Time & Work – Quantitative Aptitude (SI – Tripura Police Sub‑Inspector)
Exam Focus – IIT‑JEE / Competitive Exams
Topic – Time & Work
Target – Master the rate‑time‑work relationship, derive formulas from first principles, and solve problems that involve multiple workers, changes in workforce, and partial work.
1. Core Concepts & Definitions
| Symbol | Meaning | Units |
|---|---|---|
| (W) | Total work (job) | units of work (e.g., “tasks” or “jobs”) |
| (r_i) | Work‑rate of worker (i) | units of work per day |
| (t_i) | Time spent by worker (i) | days |
| (T) | Total time for a job | days |
| (n) | Number of workers | dimensionless |
Key Idea – Work is additive: If two workers work simultaneously, the combined rate is the sum of their individual rates.
2. Fundamental Work‑Rate Model
2.1 Work Done by a Single Worker
[ \boxed{W_i = r_i \times t_i} ]
Derivation – Work is rate × time.
2.2 Combined Rate of (n) Workers
[ \boxed{r_{\text{total}} = \sum_{i=1}^{n} r_i} ]
Proof – If each worker contributes (r_i) units/day, the total per day is the sum.
2.3 Time Required for a Job
[ \boxed{T = \frac{W}{r_{\text{total}}}} ]
Proof – Rearrange (W = r_{\text{total}} \times T).
3. Common Problem Structures
| Problem Type | Typical Question | Typical Approach |
|---|---|---|
| Single worker | A worker completes a job in 10 days. | (r = W/10). |
| Multiple workers together | 5 workers finish a job in 12 days. | (T = 12), (n = 5). Find (W) or (r_{\text{total}}). |
| Workers with different rates | A and B finish a job in 8 and 12 days respectively. | (r_A = W/8), (r_B = W/12). |
| Partial work | A worker works 4 days, then leaves. | Compute work done before leaving, subtract from (W). |
| Work replacement | One worker leaves after 2 days, another joins. | Compute work done by each period. |
| Proportion of work | A does 1/3 of job; B does 1/4. | Translate fractions to rates. |
| Non‑linear relation | Work done is inversely proportional to time. | (r \propto 1/T). |
4. Deriving Key Formulas (From First Principles)
4.1 Work Done by Two Workers
Let worker A do the job in (T_A) days, B in (T_B) days.
- Rate of A: (r_A = \frac{W}{T_A})
- Rate of B: (r_B = \frac{W}{T_B})
Combined rate: (r_A + r_B = \frac{W}{T_A} + \frac{W}{T_B}).
Time to finish together (T):
[ T = \frac{W}{r_A + r_B} = \frac{W}{\frac{W}{T_A} + \frac{W}{T_B}} = \frac{1}{\frac{1}{T_A} + \frac{1}{T_B}} = \frac{T_A T_B}{T_A + T_B} ]
Important Result – When two workers work together, the reciprocal of time is the sum of the reciprocals of individual times.