# CAT Practice : Averages, Ratios, Mixtures

A ratio proportion question that involves using a simple framework from number systems.

## Mixtures - Fun question

Q.4: 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?
1. 51.5%
2. 52.67%
3. 53%
4. Cannot be determined

Choice B. 52.67%

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## Detailed Solution

Step 1: A vs B:
Since A has a higher percentage of alcohol than B, A has to give B 200 ml of wine. Considering that A has a 60% strength mixture, B will end up having 800 ml of wine that has

60% × 1 + 48% × 3 / 4
= 51% alcohol.
Step 2: B vs C:
B has a higher percentage of alcohol than C, so B has to give 200 ml of alcohol to C now. C will end up having wine that has
51% × 1 + 50% × 3 / 4
= 50.25%
Step 3:
A gives C 200 mls of wine. We see that computing percentages is going to cumbersome. So, let us just forget the numbers and think about how the game pans out.
Step 4: A would give B 200 ml of wine. We note straight away that A will always give the wine. A had the highest percentage alcohol to start with. And no mixture can reach 60% alcohol levels. So, A will give all his wine away by this round.
Post this, we notice that, between B and C whoever has higher percentage alcohol will keep giving wine to the other. Eventually, all the wine will be with just one person. At the rate of 3 minutes per round, the game, the game will definitely be over before 1 hour.
So, the concentration of alcohol in the wine D drinks is
60% + 48% + 50% / 3
= 52.67%.

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