Time & Work – Quick Study Notes
(For ISRO Recruitment – Quantitative Aptitude)
1. Simple Explanation
Time & Work problems ask you to determine the time taken, work rate, or amount of work done when one or more workers (or machines) cooperate.
- Work (W) – Total amount of job to be finished.
- Rate (R) – How much work one worker can finish in one unit of time.
- Time (T) – How long a worker takes to finish the whole job.
Basic relationship
[
W = R \times T \quad\Longrightarrow\quad R = \frac{W}{T}\quad\text{or}\quad T = \frac{W}{R}
]
When several workers work together, their rates add up: [ R_{\text{total}} = R_1 + R_2 + \dots + R_n ]
2. Key Points / Formulae
| Concept | Formula | Meaning |
|---|---|---|
| Rate of a worker | ( R = \frac{1}{T} ) (if work (W = 1) unit) | Fraction of job finished in one hour |
| Work done by a worker in (t) hours | ( W = R \times t ) | Portion of job completed |
| Combined rate | ( R_{\text{total}} = R_1 + R_2 + \dots ) | Sum of individual rates |
| Time needed to finish job (W) | ( T = \frac{W}{R_{\text{total}}} ) | Hours required |
| Work done by one worker when another leaves | Use (T_1 = \frac{W}{R_1}) for the one who stays; subtract the portion already done by the other. | |
| Work left after a certain time | (W_{\text{left}} = W - R_{\text{total}} \times t_{\text{worked}}) |
Remember: If the problem gives “days” instead of “hours”, keep the units consistent (e.g., 1 day = 8 working hours if not specified).
3. Worked Example
Problem
A team of 4 engineers can complete a software testing project in 12 days.
If 2 of those engineers leave after 3 days, how many days will the remaining 2 engineers take to finish the rest of the project?
Step‑by‑step Solution
-
Define the total work
Let the whole project = (W = 1) (unit). -
Compute combined rate of all 4 engineers
[ R_{\text{total}} = \frac{1}{12} \text{ (jobs/day)} ] -
Work done in first 3 days by all 4 engineers
[ W_{\text{first}} = R_{\text{total}} \times 3 = \frac{1}{12} \times 3 = \frac{1}{4} ] -
Work remaining after 3 days
[ W_{\text{left}} = 1 - \frac{1}{4} = \frac{3}{4} ] -
Rate of the remaining 2 engineers
Since all 4 together did (\frac{1}{12}) per day, each engineer does (\frac{1}{48}) per day.
Thus 2 engineers rate:
[ R_{\text{remain}} = 2 \times \frac{1}{48} = \frac{1}{24} ] -
Time required for the remaining work
[ T_{\text{remain}} = \frac{W_{\text{left}}}{R_{\text{remain}}} = \frac{\frac{3}{4}}{\frac{1}{24}} = \frac{3}{4} \times 24 = 18 \text{ days} ]
Answer: The remaining 2 engineers will take 18 days to finish the project.
4. Quick Revision Summary
| Question Type | What to Look For | Quick Formula |
|---|---|---|
| Given time, find rate | “T days to finish” | (R = \frac{1}{T}) |
| Given rate, find time | “Rate = X jobs/day” | (T = \frac{1}{R}) |
| Combined work | “n workers together finish in T days” | (R_{\text{total}} = n \times \frac{1}{T}) |
| Partial work | “Some workers leave after t days” | 1. Compute work done (= R_{\text{total}} \times t) 2. Subtract from total 3. Recompute time with remaining workers |
| Work left | “After t days, how much remains?” | (W_{\text{left}} = 1 - R_{\text{total}} \times t) |
| Days vs Hours | Keep units same | Convert if needed (e.g., 1 day = 8 hrs) |
Quick Tips
- Always set total work = 1 for simplicity.
- Rates add when workers cooperate.
- Work done = rate × time; rearrange to solve for unknown.
- Check units: If time given in days, ensure rates are per day.
Practice Challenge
Five technicians can install a satellite communication system in 10 days. If two of them leave after 4 days, how many more days will the remaining three technicians need?
(Try solving before checking the answer below.)
Answer: 12 days.
These notes should give you a solid foundation for tackling Time & Work questions in the ISRO Quantitative Aptitude test. Good luck!