A tank has two inlets which can fill it in 6 hours and 8 hours, respectively. An outlet can empty the full tank in 10 hours. If all three pipes are opened together in an empty tank how much time will they take to fill the tank completely?
(a)`5(5)/(46)` hours
(b)`6 (5)/(23)` hours
(c)`6 (5)/(46)` hours
(d)`5 (5)/(23)` hours

Options

A. `5(5)/(46)` hours
B. `6 (5)/(23)` hours
C. `6 (5)/(46)` hours
D. `5 (5)/(23)` hours

Answer

To solve the problem, we need to determine the combined filling rate of the two inlets and the outlet. Here’s a step-by-step breakdown:

### Step 1: Determine the filling rates of the inlets and outlet
- **Inlet A** fills the tank in 6 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{6} \text{ tanks per hour} \]

- **Inlet B** fills the tank in 8 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{8} \text{ tanks per hour} \]

- **Outlet C** empties the tank in 10 hours. Therefore, its rate is:
\[ \text{Rate of C} = -\frac{1}{10} \text{ tanks per hour} \quad (\text{negative because it empties the tank}) \]

### Step 2: Calculate the combined rate of all three pipes
To find the combined rate when all three pipes are opened together, we add the rates of A and B and subtract the rate of C:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the values:
\[ \text{Combined Rate} = \frac{1}{6} + \frac{1}{8} - \frac{1}{10} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 6, 8, and 10 is 120. We convert each rate to have a denominator of 120:
\[ \frac{1}{6} = \frac{20}{120}, \quad \frac{1}{8} = \frac{15}{120}, \quad \frac{1}{10} = \frac{12}{120} \]

### Step 4: Combine the rates
Now we can combine the rates:
\[ \text{Combined Rate} = \frac{20}{120} + \frac{15}{120} - \frac{12}{120} = \frac{20 + 15 - 12}{120} = \frac{23}{120} \text{ tanks per hour} \]

### Step 5: Calculate the time to fill the tank
To find the time taken to fill one tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{23}{120} \text{ tanks per hour}} = \frac{120}{23} \text{ hours} \]

### Step 6: Convert to mixed fraction
Now, we convert \(\frac{120}{23}\) into a mixed fraction:
- Divide 120 by 23, which gives 5 with a remainder of 5.
- Therefore, \(\frac{120}{23} = 5 \frac{5}{23}\) hours.

### Final Answer
Thus, the time taken to fill the tank completely when all three pipes are opened together is:
\[ \text{Time} = 5 \frac{5}{23} \text{ hours} \]

### Conclusion
The correct option is:
(d) \(5 \frac{5}{23}\) hours.