Time & Work – Study Notes
Program: Chandigarh Police Sub‑Inspector (SI)
Subject: Quantitative Aptitude
Topic: Time & Work
Objective – Understand the fundamental concepts of the time‑work relationship, solve typical problems efficiently, and prepare for exam‑style questions.
1. Simple Explanation
- Work is the total amount of job that has to be completed.
- Rate of work is the amount of work done per unit time.
- If a person or a machine can finish a job in t hours, his/her rate = 1/t job per hour.
- Combined rate – when two or more workers/machines work together, their rates add up.
- Time required to finish a job when working together = (Total work) ÷ (Combined rate).
Example:
If A can finish a job in 5 hrs (rate = 1/5 job/hr) and B in 10 hrs (rate = 1/10 job/hr), together they finish 1/5 + 1/10 = 3/10 job/hr → 10/3 ≈ 3.33 hrs.
2. Key Points & Formulas
| Symbol | Meaning | Common Notation | Formula |
|---|---|---|---|
| W | Total work (jobs) | 1 job | – |
| r | Rate of a worker | job/hr | r = 1/t (if t is time to finish 1 job) |
| t | Time taken by a worker | hrs | t = 1/r |
| T | Time taken by n workers together | hrs | T = W / (Σ rᵢ) |
| n | Number of workers | – | – |
| R | Combined rate | job/hr | R = Σ rᵢ |
| Work Done | Portion completed | fraction | W_done = (Rate × Time) |
| Work Remaining | Portion left | fraction | W_remain = W - W_done |
Common Problem Types
- Individual speeds – find how long each takes alone.
- Combined speed – find total time when working together.
- Work left after some time – find remaining work or time to finish.
- Work done by one after another – sequential work.
- Work done with a new worker added/removed – change in combined rate.
- Work with varying rates – e.g., one worker works for part of time, another takes over.
3. Worked Example
Problem (exam‑style):
A and B can complete a road construction in 5 days and 8 days respectively. After working together for 2 days, B leaves. How many more days will A need to finish the remaining work?
Step‑by‑Step Solution
-
Determine individual rates
- A’s rate = 1 job / 5 days = 0.20 job/day
- B’s rate = 1 job / 8 days = 0.125 job/day
-
Combined rate (A + B)
- R₁ = 0.20 + 0.125 = 0.325 job/day
-
Work done together in 2 days
- W_done = R₁ × 2 = 0.325 × 2 = 0.65 job
-
Work remaining
- W_remain = 1 – 0.65 = 0.35 job
-
Time for A alone to finish remaining work
- t_remaining = W_remain / A’s rate
- t_remaining = 0.35 / 0.20 = 1.75 days
Answer
A will need 1.75 days (i.e., 1 day and 18 hours) after B leaves to finish the job.
Key Insight:
Work is additive; rates add when working together. Always convert to a common unit (jobs per day) before adding rates.
4. Quick Revision Summary
| Concept | Formula | Quick Tip |
|---|---|---|
| Rate of a worker | ( r = \frac{1}{t} ) | Think “job per hour/day” |
| Combined rate | ( R = \sum r_i ) | Add all workers’ rates |
| Time for combined work | ( T = \frac{W}{R} ) | (W) is usually 1 job |
| Work done in time | ( W_{\text{done}} = R \times t ) | Multiply rate by time |
| Work remaining | ( W_{\text{rem}} = W - W_{\text{done}} ) | Subtract completed part |
| Sequential work | ( W_{\text{total}} = \sum (r_i \times t_i) ) | Sum each portion |
| Changing workforce | Re‑calculate ( R ) after each change | Re‑compute combined rate |
Exam Strategy
- Read carefully – identify who works when and for how long.
- Draw a table of rates and times if multiple workers.
- Convert all times to the same unit (hours or days).
- Check units – always keep “job” as the base unit.
- Round only at the end – keep fractions until the final answer.
Good Luck!
Practice 5–10 problems per day using these steps, and you’ll master the Time & Work section.