Two pipes can fill a tank in 10 hours and 12 hours respectively, while a third pipe empties the full tank m 20 hours. If all the three pipes operate simultaneously, in how much time will the tank be full?

Options

A. 7 hrs 15 mm
B. 7 hrs 30 mm
C. 7 hrs 45 mm
D. 8 hrs

Answer

To solve the problem, we need to determine the rate at which each pipe fills or empties the tank, and then combine these rates to find out how long it will take to fill the tank when all three pipes are operating simultaneously.

### Step 1: Calculate the filling rates of the pipes

1. **First Pipe (Fills in 10 hours)**:
- Rate of filling = 1 tank / 10 hours = 1/10 tanks per hour.

2. **Second Pipe (Fills in 12 hours)**:
- Rate of filling = 1 tank / 12 hours = 1/12 tanks per hour.

3. **Third Pipe (Empties in 20 hours)**:
- Rate of emptying = 1 tank / 20 hours = 1/20 tanks per hour.

### Step 2: Combine the rates

When all three pipes are operating simultaneously, the effective rate of filling the tank can be calculated by adding the filling rates of the first two pipes and subtracting the emptying rate of the third pipe.

- Effective Rate = Rate of First Pipe + Rate of Second Pipe - Rate of Third Pipe
- Effective Rate = (1/10) + (1/12) - (1/20)

### Step 3: Find a common denominator

To add the fractions, we need a common denominator. The least common multiple (LCM) of 10, 12, and 20 is 60.

- Convert each rate:
- (1/10) = 6/60
- (1/12) = 5/60
- (1/20) = 3/60

### Step 4: Combine the rates

Now we can combine the rates:

- Effective Rate = (6/60) + (5/60) - (3/60)
- Effective Rate = (6 + 5 - 3) / 60 = 8/60 = 2/15 tanks per hour.

### Step 5: Calculate the time to fill the tank

To find the time taken to fill the tank, we take the reciprocal of the effective rate:

- Time = 1 / (Effective Rate) = 1 / (2/15) = 15/2 hours = 7.5 hours.

### Final Answer

The tank will be full in **7.5 hours** when all three pipes operate simultaneously.

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