One tap can fill a water tank in 50 minutes the filled tank empty in 75 minutes.If both the tap are open together,already half filled tank would be full in

Options

A. 60 minutes
B. 75 minutes
C. 125 minutes
D. 150 minutes

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Determine the filling rate of the tap
One tap can fill the tank in 50 minutes. Therefore, the rate at which this tap fills the tank is:
\[ \text{Filling rate of tap} = \frac{1 \text{ tank}}{50 \text{ minutes}} = \frac{1}{50} \text{ tanks per minute} \]

**Hint:** To find the filling rate, divide 1 by the time taken to fill the tank.

### Step 2: Determine the emptying rate of the tank
The tank can be emptied in 75 minutes. Therefore, the rate at which the tank empties is:
\[ \text{Emptying rate} = \frac{1 \text{ tank}}{75 \text{ minutes}} = \frac{1}{75} \text{ tanks per minute} \]

**Hint:** To find the emptying rate, divide 1 by the time taken to empty the tank.

### Step 3: Calculate the net rate when both taps are open
When both the filling tap and the emptying tap are open, the net rate of filling the tank is:
\[ \text{Net rate} = \text{Filling rate} - \text{Emptying rate} = \frac{1}{50} - \frac{1}{75} \]

To perform the subtraction, we need a common denominator. The least common multiple (LCM) of 50 and 75 is 150.

Converting the rates:
\[ \frac{1}{50} = \frac{3}{150}, \quad \frac{1}{75} = \frac{2}{150} \]
Thus,
\[ \text{Net rate} = \frac{3}{150} - \frac{2}{150} = \frac{1}{150} \text{ tanks per minute} \]

**Hint:** When subtracting fractions, find a common denominator to combine them.

### Step 4: Determine the time to fill the remaining half of the tank
Since the tank is already half filled, we need to fill the remaining half. The time required to fill half a tank at the net rate of \(\frac{1}{150}\) tanks per minute is:
\[ \text{Time} = \frac{\text{Amount to fill}}{\text{Net rate}} = \frac{\frac{1}{2} \text{ tank}}{\frac{1}{150} \text{ tanks per minute}} = \frac{1}{2} \times 150 = 75 \text{ minutes} \]

**Hint:** To find the time taken to fill a certain amount, divide the amount by the rate of filling.

### Final Answer
The time required to fill the already half-filled tank completely is **75 minutes**.