Question

Two taps ‘A’ and ‘B’ can fill a water reservoir in 8 and 6 hours respectively, a third tap ‘C’ can empty the tank completely in 24 hours. How long would it take to fill the empty tank when all the taps are open?
A) 4 hours
B) 5 hours
C) 6 hours
D) 3 hours

Answer 1

Hint: There are three taps connected to thee given reservoir, out of which two taps are for filling and one for emptying the tank timings of which are provided. We will find the fraction of the tank filled in 1 hour when all are open together. Then, using the unitary method, we can find the time required to fill the whole tank.
We denote a whole quantity by unity i.e. 1

Complete step-by-step answer:
It is given that 3 taps ‘A’, ‘B’ and ‘C’ are connected to a reservoir.
A and B can fill the tank in 8 hours respectively and C empties it in 24 hours. Tank filled or emptied by each tap in 1 hour can be calculated by the unitary method. We will consider the full tank to be equal to unity (1 volume)
Tap A:
Fills the tank in 8 hrs,
 $
   \Rightarrow 8{\text{ }}hrs = 1{\text{ }}volume \\
 \Rightarrow 1{\text{ }}hr = \dfrac{1}{8}{\text{ }}volume \\
  $
Tap B:
Fills the tank in 6 hrs,
 $
   \Rightarrow 6{\text{ }}hrs = 1{\text{ }}volume \\
 \Rightarrow 1{\text{ }}hr = \dfrac{1}{6}{\text{ }}volume \\
  $
Tap C:
Empties the tank in 24 hrs,
 $
   \Rightarrow 24{\text{ }}hrs = 1{\text{ }}volume \\
 \Rightarrow 1{\text{ }}hr = \dfrac{1}{{24}}{\text{ }}volume \\
  $
When all the taps are opened together, the reservoir filled in 1 hour would be given as:
 $ \dfrac{1}{8} + \dfrac{1}{6} - \dfrac{1}{{24}} $
We subtracted the volume of tap C because it is emptying the tank while the rest are added as they are responsible for filling the same.
Taking the LCM, we get:
\[ \to \dfrac{{6 + 8 - 2}}{{48}} = \dfrac{{12}}{{48}}\]
This shows that $ \dfrac{{12}}{{48}}th $ volume of the tank is filled in 1 hour.
Then the whole tank i.e. 1 volume will be filled in:
\[
  \dfrac{{12}}{{48}}{\text{ }}volume = 1{\text{ }}hr \\
 \Rightarrow 1{\text{ }}volume = 1 \times \dfrac{{48}}{{12}}{\text{ }}hr \\
 \Rightarrow 1{\text{ }}volume = 4{\text{ }}hrs \;
 \]
Therefore, it would take 4 hours to fill the empty tank when all the three taps are open and the correct option is A).
So, the correct answer is “Option A”.

Note: In unitary method, we can keep in mind the following example:
If value of A is x then B in terms of x can be given as:
 $
  A = x \\
 \Rightarrow B = \dfrac{x}{A} \times B ;
  $
When we found that $ \dfrac{{12}}{{48}}th $ volume of the tank was filled in 1 hour, we could either make this fraction equal to 1 as 1 denotes ‘whole’ in fractions by direct multiplication with 4 or use unitary method as we did.