Time and Work
About 7 min · Mathematics · English
Time worked out from the lesson's 453 words at 150 words a minute, plus a minute for each question.
The underlying principle is simple: Work = Rate × Time. Once you grasp that work is constant and rate is inversely proportional to time, all problems—whether about men digging canals, pipes filling tanks, or combined efforts—follow predictable patterns.
Key ideas
- Work is always taken as 1 unit (or 100%). If A completes a job in 10 days, A's one-day work = 1/10. If B does it in 15 days, B's one-day work = 1/15. Work done is cumulative and additive.
- Rate of work = 1/Time. If time decreases, rate increases proportionally. A faster worker has a higher rate. This inverse relationship is the backbone of all problems.
- Combined work: add rates, not times. If A works at 1/10 per day and B at 1/15 per day, together they work at (1/10 + 1/15) per day. Never add 10 + 15 to find combined time.
- Work efficiency is the ratio of work rates. If A is twice as efficient as B, A's rate is double B's rate. Efficiency ratios directly convert to rate ratios.
- Man-days or worker-days measure total work. If 8 men complete work in 12 days, total work = 8 × 12 = 96 man-days. This quantity stays constant; if men increase, days decrease proportionally (and vice versa).
Watch out for
- Adding times instead of rates. Students write "A takes 10 days, B takes 15 days, together 25 days" — wrong. Correct: add rates 1/10 + 1/15, then invert.
- Forgetting negative sign for outlet pipes. In pipe problems, always subtract the emptying rate. Writing 1/4 + 1/6 instead of 1/4 – 1/6 flips the answer entirely.
- Confusing work done with work remaining. If A does 1/3 of job, remaining is 2/3, not 1/3. Track cumulative work carefully.
Quick reference
- Combined time for A (a days) and B (b days) = ab/(a+b) days.
- If A does job in x days, one day work = 1/x; x days work = x × (1/x) = 1 (full job).
- Pipe filling in t₁, emptying in t₂, both open: net time = t₁t₂/|t₁–t₂| (sign determines fill or empty).
- Man-days constant: M₁D₁ = M₂D₂ when work and hours/day same.
- Efficiency ∝ 1/Time. Higher efficiency = lower time.
Check yourself: 3 questions
1. A man can complete a work in 15 days. He works for 5 days and then leaves. What fraction of the work is left?
2. Two pipes A and B can fill a tank in 12 hours and 18 hours respectively. If both pipes are opened together, how long will it take to fill the tank?
3. 15 men can complete a piece of work in 20 days. How many days will 25 men take to complete the same work?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 18 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.
Read the full noteFlashcards for this topic
Saves this lesson and its 3 questions in this browser on this phone. No sign-in; nothing about you is saved or sent. A saved copy is not updated until you save it again. Parts that load from the server when you use them (the AI tutor, adaptive practice, sign-in) still need a connection.