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?
To solve this problem, I will use the idea of counting the number of pairs of dancers in two different ways. One way is to count the pairs by men: each of the 12 men danced with 3 women, so the total number of pairs is 12 times 3, which is 36. Another way is to count the pairs by women: each woman danced with 2 men, so the total number of pairs is 2 times the number of women. Since the two ways of counting must give the same result, I can set them equal to each other and solve for the number of women: 2 times the number of women equals 36. Dividing both sides by 2, I get the number of women equals 18. 