One-Line Definition
Work equals Rate (Efficiency) multiplied by Time (Work = Rate × Time). Efficiency is the amount of work completed per unit time.
Brick Factory Mental Model
Assume a job is laying 36 total bricks:
- • Worker A takes 12 days → Rate = 3 bricks/day
- • Worker B takes 18 days → Rate = 2 bricks/day
- • Working together → 3 + 2 = 5 bricks/day → 36 / 5 = 7.2 days!
Time & Work and Pipes & Cisterns are high-speed scoring topics. Using the LCM method avoids complex fraction addition:
Man-Days Chain Rule:
SHORTCUT RULES & KEY FORMULAS
Filling vs Drain Pipe Net Rate
Filling pipes add volume (+), while drain/leak pipes subtract volume (-).
Alternate Day Work Cycles
Group work into 2-day units and divide total LCM units by 2-day output.
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: A completes a job in 12 days and B in 18 days. How long will they take together?
Step 1: Total Work = LCM(12, 18) = 36 units.
Step 2: A's rate = 3 units/day, B's rate = 2 units/day.
Step 3: Combined rate = 5 units/day → Time = 36 / 5 = 7.2 days.
Q: A can finish a job in 15 days. B is 50% more efficient than A. How many days will B take alone?
Step 1: Efficiency ratio A : B = 100 : 150 = 2 : 3.
Step 2: Time ratio is inverse = 3 : 2.
Step 3: 3 parts = 15 days → 1 part = 5 days → B's time = 2 × 5 = 10 days.
Q: Pipe A fills a tank in 8 hours, while Pipe B empties it in 24 hours. If both operate together, how long to fill the tank?
Step 1: Capacity = LCM(8, 24) = 24 units.
Step 2: Pipe A rate = +3 units/hr, Pipe B rate = -1 unit/hr.
Step 3: Net rate = +3 - 1 = +2 units/hr → Time = 24 / 2 = 12 hours.
Mistake 1
Adding Days Directly
Adding days directly (10 days + 20 days ≠ 30 days). You MUST add work rates per day (1/10 + 1/20 = 3/20 → 6.67 days).
Mistake 2
Adding Outlet Pipe Rates
Adding emptying pipe rates instead of subtracting (-). Outlet pipes drain water and reduce net filling speed.
Mistake 3
Confusing Efficiency & Time
Assuming a worker who is twice as efficient takes twice as much time. Higher efficiency means LESS time taken!
1. A can do a piece of work in 10 days and B in 15 days. How long will they take together?
Answer: LCM = 30 → A=3, B=2 → 30 / 5 = 6 days.
2. 12 men can complete a work in 9 days. How many days will 18 men take to complete the same work?
Answer: M1 × D1 = M2 × D2 → 12 × 9 = 18 × D2 → D2 = 108 / 18 = 6 days.
3. Pipe A fills a tank in 6 hours and Pipe B in 8 hours. If both pipes open together, after how many hours should Pipe B be closed so the tank fills in 4 hours?
Answer: Capacity = 24. A's rate = 4, B's rate = 3. Pipe A works 4 hrs = 16 units. Remaining = 8 units done by B → Time B = 8/3 = 2.67 hours (2h 40m).
FREQUENTLY ASKED QUESTIONS
❓ Why is the LCM method better than using fractions (1/A + 1/B)?
The LCM method converts fractional work rates into simple whole numbers. By assuming Total Work = LCM of individual days, you eliminate fraction addition and avoid calculation errors.
❓ How do you handle alternate day work questions?
Group the work into a 2-day cycle (A's daily work + B's daily work). Divide total LCM work by the 2-day output to find full cycles, then calculate remaining work for the final day.
❓ What is the sign convention for Pipes & Cisterns?
Filling (inlet) pipes add work and get positive rates (+). Emptying (outlet/leak) pipes destroy work and get negative rates (-). Net Rate = (Filling Rate) - (Emptying Rate).