Work Rates, Production and Resource Allocation
Work-rate items ask how long tasks take when effort combines, or how much output a set of resources produces. The arithmetic mirrors speed questions, but the standard error is averaging completion times.
Core rules
- Rate = work / time. A job finished in 12 days is done at 1/12 of the job per day.
- Combined rate = sum of individual rates. Time together = 1 / (1/a + 1/b).
- Work done = rate × time. For sequential phases, add the work done, then convert back to time.
- Man-hours or machine-hours = number of units × hours worked. Output is proportional to man-hours when productivity is constant.
- Two workers together always take less time than the faster worker alone - use this to eliminate options instantly.
Worked example
A completes a report in 12 days, B in 18 days. A works alone for 4 days, then both work together. What is the total duration?
Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36 per day, so together they would need 36/5 = 7.2 days. Phase 1: A alone for 4 days completes 4/12 = 1/3 of the job. Remaining work = 2/3. Phase 2: at 5/36 per day, time = (2/3) ÷ (5/36) = (2/3) × (36/5) = 24/5 = 4.8 days. Total = 4 + 4.8 = 8.8 days.
Resource allocation
Five machines produce 900 units in 6 hours, so output is 900/(5 × 6) = 30 units per machine-hour. With 8 machines over 5 hours: 8 × 5 × 30 = 1,200 units. Scaling output means scaling both the resource count and the time.
Traps
- Averaging completion times: (12 + 18)/2 = 15 days is impossible, since the answer must be below 12.
- Adding times for sequential phases instead of adding work completed.
- Assuming constant productivity when the question states a percentage efficiency change.
- Forgetting that a worker or machine joining partway contributes only for the hours present.