Time & Work – Quantitative Aptitude
Target Audience: Jharkhand Police Constable Entrance Exam
Duration: ~10 minutes of reading
1. Simple Explanation
- Time & Work deals with how long it takes to complete a job (work) when one or more workers are involved.
- A worker’s “rate” is the portion of the job they finish per unit time, usually expressed as a fraction or a percentage per hour/day.
- The key relationship:
[ \text{Rate} \times \text{Time} = \text{Work} ]
When the work is fixed (e.g., “complete a road in 5 days”), we can swap any two of the three variables (rate, time, work) to find the third.
2. Key Points & Formulas
| Concept | Symbol | Formula | Meaning |
|---|---|---|---|
| Work | (W) | – | Total job (often set to 1 unit or 100%) |
| Rate | (R) | (R = \dfrac{W}{T}) | Portion of job done per unit time |
| Time | (T) | (T = \dfrac{W}{R}) | Duration to finish the job |
| Combined Rate (workers working together) | (R_{\text{total}}) | (R_{\text{total}} = R_1 + R_2 + …) | Sum of individual rates |
| Combined Time (workers working together) | – | (T_{\text{total}} = \dfrac{W}{R_{\text{total}}}) | Time required with all workers |
| When one worker stops | – | Use remaining workers’ rate to find new time | |
| Work left | – | (W_{\text{left}} = W - \text{(work done)}) | Portion still pending |
Common Units
| Unit | Example | Conversion |
|---|---|---|
| Hours | 3 hrs | 1 hr = 60 min |
| Days | 5 days | 1 day = 24 hrs |
| Percentage | 25% | 0.25 as a decimal |
3. Worked Example
Problem
Three workers A, B and C can complete a job in 12, 15 and 20 days respectively.
Workers A and B start together. After 4 days, worker B leaves.
How many more days will worker C take to finish the remaining work?
Step‑by‑Step Solution
-
Convert each worker’s rate to “job per day”
[ \begin{aligned} R_A &= \frac{1}{12}\ \text{job/day} \ R_B &= \frac{1}{15}\ \text{job/day} \ R_C &= \frac{1}{20}\ \text{job/day} \end{aligned} ] -
Find combined rate of A and B
[ R_{AB} = R_A + R_B = \frac{1}{12} + \frac{1}{15} = \frac{5+4}{60} = \frac{9}{60} = \frac{3}{20}\ \text{job/day} ] -
Work done in first 4 days
[ W_{\text{done}} = R_{AB} \times 4 = \frac{3}{20} \times 4 = \frac{12}{20} = 0.6\ \text{job} ] -
Work remaining
[ W_{\text{left}} = 1 - W_{\text{done}} = 1 - 0.6 = 0.4\ \text{job} ] -
Now only worker C works
Rate of C: (R_C = \frac{1}{20}) job/day. -
Time required for C to finish remaining work
[ T_C = \frac{W_{\text{left}}}{R_C} = \frac{0.4}{1/20} = 0.4 \times 20 = 8\ \text{days} ]
Answer: Worker C will need 8 more days to finish the job.
4. Quick Revision Summary
- Rate = work done per unit time
- Time = work ÷ rate
- Combined Rate = sum of individual rates
- Work Left = total work – work already done
- Key Quick Checks
- Convert all times to the same unit (hours or days).
- Work is often treated as 1 unit (100 %).
- When workers change, recompute combined rate.
- Use fractions or decimals consistently; fractions avoid rounding errors.
Practice Tip
- Draw a simple table: Worker | Days | Rate | Work Done.
- For “workers working together” problems, always first find individual rates, then add them.
Use this cheat‑sheet whenever you hit a time‑and‑work problem on the exam. Good luck!