Two pipes A and B can fill a tank in 18 h and 12 h, respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

Answer

To solve the problem of how long it will take for two pipes A and B to fill a tank when opened simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Rate of Each Pipe**:
- Pipe A can fill the tank in 18 hours. Therefore, the rate of Pipe A is:
\[ \text{Rate of A} = \frac{1}{18} \text{ tank per hour} \]
- Pipe B can fill the tank in 12 hours. Therefore, the rate of Pipe B is:
\[ \text{Rate of B} = \frac{1}{12} \text{ tank per hour} \]

2. **Combine the Rates**:
- When both pipes are opened simultaneously, their rates add up. Thus, the combined rate of both pipes is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{18} + \frac{1}{12} \]

3. **Find a Common Denominator**:
- To add the fractions, we need a common denominator. The least common multiple (LCM) of 18 and 12 is 36. We can rewrite the rates:
\[ \frac{1}{18} = \frac{2}{36} \quad \text{and} \quad \frac{1}{12} = \frac{3}{36} \]
- Now we can add the two rates:
\[ \text{Combined Rate} = \frac{2}{36} + \frac{3}{36} = \frac{5}{36} \text{ tank per hour} \]

4. **Calculate the Time to Fill the Tank**:
- To find the time taken to fill the tank when both pipes are open, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{36}} = \frac{36}{5} \text{ hours} \]

5. **Convert to Mixed Fraction**:
- Now, we can convert \(\frac{36}{5}\) into a mixed fraction:
\[ 36 \div 5 = 7 \quad \text{remainder} \quad 1 \]
- So, \(\frac{36}{5} = 7 \frac{1}{5}\) hours.

### Final Answer:
The time taken to fill the tank when both pipes A and B are opened simultaneously is \(7 \frac{1}{5}\) hours.

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