Three taps can fill a cistern in 18 minute, 15 minute and 10 minute respectively. The cistern being empty, all the three taps are kept open. After 3 minute. the first tap is closed. Counting time from that moment, the cistern will be full in

Options

A. 5 minute
B. 1 minute
C. 3 minute
D. 2 minute

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the filling rates of each tap
- Tap A can fill the cistern in 18 minutes. Therefore, its rate is:
\[ \text{Rate of A} = \frac{90 \text{ liters}}{18 \text{ minutes}} = 5 \text{ liters/minute} \]
- Tap B can fill the cistern in 15 minutes. Therefore, its rate is:
\[ \text{Rate of B} = \frac{90 \text{ liters}}{15 \text{ minutes}} = 6 \text{ liters/minute} \]
- Tap C can fill the cistern in 10 minutes. Therefore, its rate is:
\[ \text{Rate of C} = \frac{90 \text{ liters}}{10 \text{ minutes}} = 9 \text{ liters/minute} \]

### Step 2: Calculate the total filling rate when all taps are open
- The combined rate when all three taps are open is:
\[ \text{Total Rate} = 5 + 6 + 9 = 20 \text{ liters/minute} \]

### Step 3: Calculate the amount filled in the first 3 minutes
- In 3 minutes, the amount filled by all three taps is:
\[ \text{Amount filled in 3 minutes} = 20 \text{ liters/minute} \times 3 \text{ minutes} = 60 \text{ liters} \]

### Step 4: Calculate the remaining volume to be filled
- The total capacity of the cistern is 90 liters. After 3 minutes, the remaining volume is:
\[ \text{Remaining volume} = 90 - 60 = 30 \text{ liters} \]

### Step 5: Determine the new filling rate after closing Tap A
- After 3 minutes, Tap A is closed, leaving Taps B and C open. Their combined rate is:
\[ \text{New Total Rate} = 6 + 9 = 15 \text{ liters/minute} \]

### Step 6: Calculate the time required to fill the remaining volume
- To find out how long it will take to fill the remaining 30 liters with the new rate:
\[ \text{Time to fill remaining volume} = \frac{30 \text{ liters}}{15 \text{ liters/minute}} = 2 \text{ minutes} \]

### Final Step: Calculate the total time taken
- The total time taken to fill the cistern is:
\[ \text{Total time} = 3 \text{ minutes} + 2 \text{ minutes} = 5 \text{ minutes} \]

### Conclusion
The cistern will be full in **5 minutes** after the first tap is closed.

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