Time and Work
Core Concepts
1. Basic Concept
If A can do work in n days, A's 1-day work = 1/n.
If A's 1-day work = 1/n, A completes work in n days.
2. Combined Work
If A does 1/a and B does 1/b per day:
Together: 1/a + 1/b per day.
Together time = ab/(a+b) days.
3. LCM Method (Efficient)
Take LCM of all given days as total work units.
Calculate per-day work for each person.
Work together per day = sum of individual work rates.
Days = Total work / combined rate.
4. Pipes and Cisterns
Inlet pipe: fills tank (positive work).
Outlet pipe: drains tank (negative work).
If inlet fills in a hours: rate = 1/a per hour.
If outlet drains in b hours: rate = -1/b per hour.
Combined rate = 1/a - 1/b per hour.
5. Work and Wages
Wages distributed in ratio of work done.
If A does work in m days and B in n days, ratio of work = n:m (inverse!).
Practice Problems
Q1: A takes 12 days, B takes 18 days. Together?
Together = 12x18/(12+18) = 216/30 = 7.2 days.
LCM method: LCM(12,18) = 36. A does 3 units/day, B does 2 units/day. Together: 5 units/day. Time = 36/5 = 7.2 days.
Q2: A and B together complete in 8 days. A alone in 12 days. B alone?
B's rate = 1/8 - 1/12 = 3/24 - 2/24 = 1/24. B alone = 24 days.
Q3: Pipe A fills in 6 hours, Pipe B empties in 10 hours. Both open, time to fill?
Net rate = 1/6 - 1/10 = 5/30 - 3/30 = 2/30 = 1/15 per hour. Time = 15 hours.
Previous Year Questions
SBI PO 2023: A can do a work in 15 days, B in 20 days. They start together, but B leaves after 5 days. How many more days for A alone?
Together 5 days: 5(1/15+1/20) = 5(7/60) = 35/60 = 7/12 of work done.
Remaining = 5/12. A alone: (5/12)/(1/15) = (5/12)x15 = 75/12 = 6.25 days.
IBPS 2022: Wages for A, B, C for 6, 8, 12 days work respectively is Rs 3300. Find A's share.
Work ratio = 1/6 : 1/8 : 1/12 = 4:3:2 (multiply by LCM=24).
A's share = 4/(4+3+2) x 3300 = 4/9 x 3300 = Rs 1466.67.