Time & Work – Study Notes
Tripura Police Sub‑Inspector (SI) – Quantitative Aptitude
(Prepared by an expert teacher – markdown format)
1. Simple Explanation
| Term | Meaning | Symbol |
|---|---|---|
| Work | Total amount of job or task to be finished | (W) |
| Rate of work | Amount of work done by a person (or group) in one unit of time | (r) |
| Time | Duration taken to finish the work | (t) |
Fundamental Relation
[ \text{Work} = \text{Rate} \times \text{Time} \qquad (1) ]
- If a person can finish a work in (x) days, his daily rate is (\dfrac{1}{x}) of the work.
- When several persons work together, their rates add up.
2. Key Points & Useful Formulas
| Scenario | Formula | Explanation |
|---|---|---|
| One person | (t = \dfrac{W}{r}) | Time needed = Work ÷ Rate |
| Two persons together | (t = \dfrac{W}{r_A + r_B}) | Combined rate = sum of individual rates |
| A works for a part, then B | (W_{\text{left}} = W - r_A t_A - r_B t_B) | Subtract work done by each segment |
| A works for (x) days, then B | (W_{\text{left}} = W - \dfrac{x}{r_A}) … | See worked example below |
| More than two persons | (t = \dfrac{W}{r_1 + r_2 + \dots + r_n}) | Add all rates |
| Ratio of work | If (r_A : r_B = a : b), then (t_A : t_B = b : a) | Inverse relation |
| Work done by a fraction of a day | (W_{\text{part}} = r \times \text{fraction of day}) | Use decimal or fraction of 1 day |
Quick Conversion Tips
- Rate from “days to do the work”:
[ r = \frac{1}{\text{days}} ] - Time from “rate”:
[ t = \frac{1}{r} ] - Fractional days: 0.5 day = 12 hours, 0.25 day = 6 hours, etc.
3. Worked Example (Step‑by‑Step)
Problem
A and B can finish a piece of work in 8 days and 12 days respectively.
A works alone for 2 days, then B works alone for 3 days.
How many days will it take them to finish the remaining work if they then work together?
Solution
-
Compute individual daily rates
[ r_A = \frac{1}{8};(\text{work per day}), \qquad r_B = \frac{1}{12};(\text{work per day}) ]
-
Work done by A in 2 days
[ W_A = r_A \times 2 = \frac{1}{8}\times 2 = \frac{1}{4} ]
-
Work done by B in 3 days
[ W_B = r_B \times 3 = \frac{1}{12}\times 3 = \frac{1}{4} ]
-
Total work already finished
[ W_{\text{done}} = W_A + W_B = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} ]
-
Work remaining
[ W_{\text{left}} = 1 - W_{\text{done}} = 1 - \frac{1}{2} = \frac{1}{2} ]
-
Combined rate when A and B work together
[ r_{AB} = r_A + r_B = \frac{1}{8} + \frac{1}{12} ]
Find common denominator (24):
[ r_{AB} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} ]
-
Time needed to finish the remaining half work
[ t_{\text{remain}} = \frac{W_{\text{left}}}{r_{AB}} = \frac{\frac{1}{2}}{\frac{5}{24}} ]
Simplify:
[ t_{\text{remain}} = \frac{1}{2} \times \frac{24}{5} = \frac{12}{5} = 2.4\ \text{days} ]
Which is 2 days and 0.4 day (i.e., 2