A tank has an inlet and outlet pipe. The inlet pipe fills the tank completely in 2 hours when the outlet pipe is plugged. The outlet pipe enpties the tank completely in 6 hours when the inlet pipe is plugged.
If both pipes are opened simultaneously at a time when the was one-third filled, when will the tank fill thereafter?

Options

A. `(3)/(2)` hours
B. `(2)/(3)` hour
C. 2 hours
D. `1(2)/(3)` hours .

Answer

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

### Step 1: Determine the filling and emptying rates of the pipes
- The inlet pipe (let's call it Pipe A) fills the tank in 2 hours.
- The outlet pipe (let's call it Pipe B) empties the tank in 6 hours.

**Hint:** To find the rate of work, use the formula: Rate = Total Work / Time.

### Step 2: Calculate the rates
- The rate of Pipe A (filling rate) = 1 tank / 2 hours = 1/2 tank per hour.
- The rate of Pipe B (emptying rate) = 1 tank / 6 hours = 1/6 tank per hour.

**Hint:** Remember that filling rates are positive and emptying rates are negative.

### Step 3: Find the net rate when both pipes are open
- When both pipes are open, the net rate = Rate of Pipe A - Rate of Pipe B.
- Net rate = (1/2 - 1/6) tank per hour.

**Hint:** To subtract fractions, find a common denominator.

### Step 4: Calculate the net rate
- The common denominator for 2 and 6 is 6.
- Convert the rates:
- (1/2) = 3/6
- (1/6) = 1/6
- Now, calculate the net rate:
- Net rate = (3/6 - 1/6) = 2/6 = 1/3 tank per hour.

**Hint:** This means that together, the pipes fill 1/3 of the tank every hour.

### Step 5: Determine how much of the tank is left to fill
- The tank is currently one-third filled, so the remaining capacity = 1 - 1/3 = 2/3 of the tank.

**Hint:** To find the remaining part, subtract the filled part from the total capacity.

### Step 6: Calculate the time to fill the remaining capacity
- Time required to fill the remaining 2/3 of the tank = Remaining capacity / Net rate.
- Time = (2/3) / (1/3) = 2 hours.

**Hint:** Dividing by a fraction is the same as multiplying by its reciprocal.

### Final Answer:
The tank will fill completely in **2 hours** after both pipes are opened.

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