📘 Time & Work – Quantitative Aptitude
Course: Odisha Forest Guard – Program
Topic: Time & Work
Audience: Fresh recruits & aspirants preparing for the written exam
1️⃣ Simple Explanation
Time & Work deals with how long it takes for one or more workers (or machines, teams, etc.) to complete a job.
- Work is a fixed amount (e.g., painting a wall, cutting bamboo, etc.)
- Rate of work is how much of that job one worker can finish in a given time (often expressed as work per hour or work per day).
- The fundamental relationship:
[ \text{Work} = \text{Rate} \times \text{Time} ]
When multiple workers are involved, their rates add up.
2️⃣ Key Points & Formulas
| Concept | Symbol | Formula | Explanation |
|---|---|---|---|
| Work | (W) | – | Total job to be completed (unit: “work”). |
| Rate of work | (r) | (r = \frac{W}{T}) | Amount of work done per unit time. |
| Time | (T) | (T = \frac{W}{r}) | Time needed to finish the work. |
| Multiple workers | (r_1, r_2, \dots, r_n) | (r_{\text{total}} = r_1+r_2+\dots+r_n) | Rates add up. |
| Work done in time | (W_{\text{done}}) | (W_{\text{done}} = r \times T) | Portion of job completed. |
| Work remaining | (W_{\text{rem}}) | (W_{\text{rem}} = W - W_{\text{done}}) | Job left after some work. |
| Worker’s days to finish | (D) | (D = \frac{W}{r}) | Common unit in exam problems. |
| Unit of rate | – | “Work per hour/day” | Often written as “1 work” or “1 job”. |
Special Cases
- One worker is twice as fast as another
- If (A)’s rate is (2r) and (B)’s rate is (r), then (A) does in half the time of (B).
- Work done in part of the job
- If 3 workers finish 1/4 of the job in 5 days, then the total job (W = 4 \times (3r \times 5)).
- Work done with different worker ratios
- When ratios of rates are given (e.g., 2:3:5), convert them to actual numbers by assigning a common multiplier (e.g., 1 unit = 1 work/day).
3️⃣ Worked Example
Problem Statement
Three forest guards, S, R and K, can clear a section of the forest in 8, 12 and 18 days respectively. They start together and work for 4 days. After that, S leaves and R and K continue alone. How many days will it take R and K together to finish the remaining work?
Step‑by‑Step Solution
| Step | Calculation | Explanation |
|---|---|---|
| 1. Find each guard’s daily rate | (r_S = \frac{1}{8}) work/day, (r_R = \frac{1}{12}), (r_K = \frac{1}{18}) | Since each completes the whole job in the given days. |
| 2. Combined daily rate of all three | (r_{\text{all}} = r_S + r_R + r_K = \frac{1}{8} + \frac{1}{12} + \frac{1}{18}) | Add rates. |
| 3. Simplify (r_{\text{all}}) | Common denominator = 72 → (r_{\text{all}} = \frac{9+6+4}{72} = \frac{19}{72}) work/day | |
| 4. Work done in first 4 days | (W_{\text{done}} = r_{\text{all}} \times 4 = \frac{19}{72} \times 4 = \frac{19}{18}) work | Work done by all three for 4 days. |
| 5. Work remaining | (W_{\text{rem}} = 1 - \frac{19}{18} = -\frac{1}{18}) | Negative? That means >1? Let's recompute: Actually (r_{\text{all}} = \frac{19}{72}) ≈ 0.2639. 4 days → (0.2639 \times 4 = 1.0556). That exceeds 1, meaning they finish the job earlier than 4 days. But the problem says they work for 4 days, implying the job is not finished. Let's re‑check the rates. |
| Re‑check rates | (r_S = 1/8 = 0.125) (r_R = 1/12 ≈ 0.0833) (r_K = 1/18 ≈ 0.0556) Sum ≈ 0.264 |
4 days → 1.056 >1, so job would finish in <4 days. The problem must have different numbers or maybe the first 4 days before S leaves? Let's adjust: Suppose S leaves after 4 days, but they finish the job after 4 days? Actually 1.056 > 1, so they finish in about 3.79 days. So the premise is wrong. Let's change the numbers: Maybe R and K alone finish in 12 and 18 days? Let's instead solve with given numbers after correction: Let’s assume the job is 1.5 times the total? But for educational purpose, we’ll illustrate the method properly using a corrected example. |
Corrected Example
Problem
*Guard A can clear the forest in 10 days,