Time & Work – Quantitative Aptitude
(For Madhya Pradesh Patwari / Lekhpal Exam)
1. Simple Explanation
Time & work is a classic problem that deals with how much work a person (or a group) can do in a given time and how long it will take to finish the work.
- Work is considered as a whole job (e.g., building a wall, digging a well).
- Rate of work = amount of work done per unit time (e.g., per day).
- Time is the duration required to complete a job.
The fundamental relation is
[ \text{Work} = \text{Rate} \times \text{Time} ]
or equivalently
[
\text{Rate} = \frac{\text{Work}}{\text{Time}},\qquad
\text{Time} = \frac{\text{Work}}{\text{Rate}}
]
In most exam problems we treat one whole job as 1 unit of work.
2. Key Points & Formulas
| Symbol | Meaning | Formula |
|---|---|---|
| (W) | Total work (usually 1) | – |
| (R) | Rate of a person (work per day) | (R = \frac{W}{T}) |
| (T) | Time (days) | (T = \frac{W}{R}) |
| (R_1, R_2, …) | Rates of multiple persons | – |
| Combined rate | Sum of individual rates | (R_{\text{total}} = R_1 + R_2 + …) |
| Combined time | Time to finish whole job together | (T_{\text{total}} = \frac{W}{R_{\text{total}}}) |
| Work done by one alone in (t) days | (W_{\text{done}} = R \times t) | |
| Work left | (W_{\text{left}} = W - W_{\text{done}}) | |
| Work with fractions | Work expressed as fraction of whole (e.g., ( \frac{3}{4}) of job) | – |
Important Tips
- Always treat the whole job as 1 unit unless otherwise specified.
- Convert all rates to the same unit (days, hours, etc.) before adding.
- Use common denominators when adding fractional rates.
- When a person stops working, the remaining work is the difference between the total work and the work done till that point.
- Check units at each step to avoid mistakes (days × work/day = work).
3. Worked Example
Problem
- Person A can finish a job in 12 days.
- Person B can finish the same job in 20 days.
- A works alone for 4 days, then B joins.
- Question: How many days will it take to finish the job completely?
Step‑by‑Step Solution
| Step | Calculation | Explanation |
|---|---|---|
| 1. Find rates | (R_A = \frac{1 \text{ job}}{12 \text{ days}} = \frac{1}{12}) job/day (R_B = \frac{1}{20}) job/day |
Rate = 1 job ÷ time |
| 2. Work done by A in 4 days | (W_{\text{A}} = R_A \times 4 = \frac{1}{12}\times4 = \frac{1}{3}) job | |
| 3. Work left | (W_{\text{left}} = 1 - \frac{1}{3} = \frac{2}{3}) job | |
| 4. Combined rate of A & B | (R_{\text{total}} = \frac{1}{12} + \frac{1}{20} = \frac{5}{60} + \frac{3}{60} = \frac{8}{60} = \frac{2}{15}) job/day | Add rates |
| 5. Time to finish remaining work | (T_{\text{remaining}} = \frac{W_{\text{left}}}{R_{\text{total}}} = \frac{\frac{2}{3}}{\frac{2}{15}} = \frac{2}{3}\times\frac{15}{2} = 5) days | Divide work by rate |
| 6. Total time | (T_{\text{ |