Multiple choice

[CAT 1998] There are two containers: the first contains 500 ml of alcohol, while the second contains 500 ml of water. Three cups of alcohol from the first container is taken out and is mixed well in the second container. Then three cups of this mixture is taken out and is mixed in the first container. Let A denote the proportion of water in the first container and B denote the proportion of alcohol in the second container. Then

  1. A > B

  2. A < B

  3. A = B

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the cup size be c. Container 1: 500ml alcohol. Container 2: 500ml water. After moving 3 cups (3c) of alcohol to container 2, container 2 has 500ml water + 3c alcohol. The mixture is 3c/(500+3c) alcohol. Moving 3c of this mixture back to container 1: container 1 now has (500-3c) alcohol + 3c * (3c/(500+3c)) alcohol and 3c * (500/(500+3c)) water. The proportion of water in container 1 is A = (1500c/(500+3c)) / 500 = 3c/(500+3c). The proportion of alcohol in container 2 is B = (3c - 3c * (3c/(500+3c))) / 500 = (1500c+9c^2-9c^2)/(500(500+3c)) = 3c/(500+3c). Thus A = B.

AI explanation

Consider the transfer of the three cups of mixture back to the first container as swapping three cups of alcohol for three cups of the mixture. Ultimately, the first container gives exactly three cups of alcohol to the second container, and receives exactly three cups of water in return because the mixture drawn from the second container displaces its alcohol. This means the amount of water in the first container exactly equals the amount of alcohol in the second container, making A equal to B.

Keep practicing — related questions