Time and Work
Overview
The core idea is simple: work done equals the product of efficiency and time. Mastering this topic requires understanding how to handle combined work, alternating work patterns, and the pipes-cisterns variation.
This topic connects directly with concepts of ratios, fractions, and LCM. Questions range from straightforward "A and B working together" problems to complex scenarios involving people leaving midway or pipes filling and leaking simultaneously. The good news: once you internalize the efficiency method, even tricky problems become mechanical calculations.
Students who score well here treat work as a fixed quantity (usually the LCM of given days) rather than as "1 unit." This approach eliminates fractions early and makes arithmetic faster—critical in time-bound exams.
Key Concepts
- Work is inversely proportional to time: If A is twice as fast as B, A takes half the time B takes for the same work.
- Efficiency = Total Work ÷ Time: When A completes work in 10 days, A's efficiency is W/10 per day.
- LCM Method: Assume total work = LCM of all given times. This gives integer efficiencies, avoiding fractions entirely.
- Combined efficiency: When A and B work together, add their individual efficiencies. Time = Total Work ÷ (Efficiency of A + Efficiency of B).
- Negative efficiency for pipes: An outlet pipe or leak has negative efficiency—it subtracts from the filling rate.
- Man-days concept: Total work = Number of workers × Number of days. Useful for problems where workforce changes.
- Fractional work: If someone leaves after completing part of the work, calculate remaining work and reassign.
Formulas / Key Facts
Basic formulas:
- If A completes work in 'a' days → A's 1 day work = 1/a
- If A and B together complete in 't' days → 1/a + 1/b = 1/t
- Time for A and B together = (a × b) / (a + b)
Efficiency method:
- Total Work = LCM(days of all workers)
- Efficiency of A = Total Work / Days taken by A
- Combined time = Total Work / Sum of efficiencies
Pipes and Cisterns:
- Inlet pipe fills in 'a' hours → Rate = +1/a (positive)
- Outlet pipe empties in 'b' hours → Rate = −1/b (negative)
- Net rate when both work = 1/a − 1/b
- Time to fill = 1 / (Net rate)
Man-days formula:
- M₁ × D₁ = M₂ × D₂ (for same work)
- M₁ × D₁ × H₁ = M₂ × D₂ × H₂ (when hours differ)
Alternating work:
- Find work done in one complete cycle (e.g., 2 days if A and B alternate daily)
- Calculate number of complete cycles, then handle remaining work
Worked Examples
Example 1: Basic combined work
A can complete a job in 12 days, B can complete it in 18 days. How many days will they take working together?
Solution:
- Total work = LCM(12, 18) = 36 units
- A's efficiency = 36/12 = 3 units/day
- B's efficiency = 36/18 = 2 units/day
- Combined efficiency = 3 + 2 = 5 units/day
- Time together = 36/5 = 7.2 days = 7 days 4 hours 48 minutes
Alternatively: (12 × 18)/(12 + 18) = 216/30 = 7.2 days
Example 2: Worker leaves midway
A and B can complete a work in 12 days. A alone can do it in 20 days. They start together but A leaves after 4 days. How many more days will B take to finish?
Solution:
- Total work = LCM(12, 20) = 60 units
- Combined efficiency = 60/12 = 5 units/day
- A's efficiency = 60/20 = 3 units/day
- B's efficiency = 5 − 3 = 2 units/day
- Work done in 4 days (together) = 5 × 4 = 20 units
- Remaining work = 60 − 20 = 40 units
- Time for B alone = 40/2 = 20 more days
Example 3: Pipes and Cisterns
Pipe A fills a tank in 6 hours, Pipe B fills it in 8 hours, and Pipe C empties it in 12 hours. If all three are opened together, how long to fill the tank?
Solution:
- Total capacity = LCM(6, 8, 12) = 24 units
- A's rate = +24/6 = +4 units/hour
- B's rate = +24/8 = +3 units/hour
- C's rate = −24/12 = −2 units/hour (negative because it empties)
- Net rate = 4 + 3 − 2 = 5 units/hour
- Time to fill = 24/5 = 4.8 hours = 4 hours 48 minutes
Common Mistakes
- Adding times instead of efficiencies: Students write "12 + 18 = 30 days together." Wrong! You must add rates (1/12 + 1/18), not times. Fix: Always convert to efficiency first.
- Forgetting negative sign for outlets: Treating an emptying pipe as positive fills nothing correctly. Fix: Outlets and leaks always subtract from net rate.
- Misapplying man-days to efficiency problems: Man-days works only when all workers have equal efficiency. Fix: Use LCM method when workers have different rates.
- Ignoring fractional cycles in alternating work: If A and B alternate and work isn't completed in full cycles, the last partial day matters. Fix: After finding complete cycles, check who works in the remaining portion.
- Confusing "days more" vs "total days": Questions often ask "how many more days" after someone leaves. Fix: Read carefully—subtract already-worked days if asking for additional time only.
Quick Reference
- Total Work = LCM of all given times → gives integer efficiencies
- Combined time = (a × b)/(a + b) for two workers with times a and b
- Outlet/leak → negative efficiency; always subtract
- M₁D₁ = M₂D₂ → only when all workers have equal efficiency
- Remaining work = Total work − Work already done
- Net rate = Sum of inlet rates − Sum of outlet rates