There are 12 pipes that are connected to a tank. Some of them are fill pipes and the others are drain pipes. Each of the fill pipes can fill the tank in 8 hours and each of the drain pipes can drain the tank completely in 6 hours. If all the fill pipes and drain pipes are kept open, an empty tank gets filled in 24 hours. How many of the 12 pipes are fill pipes?

Options

A. 6
B. 8
C. 7
D. 5

Answer

To solve the problem, let's denote:
- \( x \): the number of fill pipes
- \( y \): the number of drain pipes

From the problem, we know that:
1. The total number of pipes is 12:
\[ x + y = 12 \]

2. Each fill pipe can fill the tank in 8 hours, so the rate of one fill pipe is:
\[ \text{Rate of one fill pipe} = \frac{1}{8} \text{ tank per hour} \]
Therefore, the rate of \( x \) fill pipes is:
\[ \text{Rate of } x \text{ fill pipes} = \frac{x}{8} \text{ tank per hour} \]

3. Each drain pipe can drain the tank in 6 hours, so the rate of one drain pipe is:
\[ \text{Rate of one drain pipe} = \frac{1}{6} \text{ tank per hour} \]
Therefore, the rate of \( y \) drain pipes is:
\[ \text{Rate of } y \text{ drain pipes} = \frac{y}{6} \text{ tank per hour} \]

4. When all pipes are open, the net rate of filling the tank is given as:
\[ \text{Net rate} = \text{Rate of fill pipes} - \text{Rate of drain pipes} \]
This can be expressed as:
\[ \frac{x}{8} - \frac{y}{6} = \frac{1}{24} \text{ tank per hour} \]

Now, we have two equations:
1. \( x + y = 12 \)
2. \( \frac{x}{8} - \frac{y}{6} = \frac{1}{24} \)

### Step 1: Solve the first equation for \( y \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 12 - x \]

### Step 2: Substitute \( y \) in the second equation
Substituting \( y \) in the second equation:
\[ \frac{x}{8} - \frac{12 - x}{6} = \frac{1}{24} \]

### Step 3: Clear the fractions
To eliminate the fractions, we can multiply through by the least common multiple of the denominators (which is 24):
\[ 24 \left(\frac{x}{8}\right) - 24 \left(\frac{12 - x}{6}\right) = 24 \left(\frac{1}{24}\right) \]
This simplifies to:
\[ 3x - 4(12 - x) = 1 \]

### Step 4: Simplify the equation
Expanding the equation:
\[ 3x - 48 + 4x = 1 \]
Combining like terms:
\[ 7x - 48 = 1 \]

### Step 5: Solve for \( x \)
Adding 48 to both sides:
\[ 7x = 49 \]
Dividing by 7:
\[ x = 7 \]

### Step 6: Find \( y \)
Now, substituting \( x \) back into the first equation to find \( y \):
\[ y = 12 - 7 = 5 \]

### Conclusion
Thus, the number of fill pipes is \( \boxed{7} \).