Three taps are fitted in a cistern. The empty cistern is filled by the first and the second taps in 3 and 4h, respectively. The full cistern is emptied by the third tap in 5 h. If all three taps are opened simultaneously, the empty cistern will be filled up in?

Options

A. `1(14)/(23)h`
B. `2(14)/(23)h`
C. 2 h 40 min
D. 1 h 56 min

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the rates of work for each tap.
- **Tap A** fills the cistern in 3 hours. Therefore, the rate of work for Tap A is:
\[ \text{Rate of A} = \frac{1}{3} \text{ cisterns per hour} \]

- **Tap B** fills the cistern in 4 hours. Therefore, the rate of work for Tap B is:
\[ \text{Rate of B} = \frac{1}{4} \text{ cisterns per hour} \]

- **Tap C** empties the cistern in 5 hours. Therefore, the rate of work for Tap C (as it is emptying) is:
\[ \text{Rate of C} = -\frac{1}{5} \text{ cisterns per hour} \]

### Step 2: Calculate the combined rate of work when all taps are opened.
The combined rate of work when all three taps are opened is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the values we found:
\[ \text{Combined Rate} = \frac{1}{3} + \frac{1}{4} - \frac{1}{5} \]

### Step 3: Find a common denominator and calculate the combined rate.
The least common multiple (LCM) of 3, 4, and 5 is 60. We will convert each rate to have a denominator of 60:
- For Tap A:
\[ \frac{1}{3} = \frac{20}{60} \]
- For Tap B:
\[ \frac{1}{4} = \frac{15}{60} \]
- For Tap C:
\[ -\frac{1}{5} = -\frac{12}{60} \]

Now, substituting these into the combined rate:
\[ \text{Combined Rate} = \frac{20}{60} + \frac{15}{60} - \frac{12}{60} = \frac{20 + 15 - 12}{60} = \frac{23}{60} \text{ cisterns per hour} \]

### Step 4: Calculate the time taken to fill the cistern.
To find the time taken to fill the cistern when all taps are opened, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} \]
The total work to fill the cistern is 1 (since we want to fill one cistern):
\[ \text{Time} = \frac{1}{\frac{23}{60}} = \frac{60}{23} \text{ hours} \]

### Step 5: Convert the time into hours and minutes.
To convert \(\frac{60}{23}\) hours into hours and minutes:
1. Divide 60 by 23:
\[ 60 \div 23 \approx 2.6087 \text{ hours} \]
This means it takes 2 hours and a fraction of an hour.

2. To find the minutes, take the decimal part (0.6087) and multiply by 60:
\[ 0.6087 \times 60 \approx 36.52 \text{ minutes} \]
Rounding this gives approximately 37 minutes.

Thus, the total time taken to fill the cistern is approximately:
\[ 2 \text{ hours and } 37 \text{ minutes} \]

### Final Answer:
The empty cistern will be filled up in approximately \(2 \text{ hours } 37 \text{ minutes}\).

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