Work as a rate
If A finishes a job in a days, A does 1/a of the job per day. Rates add when people work together.
Example. A takes 10 days and B takes 15 days. Together they do 1/10 + 1/15 = 1/6 of the job per day, so they finish in 6 days.
Formula for two workers: together = ab / (a + b) days.
The LCM (total work) method
Take the total work as the LCM of the days. For A = 12 days and B = 18 days, take 36 units: A does 3 units a day and B does 2.
Example. They work together for 4 days (20 units), then A leaves. The remaining 16 units take B 16 ÷ 2 = 8 days.
This avoids fractions and is faster in exams.
Efficiency
If A is twice as efficient as B, A's rate is twice B's, and A takes half the time.
Example. A is twice as efficient as B and together they take 14 days. With rates 2x and x, 3x = 1/14, so x = 1/42 and B alone takes 42 days.
Man-days
Work = workers × days (× hours, if given). For the same job, M₁D₁ = M₂D₂.
Example. 12 men take 15 days, so the job is 180 man-days. 20 men take 180 ÷ 20 = 9 days.
For men and women, convert to one unit first. If 6 men or 10 women finish in 15 days, one man equals 10/6 women. Then 3 men and 5 women equal 10 women, who take 15 days.
Pipes and cisterns
Filling pipes add to the rate and outlets subtract.
Example. A fills a tank in 20 minutes, B in 30 minutes, and C empties it in 15 minutes. Together: 1/20 + 1/30 − 1/15 = 1/60, so the tank fills in 60 minutes.
Wages
Wages are shared in the ratio of work done, which (for the same time) is the ratio of rates.
Example. A takes 10 days and B 15 days; their rates are 3 : 2, so ₹1,500 is split ₹900 and ₹600.
Shortcuts and tricks
The standard way adds daily fractions of the work. Two shortcuts make most questions one-liners.
Product over sum
Example: A takes 12 days and B takes 24 days. How long together?
Standard way: 1/12 + 1/24 = 3/24 = 1/8 of the job a day, so 8 days.
Shortcut: for two workers, time = (a × b)/(a + b) = 288/36 = 8 days. For a filler and an emptier, use (a × b)/(b − a).
Total work as an LCM
Example: A takes 8 days, B 12 days and C 24 days. How long do all three take?
Standard way: 1/8 + 1/12 + 1/24 = (3 + 2 + 1)/24 = 6/24 = 1/4, so 4 days.
Shortcut: call the job 24 units (the LCM). A does 3 a day, B 2 and C 1: 6 a day, so 24 ÷ 6 = 4 days.
Efficiency ratios
Example: A is three times as fast as B, and together they take 9 days.
Standard way: rates 3x and x give 4x = 1/9, so x = 1/36 and B takes 36 days.
Shortcut: together they are 4 "B-workers", so B alone takes 4 × 9 = 36 days.
Exam tips
Always convert days to rates; never add days.
If one pipe empties, check whether the net rate is positive before answering.
"Works on alternate days" questions are solved by finding work done in a 2-day cycle.