A pipe can fill a water tank in 8 hours. Due to a leak in the bottom of the water tank, it is filled in 10 hours. If the water tank is full, how much time will the leak take to empty it?

Options

A. 36 hours
B. 42 hours
C. 45 hours
D. 40 hours

Answer

To solve the problem step by step, we need to find out how much time the leak will take to empty the tank. Here’s how we can approach it:

### Step 1: Determine the filling rates of the pipe and the leak.
- The pipe can fill the tank in 8 hours. Therefore, the filling rate of the pipe (P) is:
\[ P = \frac{1}{8} \text{ tank per hour} \]

- The tank is filled in 10 hours due to the leak. Therefore, the effective filling rate with the leak (E) is:
\[ E = \frac{1}{10} \text{ tank per hour} \]

### Step 2: Calculate the rate of the leak.
- The effective filling rate (E) is the rate of the pipe minus the rate of the leak (L):
\[ E = P - L \]
Substituting the values we have:
\[ \frac{1}{10} = \frac{1}{8} - L \]

### Step 3: Solve for the leak's rate (L).
- Rearranging the equation to find L:
\[ L = \frac{1}{8} - \frac{1}{10} \]

- To subtract these fractions, find a common denominator. The least common multiple of 8 and 10 is 40. Thus:
\[ L = \frac{5}{40} - \frac{4}{40} = \frac{1}{40} \text{ tank per hour} \]

### Step 4: Determine the time taken by the leak to empty the tank.
- If the leak can empty the tank at a rate of \(\frac{1}{40}\) tank per hour, then the time (T) taken to empty the full tank is:
\[ T = \frac{1}{L} = \frac{1}{\frac{1}{40}} = 40 \text{ hours} \]

### Conclusion:
The leak will take **40 hours** to empty the tank.

---