Time & Work – Study Notes
Quantitative Aptitude – Arunachal Pradesh Police Constable Exam
1. Simple Explanation
Time & Work deals with how long it takes people (or machines) to finish a job and how fast they work.
- Work is the total job to be done (e.g., building a wall).
- Rate of work is how much of the job one person can finish in a unit of time (often per day).
- If two people work together, their rates add.
- If one person works against another (e.g., painting a wall while the other is sanding), you subtract rates.
2. Key Points & Formulas
| Concept | Symbol | Formula | Meaning |
|---|---|---|---|
| Work done | (W) | (W = \text{Rate} \times \text{Time}) | Total job completed |
| Rate of work | (R) | (R = \frac{W}{T}) | Portion of job per unit time |
| Time taken | (T) | (T = \frac{W}{R}) | Duration to finish the job |
| People working together | (n) | (R_{\text{total}} = \sum_{i=1}^n R_i) | Sum of individual rates |
| One works against another | (R_{\text{net}} = R_{\text{good}} - R_{\text{bad}}) | Net progress per unit time | |
| Work left after (k) days | (W_{\text{left}} = W - k(R_{\text{total}})) | Remaining job | |
| Work done by (k) workers in (t) days | (W = k \times R \times t) | When all workers are equal | |
| Work done by a person in (t) days | (W = R \times t) | For a single worker |
Special Cases
-
Work done by one person after another leaves work
- If A does 1/3 of work in 2 days, A’s rate = ( \frac{1/3}{2} = \frac{1}{6}) job/day.
- If B does 1/4 of work in 3 days, B’s rate = ( \frac{1/4}{3} = \frac{1}{12}) job/day.
- Together, rate = ( \frac{1}{6} + \frac{1}{12} = \frac{1}{4}) job/day → 4 days to finish.
-
Work done by a person in (t) days
- If one person finishes a job in (x) days, his rate = ( \frac{1}{x}) job/day.
- The same job takes (x \times t) days for (t) people working at the same rate.
-
Work left after (t) days
- If work rate = (R) and time used = (t), then work done = (R \times t).
- Work left = (1 - R\times t) (if total work = 1 job).
3. Worked Example
Problem
Three workers, A, B, and C, can complete a road in 12, 15, and 18 days respectively.
If A and B work together for 3 days, then C joins them.
How many more days are required for all three to finish the road?
Step‑by‑Step Solution
-
Calculate individual rates.
- A’s rate (R_A = \frac{1}{12}) road/day.
- B’s rate (R_B = \frac{1}{15}) road/day.
- C’s rate (R_C = \frac{1}{18}) road/day.
-
Work done by A and B in 3 days.
[ W_{\text{AB}} = (R_A + R_B) \times 3 = \left(\frac{1}{12} + \frac{1}{15}\right) \times 3 ] Find common denominator 60:
[ \frac{1}{12} = \frac{5}{60},; \frac{1}{15} = \frac{4}{60}; \Rightarrow; R_{\text{AB}} = \frac{9}{60} = \frac{3}{20} ] [ W_{\text{AB}} = \frac{3}{20} \times 3 = \frac{9}{20} = 0.45 \text{ of the road} ] -
Work remaining after 3 days.
[ W_{\text{left}} = 1 - 0.45 = 0.55 ] -
Rate when all three work together.
[ R_{\text{total}} = R_A + R_B + R_C = \frac{1}{12} + \frac{1}{15} + \frac{1}{18} ] Common denominator 180:
[ \frac{1}{12} = \frac{15}{180},; \frac{1}{15} = \frac{12}{180},; \frac{1}{18} = \frac{10}{180} ] [ R_{\text{total}} = \frac{15+12+10}{180} = \frac{37}{180} \approx 0.2056 \text{ road/day} ] -
Time needed for the remaining 0.55 of the road.
[ t = \frac{W_{\text{left}}}{R_{\text{total}}} = \frac{0.55}{37/180} = 0.55 \times \frac{180}{37} ] Compute: [ 0.55 \times 180 = 99,; \frac{99}{37} \approx 2.6757 \text{ days} ]