Time & Work – Advanced Study Notes
(Designed for the Assam Patwari / Lekhpal quantitative aptitude course – MIT‑level)
Audience: Students who already know the basic “work‑rate‑time” formulas and wish to see the subject from a rigorous, mathematical perspective.
Goal: Build a solid foundation, expose edge cases, and develop intuition for handling complex, real‑world scenarios.
1. Foundations
1.1 Basic Definitions
| Symbol | Meaning | Units |
|---|---|---|
| (W) | Total amount of work (job) | “Work‑units” (arbitrary) |
| (R) | Rate of work | ( \text{Work},\text{units}/\text{time} ) |
| (T) | Time taken | time |
Assumption: Work can be divided into infinitesimal parts that are processed at constant rates.
1.2 From First Principles
Let a job be represented by a measurable function (f(t)) that indicates the instantaneous amount of work done per unit time (i.e., the rate) at time (t).
If the job starts at (t=0) and ends at (t=T):
[ W ;=; \int_{0}^{T} f(t),dt . ]
If the rate is constant, (f(t)=R), and the integral collapses to the familiar
[ W = R,T \quad\Longleftrightarrow\quad R=\frac{W}{T},; T=\frac{W}{R}. ]
2. Working Together – Linearity
2.1 Two Workers
Suppose worker (A) works at rate (R_A) and worker (B) at rate (R_B).
If they work simultaneously on the same job, the combined rate is
[ R_{AB}=R_A+R_B . ]
Proof:
For an infinitesimal interval (\Delta t), worker (A) completes (R_A\Delta t) units, and (B) completes (R_B\Delta t) units.
The total work is ((R_A+R_B)\Delta t).
Taking the limit (\Delta t\to 0) gives the integral above.
2.2 Generalization
For (n) workers with rates (R_1,\dots,R_n):
[ R_{\text{total}}=\sum_{i=1}^{n} R_i . ]
3. Variable Rates and Piecewise Work
3.1 Piecewise Constant Rates
If a worker’s rate changes at known breakpoints (0=t_0<t_1<\dots<t_k=T):
[ W = \sum_{i=1}^{k} R_i,(t_i-t_{i-1}) . ]
3.2 Continuous Variable Rates
When (f(t)) is a continuous function (e.g., fatigue, fatigue recovery, or machinery wear):
[ W = \int_{0}^{T} f(t),dt . ]
Example: A worker’s rate decreases linearly from (R_0) to (R_f) over (T) hours: [ f(t)=R_0-\frac{R_0-R_f}{T}t,\qquad W = \int_0^T f(t),dt = \frac{R_0+R_f}{2},T . ]
4. Edge Cases & Special Situations
| Situation | Analysis | Key Insight |
|---|---|---|
| Zero Work ((W=0)) | If (W=0), then either (T=0) or (R=0) (or both). | No job → no time or no rate. |
| Infinite Workers | (n\to\infty) with finite total rate? If each worker has rate (r/n), total rate (r) remains finite. | Scaling of rates keeps total work finite. |
| Negative Rate | (R<0) would imply undoing work. If a job can be undone, the formula still holds but yields negative work. | Important in reverse‑engineering or error correction. |
| Stochastic Rates | Rate (R(t)) is random with expectation (\mathbb{E}[R(t)]). | Expected work (\mathbb{E}[W] = \int_0^T \mathbb{E}[R(t)],dt). |
| Dependent Work | Some tasks can only start after another completes (series vs. parallel). | Need to model with DAGs or scheduling algorithms. |
5. Advanced Theory: Rate as a Function of Work Completed
Sometimes the rate is not a function of time but of the remaining or completed work, e.g., (R = k \cdot (W_{\text{remaining}})).
5.1 Exponential Decay of Work
If (R(t)=k,(W_{\text{remaining}}(t))), then
[ \frac{dW_{\text{remaining}}}{dt} = -k,W_{\text{remaining}}, ] which solves to
[ W