Four pipe A, B, C and D can fill a cistern in 20, 25, 40 and 50 hours respectively.
The first pipe A was opened at 6:00 am, B at 8:00 am, C at 9:00 am and D at 10:00 am. When will the cistern be full ?

Options

A. `4:18` pm
B. `3:09` pm
C. `12:15` pm
D. `11:09` am

Answer

To solve the problem, we need to determine how much of the cistern is filled by each pipe and when the cistern will be full. Here’s a step-by-step solution:

### Step 1: Determine the filling rates of each pipe
- Pipe A can fill the cistern in 20 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{20} \text{ cisterns per hour} = 10 \text{ units/hour} \]
- Pipe B can fill the cistern in 25 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{25} \text{ cisterns per hour} = 8 \text{ units/hour} \]
- Pipe C can fill the cistern in 40 hours. Therefore, its rate is:
\[ \text{Rate of C} = \frac{1}{40} \text{ cisterns per hour} = 5 \text{ units/hour} \]
- Pipe D can fill the cistern in 50 hours. Therefore, its rate is:
\[ \text{Rate of D} = \frac{1}{50} \text{ cisterns per hour} = 4 \text{ units/hour} \]

### Step 2: Calculate the filling done by each pipe at different times
- **From 6:00 AM to 8:00 AM (2 hours)**: Only Pipe A is open.
\[ \text{Amount filled by A} = 2 \times 10 = 20 \text{ units} \]

- **From 8:00 AM to 9:00 AM (1 hour)**: Pipes A and B are open.
\[ \text{Amount filled by A and B} = 1 \times (10 + 8) = 18 \text{ units} \]

- **From 9:00 AM to 10:00 AM (1 hour)**: Pipes A, B, and C are open.
\[ \text{Amount filled by A, B, and C} = 1 \times (10 + 8 + 5) = 23 \text{ units} \]

- **From 10:00 AM onwards**: All pipes A, B, C, and D are open.
\[ \text{Total rate of A, B, C, and D} = 10 + 8 + 5 + 4 = 27 \text{ units/hour} \]

### Step 3: Calculate the total amount filled by 10:00 AM
- Total filled by 10:00 AM:
\[ \text{Total filled} = 20 + 18 + 23 = 61 \text{ units} \]

### Step 4: Calculate the remaining capacity of the cistern
- The total capacity of the cistern is 200 units.
- Remaining capacity:
\[ \text{Remaining capacity} = 200 - 61 = 139 \text{ units} \]

### Step 5: Calculate the time required to fill the remaining capacity
- Time required to fill the remaining 139 units at a rate of 27 units/hour:
\[ \text{Time} = \frac{139}{27} \approx 5.148 \text{ hours} \approx 5 \text{ hours and } 9 \text{ minutes} \]

### Step 6: Determine the final time when the cistern will be full
- Since all pipes are open from 10:00 AM, we add 5 hours and 9 minutes to 10:00 AM:
\[ 10:00 \text{ AM} + 5 \text{ hours } 9 \text{ minutes} = 3:09 \text{ PM} \]

### Conclusion
The cistern will be full at **3:09 PM**.

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