Pipes A and B can fill a tank in 12 hours and 16 hours respectively and pipe C can empty the full tank in 24 hours. All 3 three pipes are opened together, but after 4 hours pipe A is closed. In how many hours from the beginning the tank be filled?

Options

A. 24
B. 28
C. 30
D. 32

Answer

To solve the problem step by step, we will first determine the rates at which each pipe works, then calculate how much of the tank is filled in the first 4 hours, and finally determine how long it will take to fill the remaining part of the tank after pipe A is closed.

### Step 1: Determine the capacity of the tank
Let's assume the capacity of the tank is the least common multiple (LCM) of the times taken by pipes A, B, and C to fill or empty the tank.

- Pipe A can fill the tank in 12 hours.
- Pipe B can fill the tank in 16 hours.
- Pipe C can empty the tank in 24 hours.

The LCM of 12, 16, and 24 is 48. Therefore, we assume the capacity of the tank is 48 liters.

**Hint:** To find the LCM, list the multiples of each number until you find the smallest common multiple.

### Step 2: Calculate the filling rate of each pipe
- Rate of Pipe A = 48 liters / 12 hours = 4 liters/hour
- Rate of Pipe B = 48 liters / 16 hours = 3 liters/hour
- Rate of Pipe C = 48 liters / 24 hours = 2 liters/hour (but since it empties, we will subtract this)

**Hint:** The rate of filling or emptying can be calculated by dividing the total capacity by the time taken.

### Step 3: Calculate the net filling rate when all pipes are open
When all three pipes are open, the net filling rate is:
Net filling rate = Rate of A + Rate of B - Rate of C
= 4 liters/hour + 3 liters/hour - 2 liters/hour
= 5 liters/hour

**Hint:** Remember to subtract the rate of the emptying pipe since it reduces the total amount of water in the tank.

### Step 4: Calculate how much water is filled in the first 4 hours
In the first 4 hours, with all three pipes open:
Water filled = Net filling rate × Time
= 5 liters/hour × 4 hours = 20 liters

**Hint:** To find the total amount filled over a period, multiply the rate by the time.

### Step 5: Calculate the remaining capacity of the tank
Remaining capacity = Total capacity - Water filled
= 48 liters - 20 liters = 28 liters

**Hint:** Always subtract the amount filled from the total capacity to find out how much is left.

### Step 6: Calculate the new net filling rate after pipe A is closed
After 4 hours, pipe A is closed, so we only have pipe B and pipe C working:
New net filling rate = Rate of B - Rate of C
= 3 liters/hour - 2 liters/hour = 1 liter/hour

**Hint:** When a pipe is closed, recalculate the net rate using only the remaining pipes.

### Step 7: Calculate the time required to fill the remaining capacity
Time to fill remaining capacity = Remaining capacity / New net filling rate
= 28 liters / 1 liter/hour = 28 hours

**Hint:** To find the time to fill a certain amount, divide the amount by the rate at which it is being filled.

### Step 8: Calculate the total time from the beginning
Total time = Time for first 4 hours + Time to fill remaining capacity
= 4 hours + 28 hours = 32 hours

**Hint:** Always add the time spent in different phases of the task to get the total time.

### Final Answer
The tank will be filled in **32 hours** from the beginning.

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