Time and Work — Study Notes

Overview

Time and Work is a staple 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. Expect 2–3 questions in the exam, often with real-life contexts like painting walls, digging trenches, or filling swimming pools.

You must quickly convert "days to finish" into "work done per day," combine rates for multiple workers, and handle scenarios where some workers leave midway or pipes have opposite effects (filling vs. draining). The arithmetic is straightforward once you set up the equation correctly, but the exam rewards speed and accuracy under time pressure.

Key Concepts

• **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.
• **Work left behind**: If workers complete part of a job and leave, calculate remaining work = 1 − (work done), then apply the rate of remaining workers to finish.
• **Negative time trap**: If you compute a negative time, check whether you've accidentally reversed filling/draining or miscounted the combined rate.
• **Efficiency comparison**: If A is twice as efficient as B, A's rate is double B's rate, so A takes half the time B takes.

Formulas / Key Facts

1. **Work done per day** = 1 / (days to complete alone). If A completes work in *n* days, A's rate = 1/*n* per day. 2. **Combined rate for two workers** = 1/*a* + 1/*b*, where *a* and *b* are their individual completion times. 3. **Time taken together** = 1 / (combined rate). For A and B: Time = 1 / (1/*a* + 1/*b*) = *ab* / (*a* + *b*) days. 4. **Man-days = (number of men) × (days)**. If *m* men complete work in *d* days, total work = *md* man-days. If *n* men work, days needed = *md* / *n*. 5. **Pipes formula**: Net rate = (sum of inlet rates) − (sum of outlet rates). Time to fill/empty = 1 / (net rate). 6. **Work and wage**: Wages are distributed in the ratio of work done (or man-days contributed). 7. **Fractional work**: Work completed in *t* days by a worker with rate *r* = *rt*. Remaining work = 1 − *rt*.