Time & Work – Quantitative Aptitude
(Tripura TET – Teacher Eligibility Test)
1. Simple Explanation
- Work is the amount of job that has to be done.
- Rate of work (or speed) is the amount of work done in one unit of time.
- Time is the duration needed to finish the work.
The basic relationship is
[ \textbf{Work} = \textbf{Rate} \times \textbf{Time} ]
When several workers or machines work together, their rates add up.
If someone works for a part of the work and then stops, the remaining work is done by the others.
2. Key Points
| Concept | What it Means | How to Use |
|---|---|---|
| Rate of work | Work done per unit time | ( R = \frac{W}{T} ) |
| Work done | Total job completed | ( W = R \times T ) |
| Combined rate | Sum of individual rates | ( R_{\text{total}} = R_1 + R_2 + \dots ) |
| Time for combined work | ( T = \frac{W}{R_{\text{total}}} ) | |
| Work done by one worker in part of time | ( W_{\text{partial}} = R \times T_{\text{partial}} ) | |
| Remaining work | ( W_{\text{remaining}} = W_{\text{total}} - W_{\text{partial}} ) | |
| Efficiency | Work done in a given time; used when rates differ | Compare ( R ) values |
3. Formulas (Quick Reference)
| Symbol | Meaning | Formula |
|---|---|---|
| (W) | Total work | (W = R \times T) |
| (R) | Rate of work | (R = \frac{W}{T}) |
| (T) | Time | (T = \frac{W}{R}) |
| (R_A, R_B, …) | Individual rates | (R_{\text{total}} = R_A + R_B + …) |
| (W_{\text{done}}) | Work done in a part of time | (W_{\text{done}} = R \times T_{\text{part}}) |
| (W_{\text{remaining}}) | Work left after some work | (W_{\text{remaining}} = W - W_{\text{done}}) |
4. Working Principle
- Convert all quantities to consistent units (e.g., hours, days).
- Determine each worker’s or machine’s rate (often given as “X can finish in Y days”).
- Rate (R = \frac{1}{\text{days}}) if work is considered as one unit.
- Add rates for workers working together.
- Calculate total time:
[ T = \frac{W}{R_{\text{total}}} ] - For mixed‑work problems, split into segments:
- Compute work done in first segment.
- Subtract from total to get remaining work.
- Use remaining work with the new rate to find remaining time.
5. Worked Example
Problem
A and B can finish a piece of work in 8 days and 12 days respectively.
They work together for 3 days, then B leaves.
A continues alone to finish the remaining work.
How many days does A need to finish the remaining work?
Step‑by‑Step Solution
| Step | Calculation | Result |
|---|---|---|
| 1. Rate of A | (R_A = \frac{1}{8}) work/day | (0.125) |
| 2. Rate of B | (R_B = \frac{1}{12}) work/day | (0.0833) |
| 3. Combined rate | (R_{\text{total}} = R_A + R_B = 0.125 + 0.0833 = 0.2083) | (0.2083) |
| 4. Work done in first 3 days | (W_{\text{done}} = R_{\text{total}} \times 3 = 0.2083 \times 3 = 0.625) | (0.625) |
| 5. Remaining work | (W_{\text{remaining}} = 1 - 0.625 = 0.375) | (0.375) |
| 6. Time for A alone | (T_A = \frac{W_{\text{remaining}}}{R_A} = \frac{0.375}{0.125} = 3) days | 3 days |
| 7. Total time | (3 + 3 = 6) days | 6 days |
Answer: A alone takes 3 more days to finish the remaining work; total time = 6 days.
6. Quick Revision Summary
- Rate of work = ( \frac{1}{\text{days to finish}} ).
- Combined rate = sum of individual rates.
- Total time = ( \frac{\text{Total work}}{\text{Combined rate}} ).
- For mixed‑work:
- Compute work done in the first segment.
- Subtract from total work to find remaining.
- Use remaining work with the new rate to get remaining time.
- Always keep units consistent and treat the whole job as 1 unit of work for simplicity.
7. Quick Practice Questions (Optional)
- (Easy) Two workers, P and Q, can finish a job in 10 days and 15 days respectively. They