Time & Work – Study Notes
Target Exam: Maharashtra Police Sub‑Inspector (SI) – Quantitative Aptitude
Section: Time & Work
Format: Markdown
1. Simple Explanation
Time & Work problems involve calculating how long it takes one or more workers (or machines) to complete a given job, or how much of a job is completed in a given time.
The fundamental relationship is
[ \text{Work} = \text{Rate} \times \text{Time} ]
- Rate – amount of work one worker can finish in one unit of time (e.g., per day, per hour).
- Time – number of time units required.
When more than one worker works together, their rates add up.
2. Key Points & Formulas
| Concept | Formula | Explanation |
|---|---|---|
| Single worker | (T = \frac{W}{R}) | Time needed = Work ÷ Rate. |
| Two workers together | (R_{\text{total}} = R_A + R_B) → (T = \frac{W}{R_A+R_B}) | Combine rates. |
| Three workers | (R_{\text{total}} = R_A+R_B+R_C) | Same idea. |
| Work done in a time | (W_{\text{done}} = R \times T) | Multiply rate by time. |
| Remaining work | (W_{\text{rem}} = W_{\text{total}} - W_{\text{done}}) | Subtract. |
| Work left after a part | (W_{\text{left}} = 1 - W_{\text{done}}) (if total work = 1 unit) | |
| Rate of a worker | (R = \frac{1}{T}) (if total work = 1 unit) | Inverse of time to finish whole job. |
| Difference in work rates | For workers A and B: (R_A - R_B = \frac{1}{T_A} - \frac{1}{T_B}) | |
| Work done by one worker alone after others stop | (T_{\text{alone}} = \frac{W_{\text{remaining}}}{R_{\text{solo}}}) | |
| Work done by a worker before/after another worker starts | Use the same product (R \times T) for the duration they work. | |
| Unit conversion | Always keep time in the same unit (hours, days, etc.). Convert if needed. |
3. Worked Example
Problem
A and B can paint a wall in 12 days and 15 days respectively.
A starts painting and after 3 days, B joins.
How many days will it take to finish the wall?
Step‑by‑Step Solution
-
Find individual rates (per day).
[ R_A = \frac{1}{12}\ \text{wall/day}, \qquad R_B = \frac{1}{15}\ \text{wall/day} ] -
Work done by A in the first 3 days.
[ W_{\text{first}} = R_A \times 3 = \frac{1}{12}\times3 = \frac{1}{4}\ \text{wall} ] -
Remaining work.
[ W_{\text{rem}} = 1 - \frac{1}{4} = \frac{3}{4}\ \text{wall} ] -
Combined rate when both work together.
[ R_{\text{total}} = R_A + R_B = \frac{1}{12} + \frac{1}{15} = \frac{5+4}{60} = \frac{9}{60}= \frac{3}{20}\ \text{wall/day} ] -
Time to finish the remaining work.
[ T_{\text{remain}} = \frac{W_{\text{rem}}}{R_{\text{total}}} = \frac{\frac{3}{4}}{\frac{3}{20}} = \frac{3}{4}\times\frac{20}{3}=5\ \text{days} ] -
Total time.
[ T_{\text{total}} = 3\ \text{days (A alone)} + 5\ \text{days (both)} = 8\ \text{days} ]
Answer: The wall will be finished in 8 days.
4. Types of Questions & Quick Tips
| Question Type | Quick Tip |
|---|---|
| Single worker finishing a job | Use (T = \frac{W}{R}). If (W = 1) unit, (R = \frac{1}{T}). |
| Two workers together | Add rates; if given days for each worker separately, convert to rate first. |
| Different start times | Compute work done before the second worker joins; subtract from total. |
| Worker leaves after some days | Compute remaining work; then calculate time with remaining workers. |
| One worker’s work is a fraction | If worker A does x% of the job in y days, then (R_A = \frac{x}{100y}). |
| Work left after a certain time | Use (W_{\text{done}} = R \times T). |
| Work done by a worker after another stops | Treat the remaining period as a new sub‑problem. |
| Mixed units (hours, days) | Convert all times to the same unit before applying formulas. |
| Rate comparison | For two workers A and B, if (A) does the job in (p) days and (B) in (q) days, then ( \frac{R_A}{R_B} = \frac{q}{p}). |
5. Quick Revision Summary
- Work = Rate × Time
- Rate = ( \frac{1}{\text{time to finish whole job}} )
- Combined rate = Sum of individual rates.
- Time needed = ( \frac{\text{total work}}{\text{combined rate}} ).
- Work done in a period = Rate ×