Question

A leak in the bottom of a tank can empty the full tank in 8 hours. An inlet pipe fills water at the rate of 6 liters a minute. When the tank is full the inlet is opened and due to leak the tank is empty in 12 hours. How many liters does the cistern hold?

Answer 1

Hint:In this question we have to find the net rate of water leak per min and form an equation in order to find the total capacity of the cistern/tank. In order to solve this type of problem remember that a pipe connected to a tank for filling it with water is an inlet. Pipe connected to a tank for emptying the water in it is an outlet.

Complete step by step solution:
Let us assume the capacity of water that the cistern contains when its full be ‘N’ liters.
According to the question, Leak in the bottom takes 8 hours to empty the full tank
Therefore, the rate of leak when there is no inlet pipe to empty the tank is given as:

\[
{R_1} = \dfrac{N}{8}\;litres\;per\;hour \\
= \dfrac{N}{{8 \times 60}}\;litres\;per\;\min \\
= \dfrac{N}{{480}}\;litres\;per\;\min \;\;\;\;\;\;\;\; - (1) \\
\]

Now, when the tank is full and inlet is opened, the leak takes 12 hours to empty the tank
So, the rate of leak when there is inlet pipe (open) to empty the tank is given as:
\[
= N/12\;litres\;per\;hour \\
= N/\left( {12 \times 60} \right)\;litres\;per\;\min \\
= N/(720)\;litres\;per\;\min \\
\]\[
{R_2} = \dfrac{N}{{12}}\;litres\;per\;hour \\
= \dfrac{N}{{12 \times 60}}\;litres\;per\;\min \\
= \dfrac{N}{{720}}\;litres\;per\;\min \;\;\;\;\;\; - (2) \\
\]

Since, the rate of inlet pipe filling the tank is 6 liters per minute.
Therefore, we can create the equation from (1) and (2) as:

\[Net{\text{ }}rate{\text{ to }}empty = \dfrac{N}{{720}}\]
Rate of leak to empty tank when inlet pipe (closed) - rate of inlet pipe is equals to $\dfrac{N}{{720}}$
Substituting values from Equation (1) and (2)
$\dfrac{N}{{480}} - 6 = \dfrac{N}{{720}}$

Multiplying both sides by LCM of 720 and 480 i.e., 1440, we get
$


3N - 8640 = 2N \\
\Rightarrow 3N - 2N = 8640 \\
\Rightarrow N = 8640\;litres \\
$

So, the capacity of the tank is 8640 liters.

NOTE:Students should be careful while forming an equation and calculating the answer and must cross check the answer. An inlet pipe fills water at the rate of 6 liters a minutes so find how much water it will fill in 12 hours.

While solving this question, all the main terms such as inlets, outlets etc. should be very clear. There are different ways to solve this question. But always go with the one you are pretty sure of. Always be sure of the formulas used in this type of questions.