The Bubna dam has four inlets. Through the first three inlets, the dam can be filled in 12 minutes, through the second, the third and the fourth inlet, it can be filled in 15 minutes, and through the first and the fourth inlet, in 20 minutes. How much time will it take all the four inlets to fill up the dam?

Options

A. 8 min
B. 10 min
C. 12 min
D. None of these

Answer

To solve the problem of how long it will take for all four inlets to fill the Bubna dam, we can follow these steps:

### Step 1: Define the Inlets
Let the rates of filling for the four inlets be:
- Inlet A
- Inlet B
- Inlet C
- Inlet D

### Step 2: Set Up Equations Based on Given Information
From the problem, we have the following information:
1. The first three inlets (A, B, C) can fill the dam in 12 minutes:
\[ \frac{1}{A} + \frac{1}{B} + \frac{1}{C} = \frac{1}{12} \quad \text{(Equation 1)} \]

2. The second, third, and fourth inlets (B, C, D) can fill the dam in 15 minutes:
\[ \frac{1}{B} + \frac{1}{C} + \frac{1}{D} = \frac{1}{15} \quad \text{(Equation 2)} \]

3. The first and fourth inlets (A, D) can fill the dam in 20 minutes:
\[ \frac{1}{A} + \frac{1}{D} = \frac{1}{20} \quad \text{(Equation 3)} \]

### Step 3: Add the Equations
Now, we will add all three equations together:
\[ \left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C}\right) + \left(\frac{1}{B} + \frac{1}{C} + \frac{1}{D}\right) + \left(\frac{1}{A} + \frac{1}{D}\right) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C} + \frac{1}{D}\right) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 4: Find the Common Denominator
To solve the right-hand side, we need to find a common denominator for 12, 15, and 20. The least common multiple (LCM) of these numbers is 60. Now we convert each fraction:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Adding these gives:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 5: Solve for the Total Rate
Now we have:
\[ 2\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C} + \frac{1}{D}\right) = \frac{1}{5} \]
Dividing both sides by 2:
\[ \frac{1}{A} + \frac{1}{B} + \frac{1}{C} + \frac{1}{D} = \frac{1}{10} \]

### Step 6: Find the Total Time
The total rate of filling the dam with all four inlets is \(\frac{1}{10}\). Therefore, the time taken to fill the dam is:
\[ \text{Time} = \frac{1}{\text{Rate}} = \frac{1}{\frac{1}{10}} = 10 \text{ minutes} \]

### Final Answer
It will take all four inlets 10 minutes to fill the dam.

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