At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
Because each man danced with exactly three women, there were $(12)(3)=36$ pairs of men and women who danced together. Each woman had two partners, so the number of women who attended is $36/2=\boxed{18}.$
18
Level 5
Prealgebra
Problem: At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of women at the party, we start by noting the relationships between men and women based on the dancing conditions. Each of the twelve men danced with exactly three women. Therefore, the total number of man-woman dance pairs can be calculated as follows:

Total dance pairs = Number of men × Dances per man = 12 men × 3 dances/man = 36 dance pairs.

Next, we consider the women. Each woman danced with exactly two men. Let the number of women be denoted as W. The total number of dance pairs can also be expressed in terms of the women:

Total dance pairs = Number of women × Dances per woman = W × 2 dances/woman.

Since both expressions represent the same total number of dance pairs, we can set them equal to each other:

36 dance pairs = W × 2 dances/woman.

To find W, we solve for it:

W = 36 dance pairs / 2 dances/woman = 18 women.

Thus, the reasoning leads us to conclude that the number of women who attended the party is 18.<|END_OF_LATENT|> The final answer is $18$.