Two pipes can fill a cistern in 30 and 15h respectively. The pipes are opened simultaeously and it is found that due to leakage in the botttom, 5h extra are taken for cistern to be filled up. If the cistern is full, in what time would thhe leak empty it ?

Answer

To solve the problem step by step, we will first determine the rates at which the pipes fill the cistern and then account for the leakage.

### Step 1: Determine the filling rates of the pipes
- Pipe A can fill the cistern in 30 hours. Therefore, its rate of filling is:
\[ \text{Rate of A} = \frac{1}{30} \text{ cisterns per hour} \]

- Pipe B can fill the cistern in 15 hours. Therefore, its rate of filling is:
\[ \text{Rate of B} = \frac{1}{15} \text{ cisterns per hour} \]

### Step 2: Calculate the combined filling rate of both pipes
When both pipes are opened simultaneously, their combined rate of filling the cistern is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{30} + \frac{1}{15} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 15 is 30.
\[ \frac{1}{30} + \frac{2}{30} = \frac{3}{30} = \frac{1}{10} \text{ cisterns per hour} \]

### Step 3: Determine the time taken to fill the cistern without leakage
If there were no leakage, the time taken to fill the cistern would be:
\[ \text{Time without leakage} = \frac{1 \text{ cistern}}{\text{Combined Rate}} = \frac{1}{\frac{1}{10}} = 10 \text{ hours} \]

### Step 4: Account for the leakage
It is given that due to leakage, it takes 5 extra hours to fill the cistern. Therefore, the actual time taken to fill the cistern is:
\[ \text{Actual Time} = 10 + 5 = 15 \text{ hours} \]

### Step 5: Calculate the effective rate with leakage
The effective rate of filling the cistern considering the leakage is:
\[ \text{Effective Rate} = \frac{1 \text{ cistern}}{15 \text{ hours}} = \frac{1}{15} \text{ cisterns per hour} \]

### Step 6: Determine the rate of leakage
Let the rate of leakage be \( L \) (in cisterns per hour). The effective rate can be expressed as:
\[ \text{Effective Rate} = \text{Combined Rate} - L \]
Substituting the known values:
\[ \frac{1}{15} = \frac{1}{10} - L \]

### Step 7: Solve for the rate of leakage
Rearranging the equation gives:
\[ L = \frac{1}{10} - \frac{1}{15} \]
Finding a common denominator (30):
\[ L = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} \text{ cisterns per hour} \]

### Step 8: Find the time taken by the leak to empty the cistern
If the leak empties the cistern at a rate of \( \frac{1}{30} \) cisterns per hour, the time taken to empty the full cistern is:
\[ \text{Time to empty} = \frac{1 \text{ cistern}}{L} = \frac{1}{\frac{1}{30}} = 30 \text{ hours} \]

### Final Answer
The leak will empty the cistern in **30 hours**.

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