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?
- 4 hours
- 5 hours
- 3 hours 30 minutes
- 3 hours 45 minutes
The Correct Option is D
Approach Solution - 1
To solve this problem, we need to find out the total time taken to fill the tank with water when additional taps are involved after the tank is half filled. Here's a step-by-step explanation:
1. Initial Scenario: One tap can fill the tank in 6 hours. Therefore, in one hour, the tap fills 16 of the tank.
2. Filling Half the Tank: Since half the tank needs to be filled initially, it takes the single tap 62 = 3 hours to fill half of the tank.
3. Adding More Taps: Once the tank is half filled, three more similar taps are added, making a total of 4 taps working to fill the remaining half.
4. Effect of 4 Taps: With 4 taps operating, the amount of the tank filled by all taps in one hour is 46 = 23 of the tank per hour.
5. Time to Fill the Remaining Half: To fill the remaining 12 of the tank with 4 taps, the time taken is 123 = 34 hours.
6. Calculate Total Time: Therefore, the total time taken to fill the whole tank is the time taken to fill the first half plus the time taken to fill the second half: 3+34 = 154 = 3.75 hours.
7. Convert to Minutes: 0.75 hours is equivalent to 45 minutes.
Hence, the total time taken to fill the tank completely is 3 hours and 45 minutes, which matches option: 3 hours 45 minutes.
Approach Solution -2
A tap can fill a tank in 6 hours, so it fills \(\frac{1}{6}\) of the tank in one hour.
After half the tank is filled, it takes \(\frac{1}{2} \times 6 = 3\) hours.
Three more similar taps are opened, so now there are a total of 4 taps. Each tap fills \(\frac{1}{6}\) of the tank per hour, so the four taps together fill \(4 \times \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\) of the tank per hour.
The remaining half of the tank needs to be filled.
The time it takes for the 4 taps to fill the remaining half is \(\frac{1/2}{2/3} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}\) hours, which is 45 minutes.
The total time taken to fill the tank completely is 3 hours + 45 minutes = 3 hours 45 minutes.
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