Time and Work
Overview
Time and Work is a fundamental quantitative aptitude topic that appears consistently in TNPSC Group II/IIA prelims. Questions typically involve calculating how long workers take to complete tasks individually or together, or how pipes fill/empty tanks. The underlying principle is simple: work done equals rate multiplied by time.
This topic connects directly to real-world scenarios—construction projects, manufacturing output, water management—making it relatable and frequently tested.
Students must be comfortable with LCM calculations, fraction addition/subtraction, and the ability to translate word problems into mathematical relationships. Once the foundational concept clicks, even complex problems involving multiple workers or pipes become straightforward.
Key Concepts
- Work as a unit: The total work is often assumed as 1 unit (or taken as LCM of given days for easier calculation). Individual contribution is a fraction of this whole.
- Rate of work = 1/Time: If A completes a job in 10 days, A's one-day work = 1/10 of the total job. This rate remains constant unless specified otherwise.
- Combined work: When A and B work together, their combined rate = (1/A) + (1/B). Time to complete = 1/(combined rate).
- LCM method: Taking total work as LCM of individual times converts fractions to whole numbers, making calculations faster and error-free.
- Negative work: In pipes and cisterns, outlet pipes or leaks do negative work (emptying), while inlet pipes do positive work (filling). Net rate = filling rate - emptying rate.
- Efficiency ratio: If A is twice as efficient as B, then A's rate = 2 × B's rate. Efficiency is inversely proportional to time taken.
- Man-days concept: Total work = Number of workers × Number of days. This product remains constant for the same work.
- Alternate day work: When people work on alternate days, calculate work done in a 2-day cycle, then find how many complete cycles are needed.
Formulas / Key Facts
| Scenario | Formula |
|---|---|
| A completes work in 'n' days | A's 1-day work = 1/n |
| A and B working together | Time = (A × B)/(A + B) days |
| A, B, C working together | Time = 1/[(1/A) + (1/B) + (1/C)] |
| A works for 'd' days | Work done = d × (1/n) = d/n |
| Pipe fills in 'a' hours, empties in 'b' hours | Net time to fill = (a × b)/(b - a) hours |
| If efficiency ratio is m:n | Time ratio is n:m (inverse relationship) |
| M₁ workers in D₁ days = M₂ workers in D₂ days | M₁ × D₁ = M₂ × D₂ (for same work) |
| Work with hours included | M₁ × D₁ × H₁ = M₂ × D₂ × H₂ |
Worked Examples
Example 1: Basic Combined Work
A can complete a work in 12 days, B can complete it in 18 days. In how many days will they complete it together?
Solution using formula: Combined time = (12 × 18)/(12 + 18) = 216/30 = 7.2 days = 7 days 4 hours 48 minutes
Solution using LCM method:
- Total work = LCM(12, 18) = 36 units
- A's rate = 36/12 = 3 units/day
- B's rate = 36/18 = 2 units/day
- Combined rate = 5 units/day
- Time = 36/5 = 7.2 days
Example 2: Work with Someone Leaving
A and B together can finish a work in 12 days. A alone can do it in 20 days. If A leaves after working for 4 days, how many more days will B take to finish?
Solution:
- Total work = LCM(12, 20) = 60 units
- (A + B)'s rate = 60/12 = 5 units/day
- A's rate = 60/20 = 3 units/day
- B's rate = 5 - 3 = 2 units/day
- Work done by A and B in 4 days = 4 × 5 = 20 units
- Remaining work = 60 - 20 = 40 units
- Time for B alone = 40/2 = 20 days
Example 3: Pipes and Cisterns
Two pipes A and B can fill a tank in 20 minutes and 30 minutes respectively. A drain pipe C can empty the full tank in 15 minutes. If all three are opened, how long to fill the tank?
Solution:
- Total capacity = LCM(20, 30, 15) = 60 units
- A fills = 60/20 = 3 units/min
- B fills = 60/30 = 2 units/min
- C empties = 60/15 = 4 units/min (negative work)
- Net rate = 3 + 2 - 4 = 1 unit/min
- Time to fill = 60/1 = 60 minutes
Common Mistakes
- Adding times instead of rates: Students often add 12 + 18 = 30 days when A takes 12 days and B takes 18 days. Correct approach: Add the rates (1/12 + 1/18), not the times.
- Forgetting to subtract outlet pipe rate: In cistern problems with leaks, students add all rates. Correct approach: Outlet/leak rate must be subtracted from inlet rate.
- Ignoring remaining work: When someone leaves midway, students sometimes calculate total time afresh. Correct approach: First find work done, then calculate time for remaining work only.
- Confusing efficiency with time: "A is 50% more efficient than B" doesn't mean A takes 50% more time. Correct approach: If B's efficiency is 100, A's is 150; so time ratio is 100:150 = 2:3 (A takes less time).
- LCM calculation errors: Taking wrong LCM leads to incorrect unit work distribution. Correct approach: Double-check LCM, especially with three or more values.
Quick Reference
- Work rate = 1/Time; Combined rate = Sum of individual rates
- Time together = (A × B)/(A + B) for two workers
- LCM method: Total work = LCM of days; Rate = Total work/Individual time
- Pipes filling: positive rate; Pipes emptying: negative rate
- Efficiency ∝ 1/Time — more efficient means less time
- Man-days constant: M₁D₁ = M₂D₂ for same work