Time & Work – Advanced Study Notes
Target audience: Chandigarh Police Sub‑Inspector (SI) exam candidates
Depth: MIT‑level, first‑principles, rigorous reasoning, edge cases, and a challenging problem
Tip: Keep the “work” symbol (W) universal. All rates, times, and efficiencies will be expressed in terms of (W), making it easy to compare units and track dimensions.
1. Overview
| Concept | Symbol | Unit |
|---|---|---|
| Work performed | (W) | “Work units” (e.g., person‑days) |
| Rate of a worker | (r) | (W/\text{time}) |
| Time taken | (t) | (\text{time}) |
| Efficiency factor | (e) | dimensionless (ratio of actual to ideal work) |
Fundamental Law of Work
[
\boxed{W = r \times t}
]
This relationship holds regardless of the shape of the job, whether the worker is human, machine, or team.
2. First‑Principles Derivation
-
Define a “unit of work”
For any job, choose a reference unit (e.g., one wall painted, one kilometer paved).
Let (W_{\text{ref}}) be the total amount of such units required. -
Rate as derivative of work w.r.t. time
[ r = \frac{dW}{dt} ] If (W) is linear in time (constant rate), integrate: [ W(t) = r t + C ] At (t=0), (W=0), so (C=0). Hence (W = r t). -
Dimensional consistency
([W] = \frac{[W]}{[t]} \times [t]).
The equation is dimensionally sound.
3. Work with Multiple Workers
3.1 Combined Rates
If workers A, B, C… work together:
[ r_{\text{total}} = r_A + r_B + r_C + \dots ]
Example
- A: 3 W/day
- B: 5 W/day
- Combined: (8) W/day → 1 day to finish (8) W job.
3.2 Partitioning Work Between Workers
If a job requires (W) and two workers A and B finish it together in (t) days:
[ r_A + r_B = \frac{W}{t} ]
If we know the ratio (r_A : r_B = m : n):
[ r_A = \frac{m}{m+n}\frac{W}{t}, \qquad r_B = \frac{n}{m+n}\frac{W}{t} ]
4. Edge Cases & Common Pitfalls
| Edge Case | What Happens | How to Handle |
|---|---|---|
| Zero work (W=0) | Any worker can finish instantly | Set (t=0) or treat as trivial |
| Zero rate (r=0) | Work never progresses | Infinite time, impossible job |
| Negative rate | Indicates loss of work (e.g., sabotage) | Treat as (r<0); time becomes negative → impossible |
| Partial work | Some work already done | Subtract completed work from (W) |
| Different units | Minutes vs. hours | Convert to common unit before applying formulas |
| Time-dependent rate | Rate varies with fatigue | Use integral (\int r(t) dt = W) |
| Simultaneous vs. sequential | Order matters | For sequential, add times; for simultaneous, use combined rate |
Pitfall:
- Mis‑reading “half” vs. “½.”
“Half the work” means (0.5W), not (W/2) if (W) is already a fraction of a job.
5. Advanced Topics
5.1 Inverse Proportionality
If two workers A and B can finish a job in (t_A) and (t_B) days respectively, then:
[ r_A = \frac{W}{t_A}, \qquad r_B = \frac{W}{t_B} ]
When they work together:
[ t_{\text{AB}} = \frac{1}{\frac{1}{t_A} + \frac{1}{t_B}} ; \text{(harmonic mean)} ]
Example
- A: 8 days, B: 12 days → Combined time (= \frac{1}{1/8+1/12} = 4.8) days.
5.2 Efficiency Factors
If a worker has efficiency (e) (0 < e ≤ 1), the effective rate becomes (e r).
Example:
- Ideal rate (= 5) W/day; Efficiency (= 0.8) → Effective rate (= 4) W/day.
5.3 Work vs. Distance Analogy
Think of work like distance