A cistern has a leak which would empty it is in 15 hours.A tap is turned on which admits 2 litres per hour into the cistern and it is now empited in 10 hours . How many litres des the cistern hold?

Options

A. 50 litres
B. 60 litres
C. 45 litres
D. 360 litres

Answer

To solve the problem step by step, we need to analyze the situation with the cistern, the leak, and the tap.

### Step 1: Determine the leak's rate
The leak can empty the cistern in 15 hours. This means that in one hour, the leak will empty:
\[ \text{Leak rate} = \frac{1 \text{ cistern}}{15 \text{ hours}} = \frac{1}{15} \text{ cistern per hour} \]

### Step 2: Determine the total time with the tap open
When the tap is turned on, the cistern is emptied in 10 hours. Therefore, the combined effect of the leak and the tap results in:
\[ \text{Combined rate} = \frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10} \text{ cistern per hour} \]

### Step 3: Determine the tap's rate
Let’s denote the rate at which the tap fills the cistern as \( T \) (in cisterns per hour). The tap admits 2 liters per hour.

### Step 4: Set up the equation
The combined rate of the leak and the tap can be expressed as:
\[ \text{Leak rate} - \text{Tap rate} = \text{Combined rate} \]
Substituting the values we have:
\[ \frac{1}{15} - T = -\frac{1}{10} \]
Here, \( T \) is the rate at which the tap fills the cistern in cisterns per hour.

### Step 5: Solve for the tap's rate
Rearranging the equation gives:
\[ T = \frac{1}{15} + \frac{1}{10} \]
To add these fractions, we need a common denominator, which is 30:
\[ T = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \text{ cistern per hour} \]

### Step 6: Calculate the capacity of the cistern
Now we know the tap fills at a rate of \( \frac{1}{6} \) cisterns per hour. Since the tap admits 2 liters per hour, we can find the total capacity of the cistern.

If the tap fills \( \frac{1}{6} \) of the cistern in one hour, then the capacity of the cistern (C) can be calculated as:
\[ C = \text{Rate of tap} \times \text{Time} = 2 \text{ liters/hour} \times 6 \text{ hours} = 12 \text{ liters} \]

### Step 7: Calculate the total capacity
Since the leak empties the cistern in 15 hours, we can also find the total capacity:
\[ \text{Total capacity} = \text{Leak rate} \times \text{Time} = \frac{1}{15} \text{ cistern/hour} \times 15 \text{ hours} = 1 \text{ cistern} \]
Since we established that 1 cistern equals 60 liters, the total capacity of the cistern is:
\[ \text{Total capacity} = 60 \text{ liters} \]

### Final Answer
The cistern holds **60 liters**.

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