Pipe A can fill an empty tank in 30 hours while B can fill it in 45 hours. Pipe A and B are opened and closed alternatively i.e., first pipe A is opened, then B, again A and then B and so on for 1 hour each time without any time lapse. In how many hours the tank will be filled when it was empty, initially?

Options

A. 36
B. 54
C. 48
D. 60

Answer

To solve the problem, we need to determine how long it will take for pipes A and B to fill the tank when they are opened alternately for 1 hour each.

### Step-by-Step Solution:

1. **Determine the capacity of the tank**:
Let's assume the capacity of the tank is 90 units (the least common multiple of 30 and 45).

2. **Calculate the filling rate of Pipe A**:
Pipe A can fill the tank in 30 hours. Therefore, in 1 hour, Pipe A will fill:
\[ \text{Filling rate of A} = \frac{90 \text{ units}}{30 \text{ hours}} = 3 \text{ units/hour} \]

3. **Calculate the filling rate of Pipe B**:
Pipe B can fill the tank in 45 hours. Therefore, in 1 hour, Pipe B will fill:
\[ \text{Filling rate of B} = \frac{90 \text{ units}}{45 \text{ hours}} = 2 \text{ units/hour} \]

4. **Calculate the total units filled in 2 hours**:
In the first hour, Pipe A is opened and fills 3 units. In the second hour, Pipe B is opened and fills 2 units. Therefore, in 2 hours, the total units filled is:
\[ \text{Total units in 2 hours} = 3 + 2 = 5 \text{ units} \]

5. **Determine how many complete cycles are needed to fill the tank**:
To fill the tank of 90 units, we need to find out how many complete 2-hour cycles (which fill 5 units) are required:
\[ \text{Number of complete cycles} = \frac{90 \text{ units}}{5 \text{ units/cycle}} = 18 \text{ cycles} \]

6. **Calculate the total time for complete cycles**:
Since each cycle takes 2 hours, the total time for 18 cycles is:
\[ \text{Total time for 18 cycles} = 18 \times 2 = 36 \text{ hours} \]

7. **Check if the tank is completely filled after 36 hours**:
After 36 hours, we have filled 90 units. Therefore, the tank is completely filled at the end of the 36th hour.

### Final Answer:
The tank will be completely filled in **36 hours**.

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