📚 Advanced Study Notes – Time & Work
Course: Quantitative Aptitude
Program: LIC ADO (Level‑I)
Audience: MIT‑level students who want a deep, rigorous treatment of the “Time & Work” topic.
Goal: Understand the theory from first principles, handle edge cases, and solve challenging problems.
1. Why “Time & Work” is a Mathematical Model
| Symbol | Meaning | Units |
|---|---|---|
| (W) | Total amount of work (tasks, units, jobs) | units |
| (t) | Time taken | hours (or any time unit) |
| (r) | Rate of a worker or machine | units per hour |
| (P) | Population of workers | dimensionless |
Mathematically, the relationship is
[ W = \int_{0}^{t} r(t'),dt' ]
When the rate is constant, (r(t)=r), this reduces to the simple product (W=r,t).
2. From First Principles: Constant Rates
2.1 Derivation
Assume a worker does a fixed amount (r) of work per unit time.
For a small time increment (\Delta t),
[ \Delta W = r ,\Delta t ]
Summing over (n) equal increments ((t = n\Delta t)),
[ W = r \sum_{i=1}^{n} \Delta t = r,t ]
Taking the limit (\Delta t \to 0) gives the integral form.
2.2 Basic Formulae
| Situation | Formula |
|---|---|
| One worker, one job | (t = \dfrac{W}{r}) |
| Two workers working together | (t = \dfrac{W}{r_1 + r_2}) |
| (k) workers, equal rates (r) | (t = \dfrac{W}{k,r}) |
| One worker works for (t_1), another for (t_2) | (W = r_1t_1 + r_2t_2) |
These are the “standard” textbook problems.
3. Variable Rates: Piecewise & Continuous
3.1 Piecewise Constant Rates
Suppose a worker’s rate changes after a fixed time.
| Segment | Duration (t_i) | Rate (r_i) |
|---|---|---|
| 1 | (a) | (r_1) |
| 2 | (b) | (r_2) |
| … | … | … |
Total work:
[ W = \sum_{i} r_i t_i ]
The time to finish a job (W) is found by solving for (t) in a piecewise linear equation.
3.2 Continuous Variable Rates
Let the rate be a continuous function (r(t)). Then
[ W = \int_{0}^{t} r(t'),dt' ]
Example: Diminishing returns – (r(t)=r_0 e^{-\lambda t})
[ W = \frac{r_0}{\lambda}\bigl(1-e^{-\lambda t}\bigr) ;;\Rightarrow;; t = -\frac{1}{\lambda}\ln!\bigl(1-\tfrac{\lambda W}{r_0}\bigr) ]
4. Multi‑Worker Systems: Linear Algebra View
When each worker (i) has a constant rate (r_i) and works for time (t_i), the total work is
[ W = \sum_{i=1}^{n} r_i t_i ]
This is a dot product of two vectors:
[ W = \mathbf{r} \cdot \mathbf{t} ]
Solving for the times (\mathbf{t}) given a work vector (\mathbf{W}) (if multiple jobs) leads to linear systems.
If workers have different capacities or constraints, we can model it as:
[ \mathbf{R},\mathbf{t} = \mathbf{W} ]
where (\mathbf{R})