Time & Work – Comprehensive Study Notes
(For IIT‑JEE / Competitive Exams)
1. Introduction
Time & Work problems are the bread‑and‑butter of quantitative aptitude.
They test your ability to translate everyday situations into algebraic
relationships.
The key is to express everything in terms of “work” (units of job) and
“rate” (units of work per unit time).
Definition
Work – the amount of job done.
Rate – work done per unit time.
Time – duration taken to finish a job.
2. Fundamental Relation
[ \boxed{\text{Work} = \text{Rate} \times \text{Time}} ]
or
[ \boxed{ \text{Rate} = \frac{\text{Work}}{\text{Time}} }, \qquad \boxed{ \text{Time} = \frac{\text{Work}}{\text{Rate}} } ]
- Units:
Work – a dimensionless “job” (e.g., “1 job”).
Rate – “job per hour” (or any time unit).
Time – “hours”, “days”, etc.
3. Combining Rates
When two or more workers/machines work together, their rates add.
[ R_{\text{total}} = R_1 + R_2 + \dots + R_n ]
Hence
[ T_{\text{total}} = \frac{W}{R_{\text{total}}} ]
4. Working in Ratios
If two workers A and B work together and finish a job in 12 hours,
and if A alone were to finish the same job in 20 hours, what would be B’s
time?
- Compute the individual rates
[ R_A = \frac{1}{20};\text{job/h}, \quad R_B = \frac{1}{T_B};\text{job/h} ] - Use combined rate
[ \frac{1}{12} = R_A + R_B = \frac{1}{20} + \frac{1}{T_B} ] - Solve for (T_B).
General trick – When a part of a worker’s time is not used, subtract that time from the given total.
5. Work Done by Machines / Workers with Efficiency
Sometimes the efficiency (or faulty rate) is given.
Let a machine’s nominal rate be (R), but due to a 10 % fault it actually
works at (0.9R).
Use the effective rate in all subsequent calculations.
6. Work with Breaks / Shifts
If a worker works in shifts or takes breaks, you simply subtract the unused time from the total.
Example:
Worker A works 8 h/day, breaks 2 h/day, so effective time per day = 6 h.
If he works for 5 days, total effective time = (5 \times 6 = 30) h.
7. Partially Completed Work
When a portion of a job is already done, treat the remaining work as a new problem.
- Let (W_{\text{total}}) be the whole job.
- If (x%) is done, remaining work = ( \frac{100-x}{100}, W_{\text{total}}).
8. Work in Changing Rates
A worker may start slower and get faster (or vice versa).
If the rates change in a known ratio, use the average rate.
Example:
Worker works first half at rate (R), second half at rate (2R).
Average rate = (\frac{R + 2R}{2} = 1.5R).
9. Important Formulas (Quick Reference)
| Quantity | Symbol | Formula |
|---|---|---|
| Work | (W) | |
| Rate | (R) | (R = \dfrac{W}{T}) |
| Time | (T) | (T = \dfrac{W}{R}) |
| Combined rate | (R_{\text{tot}}) | (R_{\text{tot}} = \sum R_i) |
| Work done in time (t) | (W = R \times t) | |
| Remaining work after (p%) done | (W_{\text{rem}}) | (W_{\text{rem}} = \dfrac{100-p}{100},W) |
| Effective rate (efficiency (e)) | (R_{\text{eff}}) | (R_{\text{eff}} = e \times R) |
| Work by two workers A & B in (t) hours | (\displaystyle \frac{t}{T_A} + \frac{t}{T_B}) |