Quantitative Aptitude – Time & Work
Program: Ladakh Police Sub‑Inspector (SI)
Audience: Candidates preparing for the SI exam
Format: Markdown study notes (clear, concise, exam‑oriented)
1. Simple Explanation
| Concept | What it means | How it is used in problems |
|---|---|---|
| Work (W) | The amount of job that needs to be done. Treat it as a single “unit” of work (e.g., 1 job, 1 plot, 1 building). | In calculations, we often set total work W = 1 unit so that rates become “units per hour/day”. |
| Rate (R) | How much work a person or machine can finish in a unit of time (e.g., units per hour). | Rate = Work ÷ Time. Inversely, Time = Work ÷ Rate. |
| Time (T) | The duration needed to finish a job. | Time = Work ÷ Rate. |
| Combined work | When several people work together, their rates add up. | Combined Rate = R₁ + R₂ + … + Rₙ. |
Key idea: Work is a product of Rate × Time.
If you know any two of the three quantities, you can find the third.
2. Key Points & Formulas
| Symbol | Meaning | Formula | Notes |
|---|---|---|---|
| W | Total work | 1 (unit) | Often set to 1 for simplicity |
| R | Rate of one person | (R = \frac{1}{T}) (units per hour/day) | (T) is the time taken by that person alone |
| T | Time needed | (T = \frac{1}{R}) | |
| W₁, W₂, … | Work done by each person | – | |
| R₁, R₂, … | Individual rates | – | |
| Combined Rate | Sum of all individual rates | (R_{\text{combined}} = R_1 + R_2 + … + R_n) | |
| Time to finish together | (T_{\text{together}} = \frac{1}{R_{\text{combined}}}) | – | |
| Work done by one person in t hours | (W_{\text{per}} = R \times t) | – | |
| Work left after t hours | (W_{\text{left}} = W_{\text{total}} - W_{\text{done}}) | – | |
| Difference in work or time | Use algebraic manipulation | – | Example: “A is 4 days faster than B” → (T_A = T_B - 4) |
Common Relations
- If Person X does the job in
xdays:
[ R_X = \frac{1}{x} ] - If two persons work together for
tdays:
[ (R_A + R_B) \times t = 1 \quad\text{(if total work = 1)} ] → (t = \frac{1}{R_A + R_B}) - If one person works alone for a portion of the job:
[ W_{\text{left}} = 1 - R_A \times t_A ] Then the remaining person(s) finish it in
[ t_{\text{remaining}} = \frac{W_{\text{left}}}{R_{\text{combined_remaining}}} ]
3. Worked Example
Problem (typical SI exam style)
A can complete a job in 12 days and B in 18 days.
a) How many days will A and B jointly finish the job?
b) If A works alone for 3 days, how many days will B need to finish the remaining work?
Step‑by‑Step Solution
a) Joint time
-
Find individual rates
- (R_A = \frac{1}{12}) units/day
- (R_B = \frac{1}{18}) units/day
-
Combined rate
[ R_{\text{joint}} = R_A + R_B = \frac{1}{12} + \frac{1}{18} ] Find common denominator (36):
[ R_{\text{joint}} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36};\text{units/day} ] -
Joint time
[ T_{\text{joint}} = \frac{1}{R_{\text{joint}}} = \frac{1}{\frac{5}{36}} = \frac{36}{5} = 7.2;\text{days} ] → Answer: 7.2 days (7 days 4.8 hours).