A cistern has a leak which would empty it in 6 hours. A tap is turned on which fills the cistern at the rate of10 liters per hour and then it is emptied in 15 hours. What is the capacity of the cistern?

Options

A. 100 litres
B. 166.66 litres
C. 60.66 liters
D. none of these

Answer

To solve the problem, we need to determine the capacity of the cistern based on the information provided about the leak and the tap. Let's break it down step by step.

### Step 1: Understand the problem
We have a cistern with a leak that can empty it in 6 hours. A tap fills the cistern at the rate of 10 liters per hour. When the tap is turned on, the combined effect of the tap and the leak empties the cistern in 15 hours. We need to find the capacity of the cistern.

### Step 2: Define the capacity of the cistern
Let’s assume the capacity of the cistern is \( C \) liters.

### Step 3: Calculate the leak's rate
The leak can empty the cistern in 6 hours, so the rate at which the leak empties the cistern is:
\[ \text{Leak rate} = \frac{C}{6} \text{ liters per hour} \]

### Step 4: Calculate the tap's rate
The tap fills the cistern at the rate of 10 liters per hour.

### Step 5: Calculate the combined rate when both the tap and leak are functioning
When the tap is turned on, the cistern is emptied in 15 hours. Therefore, the combined rate of the leak and the tap is:
\[ \text{Combined rate} = \frac{C}{15} \text{ liters per hour} \]

### Step 6: Set up the equation
The combined effect of the tap filling and the leak emptying can be expressed as:
\[ \text{Rate of tap} - \text{Rate of leak} = \text{Combined rate} \]
Substituting the values we have:
\[ 10 - \frac{C}{6} = \frac{C}{15} \]

### Step 7: Solve the equation
To solve for \( C \), we first eliminate the fractions by multiplying through by the least common multiple (LCM) of 6 and 15, which is 30:
\[ 30 \left(10 - \frac{C}{6}\right) = 30 \left(\frac{C}{15}\right) \]
This simplifies to:
\[ 300 - 5C = 2C \]
Now, combine like terms:
\[ 300 = 5C + 2C \]
\[ 300 = 7C \]
Now, divide both sides by 7:
\[ C = \frac{300}{7} \approx 42.86 \text{ liters} \]

### Step 8: Conclusion
The capacity of the cistern is approximately 42.86 liters.