Two pipes A & B can fill a tank in 15hr. & 20hr. respectively while pipe C can empty the completely filled tank 25hr. Three pipes are opened simultaneously. After 10hr. pipe C is closed. After what time will the tank be completely filled ?

Answer

To solve the problem step by step, we will follow these procedures:

### Step 1: Determine the total work done by each pipe.
- Pipe A can fill the tank in 15 hours, so its work rate is \( \frac{1}{15} \) of the tank per hour.
- Pipe B can fill the tank in 20 hours, so its work rate is \( \frac{1}{20} \) of the tank per hour.
- Pipe C can empty the tank in 25 hours, so its work rate is \( -\frac{1}{25} \) of the tank per hour (negative because it empties).

### Step 2: Calculate the Least Common Multiple (LCM) to find a common work unit.
- The LCM of 15, 20, and 25 is 300. This means we can consider the tank as having a capacity of 300 units.

### Step 3: Calculate the efficiency of each pipe in terms of units per hour.
- Efficiency of Pipe A:
\[ \text{Efficiency of A} = \frac{300}{15} = 20 \text{ units/hour} \]
- Efficiency of Pipe B:
\[ \text{Efficiency of B} = \frac{300}{20} = 15 \text{ units/hour} \]
- Efficiency of Pipe C:
\[ \text{Efficiency of C} = -\frac{300}{25} = -12 \text{ units/hour} \]

### Step 4: Calculate the combined efficiency of all three pipes when they are open.
- Combined efficiency when A, B, and C are open:
\[ \text{Total Efficiency} = 20 + 15 - 12 = 23 \text{ units/hour} \]

### Step 5: Calculate the amount of work done in the first 10 hours.
- Work done in 10 hours:
\[ \text{Work done} = 23 \times 10 = 230 \text{ units} \]

### Step 6: Determine how much work is left to fill the tank.
- Total work to fill the tank is 300 units, so the remaining work is:
\[ \text{Remaining Work} = 300 - 230 = 70 \text{ units} \]

### Step 7: Calculate the combined efficiency of pipes A and B after pipe C is closed.
- Combined efficiency of A and B:
\[ \text{Efficiency of A and B} = 20 + 15 = 35 \text{ units/hour} \]

### Step 8: Calculate the time required to finish the remaining work.
- Time required to fill the remaining 70 units:
\[ \text{Time} = \frac{70}{35} = 2 \text{ hours} \]

### Final Answer:
After pipe C is closed, it will take an additional 2 hours for the tank to be completely filled.

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