A tap A can fill a cistern in 8 hours while tap B can fill it in 4 hours. In how much time will the cistern be filled if both A and B are opened together?

Answer

To solve the problem of how long it will take for taps A and B to fill the cistern when opened together, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the rate of work for each tap:**
- Tap A can fill the cistern in 8 hours. Therefore, in 1 hour, it fills:
\[ \text{Work done by A in 1 hour} = \frac{1}{8} \text{ of the cistern} \]
- Tap B can fill the cistern in 4 hours. Therefore, in 1 hour, it fills:
\[ \text{Work done by B in 1 hour} = \frac{1}{4} \text{ of the cistern} \]

2. **Find the combined work done by both taps in 1 hour:**
- When both taps A and B are opened together, the total work done in 1 hour is:
\[ \text{Combined work} = \text{Work done by A} + \text{Work done by B} = \frac{1}{8} + \frac{1}{4} \]
- To add these fractions, we need a common denominator. The least common multiple of 8 and 4 is 8. Thus:
\[ \frac{1}{4} = \frac{2}{8} \]
- Now, we can add:
\[ \text{Combined work} = \frac{1}{8} + \frac{2}{8} = \frac{3}{8} \]

3. **Calculate the time taken to fill the cistern:**
- If both taps together fill \(\frac{3}{8}\) of the cistern in 1 hour, we need to find out how long it will take to fill the entire cistern (1 whole).
- The time taken to fill the cistern can be calculated by taking the reciprocal of the combined work done in 1 hour:
\[ \text{Time} = \frac{1}{\text{Combined work}} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \text{ hours} \]

4. **Convert the time into a mixed fraction:**
- To convert \(\frac{8}{3}\) into a mixed fraction:
\[ 8 \div 3 = 2 \quad \text{(remainder 2)} \]
- Thus, \(\frac{8}{3} = 2 \frac{2}{3}\) hours.

### Final Answer:
The cistern will be filled in \(2 \frac{2}{3}\) hours if both taps A and B are opened together.

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