Three friends A, B and C play a game in a pub. The rules are simple. Whenever…
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Three friends A, B and C play a game in a pub. The rules are simple. Whenever there is a contest between any two of them, the one who has a higher percentage alcohol should pour 200 ml of his wine into the one having lower percentage alcohol. The game starts as a contest between A and B, then B and C and then C and A. Post this, the game continues in the same cycle on and on. If a player has emptied all his alcohol, then the remaining two play the game with the same rules. If two players have the alcohol of the same percentage level, the younger one pours 200 ml of his alcohol into the elder one’s glass. All three of them start the game with 600 ml of wine. A’s wine has 60% alcohol, B’s has 48% alcohol and C’s has 50% alcohol. They take 3 minutes to play one round of this game. D, a fourth friend leaves the pub immediately after the game begins, returns after an hour and drinks wine from the person who has the highest alcohol percentage. What is the concentration of the alcohol that D had?
- A.
51.5%
- B.
52.67%
- C.
53%
- D.
Cannot be determined
Attempted by 6 students.
Show answer & explanation
Correct answer: B
Simulate the transfers. Each contest moves 200 ml from the higher-concentration player to the lower; the alcohol moved equals 200 × donor concentration. Each round = 3 minutes; D returns after 60 minutes (20 rounds), but the game may finish earlier.
Initial: A: 600 ml, 360 ml alcohol (60.00%); B: 600 ml, 288 ml alcohol (48.00%); C: 600 ml, 300 ml alcohol (50.00%).
Step 1 (A → B): A gives 200 ml containing 200×0.60 = 120 ml alcohol. After: A = 400 ml, 240 ml alcohol (60.00%); B = 800 ml, 408 ml alcohol (51.00%).
Step 2 (B → C): B gives 200 ml containing 200×0.51 = 102 ml alcohol. After: B = 600 ml, 306 ml alcohol (51.00%); C = 800 ml, 402 ml alcohol (50.25%).
Step 3 (A → C): A gives 200 ml containing 200×0.60 = 120 ml alcohol. After: A = 200 ml, 120 ml alcohol (60.00%); C = 1000 ml, 522 ml alcohol (52.20%).
Step 4 (A → B): A gives remaining 200 ml containing 200×0.60 = 120 ml alcohol and empties. After: A = 0 ml (out); B = 800 ml, 426 ml alcohol (53.25%).
With A out, only B and C play repeatedly. Step 5 (B → C): B gives 200 ml containing 200×0.5325 = 106.5 ml alcohol. After: B = 600 ml, 319.5 ml alcohol (53.25%); C = 1200 ml, 628.5 ml alcohol (52.375%).
Step 6 (B → C): B gives 200 ml containing 106.5 ml alcohol. After: B = 400 ml, 213 ml alcohol (53.25%); C = 1400 ml, 735 ml alcohol (52.50%).
Step 7 (B → C): B gives 200 ml containing 106.5 ml alcohol. After: B = 200 ml, 106.5 ml alcohol (53.25%); C = 1600 ml, 841.5 ml alcohol (52.59375%).
Step 8 (B → C): B gives remaining 200 ml containing 106.5 ml alcohol and empties. After: B = 0 ml; C = 1800 ml, 948 ml alcohol.
At this point all wine is with one person (C) and the game stops. This happens after 8 transfers (8 × 3 = 24 minutes), well before D returns at 60 minutes.
Total alcohol held by that person is 948 ml in 1800 ml total, so the concentration is 948/1800 = 0.526666... = 52.666...% ≈ 52.67%.
Therefore, when D returns after one hour he drinks wine at about 52.67% alcohol concentration.