A cub can be filled in 20 minutes but there is a leakage in it which can empty the full cub in 60 minutes. In how many minutes it can be filled ?

Answer

To solve the problem of how long it takes to fill a cup with a leak, we can break it down step by step.

### Step 1: Determine the filling rate of the cup
The cup can be filled in 20 minutes. Therefore, the rate of filling (F) is:
\[ F = \frac{1 \text{ cup}}{20 \text{ minutes}} = \frac{1}{20} \text{ cups per minute} \]

**Hint:** The filling rate is calculated as the total work done (filling one cup) divided by the time taken to fill it.

### Step 2: Determine the emptying rate due to the leak
The leak can empty the cup in 60 minutes. Therefore, the rate of emptying (L) is:
\[ L = \frac{1 \text{ cup}}{60 \text{ minutes}} = \frac{1}{60} \text{ cups per minute} \]

**Hint:** The emptying rate is calculated similarly to the filling rate, based on the time taken to empty the cup.

### Step 3: Calculate the net filling rate
Since the leak is emptying the cup while it is being filled, we need to subtract the emptying rate from the filling rate to find the net filling rate (N):
\[ N = F - L = \frac{1}{20} - \frac{1}{60} \]

To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 20 and 60 is 60. Thus, we convert the rates:
\[ \frac{1}{20} = \frac{3}{60} \]
Now, substituting back into the equation:
\[ N = \frac{3}{60} - \frac{1}{60} = \frac{2}{60} = \frac{1}{30} \text{ cups per minute} \]

**Hint:** When subtracting fractions, make sure to convert them to a common denominator.

### Step 4: Calculate the time to fill the cup with the leak
Now that we have the net filling rate, we can calculate the time taken to fill the cup (T) using the formula:
\[ T = \frac{\text{Total work}}{\text{Net rate}} = \frac{1 \text{ cup}}{\frac{1}{30} \text{ cups per minute}} = 30 \text{ minutes} \]

**Hint:** The time taken to complete a task is the total work divided by the effective rate of work.

### Final Answer
The cup can be filled in **30 minutes** considering the leakage.

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