Taps A and B can fill a tank in 2 and 8 hours, respectively. Tap C can empty the filled tank in 4 hours. If all the three taps are turned on simultaneously, how much time will it tank to fill the tank completely?
(a)3 hours
(b)`(9)/(2)` hours
(c)`(3)/(8)` hours
(d)`(8)/(3)` hours

Options

A. 3 hours
B. `(9)/(2)` hours
C. `(3)/(8)` hours
D. `(8)/(3)` hours

Answer

To solve the problem, we need to determine how long it will take to fill a tank when taps A, B, and C are all opened simultaneously. Here's a step-by-step breakdown of the solution:

### Step 1: Determine the filling and emptying rates of each tap.

- **Tap A** fills the tank in 2 hours. Therefore, its filling rate is:
\[ \text{Rate of A} = \frac{1}{2} \text{ tank/hour} \]

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

- **Tap C** empties the tank in 4 hours. Therefore, its emptying rate is:
\[ \text{Rate of C} = \frac{1}{4} \text{ tank/hour} \]

### Step 2: Calculate the net filling rate when all taps are open.

When all three taps are opened, the net filling rate is the sum of the filling rates of taps A and B minus the emptying rate of tap C:
\[ \text{Net Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
Substituting the values we calculated:
\[ \text{Net Rate} = \frac{1}{2} + \frac{1}{8} - \frac{1}{4} \]

### Step 3: Find a common denominator to simplify the calculation.

The least common multiple (LCM) of the denominators (2, 8, and 4) is 8. We can convert each term:
- \(\frac{1}{2} = \frac{4}{8}\)
- \(\frac{1}{8} = \frac{1}{8}\)
- \(\frac{1}{4} = \frac{2}{8}\)

Now substituting back into the equation:
\[ \text{Net Rate} = \frac{4}{8} + \frac{1}{8} - \frac{2}{8} = \frac{4 + 1 - 2}{8} = \frac{3}{8} \text{ tank/hour} \]

### Step 4: Calculate the time to fill the tank.

To find the time taken to fill the entire tank (1 tank), we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Net Rate}} = \frac{1 \text{ tank}}{\frac{3}{8} \text{ tank/hour}} = \frac{8}{3} \text{ hours} \]

### Conclusion

Thus, the time taken to fill the tank completely when all three taps are opened simultaneously is:
\[ \boxed{\frac{8}{3} \text{ hours}} \]