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. 4 hours
• B. 5 hours
• C. 3 hours 30 minutes
• D. 3 hours 45 minutes

Hint:

For identical workers, the time taken is inversely proportional to the number of workers.
If 1 tap takes 3 hours to fill the remaining half of the tank, then 4 taps will take:
\[ \frac{3\text{ hours}}{4} = 45\text{ minutes} \]
Thus, the total time is simply \( 3\text{ hours} + 45\text{ minutes} = 3\text{ hours } 45\text{ minutes} \).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A single tap can fill a tank in 6 hours. Initially, only one tap is operational until half of the tank is filled. After this, three more identical taps are opened. We need to find the total time taken to fill the entire tank.

Step 2: Key Formula or Approach:
We use the rate of work concept:
- Let the total capacity of the tank be 1 unit.
- The rate of one tap is the fraction of the tank filled per hour: \( \text{Rate} = \frac{1}{\text{Time taken}} \).
- The time taken to complete a certain amount of work is:
\[ \text{Time} = \frac{\text{Work to be done}}{\text{Rate of work}} \]

Step 3: Detailed Explanation:
1. The total time taken by a single tap to fill the tank completely is \( 6\text{ hours} \).
2. Therefore, the rate of filling of one tap is \( \frac{1}{6} \) of the tank per hour.
3. The process is divided into two distinct parts.
4. Part 1: Filling the first half of the tank.
- The amount of work to be done is \( \frac{1}{2} \) of the total capacity.
- Only 1 tap is open during this stage.
- The time taken to fill the first half of the tank is:
\[ \text{Time}_1 = \frac{1}{2} \times 6 = 3\text{ hours} \]
5. Part 2: Filling the remaining half of the tank.
- The remaining work is \( \frac{1}{2} \) of the total capacity.
- Three more identical taps are opened, making the total number of active taps equal to \( 1 + 3 = 4 \).
- Since all 4 taps are identical, their combined filling rate is:
\[ \text{Combined Rate} = 4 \times \frac{1}{6} = \frac{2}{3} \text{ of the tank per hour} \]
- The time taken to fill the remaining half of the tank is:
\[ \text{Time}_2 = \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{1/2}{2/3} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}\text{ hours} \]
6. Converting \( \frac{3}{4}\text{ hours} \) into minutes:
\[ \text{Time}_2 = \frac{3}{4} \times 60 = 45\text{ minutes} \]
7. Now, we sum the times from both parts to find the total time taken:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = 3\text{ hours} + 45\text{ minutes} = 3\text{ hours } 45\text{ minutes} \]

Step 4: Final Answer:
The total time taken to fill the tank completely is 3 hours 45 minutes, which corresponds to option (D).