One tap can fill a cistern in 2 hours and another can other can empty the cistern in 3 hours. How long will they take to fill the cistern if both the taps are opened ?

Options

A. 6 hours
B. 7 hours
C. 6 :30 hours
D. 7:30 hours

Answer

To solve the problem step by step, we need to determine how long it will take to fill the cistern when both taps are opened.

### Step-by-Step Solution:

1. **Identify the rates of the taps:**
- Tap A (filling tap) can fill the cistern in 2 hours.
- Tap B (emptying tap) can empty the cistern in 3 hours.

2. **Calculate the work done by each tap in one hour:**
- The amount of work done by Tap A in one hour is:
\[ \text{Efficiency of A} = \frac{1 \text{ cistern}}{2 \text{ hours}} = \frac{1}{2} \text{ cistern per hour} \]
- The amount of work done by Tap B in one hour is:
\[ \text{Efficiency of B} = \frac{1 \text{ cistern}}{3 \text{ hours}} = \frac{1}{3} \text{ cistern per hour} \]

3. **Convert the efficiencies to a common unit:**
- To make calculations easier, we can find a common unit. The least common multiple (LCM) of 2 and 3 is 6. Thus, we can consider the total work in terms of 6 units (the capacity of the cistern).
- The efficiency of Tap A in terms of 6 units is:
\[ \text{Efficiency of A} = \frac{6}{2} = 3 \text{ units per hour} \]
- The efficiency of Tap B in terms of 6 units is:
\[ \text{Efficiency of B} = \frac{6}{3} = 2 \text{ units per hour} \]

4. **Determine the net efficiency when both taps are open:**
- Since Tap A is filling and Tap B is emptying, we subtract the efficiency of Tap B from Tap A:
\[ \text{Net Efficiency} = \text{Efficiency of A} - \text{Efficiency of B} = 3 - 2 = 1 \text{ unit per hour} \]

5. **Calculate the time taken to fill the cistern:**
- Since the total work needed to fill the cistern is 6 units and the net efficiency is 1 unit per hour, we can find the time taken:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Net Efficiency}} = \frac{6 \text{ units}}{1 \text{ unit per hour}} = 6 \text{ hours} \]

### Final Answer:
It will take 6 hours to fill the cistern when both taps are opened.