Question

A cistern has a leak which would empty it in $8\ hours$. A tap is turned on, which admits $6\ liters$ a minute into the cistern and it is now emptied in $12\ hours$. How many liters can the cistern hold?
(a) $8000\ liters$
(b) $8400\ liters$
(c) $8640\ liters$
(d) $8650\ liters$

Answer 1

First of all, we will assume that cistern can hold $x$ liters of water. Then we will convert each and every quantity in terms of hours and liters. Then we will find the value of the amount of water the tank empties in $8\ hours$ and the tap fills per hour. Then we will take the difference of their values as, it is given that when the events take place simultaneously it will take $12\ hours$ to empty the tank. Then based on that we will find the value of the amount of water the tank can hold in terms of $\dfrac{liters}{hour}$ .

Complete step-by-step answer:
In question it is given that a cistern has a leak due to which the water will be emptied in $8\ hours$. Now, a tap is turned on which admits $6\ liters$ a minutes into the cistern and now the cistern gets emptied in $12\ hours$ and we are asked that how many liters cistern can hold, so first of all we will suppose that cistern can hold $x$ liters of water.
Now, when the tap is turned on it fills $6\ liters$ water in a minute, so we will convert it into hours. As we know that $1\ hour=60\ \text{min}$, so one minute in terms of hours can be given as,
$1\ \min =\dfrac{1}{60}hour$
So, the tap admits $\dfrac{6}{\dfrac{1}{60}}\ \dfrac{liters}{hour}\Rightarrow 360\ \dfrac{liters}{hour}$ , now it can be said that $x$ liters of water fills in $360\ \dfrac{liters}{hour}$ , which can be given mathematically as,
$\dfrac{x}{360}hours$
Now, it is given that the leak would empty the tank in $8\ hours$, so from this we can say that ${{\dfrac{1}{8}}^{th}}$ of the total will get empty each hour.
Now, if we allow the tap to admit water as well as the leak simultaneously then it will take $12\ hours$, so we can say that difference of leak and water being filled is equal to ${{\dfrac{1}{12}}^{th}}$ of the water in one hour, which can be seen mathematically as,
$\dfrac{1}{8}-\dfrac{x}{360}=\dfrac{1}{12}$
On simplifying further, we will get,
$\Rightarrow \dfrac{1}{8}-\dfrac{1}{12}=\dfrac{x}{360}$
$\Rightarrow \dfrac{12-8}{12\times 8}=\dfrac{x}{360}$
$\Rightarrow \dfrac{4}{96}=\dfrac{x}{360}$
On keeping x as subject variable on RHS, we will get
$\Rightarrow \dfrac{4\times 360}{96}=x$
$\Rightarrow x=8640\ liters$
Thus, we can say that cistern can hold $x=8640\ liters$.
Hence, option (b) is the correct answer.

Note: Here, we have converted the quantities in terms of liters and hours as our final answer is in terms of liters, but we can also solve the problem by converting them into minutes such as, $8\ hours=8\times 60\min =480\min$, by doing this there will be no change in final answer just the solution becomes longer as first we convert hours into minutes and then for final answer we again convert minutes into hours. So, this can be considered as an alternate way to solve the problem.