A pipe can fill a tank in 10 minutes while another pipe can empty it in 12 minutes. If the pipes are opened alternately each for 1 minute, beginning with the first pipe. the tank will be full after (in minutes):

Options

A. 90
B. 109
C. 108
D. 120

Answer

To solve the problem step by step, we will analyze the filling and emptying rates of the pipes and how they work together when opened alternately.

### Step 1: Determine the filling and emptying rates of the pipes.
- Pipe A can fill the tank in 10 minutes.
- Pipe B can empty the tank in 12 minutes.

**Filling rate of Pipe A**:
- In 1 minute, Pipe A fills \( \frac{1}{10} \) of the tank.
- If we consider the tank's total capacity as 60 liters (for easier calculations), then Pipe A fills \( \frac{60}{10} = 6 \) liters in 1 minute.

**Emptying rate of Pipe B**:
- In 1 minute, Pipe B empties \( \frac{1}{12} \) of the tank.
- For the same 60 liters tank, Pipe B empties \( \frac{60}{12} = 5 \) liters in 1 minute.

### Step 2: Calculate the net effect when both pipes are opened alternately.
- In the first minute, Pipe A fills 6 liters.
- In the second minute, Pipe B empties 5 liters.

**Net effect in 2 minutes**:
- In 2 minutes, the total amount filled = 6 liters (by A) - 5 liters (by B) = 1 liter.
- Therefore, in 2 minutes, the net filling is 1 liter.

### Step 3: Calculate how much time it takes to fill the remaining tank.
- The tank has a total capacity of 60 liters.
- After the first minute (when Pipe A is opened), 6 liters are filled, leaving 54 liters to fill.
- Since 1 liter is filled every 2 minutes, we need to find out how long it will take to fill 54 liters.

**Time to fill 54 liters**:
- If 1 liter takes 2 minutes, then 54 liters will take \( 54 \times 2 = 108 \) minutes.

### Step 4: Calculate the total time taken.
- The first minute is spent filling 6 liters with Pipe A.
- The remaining 54 liters take 108 minutes to fill.

**Total time**:
- Total time = 1 minute (first minute) + 108 minutes = 109 minutes.

### Final Answer:
The tank will be full after **109 minutes**.

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