A water tank has two holes. The `1^(st)` hole alone empties the tank in 8 minutes and `2^(nd)` hole alone empties the tank in 12 minutes. If water leaks out at a constant rate, bow many minutos will it take, if both the holes together empty the tank?

Options

A. `(4)/(5)`
B. `4 (4)/(5)`
C. `4 (3)/(5)`
D. `4 (2)/(5)`

Answer

To solve the problem step by step, we will first determine the efficiencies of both holes and then calculate the time taken when both holes work together.

### Step-by-Step Solution:

1. **Determine the Capacity of the Tank**:
- The first hole empties the tank in 8 minutes, and the second hole empties it in 12 minutes.
- To find a common capacity, we can take the least common multiple (LCM) of 8 and 12.
- LCM of 8 and 12 is 24.
- Therefore, we assume the capacity of the tank is 24 liters.

**Hint**: To find the LCM, list the multiples of each number until you find the smallest common one.

2. **Calculate the Efficiency of the First Hole**:
- The efficiency of a hole is defined as the amount of water it can empty per minute.
- For the first hole:
\[ \text{Efficiency of 1st hole} = \frac{\text{Capacity}}{\text{Time taken}} = \frac{24 \text{ liters}}{8 \text{ minutes}} = 3 \text{ liters/minute} \]

**Hint**: Efficiency is calculated by dividing the total capacity by the time taken.

3. **Calculate the Efficiency of the Second Hole**:
- For the second hole:
\[ \text{Efficiency of 2nd hole} = \frac{24 \text{ liters}}{12 \text{ minutes}} = 2 \text{ liters/minute} \]

**Hint**: Use the same formula for efficiency as in the previous step.

4. **Calculate the Combined Efficiency**:
- Now, we add the efficiencies of both holes to find the combined efficiency:
\[ \text{Combined Efficiency} = \text{Efficiency of 1st hole} + \text{Efficiency of 2nd hole} = 3 + 2 = 5 \text{ liters/minute} \]

**Hint**: When combining efficiencies, simply add the individual efficiencies together.

5. **Calculate the Time Taken to Empty the Tank with Both Holes**:
- To find the time taken when both holes are working together, we use the formula:
\[ \text{Time taken} = \frac{\text{Capacity}}{\text{Combined Efficiency}} = \frac{24 \text{ liters}}{5 \text{ liters/minute}} = \frac{24}{5} \text{ minutes} = 4.8 \text{ minutes} \]

**Hint**: Time taken can be found by dividing the total capacity by the combined efficiency.

### Final Answer:
The time taken for both holes together to empty the tank is **4.8 minutes**.