Step 1: Understanding the Question:
This is a mixture and replacement problem.
A tank is repeatedly drained of half its total volume and then refilled with a specific pure substance (either A or B). We need to track the exact proportion of substance A after several such operations.
Step 2: Key Formula or Approach:
When a tank is drained by half, the quantities of all individual components currently in the mixture are exactly halved.
When refilled, the added substance increases by the entire replacement volume (which is half the total tank volume in this case).
Assume the total capacity of the tank to be 100 units for easy percentage calculations.
Step 3: Detailed Explanation:
Let the total capacity of the fuel tank be 100 liters.
Initial Stage: The empty tank is entirely filled with A type petrol. The tank has 100 liters of A and 0 liters of B.
First Replacement: The tank becomes half-empty, meaning 50 liters of the mixture are removed. Since the tank was purely A, 50 liters of A are removed, leaving 50 liters of A.
It is then filled with B type petrol (50 liters of B added).
After the first replacement, the tank contains: 50 liters of A and 50 liters of B.
Second Replacement: The tank is half-empty again. Half of the current mixture is removed.
This removes half of A (25 liters) and half of B (25 liters).
Remaining volume: 25 liters of A and 25 liters of B.
It is now filled with A type petrol (50 liters of pure A added).
After the second replacement, the tank contains: \( 25 + 50 = 75 \) liters of A, and 25 liters of B.
Third Replacement: The tank is half-empty once more.
Half of the current A (\( \frac{75}{2} = 37.5 \) liters) and half of B (\( \frac{25}{2} = 12.5 \) liters) are removed.
Remaining volume: 37.5 liters of A and 12.5 liters of B.
It is then filled with B type petrol (50 liters of pure B added).
After the third replacement, the tank contains: 37.5 liters of A, and \( 12.5 + 50 = 62.5 \) liters of B.
The question asks for the final percentage of A type petrol.
Since our total assumed volume is 100 liters, the 37.5 liters of A directly correspond to 37.5%.
Step 4: Final Answer:
The percentage of A type present in the tank is 37.5%.