Time & Work – Study Notes
(For the Ladakh Patwari / Lekhpal Training Program – Quantitative Aptitude)
1. Simple Explanation
Time & Work deals with the relationship between the amount of work done, the rate at which it is done, and the time taken.
- Work – a task that can be completed (e.g., measuring a plot, preparing a ledger).
- Rate of work – the amount of work one person can do in a unit time (usually expressed as work per day).
- Time – the duration required to finish the work.
The basic equation is:
[ \text{Work} = \text{Rate} \times \text{Time} ]
Rearranging gives the three common forms:
| Formula | What it gives | Symbolic form |
|---|---|---|
| Time | How long to finish a given work | (T = \dfrac{W}{R}) |
| Rate | How fast a person works | (R = \dfrac{W}{T}) |
| Work | How much is completed | (W = R \times T) |
When more than one person works together, their rates add because each contributes to the same unit of work.
2. Key Points (Quick Reference)
| Concept | How to use | Formula |
|---|---|---|
| Single worker | Calculate time, rate, or work using one of the basic forms. | (T = \dfrac{W}{R}) |
| Multiple workers working together | Sum their rates to get the combined rate. | (R_{\text{combined}} = R_1 + R_2 + \dots) |
| Partial work | Work done by any worker for a part of the time. | (W_{\text{partial}} = R \times t_{\text{partial}}) |
| Work left after some time | Subtract partial work from total work. | (W_{\text{left}} = W_{\text{total}} - W_{\text{partial}}) |
| Inverse relationships | If one worker takes twice as long, his rate is half. | (R_2 = \dfrac{R_1}{2}) |
| Working in reverse order | When workers finish a task at the same time, find the time taken by each. | (\dfrac{1}{T} = \dfrac{1}{T_1} + \dfrac{1}{T_2}) |
| Work‐time diagrams | Helpful for visualising who works when. | — |
3. Worked Example
Problem
A patwari (P) can prepare a land‑record booklet in 8 days.
A junior clerk (J) can do the same job in 12 days.
They start together. After 3 days, P leaves. J continues alone until the job is finished.
How many days did J work after P left? How many days in total did the job take?
Step‑by‑Step Solution
-
Determine individual rates
[ R_P = \frac{1 \text{ booklet}}{8 \text{ days}} = \frac{1}{8}\text{ booklet/day} ] [ R_J = \frac{1}{12}\text{ booklet/day} ] -
Work done together in the first 3 days
[ W_{\text{together}} = (R_P + R_J) \times 3 = \left(\frac{1}{8} + \frac{1}{12}\right) \times 3 ] [ = \left(\frac{3}{24} + \frac{2}{24}\right) \times 3 = \frac{5}{24} \times 3 = \frac{5}{8}\text{ booklet} ] -
Work remaining after 3 days
[ W_{\text{left}} = 1 - \frac{5}{8} = \frac{3}{8}\text{ booklet} ] -
Time J needs to finish the remaining work alone
[ T_J = \frac{W_{\text{left}}}{R_J} = \frac{\frac{3}{8}}{\frac{1}{12}} = \frac{3}{8} \times 12 = \frac{36}{8} = 4.5 \text{ days} ] -
Total time taken
[ T_{\text{total}} = 3 \text{ days (together)} + 4.5 \text{ days (J alone)} = 7.5 \text{ days} ]
Answer
- J worked 4.5 days after P left.
- The job took a total of 7.5 days.
4. Quick Revision Summary
| Question | Answer |
|---|---|
| What is the basic equation of Time & Work? | (W = R \times T) |
| How do you find the combined rate of two workers? | Add their individual rates: (R_{\text{c}} = R_1 + R_2). |
| If one worker takes twice as long as another, how are their rates related? | The slower worker’s rate is half the faster worker’s rate. |
| How to calculate work done by a worker in part of the time? | (W_{\text{partial}} = R \times t_{\text{partial}}). |
| How to find remaining work after some time has passed? | Subtract partial work from total work: (W_{\text{ |