A water tap fills a tank in p hours and the tap of the bottom of the tank empties it in q hours. If p is less than q and when both the taps are open, the tank is filled in r hours. Then find which of the following is true
\[\begin{align}
& a)\dfrac{1}{r}=\dfrac{1}{p}+\dfrac{1}{q} \\
& b)\dfrac{1}{r}=\dfrac{1}{p}-\dfrac{1}{q} \\
& c)\text{ }r=p+q \\
& d)\text{ }r=p-q \\
\end{align}\]
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Hint: First we will calculate the net tank that is filled in 1 hour and the tank emptied in 1 hour. Hence we will get the net water in the tank after 1 hour. Also we will note that the tank is filled in r hours and hence in 1 hour this net water in the tank will be equal to $ \dfrac{1}{r} $
Complete step-by-step answer:
Now we are given that a water tap fills a tank in p hours.
Hence in 1 hour $ \dfrac{1}{p} $ part of the tank will be filled.
Also the tap of the bottom of the tank empties it in q hours.
Hence in 1 hour $ \dfrac{1}{q} $ part of the tank will be removed.
Now the water in Tank is water filled in the tank – water removed from the tank.
Hence in 1 hour the total water in the tank will be $ \dfrac{1}{p}-\dfrac{1}{q} $ ………..(1)
Now it is also given that the tank takes r hours to be filled completely.
Hence, in 1 hour $ \dfrac{1}{r} $ tank will be filled. ………….(2)
Hence from equation (1) and equation (2) we get.
$ \dfrac{1}{r}=\dfrac{1}{p}-\dfrac{1}{q} $
So, the correct answer is “Option B”.
Note: First it is very tempting to select option d as the total water will be water filled – water removed. This problem should be solved step by step as to check the amount of water in a particular time. Also the water that is being removed from the bottom top should be subtracted and not added to the water that is filled in order to get the total water in the tank.
Complete step-by-step answer:
Now we are given that a water tap fills a tank in p hours.
Hence in 1 hour $ \dfrac{1}{p} $ part of the tank will be filled.
Also the tap of the bottom of the tank empties it in q hours.
Hence in 1 hour $ \dfrac{1}{q} $ part of the tank will be removed.
Now the water in Tank is water filled in the tank – water removed from the tank.
Hence in 1 hour the total water in the tank will be $ \dfrac{1}{p}-\dfrac{1}{q} $ ………..(1)
Now it is also given that the tank takes r hours to be filled completely.
Hence, in 1 hour $ \dfrac{1}{r} $ tank will be filled. ………….(2)
Hence from equation (1) and equation (2) we get.
$ \dfrac{1}{r}=\dfrac{1}{p}-\dfrac{1}{q} $
So, the correct answer is “Option B”.
Note: First it is very tempting to select option d as the total water will be water filled – water removed. This problem should be solved step by step as to check the amount of water in a particular time. Also the water that is being removed from the bottom top should be subtracted and not added to the water that is filled in order to get the total water in the tank.