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Time and work for AFCAT: the LCM method, efficiency and men–days questions

Work questions turn into simple fractions or, faster, into total units of work. The fraction and LCM methods, efficiency, three workers, someone leaving, alternate days, men–days–hours and wages, with worked examples and a practice set.

30 Sept 2026 7 min read

In this guide
  1. Method 1: fractions
  2. Method 2: the LCM (total units) method
  3. Efficiency
  4. Men, days, hours and work
  5. Wages
  6. Worked examples
  7. Pipes and cisterns in one paragraph
  8. Common mistakes
  9. Practice set
  10. What to do next

Time and work questions look like word puzzles, but they all rest on one idea: a worker who finishes a job in n days does 1/n of it each day. Rates add when people work together. Once you see that, every question is a small sum of fractions.

The fraction method is correct but slow when the denominators are awkward. The LCM method, which treats the job as a fixed number of units, turns the fractions into whole numbers and is usually the faster choice in AFCAT, where you have about 72 seconds per question. This guide teaches both and then shows which one to use for each question type.

Method 1: fractions

If A finishes a job in 10 days, A does 1/10 of it per day. If B takes 15 days, B does 1/15. Together they do

1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 of the job per day

so together they need 6 days. The time together is always the reciprocal of the combined daily rate.

Method 2: the LCM (total units) method

Take the total work as the LCM of the individual times. For 10 and 15 days, call the job 30 units.

  • A does 30 ÷ 10 = 3 units a day.
  • B does 30 ÷ 15 = 2 units a day.
  • Together: 5 units a day, so 30 ÷ 5 = 6 days.

Why it works: the size of "one job" is arbitrary. Choosing it as the LCM just makes every daily rate a whole number. The answer does not change, because every rate scales by the same factor.

Efficiency

Efficiency is work done per day, so it is inversely proportional to time.

  • If A is twice as efficient as B, A takes half as long.
  • If the efficiency ratio of A to B is 3 : 2, their time ratio is 2 : 3.

Questions that say "A is 50% more efficient than B" are ratio questions. A : B efficiency = 3 : 2. Give A 3 units a day and B 2 units a day, and carry on with the LCM method.

Men, days, hours and work

When several people work at the same rate, the work done is proportional to the number of people × days × hours per day. So for two situations:

(M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂

where M is people, D days, H hours per day and W the amount of work. Leave out any factor the question does not mention.

Wages

Wages are shared in the ratio of work done, not time spent. If A and B work together for the whole job, the work done is in the ratio of their efficiencies.

Question typeFastest approach
Two or three workers togetherLCM method
One worker's time from a pairSubtract rates
"Twice as efficient"Assign units in the efficiency ratio
Someone leaves or joins midwayWork done so far, then remaining work
Alternate daysUnits in one 2-day cycle
Different teams, hours or job sizesM × D × H ÷ W formula
Filling and emptying tanksSame as work, with emptying as negative

Worked examples

Example 1 (one worker from a pair). A and B together finish a job in 12 days. A alone takes 20 days. How long does B take alone?

  • LCM of 12 and 20 = 60 units. A + B = 5 units a day, A = 3.
  • B = 2 units a day, so B takes 60 ÷ 2 = 30 days.

Example 2 (three pairs). A and B can do a job in 12 days, B and C in 15 days, and C and A in 20 days. How long will all three take together, and how long will each take alone?

  • LCM of 12, 15 and 20 = 60 units.
  • A + B = 5, B + C = 4, C + A = 3 units a day. Adding all three: 2(A + B + C) = 12, so A + B + C = 6.
  • Together: 60 ÷ 6 = 10 days.
  • C = 6 − 5 = 1 (60 days alone), A = 6 − 4 = 2 (30 days), B = 6 − 3 = 3 (20 days).

Example 3 (efficiency). A is twice as efficient as B. Together they finish a job in 14 days. How long would A take alone?

  • Give A 2 units a day and B 1. Together 3 units a day for 14 days = 42 units.
  • A alone: 42 ÷ 2 = 21 days.

Example 4 (someone leaves). A can finish a job in 8 days and B in 12 days. They work together for 3 days, then A leaves. How long does B take to finish?

  • LCM 24 units. A = 3, B = 2 units a day.
  • In 3 days together: 3 × 5 = 15 units. Remaining: 9 units.
  • B alone: 9 ÷ 2 = 4.5 days.

Example 5 (men, days, hours). 12 people working 8 hours a day finish a job in 10 days. How many days will 16 people take, working 6 hours a day?

  • 12 × 8 × 10 = 16 × 6 × D.
  • 960 = 96D, so D = 10 days.

Example 6 (wages). A can do a job in 10 days and B in 15 days. They do it together and are paid ₹5,000. How should the money be shared?

  • Efficiency ratio A : B = 3 : 2 (from the LCM method, 30 units).
  • A gets 3/5 × 5,000 = ₹3,000; B gets ₹2,000.

Pipes and cisterns in one paragraph

A pipe that fills a tank in 6 hours adds 1/6 of the tank per hour. A pipe that empties it in 9 hours removes 1/9. Both open: 1/6 − 1/9 = 1/18, so the tank fills in 18 hours. By LCM: 18 units, +3 and −2 per hour, net 1, so 18 hours. Our full guide to pipes and cisterns covers leaks and pipes opened at different times.

Common mistakes

  • Adding days instead of rates. A takes 10 days and B takes 15, so together they take 6 days, not 25 or 12.5.
  • Forgetting the negative sign for an emptying pipe or a worker who "undoes" work.
  • Mixing up efficiency and time. Twice as efficient means half the time.
  • Stopping at the wrong step when someone leaves: the question usually asks for the total time, or only the remaining time. Check which.

Practice set

  1. A takes 12 days and B takes 24 days. How long together?
  2. 8 people finish a job in 15 days. How long will 10 people take?
  3. Two pipes fill a tank in 4 hours and 6 hours. How long together?
  4. A, B and C take 20, 30 and 60 days alone. How long together?
  5. A is three times as efficient as B and takes 40 days less than B. How long will they take together?
  6. A can do a job in 18 days and B in 24 days. A works for 6 days and leaves. How long does B take to finish the rest?
  7. A takes 6 days and B takes 12 days. They work on alternate days, starting with A. In how many days is the job done?
  8. 15 people build a wall 60 m long in 12 days. How many days will 20 people take to build 80 m of the same wall?

Answers:

  1. 8 days. 1/12 + 1/24 = 3/24 = 1/8.
  2. 12 days. 8 × 15 = 120 person-days, and 120 ÷ 10.
  3. 2.4 hours (2 hours 24 minutes). 1/4 + 1/6 = 5/12, so 12/5 hours.
  4. 10 days. LCM 60: 3 + 2 + 1 = 6 units a day.
  5. 15 days. A takes x days and B 3x, so 2x = 40 and x = 20. B takes 60. Together 1/20 + 1/60 = 4/60 = 1/15.
  6. 16 days. A does 6/18 = 1/3. B does the other 2/3 in 2/3 × 24.
  7. 8 days. LCM 12 units: A does 2 a day, B does 1. Each 2-day cycle gives 3 units, so 4 cycles = 8 days finish all 12 units exactly.
  8. 12 days. 15 × 12 ÷ 60 = 20 × D ÷ 80, so 3 = D/4.

What to do next

  • Solve every worked example again by the method you did not use the first time.
  • Practise ten "someone leaves" and ten alternate-day questions, where most time is lost.
  • Move on to pipes and cisterns, then speed, time and distance, which uses the same rate × time idea.

A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Indian Air Force website .

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