Two pipes P and Q can fill the tank alone in 20 and 30 hours respectively. If both pipes are opened together, then in how many hours will the tank be filled?

Options

A. 12
B. 24
C. 8
D. 15

Answer

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

### Step 1: Determine the filling rates of each pipe.
- Pipe P can fill the tank in 20 hours.
- Therefore, in 1 hour, Pipe P fills \( \frac{1}{20} \) of the tank.

- Pipe Q can fill the tank in 30 hours.
- Therefore, in 1 hour, Pipe Q fills \( \frac{1}{30} \) of the tank.

### Step 2: Calculate the combined filling rate of both pipes.
- When both pipes are opened together, their combined filling rate per hour is:
\[ \text{Combined Rate} = \frac{1}{20} + \frac{1}{30} \]

### Step 3: Find the least common multiple (LCM) of the denominators.
- The LCM of 20 and 30 is 60.

### Step 4: Convert the fractions to have a common denominator.
- Convert \( \frac{1}{20} \) to have a denominator of 60:
\[ \frac{1}{20} = \frac{3}{60} \]
- Convert \( \frac{1}{30} \) to have a denominator of 60:
\[ \frac{1}{30} = \frac{2}{60} \]

### Step 5: Add the fractions.
- Now add the two fractions:
\[ \text{Combined Rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} \]
- Simplifying \( \frac{5}{60} \) gives:
\[ \frac{5}{60} = \frac{1}{12} \]

### Step 6: Determine the time taken to fill the tank.
- If both pipes together fill \( \frac{1}{12} \) of the tank in 1 hour, then to fill the entire tank (1 whole tank), it will take:
\[ \text{Time} = 12 \text{ hours} \]

### Final Answer:
Thus, the tank will be filled in **12 hours** when both pipes are opened together.

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