Time & Work – Study Notes
(For the Village Development Officer – Quantitative Aptitude)
1. Simple Explanation
- Work is any task that can be measured in units (e.g., land cleared, houses built, bricks laid).
- Rate of work is the amount of work a person/machine can do per unit time (units/day).
- Time is the period required to finish a job.
In most problems the job is fixed (say 1 whole job).
For a single worker/machine
[ \text{Rate} = \frac{\text{Work}}{\text{Time}} ]
For two or more workers working together
[ \text{Combined Rate} = \text{Rate}_1 + \text{Rate}_2 + \dots ] [ \text{Time for job} = \frac{\text{Work}}{\text{Combined Rate}} ]
If a worker stops or joins midway, the work done in the first part is subtracted from the total, and the remaining work is completed by the remaining workers.
2. Key Points & Formulas
| Concept | Symbol | Formula | Meaning |
|---|---|---|---|
| Rate of work | ( R ) | ( R = \frac{W}{T} ) | Units of work per day |
| Work done | ( W ) | ( W = R \times T ) | Total units of work in a period |
| Time needed | ( T ) | ( T = \frac{W}{R} ) | Days required to finish the job |
| Combined rate | ( R_c ) | ( R_c = R_1 + R_2 + \dots + R_n ) | Sum of individual rates |
| Time for combined work | ( T_c ) | ( T_c = \frac{W}{R_c} ) | Days for all workers together |
| Work left after ( t ) days | ( W_{\text{left}} ) | ( W_{\text{left}} = W_{\text{total}} - R_{\text{current}}\times t ) | Remaining work |
| Inverse proportion | – | ( T_1 \times R_1 = T_2 \times R_2 = W ) | If one worker takes twice as long, their rate is half |
Common Phrases
- “X can do the job in Y days” → Rate of X = ( \frac{1}{Y} ) job/day.
- “X and Y together finish the job in Z days” → Combined rate = ( \frac{1}{Z} ) job/day.
Tip: Always convert “time” phrases to rate (jobs per day) before adding.
3. Worked Example
Problem
A, B and C can complete a job in 8, 12 and 18 days respectively.
- How long will they take if they all work together?
- If C stops after 3 days, how many more days will A and B need to finish the job?
Step‑by‑Step
1. All three work together
| Worker | Days to finish alone | Rate (job/day) |
|---|---|---|
| A | 8 | ( \frac{1}{8} ) |
| B | 12 | ( \frac{1}{12} ) |
| C | 18 | ( \frac{1}{18} ) |
Combined rate
[ R_c = \frac{1}{8} + \frac{1}{12} + \frac{1}{18} ]
Find a common denominator (72):
[ R_c = \frac{9}{72} + \frac{6}{72} + \frac{4}{72} = \frac{19}{72};\text{job/day} ]
Time together
[ T_c = \frac{1;\text{job}}{19/72} = \frac{72}{19};\text{days} \approx 3.79;\text{days} ]
So all three finish the job in ( \boxed{72/19 \text{ days} \approx 3.79 \text{ days}} ).
2. C stops after 3 days
Work done by C in 3 days
[ W_C = R_C \times 3 = \frac{1}{18} \times 3 = \frac{1}{6};\text{job} ]
Remaining work
[ W_{\text{left}} = 1 - \frac{1}{6} = \frac{5}{6};\text{job} ]
Now only A and B work. Their