Two pipes A and B can fill a tank in 18 and 6h , respectively . If both the pipes are opened simultaneously , how much time will be taken to fill the tank?

Options

A. `4 (1)/(2)`
B. 7h
C. 6h
D. 10h

Answer

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

### Step 1: Determine the rate of filling for each pipe.
- Pipe A can fill the tank in 18 hours. Therefore, the rate of filling for pipe A is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{18 \text{ hours}} = \frac{1}{18} \text{ tank per hour} \]

- Pipe B can fill the tank in 6 hours. Therefore, the rate of filling for pipe B is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tank per hour} \]

### Step 2: Add the rates of both pipes.
When both pipes are opened simultaneously, their rates of filling add up:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{18} + \frac{1}{6} \]

### Step 3: Find a common denominator to add the fractions.
The least common multiple of 18 and 6 is 18. We can rewrite \(\frac{1}{6}\) as \(\frac{3}{18}\):
\[ \text{Combined Rate} = \frac{1}{18} + \frac{3}{18} = \frac{4}{18} \]

### Step 4: Simplify the combined rate.
\[ \text{Combined Rate} = \frac{4}{18} = \frac{2}{9} \text{ tank per hour} \]

### Step 5: Calculate the time taken to fill the tank.
To find the time taken to fill 1 tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{2}{9} \text{ tank per hour}} = \frac{9}{2} \text{ hours} \]

### Step 6: Convert the time into hours and minutes.
\(\frac{9}{2}\) hours can be expressed as:
\[ 4.5 \text{ hours} = 4 \text{ hours and } 30 \text{ minutes} \]

### Final Answer:
The time taken to fill the tank when both pipes A and B are opened simultaneously is **4 hours and 30 minutes**.

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