A, B and C are three pipes connected to a tank. A and B together fill the tank in 6 h, B and C together fill the tank in 10 h and A and C together fill the tank in 12h . In how much time A,B and C fill up the tank together ?

Options

A. 9h
B. `5 (3)/(7)`h
C. `5 (2)/(7)h`
D. `5(5)/(7) h`

Answer

To solve the problem step by step, we will first determine the rates at which each pipe fills the tank based on the information provided.

### Step 1: Determine the rates of filling for each combination of pipes.

1. **A and B together fill the tank in 6 hours.**
- Rate of A + Rate of B = 1 tank / 6 hours
- Rate of A + Rate of B = 1/6 tanks/hour

2. **B and C together fill the tank in 10 hours.**
- Rate of B + Rate of C = 1 tank / 10 hours
- Rate of B + Rate of C = 1/10 tanks/hour

3. **A and C together fill the tank in 12 hours.**
- Rate of A + Rate of C = 1 tank / 12 hours
- Rate of A + Rate of C = 1/12 tanks/hour

### Step 2: Set up the equations based on the rates.

Let:
- Rate of A = a
- Rate of B = b
- Rate of C = c

From the above information, we can write the following equations:
1. \( a + b = \frac{1}{6} \) (Equation 1)
2. \( b + c = \frac{1}{10} \) (Equation 2)
3. \( a + c = \frac{1}{12} \) (Equation 3)

### Step 3: Solve the equations.

To find the individual rates, we can manipulate these equations.

1. From Equation 1: \( b = \frac{1}{6} - a \)
2. Substitute \( b \) in Equation 2:
\[ \left(\frac{1}{6} - a\right) + c = \frac{1}{10} \]
Rearranging gives:
\[ c = \frac{1}{10} - \frac{1}{6} + a \]
Finding a common denominator (30):
\[ c = \frac{3}{30} - \frac{5}{30} + a = a - \frac{2}{30} = a - \frac{1}{15} \]

3. Substitute \( c \) in Equation 3:
\[ a + \left(a - \frac{1}{15}\right) = \frac{1}{12} \]
This simplifies to:
\[ 2a - \frac{1}{15} = \frac{1}{12} \]
Finding a common denominator (60):
\[ 2a - \frac{4}{60} = \frac{5}{60} \]
Thus:
\[ 2a = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \]
Therefore:
\[ a = \frac{3}{40} \]

### Step 4: Find \( b \) and \( c \).

Using \( a \) to find \( b \):
\[ b = \frac{1}{6} - a = \frac{1}{6} - \frac{3}{40} \]
Finding a common denominator (120):
\[ b = \frac{20}{120} - \frac{9}{120} = \frac{11}{120} \]

Using \( b \) to find \( c \):
\[ c = \frac{1}{10} - b = \frac{1}{10} - \frac{11}{120} \]
Finding a common denominator (120):
\[ c = \frac{12}{120} - \frac{11}{120} = \frac{1}{120} \]

### Step 5: Calculate the combined rate of A, B, and C.

Now we can find the total rate when all three pipes are open:
\[ a + b + c = \frac{3}{40} + \frac{11}{120} + \frac{1}{120} \]
Finding a common denominator (120):
\[ a + b + c = \frac{9}{120} + \frac{11}{120} + \frac{1}{120} = \frac{21}{120} = \frac{7}{40} \]

### Step 6: Calculate the time taken to fill the tank together.

The time taken by A, B, and C together to fill the tank is:
\[ \text{Time} = \frac{\text{Total Capacity}}{\text{Combined Rate}} = \frac{1 \text{ tank}}{\frac{7}{40} \text{ tanks/hour}} = \frac{40}{7} \text{ hours} \]

### Final Answer:
A, B, and C together fill the tank in \( \frac{40}{7} \) hours or approximately 5.71 hours.

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