Pipes and cisterns for RRB NTPC: methods, leaks and practice

If you have worked through time and work, you already know most of this chapter. A pipe that fills a tank is a worker; the tank is the job. The only new idea is that some pipes work against you. An outlet or a leak removes water, so its rate is subtracted instead of added.

That one sign change is where most wrong answers come from. Get it right and pipes and cisterns questions are among the quicker ones in the Maths section.

Rates with a sign

• A pipe that fills a tank in n hours fills 1/n of the tank each hour.
• A pipe or leak that empties a full tank in m hours empties 1/m each hour.
• Net rate = sum of filling rates − sum of emptying rates.
• Time to fill = 1 ÷ net rate.

Why this works: in any one hour, every open pipe acts at the same time, so the changes in water level simply add up, with outlets counting as negative. Times cannot be added, but hourly rates can.

If the net rate is negative, the tank empties instead of filling. If it is zero, the level never changes. Check the sign before you divide.

The LCM (capacity) method

1. Take the capacity of the tank as the LCM of all the times. Call the result "units".
2. Each pipe's units per hour = capacity ÷ its time. Give outlets a minus sign.
3. Add the signed rates, then divide the capacity by the net rate.

For A = 10 hours and B = 15 hours: capacity = 30 units. A fills 3 an hour, B fills 2, together 5. Time = 30 ÷ 5 = 6 hours.

Why it works: the actual size of the tank does not matter when only times are given, so you are free to pick a size that every time divides evenly. That keeps every rate a whole number.

The main question types

Worked questions at NTPC level

Q1. Pipes A and B fill a tank in 12 and 15 hours. Pipe C empties it in 20 hours. If all three are open, how long will the tank take to fill?

Capacity = LCM of 12, 15 and 20 = 60 units. A: +5, B: +4, C: −3 per hour.
Net = 6 per hour. Time = 60 ÷ 6 = 10 hours.

Q2. A tap normally fills a tank in 4 hours. Because of a leak, it takes 5 hours. How long would the leak take to empty a full tank?

Capacity = 20 units. Tap alone: 5 per hour. Tap with leak: 4 per hour. So the leak removes 1 per hour.
Leak alone = 20 ÷ 1 = 20 hours.

Q3. Pipes A and B fill a tank in 20 minutes and 30 minutes. Both are opened together, and after 6 minutes A is turned off. How much longer will B take to fill the tank?

Capacity = 60 units. A: 3 per minute, B: 2 per minute.
In 6 minutes together: 6 × 5 = 30 units. Remaining: 30 units.
B alone: 30 ÷ 2 = 15 more minutes (21 minutes in all).

Q4. Pipe A fills a tank in 6 hours and pipe B in 8 hours. They are opened alternately for one hour each, starting with A. How long will it take to fill the tank?

Capacity = 24 units. A: 4 per hour, B: 3 per hour. Each two-hour cycle fills 7 units.
After 3 cycles (6 hours), 21 units are filled and 3 are left.
Hour 7 is A's turn. A fills 3 units in 3/4 of an hour. Total = 6 3/4 hours (6 hours 45 minutes).

Q5. A leak can empty a full tank in 8 hours. An inlet pipe that fills 6 litres a minute is opened when the tank is full, and now the tank empties in 12 hours. Find the capacity of the tank.

Inlet rate = 6 × 60 = 360 litres an hour. Let the capacity be C litres.
Leak rate − inlet rate = net emptying rate: C/8 − 360 = C/12.
So C/8 − C/12 = 360, which gives C/24 = 360, and C = 8,640 litres.
Check: leak 1,080 an hour, inlet 360, net 720 an hour out; 8,640 ÷ 720 = 12 hours.

Q6. Pipes A and B together fill a tank in 12 hours. A alone takes 20 hours. How long does B take alone?

Capacity = 60 units. Together: 5 per hour. A: 3 per hour. So B: 2 per hour.
B alone = 60 ÷ 2 = 30 hours.

Common mistakes

• Adding an outlet's rate instead of subtracting it.
• Adding times instead of rates. Two pipes of 10 and 15 hours do not take 25 hours, or 12.5.
• Not checking the sign of the net rate. If outlets are faster, the tank never fills.
• Forgetting the work already done when a pipe is turned off partway.
• Mixing minutes and hours, especially when an inlet's rate is in litres per minute and the times are in hours.

Practice set

1. Pipe A fills a tank in 8 hours and B in 24 hours. How long will they take together?
2. A pipe fills a tank in 5 hours; a leak empties it in 10 hours. How long will the tank take to fill with both working?
3. A and B fill a tank in 20 and 30 hours; C empties it in 60 hours. How long will it take with all three open?
4. A and B together fill a tank in 10 hours. A alone takes 15 hours. How long does B take alone?
5. A tap fills a tank in 6 hours, but a leak makes it take 8 hours. How long would the leak take to empty a full tank?
6. Pipes A and B fill a tank in 12 and 18 minutes. Both are opened, and B is turned off after 3 minutes. How many more minutes will A take to fill the tank?
7. A full tank has an outlet that empties it in 10 hours and an inlet that fills it in 15 hours. If both are opened, how long will the full tank take to empty?
8. Two pipes fill a tank in 20 and 30 minutes. An outlet removes 3 litres a minute. With all three open, the tank fills in 24 minutes. Find its capacity.

Answers:

1. 6 hours. Capacity 24: A 3, B 1 per hour; 24 ÷ 4.
2. 10 hours. Capacity 10: +2 and −1 per hour; 10 ÷ 1.
3. 15 hours. Capacity 60: 3 + 2 − 1 = 4 per hour; 60 ÷ 4.
4. 30 hours. Capacity 30: together 3, A 2, so B 1 per hour.
5. 24 hours. Capacity 24: tap 4 per hour, with leak 3, so the leak takes 1 per hour.
6. 7 minutes. Capacity 36: A 3, B 2 per minute; 3 minutes × 5 = 15 units; the remaining 21 at 3 per minute.
7. 30 hours. Capacity 30: outlet −3, inlet +2 per hour; net −1 per hour.
8. 72 litres. C/20 + C/30 − 3 = C/24. With LCM 120: 6C + 4C − 5C = 360, so 5C = 360. Check: 3.6 + 2.4 − 3 = 3 litres a minute; 72 ÷ 3 = 24.

What to do next

• Solve 15 pipes and cisterns questions with the LCM method, writing a + or − beside every rate before you add.
• Practise five leak questions and five "closed partway" questions until each takes under a minute.
• If rates feel shaky, revise time and work, then move on to speed, time and distance.