Time & Work
Every time-and-work question becomes simple once you stop thinking in days and start thinking in work-per-day. Convert each worker to a rate, add the rates, and the problem is arithmetic. Pipes and cisterns is the same topic with one rate turned negative.
Updated 19 September 2026 · Reliably 2-3 questions, and pipes-and-cisterns is a near-certainty.
How to think about time & work
Work as a rate, not a duration
If A finishes a job in 10 days, A completes 1/10 of it per day. Rates add; days do not. Two people who each take 10 days finish in 5 days together, because their rates 1/10 + 1/10 = 1/5. Trying to average the days instead of the rates is the classic error.
The LCM shortcut
Instead of fractions, set the total work equal to the LCM of the given days. If A takes 12 days and B takes 18, let total work = 36 units. Then A does 3 units/day and B does 2 units/day. All the arithmetic becomes integers, which is much faster under time pressure.
Efficiency is inversely proportional to time
If A is twice as efficient as B, A takes half the time. So an efficiency ratio of 2:1 means a time ratio of 1:2. Questions phrased in terms of "twice as good a workman" are asking you to make this flip.
Pipes and cisterns is the same topic
A filling pipe has a positive rate, an emptying pipe a negative one. Add them as signed rates. If the sum is negative the tank never fills — which is occasionally the intended answer.
Formulas and shortcuts
- One day work
If A finishes in n days, A does 1/n per day - Two working together
Time = (a × b) / (a + b)Where a and b are the individual times.
- Finding the second worker
1/b = 1/together − 1/a - Work equivalence
(M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂M = men, D = days, H = hours per day, W = amount of work.
- Fill and drain together
Net rate = 1/fill − 1/empty
Solved examples
Q1. A can do a job in 10 days and B in 15 days. How long do they take together?
- A does 1/10 per day, B does 1/15 per day.
- Together = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 per day
- If they do 1/6 per day, the whole job takes 6 days.
Answer: 6 days
Or straight from the formula: (10 × 15) / (10 + 15) = 150 / 25 = 6.
Q2. A and B together finish a job in 12 days. A alone takes 20 days. How long would B take alone?
- Together rate = 1/12. A rate = 1/20.
- B rate = 1/12 − 1/20
- LCM of 12 and 20 is 60: = 5/60 − 3/60 = 2/60 = 1/30
- B does 1/30 per day.
Answer: B alone takes 30 days
Q3. Pipe A fills a tank in 6 hours. Pipe B empties it in 8 hours. If both are opened together on an empty tank, how long until it is full?
- A fills at +1/6 per hour. B empties at −1/8 per hour.
- Net rate = 1/6 − 1/8
- LCM of 6 and 8 is 24: = 4/24 − 3/24 = 1/24 per hour
- The net rate is positive, so it does fill.
Answer: 24 hours
If B had been faster than A the net rate would be negative and the tank would never fill — always check the sign before answering.
Practice questions with answers
1. A can do a piece of work in 12 days and B in 24 days. Working together they finish in:
Answer: B. 8 days — (12 × 24) / (12 + 24) = 288 / 36 = 8 days.
2. A is twice as efficient as B. Together they finish a job in 8 days. A alone would take:
Answer: B. 12 days — Efficiency ratio 2:1 means A does 2 parts and B 1 part, 3 parts total per day. A alone needs 3/2 of the joint time = 8 × 1.5 = 12 days.
3. If 10 men complete a job in 15 days, how long will 15 men take (same rate)?
Answer: B. 10 days — M₁D₁ = M₂D₂ → 10 × 15 = 15 × D → D = 10 days.
4. Two pipes fill a tank in 10 hours and 15 hours respectively. Together they fill it in:
Answer: B. 6 hours — (10 × 15) / (10 + 15) = 150 / 25 = 6 hours.
5. A takes 20 days for a job. He works 5 days and leaves. B finishes the rest in 9 days. B alone would take:
Answer: B. 12 days — A completed 5/20 = 1/4. Remaining = 3/4, which B does in 9 days. So the full job takes B 9 × 4/3 = 12 days.
6. A does a job in 6 days, B in 12 days. They work together for 2 days. The fraction of work left is:
Answer: B. 1/2 — Joint rate = 1/6 + 1/12 = 3/12 = 1/4 per day. In 2 days they do 1/2. Left = 1/2.
7. A tank is filled by a pipe in 5 hours but a leak empties it in 20 hours. With both active, filling takes:
Answer: A. 6 hours 40 min — Net = 1/5 − 1/20 = 4/20 − 1/20 = 3/20 per hour → 20/3 hours = 6 hours 40 minutes.
8. If 6 workers build a wall in 8 days working 6 hours a day, how many days will 4 workers take at 8 hours a day?
Answer: C. 9 days — M₁D₁H₁ = M₂D₂H₂ → 6 × 8 × 6 = 4 × D × 8 → 288 = 32D → D = 9 days.
Where marks get lost
- Averaging the days instead of adding the rates.
- Forgetting that an emptying pipe carries a negative rate.
- Flipping an efficiency ratio the wrong way (2:1 efficiency = 1:2 time).
- Working in fractions when the LCM method would keep everything in integers.
Reading is not practising.
Run a timed set on time & work and see which step you actually lose time on. Free, no card needed.
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