Time & Work – Quantitative Aptitude
(NABARD Development Assistant – Study Notes)
1. Simple Explanation
Time & Work is a classic topic that deals with how long it takes for a person or a group of people (or machines) to finish a job, and how many people (or machines) are needed to finish a job in a given time.
- Work is the total amount of job that needs to be done.
- Rate (or Productivity) of a worker = work completed per unit time (e.g., per day).
- Time = how long it takes to finish a job.
The fundamental relationship is
[ \text{Work} = \text{Rate} \times \text{Time} ]
When several workers work together, their rates add up.
2. Key Points & Formulas
| Concept | Symbol | Formula | Explanation |
|---|---|---|---|
| Rate of a worker | (r) | (r = \frac{1}{T}) | If a worker completes the whole job in (T) days, his rate is (1/T) job/day. |
| Work done by a worker | (W) | (W = r \times t) | Amount of work completed in (t) days. |
| Combined rate of (n) workers | (R) | (R = \sum r_i) | Sum of individual rates. |
| Time to finish a job | (T) | (T = \frac{W}{R}) | Total work divided by combined rate. |
| Work completed in part of a job | (W_{\text{part}}) | (W_{\text{part}} = r \times t) | Used when a worker works for only part of the job. |
| Work remaining after (k) days | (W_{\text{rem}}) | (W_{\text{rem}} = W - r_k \times k) | Subtract work done in (k) days. |
| Increase / Decrease in rate | – | – | Use percentages: (r_{\text{new}} = r_{\text{old}} \times (1 \pm %)). |
| Work done by two workers in the same time | – | (r_A + r_B = \frac{1}{T_{\text{both}}}) | If they finish together in (T_{\text{both}}) days. |
Important Rules
- Adding Rates – When workers work together, add their rates to get the combined rate.
- Subtracting Rates – When one worker leaves, subtract his rate from the combined rate.
- Work in Parts – A job can be divided into parts (e.g., 1/3, 2/3). Compute work done in each part separately.
- Unit Consistency – Keep the same time unit (days, hours, etc.) across the problem.
- Work = Rate × Time – This is the core equation; rearrange it for any missing variable.
3. Worked Example
Problem
A piece of land is to be cleared.
- Worker A alone can clear the land in 12 days.
- Worker B alone can clear the land in 18 days.
- Worker A works for 4 days, then B joins him.
- After 3 more days of joint work, the land is completely cleared.
Find the number of days B worked alone.
Step‑by‑Step Solution
| Step | What to find | Calculation | Result |
|---|---|---|---|
| 1. Individual rates | Rate of A and B | (r_A = \frac{1}{12}) job/day, (r_B = \frac{1}{18}) job/day | – |
| 2. Work done by A alone (first 4 days) | (W_{A,4}) | (W_{A,4} = r_A \times 4 = \frac{1}{12}\times4 = \frac{1}{3}) job | 1/3 of land |
| 3. Remaining work | (W_{\text{rem}}) | (W_{\text{rem}} = 1 - \frac{1}{3} = \frac{2}{3}) job | – |
| 4. Combined rate of A and B | (R_{AB}) | (R_{AB} = r_A + r_B = \frac{1}{12} + \frac{1}{18} = \frac{5}{36}) job/day | – |
| 5. Time needed to finish remaining work | (t_{\text{joint}}) | (t_{\text{joint}} = \frac{W_{\text{rem}}}{R_{AB}} = \frac{\frac{2}{3}}{\frac{5}{36}} = \frac{2}{3}\times\frac{36}{5} = \frac{24}{5} = 4.8) days | 4.8 days |
| 6. Given joint time | Check | Problem says 3 days of joint work → contradiction? | Re‑check: Problem states “After 3 more days of joint work” → So (t_{\text{joint}} = 3) days. |
| Hence compute number of days B worked alone (the remaining time after A’s 4 days and joint 3 days). | |||
| 7. Work done in 3 days of joint work | (W_{\text{joint}}) | (W_{\text{joint}} = R_{AB}\times3 = \frac{5}{36}\times3 = \frac{5}{12}) job | 5/12 of land |
| 8. Total work done | (W_{\text{done}}) | (W_{\text{done}} = W_{A,4} + W_{\text{joint}} = \frac{1}{3} + \frac{5}{12} = \frac{4}{12} + \frac{5}{12} = \frac{9}{12} = \frac{3}{4}) job | 3/4 of land |
| 9. Remaining work | (W_{\text{rem}}) |