A pool has 3 taps. The first tap takes 3 days, the second tap takes 2 days, and the third tap takes 18 hours, to fill the pool. All the three taps are opened together. After 8 hours, the second tap is closed. The total time to fill the pool completely will be:

Options

A. `128/11` hours
B. 12 hours
C. `144/13` hours
D. 4 hours

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the filling rates of each tap.
- **First tap**: Fills the pool in 3 days.
- Rate = \( \frac{1 \text{ pool}}{3 \text{ days}} = \frac{1}{72} \text{ pools per hour} \)

- **Second tap**: Fills the pool in 2 days.
- Rate = \( \frac{1 \text{ pool}}{2 \text{ days}} = \frac{1}{48} \text{ pools per hour} \)

- **Third tap**: Fills the pool in 18 hours.
- Rate = \( \frac{1 \text{ pool}}{18 \text{ hours}} = \frac{1}{18} \text{ pools per hour} \)

### Step 2: Calculate the combined filling rate of all three taps.
- Combined rate = Rate of first tap + Rate of second tap + Rate of third tap
- Combined rate = \( \frac{1}{72} + \frac{1}{48} + \frac{1}{18} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 72, 48, and 18 is 144.

- Convert each rate:
- \( \frac{1}{72} = \frac{2}{144} \)
- \( \frac{1}{48} = \frac{3}{144} \)
- \( \frac{1}{18} = \frac{8}{144} \)

- Now, add them:
- Combined rate = \( \frac{2}{144} + \frac{3}{144} + \frac{8}{144} = \frac{13}{144} \text{ pools per hour} \)

### Step 3: Calculate the amount of the pool filled in the first 8 hours.
- Amount filled in 8 hours = Combined rate × Time
- Amount filled = \( \frac{13}{144} \times 8 = \frac{104}{144} = \frac{13}{18} \text{ of the pool} \)

### Step 4: Determine the remaining capacity of the pool.
- Remaining capacity = 1 - Amount filled
- Remaining capacity = \( 1 - \frac{13}{18} = \frac{5}{18} \text{ of the pool} \)

### Step 5: Calculate the new filling rate after the second tap is closed.
- After 8 hours, only the first and third taps are open.
- New combined rate = Rate of first tap + Rate of third tap
- New combined rate = \( \frac{1}{72} + \frac{1}{18} \)

Convert to a common denominator (which is 72):
- \( \frac{1}{18} = \frac{4}{72} \)
- New combined rate = \( \frac{1}{72} + \frac{4}{72} = \frac{5}{72} \text{ pools per hour} \)

### Step 6: Calculate the time required to fill the remaining capacity.
- Time = Remaining capacity / New combined rate
- Time = \( \frac{5/18}{5/72} = \frac{5}{18} \times \frac{72}{5} = 8 \text{ hours} \)

### Step 7: Calculate the total time to fill the pool.
- Total time = Time for first 8 hours + Time for remaining capacity
- Total time = \( 8 + 8 = 16 \text{ hours} \)

Thus, the total time to fill the pool completely is **16 hours**.

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