Time and Work
About 7 min · Numerical and Mathematical Ability · English
Time worked out from the lesson's 538 words at 150 words a minute, plus a minute for each question.
The core idea is simple: if you know how long a person (or machine, or pipe) takes to complete a task alone, you can calculate how long they'll take working together or under varying conditions. Mastering this topic builds a strong foundation for related areas like pipes and cisterns, which use identical logic but in a different scenario.
Key ideas
- Work as a whole unit: Treat the entire task as 1 unit of work. If A completes it in 10 days, A does 1/10 of the work per day. This fractional rate is called A's efficiency or rate of work.
- Total work = LCM method: To avoid fractions, assume total work = LCM of all individual times. If A finishes in 12 days and B in 15 days, take total work = LCM(12,15) = 60 units. Then A's efficiency = 60/12 = 5 units/day, B's = 4 units/day.
- Combined efficiency: When A and B work together, their one-day work = (1/A's time) + (1/B's time). Joint time = 1 / (combined one-day work). Using the LCM method, just add their efficiencies and divide total work by that sum.
- Man-days concept: Work ∝ (Number of workers) × (Number of days). If 5 men complete work in 8 days, the same work needs 40 man-days. So 10 men would finish in 4 days.
- Pipes and cisterns analogy: An inlet pipe filling a tank = a worker doing work. An outlet (drain) pipe emptying a tank = negative work. Net filling rate = (sum of inlet rates) − (sum of outlet rates).
Watch out for
- Adding times instead of rates: Saying "A takes 10 days, B takes 15, together 25 days" is wrong. Always add the work rates (1/10 + 1/15), not the times themselves.
- Forgetting negative work for outlet pipes: Treat an emptying pipe or a person destroying work as negative efficiency. Students often add all efficiencies as positive, leading to incorrect answers.
- Not accounting for partial days correctly: If A and B together finish 7/10 of work in 7 days, don't assume they need 3 more days for the rest unless their combined rate stays constant. Check if anyone has left or conditions changed.
Quick reference
- **A alone in a days, B alone in b days ⇒ Together in (a×b)/(a+b) days.**
- Total work = LCM of individual times — cleanest method to avoid fraction errors.
- Work done = Efficiency × Days worked.
- Pipes: Net rate = Inlets − Outlets. Positive → fills, Negative → empties.
- Man-days: M₁D₁ = M₂D₂ for same work.
Check yourself: 3 questions
1. A can complete a piece of work in 12 days and B can complete the same work in 18 days. If they work together, in how many days will they complete the work?
2. 15 men can complete a work in 20 days. After 10 days of work, 5 more men were added. In how many more days will the work be completed?
3. A pipe can fill a tank in 6 hours. Due to a leak at the bottom, it takes 9 hours to fill the tank. If the tank is full, in how many hours will the leak empty it?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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