Pipes A and B can fill a tank in 5 hours and 20 hours respectively. If both pipes are opened then, how much time (in hours) it will take to fill the tank?

Options

A. 4.5
B. 3
C. 3.5
D. 4

Answer

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

### Step 1: Determine the filling rates of each pipe.
- Pipe A can fill the tank in 5 hours. Therefore, its rate of filling is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{5 \text{ hours}} = \frac{1}{5} \text{ tanks per hour} \]
- Pipe B can fill the tank in 20 hours. Therefore, its rate of filling is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{20 \text{ hours}} = \frac{1}{20} \text{ tanks per hour} \]

### Step 2: Add the rates of both pipes to find the combined rate.
- When both pipes are opened together, their combined filling rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{5} + \frac{1}{20} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 20 is 20. Therefore:
\[ \frac{1}{5} = \frac{4}{20} \]
So,
\[ \text{Combined Rate} = \frac{4}{20} + \frac{1}{20} = \frac{5}{20} = \frac{1}{4} \text{ tanks per hour} \]

### Step 3: Calculate the time taken to fill the tank.
- The time taken to fill the tank when both pipes are opened can be calculated using the formula:
\[ \text{Time} = \frac{\text{Work}}{\text{Efficiency}} \]
- Here, the work is 1 tank and the efficiency (combined rate) is \(\frac{1}{4}\) tanks per hour:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{4} \text{ tanks per hour}} = 1 \times 4 = 4 \text{ hours} \]

### Final Answer:
It will take 4 hours to fill the tank when both pipes A and B are opened together.

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