Taps A and B can fill a tank in 15 minutes and 10 minutes, respectively while tap C can empty the full tank in x minutes. If all the three taps are opened together, the tank is filled completely in 8 minutes then find x.

Options

A. '15' minutes
B. `10(1)/(2)` minutes
C. 24 minutes
D. `8(1)/(2)` minutes

Answer

To solve the problem, we need to find the time taken by tap C to empty the tank, denoted as \( x \) minutes. We know the filling rates of taps A and B, and we are given that when all three taps are opened together, the tank is filled in 8 minutes.

### Step-by-step Solution:

1. **Determine the filling rates of taps A and B:**
- Tap A can fill the tank in 15 minutes. Therefore, in 1 minute, it fills:
\[ \text{Rate of A} = \frac{1}{15} \text{ tank/minute} \]
- Tap B can fill the tank in 10 minutes. Therefore, in 1 minute, it fills:
\[ \text{Rate of B} = \frac{1}{10} \text{ tank/minute} \]

2. **Calculate the combined filling rate of taps A and B:**
- The combined rate of taps A and B in 1 minute is:
\[ \text{Rate of A + B} = \frac{1}{15} + \frac{1}{10} \]
- To add these fractions, find a common denominator (which is 30):
\[ \text{Rate of A + B} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \text{ tank/minute} \]

3. **Determine the filling rate when all taps are open:**
- When all three taps (A, B, and C) are opened together, they fill the tank in 8 minutes. Therefore, their combined rate is:
\[ \text{Rate of A + B + C} = \frac{1}{8} \text{ tank/minute} \]

4. **Set up the equation for the rate of tap C:**
- Let \( C \) be the rate of tap C (which empties the tank). Since C empties the tank, its rate will be negative:
\[ \text{Rate of A + B + C} = \text{Rate of A + B} - \text{Rate of C} \]
- Substituting the known rates:
\[ \frac{1}{8} = \frac{1}{6} - \frac{1}{x} \]

5. **Solve for \( x \):**
- Rearranging the equation gives:
\[ \frac{1}{x} = \frac{1}{6} - \frac{1}{8} \]
- Finding a common denominator (which is 24):
\[ \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24} \]
- Therefore:
\[ \frac{1}{x} = \frac{4}{24} - \frac{3}{24} = \frac{1}{24} \]
- Taking the reciprocal gives:
\[ x = 24 \]

### Conclusion:
The time taken by tap C to empty the tank is \( x = 24 \) minutes.