Three tapes are fitted in a cistern . The empty cistern is filled by the first and the second taps in 3 and 4 h respectively . The full cistern is emptied by the third tap in 5h . 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. 2h 40min
D. 1h 56 min

Answer

To solve the problem step by step, we will first determine the rates at which each tap fills or empties the cistern, and then calculate the combined rate when all taps are opened together.

### Step 1: Determine the filling rates of the taps
1. **Tap A** fills the cistern in 3 hours.
- Rate of Tap A = 1 cistern / 3 hours = 1/3 cistern per hour.

2. **Tap B** fills the cistern in 4 hours.
- Rate of Tap B = 1 cistern / 4 hours = 1/4 cistern per hour.

3. **Tap C** empties the cistern in 5 hours.
- Rate of Tap C = -1 cistern / 5 hours = -1/5 cistern per hour (negative because it empties).

### Step 2: Find the combined rate of all taps
Now, we combine the rates of the three taps when they are opened simultaneously:
- Combined rate = Rate of Tap A + Rate of Tap B + Rate of Tap C
- Combined rate = (1/3) + (1/4) - (1/5)

### Step 3: Calculate the combined rate using a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 3, 4, and 5 is 60.
- Convert each rate:
- (1/3) = 20/60
- (1/4) = 15/60
- (1/5) = 12/60

Now, substituting these values:
- Combined rate = (20/60) + (15/60) - (12/60)
- Combined rate = (20 + 15 - 12) / 60
- Combined rate = 23/60 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 open, we use the formula:
- Time = Total Capacity / Combined Rate

Assuming the total capacity of the cistern is 1 (or 60 units, as calculated earlier):
- Time = 1 cistern / (23/60) cisterns per hour
- Time = 60 / 23 hours.

### Final Answer
Thus, the time taken to fill the cistern when all three taps are opened simultaneously is **60/23 hours**.

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