Three pipes A,B and C can fill a tank in 6 min, 8 min, and 12 min respectively. The pipe C is closed for 4 min before the tank is filled. In what time would the tank be fill

Answer

To solve the problem step by step, we need to find out how long it takes for the tank to be filled by pipes A, B, and C, considering that pipe C is closed for the first 4 minutes.

### Step 1: Determine the rates of each pipe
- Pipe A can fill the tank in 6 minutes, so its rate is \( \frac{1}{6} \) of the tank per minute.
- Pipe B can fill the tank in 8 minutes, so its rate is \( \frac{1}{8} \) of the tank per minute.
- Pipe C can fill the tank in 12 minutes, so its rate is \( \frac{1}{12} \) of the tank per minute.

### Step 2: Set up the equation for the total work done
Let \( X \) be the total time (in minutes) taken to fill the tank. Since pipe C is closed for the first 4 minutes, it will only contribute to filling the tank for \( X - 4 \) minutes.

The total work done by the pipes can be expressed as:
\[ \text{Work done by A} + \text{Work done by B} + \text{Work done by C} = 1 \text{ (the whole tank)} \]
This can be written mathematically as:
\[ \frac{X}{6} + \frac{X}{8} + \frac{X - 4}{12} = 1 \]

### Step 3: Find a common denominator
The least common multiple of 6, 8, and 12 is 24. We will multiply each term by 24 to eliminate the denominators:
\[ 24 \left( \frac{X}{6} \right) + 24 \left( \frac{X}{8} \right) + 24 \left( \frac{X - 4}{12} \right) = 24 \]
This simplifies to:
\[ 4X + 3X + 2(X - 4) = 24 \]

### Step 4: Simplify the equation
Now, simplify the left side:
\[ 4X + 3X + 2X - 8 = 24 \]
Combine like terms:
\[ 9X - 8 = 24 \]

### Step 5: Solve for \( X \)
Add 8 to both sides:
\[ 9X = 32 \]
Now divide by 9:
\[ X = \frac{32}{9} \]

### Step 6: Convert to minutes and seconds
To convert \( \frac{32}{9} \) minutes into a more understandable format:
- \( \frac{32}{9} \) minutes is approximately 3.56 minutes.
- To convert 0.56 minutes into seconds: \( 0.56 \times 60 \approx 33.33 \) seconds.

Thus, the tank will be filled in approximately **3 minutes and 33 seconds**.

### Final Answer
The tank will be filled in \( \frac{32}{9} \) minutes or approximately 3 minutes and 33 seconds.

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