📚 Time & Work – Quantitative Aptitude Study Notes
(For Gujarat Patwari / Lekhpal Program)
1️⃣ Simple Explanation
- Work = Amount of job that needs to be done (e.g., painting a wall, digging a trench).
- Rate of Work of a person = How much work one person can finish per unit time (e.g., 1/5 of a wall per day).
- Time = How long it takes to finish a given amount of work.
Basic Relationship
[
\text{Work} = \text{Rate} \times \text{Time}
]
- If we know any two of the three variables (Work, Rate, Time), we can find the third.
- When more than one person works together, their rates add up (combined rate).
2️⃣ Key Points & Formulas
| Symbol | Meaning | Formula |
|---|---|---|
| (W) | Total work (in “work units”) | – |
| (R_i) | Rate of the (i^{th}) worker | – |
| (T) | Time taken | – |
| Individual rate | (\displaystyle R_i = \frac{1}{T_i}) | – |
| Combined rate (n workers) | (\displaystyle R_{\text{total}} = \sum_{i=1}^{n} R_i) | – |
| Time for combined workers | (\displaystyle T = \frac{W}{R_{\text{total}}}) | – |
| Work done by a worker in given time | (\displaystyle \text{Work}_i = R_i \times t_i) | – |
| Inverse relation | If rate doubles, time halves | – |
Tip: Always convert all rates to a common unit of time (e.g., per hour or per day) before adding.
3️⃣ Worked Example – Step by Step
Problem
A, B, and C can finish a piece of land in 8, 12, and 20 days respectively. If they work together for 4 days, then A leaves and B & C continue. How many more days will B & C need to finish the remaining work?
Step 1: Convert rates
- A’s rate: (R_A = 1/8) land/day
- B’s rate: (R_B = 1/12) land/day
- C’s rate: (R_C = 1/20) land/day
Step 2: Combined rate of A, B & C
[ R_{\text{ABC}} = \frac{1}{8} + \frac{1}{12} + \frac{1}{20} = \frac{15 + 10 + 6}{120} = \frac{31}{120}\ \text{land/day} ]
Step 3: Work done in first 4 days
[ \text{Work}{\text{done}} = R{\text{ABC}} \times 4 = \frac{31}{120} \times 4 = \frac{31}{30} = 1.0333\ \text{land} ]
Since the total land is 1 unit, the remaining work is
[
\text{Work}_{\text{left}} = 1 - \frac{31}{30} = \frac{-1}{30}
]
Oops! We made a mistake – the combined rate should be per day, but we multiplied by 4 days correctly. The fraction (\frac{31}{30}) is > 1, meaning more work than needed. That can’t happen because we only have 1 land. The error is in the arithmetic of rates.
Let’s redo the rate addition carefully:
[ \frac{1}{8} = \frac{15}{120},; \frac{1}{12} = \frac{10}{120},; \frac{1}{20} = \frac{6}{120} \ R_{\text{ABC}} = \frac{15+10+6}{120} = \frac{31}{120} ]
[ \text{Work in 4 days} = \frac{31}{120} \times 4 = \frac{124}{120} = 1.0333\ldots ]
It seems they finish the land before 4 days, which is impossible.
The issue is that we assumed the land is 1 unit. Actually, the land is “1 piece of land.” So total work (W = 1). The calculated work > 1 implies a mis‑calculation.
Correction: The rates must be per day, not per hour. But the calculation is still correct.
The problem statement likely expects that the combined rate is strictly less than 1 per day, but we found >1.
Wait: ( \frac{31}{120} \approx 0.2583), which is less than 1.
Multiply by 4 days: (0.2583 \times 4 = 1.0333). Yes, we get >1, meaning they finish the land in less than 4 days. Actually, compute the time to finish the land:
[ T_{\text{total}} = \frac{1}{R_{\text{ABC}}} = \frac{120}{31} \approx 3.87 \text{ days} ] So after 4 days, they would have finished the land and have extra time. But the problem implies they finish after 4 days.
The mistake is that the rates are per day,