Question

There is a leak in the bottom of the tank. This leak can empty a full tank in 8 hours. When the tank is full, a tap is opened into the tank which admits $6$ litres per hour and the tank is now emptied in 12 hours. What is the capacity of the tank?

• A. $28.8$ litres
• B. $36$ litres
• C. $144$ litres
• D. Cannot be determined

Hint:

Always carefully set up rate equations when dealing with filling and emptying problems. Use "Rate × Time = Work/Volume" logic.

Answer 1

The Correct Option is A

Let the capacity of the tank be $C$ litres.
Step 1: Rate of emptying by leak
The leak can empty the full tank in $8$ hours.
Therefore, the leak’s rate = $\frac{C}{8}$ litres per hour (negative contribution).

Step 2: Effect of tap opening
When the tap is opened, it admits water at $6$ litres per hour (positive contribution).

Step 3: Net rate when both leak and tap are working
When both work together, the tank gets emptied in $12$ hours.
So, net outflow rate = $\frac{C}{12}$ litres per hour.

Net outflow rate equation:
\[ \text{(Leak rate)} - \text{(Tap inflow rate)} = \text{Net rate} \] \[ \frac{C}{8} - 6 = \frac{C}{12} \]
Step 4: Solve for $C$
Multiply through by $24$ to eliminate denominators:
\[ 3C - 144 = 2C \] \[ 3C - 2C = 144 \] \[ C = 144 \div 5 = 28.8 \] Wait, let's check: Here, a mistake—correct solving is:
\[ \frac{C}{8} - 6 = \frac{C}{12} \] Multiply through by $24$:
\[ 3C - 144 = 2C \] \[ C = 144 \] Now, check units: A leak of $144/8 = 18$ litres/hour, tap adds $6$ litres/hour, net rate = $18 - 6 = 12$ litres/hour, which empties in $144/12 = 12$ hours — correct.
Thus, capacity is $144$ litres. This means option (C) is correct, not (A).