A pipe can fill a tank in 10 h. Due to a leak in the bottom, it fills the tank in 20 h. If the tank is full, how much time will the leak take to empty it?

Answer

To solve the problem step by step, we can follow these calculations:

### 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 the pipe and the time taken with the leak. The pipe fills the tank in 10 hours, and with the leak, it fills in 20 hours.

- **LCM of 10 and 20** = 20 units (this will make calculations easier).

### Step 2: Calculate the filling rate of the pipe
The pipe fills the tank in 10 hours. Therefore, in 1 hour, the pipe fills:

- **Filling rate of the pipe (A)** = Total capacity / Time taken by the pipe
- Filling rate of A = 20 units / 10 hours = 2 units per hour.

### Step 3: Calculate the combined filling rate of the pipe and the leak
When both the pipe and the leak are working together, they fill the tank in 20 hours. Therefore, in 1 hour, they fill:

- **Combined filling rate (A + L)** = Total capacity / Time taken with the leak
- Combined rate = 20 units / 20 hours = 1 unit per hour.

### Step 4: Set up the equation for the leak's rate
From the previous steps, we have the following equations:
- A's rate = 2 units/hour
- A + L's rate = 1 unit/hour

Now, we can set up the equation:
- 2 units/hour (A) + L's rate = 1 unit/hour

### Step 5: Solve for the leak's rate
Rearranging the equation gives us:
- L's rate = 1 unit/hour - 2 units/hour
- L's rate = -1 unit/hour.

### Step 6: Determine the time taken by the leak to empty the tank
The leak's rate indicates how quickly it empties the tank. Since the leak's rate is -1 unit/hour, it means the leak empties the tank at a rate of 1 unit per hour.

To find out how long it takes for the leak to empty the full tank, we can use the formula:

- Time taken by the leak = Capacity of the tank / Leak's rate
- Time taken by the leak = 20 units / 1 unit/hour = 20 hours.

### Final Answer:
The leak will take **20 hours** to empty the tank.

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