Time & Work
Overview
Time & Work is a high-scoring topic in Maharashtra Police Bharti Mathematics. Questions test your ability to calculate how long workers take to complete jobs, how wages should be distributed, and how pipes fill or empty tanks.
The core idea is simple: work is measured as a fraction of the total job. If a person completes a task in 10 days, they do 1/10 of the work each day. Master this concept and you can solve most problems mentally. The topic connects directly to Ratio & Proportion — wages are distributed in the ratio of work done, and combined work rates add up like fractions.
Focus areas for the exam: basic work-rate problems, two or three workers combining, workers leaving mid-way, wages distribution, and pipes & cisterns with filling and emptying pipes working together.
Key Concepts
- Work Rate: If A completes work in n days, A's one day work = 1/n. This is the fundamental building block.
- Combined Work: When A and B work together, their combined one day work = 1/a + 1/b, where a and b are their individual completion times.
- LCM Method: Assume total work = LCM of individual times. This converts fractions to whole numbers, making calculations faster.
- Inverse Relationship: More workers means less time (if efficiency is same). Time is inversely proportional to number of workers.
- Work = Rate × Time: Just like Distance = Speed × Time. This formula governs all variations.
- Wages Ratio = Work Ratio: If A does 3 units and B does 2 units, wages split in 3:2 ratio, regardless of time spent.
- Pipes & Cisterns: Filling pipe has positive rate, emptying pipe (leak) has negative rate. Net rate = sum of all rates.
- Efficiency Ratio: If A is twice as efficient as B, then A's rate = 2 × B's rate, so A takes half the time.
Formulas / Key Facts
| Situation | Formula |
|---|---|
| A's one day work | 1/A (where A = days to complete) |
| A and B together | Time = (A × B)/(A + B) days |
| A, B, C together | Time = 1/(1/A + 1/B + 1/C) days |
| Work done | W = Rate × Time |
| Remaining work | 1 − (work already done) |
| Wages distribution | Ratio of wages = Ratio of work done |
| Pipe filling tank | Time = Tank capacity / Fill rate |
| Pipe emptying tank | Time = Tank capacity / Empty rate |
| Net rate (fill + leak) | 1/Fill time − 1/Empty time |
| Man-days concept | M₁ × D₁ = M₂ × D₂ (for same work) |
Quick fact: When a filling pipe and leak work together, if filling time = F and emptying time = E, net time to fill = (F × E)/(E − F), valid only when E > F.
Worked Examples
Example 1: Basic Combined Work
Problem: A can do a work in 12 days, B can do it in 18 days. In how many days will they finish working together?
Solution:
- A's one day work = 1/12
- B's one day work = 1/18
- Combined one day work = 1/12 + 1/18 = 3/36 + 2/36 = 5/36
- Time to complete = 36/5 = 7.2 days or 7 days 4 hours
Using formula: (12 × 18)/(12 + 18) = 216/30 = 7.2 days ✓
Example 2: Worker Leaves Mid-Way
Problem: A and B can complete work in 12 days. A alone can do it in 20 days. After working together for 4 days, A leaves. How many more days will B take to finish?
Solution:
- Combined one day work = 1/12
- Work done in 4 days = 4 × 1/12 = 1/3
- Remaining work = 1 − 1/3 = 2/3
- A's one day work = 1/20
- B's one day work = 1/12 − 1/20 = (5 − 3)/60 = 2/60 = 1/30
- Time for B to finish 2/3 work = (2/3) ÷ (1/30) = (2/3) × 30 = 20 days
Example 3: Pipes & Cisterns
Problem: Pipe A fills a tank in 6 hours. Pipe B empties it in 8 hours. If both are opened, how long to fill the tank?
Solution:
- A's rate = +1/6 per hour (filling)
- B's rate = −1/8 per hour (emptying)
- Net rate = 1/6 − 1/8 = 4/24 − 3/24 = 1/24 per hour
- Time to fill = 24 hours
Example 4: Wages Distribution
Problem: A can do work in 6 days, B in 8 days. They work together and get ₹2800. Find A's share.
Solution (LCM Method):
- Total work = LCM(6, 8) = 24 units
- A's daily work = 24/6 = 4 units
- B's daily work = 24/8 = 3 units
- Work ratio = 4:3
- A's share = (4/7) × 2800 = ₹1600
Common Mistakes
- Adding times instead of rates: Wrong: "A takes 10 days, B takes 15 days, together = 25 days." Fix: Add work rates (1/10 + 1/15), not times.
- Forgetting negative sign for leaks: Wrong: Adding leak rate as positive. Fix: Emptying pipes always subtract from filling rate.
- Confusing work done vs work remaining: Wrong: Calculating time for total work after partial completion. Fix: Always find remaining work first, then apply rates.
- Equal wages for equal time: Wrong: Assuming same hours worked = same wages. Fix: Wages depend on work done, not time. Faster worker earns more per day.
- Wrong LCM application: Wrong: Using LCM as answer directly. Fix: LCM is assumed total work; still need to calculate combined rate and divide.
- Ignoring efficiency differences: Wrong: Treating "A is twice as efficient" as A takes twice the time. Fix: Twice efficient means half the time.
Quick Reference
- A alone in n days → One day work = 1/n
- A + B together → Time = (A × B)/(A + B)
- LCM method: Total work = LCM, then work per day = LCM ÷ individual time
- Wages ratio = Work done ratio (not time ratio)
- Pipes: Fill rate positive, Leak rate negative, Net rate = sum
- Man-days constant: M₁D₁ = M₂D₂ for same work