A pipe can fill an empty tank in 12 h. After it is kept open for 3 h, another pipe which can fill the tank in 8 h is opened. In how many more hours will the tank be filled?

Options

A. `17/7h`
B. `18/5h`
C. `20/9h`
D. `16/5h`

Answer

To solve the problem of how many more hours it will take to fill the tank after both pipes are opened, we can follow these steps:

### Step 1: Determine the filling rates of both pipes.
- Pipe 1 can fill the tank in 12 hours. Therefore, the rate of Pipe 1 (P1) is:
\[ \text{Rate of P1} = \frac{1}{12} \text{ tank per hour} \]
- Pipe 2 can fill the tank in 8 hours. Therefore, the rate of Pipe 2 (P2) is:
\[ \text{Rate of P2} = \frac{1}{8} \text{ tank per hour} \]

### Step 2: Calculate the volume filled by Pipe 1 in 3 hours.
- In 3 hours, Pipe 1 will fill:
\[ \text{Volume filled by P1 in 3 hours} = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \text{ of the tank} \]

### Step 3: Determine the remaining volume to be filled.
- The total volume of the tank is 1 (whole tank). After 3 hours, the remaining volume is:
\[ \text{Remaining volume} = 1 - \frac{1}{4} = \frac{3}{4} \text{ of the tank} \]

### Step 4: Calculate the combined filling rate when both pipes are open.
- The combined rate of both pipes (P1 and P2) is:
\[ \text{Combined rate} = \frac{1}{12} + \frac{1}{8} \]
- To add these fractions, find a common denominator (which is 24):
\[ \frac{1}{12} = \frac{2}{24} \quad \text{and} \quad \frac{1}{8} = \frac{3}{24} \]
- Therefore,
\[ \text{Combined rate} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24} \text{ tank per hour} \]

### Step 5: Calculate the time needed to fill the remaining volume.
- Let \( t \) be the time in hours needed to fill the remaining \(\frac{3}{4}\) of the tank. Using the formula:
\[ \text{Volume filled} = \text{Rate} \times \text{Time} \]
- We have:
\[ \frac{3}{4} = \frac{5}{24} \times t \]
- To find \( t \), rearrange the equation:
\[ t = \frac{3/4}{5/24} = \frac{3}{4} \times \frac{24}{5} = \frac{3 \times 24}{4 \times 5} = \frac{72}{20} = \frac{18}{5} \text{ hours} \]

### Step 6: Convert the time into a mixed number.
- The time \( \frac{18}{5} \) hours can be expressed as:
\[ \frac{18}{5} = 3 \frac{3}{5} \text{ hours} \]

### Final Answer:
The tank will be filled in \( \frac{18}{5} \) hours or \( 3 \frac{3}{5} \) hours after both pipes are opened.

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