Time & Work – Quantitative Aptitude
Puducherry Patwari / Lekhpal – Study Notes
1. Simple Explanation
Time & Work problems ask you to find how long it takes to complete a task (or how many people are needed) when several workers are involved.
- Work is a unit of job (e.g., “one whole plot”).
- Rate (or work‑rate) is the amount of work done per unit time.
- If a worker completes 1 job in T hours → rate = 1/T jobs/hour.
- When workers work together, their rates add up.
Key Idea:
[ \text{Total Work} = (\text{Rate of Worker A} + \text{Rate of Worker B} + \dots) \times \text{Time} ]
2. Key Points & Formulas
| Concept | Formula | Explanation |
|---|---|---|
| Rate of a worker | (r = \dfrac{1}{T}) | (T) – time taken by worker to finish 1 job alone. |
| Combined rate | (R = r_1 + r_2 + \dots + r_n) | Sum of individual rates. |
| Time to finish a job together | (t = \dfrac{1}{R}) or (t = \dfrac{W}{R}) | (W) – total work (usually 1). |
| Work done by a worker in a given time | (W = r \times t) | |
| Work done by a worker in a given time when rate is 1/T | (W = \dfrac{t}{T}) | |
| Work left after a part has been done | (W_{\text{left}} = W_{\text{total}} - W_{\text{done}}) | |
| Work after changing team | Re‑calculate combined rate and use (t = W_{\text{left}}/R_{\text{new}}). |
Common Pitfalls
- Confusing “time” with “rate” – always convert time to rate first.
- Adding times instead of rates – rates add, times do not.
- Assuming the same efficiency – always check if workers have different rates.
- Ignoring “work left” – when a worker leaves, compute the remaining work before re‑calculating time.
3. Worked Example
Problem
A and B can complete a piece of land in 12 and 15 days respectively.
A works alone for 3 days, then leaves.
B then works alone for 4 days, after which A returns and works with B.
How many more days will they need to finish the remaining work?
Step‑by‑Step Solution
-
Find individual rates
- A’s rate: (r_A = \dfrac{1}{12}) job/day
- B’s rate: (r_B = \dfrac{1}{15}) job/day
-
Work done by A in first 3 days
[ W_{A1} = r_A \times 3 = \frac{1}{12}\times 3 = \frac{1}{4} ] -
Work left after A’s first stint
[ W_{\text{left1}} = 1 - \frac{1}{4} = \frac{3}{4} ] -
Work done by B in next 4 days
[ W_{B1} = r_B \times 4 = \frac{1}{15}\times 4 = \frac{4}{15} ] -
Work left after B’s first stint
[ W_{\text{left2}} = \frac{3}{4} - \frac{4}{15} ] Convert to common denominator (60):
[ \frac{3}{4} = \frac{45}{60},\quad \frac{4}{15} = \frac{16}{60} ] [ W_{\text{left2}} = \frac{45-16}{60} = \frac{29}{60} ] -
Combined rate when A and B work together
[ R_{AB} = r_A + r_B = \frac{1}{12} + \frac{1}{15} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} ] -
Time needed to finish remaining work
[ t_{\text{remaining}} = \frac{W_{\text{left2}}}{R_{AB}} = \frac{\frac{29}{60}}{\frac{3}{20}} = \frac{29}{60} \times \frac{20}{3} = \frac{29 \times 20}{60 \times 3} = \frac{580}{180} = \frac{29}{9} \text{ days} ] [ \frac{29}{9} \approx 3.222\text{ days} ]
Answer: They need approximately 3.22 days (i.e., 3 days and 5 hours) more to finish the land.
4. Quick Revision Summary
- Rate = 1 / Time (job per hour/day).
- Rates add when workers work together.
- Time to finish a job together: (t = \dfrac{1}{\text{combined rate}}).
- Work done in a given time: (W = r \times t).
- When team changes, recompute the remaining work and the new combined rate.
- Always work with common units (hours or days).
- Check for common denominators when adding/subtracting fractions