Treat the whole job as 1. If A alone finishes it in a days, A does 1/a of it each day. Rates add when people work together, so A and B together do 1/a + 1/b per day and finish in ab/(a + b) days.
LCM method
Take total work = LCM of the days. A in 12 days and B in 18 days: work = 36 units, A does 3 and B does 2 a day, together 5 a day, so 36/5 = 7.2 days. This avoids fractions.
Standard Results
| Situation | Result |
|---|---|
| A in a days, B in b days, together | ab/(a + b) |
| A, B, C in a, b, c days together | 1 ÷ (1/a + 1/b + 1/c) |
| A + B in x days, A alone in a days; B alone | ax/(a − x) |
| A + B in x, B + C in y, C + A in z days; all three | 2xyz/(xy + yz + zx) |
Men, days and hours
M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂
Total effort (man-hours) is proportional to work. 12 men working 8 hours a day finish a job in 10 days; 16 men working 6 hours a day need 12 × 8 × 10 / (16 × 6) = 10 days.
- Efficiency is the inverse of time: if A is twice as efficient as B, A takes half B's time. If A is 25% more efficient than B, A's time = B's time × 100/125.
- Wages are shared in the ratio of work done. If A and B work the same number of days, that is the ratio of their daily rates (the inverse of their times).
- When someone leaves part-way, add up the work each person actually did and set the total equal to 1 (or to the LCM total).
- Alternate days: find the work done in one 2-day cycle, count full cycles, then finish the remainder starting with whoever works first.
Pipes and Cisterns
A filling pipe works like a worker with a positive rate; an emptying pipe or a leak has a negative rate. Pipe A fills a tank in 6 hours and pipe B empties it in 8 hours: together the net rate is 1/6 − 1/8 = 1/24, so the tank fills in 24 hours.
Leak
Leak empties a full tank in ab/(b − a) hours
a = time to fill without the leak, b = time with the leak. A pipe that fills in 4 hours takes 5 hours because of a leak: the leak alone empties it in 4 × 5 / 1 = 20 hours.
Common Mistakes
Adding days instead of rates (A in 10 and B in 15 days together is 6 days, not 25 or 12.5); forgetting the sign of an emptying pipe; and sharing wages in the ratio of days taken instead of work done.