Concept 1
Convert completion time into a daily rate
If A completes a job in x days, A's one-day work is 1/x. Work completed in d days is d/x. Time and rate are reciprocals when the total job is fixed.
LCM method: choose total work as the LCM of individual times. If A takes 12 days and B 18 days, take 36 units; their rates are 3 and 2 units/day.
Concept 2
Add simultaneous work rates
If A takes 12 days and B takes 18 days, together they do 1/12+1/18=5/36 per day, so time = 36/5 = 7.2 days.
For a worker who undoes work, subtract the rate. Never add or average completion times directly.
Concept 3
Efficiency and time move inversely
If efficiencies A:B=3:2, their times for equal work are 2:3. If A is 25% more efficient than B, efficiency ratio A:B=125:100=5:4, so time ratio A:B=4:5.
Workers × days × hours × efficiency is proportional to work when conditions remain comparable.
Concept 4
Split changing teams into time blocks
Calculate the work done before a worker joins or leaves, subtract it from one whole job, then use the new combined rate for the remainder.
For alternate-day work, find the work completed in one full cycle, count complete cycles, and handle the remaining day in the stated order.
Concept 5
Divide wages by actual work contribution
If workers operate for the same time, wages follow their efficiency ratio. Otherwise wages follow rate × time. A twice as efficient as B but working half as long contributes equally.
Do not divide wages merely by headcount unless every worker contributes equally.
Concept 6
Pipes use the same rate model
An inlet filling a tank in x hours has rate +1/x; an outlet emptying it in y hours has rate −1/y. Add signed rates, then invert the positive net rate to find filling time.
| Action | Rate sign | Example |
|---|---|---|
| Fills | Positive | +1/6 tank/hour |
| Empties | Negative | −1/9 tank/hour |
| Net | Sum | 1/18 tank/hour |
Worked methods
Examples with reasoning
Two workers together
A takes 10 days and B 15 days. Rate=1/10+1/15=1/6, so together they take 6 days.
Find an unknown worker
A+B take 8 days; A takes 12. B's rate=1/8−1/12=1/24, so B takes 24 days.
A leaves midway
A and B rates are 3 and 2 units/day for a 30-unit job. Together for 4 days complete 20 units; B needs 5 more days for the remaining 10.
Workers and days
12 workers finish in 15 days. At equal efficiency, 20 workers need 12×15/20=9 days.
Inlet with a leak
A pipe fills in 6 hours and a leak empties in 9. Net=1/6−1/9=1/18, so the tank fills in 18 hours.
Exam traps
Errors that reverse or distort rates
Adding days
Convert times to rates before combining workers.
Efficiency in the wrong direction
Greater efficiency means less time for equal work.
Ignoring completed work
When a team changes, subtract the first block before solving the remainder.
Positive leak
Outlets and leaks have negative rates while filling.
Wrong alternate-day order
The worker who starts may also handle the final partial cycle.
Equal wages by default
Wages follow contribution, not simply the number of workers.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- Rate = 1/time for one complete job.
- Combined rate = sum of individual signed rates.
- Efficiency ratio is the inverse of time ratio.
- Work ∝ workers × days × hours × efficiency.
- Wages follow actual contribution: rate × time.
- Inlets are positive; outlets and leaks are negative.
Source ledger
Official foundations
- SSC CGL official syllabus: Time and Work under Mathematical Abilities
- NCERT official mathematics exemplar directory
- NCERT official textbook portal
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 25 September 2026.
Time, speed and distance.