Time, Work and Pipes/Cisterns
What you must know
The central idea is rate: if A finishes a job in n days, A's one-day work is 1/n. Combined rates add: together they take 1/(1/a + 1/b) days. The standard identity is Work = Rate × Time, and total work is often conveniently set to the LCM of the given days to avoid fractions.
Pipes and cisterns use the same algebra with a sign convention: an inlet pipe adds a positive rate, an outlet (leak) pipe subtracts. If a cistern fills in 6 hours and a leak empties it in 12, the net rate is 1/6 − 1/12 = 1/12, so filling takes 12 hours. Efficiency problems are ratio problems in disguise: if A is twice as efficient as B, A takes half the time.
Traps
- Adding times instead of rates ("A takes 6 days, B takes 3, so together 9 days" is wrong).
- Leaving work: if A works for some days and then leaves, track the work done by each person separately.
- Wages are shared in proportion to work done, i.e., rate × time, not to time alone.
- Leaks: check whether the net rate is negative (the cistern never fills).
- Mornings-only or alternate-day work needs careful day-by-day accounting.
Rule of thumb and example
Set total work = LCM of the individual times. If A takes 12 days and B takes 18, take work = 36 units. A does 3 units/day, B does 2; together 5 units/day, so 36/5 = 7.2 days. This avoids fraction arithmetic entirely.