Question

A cistern can be filled by one tap in 4 hours and by another in 3 hours. How long will it take to fill it if both taps are opened together?

Answer 1

Hint: To solve such types of questions we will use the formula, Total work done $\times $ time taken for this particular work = the speed of this work done in 1 day. With it, we will substitute the required hours taken by both taps individually and get to the right answer.

Complete step-by-step solution:
To solve this question, we are going to consider a cistern. A cistern is actually a tank for storing water inside it.
According to the question, a cistern takes 4 hours to be filled by first tap to its fullest capacity. Therefore, the correct time taken to fill it with water is 4 hours.
Now, we will move onto its work. For this we will take the help of the formula,
Total work done $\times $ time taken for this particular work = the speed of this work done in 1 day.
Since, the speed of work done here, is taken for 1 day and time taken to fill the cistern completely is 4 hours so, we get the total work done as,
$\begin{align}
  & \text{Total work done}\times 4=1 \\
 & \Rightarrow \text{Total work done}=\dfrac{1}{4} \\
\end{align}$
Similarly, a cistern takes 3 hours to be filled by second tap to its fullest capacity so, we get
$\begin{align}
  & \text{Total work done}\times 3=1 \\
 & \Rightarrow \text{Total work done}=\dfrac{1}{3} \\
\end{align}$
Now, look at the following figure where we can see tap 1 on the left side and tap 2 on the right side.

In this figure, there are two taps which are opened simultaneously. We will now find out the total time that took to fill up this cistern. So, for this we will just add up the two work done together to get the total amount of the work done by the two taps as $\dfrac{1}{4}+\dfrac{1}{3}=\dfrac{3+4}{12}=\dfrac{7}{12}$.
After this the total time taken here for this work done can be carried out by the formula,
$\begin{align}
  & \text{Time taken = }\dfrac{1}{\text{Total work done}} \\
 & \Rightarrow \text{Time taken = }\dfrac{1}{{}^{7}/{}_{12}}=\dfrac{12}{7} \\
\end{align}$
Hence, the total timings for both the taps to fill up the cistern is $\dfrac{12}{7}$ hours.

Note: To solve this question, we will focus on some points. We will check for the units used in the question. Here, the unit for time is taken in hours. We could used a shortcut of the formula, $\text{Total Work done = }\dfrac{1}{\text{Total time taken}}\text{ = }\dfrac{1day}{\text{Time taken by tap 1}}+\dfrac{1day}{\text{Time taken by tap 2}}$. Drawing the required diagram can help to solve the question correctly. This will make us understand the problem clearly and we can use the right formulas to solve it. And at last we will focus on the calculations otherwise, we might lead to wrong answers.