Time & Work – Advanced Study Notes
Prepared for the Lakshadweep Forest Guard Quantitative Aptitude Program
MIT‑Level, First Principles to Edge‑Case Analysis
1. Introduction
The Time & Work topic is a staple of quantitative aptitude tests. Its core idea is the linear relationship between rate, time, and work. We will:
- Derive the basic formulas from first principles.
- Extend to non‑constant rates (time‑dependent, worker‑dependent).
- Discuss edge cases that often trip test‑takers.
- Present a challenging problem that synthesises multiple concepts.
Notation
- (W) : Total amount of work (a dimensionless quantity).
- (r) : Rate of work (units (W)/unit time).
- (t) : Time (unit time).
- (R_i, t_i) : Rate and time of worker (i).
2. First Principles
2.1 Definition of a Rate
Work is additive over time. If a worker completes (W_1) in (t_1) time units, then
[ r_1=\frac{W_1}{t_1}\quad\text{(units: } W/t\text{)} . ]
The rate is the derivative of work with respect to time:
[ r(t)=\frac{dW}{dt}\quad\text{(instantaneous rate)} . ]
For a constant rate, integration yields
[ W = r,t . ]
2.2 Work Done by Multiple Workers (Constant Rates)
If (n) workers start simultaneously and each works at a constant rate (r_i), the total rate is
[ R = \sum_{i=1}^{n} r_i . ]
Hence
[ \boxed{W = R,t = \left(\sum_{i=1}^{n} r_i\right)t } \quad\text{(1)} ]
Equation (1) is the work‑rate equation.
2.3 Time Required for a Task
Rearrange (1) to solve for time:
[ t = \frac{W}{\sum_{i=1}^{n} r_i} . ]
If all workers have the same rate (r),
[ t = \frac{W}{n,r}. ]
3. Derivation of Key Formulas
3.1 “Work × Time” Inverse Relationship
Let two workers, A and B, take (t_A) and (t_B) to finish the same job separately. Their rates are (r_A = W/t_A) and (r_B = W/t_B).
If they work together, the combined rate is
[ R = r_A + r_B = \frac{W}{t_A} + \frac{W}{t_B} = W\left(\frac{1}{t_A}+\frac{1}{t_B}\right). ]
Thus the time to finish together is
[ t_{AB} = \frac{W}{R} = \frac{1}{\dfrac{1}{t_A}+\dfrac{1}{t_B}}. ]
Key Insight: The reciprocal of time is additive.
Generalising to (n) workers:
[ \boxed{ \frac{1}{t} = \sum_{i=1}^{n}\frac{1}{t_i} } \quad\text{(2)} ]
Equation (2) is often the quickest route to a solution.
3.2 Work Done in Partial Intervals
If workers start at different times or work for different intervals, we sum partial works:
[ W = \sum_i r_i,t_i . ]
For a worker working partially, e.g., (t_{\text{part}}) out of a full shift (T), the effective work is (r,t_{\text{part}}).
4. Advanced Topics
4.1 Variable Rates – Time‑Dependent Work
Sometimes a worker’s rate isn’t constant (fatigue, learning, or machinery degradation). Suppose
[ r(t) = a,t^b \quad (a>0,\ b\in\mathbb{R}). ]
The total work over ([0,T]) is
[ W = \int_0^T r(t),dt = \int_0^T a,t^b,dt = a,\frac{T^{,b+1}}{b+1}. ]
Example: A machine that slows down linearly ((b=-1)) yields (W = a,\ln T).
4.2 Piecewise Constant Rates
In many problems a worker switches between two rates at a known time (t_1):
[ r(t) = \begin{cases} r_1 & 0\le t < t_1,\ r_2 & t_1\le t \le T . \end{cases} ]
The total work is
[ W = r_1,t_1 + r_2,(T-t_1). ]
4.3 Workers with Different Start Times
If worker (i) starts at (s_i) (time zero is