A cistern is normally filled in 8 hours but takes two hours longer to fill because of a leak in its bottom. If the cistern is full, the leak will empty it in ?

Options

A. A) 20 hrs
B. B) 28 hrs
C. C) 36 hrs
D. C) 36 hrs

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the filling and leaking rates
The cistern is normally filled in 8 hours. This means the filling rate of the cistern is:
\[ \text{Filling Rate} = \frac{1 \text{ cistern}}{8 \text{ hours}} = \frac{1}{8} \text{ cistern per hour} \]

### Step 2: Determine the time taken to fill the cistern with the leak
Due to the leak, it takes 2 hours longer to fill the cistern. Therefore, the total time taken to fill the cistern with the leak is:
\[ \text{Total Time with Leak} = 8 \text{ hours} + 2 \text{ hours} = 10 \text{ hours} \]

### Step 3: Calculate the effective filling rate with the leak
Since it takes 10 hours to fill the cistern with the leak, the effective filling rate with the leak is:
\[ \text{Effective Filling Rate} = \frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10} \text{ cistern per hour} \]

### Step 4: Set up the equation to find the leaking rate
Let \( x \) be the time in hours it takes for the leak to empty the full cistern. The leaking rate is:
\[ \text{Leaking Rate} = \frac{1 \text{ cistern}}{x \text{ hours}} = \frac{1}{x} \text{ cistern per hour} \]

### Step 5: Relate the filling rate, leaking rate, and effective filling rate
The effective filling rate can be expressed as the filling rate minus the leaking rate:
\[ \frac{1}{10} = \frac{1}{8} - \frac{1}{x} \]

### Step 6: Solve for \( x \)
To solve for \( x \), we first find a common denominator for the fractions on the right-hand side. The least common multiple of 8 and 10 is 40. Rewriting the equation gives:
\[ \frac{1}{10} = \frac{5}{40} - \frac{4.8}{40} \]
This simplifies to:
\[ \frac{1}{10} = \frac{5 - 4}{40} = \frac{1}{40} \]
So, we have:
\[ \frac{1}{x} = \frac{1}{8} - \frac{1}{10} \]

Finding a common denominator (which is 40):
\[ \frac{1}{8} = \frac{5}{40}, \quad \frac{1}{10} = \frac{4}{40} \]
Thus:
\[ \frac{1}{x} = \frac{5}{40} - \frac{4}{40} = \frac{1}{40} \]

### Step 7: Calculate \( x \)
Taking the reciprocal gives:
\[ x = 40 \text{ hours} \]

### Conclusion
The leak will empty the full cistern in **40 hours**.

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