Pipes A and B can fill an empty tank in 10 hours and 15 hours respectively. If both are opened together in the empty tank, how much time will they take to fill it completely?

Answer

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

### Step 1: Determine the work done by each pipe in one hour.
- Pipe A can fill the tank in 10 hours. Therefore, the work done by Pipe A in one hour is:
\[ \text{Work done by A in 1 hour} = \frac{1}{10} \text{ of the tank} \]

- Pipe B can fill the tank in 15 hours. Therefore, the work done by Pipe B in one hour is:
\[ \text{Work done by B in 1 hour} = \frac{1}{15} \text{ of the tank} \]

### Step 2: Add the work done by both pipes when opened together.
- When both pipes are opened together, the total work done in one hour is:
\[ \text{Total work in 1 hour} = \frac{1}{10} + \frac{1}{15} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple (LCM) of 10 and 15 is 30. We can rewrite the fractions:
\[ \frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{1}{15} = \frac{2}{30} \]
- Now, add the two fractions:
\[ \text{Total work in 1 hour} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \]

### Step 4: Calculate the time taken to fill the tank completely.
- If both pipes together can fill \(\frac{1}{6}\) of the tank in one hour, then the time taken to fill the entire tank is the reciprocal of \(\frac{1}{6}\):
\[ \text{Time taken} = 6 \text{ hours} \]

### Final Answer:
Thus, if both pipes A and B are opened together, they will take **6 hours** to fill the tank completely.

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