Time & Work – Advanced Study Notes
(quantitative aptitude, “Time & Work”)
Target audience – Undergraduate students (MIT‑level or equivalent).
Goal – Build a solid mathematical foundation, explore edge cases, and develop the ability to solve and design advanced problems.
1. Introduction
“Time & Work” problems ask us to determine how long it takes for one or more workers (or machines, processes, etc.) to finish a task, based on their rates of work.
The problems are ubiquitous in engineering, operations research, and applied mathematics.
At the university level we treat the following general framework:
| Symbol | Meaning | Units | Notation |
|---|---|---|---|
| (W) | Total amount of work (tasks, units, distance, etc.) | unit | (W) |
| (r_i) | Rate of worker (i) | unit/time | (r_i) |
| (t) | Time elapsed | time | (t) |
| (n) | Number of workers | – | (n) |
| (A) | Work area under a rate‑vs‑time curve | unit | – |
Fundamental law (the rate rule):
[
\text{Work done} = \int_0^t r(\tau), d\tau
]
If each worker works at a constant rate (r_i) over the interval ([0,t]), the total work is simply
[
W = t\sum_{i=1}^n r_i.
]
This is the starting point for all “time & work” problems.
2. Basic Algebraic Treatment
2.1 Single worker
If worker (A) can finish a job in (T) hours, his rate is [ r_A = \frac{W}{T}. ] If we know (r_A), the time it takes to finish the same job is [ T = \frac{W}{r_A}. ]
2.2 Two workers working together
Assume worker (A) works at rate (r_A) and worker (B) at rate (r_B).
When both work together, the combined rate is
[
r_{AB} = r_A + r_B.
]
Thus the time to finish the job is
[
T_{AB} = \frac{W}{r_A + r_B}.
]
[
\boxed{\frac{1}{T_{AB}} = \frac{1}{T_A} + \frac{1}{T_B}}
]
The reciprocal of time is linear in the reciprocal of individual times.
2.3 Many workers
For (n) workers, [ T = \frac{W}{\sum_{i=1}^n r_i} = \frac{W}{\sum_{i=1}^n \frac{1}{T_i}} ] where (T_i = W/r_i) is the time worker (i) would take alone.
3. Edge Cases & Common Pitfalls
| Scenario | What to Watch Out For | Typical Mistake | Correct Handling |
|---|---|---|---|
| Zero rate | A worker who cannot work (e.g., due to injury) | Counting them as “working” | Exclude from the sum; set (r_i = 0). |
| Infinite time | Work cannot be done (no one works) | Forcing a division by zero | Declare the job impossible. |
| Negative rate | A worker undoes work (e.g., a mistake or sabotage). | Treating as positive | Allow negative terms in the sum; time may become negative, indicating the job will unfinish itself. |
| Partial work | Workers don’t start at the same instant. | Adding rates before aligning start times | Use piecewise integration or add delayed work: (W = r_A t_A + r_B t_B + \dots). |
| Non‑constant rates | Workers’ efficiency changes over time. | Using a single rate | Integrate: (W = \int_0^t r_i(\tau) d\tau). |
| Task switching | Workers switch between tasks or take breaks. | Assuming continuous work | Model break periods with a piecewise function or separate integrals. |
4. Advanced Theory
4.1 Continuous Time & Variable Rates
When a worker’s efficiency changes, we treat (r_i(t)) as a rate function.
The total work done by worker (i) over ([0,T]) is
[
W_i = \int_0^T r_i(t), dt.
]
If (r_i(t)) is linear, piecewise linear, or exponential, we can integrate analytically.
Example: a worker who gets tired, with rate (r(t) = r_0 - \alpha t).
Then
[
W = \int_0^T (r_0 - \alpha t), dt = r_0 T - \frac{\alpha}{2}T^2.
]
4.2 Stochastic Work Rates
In real life, rates may be random due to variability.
We model (r_i) as a random variable and work as a random variable:
[
W = \sum_{i=1}^n \int_0^T r_i(\tau), d\tau.
]
We can compute expectations:
[
\mathbb{E}[W] = \sum_{i=1}^n \int_0^T