Pipes and Cisterns Aptitude Formulas, Definitions, & Examples:

Overview:

Pipe:

A pipe is a tubular section or hollow cylinder used to fill or empty the cistern or Tank.

Inlet Pipe:

It is a pipe that allows a liquid to enter the tank or cistern.

Outlet Pipe:

It is a pipe that allows a liquid to withdraw from the tank or cistern.

Pipes and Cisterns Case (1):

If an inlet pipe can fill a tank in \(x\) hours, then a certain part of the tank can fill the inlet pipe in one hour = \(\frac{1}{x}\)

Pipes and Cisterns Case (2):

If an outlet pipe can empty a tank in \(y\) hours, then a certain part of the tank can empty the outlet pipe in one hour = \(\frac{1}{y}\)

Pipes and Cisterns Case (3):

If an inlet pipe is filling the tank and an outlet pipe is emptying the tank at the same time, then a certain part of the tank can fill the inlet pipe in one hour = \(\frac{1}{x} - \frac{1}{y}\)

Example: A inlet pipe can fill a tank in two hours and a outlet pipe can empty the tank in four hours, then find the required time to fill the tank if both the pipes are running simultaneously?

Solution: Given, \(x = 2 \ hours\), \(y = 4 \ hours\), then a certain part of the tank filled in one hour $$ = \frac{1}{x} - \frac{1}{y} $$ $$ = \frac{1}{2} - \frac{1}{4} $$ $$ = \frac{1}{4} $$ Hence, tank will be filled in \(4\) hours.

Pipes and Cisterns Case (4):

If there are \(n\) inlet pipes filling the tank and \(n\) outlet pipes are emptying the tank at the same time, then a certain part of the tank can fill the inlet pipes in one hour $$ = \left[\frac{1}{x_1} + \frac{1}{x_2}...\right] - \left[\frac{1}{y_1} + \frac{1}{y_2}...\right] $$

Example: Two inlet pipes can fill a tank in \(5\) and \(10\) hours respectively and two outlet pipes can empty the tank in \(15\) and \(20\) hours, then find the required time to fill the tank if all four pipes are running simultaneously?

Solution: Given, \(x_1 = 5 \ hours\), \(x_2 = 10 \ hours\), \(y_1 = 15 \ hours\), and \(y_2 = 20 \ hours\) then a certain part of the tank filled in one hour $$ = \left[\frac{1}{x_1} + \frac{1}{x_2}\right] - \left[\frac{1}{y_1} + \frac{1}{y_2}\right] $$ $$ = \left[\frac{1}{5} + \frac{1}{10}\right] - \left[\frac{1}{15} + \frac{1}{20}\right] $$ $$ = \frac{3}{10} - \frac{7}{60} $$ $$ = \frac{11}{60} $$ Hence, tank will be filled in \(\frac{60}{11}\) hours or \(5.45\) hours.