Every work problem becomes simple the moment you stop thinking in "days" and start thinking in "work done per day" — a rate that can be added directly.
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If someone finishes a job in "n" days, they complete exactly 1/n of the job every single day — and rates from different people simply add.
📌 The one-day-rate method
Convert "days to finish" into a daily rate: A takes 12 days → A does 1/12 of the job per day.
Add the rates to combine workers: A(1/12) + B(1/18) = 3/36 + 2/36 = 5/36 per day together.
Flip the combined rate to get the time: 36/5 = 7.2 days.
✨ Finding one worker's time from the pair
Given
A alone: 20 days. A+B together: 12 days
B's rate = 1/12 − 1/20 = 1/30
Answer
flip the rate
B alone takes 30 days
When two people work on alternate days, calculate how much gets done in each 2-day CYCLE, then see how many full cycles are needed.
Worked example: A takes 10 days alone, B takes 15 days alone. Working on alternate days (A on day 1, B on day 2, and so on), each 2-day cycle completes 1/10 + 1/15 = 1/6 of the work. After 6 such cycles (12 days), the work is exactly finished.
A pipe that DRAINS a tank works exactly like a worker with a NEGATIVE rate — it undoes the work the filling pipes do.
🏆 Pipes and cisterns rules
| Situation | Formula |
|---|---|
| Two filling pipes together | Add both rates: 1/a + 1/b |
| One fills, one drains (leak) | Subtract the drain rate: 1/a − 1/b |
| If the net rate is negative | The tank never fills — it empties instead |
Worked example: a pipe fills a tank in 6 hours; a leak would empty it in 8 hours. Net rate = 1/6 − 1/8 = 1/24, so the tank fills in 24 hours with the leak present — much slower than 6 hours alone.
12 questions on time and work, alternate-day working, and pipes and cisterns.
