Pipe A can fill the tank in 80 minutes and pipe B in 120 minutes. Then in how much time both the pipe can together fill the tank?

Answer

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

### Step 1: Determine the filling rates of each pipe
- **Pipe A** can fill the tank in **80 minutes**. Therefore, its rate of work is:
\[ \text{Rate of Pipe A} = \frac{1 \text{ tank}}{80 \text{ minutes}} = \frac{1}{80} \text{ tanks per minute} \]

- **Pipe B** can fill the tank in **120 minutes**. Therefore, its rate of work is:
\[ \text{Rate of Pipe B} = \frac{1 \text{ tank}}{120 \text{ minutes}} = \frac{1}{120} \text{ tanks per minute} \]

### Step 2: Find the combined rate of both pipes
To find the combined rate of both pipes working together, we add their individual rates:
\[ \text{Combined Rate} = \text{Rate of Pipe A} + \text{Rate of Pipe B} = \frac{1}{80} + \frac{1}{120} \]

### Step 3: Calculate the least common multiple (LCM)
To add the fractions, we need a common denominator. The LCM of 80 and 120 can be calculated:
- The prime factorization of 80 is \(2^4 \times 5\).
- The prime factorization of 120 is \(2^3 \times 3 \times 5\).
- The LCM is \(2^4 \times 3 \times 5 = 240\).

### Step 4: Convert the rates to a common denominator
Now we can convert the rates to have a common denominator of 240:
\[ \text{Rate of Pipe A} = \frac{1}{80} = \frac{3}{240} \]
\[ \text{Rate of Pipe B} = \frac{1}{120} = \frac{2}{240} \]
Thus, the combined rate becomes:
\[ \text{Combined Rate} = \frac{3}{240} + \frac{2}{240} = \frac{5}{240} \]

### Step 5: Calculate the time taken to fill the tank together
The time taken to fill the tank when both pipes are working together is the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Combined Rate}} = \frac{1}{\frac{5}{240}} = \frac{240}{5} = 48 \text{ minutes} \]

### Final Answer
Thus, both Pipe A and Pipe B together can fill the tank in **48 minutes**.

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