Goa PSC – Quantitative Aptitude
Topic: Time & Work
Time & Work problems are a staple of the Goa PSC exam.
They test your ability to convert between time, rate, and amount of work, and to solve linear equations that involve rates of work.
1. Simple Explanation
- Work – Any job that can be measured in “units” (e.g., cubic metres of concrete poured, pages written, kilometres travelled).
- Rate of Work – How fast a person/thing can do the job, expressed as units per hour (or day, minute, etc.).
- Relation
[ \text{Rate} \times \text{Time} = \text{Work} ] or, equivalently,
[ \text{Time} = \frac{\text{Work}}{\text{Rate}} ] - Combined Work – When several people work together, their rates add.
[ R_{\text{total}} = R_1 + R_2 + \dots + R_n ]
2. Key Points / Formulae
| Concept | Formula | Example |
|---|---|---|
| Rate of a single worker | ( R = \frac{W}{T} ) | 5 workers finish a job in 8 hrs → (R = \frac{1}{8}) job/hr per worker |
| Combined rate | ( R_{\text{total}} = \sum R_i ) | 3 workers: (R_1 = 0.2), (R_2 = 0.1), (R_3 = 0.15) → (R_{\text{total}} = 0.45) |
| Time to finish a job | ( T = \frac{W}{R_{\text{total}}} ) | Job = 1 unit, (R_{\text{total}} = 0.45) → (T = \frac{1}{0.45} = 2.22) hrs |
| Work done in a given time | ( W = R \times T ) | A worker with rate 0.08 job/hr works 5 hrs → (W = 0.4) job |
| Work left after a period | ( W_{\text{left}} = W_{\text{total}} - W_{\text{done}} ) | 1 job – 0.3 job done = 0.7 job left |
| Ratio of rates | ( \frac{R_1}{R_2} = \frac{T_2}{T_1} ) | If A finishes a job in 5 hrs and B in 8 hrs, ( \frac{R_A}{R_B} = \frac{8}{5} ) |
Common Problem Types
- Two workers together – Find time to complete a job or find the time if one starts later.
- Work done by a worker after a break – Compute work done before and after the break.
- Adding/removing workers – Find how many workers are needed to finish a job in a given time.
- Work done by a group and an individual – Combine rates or find the individual’s rate.
- Work with partial days – Convert hours to fractions of a day (e.g., 3 hrs = 0.125 day).
3. Worked Example (Typical Goa PSC Question)
Question
A and B can paint a wall together in 12 hrs. B alone can paint the same wall in 20 hrs.
How long will it take for A to paint the wall alone?
Hint: Work = 1 wall.
Step‑by‑step Solution
| Step | Explanation | Calculation |
|---|---|---|
| 1. Write individual rates | Let (R_A) = rate of A (walls/hr), (R_B) = rate of B. | (R_B = \frac{1}{20}) walls/hr |
| 2. Combined rate | Together they finish 1 wall in 12 hrs → rate = ( \frac{1}{12}). | (R_A + R_B = \frac{1}{12}) |
| 3. Substitute (R_B) | (R_A + \frac{1}{20} = \frac{1}{12}). | |
| 4. Solve for (R_A) | (R_A = \frac{1}{12} - \frac{1}{20}). | (R_A = \frac{5-3}{60} = \frac{2}{60} = \frac{1}{30}) walls/hr |
| 5. Find time for A alone | (T_A = \frac{1}{R_A} = \frac{1}{1/30} = 30) hrs | Answer: 30 hrs |
Result – A alone takes 30 hours to paint the wall.
4. Quick Revision Summary
| Topic | What to Remember |
|---|---|
| Rate & Time | (R = \frac{W}{T}); (T = \frac{W}{R}) |
| Combined Work | Add rates when working together. |
| Partial Work | Work done = rate × hours worked. |
| When a worker leaves/joins | Adjust the combined rate accordingly. |
| Units | Consistency is key – convert hours to days if needed. |
| Ratio of rates | Inversely proportional to time taken. |
| Common pitfalls | 1) Mixing up “time” and “rate”. 2) Forgetting to convert minutes/hours to |