A tank is fitted with two taps. The first tap can fill the tank completely in 45 minutes and the second tap can empty the full tank in one hour. If both the taps are opened alternately for one minute, then in how many hours the empty tank will be filled completely?

Options

A. 2 hours 55 minutes
B. 3 hours 40 minutes
C. 4 hours 48 minutes
D. 5 hours 53 minutes

Answer

To solve the problem step by step, we need to analyze the filling and emptying rates of the two taps and how they work when opened alternately.

### Step 1: Determine the filling and emptying rates of the taps.
- **Tap A** fills the tank in 45 minutes.
- **Tap B** empties the tank in 60 minutes.

To find their rates:
- The rate of Tap A (filling rate) = 1 tank / 45 minutes = \( \frac{1}{45} \) tanks per minute.
- The rate of Tap B (emptying rate) = 1 tank / 60 minutes = \( \frac{1}{60} \) tanks per minute.

### Step 2: Calculate the effective rate when both taps are opened alternately.
When both taps are opened alternately for one minute:
- In the first minute, Tap A fills the tank: \( \frac{1}{45} \) tanks.
- In the second minute, Tap B empties the tank: \( \frac{1}{60} \) tanks.

The net effect after 2 minutes:
- Amount filled in 1 minute by Tap A = \( \frac{1}{45} \)
- Amount emptied in 1 minute by Tap B = \( \frac{1}{60} \)

To find the net amount filled in 2 minutes:
\[ \text{Net amount filled in 2 minutes} = \frac{1}{45} - \frac{1}{60} \]

### Step 3: Find a common denominator to calculate the net amount.
The least common multiple (LCM) of 45 and 60 is 180.
- Convert \( \frac{1}{45} \) to have a denominator of 180:
\[ \frac{1}{45} = \frac{4}{180} \]
- Convert \( \frac{1}{60} \) to have a denominator of 180:
\[ \frac{1}{60} = \frac{3}{180} \]

Now, calculate the net amount:
\[ \text{Net amount filled in 2 minutes} = \frac{4}{180} - \frac{3}{180} = \frac{1}{180} \]

### Step 4: Determine how much time it takes to fill the entire tank.
Since \( \frac{1}{180} \) of the tank is filled in 2 minutes, we can find out how long it takes to fill the entire tank:
- To fill 1 tank, it will take:
\[ \text{Total time} = 180 \text{ units} \times 2 \text{ minutes/unit} = 360 \text{ minutes} \]

### Step 5: Account for the last minute of filling.
Since the last minute will be filled by Tap A, we need to add 1 more minute to the total time:
\[ \text{Total time} = 360 + 1 = 361 \text{ minutes} \]

### Step 6: Convert minutes to hours and minutes.
To convert 361 minutes into hours and minutes:
- 361 minutes = 6 hours and 1 minute.

### Final Answer:
The empty tank will be filled completely in **6 hours and 1 minute**.

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