Tap 'A' can fill a water tank in 25 minutes , tap 'B' can fill the same tank in 40 minutes and tap 'C' can empty that tank in 30 minutes. If all the three taps are opened together , in how many minutes will the tank the completely filled up or emptied ?

Options

A. 3
B. 9
C. 12
D. none of these

Answer

To solve the problem, we need to determine how long it will take for the tank to be filled or emptied when all three taps are opened together. Let's break it down step by step.

### Step 1: Determine the rates of each tap
1. **Tap A** fills the tank in 25 minutes.
- Rate of Tap A = 1 tank / 25 minutes = 1/25 tanks per minute.

2. **Tap B** fills the tank in 40 minutes.
- Rate of Tap B = 1 tank / 40 minutes = 1/40 tanks per minute.

3. **Tap C** empties the tank in 30 minutes.
- Rate of Tap C = 1 tank / 30 minutes = 1/30 tanks per minute.

### Step 2: Calculate the combined rate of all taps
When all taps are opened together, we need to consider the filling rates of taps A and B and the emptying rate of tap C.

- Combined rate = Rate of A + Rate of B - Rate of C
- Combined rate = (1/25) + (1/40) - (1/30)

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 25, 40, and 30 is 600.

- Convert each rate:
- Rate of A = (1/25) = (24/600)
- Rate of B = (1/40) = (15/600)
- Rate of C = (1/30) = (20/600)

### Step 4: Calculate the combined rate
Now we can add the rates:
- Combined rate = (24/600) + (15/600) - (20/600)
- Combined rate = (24 + 15 - 20) / 600 = (19/600) tanks per minute.

### Step 5: Determine the time to fill the tank
To find out how long it will take to fill the tank, we use the formula:
- Time = Total capacity / Combined rate
- Assuming the tank capacity is 600 units (as we calculated earlier):
- Time = 600 / (19/600) = 600 * (600/19) = 36000 / 19 minutes.

### Step 6: Calculate the final time
Now we can calculate the approximate time:
- Time ≈ 189.47 minutes.

### Conclusion
If all three taps are opened together, the tank will take approximately **189.47 minutes** to fill.