Class Strength

In a certain algebra lecture class, Chris and Pat count the students and compare notes. "Hmm, 12/17 of my classmates in here are women," notes Pat. "Funny," recounts Chris, "5/7 of my classmates are women. "They were both right. How many students, men and women, were in the class, and what were the genders of Chris and Pat? Men and women are mutually exclusive sets for the purposes of this problem. Explain your reasoning.

Solution:

Let n be the total number of students in the class, w = the number of women. Now 12 / 17 is less than 5 / 7 , so Pat has to be a woman, as Chris sees more women classmates from his perspective. Then Pat sees that (w-1) / (n-1) = 12 / 17 , and Chris sees w / (n-1) = 5 / 7. The denominator (n-1) can be the product 7*17 = 119, because 12/17 = 84/119 and 5/7 = 85/119 , so since 84 and 85 are 1 apart, we're there. Pat has 84 women classmates, and Chris has 85. Don't forget the 119 doesn't count the observer, so there are 120 students in the class.