Tap A can fill a tank in 20 hours, B in 25 hours but tap C can J empty a full tank in 30 hours. Starting with A followed, B and C each tap opens alternatively for one hour period .till the tank gets filled up completely. In how many hour the tank will be filled up completely?

Options

A. `51 11/15`
B. `52 2/3`
C. `24 4/11`
D. none of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the filling and emptying rates of taps A, B, and C.

- **Tap A** can fill the tank in 20 hours.
- Rate of A = 1 tank / 20 hours = 1/20 tanks per hour = 15 units per hour (if we assume the tank capacity is 300 units).

- **Tap B** can fill the tank in 25 hours.
- Rate of B = 1 tank / 25 hours = 1/25 tanks per hour = 12 units per hour.

- **Tap C** can empty the tank in 30 hours.
- Rate of C = 1 tank / 30 hours = 1/30 tanks per hour = 10 units per hour.

### Step 2: Calculate the net effect of one complete cycle of A, B, and C.

In one cycle (which takes 3 hours), the taps operate as follows:
- **Hour 1:** Tap A fills 15 units.
- **Hour 2:** Tap B fills 12 units.
- **Hour 3:** Tap C empties 10 units.

Net units filled in one cycle:
\[ \text{Net units} = 15 + 12 - 10 = 17 \text{ units} \]

### Step 3: Calculate how many complete cycles are needed to fill the tank.

The total capacity of the tank is 300 units. We can find out how many full cycles are needed to fill the tank:
\[ \text{Number of cycles} = \frac{300 \text{ units}}{17 \text{ units per cycle}} \approx 17.65 \text{ cycles} \]
Since we can only have complete cycles, we take 17 cycles.

### Step 4: Calculate the total time for these complete cycles.

Each cycle takes 3 hours, so:
\[ \text{Total time for 17 cycles} = 17 \times 3 = 51 \text{ hours} \]
In 51 hours, the total units filled will be:
\[ \text{Total units filled} = 17 \times 17 = 289 \text{ units} \]

### Step 5: Determine how many units are left to fill the tank.

After 51 hours, the tank has:
\[ 300 - 289 = 11 \text{ units left to fill} \]

### Step 6: Calculate how long it takes to fill the remaining units.

In the next hour, Tap A will operate again and fill 15 units. However, we only need 11 units. The time taken to fill the remaining 11 units by Tap A is:
\[ \text{Time} = \frac{11 \text{ units}}{15 \text{ units/hour}} = \frac{11}{15} \text{ hours} \]

### Step 7: Calculate the total time taken to fill the tank completely.

Adding the time for the complete cycles and the additional time:
\[ \text{Total time} = 51 \text{ hours} + \frac{11}{15} \text{ hours} \]
To convert this into a single fraction:
\[ 51 = \frac{765}{15} \text{ hours} \]
So,
\[ \text{Total time} = \frac{765 + 11}{15} = \frac{776}{15} \text{ hours} \approx 51.73 \text{ hours} \]

### Final Answer:
The tank will be filled completely in \( 51 \frac{11}{15} \) hours.

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