Two taps A and B can fill a tank in 15 minutes and 20 minutes respectively.If both the taps are opened simultaneouly, then in how much time can the empty tank be filled ?

Options

A. `8(4)/(7)`hours
B. 8 minutes
C. `8(4)/(7)` minutes
D. 16 minutes

Explanation

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

### Step 1: Determine the filling rates of taps A and B.
- Tap A can fill the tank in 15 minutes.
- Tap B can fill the tank in 20 minutes.

### Step 2: Calculate the filling rate of each tap.
- The rate of tap A is \( \frac{1 \text{ tank}}{15 \text{ minutes}} = \frac{1}{15} \text{ tanks per minute} \).
- The rate of tap B is \( \frac{1 \text{ tank}}{20 \text{ minutes}} = \frac{1}{20} \text{ tanks per minute} \).

### Step 3: Find a common volume for the tank.
- To make calculations easier, we can assume the total volume of the tank is the least common multiple (LCM) of 15 and 20.
- The LCM of 15 and 20 is 60. Thus, we can assume the tank's volume is 60 units.

### Step 4: Calculate the filling capacity in units per minute.
- For tap A:
\[ \text{Units filled by A in 1 minute} = \frac{60 \text{ units}}{15 \text{ minutes}} = 4 \text{ units per minute} \]
- For tap B:
\[ \text{Units filled by B in 1 minute} = \frac{60 \text{ units}}{20 \text{ minutes}} = 3 \text{ units per minute} \]

### Step 5: Combine the filling rates of both taps.
- When both taps are opened together, the total filling rate is:
\[ \text{Total rate} = 4 \text{ units per minute} + 3 \text{ units per minute} = 7 \text{ units per minute} \]

### Step 6: Calculate the total time to fill the tank.
- The total time to fill the tank can be calculated by dividing the total volume of the tank by the combined filling rate:
\[ \text{Time} = \frac{\text{Total Volume}}{\text{Total Rate}} = \frac{60 \text{ units}}{7 \text{ units per minute}} = \frac{60}{7} \text{ minutes} \]

### Step 7: Simplify the time.
- Dividing 60 by 7 gives:
\[ 60 \div 7 = 8 \text{ minutes} + \frac{4}{7} \text{ minutes} \]
- Thus, the total time to fill the tank is approximately \( 8 \frac{4}{7} \) minutes.

### Final Answer:
The tank will be filled in \( 8 \frac{4}{7} \) minutes when both taps A and B are opened simultaneously.

---