📚 Study Notes – Time & Work
Program: Chhattisgarh Forest Guard
Subject: Quantitative Aptitude
1️⃣ Simple Explanation
- Work = The amount of job that has to be done (e.g., clearing a forest area, planting trees, etc.).
- Rate of work = How much work one person/vehicle/tractor can finish in a given time (usually expressed as work per day).
- Time = How long it takes to finish the work at a given rate.
Formulae
| Symbol | Meaning | Formula |
|---|---|---|
| (W) | Total work (in work‑units) | – |
| (R) | Rate of work (work‑units per day) | – |
| (T) | Time required (days) | (T = \frac{W}{R}) |
| (R_i) | Individual rate of person (i) | – |
| (R_{sum}) | Combined rate of several persons/units | (R_{sum} = R_1 + R_2 + \dots + R_n) |
| (T_{sum}) | Time for a group to finish the job | (T_{sum} = \frac{W}{R_{sum}}) |
| (T_1, T_2) | Time taken by two persons separately | – |
| (R_{1,2}) | Combined rate | (R_{1,2} = \frac{1}{T_1} + \frac{1}{T_2}) (if work is same) |
Key Idea:
Rate of work = work completed in a unit time.
Time needed = Total work ÷ Rate.
2️⃣ Key Points
| # | Concept | Quick Note |
|---|---|---|
| 1 | Work Units | Treat the whole job as 1 unit or 100 units; keep consistent. |
| 2 | Rate Conversion | If a worker does 1/4 of the work in a day ⇒ rate = 0.25 work‑units/day. |
| 3 | Working Together | Add individual rates to get combined rate. |
| 4 | Working Together & Apart | If one works alone for some time, subtract that portion before adding the other’s rate. |
| 5 | Opposite Rates | When one worker’s rate is k times the other’s, use (R_1 = kR_2). |
| 6 | Partial Work | Calculate work done in given time, then subtract from total to find remaining work. |
| 7 | Common Mistakes | • Mixing up time and rate. • Forgetting to convert rates to same unit. • Assuming “time taken” = “time left” incorrectly. |
| 8 | Useful Tricks | • Use “work per day” instead of “days per work”. • Convert fractional days to hours if needed. • For two persons, if one takes (x) days, the other takes (y) days, the combined time is (\frac{xy}{x+y}). |
3️⃣ Worked Example
Problem (Typical Forest Guard Question)
A team of 5 forest guards can clear a forest area in 12 days. If 2 additional guards join the team, how many days will it take to clear the area?
Step‑by‑Step Solution
| Step | Explanation | Calculation |
|---|---|---|
| 1. Define Total Work | Let the entire area = 1 work‑unit. | (W = 1) |
| 2. Compute Initial Rate | 5 guards finish 1 unit in 12 days ⇒ rate per day: | (R_{5} = \frac{1}{12}) work‑units/day |
| 3. Find Rate of One Guard | Since all guards are equal, divide by 5. | (R_{guard} = \frac{R_{5}}{5} = \frac{1}{12 \times 5} = \frac{1}{60}) |
| 4. New Team Size | 5 + 2 = 7 guards. | – |
| 5. Compute New Combined Rate | 7 × (R_{guard}). | (R_{7} = 7 \times \frac{1}{60} = \frac{7}{60}) |
| 6. Find New Time | (T = \frac{W}{R_{7}}). | (T = \frac{1}{\frac{7}{60}} = \frac{60}{7} \approx 8.57) days |
| 7. Express Answer | 8 days and 0.57 day ≈ 8 days 13.7 hours. | ≈ 8.57 days |
Answer: About 8.6 days (or 8 days 13½ hours).
4️⃣ Quick Revision Summary
- Rate = work done per unit time.
- Time = total work ÷ rate.
- Combined rate = sum of individual rates.
- For (n) persons: (T = \frac{W}{n \times \text{(rate of one)}}).
- If two people take (x) and (y) days separately: Combined days (= \frac{xy}{x + y}).
- Partial work: Work done = rate × time; remaining = total – done.
- Check units: keep days, hours, and fraction consistent.
- Common pitfall: Mixing “time” and “rate”; always verify the direction of the formula.
🏆 Quick Self‑Check Questions
- If 3 workers can finish a job in 10 days, how many days will 5 workers take?
- A machine does 1/5 of the job in 2 days. How many days to finish the job alone?
- Worker A takes 8 days, Worker B takes 12 days. How long together?
Take a minute to solve them before moving on. Good luck with your preparation!