units take A+C another 18/(3 + 1) = 4 1/2 days. A and C each work 7 1/2 days; B works 3. Their contributions are 3 × 7.5 = 22.5, 2 × 3 = 6, and 1 × 7.5 = 7.5 units. Total = 36 units; payment per unit = 36,000/36 = ₹1,000. A receives 22.5 × 1,000 = ₹22,500; B receives ₹6,000; C receives ₹7,500.
Common mistakes
Dividing by days despite unequal rates; reversing efficiency ratios; crediting absent workers; rounding up attendance; treating this pool as fixed daily wages.
Practice questions
- Equally efficient A and B work 7 and 5 days respectively to finish a job. Divide ₹18,000.
- A:B efficiency is 4:3. A works 6 days and B 8 days to complete a job. Divide ₹24,000.
- A:B:C efficiency is 2:3:5. Their actual working durations are 6, 4 and 3 days respectively. Divide ₹39,000 for the completed job.
- A alone needs 18 days and B 24 days. With C helping from the start, all three finish in 6 days. Divide ₹36,000 according to actual work.
Worked solutions
- Contributions are in ratio 7:5. A receives 18,000 × 7/12 = ₹10,500; B receives 18,000 × 5/12 = ₹7,500.
- Weights = 4 × 6 : 3 × 8 = 24:24 = 1:1. Equal work earns equal shares: 24,000/2 = ₹12,000 each.
- Weights = 2 × 6 : 3 × 4 : 5 × 3 = 12:12:15 = 4:4:5. One part = 39,000/13 = ₹3,000. A and B receive ₹12,000 each; C receives ₹15,000.
- A completes 6/18 = 1/3; B completes 6/24 = 1/4. C completes the remainder, 1 − 1/3 − 1/4 = 5/12. Ratio = 4:3:5. One part = 36,000/12 = ₹3,000. Payments are ₹12,000, ₹9,000 and ₹15,000 respectively.