Time and Work
About 7 min · Everyday Mathematics · English
Time worked out from the lesson's 511 words at 150 words a minute, plus a minute for each question.
Time and Work is a topic in SOF IMO's Everyday Mathematics section. Problems test your ability to relate how much work a person or machine completes in a given time and how multiple workers combine their efforts. The key insight is treating work as a unitary quantity (usually "1 complete job") and expressing rates of work per unit time. Mastery requires understanding individual efficiency, joint work rates, and the special case of pipes filling or emptying tanks. Questions appear as single-worker problems, joint-work scenarios, or pipes-and-cisterns variants.
Key ideas
- Work is one unit: Treat the entire task as 1 job. If A finishes in 10 days, A's one-day work = 1/10 of the job.
- Rate = Work / Time: A person's efficiency or work rate is the fraction of the job completed per unit time.
- Combined work rate: When A and B work together, their joint rate = (A's rate) + (B's rate). Time to finish together = 1 / (combined rate).
- Man-days concept: Total work = (number of workers) × (days worked). Use this to scale problems when the number of workers changes.
- Pipes and cisterns: Inlet pipes fill (positive rate), outlet pipes drain (negative rate). Net rate = sum of all pipe rates with correct signs.
Watch out for
- Adding times instead of rates → Wrong: "A takes 10 days, B takes 15 days, together they take 10+15=25 days." Correct: Add rates (1/10 + 1/15), then invert to get time.
- Forgetting to subtract outlet rates → When a drain is open, subtract its rate. Treating all pipes as positive leads to wrong net rate.
- Ignoring partial work done → If A works for 3 days then leaves, compute work done = 3 × (A's rate), then apply B's rate to the remainder. Don't restart from "1 job."
Quick reference
- One worker's rate = 1 / (time to complete alone). Invert to get time from rate.
- Joint work time = ab / (a + b) when A takes a days, B takes b days.
- Pipes: inlet (+), outlet (−). Net rate = sum with signs. Time = 1 / net rate.
- Man-days product is constant: *m*₁*d*₁ = *m*₂*d*₂. Use to scale workforce or duration.
- Check units: All rates in same time unit (days or hours). Convert before computing.
Check yourself: 3 questions
1. A can complete a work in 12 days and B can complete the same work in 18 days. If both A and B work together, in how many days will they complete the work?
2. A pipe can fill a tank in 15 hours. Due to a leak at the bottom, it takes 20 hours to fill the tank. If the tank is full, how long will the leak take to empty it?
3. A and B can complete a work in 10 days. B and C can complete the same work in 15 days. A and C can complete it in 12 days. In how many days can A, B and C together complete the work?
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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