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**Introduction: Pipes & Cisterns**

Pipes and Cisterns problems are almost the same as those of Time and Work problems. Thus, if a pipe fills a tank in 6 hrs, then the pipe fills 1/6th of the tank in 1 hour. The only difference with Pipes and Cisterns problems is that there are outlets as well as inlets. Thus, there are agents (the outlets) which **perform negative work **too. The rest of the process is almost similar.

**➢ Inlet**

- A pipe connected with a tank (or a cistern or a reservoir) is called an inlet, if it fills the tank. The work done by this is taken as
**P****ositive work**.

**➢ Outlet**

- A pipe connected with a tank is called an outlet if it empties the tank and the work done by this is taken as the
**Negative work.**

**➢ Basic Concept**

- The basic concept of Time and Work is similar to that across all arithmetic topics, i.e. the concept of proportionality.
- Efficiency is inversely proportional to the Time taken when the amount of work done is constant.
- This can be used to compare efficiencies and time taken across different groups in Time-Speed-Distance, efficiency is replaced by Speed, i.e. Speed is inversely proportional to Time when the Distance is constant.
- Pipes and Cisterns are just an application of Time and Work. Concept wise, it is one and the same.
- In the above proportionality, Efficiency is replaced by Rate of filling.
__The equation, in this case, becomes:__

➢ **Introduction: Negative Work**

- Negative work
**increases the time**in which work is to be completed. This application can be extended to cases involving Pipes and cisterns. - Suppose there are two pipes in a Cistern. Pipe A is used to fill the Cistern and Pipe B is used to empty the Cistern. Here we say that Pipe B and Pipe A are working against each other. When a leak is developed in the Cistern, the leak forms the component of negative work, which slows down the completion of the task (in this case, the filling of the Cistern).

**Example 1. ****A Cistern has three pipes A, B, and C. Pipe A can fill a Cistern in 10 hrs, Pipe B can fill a Cistern in 5 hrs while Pipe C can empty the Cistern in 20 hrs. If they are switched on at the same time; in how many hours will the Cistern be filled?****Solution:**

► In one hour, Pipe A can fill 100/10 = 10% of the Cistern.

► In one hour, Pipe B can fill 100/5 = 20 % of the Cistern.

► In one hour, Pipe C can empty 100/20 = 5 % of the Cistern.

► If all three are working together, (10 + 20 – 5) = 25% of the Cistern will get filled in one hour, so it will take **4 hrs **for the Cistern to fill.

**Example 2. ****A tank can be filled by tap A in 6 hrs and by tap B in 3 hrs. But when they are open simultaneously to fill an empty tank, they take 3 hrs more than their normal Time. A hole is later discovered as the reason for the delay. Find the Time taken by the hole to empty the tank if it is completely filled.****Solution:**

► Tap A takes 6 hrs to fill, so in one hour it will fill 16.67 % of the tank.

► Tap B takes 3 hrs to fill, so in one hour it will fill 33.33% of the tank.

► Together they will fill 16.67% + 33.33% = 50 % of the tank.

► They should take 100/50= 2 hrs to fill the tank. But they take 3 hrs more, because of the hole; they totally take 5 hrs to fill, i.e. they fill 20% in an hour.

► This is possible if the hole empties 50-20 = 30% of the tank in an hour.

► So, to completely empty the full tank, the hole will take 100/30 hrs = 3.33 hrs = **3 hrs and 20 mins.**

**Example 3****. Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?****Solution:**

► Part filled by A alone in 1 hour = 1/36

► Part filled by B alone in 1 hour = 1/45

∴ Part filled by (A + B) in 1 hour =

Hence, both the pipes together will fill the tank in **20 hours.**

Question 1:Three pipe P, Q and R can fill a tank In 12 minutes, 113 minutes and 29 minutes respectively. The pipe R is closed 12 minutes before the tank Is filled. In what time the tank is full?

Question 2:Pipe A can fill a tank in ‘an’ hours. On account of a leak at the bottom of the tank, it takes thrice as long to fill the tank. How long will the leak at the bottom of the tank take to empty a full tank, when pipe A is kept closed?

Question 3:Three pipes A, B and C can fill a tank from empty to full in 30 minutes, 20 minutes and 10 minutes respectively. When the tank is empty, all the three pipes are opened. A, B and C discharge chemical solutions P, Q and R respectively. What is the proportion of solution R in the liquid in the tank after 3 minutes?

Question 4:**Pipe A can fill a tank in 36 hours and pipe B can empty it in 45 hours. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?**

**Solution****.**

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