Time and Work
Overview
The core concept is simple: work is measured by how much a person or machine completes per unit time (efficiency), and total work equals efficiency multiplied by time.
This topic connects directly with Pipes and Cisterns (filling/emptying tanks) and often combines with Ratio and Proportion. Mastering the efficiency method—where you assume total work as the LCM of given days—drastically speeds up calculations. Most Clerk-level questions are straightforward applications of combined work or alternating work patterns.
Students who build strong mental math around LCM and fraction addition will find these questions among the easiest marks in the Numerical Ability section.
Key Concepts
- Efficiency = Work done per day. If A completes a job in 10 days, A's efficiency = 1/10 of the work per day.
- Total Work = Efficiency × Time. When using the LCM method, total work becomes a concrete number (like 60 units), making calculations cleaner.
- Combined Work: When A and B work together, their efficiencies add up. If A does 1/10 and B does 1/15 per day, together they do 1/10 + 1/15 = 1/6 per day.
- LCM Method: Assume total work = LCM of all given days. This converts fractions to whole-number efficiencies, eliminating fraction arithmetic.
- Negative Efficiency: When someone destroys work or a pipe empties a tank, treat their efficiency as negative.
- Work Equivalence: Man-days (or person-days) measure total work. 10 men working 5 days = 50 man-days of work.
- Alternating Work: When A and B work on alternate days, calculate work done in a 2-day cycle and find how many complete cycles fit.
Formulas / Key Facts
Basic Formula
- Time = Total Work / Efficiency
- Efficiency = Total Work / Time
- Work = Efficiency × Time
Combined Work
- If A takes 'a' days and B takes 'b' days alone:
- Together: Time = (a × b) / (a + b) days
LCM Method Setup
- Total Work = LCM(days of A, days of B, ...)
- Efficiency of A = Total Work / Days of A
Man-Days Formula
- M₁ × D₁ × H₁ = M₂ × D₂ × H₂ (for same work)
- M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂ (for different work amounts)
Wages Distribution
- Wages are distributed in the ratio of work done (i.e., ratio of efficiencies × days worked)
Pipes and Cisterns
- Filling pipe = positive efficiency
- Emptying pipe/leak = negative efficiency
- Net efficiency = Sum of all efficiencies (with signs)
Worked Examples
Example 1: Basic Combined Work
A can do a work in 12 days, B can do it in 18 days. In how many days will they finish working together?
LCM Method:
- 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 = 36/5 = 7.2 days = 7 days 4 hours (or 36/5 days)
Formula Method:
- Time = (12 × 18)/(12 + 18) = 216/30 = 7.2 days ✓
Example 2: Work Left After Partial Contribution
A can complete a work in 20 days. He works for 8 days and leaves. B finishes the remaining work in 9 days. How long would B take alone?
- Work done by A in 8 days = 8/20 = 2/5
- Remaining work = 1 - 2/5 = 3/5
- B completes 3/5 work in 9 days
- B's rate = (3/5)/9 = 3/45 = 1/15 per day
- B alone takes 15 days
Example 3: Alternating Work
A can do a work in 10 days, B in 15 days. They work on alternate days starting with A. In how many days will the work be completed?
- Total Work = LCM(10, 15) = 30 units
- A's efficiency = 3 units/day
- B's efficiency = 2 units/day
- Work in 2-day cycle (A then B) = 3 + 2 = 5 units
- Complete cycles needed = 30/5 = 6 cycles = 12 days
Since 6 complete cycles finish exactly 30 units, work completes in 12 days.
Note: If work didn't divide evenly, check who works on the final day and calculate partial day.
Common Mistakes
Wrong: Adding times instead of efficiencies → Correct: If A takes 10 days and B takes 15 days, together they DON'T take 25 days or 12.5 days. Add rates: 1/10 + 1/15 = 5/30 = 1/6, so 6 days together.
Wrong: Forgetting to subtract when someone leaves or destroys work → Correct: If A works for some days and leaves, first calculate work done, then find remaining work. For pipes/leaks, subtract the leak's rate.
Wrong: Using wrong LCM leading to calculation errors → Correct: Double-check LCM before proceeding. LCM(12, 18) = 36, not 72 or 24.
Wrong: In alternating work, assuming work divides exactly into cycles → Correct: Always verify. If 28 units remain after 5 cycles of 5 units each, the 6th cycle may end mid-way.
Wrong: Distributing wages by time worked instead of work done → Correct: Wages ∝ (Efficiency × Days worked). If A is twice as efficient and works same days, A gets double the share.
Quick Reference
- LCM method: Total work = LCM of days; Efficiency = Total work ÷ Days
- Together formula: Time = (a × b)/(a + b) days
- Man-days constant: M₁D₁ = M₂D₂ for same work
- Efficiency adds when working together, subtracts for leaks/destroyers
- Wages ratio = Work done ratio = Efficiency × Time ratio
- Alternating work: Find 2-day cycle work, then count complete cycles + remainder