Two pipes A and B can fill an empty tank in 10 hours and 16 hours respectively. They are opened alternately for 1 hour 30 each, starting with pipe A first. In how many hours, the empty tank will be filled?

Options

A. `12 1/3`
B. `12 1/8`
C. `12 1/4`
D. `12 1/6`

Answer

To solve the problem step by step, we will analyze the filling rates of pipes A and B, calculate how much water they fill when opened alternately, and determine the total time taken to fill the tank.

### Step 1: Determine the filling rates of pipes A and B
- **Pipe A** can fill the tank in 10 hours.
- **Pipe B** can fill the tank in 16 hours.

To find the rate at which each pipe fills the tank:
- Rate of Pipe A = Total Capacity / Time taken by A = 1 tank / 10 hours = 1/10 tanks per hour.
- Rate of Pipe B = Total Capacity / Time taken by B = 1 tank / 16 hours = 1/16 tanks per hour.

### Step 2: Calculate the amount filled in 1 hour and 30 minutes by each pipe
Since they are opened alternately for 1 hour and 30 minutes each:
- In 1 hour and 30 minutes (which is 1.5 hours), the amount filled by Pipe A:
\[ \text{Amount filled by A} = \text{Rate of A} \times 1.5 = \frac{1}{10} \times 1.5 = \frac{1.5}{10} = \frac{3}{20} \text{ tanks} \]

- In the next 1 hour and 30 minutes, the amount filled by Pipe B:
\[ \text{Amount filled by B} = \text{Rate of B} \times 1.5 = \frac{1}{16} \times 1.5 = \frac{1.5}{16} = \frac{3}{32} \text{ tanks} \]

### Step 3: Calculate the total amount filled in 3 hours
Now, we add the amounts filled by both pipes in 3 hours:
- Total amount filled in 3 hours:
\[ \text{Total filled} = \frac{3}{20} + \frac{3}{32} \]

To add these fractions, we need a common denominator. The least common multiple of 20 and 32 is 160.
- Convert the fractions:
\[ \frac{3}{20} = \frac{3 \times 8}{20 \times 8} = \frac{24}{160} \]
\[ \frac{3}{32} = \frac{3 \times 5}{32 \times 5} = \frac{15}{160} \]

- Now add them:
\[ \text{Total filled} = \frac{24}{160} + \frac{15}{160} = \frac{39}{160} \text{ tanks} \]

### Step 4: Determine how many cycles are needed to fill the tank
The tank's total capacity is 1 tank. We need to find out how many cycles of 3 hours are required to fill the tank:
- In 3 hours, \(\frac{39}{160}\) of the tank is filled.
- To find out how many cycles (3-hour intervals) are needed to fill the tank, we set up the equation:
\[ n \times \frac{39}{160} = 1 \implies n = \frac{160}{39} \approx 4.1 \]
This means we need 4 complete cycles (12 hours) and a bit more.

### Step 5: Calculate how much is filled in 12 hours
- In 12 hours (which is 4 cycles):
\[ \text{Total filled in 12 hours} = 4 \times \frac{39}{160} = \frac{156}{160} = \frac{39}{40} \text{ tanks} \]

### Step 6: Calculate the remaining amount to fill
- Remaining amount to fill:
\[ 1 - \frac{39}{40} = \frac{1}{40} \text{ tanks} \]

### Step 7: Determine how long it takes to fill the remaining amount
- Now, we will use Pipe A to fill the remaining \(\frac{1}{40}\) of the tank:
- The rate of Pipe A is \(\frac{1}{10}\) tanks per hour.
- Time taken by Pipe A to fill \(\frac{1}{40}\) of the tank:
\[ \text{Time} = \frac{\text{Amount}}{\text{Rate}} = \frac{1/40}{1/10} = \frac{1}{40} \times 10 = \frac{1}{4} \text{ hours} = 15 \text{ minutes} \]

### Final Calculation: Total time taken to fill the tank
- Total time = Time for 4 cycles + Time to fill remaining amount
\[ \text{Total time} = 12 \text{ hours} + \frac{1}{4} \text{ hours} = 12 \text{ hours} + 15 \text{ minutes} = 12 \text{ hours and } 15 \text{ minutes} \]

### Answer
The empty tank will be filled in **12 hours and 15 minutes**.

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