Question

A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?
(a) 3 hrs 15 min
(b) 3 hrs 45 min
(c) 4 hrs
(d) 4 hrs 15 min

Answer 1

Hint:For solving this problem we will use a simple unitary method and first find how much time is taken by one tap to fill half of the tank and after three more taps are opened then calculate total time taken by these four taps to fill remaining half of the tank.Later we add all total time taken by taps of filling first half and 2nd half of tank we will get required answer.

Complete step-by-step answer:
Given:
It is given that one tap takes 6 hours to fill the tank when used alone. After half the tank is filled, three more similar taps are opened, and we have to find the total time taken to fill the tank.
Let, the capacity of the tank is $L$ litres. Then,
In 6 hours one tap fills the $L$ litres of water. So, in 1-hour one tap can fill $\dfrac{L}{6}$ litres of water. Then,
In a 1-hour amount of water, one tap can fill $=\dfrac{L}{6}\text{ litres}$ .
For filling half tank one tap will consume $=\dfrac{\dfrac{L}{2}}{\dfrac{L}{6}}\text{=}\dfrac{6}{2}\text{=3 hours}...................\left( 1 \right)$ .
Now, as it is given that after half the tank is filled, three more similar taps are opened. So, there will total 4 taps and after 3-hours 4 taps will fill $\dfrac{L}{2}$ litres of water
As in 1-hour amount of water one tap can fill $\dfrac{L}{6}$ litres of water. So, in 1-hour 4 taps can fill $4\times \dfrac{L}{6}=\dfrac{2L}{3}$ litres of water. Then,
For filling remaining half tank 4 taps will consume $=\dfrac{\dfrac{L}{2}}{\dfrac{2L}{3}}=\dfrac{3}{2\times 2}=\dfrac{3}{4}\text{ hours}$ .
Now, as we know that, there are 60 minutes in 1 hour. Then,
For filling remaining half tank 4 taps will consume $=\dfrac{3}{4}\text{ hours}=\dfrac{3}{4}\times 60=45\text{ minutes}.............\left( 2 \right)$ .
Now, add the results from the equation (1) and equation (2) to get the total time taken to fill the tank. Then,
Total time taken to fill the tank completely $=3\text{ hours}+45\text{ minutes}=3\text{ hrs }45\text{ min}$
Now, from the above result, we conclude that it will take 3 hrs 45 min to fill the tank.
Thus, our final answer will be 3 hrs 45 min.
Hence, option (b) will be the correct option.

Note: The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value. In essence, this method is used to find the value of a unit from the value of a multiple, and hence the value of a multiple.Here, the student should first understand what is asked in the question and then proceed in the right direction to get the correct answer. Moreover, one should not directly add the time taken by each tap to fill the other half of the tank when used alone directly to get the answer, it would be a wrong approach.