Time & Work – Study Notes
Manipur PSC – Quantitative Aptitude
1. Simple Explanation
- Time & Work problems ask you to determine how long it takes for one or more workers (or machines, people, etc.) to complete a job, or how many workers are required to finish a job in a given time.
- Think of work as a quantity (e.g., a job, a task, a length of work) that can be divided into work units (like “hr‑work”).
- The rate of a worker is work done per unit time (e.g., “1 job in 4 hrs” → rate = ¼ job/hr).
- For multiple workers, rates add up (if they work together).
- Use the basic relation
[ \text{Work} = \text{Rate} \times \text{Time} ]
2. Key Points & Formulas
| Concept | Symbol | Formula | What it means |
|---|---|---|---|
| Total work | (W) | — | Usually “1 job” or a given quantity |
| Rate of one worker | (R) | (R = \frac{1}{T}) | Work done per hour |
| Time taken by one worker | (T) | (T = \frac{1}{R}) | Hours needed by one worker |
| Combined rate of (n) workers | (R_{\text{total}}) | (R_{\text{total}} = \sum_{i=1}^{n} R_i) | How fast all workers together finish |
| Work done by one worker in (t) hours | (W_i) | (W_i = R_i \times t) | Portion of job completed |
| Work done by all workers together in (t) hours | (W_{\text{total}}) | (W_{\text{total}} = R_{\text{total}} \times t) | Portion of job completed |
| Workers needed to finish a job in (T) hours | (n) | (n = \frac{W}{R \times T}) | When workers have same rate |
Common Tricks
- Work left after a period: (W_{\text{left}} = W_{\text{total}} - W_{\text{done}}).
- Work done by multiple workers in different periods: Break the job into segments, compute each segment separately, then sum.
- Ratio method: If two workers work together for some time, the ratio of their rates equals the ratio of work they individually finish in the same time.
- Work‑time graph: Visualize cumulative work vs. time; slope = rate.
3. Worked Example
Problem
A, B, and C can complete a job in 10, 12, and 15 days respectively. They work together for 4 days. After that, B and C continue alone. How many more days will B and C need to finish the job?
Step 1: Convert days to rate (jobs per day)
| Worker | Time to finish 1 job (days) | Rate (R) (jobs/day) |
|---|---|---|
| A | 10 | (R_A = \frac{1}{10}) |
| B | 12 | (R_B = \frac{1}{12}) |
| C | 15 | (R_C = \frac{1}{15}) |
Step 2: Combined rate of all three
[ R_{\text{ABC}} = R_A + R_B + R_C = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} ]
Compute common denominator (60):
[ R_{\text{ABC}} = \frac{6+5+4}{60} = \frac{15}{60} = \frac{1}{4}\text{ job/day} ]
So together they finish ¼ of the job per day.
Step 3: Work done in first 4 days
[ W_{\text{done}} = R_{\text{ABC}} \times 4 = \frac{1}{4} \times 4 = 1 \text{ job} ]
Wait – that would be the whole job, but we mis‑calculated:
Actually (R_{\text{ABC}} = \frac{1}{4}) job/day → in 4 days they finish (1) job. But that can't be; check calculation.
Hold on! Recalculate:
[
\frac{1}{10} = 0.1,\quad \frac{1}{12}\approx0.0833,\quad \frac{1}{15}\approx0.0667
]
Sum ≈ 0.25. Yes, indeed they finish the whole job in 4 days.
But the problem expects them to finish partially. Let's adjust the example to avoid confusion.
New Example (corrected)
A, B, C finish a job in 8, 12, and 24 days respectively. A and B work together for 2 days. Then B and C work together until the job is finished. How many days did B work in total?
Revised Steps
- Rates
[ R_A = \frac{1}{8},; R_B = \frac{1}{12},; R_C = \frac{1}{24} ]
- Combined rate of A & B
[ R_{AB} = \frac{1}{8} + \frac{1}{12} = \frac{3+2}{24} = \frac{5}{24}\text{ job/day} ]
- Work done in first 2 days
[ W_{\text{done}} = R_{AB} \times 2 = \frac{5}{24} \times 2 = \frac{5}{12}\text{ job} ]
- Work left
[ W_{\text{left}} = 1 - \frac{5}{12} = \frac{7}{12}\text{ job} ]
- Combined rate of B & C
[ R_{BC} = \frac{1}{12} + \frac{1}{24} = \frac{2+1}{24} = \frac{3}{24} = \frac{1}{8}\text