Time & Work – Quantitative Aptitude
Program: Lakshadweep Forest Guard
Topic: Time & Work
“Time & Work” problems ask you to find how long it takes to finish a job, or how much work is done in a given time, when two or more people (or machines) are working together or separately.
1. Simple Explanation
| Concept | What it means | Example |
|---|---|---|
| Work | The total amount of job that must be done. It can be thought of as “1 job” or “100 % of the job”. | W = 1 (one whole task) |
| Rate of work | How much of the job a person (or team) can finish in one unit of time. | Person A does 1/8 of the job in one day → Rateₐ = 1/8 job/day |
| Time | How long it takes to complete the job. | If A alone takes 8 days → Tₐ = 8 days |
| Work‑Rate‑Time relation | Work = Rate × Time | For A: 1 = (1/8) × 8 |
When multiple people work together, their rates add up.
When someone leaves earlier, you adjust the time they actually worked.
2. Key Points (Quick Reference)
| # | Formula | What to calculate | Notes |
|---|---|---|---|
| 1 | Rate = Work / Time | Rate of a person | If Work = 1 → Rate = 1/Time |
| 2 | Time = Work / Rate | How long it takes for one person | Time = 1 / Rate |
| 3 | Work = Rate × Time | Amount of work done in a given time | Useful for partial work |
| 4 | Combined Rate = Rate₁ + Rate₂ + … | Rate when several people work together | Add the individual rates |
| 5 | Total Work = Combined Rate × Time | Work done by a group in a period | If they work for t days |
| 6 | Work Left = Total Work – Work Done | Remaining work | Usually Work Left = 1 – Work Done |
| 7 | Fraction of work done | Rate × Time (e.g., 0.25 of job) |
Express as a decimal or fraction |
Approach to Solve a Problem
- Identify the total work – usually 1 job (100 %).
- Convert given times or rates into a common unit (days, hours, etc.).
- Compute rates for each worker using
Rate = 1 / Timeif only time is given. - Add rates if workers work together.
- Use the total time worked to find work done (
Rate × Time). - Subtract from 1 to find the remaining work.
- If a worker leaves, multiply his/her rate by the actual time worked.
- Solve for unknown time using
Time = Work / Rate.
3. Worked Example
Problem (Typical of Forest Guard exam)
A can finish a piece of work in 8 days. B can finish the same work in 12 days.
A and B work together for 2 days. After that, A leaves.
How many more days will B need to finish the remaining work?
Step‑by‑Step Solution
| Step | Calculation | Result |
|---|---|---|
| 1 | Rate of A: (R_A = \dfrac{1}{8}) jobs/day | (0.125) jobs/day |
| 2 | Rate of B: (R_B = \dfrac{1}{12}) jobs/day | (0.0833) jobs/day |
| 3 | Combined rate for first 2 days: (R_{AB} = R_A + R_B = 0.125 + 0.0833 = 0.2083) jobs/day | |
| 4 | Work done in 2 days: (W_{done} = R_{AB} \times 2 = 0.2083 \times 2 = 0.4166) jobs | ≈ 41.67 % of the job |
| 5 | Work remaining: (W_{left} = 1 - 0.4166 = 0.5834) jobs | ≈ 58.34 % of the job |
| 6 | B’s rate alone: (R_B = 0.0833) jobs/day | |
| 7 | Time for B to finish remaining work: (T_B = \dfrac{W_{left}}{R_B} = \dfrac{0.5834}{0.0833} ≈ 7.0) days | 7 days |
Answer: B needs 7 more days to finish the remaining work.
4. Quick Revision Summary
- Work = 1 job (or 100 %)
- Rate (per day) =
1 / Timeof the worker alone - Combined Rate = Sum of individual rates
- Work done =
Rate × Time - Time needed =
Remaining Work / Rate - Remember:
- Convert all times to the same unit (days usually).
- Add rates when people work together.
- Subtract the work done from 1 to get the remaining work.
- When someone leaves, use the actual time they worked, not the full schedule.
Quick Practice Question
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