Pipes & Cisterns • Topic 2 of 3

Combined Filling

When several inlets work together, add their rates; the tank fills in capacity ÷ (sum of rates). For two pipes filling in a and b hours, the together-time is the product over the sum: ab/(a+b) — the same harmonic shortcut as Time & Work. The LCM method scales to any number of pipes and is much safer than juggling fractions. CAT loves the staggered version: a pipe runs alone for a while, then a second joins, or one is shut partway. Handle it as units of water — compute how much each pipe pours in its open window and total to one tank (the capacity). Another classic asks how long the second pipe alone would take, given the combined time and the first pipe’s time: rate(B) = combined rate − rate(A), then invert. Keeping everything in integer units of the LCM tank makes these multi-pipe stories almost arithmetic-free.

✅ Solved examples

1. Pipes A and B fill a tank in 12 h and 18 h. Together, how long?
Shortcut ab/(a+b) = (12×18)/(12+18) = 216/30 = 7.2 hours. (LCM check: cap 36, +3 and +2, net 5, 36/5 = 7.2.)
2. A, B, C fill in 10, 12, 15 hours. All open — fill time?
Capacity = LCM(10,12,15) = 60. Rates +6, +5, +4 = +15/h. Time = 60/15 = 4 hours.
3. A fills in 20 h, B in 30 h. A runs alone 5 h, then B joins. Total time to fill?
Capacity = 60. A = +3, B = +2. A alone 5 h pours 15 units; 45 left at +5/h = 9 h. Total = 5 + 9 = 14 hours.
4. Two pipes together fill a tank in 6 h. Alone, the first takes 10 h. How long for the second alone?
Capacity = LCM(6,10) = 30. Combined +5, first +3, so second = 5 − 3 = +2/h ⇒ 30/2 = 15 hours.

✏️ Practice — try these, take hints as needed

1. Pipes fill in 15 h and 10 h. Together, fill time?
ab/(a+b).
(15×10)/25.
150/25.
6 hours
2. A, B, C fill in 6, 8, 12 hours. All open — time?
Capacity = LCM = 24.
Rates +4, +3, +2 = +9.
24/9.
8/3 hours (2 h 40 min)
3. A fills in 16 h, B in 24 h. B opens 4 h after A. Total fill time from when A started?
Capacity = 48. A +3, B +2.
A alone 4 h = 12 units; 36 left.
36 at +5/h = 7.2 h, add 4.
11.2 hours
4. Together two pipes fill in 4 h; the slower alone takes 12 h. Faster pipe alone?
Capacity = 12. Combined +3.
Slower +1.
Faster = 3 − 1 = +2.
6 hours
5. Three identical pipes each fill in 9 h. All open — fill time?
Rates add: 3 × (1/9).
= 3/9 = 1/3 per hour.
Invert.
3 hours

📝 Topic test — 8 questions

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