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 for two taps A and B to fill a tank when opened simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Filling Rates of Each Tap:**
- Tap A can fill the tank in 15 minutes. Therefore, its filling rate is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{15 \text{ minutes}} = \frac{1}{15} \text{ tanks per minute} \]
- Tap B can fill the tank in 20 minutes. Therefore, its filling rate is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{20 \text{ minutes}} = \frac{1}{20} \text{ tanks per minute} \]

2. **Calculate the Combined Filling Rate:**
- When both taps are opened simultaneously, their rates add up:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{15} + \frac{1}{20} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 20 is 60.
\[ \frac{1}{15} = \frac{4}{60} \quad \text{and} \quad \frac{1}{20} = \frac{3}{60} \]
- Therefore:
\[ \text{Combined Rate} = \frac{4}{60} + \frac{3}{60} = \frac{7}{60} \text{ tanks per minute} \]

3. **Calculate the Time to Fill the Tank:**
- The time taken to fill one tank can be calculated using the formula:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Combined Rate}} = \frac{1}{\frac{7}{60}} = \frac{60}{7} \text{ minutes} \]
- Converting this to a mixed number:
\[ \frac{60}{7} = 8 \frac{4}{7} \text{ minutes} \]

### Final Answer:
The time taken to fill the tank when both taps A and B are opened simultaneously is \( 8 \frac{4}{7} \) minutes.

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