Time & Work – Advanced Study Notes
(For West Bengal Patwari / Lekhpal – Quantitative Aptitude)
A Patwari or Lekhpal often deals with schedules, man‑power allocation, and time‑based budgeting. Mastery of the “time & work” concept is therefore essential for civil‑service exams and for practical field work.
1. Foundations – From First Principles
| Symbol | Meaning | Unit |
|---|---|---|
| (W) | Work (amount of job) | “job‑units” (e.g., acres surveyed) |
| (r) | Rate of work | job‑units per unit time |
| (t) | Time | days, hours, etc. |
Basic Relation
[ W = r,t \tag{1} ]Proof: By definition, rate is the amount of job completed per unit time. Multiply both sides by (t) to obtain the total job.
Rearrangements
[ r = \frac{W}{t},\qquad t = \frac{W}{r} \tag{2} ]
These three formulas are the building blocks for every “time & work” problem.
2. Combined Work – Multiple Workers
When (n) workers work together, rates add:
[ r_{\text{total}} = \sum_{i=1}^{n} r_i \tag{3} ]
Hence, if all workers are required to finish a job (W),
[ t = \frac{W}{\sum r_i} \tag{4} ]
Example
A & B finish 30 acres in 4 days.
(r_A + r_B = \frac{30}{4}=7.5) acres/day.
If A alone can finish 20 acres in 5 days ⇒ (r_A = \frac{20}{5}=4) acres/day.
Therefore (r_B = 3.5) acres/day and (t_B = \frac{30}{3.5}\approx 8.57) days.
3. Partial Work – When Workers Join/Leave
3.1 Worker leaves after completing a part
Let worker A work for (t_a) days at rate (r_a).
Remaining work (W_{\text{rem}} = W - r_a t_a).
If B works at rate (r_b) for the rest,
[ t_b = \frac{W_{\text{rem}}}{r_b} \quad\Rightarrow\quad t_{\text{total}} = t_a + t_b ]
3.2 Worker joins after a delay
If B starts after (d) days, then
[ t_{\text{total}} = d + \frac{W - r_a d}{r_a + r_b} ]
Edge Case – If (r_a + r_b = 0) (both ineffective), the job never finishes.
4. Variable Rates – Work as a Function of Time
In real life, a worker’s efficiency may change (fatigue, weather, learning).
Let (r(t)) be the instantaneous rate at time (t).
Total work done in an interval ([0,T]) satisfies
[ W = \int_{0}^{T} r(t),dt \tag{5} ]
4.1 Piecewise Constant Rate
If the rate changes at known times:
[ W = \sum_{k=1}^{m} r_k ,\Delta t_k ]
4.2 Linear Decrease (example)
Suppose a surveyor starts at 10 acres/day and loses 0.5 acres/day due to fatigue.
[ r(t) = 10 - 0.5 t \quad (t \text{ in days}) ]
Work done in (T) days:
[ W = \int_{0}^{T} (10 - 0.5t),dt = 10T - 0.25T^2 ]
If the job is 75 acres, solve (10T - 0.25T^2 = 75) ⇒ (T = 10) days.
5. Harmonic Mean of Rates – When Each Worker Completes the Same Amount
If (n) workers each finish the same job (W) in times (t_i), their rates are (r_i = W/t_i).
The effective rate when all work together equals the harmonic mean of the individual rates:
[ r_{\text{eff}} = \frac{n}{\displaystyle\sum_{i=1