A tap can completely fill a water tank in $8$ hours. The water tank has a hole in it through which the water leaks out. The leakage will cause the full water tank to get empty in $12$ hours. How much time will it take for the tap to fill the tank completely with the hole?

### Concept & Net Filling Rate with Leakage When a tank is being filled by a tap but simultaneously drained by a leak, the effective filling rate is the difference between the tap's filling rate and the leak's emptying rate. $$ \text{Net Rate} = \text{Filling Rate} - \text{Emptying Rate} $$ ### Step-by-Step Solution * **Given Rates:** Time taken by the tap to fill the tank = $8$ hours. Part filled by the tap in $1$ hour = $\frac{1}{8}$. Time taken by the leak to empty the tank = $12$ hours. Part emptied by the leak in $1$ hour = $\frac{1}{12}$. * **Calculate Net Rate:** Net part filled in $1$ hour = $\frac{1}{8} - \frac{1}{12}$ * **Solve the Fraction:** Find the LCM of $8$ and $12$, which is $24$. Net part filled = $\frac{3}{24} - \frac{2}{24} = \frac{1}{24}$ * **Find Total Time:** Since $\frac{1}{24}$ of the tank is filled in $1$ hour, it will take exactly $24$ hours to fill the entire tank. ### Exam Strategy & Shortcut Use the direct formula for one inlet and one outlet: $\frac{xy}{y - x}$. Here, $x = 8$ and $y = 12$. Calculation: $\frac{8 \times 12}{12 - 8} = \frac{96}{4} = 24$ hours. This bypasses fraction arithmetic entirely. ### Common Pitfall A common mistake is adding the two rates ($\frac{1}{8} + \frac{1}{12}$) instead of subtracting them, which assumes both are filling the tank, leading to an incorrect answer of $4.8$ hours. ### Final Answer Therefore, the correct answer is **$24$ hours**.