Pipes P and Q can fill a tank in 12 hr and 36 h respectively. If both pipes are opened then, how much time (in hours) it will take to fill the tank?

Options

A. 9
B. 8
C. 10
D. 7

Explanation

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

### Step 1: Determine the rate of work for each pipe.
- Pipe P can fill the tank in 12 hours. Therefore, the rate of work for pipe P is:
\[ \text{Rate of P} = \frac{1 \text{ tank}}{12 \text{ hours}} = \frac{1}{12} \text{ tanks per hour} \]

- Pipe Q can fill the tank in 36 hours. Therefore, the rate of work for pipe Q is:
\[ \text{Rate of Q} = \frac{1 \text{ tank}}{36 \text{ hours}} = \frac{1}{36} \text{ tanks per hour} \]

### Step 2: Combine the rates of work.
When both pipes are opened together, their combined rate of work is the sum of their individual rates:
\[ \text{Combined Rate} = \text{Rate of P} + \text{Rate of Q} = \frac{1}{12} + \frac{1}{36} \]

### Step 3: Find a common denominator to add the rates.
The least common multiple (LCM) of 12 and 36 is 36. We can rewrite the rates with a common denominator:
\[ \frac{1}{12} = \frac{3}{36} \]
Now, we can add the two rates:
\[ \text{Combined Rate} = \frac{3}{36} + \frac{1}{36} = \frac{4}{36} \]

### Step 4: Simplify the combined rate.
\[ \frac{4}{36} = \frac{1}{9} \text{ tanks per hour} \]

### Step 5: Calculate the time to fill the tank.
To find the time taken to fill one tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{9} \text{ tanks per hour}} = 9 \text{ hours} \]

### Final Answer:
It will take 9 hours to fill the tank when both pipes P and Q are opened together.

---