There are 7 pipes attached with a tank out of which some are inlets and some are outlets. Every inlet can fill the tank in 10 h and every outlet can empty the tank in 15 h. When all the pipes are opened simul- taneously, the tank is filled up in `2 (8)/(11)` h. find the numbers of inlets and outlets .

Options

A. 5,2
B. 6,1
C. 4,3
D. 3,4

Answer

To solve the problem step by step, we will define variables and set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = number of outlet pipes
- \( 7 - x \) = number of inlet pipes (since there are a total of 7 pipes)

### Step 2: Determine the Rate of Work for Inlets and Outlets
Each inlet can fill the tank in 10 hours, so the rate of one inlet is:
\[ \text{Rate of one inlet} = \frac{1}{10} \text{ tank/hour} \]
Thus, the rate of \( (7 - x) \) inlets is:
\[ \text{Rate of inlets} = (7 - x) \times \frac{1}{10} = \frac{7 - x}{10} \text{ tank/hour} \]

Each outlet can empty the tank in 15 hours, so the rate of one outlet is:
\[ \text{Rate of one outlet} = -\frac{1}{15} \text{ tank/hour} \quad (\text{negative because it empties the tank}) \]
Thus, the rate of \( x \) outlets is:
\[ \text{Rate of outlets} = x \times -\frac{1}{15} = -\frac{x}{15} \text{ tank/hour} \]

### Step 3: Combine the Rates
When all pipes are opened simultaneously, the combined rate of filling the tank is:
\[ \text{Combined rate} = \text{Rate of inlets} + \text{Rate of outlets} = \frac{7 - x}{10} - \frac{x}{15} \]

### Step 4: Convert Time to Hours
The tank is filled in \( 2 \frac{8}{11} \) hours, which can be converted to an improper fraction:
\[ 2 \frac{8}{11} = \frac{30}{11} \text{ hours} \]
Thus, the rate of filling the tank when all pipes are open is:
\[ \text{Rate} = \frac{1 \text{ tank}}{\frac{30}{11} \text{ hours}} = \frac{11}{30} \text{ tank/hour} \]

### Step 5: Set Up the Equation
Now we can set up the equation based on the combined rate:
\[ \frac{7 - x}{10} - \frac{x}{15} = \frac{11}{30} \]

### Step 6: Solve the Equation
To solve this equation, we will first find a common denominator for the left side:
The common denominator of 10 and 15 is 30. So we rewrite the equation:
\[ \frac{3(7 - x)}{30} - \frac{2x}{30} = \frac{11}{30} \]
Combining the fractions:
\[ \frac{21 - 3x - 2x}{30} = \frac{11}{30} \]
This simplifies to:
\[ \frac{21 - 5x}{30} = \frac{11}{30} \]

### Step 7: Eliminate the Denominator
Multiply both sides by 30:
\[ 21 - 5x = 11 \]

### Step 8: Solve for \( x \)
Rearranging gives:
\[ 21 - 11 = 5x \\ 10 = 5x \\ x = 2 \]

### Step 9: Find the Number of Inlets
Now, substitute \( x \) back to find the number of inlets:
\[ \text{Number of inlets} = 7 - x = 7 - 2 = 5 \]

### Final Answer
- Number of inlets = 5
- Number of outlets = 2