Time & Work – Puducherry TET Quantitative Aptitude
(Study Notes – Easy to Understand, Quick Revision)
1. Simple Explanation
Time & Work problems ask you to find how long it takes (time) for a person/worker to finish a job (work) or how many people are required to finish a job in a given time.
- Work = amount of job (e.g., a house, a road).
- Rate of Work = how much work is done in a unit time (e.g., work per hour).
- Work Done = Rate × Time.
If a worker can finish a job in t hours, his/her rate is 1/t of the job per hour.
2. Key Points & Formulas
| Concept | Formula | Explanation |
|---|---|---|
| Rate of Work | ( R = \dfrac{1}{t} ) | Work per hour, where t = hours to finish alone. |
| Work Done | ( W = R \times T ) | Total work completed in T hours. |
| Combined Rate | ( R_{\text{total}} = R_1 + R_2 + \dots + R_n ) | Add individual rates for simultaneous work. |
| Time for Combined Work | ( T = \dfrac{1}{R_{\text{total}}} ) | Inverse of combined rate. |
| Work–Time Relation | ( \text{Work} \times \text{Time} = \text{Constant} ) | If rate is constant, work × time = constant. |
| Part‑Work (Percent) | ( \text{Part} = \dfrac{\text{Work}}{\text{Total Work}} \times 100% ) | Useful for “x% of work” questions. |
| Work–Man Hours | ( \text{Man‑hours} = \text{Workers} \times \text{Hours} ) | Common in “how many workers needed” problems. |
| Work Reduction/Increase | Adjust rate when a worker is replaced or added. | Example: If A’s rate increases by 20%, new rate = 1.2 × old rate. |
3. Types of Problems in TET
| Type | Typical Question | Strategy |
|---|---|---|
| Single Worker, Single Time | “A can finish a job in 8 days.” | Find rate, then use (W=R\times T). |
| Multiple Workers, Same Time | “A and B together finish a job in 5 days.” | Find combined rate, then split as needed. |
| Multiple Workers, Different Times | “A alone takes 10 days, B alone takes 15 days.” | Compute individual rates, add. |
| Replacement / Addition | “After 3 days, C joins A and B.” | Compute work done before C joins, then remaining work with new rate. |
| Work–Time Inverse | “If 6 workers can finish in 4 days, how many days for 12 workers?” | Use ( \text{Work} = \text{Workers} \times \text{Days} ). |
| Percentage of Work | “After 50% of work, B leaves.” | Compute remaining work as 50% of total. |
| Rate Change | “A’s rate increases by 25%.” | Update (R_A = 1.25 \times (1/t_A)). |
4. Worked Example (TET Style)
Question
A can finish a job in 12 days. B can finish the same job in 20 days. After working together for 3 days, A leaves. How many more days will B need to finish the remaining work?
Step‑by‑Step
| Step | Calculation | Result |
|---|---|---|
| 1. Find rates | (R_A = \frac{1}{12}) job/day, (R_B = \frac{1}{20}) job/day | (R_A = 0.0833,; R_B = 0.05) |
| 2. Combined rate | (R_{AB} = R_A + R_B = 0.0833 + 0.05 = 0.1333) job/day | |
| 3. Work done in 3 days | (W_{\text{done}} = R_{AB} \times 3 = 0.1333 \times 3 = 0.4) job | 40 % of job finished |
| 4. Remaining work | (W_{\text{rem}} = 1 - 0.4 = 0.6) job | 60 % of job left |
| 5. Time for B alone | (T_{\text{B}} = \frac{W_{\text{rem}}}{R_B} = \frac{0.6}{0.05} = 12) days | |
| Answer | B needs 12 more days after A leaves. |
Key Insight: After A leaves, only B’s rate matters. The remaining work is a fraction of the whole job, so simply divide that fraction by B’s rate.
5. Quick Revision Summary
- Rate = ( \frac{1}{\text{Time to finish alone}} ).
- Work Done = Rate × Time.
- Combined Rate = Sum of individual rates.
- Time for Combined Work = ( \frac{1}{\text{Combined Rate}} ).
- Work–Time Inverse: Work × Time = constant (if workers unchanged).
- Man‑Hours: Workers × Days = constant for a fixed job.
- When a worker leaves or joins, recompute the combined rate for the remaining period.
- For “x % of work” problems, convert percent to fraction (0.5 for 50 %) and apply it to the total work.
- Always