A tap can fill a cistern in 8 hours and another can empty it in 16 hours.If both the taps are opned simultaneously,the time to fill the tank is------

Options

A. 8
B. 10
C. 16
D. 24

Answer

To solve the problem of how long it takes to fill a cistern with one tap filling it and another tap emptying it, we can follow these steps:

### Step 1: Determine the rates of the taps
- **Filling tap (A)**: It can fill the cistern in 8 hours.
- **Emptying tap (B)**: It can empty the cistern in 16 hours.

### Step 2: Calculate the filling rate of tap A
- If tap A fills the cistern in 8 hours, then in 1 hour, it fills:
\[ \text{Rate of A} = \frac{1 \text{ cistern}}{8 \text{ hours}} = \frac{1}{8} \text{ cistern per hour} \]

### Step 3: Calculate the emptying rate of tap B
- If tap B empties the cistern in 16 hours, then in 1 hour, it empties:
\[ \text{Rate of B} = \frac{1 \text{ cistern}}{16 \text{ hours}} = \frac{1}{16} \text{ cistern per hour} \]
Since this is an emptying tap, we will consider this rate as negative:
\[ \text{Rate of B} = -\frac{1}{16} \text{ cistern per hour} \]

### Step 4: Combine the rates of both taps
- When both taps are opened simultaneously, the net rate of filling the cistern is:
\[ \text{Net rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{8} - \frac{1}{16} \]

### Step 5: Find a common denominator to combine the rates
- The least common multiple of 8 and 16 is 16. Therefore, we convert the rates:
\[ \frac{1}{8} = \frac{2}{16} \]
Now, substituting back:
\[ \text{Net rate} = \frac{2}{16} - \frac{1}{16} = \frac{1}{16} \text{ cistern per hour} \]

### Step 6: Calculate the time to fill the cistern
- If the net rate of filling the cistern is \(\frac{1}{16}\) cistern per hour, then the time taken to fill 1 cistern is:
\[ \text{Time} = \frac{1 \text{ cistern}}{\frac{1}{16} \text{ cistern per hour}} = 16 \text{ hours} \]

### Conclusion
The time taken to fill the tank when both taps are opened simultaneously is **16 hours**.

---