1. Time and Work, Pipes and Cisterns
Time and work questions ask how long people or machines take to finish a job, alone or together. Pipes and cisterns are the same idea: a filling pipe does positive work and an emptying pipe does negative work. The easiest method is the LCM (total work) method, which avoids fractions.
The LCM method
- Step 1: Take total work = LCM of the days given. Step 2: Find each person's work per day. Step 3: Divide total work by combined work per day.
- Worked example: A can finish a job in 12 days and B in 18 days. LCM of 12 and 18 = 36 units. A does 36 ÷ 12 = 3 units/day; B does 36 ÷ 18 = 2 units/day. Together = 5 units/day. Time = 36 ÷ 5 = 7.2 days.
- Shortcut for two people: together time = (a × b) ÷ (a + b). Example: 15 and 10 days → 150 ÷ 25 = 6 days.
- If A and B together take t days and A alone takes a days, then B alone takes (a × t) ÷ (a − t). Example: together 8, A 12 → 96 ÷ 4 = 24 days.
- Worked example (one person leaves): A takes 20 days, B takes 30 days. They work together for 6 days, then A leaves. LCM = 60 units; A = 3, B = 2 units/day. In 6 days they do 6 × 5 = 30 units. Remaining 30 units by B alone = 30 ÷ 2 = 15 days.
Men, days and hours
- Formula: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where M = men, D = days, H = hours per day, W = work.
- Worked example: 12 men can finish a work in 20 days. How many days will 15 men take? 12 × 20 = 15 × D₂ → D₂ = 240 ÷ 15 = 16 days.
- More men means fewer days (inverse relation). More work means more days (direct relation).
- Trap: if the hours per day change, include H in the formula. Do not ignore it.
Pipes and cisterns
- A filling pipe adds water (positive work). An emptying pipe or a leak removes water (negative work).
- Worked example: Pipe A fills a tank in 10 hours. Pipe B empties it in 15 hours. Both are opened. LCM = 30 units. A = +3 units/hour, B = −2 units/hour. Net = +1 unit/hour. Time = 30 hours.
- Shortcut: fill time with one filler (a hours) and one emptier (b hours) = (a × b) ÷ (b − a), when b is more than a.
- Trap: if the emptying pipe is faster than the filling pipe, a full tank will be emptied, and an empty tank will never fill.
Quick facts
- Total work
- LCM of the days
- Two people together
- ab/(a + b) days
- B alone (given A and together)
- at/(a − t)
- Man-days
- M₁D₁H₁/W₁ = M₂D₂H₂/W₂
- Filler a, emptier b
- ab/(b − a) hours
- Efficiency
- Work per day = Total work ÷ Days
💡 Always use the LCM method in the exam. Working with whole units is faster and safer than adding fractions like 1/12 + 1/18.
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